Microwave Mixer Technology and Applications [1 ed.] 9781608074907, 9781608074891

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Microwave Mixer Technology and Applications

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For a complete listing of titles in the Artech House Microwave Library, turn to the back of this book.

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Microwave Mixer Technology and Applications Bert Henderson Edmar Camargo

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Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the U.S. Library of Congress. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library. Cover design by Vicki Kane

ISBN 13: 978-1-60807-489-1

© 2013 ARTECH HOUSE 685 Canton Street Norwood, MA 02062

All rights reserved. Printed and bound in the United States of America. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the publisher. All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized. Artech House cannot attest to the accuracy of this information. Use of a term in this book should not be regarded as affecting the validity of any trademark or service mark.

10 9 8 7 6 5 4 3 2 1

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To our wives, Marcia and Ann

Contents Preface

xi

CHAPTER 1

Origins of Electronic Mixers 1.1 History of Radio Development 1.2 Single Ended Circuits 1.3 Singly Balanced Mixers 1.4 Doubly Balanced Star Circuit 1.5 Special Receiver Architectures 1.6 Harmonic Mixers 1.7 Self-Oscillating Balanced Mixers 1.8 Distributed Mixers 1.9 Summary

CHAPTER 2

System Parameters and Performance 2.1 System Overview 2.2 Digital Modulation 2.3 Error Performance 2.4 Receiver Architectures 2.5 Mixer Linearity 2.6 Noise 2.7 Noise and Distortion in Communication Subsystems 2.8 Dynamic Range 2.9 Summary

49 49 54 64 68 74 76 93 102 103

CHAPTER 3

Semiconductor Modeling 3.1 Modeling Schottky Diodes 3.2 Modeling Bipolar Transistors 3.3 Modeling Field Effect Transistors 3.4 Summary

105 105 109 121 151

CHAPTER 4

Passive and Active Coupling Structures 4.1 Balun Structure 4.2 Marchand Balun 4.3 Microstrip Baluns 4.4 Lumped Elements 4.5 Slotline Type 4.6 Active Approach 4.7 180 Hybrid Couplers (Magic-T) 4.8 Quadrature Hybrids

155 155 164 169 187 191 194 203 221

vii

1 1 18 23 29 30 42 44 45 46

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Microwave Mixer Technology and Applications

4.9 Summary Appendix 4A — Guanella 4:1 Transformer Appendix 4B — Compensated Balun Appendix 4C — Active 180 FET Power Divider Appendix 4D — Alternative 180 FET Power Divider Appendix 4E — Active 180 FET Combiner

225 230 232 235 237 239

CHAPTER 5

Diode Mixer Theory 5.1 History of Linear and Nonlinear Analysis 5.2 Linear Mixer Analysis 5.3 Frequency Conversion Matrix 5.4 Computer Simulation Example 5.5 Large Signal Conversion Analysis 5.6 Subharmonic Mixer 5.7 Balanced Diode Circuits 5.8 Mixer Circuit Synthesis 5.9 Summary Appendix 5A — Parasitic Losses in Diode Mixers Appendix 5B — Conversion Matrix Including Parasitics Appendix 5C — Image Imp. and RF-Image Conversion Appendix 5D — Saleh Exp. Diode Mixer Performance

241 242 243 247 276 289 300 313 323 326 329 333 340 341

CHAPTER 6

Diode Applications 6.1 Single Ended 6.2 Singly Balanced 6.3 Doubly Balanced 6.4 Triply Balanced 6.5 Quadrature Mixers 6.6 Subharmonic Mixers 6.7 Summary

345 345 353 384 400 413 416 424

CHAPTER 7

BJT Mixer Theory 7.1 Low Frequency Mixer 7.2 Conversion Matrix 7.3 Mixer Properties 7.4 Design Study: CDMA Down-Converter 7.5 Cascode Approach 7.6 Singly Balanced Mixer 7.7 Singly Balanced Subharmonic 7.8 Doubly Balanced Mixer 7.9 Design Study: WiFi 2.45 GHz Gilbert Mixer 7.10 Differential Triple Level 7.11 Doubly Balanced Subharmonic 7.12 Subharmonic Triple Level

431 433 441 449 453 457 461 473 475 490 495 500 504

Contents

ix

7.13 Summary Appendix 7A — 2SC5006 Gummel Poon Parameters Appendix 7B — AT305 Spice Parameters

508 511 512

CHAPTER 8

Bipolar Junction Transistor Applications 8.1 Single Ended 8.2 Parallel Combined Mixers 8.3 Integrated Circuit Topologies 8.4 Doubly Balanced 8.5 Image Reject 8.6 Subharmonic Topologies 8.7 Summary

513 513 517 531 543 572 580 585

CHAPTER 9

FET Mixer Theory 9.1 Gate LO Injection 9.2 Source LO Injection 9.3 Drain LO Injection 9.4 Resistive Approach 9.5 Cascode Mixer 9.6 Singly Balanced 9.7 Doubly Balanced 9.8 Subharmonic Mixing 9.9 Distributed Mixers 9.10 Summary Appendix 9A — NE67300 Parameters Appendix 9B — Transistor NMOS 0.13 µm Tech.

589 591 614 616 624 637 642 659 665 669 674 679 685

CHAPTER 10

Passive FET Applications 10.1 Single Ended 10.2 Floating Approach 10.3 Singly Balanced 10.4 Doubly Balanced 10.5 Distributed GaAs Applications 10.6 Summary

687 687 697 707 725 751 753

CHAPTER 11

Active FET Applications 11.1 Single Ended 11.2 Singly Balanced 11.3 Doubly Balanced 11.4 Subharmonic Approach 11.5 Self-Oscillating FET Mixer 11.6 Distributed Applications 11.7 Summary

757 757 775 791 814 820 826 838

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Microwave Mixer Technology and Applications

Appendix Sampling Mixers About the Authors Index

843 857 859

PREFACE In view of the extensive coverage available in existing books on the topic of microwave mixers for graduate and professional engineers, another book in this area warrants some explanation. This work is intended to provide the new generation of design engineers with basic theory, a glimpse into the history of mixer technology development, and to professional engineers it offers a wealth of material on important publications and patents. Therefore we expect this text will improve the understanding of frequency conversion circuits and provide a platform for those who want to go further and extend the state of the art. Another aspect of this book is its emphasis on practical applications, counterbalancing the available resources that tend to focus on theoretical aspects more so than the basic concepts of how devices are used for mixing. Over one thousand patents on mixers and frequency conversion were reviewed, with many of the important and interesting ones discussed in the application-oriented chapters. In addition, important contributions from the technical literature are included to provide a solid theoretical foundation. This text represents the authors’ efforts to introduce the basic theory of mixing using diodes, bipolar transistors, and FETs. Discussion of the theory is followed by numerous circuit applications based on patents and other publications from the literature. Very often the original patent manuscripts are incomplete or unclear, so their language is clarified, and in many cases circuit simulations are done to verify the patents’ statements. Therefore we feel the material in this book is useful for undergraduate students who desire to learn basic concepts and to improve their ability and confidence in developing new circuits. And it is also a book for professionals looking for clues for specific applications which are available in most chapters. We see this book as a kind of handbook, and we hope its material will warrant a place in the bookshelf of professional designers. The book is organized into eleven chapters as follows: Chapter 1 introduces the origins of mixer technology, starting with the prevacuum tube era up to its golden age in the late 1940s and early 1950s. Most of the basic topologies were developed within this time frame and formed the basis for analog radio development until the late 1970s. The application of mixers in systems is discussed in Chapter 2 along with the parameters associated with mixers. A few spreadsheet examples analyzing the role of mixers in receivers are presented, including cascaded analyses of gain, IP3, noise figure including image and LO noise, and phase noise. Chapter 3 discusses the modeling of diode, bipolar, and FET devices that apply particularly to use in frequency conversion circuits. Chapter 4 discusses the basic building blocks in mixers, with emphasis on balun theory, one of the most important components in a mixer. The remaining chapters are more specific to the various mixing devices, each followed by a chapter on application circuits. Chapter 5 introduces the basic theory of diode mixers, including the conventional theory by Torrey and Whitmer xi

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Microwave Mixer Technology and Applications

in the 1940s, the work of Saleh in the 1970s, and the work of Maas in recent years. Chapter 6 summarizes patents and major publications on the application of diodes to various topologies and classes of mixers. The theory of bipolar transistor mixers is dealt with in Chapter 7, followed by applications in Chapter 8. The theory of FET mixers is in Chapter 9, detailing the principles of operation of the FET family for active and passive approaches. Chapter 10 discusses applications of passive FET mixers, and Chapter 11 presents applications of active FET mixers. This text assumes as minimum prerequisites: familiarity with basic electronics and simple circuit analysis, familiarity with the Fourier representation of simple waveforms, and knowledge of microwave fundamentals and communication theory. With this background the reader will certainly have a better appreciation and ability to absorb the technical content. We would like to thank and acknowledge the support of Agilent and AWR in providing circuit simulation software: Advanced Design System (ADS) by Agilent, and Microwave Office by AWR; and recognize Regis Camargo for providing the book cover design concept. It is with deep appreciation that we acknowledge the support of our wives, Ann Henderson and Marcia Camargo, who helped make the writing of this book a pleasant task. We also would like to honor our parents who put us onto the right path, Tiburcio and Terezinha França Camargo, and Connie and Lillian Henderson.

Chapter 1 Origins of Electronic Mixers This chapter presents the historical context for the emergence of the mixer as a critical component in radio communication systems. The origins of electronic mixers are closely associated with the origins of radio communications, starting with the first major advance, which was detection of RF signals in receivers. The quality of detectors improved considerably with the introduction of diode tubes for simple detection of continuous wave (CW) and amplitude modulated (AM) radio signals. The introduction of heterodyning and super-heterodyning receivers was the second major advance in the development of radio, which increased the importance of the mixer by its ability to provide frequency conversion. Increased gain was also obtained using regeneration and super regeneration methods. Clever implementation of frequency conversion, as well as improving tube technology itself, led to dramatic performance gains. Improvements in tube technology included the introduction of high-quality diode, triode, tetrode, and pentode tubes. Numerous patents were developed for tube-based mixer circuits, some related to tube technology itself and others related to improved heterodyne operation. Valuable insights can be derived from these early developments in mixer technology. Many of the early tube design ideas were simply transferred to semiconductors such as diodes, and in particular FETs, which are a natural replacement for the tube due to similarities in their electrical properties. 1.1 HISTORY OF RADIO DEVELOPMENT At the turn of the 20th century, wireless communications technology consisted of using a spark gap in the transmitter to generate electromagnetic waves, and using a coherer in the receiver for detection. The spark gap comprised two electrodes terminated by metallic spheres and separated by a gap while surrounded by a gas atmosphere. When a high voltage pulse was applied between the electrodes, a spark was formed by ionizing the gas, which in turn reduced its electrical resistance, producing a current pulse. This generator was connected to an antenna and capacitors that constituted a tank circuit. Damped oscillations were produced

1

2

Microwave Mixer Technology and Applications

by the “spark” and a portion of this spectrum would be selected by the tank’s resonant frequency, with the resulting EM energy being radiated. An example of this transmitter used for Morse code transmission is shown in Figure 1.1. It consisted of a Morse code key and a battery connected to the primary of a high voltage transformer, whose secondary coil was connected to the spark gap, tank circuit, and antenna. When the key was closed, a high voltage appeared at the secondary of the transformer that would charge the capacitor. After reaching a certain voltage, a spark would jump between the electrodes causing damped oscillations in the tuned circuit. The high peak voltage reaching the antenna caused energy to radiate into free space. Transformer

Spark Gap Capacitor

Aerial Coil

Tuned Circuit

Battery Figure 1.1

Morse Key

Earth

A simple spark gap transmitter consisting of a Morse key and a high voltage transformer connected to a spark gap in between a tank circuit. The filtered signal is delivered to the aerial.

On the receiver side, a resonant circuit was connected to the antenna and tuned to the transmitting frequency. The filtered RF signal was detected by a device called a “coherer,” [1], which comprised metal fillings enclosed in a glass capsule and kept at low pressure between two electrodes. With no signal present, the filings were slightly oxidized, causing the impedance of the coherer to be high. Once an RF signal incident at the coherer reached a certain threshold level, the fillings would become aligned due to electro-static interaction and become slightly welded due to microscopic sparks between them. The resistance between the electrodes then would fall to a few ohms. However, after being subjected to potential differences set up by the impressed signal, the coherer would not by itself resume its initial high resistance state, so a small hammer was provided to gently tap the glass capsule and mechanically shake the fillings to restore the initial high resistive state. A diagram of a typical receiver of the time is shown in Figure 1.2 [2]. The electromagnetic wave picked up by the antenna generated a current to ground. The resulting voltage in the transformer, tuned by the capacitor, is applied to the coherer that would become a low resistance device allowing current to be developed on the resistor R.

Origins of Electronic Mixers

Aerial

Tank

3

DC block Capacitor

R Ground Figure 1.2

Tuning Capacitor

+

Coherer

The received signal is filtered by the tank circuit and applied to the coherer. This transient is sensed by the head phone.

The operation of this “on-off” system sounds very crude nowadays, but it met the requirement at that time of transmitting and receiving Morse code without wires. Many engineers and scientists contributed to the further development of wireless communications, but two inventors stand out as major contributors: Marconi [3] who filed a patent for his “Apparatus for Wireless Telegraphy” in November 1900, and Tesla [4] who filed a very similar system in July 1900, “System of Signaling.” They proposed a wireless transceiver system that was reliable for the transmission of Morse code. However, it was not capable of transmitting voice over electromagnetic waves. A few more years would pass before a cost efficient system would be developed to transmit voice. Reginald Fessenden [5] was the scientist pursuing the transmission of voice over a pure sinusoidal carrier or a “continuous wave.” He tried to improve the spark system for that purpose but was not successful. The only source of pure electrical sinusoidal signal with high power was from an alternator (dynamo), normally used to generate electricity. The problem was its inability to generate high frequencies. Eventually, he teamed up with another engineer, Alexanderson [6], who was able to build a high-speed alternator operating at 100 kHz and generating 1kW of power. He added a carbon microphone to the alternator circuit, providing amplitude modulation to the carrier. In spite of the success of his invention, it soon became obsolete with the advent of the diode vacuum tube in 1905 by Fleming [7] and the triode vacuum tube in 1908 by Lee de Forest [8]. The device invented by Fleming, depicted in Figure 1.3, consisted of a filament b surrounded by a cylinder c isolated from each other, all contained in a vacuum glass bulb. By heating the filament it becomes a source of electrons inside the tube. Applying a positive potential to the cylinder, electrons flow from the filament to the cylinder creating a current in the external circuit. If the potential applied to the cylinder is negative, then the electrons are repulsed and there is no current in the circuit. This device allowed rectification of high frequency

4

Microwave Mixer Technology and Applications

alternating signals and replaced the coherer due to its many advantages. This vacuum tube diode is considered to be the device that initiated the field of electronic science. The hot filament became known as cathode and the cold cylinder as the anode.

I V

Figure 1.3

Vacuum tube diode where a current is detected by the galvanometer if a positive voltage is applied to the cylinder with respect to the filament. If the voltage is reversed, then there is no current detected in the galvanometer.

The description of current as a function of voltage is given by the ChildLangmuir equation [9], valid for low anode voltages and before a space charge is built up between the electrodes, which limits the current. The constant K takes into account the geometry of the tube, also called a valve. 3

I  KV 2

(1.1)

The invention of the triode, called the audion by Lee De Forest, was the second most important milestone in the new field of electronics. As observed in Figure 1.4, De Forest added a grid, a, between the cathode, F, and the anode, b. The application of a small voltage on the grid allowed control of the flow of electrons between the heated cathode to the cold anode, which provide a detected signal to the telephone terminal, T. His objective was to detect high frequency signals achieved by coupling his device to an antenna by means of a transformer. The capacitor C’ is part of the tank circuit and C” is used to block DC current from flowing from grid to cathode. The circuit showed improved sensitivity, due to the voltage gain impressed by the tube. It is an example of an active rectifier for RF signals. In this circuit the grid remained unbiased, but it could become biased by the charge stored on the blocking capacitor.

Origins of Electronic Mixers

5

Vpk Ip Vgk

Figure 1.4

Triode applied as wireless telephony receiver. Originally proposed as an active rectifier it can also operate as a signal amplifier and as a generator and beat receiver.

Later it became usual practice to bias the grid negatively to control the current. If the grid potential becomes positive, then current would start flowing to the grid, which degraded performance and could destroy the device by melting the fine wires comprising the grid. The I,V description of this device follows a similar equation developed for diodes [9] with K1, K2 constants. 3

I p  K1 (Vgk  K 2V pk ) 2

(1.2)

The current is a stronger function of the control voltage, V pk, so K2 is a small fitting parameter. The fact that the device can provide gain allowed the development of sinusoidal sources, amplifiers, and a myriad of new applications. A clear explanation of the triode operation as an amplifier/rectifier was given by Armstrong [10]. Figure 1.5 shows a transfer characteristic Iplate versus Vgrid of a typical audion of the time. If the grid is biased near point M, the transfer characteristic is very close to a diode characteristic, where plate current will be amplified for a positive grid voltage swing and will be nearly zero on the negative swing. The same effect can be obtained if bias is shifted to point N, except that the positive part of the incoming wave train would be rectified. The circuit of Figure 1.6 uses such a characteristic to detect amplitude modulated (AM) information contained in high frequency signals. The potentiometer P is adjusted to set the grid bias to point M in the transfer characteristic. The initial wave train sets up oscillations in the parallel LC circuit and the positive half cycles generate a high “wing” (plate) current while the negative ones are heavily compressed. B1, B2, B3 are voltage supply to filament, plate and grid, respectively.

6

Microwave Mixer Technology and Applications

1.0 Wing (plate) current, Ip (mA)

N P

0.5

M 0 -5 -3 0 3 5 Figure 1.5

Vgk -Volts

Plate current function of grid potential with respect to filament. The transconductance of this device is about 0.2 mA/V. After [10].

This asymmetrical characteristic is similar to one produced by a diode, the difference here is that the positive half cycles are amplified by the wing (plate) current circulating in the load. The load is composed of a capacitor C in parallel with a microphone that can be represented by a resistor. The high frequency charge on the capacitor, which discharges through the telephone, has a long time constant. This arrangement constitutes a low pass filter detecting the low frequency signal modulating the carrier. The high frequency signal is therefore rectified and amplified simultaneously using the same device.

L

C P B3

Figure 1.6

T

CT

B1 B2

Headphones

Active rectifier proposed by Armstrong using an audion valve. After [10].

The stability of a triode is dependent on the load, which can be evaluated from the circuit representing a triode amplifier shown in Figure 1.7. The model

Origins of Electronic Mixers

7

includes parasitic capacitances, Cg = grid to ground, Cag = anode to grid, Ca = anode to ground, and the voltage generator is dependent on gate voltage, with gain µ. The plate (anode) resistance, rp, is much lower than the load. Cag Vg

Eg

rp Cg

Rg

Figure 1.7

µVg

Ca

ZL = RL + j XL

Equivalent circuit for a triode amplifier.

The input admittance for this circuit is approximately given by:

Yin 

  C ag  C ag  1 j X L  j  C g  RL    Rg rp rp  

(1.3)

From this equation it is found that if the load is capacitive, CL, then the real part of input admittance, Yin, is equal to:

Re(Yin ) 

1 C ag   Rg C L rp

(1.4)

In this case Yin is positive and the circuit is stable. On the other hand if the load is inductive, LL, then the real part of input admittance is equal to:

Re(Yin ) 

2 1  Cag LL  Rg rp

(1.5)

This shows there is a combination of parameters and frequency where the real part of the input conductance is negative, rendering the circuit unstable. 1.1.1 Heterodyne Concept The application of the heterodyne principle emerged as an application to radio communications and was proposed by Reginald Fessenden [11], in his patent “Wireless Signaling” granted in 1902. In his proposal, shown in Figure 1.8, the

8

Microwave Mixer Technology and Applications

sending station contains two antennas of different length transmitting two signals simultaneously with a small difference in frequency. The inventor does not describe how the oscillations are generated, but one can think of a spark gap generator coupled to the magnetic coil 5. Once the spark passes between terminals 3 and 4, signals of different frequencies are generated and the antennas will select oscillations that fall within its own resonating frequency. Therefore, at each spark two signals are radiated simultaneously. At the receiver side two antennas, 6 and 7, are tuned to the corresponding transmitter frequencies and both are connected to a coherer or any other type of detector. The first oscillations reaching the coherer are of same amplitude, as explained in Figure 1.9, but after a few periods there is a difference in the potential applied to the coherer resulting from the beating signal. This signal is sensed by the coherer and activates a Morse code relay.

Figure 1.8

Dual frequency transmitter and receiver antenna, the origin of heterodyne system.

Figure 1.9 Beating of two sinusoids, whose beat or frequency difference is the desired signal.

This invention paved the way to modern communication systems, by indicating the convenience of beating two sinusoids to extract the frequency difference signal. Later, one of the transmitted signals became a local oscillator at the receiver, a common practice in any heterodyne or super-heterodyne receiver.

Origins of Electronic Mixers

9

1.1.2 Regenerative Receiver The next major advance in receiver development was the introduction of the regenerative circuit by Armstrong [12] in 1914, which is essentially an AM detector, converting an RF signal to audio. His invention, reproduced in Figure 1.10, contained two interlinked circuits: a tuned receiving circuit coupling the antenna to the audion grid, “tuned grid circuit,” and the “wing circuit” containing the remaining high frequency and DC circuitry. The linking between both circuits is through the impedance connected from cathode (filament) to a common point. This introduces a positive feedback from wing to grid and the amount of feedback is controlled by an auto-transformer. If the amount of feedback is kept below the onset of oscillations, very high gain can be obtained from a tuned circuit, increasing the sensitivity of a receiver. The valve therefore can simultaneously act as a tuned RF amplifier and as a local oscillator. Armstrong added capacitor C’ in series with the grid, which moves the bias somewhere near the center of the valve transfer characteristic. The result is the grid to filament circuit acts as a rectifier whose detection threshold is dependent on how much charge is accumulated on the grid as a result of signal applied to the terminals.

Figure 1.10

The regenerative receiver introduced innovative concepts in the design, including impedance matching, positive feedback, and active mixer.

More innovations were provided by this invention, like the addition of a capacitance in parallel to the grid circuit and an inductance in series to boost the gain. In modern terms this arrangement performs impedance matching from source to grid, to improve the power transmission. The detected audio signal is also coupled back to the grid and is amplified. Due to high gain, regenerative circuits use fewer components and their architecture is simpler. In this single stage

10

Microwave Mixer Technology and Applications

the level of distortion is very low, resulting in a very high linearity receiver. However, it is not practical for high volume commercial usage because it requires careful adjustment by the user. It is impressive that this invention introduced the use of positive feedback, impedance transformation, signal amplification, active rectification, and the concept of active mixer, all with a single tube.

1.1.3 Super-Regenerative Receiver The super-regenerative amplifier circuit was developed based on the principles of the regenerative amplifier, using positive feedback. In such a circuit, the amplifier gain increases and becomes very high right before it enters into an oscillatory state. Right at this point the system is at its most sensitive state, and any disturbance can make it oscillate. This high sensitivity, however, is transient because once oscillations start, the tube compresses and the system becomes insensitive to any impressed signal. In Armstrong’s proposal [13], by controlling the amount of natural damping of the system and the amount of positive feedback or regeneration, the transient state can be made constant and controllable so that the system may be maintained at all time in the super-regenerative state. In order to do that, he used two oscillators: one oscillator constitutes the regular regenerative receiver, and the second is used to alternatively apply positive feedback bringing the circuit on the verge of self oscillation and switch on dampening to bring the circuit back to stable condition. Another way of stating this effect is to say one tube acts as an RF oscillator whose oscillations are periodically quenched by a second oscillator operating at a lower frequency. At the end of the quench period, oscillations in the RF tube build up in response to initial conditions imposed by the incoming radio signal. Thus, after a fixed elapsed time imposed by the low frequency oscillator, the RF oscillations build up to a level whose amplitude is proportional to the instantaneous amplitude of the received radio signal at the moment that oscillations began. The longer the time between the quenching periods, the greater is the achieved gain. Among the various topologies proposed in the patent, the one shown in Figure 1.11 is the simplest. The RF signal from the antenna is coupled to a resonant LC circuit and then coupled to the grid of tube 23. The RF signal is amplified, and part of it is fed back to the grid by the transformer connected to the plate. The feedback is made positive by the transformer coupling circuit. Part of the amplified signal is coupled by another transformer to a detector. The second tube oscillates at the frequency determined by the resonant circuits, generating a low frequency voltage that modulates the plate of the first tube. Adjusting the circuit the low frequency signal controls the amount of feedback and dampening creating a very high RF gain. In fact the maximum gain depends exponentially on the relative frequencies of the two oscillators. The resulting output signal from the RF oscillator is a series of oscillation bursts whose amplitudes are proportional to the RF amplitude. The

Origins of Electronic Mixers

11

output can then be demodulated with a simple AM detector. The low frequency generator may generate an audible signal after detection, similar to a whistle that is annoying. By adjusting the low frequency generator to a higher frequency, the interfering signal falls into the supersonic region and cannot be heard by the human ear.

Figure 1.11

The super-regenerative circuit is capable of very high gain by using controlled positive feedback.

An interesting effect of this circuit is its capacity to operate as an FM demodulator as well. If a narrow band FM signal is applied to a super-regenerative radio, mixing products are produced. If the difference frequency is close to the quenching period of oscillations, they will lock and FM modulates the quenching frequency. This produces a variation in the gain that exactly follows the FM modulation of the received signal. The gain variation contains the extracted FM modulation in terms of amplitude, which is applied to a speaker.

1.1.4 Super-Heterodyne Concept The most important contribution from Armstrong is the super-heterodyne principle, which he developed while in Europe during World War I. By that time, the detection of high frequency signals presented several challenges: the propagated signal strength would generally decay as the carrier frequency increased, making direct detection impractical; and direct amplification of short wave signals was difficult due to the poor frequency response of vacuum tubes. Heterodyning improved sensitivity of receivers, but the poor stability of local oscillators made it difficult to implement. The use of a low frequency amplifier after the detector would improve sensitivity to a certain extent, but the noise of the low frequency amplifier was a serious limiting factor. Armstrong recognized that the sensitivity of the detector for weak signals lies in the amplification of the radio frequency currents before applying them to the detector. So the obvious solution

12

Microwave Mixer Technology and Applications

was to add a cascade of tuned amplifiers to bring the signal up to a reasonable level for detection. This solution worked successfully in certain frequency ranges. However, this solution faced serious drawbacks: (1) inter stage capacitance placed severe limits on maximum frequency of operation; (2) the high gain circuit became unstable and difficult to stabilize; and (3) it was not practical to tune the amplifiers to different frequencies. In his patent “Method of Receiving High Frequency Oscillations” [14], filed in 1919 and granted in 1920, Armstrong proposed a new radio architecture that overcomes these problems while adding many advantages. He proposed an indirect method of amplification that operates independently of RF frequency and provides high gain, selectivity, and sensitivity compared to previous solutions. The method consists of converting the frequency of incoming signal down to a pre-determined lower frequency. At that intermediate frequency a fixed amplifier tuned at IF can provide high gain without danger of oscillations, since the output frequency is different than the RF frequency. After the signal is amplified to a desirable high level, it is then detected by a diode detector. The gain is now stable and may be high, thus improving the RF performance. His original diagram is reproduced in Figure 1.12. The first diode, 4, operates as a single ended receiver mixer. The IF signal is amplified by the multistage high gain amplifier and the second diode, 9, is either an amplitude peak detector or a synchronous demodulator if a signal at IF frequency is applied to the diode. This conceptual diagram gave birth to the signal mixers used in all frequency translation applications.

Figure 1.12

Single converter block diagram used to explain the super-heterodyne process.

The circuit schematic of Figure 1.13 shows the first proposal for a superheterodyne receiver, where the first tube is a regenerative stage, operating

Origins of Electronic Mixers

13

simultaneously as a local oscillator and mixer. In Figure 1.14 Armstrong proposes the “dual conversion system,” which provides exceptional selectivity, stable gain, and very high rejection of image signals. This is a more complex system requiring two LOs and two IF amplifiers, which translates into higher cost. Armstrong mentioned that theoretically there is no limit to how many conversions can be employed but in practice, most modern communication systems use dual conversion.

Figure 1.13

Self oscillating converter design with a high gain IF amplifier.

Figure 1.14

Example of the dual conversion system to improve isolation between output/input and to overcome the limits on amplification from the tubes.

1.1.5 Continuous Wave Receiver This invention conceived by Armstrong [15] is the precursor of “direct conversion receivers.” In his proposal, illustrated in Figure 1.15, there are two resonant circuits, one coupled to the antenna that he called the resonant receiving circuit tuned to the incoming signal. The second, called the resonant detector circuit, is coupled with the resonant receiving circuit and includes the detector. The former is similar to the circuit used in the regenerative circuit and the latter is a series resonant circuit. The capacitor C2 is part of both high frequency circuits coupling

14

Microwave Mixer Technology and Applications

energy from the wing to the grid. Therefore, a current induced in the receiving circuit will circulate in the valve producing currents of the same frequency in the detector circuit that are coupled back to the receiving circuit, reinforcing the original current. This positive feedback is adjusted to sustain oscillations in the circuit. Any DC current in the grid is blocked by capacitor C1, which is charged to a potential determined by the oscillations and biases the valve at a high gain point. If the oscillations generated in the circuit are of the same frequency as the received RF signals, the difference is null and no signal will be heard in the telephone, R. Adjusting the frequency of oscillations to differ slightly from the RF signal, a low frequency beating signal is produced and is audible in the headphone. The circuit is most effective when the electromagnetic coupling between the antenna and the resonant receiving circuit is extremely loose, facilitating the tuning of the LO frequency to that of the incoming signal. The functions of RF amplifier, local oscillator, and detector are all combined in a single valve.

C1

C2 C5 C3

Figure 1.15

C4

Continuous wave receiver is similar to a self oscillating mixer with LO frequency set close to the incoming RF signal.

1.1.6 Frequency Modulated Transceiver Moved by the need to make the radio systems more immune to the effects of fading and static noise, Armstrong [16] conceived the idea of modulating the carrier frequency instead of its amplitude. He correctly noticed the degrading effects of fading and static noise have a great impact on the carrier amplitude, but not on its frequency. Figure 1.16 depicts the schematic for the FM transmitter. The RF frequency is determined by the oscillator composed of tube 5 that contains a transformer to feedback the wing to grid signal in the proper phase for oscillation.

Origins of Electronic Mixers

15

The modulator system contains a microphone modulating the grid of valve 7, which in turn varies the reactance of the transformer connected to the oscillator, modulating its frequency. The oscillator is connected to a high gain amplifier and limiter which eliminates all amplitude variations. A bandpass filter 20 is used to reject all harmonics generated in the limiting process. The RF power amplifier 23 delivers the signal to the antenna for transmission.

Figure 1.16

FM transmitter where the LO frequency is modulated by a variable reactance tube.

The FM receiver shown in Figure 1.17 contains an RF amplifier, a limiter, and a bandpass filter to eliminate amplitude modulations introduced by fading and other propagation effects. Once the signal is received, the next step is to transform the frequency variations into amplitude variations to be sensed by a telephone or a loudspeaker. The circuit proposed by Armstrong is a signal divider consisting of two outputs arranged in such a way that amplitude variations are common to both outputs and frequency variations are differential. The principle of operation can be summarized by applying a generator V in with an internal resistance R between terminals A and C.

V AC

 1 1  Vin  jL41    jC 40 jC 39    1 1 R  jL41   jC 40 jC 39

(1.6)

V BC

 1  Vin  jL41   jC 40    1 1 R  jL41   jC 40 jC 39

(1.7)

Armstrong devised two main frequencies, the low end of the band where L41, C40 resonates and the high end of the band where C40, C39, L41 resonates.

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Microwave Mixer Technology and Applications

Therefore, VAC and VBC have two defined values, low end where L41, C40 resonates and high end where L41, C40, C39 resonates.

V AC 

Vin at ω = ω1 and VAC = 0 at ω = ω2 1  jRC 39  Vin VBC  at ω = ω2 and VBC = 0 at ω = ω1 jRC 39

(1.8) (1.9)

C39 C40 L41

Figure 1.17

FM receiver containing a tuned RF receiver and a frequency discriminator circuit.

Origins of Electronic Mixers

17

At the low frequency end, voltage V AC is high and decreases with increasing frequency, reaching zero at the high end. At the low end V BC starts at zero volts, and reaches a negative value at the high end. The equations indicate that |VAC| < |VBC| at the high voltage points. To correct this inequality, elements 36 and 38 move the zero crossings to the band center, transforming the denominators of both equations to make the amplitudes more similar to each other. A plot of voltage as a function of frequency is shown in Figure 1.18 for a typical circuit, operating between 1.20 and 1.25 MHz.

Figure 1.18

Performance of frequency to voltage converter circuit.

The resulting converted voltages are amplified by 42 and 43 and delivered to the detectors 48 and 49. If the detectors are linear, any amplitude variation at a fixed frequency affects equally both outputs of the signal divider that is cancelled in the output headphone. However, any frequency variation results in different voltages applied to the detectors, resulting in an output current at the headphone. The headphone acts to transform two balanced signals into an unbalanced signal, giving the frequency voltage relation depicted in Figure 1.19.

Figure 1.19

Ideal frequency to amplitude conversion.

The whole frequency detecting system is floating so that any amplitude variations external to the system will affect both balanced lines and will not be sensed by the headphone.

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Microwave Mixer Technology and Applications

1.2 SINGLE ENDED CIRCUITS With the consolidation of the super-heterodyne architecture by mid 1920, the mixer became the most important element in the receiver chain. In consequence, several patents found in the literature for single ended mixers use either diode or triode tubes. The patents in this section were selected for specific important features offered by single ended mixers. The first uses a special bridge circuit to match the antenna to the mixer tube, providing properties similar to a balanced configuration. The second invention proposes a feedback arrangement to neutralize the tube parasitic feedback and consequently improve the frequency of operation. The third invention was targeted for noise reduction but the key feature is the feeback at the IF band. 1.2.1 Bridge Combiner Applied to Mixer The circuit of Figure 1.20 originated in 1929 and contains an LO, the input bridge, and the mixer circuits [17]. The LO signal is generated by a Hartley oscillator circuit, enclosed within a shielded cavity 10. The RF and LO are both applied to the grid of a triode mixer tube 16, and the plate circuit is connected to an output transformer that couples IF energy to the load. Further filtering of RF and LO signals are assumed to be part of the IF circuit.

Figure 1.20

Bridge mixer circuit with LO generator.

Origins of Electronic Mixers

19

The bridge circuit is better explained by means of schematic in Figure 1.21, composed of input coupling coils 24, 26, condenser 30, and the grid to cathode capacitance C. RF signal from the antenna couples energy to coils 24, 26, which resonate with capacitor 28. The induced current develops voltage at node A feeding the grid. By balancing the bridge, the RF voltage at the junction of 24, 26 has the same potential as node F, so a virtual ground is imposed. The LO signal is inserted at the junction of 24, 26 so that current flows in opposite directions to ground. The LO voltages at nodes A and B are the same, so that both RF and LO are applied to the grid. Due to the direction of the LO current, there is ideally no LO signal coupled to the antenna. LO to RF isolation is therefore obtained with a single ended mixer.

Figure 1.21

The bridge circuit is formed by input inductors, input tube capacitance, and external variable capacitors.

1.2.2 UHF Triode Converter During World War II there was an effort to develop high frequency radars resulting in the discovery that at UHF bands triode tubes offered better signal to noise ratio because of their shorter cathode to plate electron transit times, compared to more complex tubes such as tetrodes and pentodes. The tradeoff is that triode tubes are less stable due to parasitic voltage feedback between plate to ground, plate to grid capacitance and inductances between cathode to ground. A patent was disclosed in 1944 [18], proposing a circuit to neutralize this feedback. The goal is to provide in the path between the plate and cathode a low impedance at the signal frequency and a high impedance at the IF. Referring to Figure 1.22, to achieve this effect, both the plate and anode of the converter triode were fitted with a second lead 7 and 8. The leads were connected to an open circuited stub quarter wave long at the signal frequency. This provided the low impedance at

20

Microwave Mixer Technology and Applications

signal frequency and the high impedance at IF needed to overcome the parasitic feedback. The LO was generated by a separate triode tube, V2. In this case it is convenient to use the inter-electrode capacitances to generate oscillations. The tube is connected in common grid with a coaxial short stub on the bias line for high impedance at the signal frequency and low impedance for other frequencies. The coaxial tank circuit is connected to the cathode providing the oscillation conditions. Power is lightly coupled from the cathode tank circuit.

Figure 1.22

A UHF triode converter merging coaxial transmission lines with vacuum tube components.

1.2.3 Noise Reduction Technique Noise reduction using feedback is addressed in the patent filed in 1946 by Strutt et al [19]. Thermal noise is introduced by the components connected to the frequency conversion tube. Noise is also produced by current fluctuations within the tube itself, which is the subject of the patent. The current fluctuations in the tube are categorized as emission-fluctuations and subdivision-fluctuations. Emission-fluctuations, also called cathode noise, are due to irregularities in the flow of electrons from the cathode. Subdivision-fluctuations are due to irregularities in the subdivision of the current between positively biased electrodes. For tubes having screen grids, the subdivision-noise current in the anode and screen-grid circuits, respectively, are equal and opposite to each other. The subdivision-noise currents run between the anode to the screen-grid, without affecting the cathode. Thus the signal-to-noise ratio (SNR) in screen-grid tubes is

Origins of Electronic Mixers

21

higher in the cathode circuitry than in the anode circuitry. Based on this, it has been proposed to introduce regenerative feedback at the input, increasing the signal current and improving the overall SNR. However, this arrangement makes it difficult to avoid self-excitation over a wide frequency band.

Figure 1.23

SNR improvement with positive/negative feedback.

The invention proposes regenerative feedback at the IF band where it is easier to control the circuit stability. It also proposes using negative feedback at IF as a means to trade off stability and SNR improvement. Referring to Figure 1.23, positive IF feedback is obtained by coupling screen-grid 8 to inductor 20 via capacitor 18. The tank circuit 21 is tuned to the IF frequency and coupled to the tank circuit 2 that is tuned to the signal frequency. Negative IF feedback is obtained by coupling the IF output through capacitor 19 to inductor 22. 1.2.4 Cascode Mixer-Amplifier Efforts during the development of radars in World War II included the focus on reducing noise and increasing gain while maintaining stability. This work led to the use of the cascode amplifier circuit that employed two triode tubes. The cascode circuit has the noise figure of a triode tube, and the gain and isolation of a pentode tube. The noise figure of the cascode is essentially that of the input triode. The gain is proportional to the trans-conductance of the first triode and the load resistance on the anode of the second triode. A frequency converter would use

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Microwave Mixer Technology and Applications

such a cascode amplifier followed by a separate frequency conversion circuit. In contrast, the circuit proposed in 1955 [20] uses the first triode for gain and the second triode for frequency conversion, combining the amplification and frequency conversion in one circuit, thus improving economy and efficiency. Previously, the problem with this arrangement was the introduction of LO signal into the first triode, which degraded performance and allowed LO leakage out the antenna. Referring to Figure 1.24, the patented circuit solves this problem by using a trap 45 between first triode 15 and second triode 25 to peak the RF signal voltage, and short the LO voltage thus isolating it from triode 15.

Figure 1.24

The cascode amplifier-mixer with trap to prevent local oscillator in driven stage from affecting driving stage.

1.2.5 Frequency Translator Circuit Another problem encountered in frequency conversion is variation in the conversion gain, caused by variations between production lots of tubes and by inexact tuning of the LO resonant circuit. A circuit was disclosed in 1941 [21] having goals to maximize SNR and increase signal gain, while reducing the effect of variations in tubes and circuitry. It also solved the problem of variation in LO tuning voltage over a wide frequency range. The patent’s design accomplishes this by keeping a constant ratio of LO injection voltage to DC voltage at the signal grid. Referring to Figure 1.25, the LO signal from triode 24 is applied to an autotransformer 20. Conductor 8 taps the LO voltage and impresses it between cathode 5a and grid 5c. Part of this voltage is rectified by diode 12 and the resulting DC voltage on resistor 10 is applied to the grid of the mixer tube 5. This bias is proportional to the LO peak voltage impressed on the translator. If the LO voltage is higher, then the mixer is biased more negative, maintaining the ratio LO

Origins of Electronic Mixers

23

peak voltage/DC voltage constant, and vice versa. The mixer output signal is partially shared by the inductance 8 due to the common grid connection of the mixer tube. Therefore, a large signal on the plate is partially rectified by the same diode protecting the mixer against overloading.

Figure 1.25

The mixer is used as a frequency translator system with feedback to maintain a high conversion gain over the tuning range.

1.3 SINGLY BALANCED MIXERS This topology contains two similar mixers with LO applied in phase and the RF applied in counter phase, or vice versa. Therefore, LO and RF signals are out of phase and isolated from each other. 1.3.1 An Ultra-High Frequency Mixer The schematic of Figure 1.26 was realized in two approaches using a dual anode tube [22]. The LO signal is applied between the center tap of inductance 6b and the grounded cathode, causing the diodes to switch on-off in phase with each other at the LO frequency. The incoming RF signal excites the antenna that is tuned by tank circuit 6a, 6b, and which in turn couples the RF differentially to the diodes.

24

Microwave Mixer Technology and Applications

The fact that the diodes are switched in phase by the LO creates an equivalent impedance between anode and cathode such that the RF current flows between the two anodes. The grounded cathode has no effect on these currents in a well balanced circuit. A trap LC filter was added in the IF circuit to block high frequencies from reaching the output transformer that combines the push-pull IF currents.

Figure 1.26

Schematic of a UHF receiving system with a special two-anode tube using lumped elements.

The first application of this circuit makes use of Lecher lines as depicted in Figure 1.27. The LO signal is fed from left to right using a pair of lines, one of which splits into two additional lines. All three lines terminate onto a common ground plane. The line splitting actually divides the signal into two half power signals of the same phase, both referenced to the common conductor. The length of line from the split point to ground is approximately half wavelength at the RF frequency; therefore, it comprises a resonant circuit at that wavelength. The RF signal coming from aerial conductors are fed through the ground plane apperture and are made to run parallel to each one of the resonator lines before being grounded. The RF signal therefore induces current on the Lecher resonator,

Origins of Electronic Mixers

25

transferring RF power. The diodes are located at the middle of the resonator where the RF and LO voltages are highest.

Figure 1.27

Lecher line version of a UHF mixer.

The second application uses a shielded Lecher system, within a coaxial structure, as shown in Figure 1.28. A Lecher system is observed in the two conductors 3,4 and plays a similar function to that in the previous version. It is shorted to the center conductor of a coaxial line on the right and terminated on the left on a current loop. The Lecher line is contained within a cavity measuring half wavelength, and contacts are made at its midpoint to the anodes of the diodes. The dual diode tube is installed externally to this cavity.

Figure 1.28

Coaxial version of the UHF mixer.

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Microwave Mixer Technology and Applications

The LO signal is applied to the coaxial line, and it splits in two with equal amplitude and phase when it reaches the Lecher system. Due to the dimensions of the cavity the fields at its left end are low, and it behaves essentially as a short circuit for the LO, preventing LO from propagating into the receiver cavity. The incoming RF signal from the antenna is applied to a coaxial receiving cavity, which measures half wavelength at the RF frequency, and is perpendicularly attached to the LO cavity. At the mid point from the back short, 20, the current loop gain protruding from the LO cavity is placed in close contact with the center conductor. The RF signal induces current into the probe that is fed into the LO cavity in counter phase on terminals 3,4. The contact from the Lecher line to the diodes contains a capacitor for DC and IF signal blocking. The IF signal is extracted from the anode with an inductive wire, that presents a high impedance to the RF and LO signals for IF isolation. 1.3.2 Differential Mixer A precursor to the modern differential mixer circuit is found in a patent filed in 1946 [23]. It provides isolation between the signal and LO inputs, and between these inputs and the IF output, by using circuit balance without filtering. Referring to Figure 1.29, the signal or LO input is applied at port 10, with the other input applied to port 21. The IF output is taken at port 25. Isolation is achieved as follows. The polarity of signal B at anode of tube 12 is 180° opposite that of signal C at cathode of tube 12. Signal C is coupled to the cathode of tube 15 via resistor 17 and capacitor 18 to produce signal D. Signal B also couples to the cathode of tube 15 via capacitor 19 to produce signal E.

Figure 1.29

An example of an early differential mixer using a phasing circuit to improve R – L isolation.

Origins of Electronic Mixers

27

Signals D and E cancel each other. The same process causes cancellation at the cathode of tube 12. Resistors 13 and 16 are adjusted for balance to maximize the cancellation. This cancellation at the two cathodes effects the isolation between ports 10 and 25, and ports 21 and 25. Isolation between ports 10 and 21 presumably is effected by isolation in the tubes. The mixing is obtained from the application of signal, for example, from node C to node F that is the plate of device 15. Application of a signal voltage at the plate modulates the bias voltage and the bias current as well, creating a beat note with the signal applied from terminal 21. The IF current circulating from cathode to anode passes through resistor 17, generating the output IF voltage. The same applies to the device 11, generating an IF current that adds up in phase to current generated by device 15, improving the conversion efficiency. The capacitor 18 is used to bypass the sum of input frequencies and is transparent to the desired difference frequency. 1.3.3 Push-Push/Push-Pull Operation A mixing arrangement was proposed [24] where the RF signal is fed in push-push to the mixing tube and the local oscillator is fed in push-pull. This approach, represented in Figure 1.30, has been demonstrated to show a better signal-to-noise ratio compared to other alternatives. The incoming signal is picked up by antenna D and tuned by the L1, C1 tank circuit. After filtering, the RF signal is split into two signals of equal phase and amplitude by means of the transformer with inductances L2’, L2” and parallel tube capacitances, C3’, C3”. The capacitance C2 is connected to points of the same potential so it is transparent in this mode. On the anode side, the incoming signal is filtered by a low pass circuit L7’, L7” and C4’, C4”. The LO voltage is magnetically coupled from winding L3 to the transformer containing L2’, L2”, and is split into two signals with the same amplitude and 180º phase relative to each other before being applied to the grid. The capacitance C2 comprises with L2’, L2” a tuned circuit at the LO frequency. The large LO signal voltage modulates the tube transconductance, and at the anode side is filtered by the same filter L7, C7 used by the RF signal. The IF signals at both anodes follow the phase of LO, so they are in counter phase and combined by the output transformer, L4’, L4” and L6. The capacitances C4’, C4” are in resonance with L4’, L4”. The coils L5’, L5” filter the high frequency signals preventing them from reaching the output load. The capacitances C4’, C4” can be adjusted to make the anode impedance more capacitive at the RF and LO frequencies, improving stability or more inductive introducing positive feedback into the circuit improving conversion gain. If the amount of inductance in the anode is sufficient, then the circuit will start self oscillating, removing the need for an external signal source.

28

Figure 1.30

Microwave Mixer Technology and Applications

Dual push-pull/push-push mixer. The LO signal, O, is magnetically coupled to the input splitting transformer.

1.3.4 Bifilar Line Mixer The use of bifilar wire wound on a rod of magnetic material is disclosed in this invention with applications in the VHF and low UHF frequency bands [25]. The schematic in Figure 1.31 displays two rods each with two windings, connected at one side to a source of RF signals, say an antenna, and on the other side to a pair of diode tubes and a LO source. The IF load is connected to the point of junction of both anodes, represented by resistor 22.

Figure 1.31

Mixer with diodes and magnetic rods wound with bifilar wire transmission lines.

Origins of Electronic Mixers

29

The operating principle of this mixer is best described by Figure 1.32, where it is arranged in a bridge format. The resistors 34 represent the diode in series with resistor 22. The LO source 21 applies current that is split equally into each rod and then applied to the diodes. The inductances 15, 18 and 16, 19 are in series with respect to the LO. If the circuit is balanced, the potential at the RF terminals, 15, 16 are the same, so there is no LO current circulating on the RF source. The currents originating from the RF source 31 flows in parallel to coils 15, 18 and return from coils 16, 19, so both circuits are in parallel with respect to the RF source. The RF signal is applied to the diodes in differential form and the node 37 is a virtual ground so that LO is isolated from RF.

Figure 1.32

Schematic of an equivalent circuit for the bifilar mixer.

1.4 DOUBLY BALANCED STAR CIRCUIT The star type of mixer was explored with by Hahnle in 1937 [26], where four diodes are connected to a common point. The other terminals are connected to identical transformers that couple power from LO, RF and deliver the converted signal to a load. The mixer diagram contained in Figure 1.33 shows all cathodes connected to the star point and the transformers have been split to simplify the figure. Therefore, transformer U1 actually comprises a transformer with one primary magnetically coupled to two secondary windings; the same is true for transformers U2 and U3. The connections are such that the signals applied to one transformer are orthogonal to the other two transformers, thus providing isolation and differential application to the diode tubes. Also, it is immaterial if the LO and RF, respectively, are applied to U1 and U2 or to U2 and U1. The mixer is proposed to reduce the undesired harmonics of the modulating signal (i.e., fh-2fn, fh+3fn, where fh = carrier frequency and fn =

30

Microwave Mixer Technology and Applications

modulating frequency). One approach to reduce the harmonics is to reduce the magnitude of modulating voltage so that the rectifier resistance is a linear function of current. The other is to have the carrier strongly modulate the tubes resulting in a rectangular waveform with constant on and constant off resistances. The tube has an internal resistance that varies from 5,000 ohms to practically infinite, 50,000,000 ohms. Making the load resistance the geometrical average of those two values, 500,000 ohms results in small variations in the on and off resistances. Therefore, the sidebands, fh + 2fn and above are greatly reduced.

Figure 1.33

Vacuum tube diode star mixer with the star center grounded. RF and LO signals are magnetically coupled to the diodes as well the load collecting the IF signal.

1.5 SPECIAL RECEIVER ARCHITECTURES Special architectures were developed in the early days of radio and are still in use today, such as quadrature mixers used for image rejection and SSB generation. They are the object of this section along with circuits specifically invented for TV receivers. 1.5.1 SSB Concept The concept of SSB transmission started with the work of Carson [27], who noted that in normal AM systems most of the energy is located in the carrier, which does not transmit any information. With or without a signal at the input of the system

Origins of Electronic Mixers

31

the same energy is wasted in transmitting the carrier. His basic patent disclosed the key process to generate multiplex systems. He assumed the transfer characteristic of a modulator follows a nonlinear relationship given by the equation.

V  av  bv 2  cv 3  dv 4  ...

(1.10)

Hence, if the input signals are the carrier and the modulating signal, the total input voltage is given by:

v  A cos( LO t )  B cos(IF t )

(1.11)

Applying v into equation for V, the output will contain a series of terms that can be filtered and only the signals at the vicinity of LO are retained. Therefore:

v  aA cos( LO t )  2bAB cos( IF t ) cos( LO t )  cv 3

(1.12)

Rearranging the terms, the equation modifies to:

v  aA cos( LO t )  bAB cos( LO   IF t )  bAB cos( LO   IF t )

(1.13)

The first term represents the carrier that is unmodulated and the other two represent the modulated carrier that contains the desired information. To eliminate the carrier, the inventor proposed making a = 0, which can be done by using two identical modulators 180° out of phase. One modulator follows the equation for V and the other has its input phase reversed by 180 and follows V’. Therefore,

V '  a(v)  b(v) 2  c(v) 3  d (v) 4  ...

(1.14)

If the output signal of both modulators is added, then the total output voltage is equal to:

V ' '  V  V '  2b(v) 2  2d (v) 4  ...

(1.15)

Assuming coefficient d > nVT, the current source generates a high current according to (3.1), equivalent to a low resistance shorting the capacitance. Notice the nonlinear parameters are defined in terms of junction voltage that is not accessible, as voltage is measured at the external terminals including the intrinsic series resistance, Rs. The thermal voltage VT is a temperature dependent parameter having a value of 26 mV at room temperature. The ideality factor, n, denotes how much a junction deviates from the ideal exponential I,V, and equals unity for an ideal diode. The saturation current, IS is defined by (3.2). Id

Ls

Rs

A Vd C Figure 3.1

Cp

Cj(Vj)

Vj

I(Vj)

A (+)

C (-) (a) Circuit schematic. (b) Symbol. Model for a Schottky diode. A refers to anode and C to cathode. Vj is the voltage at the junction and Vd is the voltage at the external terminal. After [2].

The parasitic elements Ls, Cp are dependent on package configuration and assembly. The beam lead packaging approach [3] provides lower Ls and Cp than a wire bonded die, making it ideal for microwave and millimeter-wave applications. If diodes are inserted into plastic packaging, the parasitics limit the frequency of operation.

Semiconductor Modeling

 Vj  I  I S  e VT  1    

107

(3.1)

Where, VT = kT/q k = 1.38 x10-23 J/ºK = Boltzmann constant T = temperature, ºK q = 1.6x10-19 C = electron charge IS = diode saturation current  = ideality factor

I S  Ad A**T 2e

 q B KT

(3.2)

It is interesting to observe that Is is proportional to the diode area, Ad, and to the square of temperature, T. The additional constants are the Richardson constant, A**, which equals 120 A/cm2/K2 for free electrons, and the barrier height, B, which is approximately equal the built in potential, Vbi. If the diode is operating with a reverse voltage applied to its terminals, then the current given by (3.2) is reduced to zero. If the reverse voltage is higher than VB, then avalanche breakdown can occur and the current equation is modified to (3.3). The term VB represents the avalanche breakdown of the diode and mb is a fitting parameter.

I

IS Vj 1    VB

  

(3.3)

mb

The junction capacitance is the important nonlinear element in a reverse biased junction and is described by (3.4). The parameter Cj(0) represents the capacitance with diode biased at zero volts, and m represents the type of junction and is in general equal to 1/2.

C j (V ) 

C j (0)  Vj  1    Vbi 

m

(3.4)

If the diode is forwarded biased with Vj close to Vbi, then capacitance approximates infinity causing numerical problems in simulators. Due to this

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Microwave Mixer Technology and Applications

model limitation, the capacitance model is changed to a linear C(V) when Vj is near Vbi. This capacitance description works well in time domain simulators but in harmonic balance simulators, using Q(V) works better. 3.1.2 Model Extraction The relation between voltage and current for the circuit in Figure 3.1(a) is given by (3.5). At low currents the voltage drop in the series resistor can be disregarded and the convenient (3.6) results. Based on this equation one can measure the diode I/V and plot the results in the form of log(Id) versus Vd.

I  Vd  RS I d  VT ln  d   IS  V ln( I d )  ln( I S )  d VT

(3.5) (3.6)

Log(Id) Vd

Slope=1/RS

Slope=1/(nVT)

IS Vd Figure 3.2

Plot used to determine the Schottky diode to determine, IS, VT, Vbi and RS.

From this plot, Is, is obtained by extrapolating the line to the y-axis crossing Vd = 0. At this voltage the Vbi is also obtained by applying (3.7). The slope of the straight line provides the ideality factor, .

Vbi 

1  A**T 2  ln  Ad  VT  I S 

(3.7)

The series resistance RS is found from the difference in voltage Vd between ideal I/V and the slope caused by the series resistance, RS =Vd/Id at high

Semiconductor Modeling

109

S(1,1)

currents. A better evaluation of series resistance is obtained from S-parameters at high frequency with an unbiased diode. The equivalent circuit is then the series resistance and a capacitance to ground, Cj(0). The S11 plot for a typical GaAs diode is in Figure 3.3 showing a resonance at 29 GHz between the equivalent series capacitance and parasitic inductance. The series resistance obtained from RF measurements maybe a little different than the one measured at DC, due to Reflection temperature effects on the junction,Input which doCoefficient not exist in the unbiased diode.

m2

m2 freq=29.00GHz S(1,1)=0.852 / -179.539 impedance = 4.000 - j0.200

freq (100.0MHz to 30.00GHz)

Figure 3.3 Smith chart plot of diode S11 for a typical GaAs diode. Determination of RS, Cj(0), and Lp.

3.2 MODELING BIPOLAR TRANSISTORS The bipolar junction transistor (BJT) mixer, depending on circuit conditions can provide power gain in the conversion process, in contrast with Schottky diodes where conversion is always lossy. Since the early 1960s silicon technology has dominated the market for BJTs: silicon devices have become capable of operating at low microwave frequencies and are still employed in active mixers up to the UHF band. Recently, gallium arsenide (GaAs) and silicon germanium heterojunction bipolar transistors (HBTs) were developed with superior performance at high frequencies compared to silicon devices. Therefore, it became an option for mixer design in microwave/mmWave frequencies. HBTs are available only in npn, while silicon bipolar devices are available both in npn and pnp, even though for RF applications only npn is employed due to its higher mobility resulting in more gain at higher frequencies. 3.2.1 Ebers-Moll and T-Model The Ebers-Moll model, [4], was developed in the early 1950s with the purpose of representing the bipolar transistor I/V characteristic. The circuit is in Figure 3.4 consisting of two diodes and two current sources controlled by current. The currents through the diodes are described by (3.8) and (3.9) and the terminal currents are on (3.10) to (3.12). Only four parameters are necessary to be determined, IES, ICS,  F, and R. The first two can be determined by measuring the

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Microwave Mixer Technology and Applications

device IC as a function of VBE and IE as a function of VBC. The other two parameters, are the common emitter forward and reverse current gain, which are assumed to be independent of bias in this model. The i parameters are obtained from (3.13) with i = F for forward and i = R for reverse operation. C

IB

IC

IR

FIF

IF

RIR

B

C

B E

E

IE

(a) Circuit schematic. Figure 3.4

(b) Symbol.

Nonlinear Ebers-Moll model consisting of four resistive parameters. After [4].

 qVKTBE  I F  I ES  e  1   qVBC   I R  I CS  e KT  1   IC   F I F  I R I E  I F   R I R I B  1   F I F  1   R I R i   i /(1   i )

(3.8)

(3.9) (3.10) (3.11) (3.12) (3.13)

The family of input and output DC I/V characteristics for a common emitter configuration provided by the model is shown in Figure 3.5. The transfer characteristics ICE(VBE) is not shown but it is similar to the input exponential characteristics. The ICE(VCE) plot depicts the normal operating region where the emitter base junction is forward biased and the base collector junction is reverse biased. This is called the first quadrant of operation, where ICE and VCE are positive. The inverse region of operation corresponds to operation in the third quadrant with base emitter junction reverse biased and base collector forward biased. The cut-off region is defined when both junctions are reverse biased.

Semiconductor Modeling

ICE

IBE

111

IBE1 IBE2 IBE3

VCE

VBE (a) Input IE BE(VBE) plot. Figure 3.5

(b) Output ICE(VCE) with IBE as a parameter.

DC I/V for a bipolar device modeled after Ebers-Moll.

This simple model can provide reasonable accuracy if the access resistances and capacitances are added to the model and if device operation is within a narrow voltage range relative to bias. The T-model is obtained from Ebers-Moll model by making IR = 0 so that currents are described by (3.14) – (3.16). A base resistor, rB and capacitances are added to the circuit model, making it useful for manual calculations and is very efficient in a nonlinear simulator.

 qVBE  I E  I ES  e KT  1   IC   F I E

(3.14) (3.15)

I B  IC / 

(3.16) C

IC

CB B Ib

rB

FIE

C

B’

IE

CJE CD

Figure 3.6

E

Nonlinear T-model consisting of three resistive parameter and three reactances.

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Microwave Mixer Technology and Applications

The base emitter capacitance includes two terms: diffusion, CD, and depletion, CJE. The diffusion term corresponds to capacitance created by charges injected by the emitter into the base. The charge in the base, Q B, depends on the magnitude of collector current and on the time the carriers take to transit through the base, F, defined by (3.17). The diffusion capacitance is then obtained by the derivative of the charge defined by (3.18). The depletion term is given by the capacitance of a reverse biased junction, (3.19). The base collector capacitance CBC, is also caused by depletion and is defined by a similar equation.

QB  I C F dI C CD   F  g m F dVBE C JE 0 CJ  V 1  BE Vbi

(3.17) (3.18) (3.19)

As long as the signals swinging in the I/V planes are contained in the first I/V quadrant, where both I and V are positive, the collector junction is reverse biased and emitter junction is forward biased. Under these conditions, CBC is much lower in value compared to the emitter capacitance, CBE = CJE + CD. 3.2.2 Gummel-Poon If more accurate results are required, then a more complete analysis of the device under large signal operation is needed, which can be provided by the popular GP Gummel-Poon model [5]. This model is derived from the Ebers-Moll with added circuit elements and modified equations, based in the device physics. The circuit schematic in Figure 3.7 shows the GP model has more parameters to be determined, for a complete characterization. The first current, ICT, called the transport (3.20) is defined by the difference between forward and reverse currents, (3.21) and (3.22). The coefficients F, R are ideality factors of base emitter and base collector diodes, respectively, and C, E are called base-collector and baseemitter leakage emission coefficients. One can notice the saturation current is no longer constant; a dimensionless coefficient qbb has been introduced that takes into account two effects. The first corresponds to the base width modulation, also called Early effect, (3.24). This effect appears in the I/V characteristic as a slope on the current function of voltage in the manner indicated in Figure 3.8.

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113

IBC B'

RB

B

CJ C

C

R

C'

C

VCB CJ

VB'

E

E

E' Figure 3.7

CD

VBE

ICT=ICC-IEC

VC'

E

IBE

RE

E

E'

Gummel-Poon model for a BJT including access resistances and additional diodes to account for base current leakages at low currents. After [5].

I CT  I CC  I EC I CC 

I EC

VBC VBE I S   FVT  e RVT e qbb 

(3.20)

  

VBC  VBCV   I S   RVT   e  1  I SC  e C T  1     R    



qb 1  1  4q S 2 1 qb  VBE VBC 1  V AR V AF qbb 

qS 

IS I KF



 VBC   nVBEV   e F T  1  I S  e nRVT  1   I KR      

(3.21)

(3.22)

(3.23) (3.24)

(3.25)

Where qb, qS and qbb are dimensionless coefficients. The second effect on qbb corresponds to high level injection where qs >> qb, so that (3.21) becomes approximately described by (3.26). The ideality factor in this condition is doubled reducing collector current at high currents. Those equations were developed for silicon BJT and not all of them are relevant for HBT modeling. Actually the important terms for HBTs are the saturation current, IS,

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Microwave Mixer Technology and Applications

forward and reverse Early voltages, VF, VR,and forward and reverse corner currents, IKF, IKR.

Figure 3.8

I/V plots for a typical BJT device modeled by Ebers-Moll, (black) and Gummel-Poon (gray), showing a positive slope.

I CC  I S I KF e

VBE 2 nFVT

(3.26)

The BJT base currents are represented in the model by splitting the current into two components, a term that is a fraction of collector current and another related to leakage currents in the device. The diode base-emitter current, IBE, and base-collector current, IBC, are given by (3.27) and (3.28), with ISE and ISC defined as the base-emitter and base-collector leakage saturation current. The two diode models are still used for III-V HBTs, with different saturation current and ideality factor for each term [6]. VBE   VBE  I S   FVT e  1  I SE  e EVT  1     F     VBC VBC    I   S  e RVT  1  I SC  eCVT  1     R    

I BE 

(3.27)

I BC

(3.28)

The common emitter DC current gain defined by  = IC/IB is a function of bias current and its representation simulated on ADS using the GP model is in Figure 3.9. Gain is relatively constant within a certain range of bias and it drops at low and high currents. The low current gain drop simulates the effect of leakage currents at the device surface that are not controlled by base voltage. The gain reduction at high current levels simulates the emitter crowding and Kirk effects. The emitter crowding occurs at high levels due to nonuniform current density within an emitter finger. The term is usually used to describe the situation in

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115

 = IEC/IBE,

which high current density appears at the edge of the emitter stripe, resulting from voltage drop across the base. Consequently, the device becomes unevenly biased with reduced collector current for the same base current. This effect is not relevant in HBTs, where the base has low resistance compared to silicon bipolars, and the voltage along the emitter surface is evenly distributed.

Figure 3.9

Current gain, , as a function of collector current for a typical BJT modeled after Gummel-Poon. The same plot using Ebers-Moll model would give a flat gain from low to large currents.

The Kirk effect occurs in both homo and heterojunction devices and results from the increased charge density associated with high current flowing through the base-collector region. In this region of high and low fixed charge density, respectively, an excess charge builds up at forming a dipole at the collector side of the collector-base region, increasing the electric field in that area, making the base extend over the collector. In this process a large current of holes is injected into the base to maintain charge balance and the base transit time increases. These effects reduce the device current gain. The capacitances in the GP model includes both diffusion and depletion terms similar to the T-model, (3.29) – (3.30). The diffusion capacitance is in the first term of the equation, where high level current effects are included in the GP model through the coefficient qbb. VBE

CBE

 V  I S V  F e  C JE 0 1  BE  FVT qbb  Vbi

CBC

 V  I  R S e V  C JC 0 1  BC  RVT qbb  Vbi

T

VBC

R T

  

 m je

  

(3.29)  m jc

Where F = represents the forward base transit time R = represents the reverse base transit time

(3.30)

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Microwave Mixer Technology and Applications

Recent studies [7, 8], revealed the diffusion charges are determined from the integral of current by integrating the small signal transit time, F, times current, Qin = F(Vbc,I)dI instead of simple multiplication as indicated in the previous equations. Those effects are in place in more advanced models and the GP equations can be modified by replacing them with the equations provided in the references. 30

fT(GHz)

20

10 VC E

0 10-3 Figure 3.10

10-2

10-1 JC[mA/µm2]

100

101

Transit frequency dependency on current density and collector voltage for bipolar devices for a Si bipolar measuring 0.4X14µm2. After [10].

Two figures of merit frequently used for transistors are the fT and fmax. The fT is defined as the frequency where the common emitter gain is equal to unity, (3.31), [9]. The resulting equation shows fT as a function of device gm and capacitances. The parameter fmax is defined as the maximum frequency of oscillation of a device, and according to (3.32) is a function of base resistance and device capacitances.

fT 

gm 2 (C D  C JE  C JC )

f max 

fT 8 rBB C JC

(3.31)

(3.32)

The limiting frequency of operation is seen in (3.31) to be dependent on collector current. This is observed in the plot of Figure 3.10 showing fT increasing more or less monotonically with current density, up to a peak value, [10]. As already discussed the base transit time increases at high currents with resulting fT

Semiconductor Modeling

117

reduction. Those parameters are small signal derived and strongly dependent on biasing conditions. Even though they are not very useful in large signal applications or in digital circuits, they are used as a figure of merit for technology performance. The GP model was developed for silicon bipolar devices and successfully provides gain, power and efficiency information and works well for such devices at moderate frequencies. The temperature models contained in the GP model are designed for silicon bipolars and are not capable of taking into account self heating effects which is common in HBTs. But in spite of differences in device physics and technology between BJT and HBT, the GP has been used to model HBTs with relative success. In a recent publication, [6], it is shown how modifications in the GP model can improve its representation of HBTs in PA applications. Other advanced models were developed to improve simulation of real devices, including linearity and temperature effects, namely, Mextram, Hicum, and VBIC models. Mextram was developed by Philips and Hicum, High Current Model, was jointly developed by Conexant and Dresden University. The VBIC or Vertical Bipolar Inter Company is a public domain model receiving contribution from many industries. An additional model important for HBT is the UCSD or University of California in San Diego model. 3.2.3 Model Extraction Automated equipment and related software capable of acquiring the necessary data for extracting the parameters of most common models are available in the market. However, the Gummel-Poon model can be extracted with simple DC supply, multimeters, and a network analyzer to measure S-parameters. The parameters to describe I/V are obtained from the DC plot log(I) versus VBE in the forward and reverse mode, Figure 3.11. IC

Log(I) IK

IB

Slope=1/RE

 Slope=1/nFVT

ISE IS

Slope=1/nLVT VB E

Figure 3.11

Plot of IC, IB as a function of VBE to determine the DC parameters of GP model in the normal mode. A similar plot can be carried out in the reverse mode.

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Microwave Mixer Technology and Applications

Forward mode is defined for operation with V BE > 5FVT and VBC < 5RVT also denoted as the normal operating region. The reverse mode corresponds to VBE < -5FVT and VBC > 5RVT. As already discussed in the diode section, the plot corresponding to the collector current provides, IS, F. The same parameters are determined from the base current plot. IK is also determined from same plot and is defined as the current at the crossing of two asymptotes from low and high injection base currents. The DC current gain, defined by the ratio IC/IB is linear within the region where both plots IC and IB are parallel to each other. The Early voltage VAF is obtained from the extrapolation of IC(VCE) curve in the linear range at the point where it crosses the horizontal axis, i.e. IC = 0, Figure 3.12. In the reverse mode the Early voltage is denoted VAR. IC

VAF Figure 3.12

VCE

Plot to determine the Early voltage.

The emitter resistance can be determined by different methods [11, 12], a straightforward procedure is to use the open-collector measurement, based on (3.33). By applying a current source to the base of a grounded emitter device, the approximate voltage on the emitter resistance is measured by applying a voltage meter at the collector, Figure 3.13. A plot of VCE ( VRE) versus IE = IB provides a nearly straight line whose slope is equal to RE. The collector resistance is determined by a similar process with open emitter and grounded collector. VCE

VCE Slope = 1/RE

IB

VBE

Figure 3.13

IB

Schematic of measurement of VCE (IB) with IC = 0 to determine RE. After [11, 12].

 1  VCE  VT ln    RE I B  R 

(3.33)

The base resistance uses a similar measurement set up with an additional voltmeter to monitor VBE as well, [13]. Another equation applicable to the circuit provides the relation between base voltage and base resistance

Semiconductor Modeling

119

 I  VBE  VT ln   F B   RE I B  RB I B IS  

(3.34)

Subtracting (3.34) from (3.33) and dividing by IB, gives an equation relating the difference of collector and base voltage to the base current, (3.35). This ratio when plotted as a function of 1/IB results in approximately a straight line, whose extrapolation to 1/IB = 0 gives the value of RB depicted in Figure 3.14.

VBE  VCE VT   IB  1   ln   F   ln   RB IB IB   IS  R 

(3.35)

(VBE-VCE)/IB RB Figure 3.14

1/IB

Plot of linear equation (3.35). The term in the bracket is dominated by ln(IS). After [13].

The access resistances can also be determined from S-parameters at a range of frequencies low enough to allow disregarding capacitive effects [5]. The method consists in the application of a base current and DC shorting the emitter and collector to ground. The bias current should be sufficiently high to induce a small dynamic resistance, Rd, in the base-emitter and base-collector junctions. The equivalent circuit is in Figure 3.15 showing the access resistances and parasitic inductances.

Rd 

VT IB / 2

(3.36)

Lb 50 

Ib

Rb

Rd Vc RC Rd

Rce Ve Re Le

Figure 3.15

LC Lbias Ib/2

50  Cbias

Schematic of bipolar device with base overdrive current with emitter and collector grounded. After [5].

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Microwave Mixer Technology and Applications

The extraction process requires Vc = Ve which is ensured by having similar current in both base-emitter and base-collector junctions. Since RC is different than Re a current source is inserted into the DC collector circuit to force equal current Ib/2 in both junctions. The measured S-parameters are converted to Z-parameters and compared with the Z-parameters for the circuit in Figure 3.15, described in (3.37).

Re  jLe  Rd / 2  Rb  Re  j ( Le  Lb )  (3.37)  Re  jLe Rc  Re  j ( Le  Lc )  The junction capacitances are obtained from unbiased device (cold BJT or HBT), [14], whose equivalent circuit for small devices is represented in Figure 3.16. A transistor cell with few emitters is considered small in this context. The unbiased circuit refers to the absence of collector current. Bias is however applied to the base to determine capacitance as a function of voltage. The pad capacitances C11p and C22p are determined separately. The access resistances are de-embedded from the measured S-parameters. Then a two port Y-parameter matrix is built for the remaining circuit, from which the capacitances are easily extracted. Rbc

Rb

RC

C11p

50 

Cbc

Rbe

C22p

50 

Cbe Re

Figure 3.16

Schematic of a BJT or HBT to extract the junction capacitance as a function of bias. After [14].

From the plot of capacitance versus voltage, CJE0, CJC, MJE and MJC are determined. The diffusion capacitance was demonstrated in [15], to be defined as a function of the GP parameters indicated in (3.38).

 F

VAR C JE I KF

(3.38)

The standard procedure in device modeling is to gather as many as possible parameters from DC and RF/microwave measurements. Those are considered the initial parameters or initial guess of preliminary model that are then

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121

submitted to a curve fitting process where the parameters are numerically fine tuned to optimize the match between measured and model parameters. 3.3 MODELING FIELD EFFECT TRANSISTORS As discussed previously, the I/V characteristic of diodes and bipolar transistors are described using exponential functions. Approximating that relation in terms of a power series results in square, cubic and higher order terms, which translate into spurious signals when those devices are employed as mixers. The initial application of silicon junction field effect transistors (JFET), was very successful due to its square law relation between drain current and gate voltage. The mixing properties exhibited lower distortion content allowing for operation at higher signal levels compared with exponential I/V devices. The next FET type introduced was the silicon MOS, or metal oxide semiconductor that also exhibits a square law characteristic and became used as a low frequency mixing element. The MOS technology including CMOS, which stands for complementary metal oxide semiconductor, was initially applied to RF applications and to analog circuits. Later CMOS entered into the digital world after its speed became attractive to digital designers and in recent years became applied to high frequency analog circuits. The development of this technology was followed closely by sophisticated physics based models capable of predicting manufacturing variables relating to the device performance. One of the first high frequency FETs appeared in 1967 as a gallium arsenide MESFET, [16], short for metal semiconductor FET, showing a fT of 1 GHz. Microwave silicon MESFETs were more advanced at the time, in 1970 a fmax of 12 GHz and 12 dB gain at 2 GHz was reported, [17]. The GaAs FET devices provided impressive breakthroughs for signal processing components and later evolved into other types of devices including the high electron mobility transistor (HEMT), and pseudomorphic high electron mobility transistor (PHEMT) in depletion and enhancement forms. In depletion mode there is current flowing between source and drain with no bias applied to the gate. Application of a negative potential on the gate decreases the flow of current due to smaller channel thickness controlled by a "depletion" region under the gate. In enhancement mode the device is constructed so that the channel is already depleted at zero gate bias. A positive potential on the gate reduces the depletion thickness allowing more current to flow between source and drain. The enhancement devices became attractive in many applications because both drain and gate are biased positively compared to depletion where a symmetrical supply voltage is required. The analytical models initially developed for JFETs were modified to represent these devices. They are sufficient to predict conversion loss and impedance matching given variations in frequency and bias. However, they are not adequate for representing certain large signal behavior. The following

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Microwave Mixer Technology and Applications

section reviews a few analytical FET models that are adequate for mixer analysis including distortion effects. 3.3.1 Curtice One of the first nonlinear GaAs FET models was proposed by Curtice [18] in 1980, where the theoretical transfer characteristic of a depletion mode MESFET has a quadratic relation between drain current and gate voltage. It is common to refer to this model as quadratic Curtice. The schematic of the large signal n-type model is represented in Figure 3.17. The equations describing the current are given by the set of (3.39) – (3.42), separated into two parts, one for gate voltage above threshold, and the other for gate voltage below threshold where the current is ideally zero. The threshold voltage is therefore defined as the gate-source voltage that causes drain current to equal zero. Additionally the active area is defined as the product of two functions, one is a function of gate voltage, F(VGS), and the other a function of drain voltage, G(VDS). The nonlinear equations are described as a function of internal gate-source and gate-drain voltages. The model shows access resistances RG, RD, RS that represent the effect of materials employed in the device structure to make the connection between the internal gate, source and drain to the external circuitry.

I DS (VGS ,VDS )  F (VGS )G(VDS ) for VGS > Vth

(3.39)

F (VGS )   (VGS  Vth ) G(VDS )  (1  VDS ) tanh(VDS )

(3.40)

I DS (VGS ,VDS )  0 for VGS  Vth

(3.42)

2

(3.41)

Where VGS = gate source voltage, V VDS = drain source voltage,V Vth = threshold voltage, V The drain current at zero gate voltage is equal to IDSS, and  = IDSS/Vth2 at this point. The threshold is defined as the gate voltage where drain current transitions from zero and starts flowing in the circuit. Another form of (3.39) normally found in text books is (3.43) defined in terms of pinch off voltage, VP, the voltage that pinches off the channel making drain current zero. Since this transition region is not well defined, semiconductor manufacturers prefer to define VP as a percentage value of IDSS. The current in the region between VD'S' = 0 and VD'S' = VDsat is given by (3.41) in the form of a hyperbolic tangent. The saturation drain voltage, VDsat expresses the point above which the hyperbolic term becomes constant, reflecting a nearly constant current. The drain current is therefore said to

Semiconductor Modeling

123

be saturated when VDS > VDsat. The drain source conductance is defined by the parameter, . G'

RG

DGS

S'

Cds

VGS S RS

RD

D

CGS

VG'S'

Figure 3.17

CGD

G

D'

VD'S'

ID(VGS,VDS) S'

Curtice_I nonlinear MESFET model. After [18].

 V F (VGS )  I DSS 1  GS  VP

  

2

(3.43)

Equations (3.39) – (3.42) do not describe device behavior beyond the gate voltage limits, determined on the positive side by the Schottky junction conduction and on the negative side by the breakdown. Positive gate voltage: A Schottky diode represents the gate source junction, so the gate-to-source voltage, VGS, can become positive as long as there is no gate conduction. The maximum voltage is limited by the forward voltage of the Schottky junction, VF, and in most MESFETs this limit is around 0.6 V. The drain current from VGS = 0 to 0.6 V increases above IDSS and limits out to a value called Imax provided by the manufacturer. Voltages above this limit should be avoided because RF losses are introduced in the gate circuit and cause higher order circuit nonlinearities and the device may be destroyed. Negative gate voltage: When the reverse gate voltage exceeds the threshold voltage, the conducting channel within the device is pinched off and ceases to conduct. The limit is the breakdown of the gate source Schottky diode, ranging from a few negative volts for a small signal device to more than -5 V for power devices. The limit on the drain side is set by the breakdown of the drain-to-gate junction. This junction is reverse biased in normal operation, but if the drain voltage reaches the breakdown value, VB, then drain current starts flowing into the gate circuitry, degrading performance, and depending on temperature and voltage it can destroy the device. The breakdown effects are not taken into account in the original Curtice model. The red traces plot in Figures 3.18(a) and (b) illustrate the I/V of device NE673 and the Curtice model in black traces are superimposed. If the circuit operation is constrained to operate within the dotted line depicted in Figure 3.17(a), the model is useful for estimating certain classes of mixers as will be detailed in further chapters.

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Microwave Mixer Technology and Applications

0.100

0.120

0.075

test..IDS.i, A IDS.i

test..IDS.i, A IDS.i

0.096

0.072

0.048

0.050

0.025

0.024

0.000

0.000 0

VDsat

1

2

3

4

VDS

5

6

VB

0.00

0.25

0.50

VGS

(a) IDS(VDS, VGS) Figure 3.18

-2.00 -1.75 -1.50 -1.25 -1.00 -0.75 -0.50 -0.25

(b) IDS (VGS)VDS= 3V

I/V for NE673 device for VGS between -1.6V and = 0.4V.

The capacitances in the model follows the "division by capacitance" principle, [19], where the charge in the gate is controlled by the gate-source and gate-drain voltage, therefore:

Ig  Is  Id 

dQg dt



Qg dVg Vg dt

Qg dVg Vg dt Qg dVgd Vgd dt



Qg dVgd Vgd dt

(3.44)

(3.45)

(3.46)

This is consistent because Ig = Id + Is, and the capacitances are defined as in the conventional manner:

C gs  C gd 

Qg Vgs Qg Vgd

(3.47)

(3.48)

In this model the author assumed the gate-drain capacitance to be constant, which is a good approximation if the signal swings within the saturated area. For the gate-source diode the model uses the capacitance of a reverse biased Schottky junction defined by (3.4). Comparing this capacitance model with

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125

measured data from device NE67300 MESFET the error can be as high as 15 % if the gate voltage swing is within pinch off and zero, Figure 3.19. If gate voltage goes above zero the real capacitance becomes approximately constant, while the capacitance from the Schottky equation rises sharply. Therefore, for signals giving a positive gate voltage the error in the capacitance becomes too high. In spite of this limitation, many designers still use this capacitance model due to simplicity of model extraction and reasonable estimate of certain circuits. One problem with this model is the nearly infinity value reached when gate voltage is on the order of the built in voltage, creating convergence problems in the electrical simulator.

Figure 3.19

Gate-source capacitance from device NE673 in black plot compared with Schottky junction capacitance in red plot for VGS between -1.8 to 0 V.

Instead of using hyperbolic tangent in (3.41), some authors [20] prefer the cubic expression of (3.49). The resulting current description is similar and it is easier to calculate trans-conductance derivatives. Table 3.1 shows typical values for the MESFET NE673 by NEC, device with 0.3 µm gate length and 280 µm width.

  V 3  G (VDS )  (1  VDS ) 1  1  DS   for 0 < VDS < 3/ 3     G(VDS )  (1  VDS ) for VDS > 3/ Table 3.1 Curtice Model DC Parameters, NE673

DC Parameter β α λ Vp

Value 0.038 7.8 0.102 -1.55 volt

DC Parameter Rs Rd Rg

Value 3.0 Ohm 1.5 Ohm 4.0 Ohm

(3.49) (3.50)

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Microwave Mixer Technology and Applications

Equation (3.39) can also be used to represent enhancement mode devices simply by changing the gate and threshold voltage polarities. The concept is still the same: gate voltage above which drain current starts flowing in the channel. Notice the term IDSS is defined at zero gate voltage for depletion devices and has no meaning for enhancement mode. The symbol used to represent GaAs MESFETs, Figure 3.20, was borroughed from JFETs, with an arrow at the gate indicating type of channel. If type N, the arrow is from left to right, if type P, the arrow is from right to left. Since there is no type P MESFET devices, there is no need for an arrow and with time it was eliminated. In spite of wide spread usage of the MESFET symbol, some publications still use the JFET symbol to represent MESFETs. D G

D G

S

(a) Figure 3.20

S

(b)

FET symbols: (a) JFETs has an inward arrow representing n type; (b) MESFET has no arrow because only n type are manufactured.

3.3.2 Curtice-Ettenberg The transfer characteristic of real MESFETs is not quite quadratic depending on gate length, doping profile of the transistor active layer, device size, and other parameters. In 1985 Curtice and Ettenberg noticed that it is more accurate to represent the same relation by a cubic approximation [21], represented by the voltage series in (3.51). This approach is also called cubic Curtice by some authors. The voltage V1 is represented in (3.52) as a function of drain voltage that invalidates the assumption for the function F(VGS) to be a function of gate voltage alone. This is an important correction required for power amplifiers; however, for small signal devices used in mixer applications c can be made equal to zero making F(VGS) a function of VGS only. 2

3

F (VGS )  A0  A1V1  A2V1  A3V1

V1  VGS e

 j 

1  c VDS 0  VDS 

(3.51) (3.52)

Where A0,…A3 = fitting parameters  = A4VDS - the time delay accounts for the time signal takes to transit between source and drain VDS0 = drain voltage where the An coefficients are measured. c = pinch-off voltage coefficient

Semiconductor Modeling

127

The equivalent circuit represented in Figure 3.21 shows the following changes compared to the previous Curtice model: introduction of resistor Ri to represent the time constant to charge CGS and resistor RDS to adjust the output conductance. The gate-source diode was eliminated and a current source was added to simulate the Schottky gate-source junction, (3.53). Gate current starts to flow for gate voltage amplitudes larger than Vbi in the Schottky junction IDG(Vout,Vin) G'

RG

CGS

RDS VGS

Vin(t)

Ri IGS(Vin)

S' Figure 3.21

D'

D RD

CGD

G

RS

CDS

S

ID(VGS,VDS)

Vout(t) S'

Curtice-Ettenberg nonlinear MESFET model. After [21].

A current source was introduced between drain and gate to represent the breakdown effect; the current is defined by (3.54). The resistance R1 represents the breakdown resistance and R2 is the resistance relating breakdown voltage to channel currents.

Vin (t )  Vbi , forVin (t )  Vbi  RG I gs    0, forVin (t )  Vbi

I dg

Vdg (t )  VB , forVdg (t )  VB  R1   0, forVdg (t )  VB

VB  VB 0  R2 I DS

(3.53)

(3.54)

(3.55)

A different model for the gate-source capacitance was proposed, which is a function of both drain and gate voltages, extracted from measured data. The

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Microwave Mixer Technology and Applications

coefficient C0 is the value of CGS evaluated at VGSDC and biased at VDS. The gatedrain capacitor was assumed to be linear. DC

3.75  VGS CGS  0.909[1  0.0125VDS ] C0 3.75  VGS

(3.56)

The model can be applicable to the enhancement mode by proper change of gate and threshold voltages from negative to positive. In spite of certain drawbacks this model is still often used by designers. One of the disadvantages of this model is the fact it may not be able to model a zero transconductance when the gate is pinched off. S. A. Maas [22] provided a simple correction to this equation for these cases. The corrections are as follows:

A0  A0  I DS

Vmax Vmax  VP

(3.57)

A1  A1  I DS

Vmax Vmax  VP

(3.58)

Where Vmax = maximum gate voltage, can be zero volts IDS = error of IDS at VGS=-VP The plot of Figure 3.22(a) shows the simulated I/V (gray) compared to measured data (black) with corrections to provide zero current and transconductance, simulated in part (b) of figure. The parameters used to generate the plots are listed in Table 3.2. 0.100

100

0.086

80

IDS.i, A gm_mS

test..IDS.i, mA IDS.i, mA

0.071

60

40

0.057 0.043 0.029

20 0.014

0

0.000

0

1

2

3

4

VDS

(a) IDS(VGS)VDS=3V Figure 3.22

5

6

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

-0.0

0.2

VGS

(b) gm(VGS) VDS=3V

Curtice model with cubic transfer characteristic simulated in gray compared with the measured data in black traces for the device NE67300.

0.4

Semiconductor Modeling

129

Table 3.2 DC Parameters Cubic Model

Parameter A0 A1 A2 A3 β

Value 0.0709 0.0605 0.0052 -0.0036 0.074

Parameter Vds0 γ Rds0 Rd Rg

Value 4.0 3.2 3000 1.4 2.0

3.3.3 Chalmers (Angelov) The previous FET models are reasonable for estimating conversion gain and impedance for a mixer. If more accurate results are required and there is a need for linearity information one has to look for a model capable of representing in addition, the distortion levels, IMD2, IMD3. This is possible if the first, second and third derivatives of d(ID)/d(VG) are not discontinuous over the range of device operating voltages and currents. The Chalmers model, also known as Angelov, whose circuit schematic is illustrated in Figure 3.23 is capable of providing this feature [23]. In this model the gate dependent part of current is described by a hyperbolic represented in (3.59) and the argument is given by the series in (3.60).

I DS (VGS ,VDS )  F (VGS )G(VDS ) I DS (VGS ,VDS )  I pk (1  tanh( ))(1  VDS ) tanh(VDS )

  P1 (VGS  Vpk )  P2 (VGS  Vpk ) 2  P3 (VGS  Vpk )3  ...  g m  I pk sec h 2 ( ) G(VDS ) VGS

(3.59) (3.60) (3.61)

Its derivative in (3.61), contains a squared hyperbolic secant whose shape accounts for a Gaussian bell shape type, which, if properly adjusted can represent various shapes of FET transconductance. The parameter Ipk is the drain current and Vpk is the gate current at the gate voltage where the transconductance is maximum, gmpk. The transconductance gmpk is intrinsic which means the series access resistances have to be taken into account. If one considers the first derivative of current in relation to VGS, the term P1 defines the transconductance at Vpk; the term P2 makes the derivate of the drain current asymmetrical and P3 adjusts the drain current at gate voltage close to pinch off. If the accuracy at low drain voltages is important then P1 and Vpk dependence on drain voltage needs to be taken into account, (3.62) – (3.63). The additional parameters are determined from a transconductance plot at low drain voltages, defining g mpk0, Vpk0 and P10.

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Microwave Mixer Technology and Applications

Vpk  Vpk0  (Vpks  Vpk0 ) tanh(VDS )

(3.62)

 P   1 P1 (VDS )  P1 1   10  1  2  cosh ( BVDS )    P1

(3.63)

Therefore, P1 = gmk/ Ipk is defined in the saturation region and P10 = gm0/Ipk0 is defined at low VDS. The fitting parameter B is on the order of 1.5 and its dependence in terms of gate voltage is taken into account in the expression:  = 0/(1-KgVGS), with 0 determined at zero gate bias and Kg is a fitting parameter. It has been found experimentally [24] that the transconductance and drain resistance measured from S-parameters in the MHz to GHz range are different than the values determined from DC measurements. This phenomena is called dispersion and is attributed to trapping of electrons at the FET surface and within the device and occurs in the KHz range. This problem can be minimized if measurements to determine the device parameters are made under pulsed conditions, (i.e., the bias is applied during a small interval of time sufficient to be able to make the evaluations but not enough for the dispersion effects take place and for temperature to change). DGD RG

G'

Rgd

G CGD

CGS DGS

D

CF

S Ri S'

Figure 3.23

RS

D'

RF

VGS

VGS

RD

ID(VGS,VDS)

VDS

CDS

S'

Chalmers equivalent circuit of the MESFET. After [23].

The approach from the authors, [25], is to take this effect into account by multiplying the term P1d to obtain (3.65). The equation P1d takes into account the effect of frequency on transconductance and is added as a pole in ftr, the corner frequency where the dispersion effect occurs. The term P1dc is the value of P1 measured in the saturation region at DC and P1rf is the extraction of P1 under RF conditions.

P1d 

P1srf P1sdc



P1sdc  P1srf P1sdc

1 2 1   f / f tr 

(3.64)

Semiconductor Modeling

 P   1 P1 (VDS )  P1d P1 1   10  1  2  cosh ( BVDS )    P1

131

(3.65)

The RF, CF parameters in the equivalent circuit are added to fit the static output conductance to that measured by S-parameters. This correction only works for small signal applications. For large signal, the resistor is replaced by a resistance that is a function of gate voltage, (3.66), with RF0 the minimum value of RF and RFp0 is the value of RF at the pinch off voltage. The forward current on the gate junctions are modeled by Schottky diodes, (3.1), with Is replaced by Igs and Vj replaced by (Vgs – Vbi) for the gate-source junction and by Igd and (Vgd – Vbi) for the gate-drain junction. Breakdown effects in the model considers two currents in the circuit when drain voltage is close to the breakdown voltage, VB.

RF  RF 0 

RFp 0

(3.66)

1  tanh( )

There is a rise in the drain current that was modeled by the author as a decrease in output conductance and shows up in the I/V plot as an increase in current, (3.67). This effect is more noticeable when gate voltage in near pinch off, which makes total voltage on the drain junction the sum of VB and pinch-off voltages. The second current is a reverse current on the drain-gate junction that builds up with time and may quickly drive the device into destruction. This current is represented by an equation similar to the one used for forward conduction, (3.68) – (3.69).

GDS (VDS )  (1  VDS  sb e Vgs VB   I gsb  I gs 1  K gbe VT      Vgd VB   I gdb  I gd 1  K dbe VT     

kb (Vdg VB )

) tanh(VDS )

(3.67)

(3.68)

(3.69)

An illustration of DC currents and transconductance for a MESFET modeled after the Chalmers model is found in Figure 3.24, where the peak of transconductance occurs around VGS = + 0.35 V. Gray traces are measured and black traces are from Angelov model.

132

Microwave Mixer Technology and Applications 100

100 90 80

80

tes..IDS.i, mA IDS.i, mA

70

gm2_mS gm_mS

60

40

60 50 40 30

20

20 10

0

0

0.0

0.5

1.0

1.5

2.0

2.5

-2.0

-1.5

-1.0

VDS

a) IDS(VGS,VDS) Figure 3.24

-0.5

0.0

0.5

VGS

b)gm(VGS)VDS=2V

Chalmers model for NE67300 for: Ipk = 58 mA; Vpk=-0.25V; P1 = 1.2; P2 = 0.24; P3 = 0.64;  = 4.2; = 0.04. Gate voltage swept from -1.6 to +0.4V.

An additional current correction is the introduction of a linear function to account for temperature dependency. This effect is important for large devices operating at high power but not so much for the smaller devices (< 300 µm gate width) used in mixers. The new family of FET devices HEMTs and PHEMTs appeared in the mid 1980s with several advantages in terms of low noise, gain and frequency of operation compared to MESFETs. The mechanism of operation is different, which reflects in different properties, the most obvious is a higher transconductance. Also the transconductance in general peaks at a gate voltage equal to or lower than zero volts and its profile as a function of gate voltage resembles a Gaussian shape. The device turn on is much more abrupt and the threshold voltage is closer to zero, which are desirable features for mixer designs. For these reasons PHEMTs or HEMTs became an attractive option in new mixer designs. In spite of the different mechanism of operation, in terms of electrical circuit, similar models applied to MESFETS are equally applicable to HEMTs. A simulation of I/V and transconductance for a low noise HEMT device is in Figure 3.25 showing the Gaussian profile for gm, illustrating the model versatility. 50

α

gmpk

40

100

80

90

70

80 60



20

gm_mS

30

50

60 50

40

40

30

30

10

20

20

0

0.0

Vknee Figure 3.25

0.5

1.0

1.5

VDS

a) IDS(VGS,VDS)

2.0

2.5

Ipk

10

10

0

IDS.i, mA

IDS.i, mA

70

0 -1.0

-0.8

-0.6

-0.4

-0.2

VGS

b)IDS(VGS)VDS=2V

-0.0

0.2

0.4

Vpk

Chalmers model for NE32100 for the following parameters: Ipk = 20 mA; Vpk=-0.25V; P1 = 3.2; P3 = 0.5;  = 2;  = 0.16.

Semiconductor Modeling

133

Some of the model parameters are extracted from I/V and gm,VG plots indicated in the figure. A similar hyperbolic tangent with an argument represented by a power series with gate-source voltage as a variable, was applied to model the capacitances due to similarities of the IDS(VGS, VDS) and CGS (VGS, VDS) profiles. The functions are expressed by (3.70) for the gate-source capacitance and (3.71) for the gate-drain capacitance.

CGS  CGS 0 [1  tanh(1 )][1  tanh(2 )] CGD  CGD 0 [1  tanh(3 )][1  tanh(4 )]

(3.70) (3.71)

The functions i, for i=1 - 4 are expressed by four independent series represented by (3.72) – (3.75).

1  P0 gsg  P1gsgVGS  P2 gsgVGS 2  P3 gsgVGS 3

(3.72)

2  P0 gsd  P1gsdVDS  P2 gsdVDS 2  P3 gsdVDS 3

(3.73)

3  P0 gdg  P1gdgVGS  P2 gdgVGS 2  P3 gdgVGS 3

(3.74)

4  P0 gdd  ( P1gdd  P1ccVGS )VDS  P2 gddVDS 2  P3 gddVDS 3

(3.75)

The author provided simplified expressions usable if a maximum accuracy of 10% is sufficient.

CGS  CGS 0 [1  tanh(P1gsgVGS )][1  tanh(P1gsdVDS )]

CGD  CGD 0 [1  tanh(P1gdgVGS )][1  tanh(P1gdd  P1ccVGS )VDS ]

Figure 3.26

(3.76) (3.77)

Capacitance from Chalmers model for the NE67300 device biased at VDS = 2V and gate voltage ranging from -1.8 to +0.4 V in 0.4 V increments.

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Microwave Mixer Technology and Applications

An example of a capacitance model for a MESFET is observed in Figure 3.26 for the gate source capacitance. The model capacitance in the gray plot compares well with the measured capacitance in the black plot. The capacitance was obtained from S-parameter measurements at low frequency (in the hundredths of MHz) as a function of voltage. The parameter values are as follows: P0gsg = 0; P1sgs = 0.1; P2gsg = 0.072; P3gsg = 0.028. One limitation of this model, considering the maximum value of tanh() is a maximum drain current equal to 2Ipk. In case the drain current goes above this limit, the model may not provide a good fit to transconductance, limiting model accuracy. 3.3.4 Model Extraction The extraction of parameters for MESFET models follows a similar strategy applied to BJTs. The first step is the de-embedding of the series resistances, which can be obtained from DC measurements. The conventional Fukui [26], method is still one of the most widely. It consists in making three measurements with a forward current Ig applied to the gate, one with drain open and source grounded, one with drain grounded and source open and one with source and drain grounded. Equation (3.78) describes this circuit in all three situations where the difference is in Req, which can be Rs, Rd, or Rs in parallel with Rd respectively.

 Ig  VGS  ( Rg  Req ) I g  VT ln    Is 

(3.78)

Three equations are developed from basic (3.77), namely,

R1  VGS (opendrain)  VGDS (drainsourceshort)/ I g

R2  VGS (opensource)  VGS (opendrain)/ I g

(3.79) (3.80)

The solution for the system results in the following: 2

Rs  R1  R1  R1R2

(3.81)

Rd  R2  Rs

(3.82)

The gate resistance is found by applying two different currents to the diode, monitor the difference in voltages and calculate the resistance by means of (3.83). The currents should be close to maximum gate current to guarantee the logarithm term is nearly constant at two close currents.

Semiconductor Modeling

Rg 

135

(VGS1  VGS 2 )  Req ( IGS 1  I GS 2 )

(3.83)

Another alternative to determine the source or drain resistance is to apply a voltage at the gate and monitor the voltage over Rs and Rch with a multimeter connected to the drain. The evaluation is valid if the gate current is sufficiently high, so that Rch 3b. This design criteria is similar to that of the coaxial balun.

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Microwave Mixer Technology and Applications

Another version of the coaxial balun adapted to suspended micro-strip lines is shown in Figure 4.24, [29]. In this third order Marchand balun, the top metal contains the input unbalanced line, the output lines, and the series compensating stub. The input line and stub are separated by one line width from the output balanced lines, except near the junction where they are closer to each other for a short distance. The middle conductor contains the ground plane for the balun lines. To minimize the even-mode coupling of the top side lines to the fixture ground, the widths of the suspended ground lines are five times the widths of the top lines for isolation. In this condition the fields are largely contained in the dielectric between the top lines and the suspended ground lines.

Figure 4.23

Suspended microstrip balun with unbalanced input at bottom-side at H, represented by the dotted line. The signal from the input microstrip line is transferred to the two lines at the top side with outputs G and F and grounded at the input H reference. From [28].

Active and Passive Coupling Structures

Figure 4.24

Layout for suspended microstrip balun, with two substrates attached on top of each other. From [29].

b 

Za

-b



 

Yb

Input

Z=1

Z0=1

a Figure 4.25

181

-a

c

Z=R

ZC

Z=1

-c

 f 2 f0

d

R

-d

Equivalent circuit for the third order suspended microstrip balun, with added matching section, Zc, between the compensation junction and the load. a = (-jZacot/(2-jZacot); b = (-jYbcot/(2-jYbcot); c = (Zc-1)/(Zc+1); d = (R-Zc)/(R+Zc). After [29].

The equivalent circuit is similar to the coaxial type of balun, except the balanced line is lumped into a single transformer Zc. In the model the impedance from junction to load corresponds to twice the balun impedance. Equations were developed for that structure using the theory of small reflections at junctions, and the design equations are given in [29], where  is the reflection coefficient within the band, and ωq is fractional bandwidth, (f2-f1)/f0. The paper applies the theory to

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Microwave Mixer Technology and Applications

a coupler designed for a 100 to 50 ohms transformation, with ω q = 0.67, and  = 0.03. The impedance values are: Za = 46.2 ohms, Zb = 39.0 ohms and Zc = 70.7 ohms. The results obtained are in the plots contained in Figure 4.26.

Figure 4.26

Performance of the balun from figure 4.25. It covers nearly one octave band with input return loss in the order of 10 dB. From [29].

4.3.5.1 Tapered Coupled Line Balun A popular broadband balun uses suspended substrate to transition between unbalanced and balanced microstrip lines. The microstrip line at the unbalanced end tapers to a different width at the balanced end. And the ground plane metal at the unbalanced end tapers in width to equal the same width as the top strip at the balanced end. It is desirable to maintain the same characteristic impedance along the line, which is accomplished by specially tapering the top and bottom lines. Various types of tapers have been used, including Klomfpenstein, Hecken, and Tchebycheff [30]. An early tapered microstrip balun was found to have improved performance with the first half of its top line length forming the taper, and the second half having constant width [31], as in Figure 4.27. Another early tapered microstrip balun was used in a wideband doubly balanced mixer [32]. Since the tapered balun is suspended above ground, normally in a metal enclosure with

Active and Passive Coupling Structures

183

ground metal above, below, and on the sides, a parasitic even-mode exists. As discussed previously, this causes in-band resonances, the effect of which can be reduced by designing to achieve a high even-mode impedance. In the case of the doubly balanced mixer [32], ferrite absorber was placed inside the enclosure to reduce effects from the parasitic mode. It also attenuates cavity resonances from the enclosure. The even-mode impedance ideally is at least ten times that of the odd-mode impedance. In contrast, the Marchand balun is much more tolerant of the even-mode, requiring even mode impedance to be only about three times the odd-mode impedance [33].

(a) Figure 4.27

(b)

Tapered microstrip balun, unbalanced at left, balanced at right: (a) top metal; (b) bottom metal.

4.3.6 Broadside Coupled Balun A broadside coupled balun using three metal layers appeared in a patent issued in 1988, [34]. It is an implementation of a planar Marchand balun, comprising top, middle, and bottom (ground) metal layers as shown in Figure 4.28. The top and middle metal layers comprise a transmission line, and the middle and ground layers form another transmission line. A third one is created by the top and bottom layers, and constitutes a parasitic impedance. The unbalanced signal is applied in the middle layer, and then broadside coupled into the top metal layers, which in turn are connected to the balanced load. The signal voltage is balanced and at a maximum at the mid-point where there is a gap between the two short circuited stubs. The design was proposed as being implemented using thin film technology with alumina for the bottom dielectric, and the top dielectric layer having similar dielectric constant and about 1/5 the thickness compared with the bottom dielectric. The inventor also proposed applying this in GaAs technology. Additional variations are also proposed having different open and short circuit configurations.

184

Figure 4.28

Microwave Mixer Technology and Applications

Broadside coupled balun.

A version of this planar Marchand balun applicable to GaAs MMIC technology was patented in 1991 [35, 36] for wide band microwave applications. The balun is depicted in Figure 4.29(a), with a cross section of transmission lines TL1 and TL2 shown in Figure 4.29(b).

(a) 3D Layout view. Figure 4.29 MMIC balun constructed on top of GaAs substrate.

(b) Cross section.

The oxide separating the top and middle metal is thin compared to the GaAs substrate, resulting in a tightly coupled TL1 transmission line, reducing the odd mode impedance so the parasitic even mode of TL3 is less critical. A series/shunt Marchand balun configuration is created at the junction where the gap exists in the middle metallization. The schematic for this circuit is in Figure 4.30(a) and the impedances Z1 and Z2 describe the input line and the open stub, respectively. The short stub Z3 comprises two parallel short stubs ZS1 and ZS2. The output balanced line Z4 is replaced by ZB. The inventors used a polyimide material with thickness between 2.5 and 10 µm for TL1, with r = 5.5. The GaAs substrate is about 100 µm thick,

Active and Passive Coupling Structures

185

with r = 12.9. A plot of the balun performance is in Figure 4.30(b), showing equal power splitting over the 6 to 18 GHz band with 1.5 to 2.0 dB insertion loss.

Figure 4.30

(a) Equivalent TL model. (b) Insertion loss. MMIC balun circuit and performance. Source for (b): [36].

A similar circuit was built by the patent authors on 0.125 mm thick alumina substrate using a polyimide dielectric 0.05 mm thick in place of GaAs oxide. The measured performance is similar to the one obtained with the MMIC version.

4.3.7 Planar Spiral Transformer—Marchand Balun The planar spiral balun of Figure 4.31(a) is a direct mapping from the planar micro-strip balun of Figure 4.21, but where each 90 degree coupled line is wound on itself, [37]. The balun comprises two spiral transformers, one of which is depicted in Figure 4.31(b). Coiling the transmission lines increases the even mode impedance that improves balun performance, without increasing odd mode impedance. The mutual capacitance and inductance of the lines also increase, so the resonant frequencies are much lower for the same length of line, and for a given frequency the line dimensions are smaller. That makes the structure more compact, with lower resistance and lower insertion loss. Operation of the balun can be approximated by considering a cascade of multiple coupled lines, [38], in the manner depicted in Figure 4.31(b). The line length from port a to c is calculated from (4.43). The line length from b to d is assumed to be of the same length. n 1   Ltot  4( ID  W )n  2(W  S ) k  k 0  

(4.43)

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Microwave Mixer Technology and Applications

(a) Equivalent circuit. (b) Multicoupled line model. Figure 4.31 Marchand balun realized using spiral multiply coupled lines. From [37, 38]

Where ID = Inner square width; n is the number of coupled lines on each side; W is line width; and S is line spacing. The area A of one coil is obtained from:

A  [ ID  2n(W  S )]2

(4.44)

The analytical description is complex, but fortunately most electrical simulators, have models for 2, 3, and 4 coupled lines. A good methodology for the design of spiral transformers on MMIC following this approach is described as follows: a. b. c. d. e. f.

Calculate the /4 wavelength at the center frequency. Line widths and line gaps are selected to minimize chip area. Determine the inner square width (ID), for both size and coupling. Let Ltot = /4, then calculate the number of coil turns, n, from the equation. Use a multicoupled micro-strip model to synthesize the initial design of this transformer. Finally, a planar EM simulator is utilized to determine the final layout.

A spiral balun using a similar circuit to Figure 4.31(a) was patented [39], and the layout is depicted in Figure 4.32(a). It was designed for MMIC applications on GaAs substrates. There is no construction detail so one can assume it is built using conventional technology. The simulated performance shown in Figure 4.31(b) covers roughly 12 to 24 GHz with 1 dB insertion loss and 12 dB return loss at the worst point within the band.

Active and Passive Coupling Structures

Figure 4.32

(a) Layout. Planar spiral Marchand balun over a GaAs substrate.

187

(b) Results.

4.4 LUMPED ELEMENTS Lumped element baluns are widely used in MMICs due to their considerable space savings compared with distributed ones. The simplest application corresponds to the use of high-pass and low-pass networks connected in parallel as shown in Figure 4.35. The elements can be designed according to filter theory. If the low- and high-pass filters are Butterworth, then their normalized reactances equal 2 for each element. The source and load are normalized to the reference impedance Z0.

L C

2Z 0

(4.45)

0

1 2 Z 00

(4.46)

Calculating the relation between V2 and V3, a 180 degree phase difference if found at ω = ω0, (4.47). The phase is frequency independent; however, the amplitude, given by the next equation, shows V2 decreases and V3 increases with frequency and a crossover occurs where V2 = V3 at resonance. At this particular frequency, equal power division occurs. The circuit is therefore narrow band and its usability depends on how much amplitude mismatch can be accepted.

V2     0  V3 

2

(4.47)

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Microwave Mixer Technology and Applications

V2  V1

1

(4.48)

2

 1     2  0  V2

2

2

V1 1

V3 2

Figure 4.33

1

1

2

First order lattice lumped balun with normalized reactances equal to 2 for inductors and capacitors.

An alternative approach that increases the bandwidth of a lumped element balun was published recently, [40], where additional cells are employed to increase bandwidth. Figure 4.34 shows first and second order lattice baluns; the second order balun includes two low-pass and two high pass T-sections. PORT P=1 Z=50 Ohm PORT P=1 Z=50 Ohm

IND ID=L1 L=L nH

CAP ID=C1 C=C pF

PORT P=3 Z=50 Ohm

IND ID=L1 L=LL nH

CAP ID=C1 C=CH pF

IND ID=L2 L=LL nH

CAP ID=C5 C=CH pF

CAP ID=C3 C=CL pF IND ID=L3 L=LH nH

CAP ID=C3 C=C pF

IND ID=L4 L=LH nH

IND ID=L3 L=L nH

CAP ID=C6 C=CH pF

CAP ID=C4 C=CL pF IND ID=L5 L=LL nH

IND ID=L6 L=LL nH

PORT P=2 Z=50 Ohm

(a) Figure 4.34

LL=2.28 LH=5.3

CAP ID=C2 C=CH pF

PORT P=2 Z=50 Ohm

PORT P=3 Z=50 Ohm

(b)

Lumped lattice balun: (a) first order; (b) second order. From [44].

CL=.76 CH=1.41

Active and Passive Coupling Structures

189

If the same reactance values are employed for the low-and high-pass elements, then the amplitude imbalance essentially is the same, and there is no benefit to using more elements. The reference proposes shifting the resonance frequency of the low-pass and high-pass filters to increase bandwidth. The resonance of the low-pass filter is increased by a factor n, and the highpass is decreased by same factor 1/n. This affects impedance match, which is corrected by reducing the characteristic impedance of the T-sections. This is done by replacing 2 by a constant m < 2. The design equations are as follows:

L LP 

mZ 0 n 0

C LP 

1 mZ0 n0 mZ0 n

LHP  C HP

(4.49)

(4.50)

(4.51)

0 n  mZ 00

(4.52)

The ratio between the voltages, V2/V3, given in the following equation, shows the voltages are in counter phase and the frequency response is obtained by calculating the zeros and poles as a function of factor n.

 (2n 2 2  0 ) V2   04 V3  (2n 20 2   2 ) 1 0    2n0 2n 4

2

(4.53)

(4.54)

The authors proposed to differentiate (4.53) with respect to voltage to optimize bandwidth. The point for maximally flat is found for n = 1.22, with larger values resulting in symmetrical amplitude deviations around the resonance frequency. Several values of n and corresponding performance are summarized in Table 4.5. The values are derived from filter theory using Butterworth max-flat equations. The authors proposed a test circuit with ideal components having values for the circuit on Figure 4.34(b), [40]. Simulated performance of the circuit with ideal inductors and capacitors is shown in Figure 4.35. The inductors employed are 2.28 and 5.3 nH, and capacitors are 0.74 and 1.41 pF, respectively,

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Microwave Mixer Technology and Applications

for low-pass and high-pass values. Amplitude balance is within 1 dB and the phase difference is close to 180 deg over about 2.8 to 4.3 GHz. Table 4.5 Design Parameters for Chosen Specifications

Amplitude Balance Max Flat +/- 0.05 +/- 0.1 +/- 0.2 +/- 0.5 +/- 1 +/- 1.25

n 1.22 1.26 1.28 1.31 1.39 1.51 1.56

m 1.3 1.3 1.3 1.26 1.25 1.2 1.14

Reflection Bandwidth loss - dB 1:1.6 1:1.8 1:2.1 1:2.7 1:3.7 1:4.0

35 35 28 22 17 14

Source: [40].

Magnitude and Phase 2

120

1

90 DB(|S(2,1)|) (L) Schematic 1

-1 Gain dB

60

DB(|S(3,1)|) (L) Schematic 1

30

Ang(S(2,1)) (R, Deg) Schematic 1 Ang(S(3,1)) (R, Deg) Schematic 1

-2

0

-3

-30

-4

-60

-5

-90

-6

-120

-7

-150

-8

-180 2500

Figure 4.35

Phase deg

0

2750

3000

3250 3500 3750 Frequency (MHz)

4000

4250

4500

Example lumped lattice balun of Figure 4.34(b), with amplitude and phase performance over about 2.8 to 4.4 GHz.

Active and Passive Coupling Structures

191

4.5 SLOTLINE TYPE A slot line arranged as a power splitter can function as a balun, as shown in Figure 4.36, by having output ports that are 180 degree apart in phase. The input signal is coupled into port 1 by some means at a point located a quarter wave length (up) from the short. The energy propagates (upward) within the slot to the point where it splits into two new slots connected in series across the first slot. One can notice the fields are distorted so that left and right slots have the fields reversed. metal slot

2

3 1

Figure 4.36

Slot line “balun” has reversed E fields at ports 2 and 3.

The E fields are maximum at the point located a quarter wave length away from the respective shorts for ports 2 and 3, which are convenient locations for attaching a mixing device. Thus, the applied signal is split into two parts with equal amplitude and in counter phase. Based on this principle several applications appear in the literature using slot lines for balun construction. The balun presented in Figure 4.37, makes use of an asymmetrical coplanar line. A detailed crosssection is given in Figure 4.38(b), [41].

Figure 4.37

Asymmetrical coplanar balun, top view. From [41].

192

Figure 4.38

Microwave Mixer Technology and Applications

Cross section views of Figure 4.37: (top) slotline; (bottom) CPW. From [41].

The circuit consists of an input slot line on the left, a transition from slot line to coplanar ring, a mode suppression resistor, and two coplanar lines at the output. A signal applied to port 1 on the slot is split in two at the coplanar interface. Both signals travel the quarter wave long ring in counter phase, then each couple into another coplanar line that in turn connects to ports 2 and 3. The signals at each output port are equal in magnitude and opposite in phase. The operation of the circuit can be understood by considering the even and odd modes of propagation in the symmetrical circuit. This assumes that slotline supports both the even and odd modes. Figure 4.39 contains two circuits that represent half the total circuit. The left (a) assumes voltages are applied to ports 2 and 3 in phase, and the right (b) assumes voltages are applied in counter phase. In both cases the circuit is symmetric and can be divided in two with terminations depending on the mode of propagation. The even mode equivalent half-circuit comprises an open circuit transmission line cascaded with the series combination of a resistor having twice the isolating resistor value and the port 2 (or 3) impedance, Z 0. The open circuit is due to the voltages having equal phase on each side of the coplanar line, so no energy couples into the slot; thus, it is terminated in an open circuit. The resistor equals 2R resulting in the final value of R when two instances the equivalent half-circuit (a) are paralleled to form the full circuit. In the odd mode circuit (b), the signals are applied in counter phase. Therefore, at the joining point 2,3 the voltages add, and transfer from the coplanar lines to the circular slot lines, where they split into two with opposing phase. Hence the potential at both ends of the resistor, R, are the same. The resistor is then short circuited for odd mode signals, so it does not appear in (b). The terminating impedance at the left of equivalent half-circuit (b) is Z0/2. The full circuit comprises two instances of this equivalent half-circuit connected in series, resulting in Z0 terminating the left end of the full circuit in (b).

Active and Passive Coupling Structures

193

The impedance of both slot and coplanar lines were designed for 100, so that the impedance of the circular slot-line was 70.7, and the isolating resistor was 50. The gap the for slot line impedance was 1.2 mm and the width and gap for the coplanar line were 0.3 and 0.4 mm, respectively. If the impedance of the lines were chosen to be 50, then the resulting dimensions would be impractical to realize with this substrate.

(a) Figure 4.39

(b)

Fundamental modes of operation: (a) even; (b) odd. From [41].

The performance reported in the paper is depicted in Figure 4.40, showing good amplitude and phase balance over 2.5 to 3.5 GHz. If the performance is limited to insertion loss of 0.5 dB, phase imbalance of +/- 1º, and return loss to 20 dB, then the bandwidth is reduced to nearly 10%.

(a) Coupling, isolation, and return loss.

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Microwave Mixer Technology and Applications

(b) Amplitude and phase balance. Figure 4.40

Experimental results for the slot line balun. The circuit was built on a RT/Duroid 6010 substrate, with r = 10.8, substrate thickness of 1.5 mm and metal thickness of 18 µm. The center frequency was 3.0 GHz. From [41].

4.6 ACTIVE APPROACH The active 180º power splitter was originally used in audio circuits, with signals at the emitter and collector in counter phase and having equal voltage amplitudes. The same approach can be implemented with a FET, such as the one depicted in Figure 4.40. S21 is the gain from the gate to drain, and S31 is the gain from the gate to source. S21 and S31 are derived in Appendix 4C using a simple FET model. Rg VG

VD VS RS

RD

Figure 4.41 Single device balun using the device as a three port component.

S 21 

2 g D g 0  jC gs g ds  g S g m 

g 0 g D ( g S  g m  g ds )  g 0 g S g ds  jC gs g S  g 0 g D  g ds   g D g ds  (4.55)

Active and Passive Coupling Structures

S 31 



195



2 g S g 0 g D g m  jC gs g ds  g D 

g 0 g D ( g S  g m  g ds )  g 0 g S g ds  jC gs g S  g 0 g D  g ds   g D g ds  (4.56)

The output conductance, gds, is included as it assumed to be much lower than the gate and source conductances. One can observe that at low frequencies a 180º phase shift exists between source and drain voltages, and their magnitudes are equal if RS = RD. When frequency starts to increase, a difference from 180 degree phase difference arises due to the gate-source and gate-drain capacitances, limiting usability. Additionally, the impedance connected to source and drain has to be high enough to avoid mismatch and minimize power imbalance between ports. An alternative option is to use the properties of a CS-common source and CG-common gate device [42], as in Figure 4.42(a), where the gate circuits are tied to a common port and the two drains are independent. Employing a simplified equivalent circuit for the FETs, and assuming both terminations have the same output conductance, g = g2 = g3, with input termination, g1, the S21 and S31 are developed in Appendix 4D and given below. S21 is the gain from input to CS drain, and S31 is the gain from input to CG drain.

S 21 

S 31 

 2 g m1 gg1

g g1  g m 2  g ds   g1 g ds  2 jC gs g  g ds  2g m 2  g ds  gg1

g g1  g m 2  g ds   g1 g ds  2 jC gs g  g ds 

(4.57)

(4.58)

Both devices will have the same magnitude and opposed phase if the trans-conductance is much higher than the output conductance and as long as the only important reactance in the circuit is C gs. Notice there are two Cgs capacitances in parallel at the input port, which reduces the maximum operating frequency by a factor of two. Notice the CG device is usually much lower impedance compared with CS and can be adjusted to be close to 50 ohm, determining the input match. On the other hand, the output impedance of the CS device is lower than that of the CG device. Experience has shown obtaining good amplitude and phase balance is difficult for this circuit. An alternative balun circuit, [43], is obtained by rearranging the circuit of Figure 4.42(a), and replacing the drain bias resistors with active loads, as in Figure 4.42(b). The presence of the inductor and capacitor in the circuit establish the frequency range of operation. The bias of each device is provided by an additional circuit not shown.

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Microwave Mixer Technology and Applications

VDD2 M2 VDD1

M1

(a)

Conventional bias. After [42].

VDD OutN

OutP M1

In M2

Bias Circuit

VSS Figure 4.42

(b) Modified bias. Example of active common-source/common-gate balun.

4.6.1 Differential Active Balun The differential topology can operate as a balun if one of the terminals is AC grounded. For example, in the circuit of Figure 4.43, if the base of Q2 is tied to ground with a capacitor, and a signal is applied to the base of common emitter Q 1, then the signal at the collector of Q1 is 180 degrees out of phase with the signal applied to the base of Q1. The signal at the Q1 emitter is fed in phase to the emitter of common base Q2 and appears with the same phase at the collector of Q 2. Thus, the phase difference between the two collectors is 180°, and the collectors have similar amplitudes if the load impedances are equal. A metric for evaluating the quality of active baluns, often used for opamps, is the common mode rejection ratio (CMRR), which compares the differential gain with common mode gain. Differential gain, gm, applies when both

Active and Passive Coupling Structures

197

inputs are equal in amplitude but opposed in phase, as depicted in Figure 4.43. This is equivalent to the gain of half the differential amplifier with the emitter connected to ground, and is given by (4.59), assuming a T-model for the devices and disregarding the Miller impedance, Zcb. In contrast, common mode gain gmm is ideally zero and applies when both input voltages are equal in amplitude and phase. In common mode the amplifiers are in parallel and the gain is the gain of a common emitter device with twice the emitter impedance, ZE, given by (4.60). The CMRR per definition is given by (4.61).

gm 



re  (1   )rb 

g mm 

gm 1  2g m Z E

CMRR  g m /

gm  1  2gm Z E  2gm Z E 1  2gm Z E

(4.59)

(4.60)

(4.61)

Therefore high CMRR is obtained by either replacing the emitter impedance by a current source or by realizing ZE with a parallel tuned LC circuit that is high impedance at the resonance frequency. If a current source is used, the collector is attached to the emitters of Q1, Q2, and ground. However there is a frequency limitation to the operation of this current source. At microwave frequencies, the collector-base reactance of the current source decreases, causing CMRR to reduce. An improved method was proposed for minimizing the CMRR degradation, [44]. The idea is to represent the differential mode and common mode amplifiers with Z-parameters as in (4.60) and impose cancellation of CMRR. Therefore, making the denominator equal to zero provides the solution indicated in (4.63). The transimpedance parameter, Z21, for a current source, assuming a simplified T-model with Miller capacitance is in (4.64), confirming the degradation effect on CMRR caused by the collector to base capacitance.

CMRR 

( Z  2Z E )( Rc  Z 22  2Z E )  ( Z12  2Z E )( Z 21  2Z E ) Z 21 . 11 Z 21  2Z E Z11( Rc  Z 22 ))  Z12 Z 21 (4.62)

Z 21 2  re  Z cb

Z Eopt  

(4.63)

Z 21

(4.64)

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Microwave Mixer Technology and Applications

+VCC Rc

Rc

(-)Vout/2

IT

(+)Vout/2

IS

Q1

Q2 (-)Vin/2

(+)Vin/2 IE ZE

Figure 4.43

Differential amplifier topology with impedance ZE.

According to (4.63), the optimum impedance for the differential common emitter amplifier may result in negative resistance that must be avoided. An emitter impedance should be selected that trades off best CMRR and circuit stability. In the reference, the author proposed adding capacitance in parallel with the current source. The improved balun is depicted in the schematic of Figure 4.44(b), consisting of a cascade of common emitter and common collector differential amplifiers. The capacitance Ccp and inductance Lcp were optimized for CMRR up to 20 GHz. A similar approach is also applicable to FETs.

Lcp L1

2

C1 Q1

Q2

Le

Le

Q3

Q4 4

OUT-

OUT+

Ccp1 (a)

Schematic of a cascade of two baluns, using RC in the emitter in first stage and an inductance in the collector of the second stage.

Active and Passive Coupling Structures

199

40

CMRR (dB)

30 Stage1 Stage2 Overall

20 10 0 -10 0

5

10

15

20

25

30

Frequency GHz Figure 4.44

(b) CMRR over frequency for each stage independently and for the cascade. Improved cascaded balun and respective CMRR. From [44].

4.6.2 Active 180 Combiner The active circuits described so far, generate a balanced signal from a single ended input, but they cannot generate a single ended output from a balanced input signal. Transforming a balanced signal into a single ended output requires different circuitry. One option is to use a differential amplifier and extract the signal from only one of the devices, terminating the unused collector (or drain) with a dummy resistor. The amplifier gain is then g m/2 and the CMRR is half the value provided by (4.60). Using a differential pair with a current mirror load, as displayed in Figure 4.45(a), improves gain and CMRR [45]. Due to the current mirror the collector currents of Q3 and Q4 are equal, which makes I4 = I3 = -I1 and Iout = -I4 - I2 = I1 - I2. Therefore, when a base current is applied to Q1 with a positive phase it will appear at the output also with a positive phase. Since this is a differential circuit, a base current is simultaneously applied to Q 2 with a negative phase, so that Iout equals the sum of both currents. If both input signals are at the same phase, then Iout = 0. The gain of this configuration can be found by considering the collector currents given by (4.65) and (4.66). By manipulating the equations, the output current Iout is given by (4.67), and transconductance by (4.68) that resembles a bell shape function, showing transconductance is maximum for low input voltages.

I1  I2 

 F I EE 

Vid VT

1 e  F I EE 1 e

Vid VT

(4.65)

(4.66)

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Microwave Mixer Technology and Applications

V  I out   F I EE tanh id   2VT  V   I g md  F EE sec h 2  id  2VT  VT  With

R2

+VCC

Q5

Q6

Q4

Q3

I4 I1 Q1

Q2

V2

IEE

I3

+VCC

Q4

Q3

Vout

Iout

I2

V1

-VEE

(4.68)

Vid = the differential input current gmd = small signal transconductance.

R1

I3

(4.67)

I4 I1 Q1

Iout

I2 Q2

V1

-VEE

Vout

V2

IEE

(a) Conventional (b) Improved Figure 4.45 Current mirror to convert differential to single ended output.

There are many variations of this circuit in the literature, each designed to meet a specific need. The additional current mirror Q5, Q6 in Figure 4.45(b) minimizes offset of the output current [45]. As already mentioned, the common mode gain of the ideal circuit is zero, but in real circuits imbalances exist in the differential and common mode currents, given by (4.69) and (4.70) in terms of imbalance errors d and m in the differential and common mode, respectively, [46].

I 1  I 2 (1   d ) I 4  I 3 (1   m ) I out   I 4  I 2  I1 (1   m )  I 2 I out  I 2 (1   d )(1   m )  I 2 I out   I 2 ( m   d )

(4.69) (4.70)

(4.71)

Active and Passive Coupling Structures

201

The common mode transconductance, gmm, is then given by the ratio of output current to input common mode voltage, Vic, (4.72). In the ideal case m = d = 0 and gmm = 0. The ratio I2/Vic is the same as used in the definition of (4.60), so that it has been replaced in (4.72) and the CMRR is given by (4.73), [46].

gm I2 ( m   d )  ( m   d ) Vic 1  2gm Z E 1  2 g m AEl CMRR  m  d g mm 

(4.72)

(4.73)

This configuration works well with silicon bipolars at VHF and UHF but it will not operate well at microwave frequencies, where PNP devices have inferior mobility affecting gain and other parameters. A third option, shown in Figure 4.46, is a concept long used by audio amplifier designers, which uses two FETs connected in cascode, with the source of Q1 connected to the drain of Q2. The output voltage, in terms of input voltages V1 and V2, is given by (4.74) for the case of all equal terminations. The equation, derived in Appendix 4E, shows that for input voltages of same magnitude and phase, the output voltage is approximately zero. If they are equal in magnitude and opposed in phase, then they add constructively. V3 

g 0 g m (V1  V2 )  jCgs g 0 g 0 ( g 0  g m  2 g ds )  j 2Cgs ( g 0  g ds )

(4.74)

+VDD V1

Q1 V3

V2

Figure 4.46

Q2

Modified cascode active balun with resistive bias.

The circuit performs well at low frequencies where leakage current to the FET gates is small, and load impedance at port 3 does not affect the port

202

Microwave Mixer Technology and Applications

impedance. As frequency increases, the output voltage is no longer a perfect combination of two input voltages.

4.6.3 Active Current Based Balun The active examples shown so far are voltage baluns. An example of a current based balun is found in a patent [47] and reproduced in Figure 4.47. The input for the balun is the base of T 1, terminal 1, and the two outputs are at the collector of T2 and T3, terminals 2 and 3. A constant voltage supply is connected to the base of T2, making it a common base amplifier.

Figure 4.47

Current based balun consisting of a current mirror, devices T4 and D1 plus common base T3 and common emitter T3 amplifiers.

The diode (a transistor connected as a diode) and transistor T 4 form a current mirror that biases T 1. Both T1 and T4 have the same area so their bias currents are the same. A voltage applied to terminal 1 creates signal current i that appears at terminal 2 with the same phase as the voltage. The signal current develops a voltage drop on R1, decreasing the T3 emitter potential by the same amount. The current in R2 is given by -iR1/R2, having the same amplitude as the current in R1 if R1 = R2 but in counter phase.

Active and Passive Coupling Structures

203

4.7 180º HYBRID COUPLERS (Magic-T) The magic-T, also known as a 180° hybrid coupler, is a very useful component that finds application in many types of devices including mixers and antennas. The name magic-T is likely derived from its implementation early in the 20 th century in waveguide resembling a T shape. It is depicted functionally in Figure 4.48, as having four ports with inputs at P1 and P2 and outputs at Σ and Δ. Signal power incident at port P1 splits, with half delivered to port Σ and half delivered to port Δ with a 180 degree difference between the two transfer phase angles. In contrast, signal power incident at port P2 splits, with half delivered to port Σ and half delivered to port Δ, with equal transfer phase angles. Ports P1 and P2 are isolated from each other because, for example, signal power incident at P1 splits between the 0°-0° and 0°-180° paths, causing cancellation when the two halves combine at P2. The opposite is true for power incident at P2: it cancels at P1. There are many implementations of the magic-T, perhaps the most widely known is the ratrace coupler commonly implemented in microstrip. Critical performance metrics for the magic-T are bandwidth, insertion loss, isolation, and amplitude and phase balance. The Σ and Δ ports, respectively, are commonly called the sum and difference ports.

Σ P1 0

0 P2

π

0

Δ Figure 4.48

Functional diagram of the 180° hybrid.

4.7.1 Waveguide 180° An early description of a waveguide magic-T is given in a 1942 patent [48] that includes descriptions of the waveguide implementation shown in Figure 4.49. The waveguide ports that face toward the upper left and lower right corners of the figure correspond to ports P1 and P2 of Figure 4.48. The port facing the lower left corner corresponds to the difference port, and the remaining port that faces up corresponds to the sum port. The field lines for the sum and difference connections are depicted in Figure 4.49 (a) and (b).

204

Microwave Mixer Technology and Applications

Σ

P1

Δ

Figure 4.49

Figure 4.50

P2

Waveguide implementation of the 180° hybrid, illustrating the points P1,P2 and the  and  ports.

(a) Δ port excitation (b) Σ port excitation Electric field lines for the waveguide 180° Hybrid.

4.7.2 Bifilar Hybrid Bifilar coupled-lines can also be used to form a 180° hybrid, [14]. For example, a 1:4 Ruthroff transformer combined with a 1:1 balun becomes a 180° hybrid, as depicted in Figure 4.51. Assuming lossless lines, the sum and difference port input impedances, respectively, are Rin,Σ = RL /2 Rin,Δ = 2RL. The input voltage, Vg1, is applied to T1, which divides Vg1 into two differential voltages V A, VB that in turn are applied to load 2RL and to T2. The voltage across the two series windings of T2 is given by (4.75). The impressed differential VA and VB voltages cancel each other at port Σ, so VT = 0. If a voltage is applied to port Σ, then the resulting V A and VB are in phase and applied to the load resistors RL that are in parallel to ground. The high common mode impedance of T1 isolates the in-phase signals VA and VB from generator Vg1.

V A  VS  VB  VS

(4.75)

Active and Passive Coupling Structures

VS 

205

V A  VB 2

(4.76)

Therefore, both generators are isolated from each other and both apply signal to the load. Vg1 applies signal to the load resistors in series, and V g2 applies signal to the load resistors in parallel. IL VA Rg1

Δ

I3

I1

Σ

Vg1 V1

T1

T2

RL

VS RL

I4

I2 IL

VB

Vg2 Rg2

Figure 4.51

Bifilar transformer as a hybrid. After [14].

4.7.3 Ratrace Hybrid The ratrace hybrid, illustrated in Figure 4.52, is a popular 180 hybrid coupler [49]. It can be analyzed by using the line of symmetry, aa, to take advantage of even- and odd-mode analysis, similarly to the that for coupled-lines. The circuit is first solved for two ports, and then the full four port parameters are derived by superposition. The angle 1 is designed for center frequency at 90º, equivalent to /4.The resulting S-parameters for the ratrace hybrid are expressed below as a function of ratrace line admittance, Yr = 1/Zr and reference admittances, Y0 = 1/Z0.

Y0  2Yr

2

Y0  2Yr

2

2

S11  S 22  S 33  S 44 

2

(4.77)

S 31  S 42  0

S12  S 34   S 41  S 32 

(4.78)

 j 2Y0Yr Y0  2Yr 2

2

(4.79)

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Microwave Mixer Technology and Applications 2

3 1

Z0

Z0

1

1

a Zr 1

4

a

2=31 Figure 4.52

Ratrace hybrid.

The ports 3 and 1 are uncoupled, according to (4.76), and the same holds true for ports 2 and 4. For matched conditions, Y 02 = 2Yr2, and inserting this condition into the equations, one finds that S21 = -S41 = 1/2, which results in 3 dB power split and ports 2 and 4 being 180 out of phase. The parameters S32 = S34 indicate that signal applied at port 3 splits in half and appears at ports 2 and 4 with equal amplitude and phase. This circuit operates over a narrow band due to the ¾ wavelength long line. To decrease the sensitivity of the long line with frequency, [50] proposes to replace that line with a coupled line, with both ends shorted, as shown in Figure 4.53. This coupled line is known to reverse the phase by 180º with a quarter wavelength long line. The characteristic impedance of the coupled line is in the following equation showing Zc as a function of even an odd impedance, with line length expressed by angle . The even and odd impedances are obtained from simplified equations

Zc 

Z  Z    2  1Z   2  1Z oe

Z oo Z oo

2Z oe Z oo sin 

2

oo

 Z oe  Z oo  cos 2  2



1/ 2

(4.80)

r

(4.81)

r

(4.82)

For matched operation, the impedance of the coupled line section is made equal to the impedance of the ratrace, Zr. For a 50 ohms reference impedance and 3.0 dB coupling, the even and odd mode impedances, respectively, equal 176.2  and 30.2. These values provide a balun with 180º +/- 15º over a 2:1 frequency range. Such a high even mode impedance is difficult to obtain, but adequate bandwidth performance is still achieved with realizable values.

Active and Passive Coupling Structures

Figure 4.53

207

Ratrace with extended bandwidth. From [50].

The effect of frequency variation is observed by scaling the angle  = 0f/f0 to the mid band frequency in the characteristic impedance (4.78). The impedance increases as a function of frequency and becomes maximum at the extreme of the band, determined by the angle that makes Zc infinite, (4.83) – (4.84). The author concludes that a better result can be obtained by mismatching the impedance at mid band by decreasing Zoe, Zoo by 10%, resulting in lower overall mismatch within a 2:1 frequency range.

cos 1 

Z oe  Z oo Z oe  Z oo

 2  180  1

(4.83) (4.84)

An alternate means of extending the bandwidth of the ratrace coupler is described [51] using coplanar lines as depicted in Figure 4.54. The 180 degree phase shift is obtained by swapping the positive and negative sides of the coplanar strip transmission line. An equivalent circuit is derived for which a Chebychev response is fitted resulting in a multioctave response. The ratrace coupler is used in the singly balanced mixer shown in Figure 4.54, which has typically 6 to 7 dB conversion loss over a 10 – 40 GHz RF frequency range. Another example of a ratrace hybrid implementing the 180 degree phase shift by swapping positive and negative lines uses suspended substrate with wideband microstrip to CPW transitions [52], depicted in Figure 4.55, with measured performance in Figure 4.56. Usable performance is achieved over about 4 to 12 GHz, with 4 to 6 dB insertion loss, and 25 to 40 dB isolation between ports.

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Microwave Mixer Technology and Applications

Figure 4.54

Singly-balanced mixer using coplanar strip multioctave ratrace coupler. From [51].

Figure 4.55

Ratrace hybrid using wideband microstrip to CPW transitions to achieve 180 phase shift by interchanging positive and negative lines. From [52].

Figure 4.56

Simulated and measured results for the ratrace hybrid. From [52].

Active and Passive Coupling Structures

209

4.7.4 Lumped Ratrace The derivation of this hybrid as reported in the literature, [53], is done by equating the ABCD parameters for the transmission lines of a distributed ratrace coupler, to the ABCD parameters for a low-pass filter composed of 90º lines and using a high-pass filter to represent the 270º line. Comparing the two results in the following high-pass and low-pass networks:

X C  Z0

X L BC  1 X L  Z 0 and BC  Y0

Figure 4.57

BL X C  1 and BL  Y0

(a) Low pass (b) High pass Derivation of lumped ratrace by correlation. After [53].

Replacing the lines from the ratrace by the equivalent lumped elements, one obtains the circuit of the Figure 4.58, where the difference signal is at terminal 4 and the sum is at terminal 3. The insertion loss performance plots are in Figure 4.59 complemented by the plots on Figures 4.60 and 4.61 for phase and return loss.

2C

2C

L

3,Σ

1 L C

C

L 4,

2 C

Figure 4.58

L

Δ

C

Lumped ratrace built with low pass and high pass filters. After [53].

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Microwave Mixer Technology and Applications

Figure 4.59

Lumped ratrace insertion loss between ports 2-4, 1-2 and 1-4. From [53].

Figure 4.60

Lumped ratrace insertion phase between ports 1-3,1-4, 2-3 and 2-4. From [53].

A variation of this publication is found in a patent, [54], for a lumped ratrace layered on top of GaAs. The proposal is to replace the ratrace transmission line sections with lumped equivalents found through application of ABCD matrices. In another patent, [55], lumped elements are used to replace a Marchand balun prototype with lumped elements using LTCC technology.

Active and Passive Coupling Structures

Figure 4.61

211

Lumped ratrace return loss of each port. From [53].

4.7.5 Coupled Line Balun Converted to Hybrid A 180 degree hybrid can be formed by combining a simple transmission line balun with an in-phase power splitter. Following this idea, a publication, [56], proposed adding the balun built with coupled lines to an in–phase coupled line power splitter. The schematic of the proposed hybrid is in Figure 4.62, showing the details of the combination. A similar version of this 180° hybrid was patented in 1970 [57]. The S-parameters of an ideal balun and power divider are given by the following equations. The hybrid matrix combining both circuits is given by the Sparameter hybrid as indicated. The objective is therefore to combine the S parameter matrix from a balun with input impedance Z0 and output Z1 with the S parameter matrix from a power divider with input impedance Z0 and output impedance Z1. The matrices are combined after the multiport connection method found in the reference. For S balun, S12 refers to port 2 and Δ-port 1; for Sdivider S12 refers to port 2 and Σ-port 1 of Figure 4.62.

0 S12  S12 S balun  S12 1 / 2 1 / 2  S12 1 / 2 1 / 2

(4.85)

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Microwave Mixer Technology and Applications

S diivider

0  S12 S12

S hybrid

0 S  12  S12 0

S12  S12 1/ 2  1/ 2  1/ 2 1/ 2 S12 0 0 S12

 S12 0 0 S12

(4.86)

0 S12 S12 0

(4.87)

By combining the balun and in-phase power divider, their individual antiphase and in-phase power splitting characteristics are maintained, with the balun isolated from the power divider input under matched conditions. The Shybrid can be identified as the S matrix of an ideal 180 degree hybrid. The circuit in Figure 4.62, besides operating as a hybrid, also transforms the source impedance Z0 at ports 1 and 4 to match the load impedances at ports 2 and 3. This is obtained by designing the in-phase and balun independently. The matching input impedance for the divider is in matrix (4.85), SCdivider, by imposing the S11 = 0, resulting Z a 

2 Z 0 Z1 .  - Port 4

In phase divider

Port 3 1a1

/4 Za

Z0

Z0 Port 2 1a1

Z1

Z1

 - Port 1 Z0

Figure 4.62

/4 Zoe, Zoo Impedance transforming Marchand Balun Hybrid with coupled line balun in-phase divider. From [56].

The input matching condition for the balun is obtained by making S 11 = 0 and S22 = S33 = 0.5 in the equation for the coupled line balun, (4.38). The required even and odd mode impedances for the coupled line sections are given by (4.87) and (4.88) and the coupling required for transforming impedance Z0 to Z1 is in (4.89).

Active and Passive Coupling Structures

Z a  2Z 0 Z1 2

Z a  2Z 0 Z1 2Z Z Z   j 2a 0 1 Z a  2Z 0 Z1 2Z Z Z  j 2a 0 1 Z a  2Z 0 Z1 2

S Cdivider

j

2Z a Z 0 Z1 Z a  2Z 0 Z1 2 Za 2

j

213

2Z a Z 0 Z1 Z a  2Z 0 Z1  2Z 0 Z1 2

Z a  2Z 0 Z1  2Z 0 Z1

Z a  2Z 0 Z1 2 Za

Z a  2Z 0 Z1

Z a  2Z 0 Z1

2

2

2

(4.88)

2

Z oe  Z 0

1 C 1 C

(4.89)

Z oo  Z 0

1 C 1 C

(4.90)

C

Z0 Z 0  2Z1

(4.91)

The proposed layout is indicated in Figure 4.63. It was built on a 1.6 mm thick FR-4 substrate having dielectric constant equal to 4.4. The coupled line section was realized using three-conductor coupled lines [56].

Figure 4.63

Hybrid with coupled lines. From [56].

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Microwave Mixer Technology and Applications

The measured results for the balun alone gave a bandwidth from 1 to 3.2 GHz with an insertion loss of about 0.5 dB within the band. The phase imbalance with respect of 180 degrees difference is 5 at the low end of the band and 10 at the higher end. Insertion loss of the power divider alone over 1.25 to 3.0 GHz is ≈ 0.5 dB. The measured and simulated performance for the sum and difference port of the hybrid are in Figure 4.64 (a) and (b), respectively. Total phase variation is 5º and + 5º respectively within the aforementioned band.

`` Figure 4.64

(a)  - port Coupled line hybrid performance. From [56].

(b)  - port

4.7.6 Slotline Microstrip Hybrid Junction Using the properties of propagation on slotline, micro-strip line and the coupling between them, a 180 degree hybrid junction can be realized with those transmission medium, [58]. A suspended substrate having a microstrip line 27 on the bottom side and ground plane with slotlines 35, 36, 37 on the top side is depicted in Figures 4.65(a) and (b). Figure 4.65(a) shows excitation of the slotline 35 at port 32, which causes an equal division of power into slotlines 36 and 37, with a 180 degree phase difference between E fields at slotline ports 33, 34. This 180 degree phase difference causes cancellation of coupled fields into the microstrip 27, thus the microstrip and slotline 35 are isolated from each other. In Figure 4.65(b) excitation occurs at port 31 of the microstrip 27, causing equal power division into slotlines 36 and 37 with equal phase. Isolation between the microstrip and slotline 36 is caused by cancellation of the E fields in slotlines 36 and 37 at the slotline Y junction. The microstrip port 31 is the sum port, and the slotline port 32 is the difference port. Alternative slotline and microstrip configurations are given in the reference.

Active and Passive Coupling Structures

215

(a) Power division with 180º phase (b) Equal phase power division Figure 4.65 Coupling of E fields in microstrip to slotline medium. Dashed lines are microstrip line on bottom side; solid lines are slot lines in top side ground plane metal and arrows depict E-fields.

4.7.7 Broadband Hybrids The 180 degree hybrid combiners discussed so far have frequency limitations largely caused by the use of quarter wave transmission line lengths. A class of 180° hybrids exists that are not limited in this way and thus can operate over extremely wide, multi-octave frequency ranges. A wideband 180 degree hybrid was described by Mouw in 1968 in conjunction with a doubly balanced mixer in a “ star” configuration [59]. A simplified depiction of it is found in Figure 4.66, which has four balanced ports represented by resistors R1, R2, R3, and R4. TL1A(Z01,θ) 4

3'

+ -

1 +

R4 3

-

+

TL2A(Z01,θ)

TL1B(Z01,θ)

-

R3

2+ -

4'

TL2B(Z01,θ)

R1

1'

2'

Figure 4.66

R2

Wideband four-port hybrid using four equal transmission lines. It has four balanced ports at R1, R2, R3, and R4. In practice ports R1 and R2 are unbalanced by grounding at points 1’ and 2’. From [59].

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Microwave Mixer Technology and Applications

Ports 3 and 4 connect to port 1 via transmission lines TL1A and TL1B; ports 3 and 4 also connect to port 2 via transmission lines TL2A and TL2B. These four transmission lines all have equal characteristic impedance, Z01, and phase shift θ. Signal power incident at port 1 is delivered with equal amplitude and phase to ports 3 and 4; and power incident at port 2 is delivered with equal amplitude but in opposed phase to ports 3 and 4. Thus, ports 1, 2, 3, and 4 of Figure 4.66, respectively, correspond to ports Σ, Δ, 2, and 1 of the magic-T in Figure 4.48. It is also easily seen that ports 1 and 2 are isolated from each other by cancellation. Figure 4.67 shows the equivalent circuit of the portion of the hybrid that is associated with port 1. Since port 2 is a virtual ground relative to port 1, R 2 does not affect S11, S31, or S41. Resistors R3 and R4 can be halved, and connected at the center point since that point is also a virtual ground relative to port 1. The circuit described thus far has no frequency limitation, since all four ports are assumed balanced. For example, when the circuit is evaluated in a linear simulator, using ideal transformers to interface the four balanced ports to single ended ports, the resulting S-parameters show no frequency dependence. However, when this ideal circuit is realized in a test circuit, three frequency limiting factors come into play. One is that the ports 1 and 2 are actually unbalanced, for example with nodes 1’ and 2’ grounded. As discussed previously for the coupled-line baluns, an even mode impedance exists between ground and the grounded lines of TL1A, TL1B, TL2A, and TL2B that adds frequency dependence. Also, the low frequency end of operation is limited by the inductance of the grounded lines of TL1A, TL1B, TL2A, and TL2B; a high reactance relative to the load resistances is required to maintain amplitude and phase balance at ports 33’ and 44’, and isolation between ports 1 and 2, at low frequencies. TL1A(Z01,θ) 4 +

R1

1' -

1 +

3 TL1B(Z01,θ) +

Figure 4.67

R3 2

R4 2 R3 2

3' TL 2C (Z0

2,φ

)

R4 2

4' TL 2D (Z0

2,φ

)

Equivalent circuit of port 1 with parasitic short circuit TL2C and TL2D. TL2C comprises lines of TL2A and TL2B connecting to 2’. TL2D comprises lines of TL2A and TL2B connecting to 2. From [59].

The third factor limiting bandwidth is the addition of the parasitic transmission lines TL2C and TL2D, depicted in Figure 4.67. TL2C comprises the lines of TL2A and TL2B that connect to node 2’. And TL2D comprises the lines

Active and Passive Coupling Structures

217

of TL2A and TL2B that connect to node 2. TL2C and TL2D have characteristic impedance Z02, and phase shift φ. The lines present a short circuit across nodes 4 and 3’, and across nodes 3 and 4’ at φ = nπ/2, n=0,1,2... Ideally Z02 is infinite, but in practical terms it should at least be significantly higher than Z01 and R1 through R4. Z02 can be thought of as the even-mode impedance, and its value varies depending on the implementation, for example in balanced microstrip or bifilar coupled lines wrapped around magnetic material. Due to symmetry, the equivalent circuit of Figure 4.67 can be divided into two identical half circuits— one half is shown in Figure 4.68. TL1A retains characteristic impedance, Z01, and phase shift θ. Resistor r = R1 = R2, and resistor R = R3 = R4. The same analysis can be done for the port 2 portion of the hybrid, resulting in the same equivalent circuits as shown in Figures 4.67 and 4.68. TL1A(Z01,θ) 4 1'

(Z0

1

2r Figure 4.68

TL2 C

3' 2,φ )

R

Equivalent half circuit for port 1. From [59].

The input impedance at ports 1 is given by (4.92), with Z01 and θ referring to TL1A, and Z02 and Φ to TL2C [59]. If the parasitic transmission line TL2C is ignored, then the simplified input impedance is given by (4.93), in which case the maximum bandwidth condition is given by (4.94). The same analysis and results apply to the analysis of input impedance at port 2.

  Z 01 Z  j cot   01 cot   R Z 02 Z   Z i1  Z i 2  01 Z Z 2 1  01 cot  cot   j 01 cot  Z 02 R Z 01  jcot   Z 01 R Z i1  Z i 2  2 1  j Z 01 cot  R 2r  R  Z 01

(4.92)

(4.93)

(4.94)

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Microwave Mixer Technology and Applications

The input impedance at ports 3 and 4 is given by (4.95). If the parasitic transmission lines are ignored, then the simplified input impedance is given by (4.96).

Z 01  j cot  2r Z i 3  Z i 4  Z 01 Z Z Z 1  01 cot  cot   j 01 cot   01 cot  Z 02 2r Z 02 Z 01  j cot  Z i 3  Z i 4  Z 01 2r Z 1  01 cot  2r

(4.95)

(4.96)

S-parameters can also be derived, and simplified as in (4.97) – (4.98) by neglecting the parasitic transmission lines. The simulation results based on these equations are in Figure 4.69 for the return loss and Figure 4.70 for the insertion loss performance.

Z 01 R   2r Z 01 S11  S 22  Z 01 R   2r Z 01

R  j   1 cot   2r  R  j   1 cot   2r 

(4.97)

Z 01 2r   R Z 01 S33  S 44  Z 01 2r   R Z 01

 2r  j   1 cot  R  2 r   j   1 cot  R  

(4.98)

R 1 2r S13  S14  S 23   S 24   2 1   R Z 01  R  sin    1   cos   j  2  2r   Z 01 2r   (4.99) The wideband hybrid circuit can also be described as a Wheatstone bridge, with transmission lines TL1A, TL1B, TL2A, and TL2B replacing the four resistors of the bridge, [60]. The two resistors between the opposite corners of the bridge become ports 1 and 2. The author also discusses use of the hybrid in the so

Active and Passive Coupling Structures

219

called termination insensitive mixer. It is also used in triple balanced mixers as will be discussed in Chapter 6.

Figure 4.69

VSWR of the hybrid of Figure 4.66: (top) given Z01=R=2r, (bottom) given Z01= (2Rr)0.5, R=2r. From [59].

Figure 4.70

Insertion loss of the hybrid of Figure 4.66: (top) given Z01=R=2r, (bottom) given Z01=(2Rr)0.5, R=2r; in all cases θ=φ. From [59].

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Microwave Mixer Technology and Applications

4.7.8 Suspended Microstrip Line Wideband Hybrid A suspended microstrip line implementation of the wide band hybrid discussed earlier was proposed in a patent by Mouw, [61], comprising two double sided substrates as shown in Figure 4.71. The hybrid was used in a doubly-balanced star mixer, with the diodes located between the top substrate and bottom substrate. Ports 1 and 2 are the RF and LO ports of the mixer, corresponding to the ports P1 and P2 of the magic-T in Figure 4.48. One can observe in the top substrate, the top metallization contains an input line at port 2 that splits into two lines terminated by the compensating open-circuit stub.

(a) Top substrate (b) Bottom substrate

Figure 4.71

The two substrates comprising Mouw’s 180° hybrid. The connection between the substrates is the diode in star configuration connected at 56 – 59 at the top and 60 – 63 terminal at the bottom.

The two metal lines on the bottom side are grounded at their extremes and open at the mid-point gap, where they couple energy from the top lines. The

Active and Passive Coupling Structures

221

same arrangement is true for the bottom substrate of Figure 4.71(b). Terminals 56 – 60 connect together to make port 4; similarly, terminals 59 – 63 make port 4’, terminals 57 – 61 make port 3’, terminals 58 – 62 make port 3. These four points connect to the four diodes, as discussed in Chapter 6. An alternate form of this balun is realized using just one suspended substrate having four grounded metal lines on the bottom side to form the four diode ports is depicted in Figure 4.72, [62]. A MMIC version of the mixer circuit using a fully planar version of the magic-T is shown in Figure 4.73 [63].

Figure 4.72

A planar magic-T with suspended microstrip using two metal layers. From [62].

Figure 4.73

A MMIC star mixer using a planar version of the magic-T.

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Microwave Mixer Technology and Applications

4.8 QUADRATURE HYBRIDS 4.8.1 Branchline Coupler Quadrature 3 dB couplers can provide the important function of signal combining in mixers [64]. The branch-line quadrature coupler is represented in Figure 4.74 in both rectangular and the circular forms. The analysis of operation can be simplified by noting the coupler’s horizontal and vertical symmetry. Referring to the rectangular layout, the symmetry runs along the A and B lines, making it possible to characterize the whole by analyzing only one quarter of the circuit. The symmetry planes crossing at these lines are assumed to be either electric or magnetic walls as discussed earlier in this chapter for coupled-lines, so that four possibilities are possible: dual open circuit at Y1 and Y2 for a magnetic wall, dual short circuit for the same line sections for an electric wall, and mixed electric/magnetic walls (short-open, open-short). The analysis is based on the reflection coefficients for each of these conditions. Yc

Y1

Figure 4.74

Yc Y2

Y2 B

Yc

A

Y1

Yc

Y1

A

Yc

Y2 B

Yc

(a) Rectangular version. Branchline quadrature coupler.

Y2

Yc

Y1

Yc

(b) Circular version.

If the through lines and branch lines are a quarter wavelength long at the center frequency of operation, and the impedance of the lines are related by the expression, Y12 - Y22 = Yc2, then the coupler is matched at all ports. The Sparameter matrix for the coupler is given in (4.100) as a function of the reference Yc and line admittance Y1. For a 3 dB coupler, a solution is Y2 = Yc and Y1 = 2Yc. Therefore, given signal excitation at port 1, the quadrature coupler provides equal power division with 90 phase difference at ports 2 and 3. If the ports are terminated in 50 ohms then port 4 is isolated from port 1. This circuit is limited to narrow band operation around the frequency where the lines are quarter wave.

Active and Passive Coupling Structures

  0   jYc  Y1 S Y  2  Y1   0 





jYc Y1



Y2 Y1

0

0

0

0

Y2 Y1



jYc Y1

 0   Y2   Y1  jYc    Y1   0  

223

(4.100)

4.8.2 Lange Hybrid Coupler This type of coupler performs well over an octave or more of bandwidth. It equates to two coupled lines, but uses four or more to obtain the same coupling but with wider gaps that are realizable. The physical layout for the four line version of the coupler is in Figure 4.75. Equations for the design are available in the literature, [65], and are derived in terms of the desired coupling, K, and characteristic impedance, Z0. The line length, l 2 , is a quarter wavelength at the center frequency of operation. The design problem is to find the gap between the lines, S, and the line width, W, for a given substrate thickness, h, and dielectric constant, r. The set of design equations are given next for an N-line coupler (N even).

Figure 4.75

Lange coupler as an association of four coupled lines. From [65].

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Microwave Mixer Technology and Applications

K

( N  1)(1  R 2 ) ( N  1)(1  R 2 )  2 R

Z

Z oo Z0

R

Z oo Z oe

R[( N  1)  R][( N  1) R  1] 1 R

(4.101)

(4.102)

(4.103)

Where, Zoo and Zoe, respectively, are the odd and even mode impedances of two adjacent coupled lines, of which all pairs in the structure are identical. The determination of impedance ratio, R, and normalized odd impedance are available from standard design charts, reproduced in Figure 4.76(a) and (b). These values can then be used with tables of coupled line data to determine S/h and W/h to determine dimensions for the coupler. More accurate results can be obtained from EM analysis of a coupled line structure. For the particular case of 50 ohms and 3 dB coupling, which is the object of the present quadrature hybrids, the desired dimensional relations are obtained from the plot on Figure 4.77 for dielectric constant ranging from r = 2 to 16.

Impedance ratio versus coupling. Figure 4.76

Normalized odd mode impedance versus coupling. Design charts for four lines Lange couplers. From [65].

Active and Passive Coupling Structures

Figure 4.77

225

Gap and line width normalized to substrate height versus dielectric constant. From [65].

4.9 SUMMARY This chapter included extensive material on various types of coupling structures required to apply LO and RF to a mixer and extract the converted IF signal. The passive components ranged from waveguide, to coaxial, to various forms of transmission lines using solder, chip and wire, and monolithic integrated circuits on a GaAs or Si die. The bifilar components, which are still useful today in lower frequency applications, operate over multi-octave bandwidths and allow size reduction. Active approaches with bipolar or FETs were also introduced, and are useful to build mixer circuits in a monolithic semiconductor die.

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Microwave Mixer Technology and Applications

REFERENCES [1] J. S. McLean, “Balancing Networks for Symmetric Antennas: Part 1Classification and Fundamental Operation,” IEEE Transactions on Electromagnetic Compatibility, Volume 44, No. 4, November 2002, pp. 503-514. [2] C. Trask, “A Single-Core 4:1 Current Balun of Improved Performance,” Sonoran Radio Research, P.O. Box 25240, Tempe, AZ. [3] C. T. Rauch, “Balun for an Antenna,” US Patent 7,319,435, issued January 15, 2008. [4] E. Rotholz, “Transmission-Line Transformers,” IEEE Transactions On Microwave Theory and Techniques, Volume MTT-29, No. 4, April 1981, pp. 327331. [5] C. Trask, “Transmission Line Transformers: Theory, Design and Applications – Part 2,” High Frequency Electronics, January 2006, pp. 26-33. [6] J. Sevick, “A Simplified Analysis of the Broadband Transmission Line Transformer,” High Frequency Electronics, February 2004. [7] P. Gomez-Jimenez, P. Otero, and E Marquez-Segura, “Analysis and Design Procedure of Transmission-Line Transformers,” IEEE Transactions on Microwave Theory and Techniques, Volume 56, No. 1, January 2008, pp. 163171. [8] G. Guanella, “Device for Intercoupling Single-Ended and Double-Ended Circuits,” US Patent 2,509,057, issued May 23, 1950. [9] G. Guanella, “High-Frequency Matching Transformer,” US Patent 2,470,307, issued May 17, 1949. [10] J. Sevick, “Design of Broadband Ununs With Impedance Ratios Less than 1:4,” High Frequency Electronics, November 2004. [11] C. L. Ruthroff , “Some Broad-Band Transformers,” Proceedings of the IRE, Volume 47, August 1959, pp. 1337-1342. [12] J. Sevick, “Magnetic Material for Broadband Transmission Line Transformers,” High Frequency Electronics, January 2005. [13] P. Lefferson, “Twisted Magnet Wire Transmission Line,'' IEEE Transactions on Parts, Hybrids, and Packaging, December 1971, Volume PHP-7, No. 4, pp. 148-154. [14] O. Pitzalis, Jr., Robert E. Horn, and Ronald J. Baranello, “Broadband 60-W HF Linear Amplifier,” IEEE Journal of Solid-State Circuits, June 1971, Volume SC-6, No. 3, pp. 93-103. [15] N. Marchand, “Transmission Line Conversion Transformers,” Electronics, December 1944, Volume 17, No. 12, pp. 142-145. [16] George Oltman, “The Compensated Balun,” IEEE Transactions on Microwave Theory and Techniques, March 1966, Volume MTT-14, No. 3, pp. 112-119. [17] Willmar K. Roberts, “A New Wide-Band Balun,” Proceedings of the IRE, December 1957, pp. 1628-1631.

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[18] J. H. Cloete, “Exact Design of the Marchand Balun,” Microwave Journal, May 1980, pp. 99-102. [19] R. Knochel, B. Mayer, and U. Goebel, “Unilateral Microstrip Balanced and Doubly Balanced Mixers,” 1989 IEEE MTT-Symposium Digest. [20] Ross Lumpe Jr., “Wideband Balun Realized by Equal Power Divider and Short Circuit Stubs,” US Patent 4,725,792, issued February 16, 1988. [21] H. R. Ahn and B. Kim, “Toward Integrated Circuit Size Reduction,” IEEE Microwave Magazine, February 2008, pp. 65-75. [22] G. L. Matthaei et al, Microwave Filters, Impedance Matching Networks and Coupling Structures, New York, McGraw-Hill, 1964, pp. 221. [23] R. Jacques and D. Meignant, “Novel Wide Band Microstrip Balun,” European Microwave Conference, September 1981. [24] C. Cho and K. C. Gupta, “A New Design Procedure for Single Layer and Two-Layer Three-Line Baluns,” IEEE Transactions on Microwave Theory and Techniques, Volume 46, No. 12, December 1998, pp. 2514-2519. [25] H. Tanaka, Y. Sasaki and T. Hashimoto, “Unbalanced-to-Balanced Converter,” US Patent 6,040,745, issued March 21, 2000. [26] K. S. Ang and I. D. Robertson, “Analysis and Design of ImpedanceTransforming Planar Marchand Baluns,” IEEE Transactions on Microwave Theory and Techniques, Volume MTT- 49, No. 2, February 2001, pp. 402-406. [27] G. E. Bodway, “Two Port Power Flow Analysis Using Generalized Scattering Parameters,” Microwave Journal, Volume 10, May 1967. [28] R. Bawer and J. J. Wolfe, “A Printed Balun for Use with Spiral Antennas,” IRE Transactions on Microwave Theory and Techniques, May 1960, pp. 319-325. [29] G. J. Laughlin, “A New Impedance-Matched Wide-Band Balun and Magic Tee,” IEEE Transactions on Microwave Theory and Techniques, Volume MTT24, No. 3, March 1976, pp. 135-141. [30] Carro, P.L., de Mingo, J., Garcia-Ducar, P., and Sanchez, C., ”Synthesis of Hecken-Tapered Microstrip to Parallel-Strip Baluns for UHF Frequency Band,” Microwave Symposium Digest (MTT), 2011 IEEE MTT-S. [31] B. F. Gunshinan, “Microstrip Balun,” US Patent 3,523,260, issued August, 4, 1970. [32] D. Neuf, “Double Balanced Microwave Mixer Using Balanced Microstrip Baluns,” US Patent 3,652,941, issued March 28, 1972. [33] S. A. Maas, Microwave Mixers, Second Edition, Artech House, Inc., 685 Canton St., Norwood, MA 02062, 1993. [34] W. Marcczewski and W. Niemyjski, “Microwave Balun for Mixers and Modulators,” US Patent 4,755,775, issued July 5, 1988. [35] A. M. Pavio, “Monolithic Multilayer Planar Transmission Line,” US Patent 5,025,232, issued June 18, 1991. [36] A. M. Pavio and A. Kikel, “A Monolithic or Hybrid Broadband Compensated Balun,” IMS-1990 Symposium Digest, pp. 483-486.

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[37] K. S. Ang, S. B. Economides, S. Nam, and I. D. Robertson, “A Compact MMIC Balun Using Spiral Transformers,” IMS-1999 Symposium Digest, pp. 655658. [38] P.S. Wu, C. H. Wang, T. W. Huang, and H. Wang, “Compact and BroadBand Millimeter_Wave Monolithic Transformer Balanced Mixers,” IEEE Transactions on Microwave Theory and Techniques, Volume MTT-53, No. 10, October 2005, pp. 3106-3114. [39] J. G. Padilla, “Ultra-Wideband Planar Coupled Spiral Balun,” US Patent 6,683,510, issued January 27, 2004. [40] D. Kuylenstierna and P. Linner, “Design of Broadband Lumped Element Baluns,” IMS-2004 Symposium Digest, pp. 899-902. [41] L. Fan and K. Chang, “A 180º out-of-Phase Power Divider Using Asymmetrical Coplanar Stripline,” IEEE Microwave and Guided Wave Letters, Volume 6, No. 11, November 1996, pp. 404-406. [42] M. C. Tsai, M. J. Schindler, W. Struble, M. Ventresca, R. Binder, R. Waterman, and D. Danzilio, “A Compact Wideband Balanced Mixer,” IMS-1994 Symposium Digest, pp. 5-8. [43] P. R. Andrys and P. H. Thompson, “Double Balanced Differential Active Ring Mixer With Current Shared Active Input Balun,” US Patent 6,057,714, issued May 2, 2000. [44] Christophe Viallon and Thiery Parra, “Microwave Differential Structures Optimization: Application to a Double Balanced SiGe Active Down-converter Design,” IEEE International Workshop on Radio Frequency Integration Technology, November 30 – December 2, 2005, Singapore, pp. 91-94. [45] Tatsuo Tanaka, "Differential Amplifier," US Patent 4,555,670, issued November 26, 1985. [46] P. R. Gray, P. J. Hurst, S. H. Lewis, and R. G. Meyer, Analysis and Design of Analog Integrated Circuits, John Wiley & Sons, Inc., 2001, pp. 296. [47] A. Prins and H. J. Velo, “Single-Ended/Push-Pull Converter,” US Patent 4,001,706, issued January 4, 1977. [48] W. A. Tyrrell, “Coupling Arrangement for Use in Wave Transmission Systems,” US Patent 2,445,895, issued July 27, 1948. [49] R. E. Collin, Foundations for Microwave Engineering, IEEE Press, 2001, pp. 437-442. [50] S. March, “A Wideband Stripline Hybrid Ring,” IEEE Transactions on Microwave Theory and Techniques, June 1968, pp. 361. [51] Chi-Yang Chang, Chu-Chen Yang, Dow-Chih Niu, “A Multioctave Bandwidth Rat-Race Singly Balanced Mixer,” IEEE Microwave and Guided Wave Letters, Volume 9, No. 1, January 1999, pp. 37-39. [52] Kim, Y.G., Song, S.Y, Kim, K.W., “A Compact Wideband Ring Coupler Utilizing a Pair of Transitions for Phase Inversion,” IEEE Microwave and Wireless Components Letters, Volume 21, No. 1, January 2011, pp. 25-27. [53] S. J. Parisi, “180º Lumped Element Hybrid,” IMS-1989 Symposium Digest, pp. 1243-1246.

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[54] M. Dydyk, R. F. Keilmeyer, Jr, J. K. Lauchner, “Lumped Element Realization of Ring Hybrids Including  Circuit and Tank Circuit Means,” US Patent 5,175,517, issued December 29, 1992. [55] S. Ohi, H. Matsubara and N. Ohyama, “Balun,” US Patent 7,116,185, issued October 3, 2006. [56] K. S. Ang and Y. C. Leong, “Converting Baluns Into Broad-Band Impedance-Transforming 180º Hybrids,” IEEE Transactions on Microwave Theory and Techniques, Volume MTT- 50, No. 8, August 2002, pp. 1990-1995. [57] R. E. Blight, “Broadband Microstrip Hybrid Tee,” US Patent 3,530,407, issued September 22, 1970. [58] H. G. Oltman Jr, N. Pelner, and E. N. Torgow, “Slot Line/Microstrip Hybrid”, US Patent 3,946,339, issued March 23, 1976. [59] R.B. Mouw, “A Broadband Hybrid Junction and Application to the Star Modulator,” IEEE Trans. MTT, November 1968. [60] David E. Norton, “Three Decade Bandwidth Hybrid Circuits,” Microwave Journal, November 1988, pp.117-126. [61] R. B. Mouw, “Hybrid Junction and Mixer or Modulator,” US Patent 3,818,385, issued June 18, 1974. [62] S. A. Maas and K. W. Chang, “A broadband, planar, doubly balanced monolithic Ka-band diode mixer,” IEEE Transactions on Microwave Theory and Techniques, December 1993, Volume 41, pp. 2330-2335. [63] K. W. Chang, T-Hung Chen, L. C. T. Lui, and S. B. T. Bui, "Diode Mixer Implemented in a Planar Mononolithic IC", US Patent 5,428,838, issued June 27, 1995. [64] R. E. Collin, Foundations for Microwave Engineering, IEEE Press, 2001, pp. 437-442. [65] A. Presser, “Interdigitated Microstrip Coupler Design,” IEEE Transactions on Microwave Theory and Techniques, Volume MTT- 26, No. 10, October 1978, pp. 801-805.

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Appendix 4A – Guanella 4:1 Transformer While two port Z and Y parameters, respectively, are useful for connecting networks in series and parallel, H parameters are useful for connecting them in parallel at one port and in series at the other. For two networks connected in series at the input, and in parallel at the output, the H parameters for the respective networks are simply added. Since the Guanella 1:4 balun has two transmission lines connected in parallel at the input and in series at the output, the ABCD matrix for the lossless transmission line is transformed to the H matrix, and then multiplied by 2 since the transmission lines are identical. Rg

I1+I2

I2

+

V

-

I1

Vg

V2 RL = 4 R G

V1 V2

Figure 4A.1

Guanella 1:4 transformer (see Figure 4.3).

The ABCD matrix of a lossless transmission line is:

ABCDT  LINE

 cos( l ) 1    jZ 0 sin( l )

jZ 0 sin( l )  cos( l )  

(4A.1)

The H matrix in terms of the ABCD matrix is:

H

1  B det A C  D  1

(4A.2)

The H matrix of the lossless transmission line is obtained and then multiplied by 2 to represent the complete balun:

Active and Passive Coupling Structures

  jZ 0 tan(l ) H T  LINE   1   cos( l ) H BALUN  2H T  LINE

1  cos( l )   j tan(l )  Z 0 

231

(4A.3)

The input impedance on the high impedance side of the Guanella transformer in terms of H parameters, given source resistance, Rg, is Zin,HI:

Z in, HI 

(det(H ) R g  h11 1  R g h22

2 jZ 

2 0



 4 R g Z 0 tan(l ) cos 2 l  tan(l )  4 R g Z 0 cos 2 ( l )2 jR g tan(l )  Z 0 ) 

(4A.4) With Rg =25 ohms, Z0=50 ohms, and βl=π/4 the resulting impedance is Zin,HI = 100 ohms. The input impedance on the low impedance side of the Guanella transformer in terms of H parameters, given load impedance, Z L, is Zin,LOW:

Z in, LOW 

Z 0 cos 2 l ( Z L  j 2 tanl Z 0 Z L  h11  det H  Z L h22 2 tan(l ) cos 2 ( l ) jZ L  2Z 0 tan(l )   4Z 0 (4A.5)

With ZL = 100 ohms, Z0 = 50 ohms, and βl = π/4 the resulting impedance is Zin,LOW = 25 ohms. These equations for Zin,HI and Zin,LOW give the same numerical results as (4.1) and (4.2). The available gain for the balun going from the series end to the parallel connected end in terms of Z parameters is:

R Z 21 GA  G RL Z11  Z G

2

(4A.6)

Converting the HBALUN matrix to a Z matrix:

Z

1 h22

det H h  21

h12  1 

(4A.7)

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Microwave Mixer Technology and Applications

  2Z 0 tan(l ) 2 1  cos(2l )  4Z 0  j sin(2l ) Z  Z0   j sin( l )

 Z0  j sin( l )  Z 0 cot(l )   2j

(4A.8)

The available gain from high impedance to low impedance is then: 2

R Z0 G A, HighToLow  G RL RG 2 sin 2 ( l )  4Z 0 2 cos 2 ( l )

(4A.9)

In this case RG = 100 ohms, and RL = 25 ohms and gain is equal to 1, or 0 dB. The available gain in the opposite direction, from low impedance to high impedance, is obtained by swapping RG and RL, and is also given in (4.3): 2

Z0 R G A, LowToHigh  L RG RL 2 sin 2 ( l )  4Z 0 2 cos 2 ( l ) In this case RG = 25 ohms, RL = 100 ohms, Z0=50 ohms, and βl=π/4. These give GA = 1 = 0 dB. Appendix 4B – Compensated Balun The balun shown in Figure 4B.1 can be analyzed by first writing the Y matrix for the circuit. The Y matrix is inverted to solve the Z matrix that gives node voltages in terms of driving current. Then the S-parameters are easily obtained from Z parameters. Terminations G1, G2, and G3, respectively, are at ports 1, 2, and 3. The ABCD matrix for the lossless transmission line is in (4B.1) and the equivalent two port Y matix is in (4B.2).

ABCDT  LINE

 cos( l ) 1    jZ 0 sin( l )

jZ 0 sin( l )  cos( l )  

(4B.1)

Active and Passive Coupling Structures

YT  LINE

 cos( l )  jZ sin( l )  0 1   jZ 0 sin( l )

1  jZ 0 sin( l )   cos( l )  jZ 0 sin( l ) 

233

(4B.2)

2 T2

T1

3 T3

T1: L1 = /2 Zo = R2 T2: L2 = 3/4 Zo = R2 T3: L3 = /4 Zo = R2

1 Figure 4B.1 Compensated microstrip balun (see Figure 4.13).

By inspection, the YBALUN matrix for the compensated balun is:   cos(l3 ) cos(l2 ) 1 1  G1   jZ sin(  l ) jZ sin(  l ) jZ sin(  l ) jZ sin(  l ) 3 3 2 2 2 2 3 3   1 cos(l1 ) cos(l2 ) 1    G2     jZ 2 sin( l2 ) jZ1 sin( l1 ) jZ 2 sin( l2 ) jZ1 sin( l1 )    1  1 cos(  l ) cos(  l ) 1 3   G3   jZ 3 sin( l3 ) jZ1 sin( l1 ) jZ1 sin( l1 ) jZ 3 sin( l3 )  

(4B.3) To simplify the matrix inversion, the parameters are assigned values: βl1 = π βl2 = 3π/2 βl3 = π/2 G1 = G2 = G3 = 1/R Z1 = Z2 = Z3 = R√2 These values are substituted in to simplify the Y matrix, except for βl1 = π in the denominator sin(βl1) terms that instead are substituted after inverting to get the Z matrix. After substitution the Y matrix becomes:

234

YBALUN

Microwave Mixer Technology and Applications

 1   R  1  j 2R   1  j 2 R

1 j 2R 1 1  R j 2 R sin( l1 ) 1 j 2 R sin( l1 )

 1  j 2R  1  j 2 R sin( l1 )   1 1   R j 2 R sin( l1 ) 

(4B.4)

Inverting the Y matrix and substituting in the βl1 = π terms gives the following Z parameters:

Z11  R / 2

Z 21   2 R /( 4 j ) Z 31  2 R /( 4 j )

Z 22  R / 4 Z 33   R / 4 Once the Z matrix including port terminations is obtained, S-parameters are obtained as follows, with Rn as the nth port terminating resistance.

S nn 

2Z nn  1 ; on diagonal Rn

2Z nm

S nm 

Rn Rm

 1 ; on diagonal

With all termination resistances equal to R, the resulting S parameters are:

S11  0 S 21 

S 21 

S 22

1

j 2 1

j 2  S 33  S 23  1 / 2

Active and Passive Coupling Structures

235

Appendix 4C – Active 180 FET Power Divider 1

I1

2

Cgs

g0

gds

gD

3 gm(V1-V3) gS

Figure 4C.1

Active power divider with simplified FET model. Node 1 corresponds to gate, node 2 corresponds to drain, and node 3 corresponds to source.

The Y matrix including the termination resistors, gD and gS is in (4C.1). Replacing the current source relation with input voltages, there results (4C.2).

I1   g 0  jC gs  g (V  V )   0 3    m 1  g m (V1  V3 )    jC gs

0 g D  g ds  g ds

 V1   jC gs    g ds  V2  g S  g ds  jC gs  V3  (4C.1)

 I1   g 0  jC gs 0    gm    0   jC gs  g m

0 g D  g ds  g ds

 V1   jC gs    g ds  g m  V2  (4C.2) g m  g S  g ds  jC gs  V3 

Inverting the Y matrix gives the Z matrix. Since the driving current is only at port 1 (gate), the second and third columns of the Z matrix are not needed.

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Microwave Mixer Technology and Applications

 g D ( g S  g m  g ds )  g S g ds  jC gs  g ds  g D  x  g g ( g  g m  g ds )  g 0 g S g ds  jC gs g S  g 0 g D  g ds   g D g ds  V1   0 D S jC gs g ds  g S g m V     2   g g ( g  g  g )  g g g  jC g  g g  g   g g  x m ds 0 S ds gs S 0 D ds D ds  0 D S V3   g D g m  jC gs  g ds  g D  x   g 0 g D ( g S  g m  g ds )  g 0 g S g ds  jC gs g S  g 0 g D  g ds   g D g ds 

 x I   1 x  0   0    x 

(4C.3) Once the Z matrix including port terminations is obtained, S- parameters are obtained as follows, with Rn as the nth port terminating resistance.

S nn 

S nn 

2Z nn  1 ; on diagonal Rn

2Z nm Rn Rm

; off diagonal

jC gs g ds  g S g m

S 21  2 g D g 0

g 0 g D ( g S  g m  g ds )  g 0 g S g ds  jC gs g S  g 0 g D  g ds   g D g ds  (4C.4)

S 31  2 g S g 0

g 0 g D ( g S  g m  g ds )  g 0 g S g ds  jC gs g S  g 0 g D  g ds   g D g ds  (4C.5)

g D g m  jC gs g ds  g D 

Active and Passive Coupling Structures

237

Appendix 4D – Alternative 180 FET Power Divider The schematic for the circuit of Figure 4.42 using a simplified FET model is in Figure 4D.1. Notice there is DC current at port 1 from bias of common gate device, which means there is a need for a DC block and feed circuit at node 1 not shown in the figure. The Y matrix for this circuit including the termination resistors is in (4D.1). 2 Gate1

Drain1

Cgs1 Source2

1

I1

gds1

g1

g2

gm1

gds

3

Source2 Drain2 gm2

g3

Cgs2 Gate2 Figure 4D.1

Active power divider with simplified FET model. Node 1 corresponds to connection of gate of common source device and source of common gate device, node 2 corresponds to drain of common source device, and node 3 corresponds to drain of common gate device.

 I 1   g m 2  g1  g ds  2 jC gs 0    g m1    0    g m 2  g ds

0 g 2  g ds 0

 g ds  V1  0  V2  g 3  g ds  V3 

(4D.1)

To simplify results let’s make the output terminations the same g = g3 = g2, and leave the input termination as g1. Inverting the Y matrix gives the Z matrix. Since the driving current is only at port 1 (gate), the second and third columns of the Z matrix are not needed.

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Microwave Mixer Technology and Applications

V1   Z11 V    Z  2   21 V3   Z 31

x x x

x   I1  x  0  x  0 

The Z parameters for this circuit is given by (4D.2) to (4D.4) as follows:

Z11 

Z 21 

Z 31 

g g1  g m 2

( g  g ds )  g ds   g1 g ds  2 jC gs g  g ds 

(4D.2)

g g1  g m 2

 g m1  g ds   g1 g ds  2 jC gs g  g ds 

(4D.3)

g g1  g m 2

g m 2  g ds  g ds   g1 g ds  2 jC gs g  g ds 

(4D.4)

Once the Z matrix with port termination included is obtained, Sparameters are derived as follows, with Rn as the nth port terminating resistance.

S nn 

S nn 

S 21 

S 31 

2Z nn  1 ; on diagonal Rn

2Z nm Rn Rm

; off diagonal

 2 g m1 gg1

g g1  g m 2  g ds   g1 g ds  2 jC gs g  g ds 

2g m 2  g ds  gg1

g g1  g m 2  g ds   g1 g ds  2 jC gs g  g ds 

(4D.5)

(4D.6)

Active and Passive Coupling Structures

239

Appendix 4E – Active 180 FET Combiner Using the simplified FET model, the circuit from Figure 4.46 can be represented as shown in Figure 4E.1. The DC biasing circuitry was suppressed for ease of calculation. The Y matrix for this circuit including the termination resistors is in (4E.1), assuming the common drain and common source devices are similar. 1 g1

I1

Cgs

gds gm 3

2 g2

I2

Cgs

gds

g3

gm Figure 4E.1

Active power combiner. Node 1 corresponds to connection to the gate of common drain device, node 2 corresponds to connection to the gate of common source device, and output node 3 corresponds to the connection point of the common-source drain with common-drain source.

 I 1   g1  jC gs I    0  2  0   g m  jC gs

0 g 2  jC gs gm

 V1   jC gs   0  V2  g 3  2 g ds  jC gs  V3 

(4E.1)

Inverting the Y matrix we obtain the Z matrix that provides the required voltage relations. Similar to previous manipulation, let’s make the following terminations the same g2 = g3 = g1= g0. The Z-parameters of interest are in (4E.2) and (4E.3). Z 31 

g m  jCgs g 0 ( g 0  g m  2 g ds )  j 2Cgs ( g 0  g ds )

(4E.2)

Z 32 

 gm g 0 ( g 0  g m  2 g ds )  j 2Cgs ( g 0  g ds )

(4E.3)

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The S-parameters derived from the Z parameters are in (4E.4) and (4E.5). The output voltage as a function of the two input voltages is in (4E.6), the final result for output voltage is in (4E.7). S31 

2 g 0 ( g m  jCgs ) g 0 ( g 0  g m  2 g ds )  j 2C gs ( g 0  g ds )

(4E.4)

S32 

 2 g m g0 g 0 ( g 0  g m  2 g ds )  j 2C gs ( g 0  g ds )

(4E.5)

V3  Z 31 I 1  Z 32 I 2  Z 31V1 g1  Z 32V2 g 2 V3 

g 0 g m (V1  V2 )  jC gs g 0V1 g 0 ( g 0  g m  2 g ds )  j 2C gs ( g 0  g ds )

(4E.6) (4E.7)

Chapter 5 Diode Mixer Theory Mixing is the process used to translate the center frequency of an RF signal to an intermediate frequency (IF). Ideally, any modulation present on the RF signal transfers perfectly to the IF without the introduction of noise or undesired signals. Frequency translation is accomplished by combining, or “mixing,” the RF signal with a local oscillator signal, LO, to create the IF signal. The IF equals the sum or difference between the RF and LO frequencies. The RF level is usually much lower than that of the LO, so it can be approximated as a small signal that converts to the IF using a linear process. The small signal RF is modulated by the time varying admittance (or impedance) created by the large signal LO “pumping” a nonlinear device, which is analyzed as a nonlinear process. Various nonlinear devices are used including diodes, FETs and BJTs. Frequency conversion from RF to IF can be described mathematically by multiplying the Fourier series of the RF signal with that of the admittance or impedance waveform, resulting in the Fourier series of the IF output. The terms RF and IF are not used consistently in the literature; for example, in an up-converter RF sometimes refers to the output, and IF to the input. However, in this chapter the RF is the small signal input and the IF is the desired output regardless of which frequency is higher. This goals of this chapter are to provide an overview of mixer theory in a simple yet sufficiently detailed manner to allow a solid understanding of the diode mixer circuits presented in Chapter 6, as well as provide insights for understanding the transistor based mixers of the later chapters. The method of presentation is to use the most basic of mixer circuits with a detailed presentation of the theory. We begin with a discussion of the basic square wave mixer and the concept of the conduction waveform whose Fourier coefficients are derived using a nonlinear analysis, which are then used to calculate small signal mixer performance using linear analysis. The conversion matrix is introduced using the classic analysis of the Y-mixer to illustrate the inter-relationships between mixer performance, port impedances and terminations, and the diode I/V equation. Closed form equations are derived for conversion loss and port impedances, and then verified by harmonic balance simulations. The classic nonlinear analysis approaches are described for estimating gain compression and intermodulation (IM) suppression, followed by a detailed discussion of subharmonically pumped 241

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diode mixers. The effects of diode series resistance, junction capacitance, and lead inductance are given using the standard approach that ignores terminations to higher order mixing products, and another that uses a five term conversion matrix that is solved numerically and includes diode parasitics. Next, a simple method is described for determining which mixing products will be present at the various ports of a balanced mixer, to help select the best circuit for a given mixer application. The chapter ends with a discussion of mixer synthesis using an existing linear approach that is applied to the conversion matrix. 5.1 HISTORY OF LINEAR AND NONLINEAR ANALYSIS If the RF input signal is a complex modulated waveform, the IF output ideally will have an identical modulation spectrum shifted to the new center frequency. As the incident RF level increases, gain compression and intermodulation occur. To capture these effects, the analysis must abandon the linear-conversion assumption and use a fully nonlinear approach in which the RF signal and LO are both included in the large signal analysis. Linear analysis is useful to calculate conversion loss, noise figure, port impedances, and optimal port terminations. It requires a single integration to calculate Fourier coefficients of the admittance (or impedance) waveform associated with LO harmonics. In contrast, a full nonlinear analysis provides the same results plus information about gain compression and intermodulation. The tradeoff is the full nonlinear analysis is significantly more complicated, with the easiest part being the inclusion of a double integration to calculate the Fourier coefficients associated with both LO and RF harmonics. Over the years many published analyses have used the linear approach, including the work by Torrey and Whitmer in 1948 [1], Barber in 1967 [2], Saleh in 1971 [3], Egami in 1973 [4], and Held and Kerr in 1978 [5]. These include the “conversion matrix” discussed later in this chapter that includes effects of arbitrary terminations to RF, IF, image, and other mixing products. The early linear analyses limited the conversion matrix to include only the RF, IF, and image, with all other mixing products assumed short or open circuited. This resulted in closed form approximations for conversion loss and optimal port impedances that gave valuable insight into mixer behavior. The later analyses included additional mixing products in the conversion matrix that provided results needed for modern mixer design purposes, and required computer simulation. The early nonlinear analyses were carried out without the conversion matrix by approximating the diode as an ideal switch or exponential diode resulting in closed form equations [6, 7, 8, 10]. In light of their simplicity, they provided surprisingly valuable estimates for compression and intermodulation. One such approach is used in modern system simulators to estimate IM suppression [9]. These simplified approaches do not include effects from terminations to image, sum, and higher order mixing products, which are required for modern mixer design purposes. In contrast, nonlinear analyses that include the conversion

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243

matrix are carried out numerically. The first report of accurate simulation of intermodulation in a diode mixer including nonlinear capacitance was given in 1987 [11]. The analysis showed that terminating the image in a high-impedance inductive load reduced conversion loss as expected, but could also increase IM and noise levels. Providing a short circuit termination to the image port reduced conversion loss without degrading noise or IM levels. The distinction between linear and nonlinear analysis methods in recent years has largely become academic. With the advent of SPICE and harmonic balance simulation software, and high quality nonlinear device models, mixers can now be analyzed and designed without the need for limiting assumptions such as small signal RF, exclusion of nonlinear junction capacitance, and all mixing products shorted except RF, LO, and IF. Having said this it is instructive to review the development of linear and nonlinear analysis methods and the resulting closed form approximations to understand the underlying relationships that drive mixer performance and behavior. 5.2 LINEAR MIXER ANALYSIS The periodic time varying impedance, generated by the LO pumping the nonlinear device, can be approximated as a switching function having a rectangular or a square waveform [2]. Both linear and nonlinear analyses have been done using this ideal approach. Alternately one can model the mixing process by applying the sum of the RF and LO voltages to the equation relating the current and voltage of the mixing device; for example, using the exponential diode equation. This approach also yields a time varying modulation waveform, and both linear and nonlinear versions of this approach have been published. The simplest equation relating the current and voltage at the mixing device is a quadratic. A practical device that provides a true quadratic current-voltage relationship does not exist [12]; however, it is most closely approximated by the FET. On the other hand, the simplest mixing device is the two terminal diode rectifier, whose current-voltage transfer function is exponential, which can be viewed as a high degree polynomial. Regardless of whether the mixing process is characterized as a small signal voltage or current being modulated by a switching waveform, or as the RF and LO voltages applied to a nonlinear I/V transfer function, at the most basic level the math is the same. Given sinusoidal RF and LO input signals, the mixing process is characterized by multiplying the two sinusoids as in (5.1). The familiar trigonometric identity shows the multiplication results in two new sinusoids having frequencies ωLO + ωRF and ωLO – ωRF, respectively, referred to as the sum IF and difference IF.

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Microwave Mixer Technology and Applications

1 cos( LO   IF )t  cos( LO   IF )t  2

cos LO t  cos IF t  

(5.1)

Figure 5.2

3fL 2fL + fR

fL + fR f+2 sum

4fL - fR

2fL 3fL - fR

fR

2fL - fR

f-2

f-3

f+3

f-3

f+3

Mixer output spectrum with RF included in the nonlinear analysis, with fIF = fRF-fLO includes terms at pfIF and nfLO±mfIF for n=1,2,3 and p=m=1,2.

fL + 2fR

2fL + fR

3fL 4fL - fR

f+2

5fL - 2fR

fL +fR

f-2

2fR

3fL - fR

f+1

4fL - 2fR

f-1

-fL + 2fR

fL

2fL

Mixer output spectrum given a linear analysis with fIF = fRF-fLO includes terms at fIF and nfLO±fIF for n=1,2,3.

fR

fIF

f+1 signal

2fL - fR

fR - fL 2(fR - fL)

Figure 5.1

f-1 image

3fL -2fR

fIF

fL

fR - fL

Modulating the RF signal produces an ensemble of IF output signals. Given a linear analysis with fIF = fRF-fLO, the output ensemble comprises terms at fIF and nfLO±fIF as depicted in Figure 5.1 for n=1, 2, 3. The frequency notation is f±n = nfL±fIF, which is useful in the conversion matrix. In contrast, given a fully nonlinear analysis, the output ensemble comprises terms at pfIF and nfLO±mfIF as depicted in Figure 5.2, for n=1,2,3; p=1,2; and m=1,2.

Diode Mixer Theory

245

If two small signal tones are applied, fRF1 and fRF2, then their resulting intermodulation products will also be present, most notably the second and third order ones. 5.2.1 Simple Mixer Perhaps the simplest and most popularized method of characterizing the linear mixing process is to represent it as the product of the RF signal and a rectangular or square wave switching function. If the switching function varies between 0 and 1 at the LO frequency, then the product between it and the RF signal produces an IF output consisting of the RF signal switched on and off at the LO frequency. The IF spectrum contains the RF fundamental, and the sum and difference frequencies of (5.1). This simple case of the switching waveform varying between 0 and 1 corresponds to the single-ended mixer, which uses a single mixing device to switch the RF signal on and off at the LO frequency, and illustrates mixing as a process of multiplication. If the switching function instead varies between +1 and -1 as depicted in Figure 5.3, and as represented by the Fourier series in (5.2), then the product of the RF multiplying the switching function results in the voltage given by (5.3). During the interval 0 to π, the RF signal multiplies 1 and thus is unchanged. But between π to 2 π, the RF multiplies -1. Thus the IF output equals the RF input signal shifted in phase by 180 degrees at the LO frequency. While this may appear to be a phase modulation process, it is fundamentally multiplicative. The spectrum of Vout for (5.3) corresponds to the doubly balanced mixer, in which the RF fundamental is suppressed, and the output signals have frequencies equal to nfLO±fRF where n=odd. For example, referring to the ring doubly balanced mixer of Figure 5.4, it can be seen that the IF output equals the RF input switched 180 degrees in phase at the LO frequency. This is accomplished as the LO alternately switches the diodes D1/D2 on and D3/D4 off, and vice versa, alternately grounding the opposite polarity ends of the RF balun through the grounded center tap of the LO balun.

1 S(t)

ωLt -1

Figure 5.3

π

2π

Doubly balanced mixer switching function.

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Microwave Mixer Technology and Applications

S (t ) 

4



1 sin(n LO t )  n 1,3,5,... n



4  1  VOUT  VRF sin( RF t )  sin(n LO t )    n 1,3,5,... n  VOUT  VRF

2



(5.2)

(5.3)

cos( LO   RF )t  cos( LO   RF ) 

1 cos(3 LO   RF )t  cos(3 LO   RF )  3 1 cos(5 LO   RF )t  cos(5 LO   RF )  ... 5 

V  LC  20 log  OUT   VRF   2   20 log   at nωL±ωR.  n 

(5.4)

(5.5)

= 3.92 dB; at ωL±ωR Taking the desired IF output to be the sum or the difference product with n=1, ωIF = ωLO ± ωRF, conversion loss is the familiar value of 3.92 dB. This analysis so far assumes no losses in the switching device, and the remaining RF signal power converted to nωLO ± ωRF is lost. Conversion loss can be reduced to theoretically 0 dB by using reactive terminations at the nωLO ± ωRF mixing frequencies. One may assume the power at the nωLO ± ωRF frequencies reflects back into the mixing device and remixes to add constructively or destructively with the desired IF, depending on the reactance of each termination. In practical circuits use reactive terminations to affect mixer performance is limited to one or a few of the mixing terms, generally the image and sum, and useful over only a narrow frequency range. An important point about the doubly balanced diode mixer is illustrated by Figure 5.4 concerning IM distortion, which is mentioned here briefly and developed more fully later in this chapter and Chapter 6. At a particular instant in time the phase of RF and LO signals relative to the four diode currents are as depicted in the figure. The LO forward biases diodes D1 and D2, and reverse biases diodes D3 and D4. And the RF adds a slight forward bias to diodes D2 and D3, and a slight reverse bias to diodes D1 and D4. Ideally, diodes D1 and D2 turn on at the same instant in time, and diodes D3 and D4 turn off at the same instant,

Diode Mixer Theory

247

controlled by solely the LO. However, the effect of the RF current voltage is to cause them to turn on and off at slightly different times; this is also true for D 3 and D4. This effect has been described as phase modulation of the zero crossings of the respective conductance waveforms of the four diodes [13]. The effect is to introduce IM distortion that increases with increasing RF signal level. It is assumed for the small signal linear mixing analysis that this effect is negligible, thus conductance is controlled only by the LO. But for the large signal analyses, both LO and RF signals are included in analyzing the diode conductance. This discussion so far concerns the diode mixing device, but the same problem occurs for balanced mixers using FET and BJT devices, with the level of distortion produced being dependant on the circuit implementation.

LO

+VLO i4

i1 D1

D4

D2 D 3 -VLO

i2

i3 IIF

+VRF

-VRF

IF

RF Figure 5.4

Doubly balanced mixer schematic.

In this discussion so far, the linear mixer has been described as having an RF input port, an IF output port, and also ports for higher order products. These virtual ports are at the frequencies of the various mixing products, and share the same physical connection point. In the ensuing discussion, the conversion matrix is introduced that represents a linear mixer having three virtual ports: the RF input, IF output, and the image. The “Y,” “Z,” “H,” and “G” mixers are introduced, and the “Y” mixer is examined in detail.

5.3 FREQUENCY CONVERSION MATRIX During the 1940s significant work was done to analyze and design crystal rectifier mixers for radar applications [1]. The analysis process assumed the crystal rectifier was pumped by a large signal LO that controlled the switching characteristics. The RF signal was applied at a level much below that of the LO, allowing for the approximation of linear mixing. This in turn allowed the use of a conversion matrix relating the currents and voltages of the RF, IF, and other

248

Microwave Mixer Technology and Applications

mixing products. And to simplify the analysis to obatin closed form equations, this early linear analysis ony considered the RF, IF and image signals. Three virtual ports were involved that shared the same physical point. In modern linear analyses, more ports can be analyzed using computer simulation, but the underlying approach is the same. The assumption of small signal RF allows separation of the analysis into large signal and small signal parts. The large signal part concerns the LO and bias, and the small signal part uses conventional linear network analysis to relate the RF, IF, image, and other mixing products to each other. The conductance waveform of the nonlinear element is determined in the large signal analysis for a given LO power, and then used in the small signal analysis. The conductance waveform is determined by taking the derivative of the large signal current in the nonlinear device with respect to the voltage across it. The Fourier series of the conductance waveform is obtained, and its coefficients are applied to a conversion matrix, relating the currents and voltages of the small signal mixing products to each other. The conversion matrix can be expressed in an impedance or admittance form, as is convenient, depending on the mixing device. Usually (but not always) one form can be obtained from the other by inversion. If the conversion matrix is an admittance matrix, then it multiplies the array of unknown node voltages, equaling the array of known driving currents at the nodes. The admittance matrix is inverted, and then multiplied with the array of known currents to solve for the unknown voltages. Port impedances and conversion factors for the three signals are thus obtained, as well as the effects of image termination on conversion loss and RF and IF port impedances. 5.3.1 Conversion Matrix The conversion matrix approach was introduced by Petersen and Llewellyn in 1945 [14], and enlarged upon in major works by Torrey and Whitmer in 1948 [1], and by Saleh in 1971 [3]. The method starts with the basic relationship that current through the device is a function of the voltage controlling the device.

i  f (v)

(5.6)

The current and voltage are assumed to comprise (a) dc, (b) large signal, and (c) small signal components: 

i(t )  I 0   2 I n e jnLOt  n 1 



k   

v(t )  V0   2Vn e jnLOt  n 1

(a)

(b)

i e  j

k

kt

v e 

k  

j

k

(5.7) kt

(c)

(5.8)

Diode Mixer Theory

249

The large signal LO fundamental and harmonics are given by (5.7b) and (5.8b) for n≠0, and the DC terms in (5.7a) and (5.8a) for n=0. The small signal sidebands centered around the LO fundamental and harmonics are given by (5.7c) and (5.8c), and have frequency ωk = (kωLO + ωIF) and ω-k = (kωLO - ωIF), as in Figure 5.1. It should be noted that (5.7c) and (5.8c) are not Fourier series because -ωk ≠ ω-k. The nonlinear device is treated as a conductance g(t), where i(t) = g(t)v(t), which is analogous to the switching waveform already introduced. The conductance is represented by the Fourier series in (5.9), where g(t) is the derivative of the I/V characteristic of the nonlinear device. As we will see, the gn Fourier coefficient values also populate the admittance matrix of (5.12). For the conductance waveform represented by (5.9), g0 is the average value, g1 is at the fundamental LO frequency, g2 is at the LO second harmonic frequency, and so forth.

g (t ) 



g e

n  

n

jn LOt

(5.9)

Figure 5.5 illustrates that harmonic currents are generated by a sinusoidal voltage pumping a nonlinear device. The currents include DC, fundamental, and harmonics with frequencies equal to nωLO (n = 0, 1, 2,...). I BPF ωLO

LPF 0

G0

VDC

Figure 5.5

BPF 2ωLO

G1

G2

BPF 3ωLO

G3

V

VLO

LO voltage and harmonics generated by the nonlinear conductance.

Small signal voltages vk and currents ik in (5.7c) and (5.8c), respectively relate to each other by the admittance matrix defined in (5.12) that contains the gn values from (5.9) for n = 0 to ±4. Recalling the frequency nomenclature of Figure 5.1, we have:

k  k LO   IF k  0,1,2,3...

(5.10)

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Microwave Mixer Technology and Applications

For k = 1, ω1 = ωLO + ωIF. If we assign the IF as ω IF = ωRF - ωLO, then ω1 = ωRF. And for k = -1, ω-1 = ωLO - ωIF = 2ωLO-ωRF, which is the image frequency, thus ω-1 = ωIM. Similarly, for k=2, we find that ω +2 = ωLO + ωRF and ω-2 = 3ωLOωRF, comprising sidebands about the LO second harmonic frequency. Thus mixing products of order k comprise sidebands about the kth LO harmonic frequency, as depicted in Figure 5.1. For example, for k = -1, 0, +1 the voltages are given by (5.11), and the currents have the same form.

v1 '  v1e j (LO IF )t0

= RF

(5.11a)

v0 '  v0 e jIFt0

= IF

(5.11b)

v1 '  v1e j (LO IF )t0

= Image

(5.11c)

Since ik and vk of (5.12) are small signal, the approximation can be made that yk = gn for n=k. Equation (5.12) comprises what is known as the conversion matrix.

i 2   y 0 i    1   y 1 i0    y 2    i1   y 3 i2   y 4  

y1 y0 y 1 y 2 y 3

y2 y1 y0 y 1 y 2

y3 y2 y1 y0 y 1

y 4  v  2    y3  v 1  y 2  v 0    y1  v 1  y 0  v  2 

(5.12)

The meaning of (5.12) is readily understood. For example, taking the top row we get (5.13). The current i+2 and voltage v+2 are both at frequency ω LO + ωRF, so they are related by the average (non time varying) conductance term y0. Next, the voltage v+1 is at frequency ωRF, which converts to current i+2 at frequency ωLO + ωRF by multiplication with y1, which is the Fourier coefficient of the conduction waveform at ωLO. Next, the voltage v+0 is at frequency ωRF - ωLO, which converts to current i+2 at frequency ωLO + ωRF by multiplication with y2, the Fourier coefficient of the conduction waveform at 2ω LO. The same pattern holds for the remaining current and voltage terms of (5.12) and (5.13).

i 2  y0v 2  y1v1  y2v 0  y3v1  y4v 2

(5.13)

Diode Mixer Theory

251

The yk coefficients could be complex to represent resistive and reactive nonlinearities of the mixing element. For this discussion they are limited to resistive nonlinearities, which is a valid approximation for mixers whose mixing element has little or no capacitive effects. Under these assumptions, the I/V relation and respective derivatives are real. The fact still remains that the voltages and currents at the various frequencies have relative phase between them, which are a function of time. The phase due to time can be eliminated by selecting the origin of time to be zero in the equations. With these assumptions the admittance matrix can be greatly simplified. First, the terms with negative index are the complex conjugate of the terms with positive index, (5.14). Secondly, the matrix represents the I/V relation for a linear passive circuit, so the matrix is symmetrical, and consequently is reciprocal, (5.15).

y n  yn * y n  yn

(5.14) (5.15)

Under these conditions, the coefficients yn will be real, and di/dv is an even function of time, which implies v is also an even function of time, and the coefficients yn are real. This assumption causes no limitations, as there is no clear time t = 0 reference, so the LO waveform can be shifted in time arbitrarily. In the classical analysis [1] the admittance matrix is reduced to three ports by assuming Vn = 0 for all n, except n = -1, 0, 1 relating the IM, IF and RF signals in (5.16). This simplifying assumption that all other voltages are shorted defines what is called the “Y” mixer. The shorted voltages all have frequencies centered about the LO second harmonic or higher; the short circuits are implemented within the circuit structure. Other versions have been studied: the “Z” mixer having open circuited ports, and the “H” and “G” mixers having a mix of open and short circuits [3].

I RF I IF I IM

g0  g1 g2

g1 g0 g1

g 2 VRF g1 VIF g 0 VIM

(5.16)

The three-port “black-box” circuit can be extracted from (5.16) and represented by Figure 5.6. The three ports may share the same physical connection point, but their frequencies are harmonically related by the Fourier series, so they are orthogonal to each other and obey superposition within the linearized network.

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Microwave Mixer Technology and Applications

Including the terminations at each port in the matrix equation, the resulting matrix relates the voltages at each port to the input drive current, IRF. Note that port terminations RIM = 1/GIM, RIF = 1/GIF, RRF = 1/GRF.

I IM  GIM VIM I IF  GIFVIF I 0  I RF  GRFVRF

(5.17a) (5.17b) (5.17c) IIM

I0 VIM

GIM

VRF

IIF

GRF

IRF

GIF

VIF

Figure 5.6

gd(t )

Nonlinear conductance mixer circuit.

The black box concept, and (5.16), can be represented schematically by Figure 5.7, where G0 = G2 = g0 + g1 + g2 , and G1 = g0 + 2g1. The negative conductance values indicate a lossless network to the RF, image, and IF. One consequence of a lossless three port circuit is it is impossible to conjugately match all three ports simultaneously.

I0 VRF IRF

GRF

IIF

- g1

G0

- g1

G1

GIF

VIF

- g2 IIM G2

Figure 5.7

VIM

Equivalent circuit representation of conductance mixer.

GIM

Diode Mixer Theory

253

The conductance coefficients given in complex form by (5.9) can alternately be written in terms of cosines in (5.18). The gn coefficients from this equation equals those used in (5.16). Note the factor of 2 in (5.18). 

g (t )  g0   2 g k cos(kLOt )

(5.18)

k 1

The port terminations can be added into (5.16) to simplify later calculation of S-parameters. Noting that only the RF port is driven, the conversion matrix is given by (5.19). The gk values describe the conductance waveform, and are dependent on the characteristics of the nonlinear device, LO pumping power, and terminations to DC and the LO fundamental and harmonics. And, GRF, GIF, and GIM are the load conductances at the RF, IF, and image frequencies. This analysis applies to any nonlinear device used to generate the conduction waveform, including diodes, and FET channel conductance. In the case of active operation with FETs and BJTs, the conductance waveform describes the mixing effects with the added advantage of providing conversion gain instead of loss.

I RF 0 0

g 0  GRF  g1 g2

g1 g 0  GIF g1

VRF g2 g1 VIF g 0  GIM VIM

(5.19)

Inverting the conductance matrix of (5.19), one obtains the impedance matrix in (5.20). This matrix multiplies the known (independent) currents to obtain the unknown (dependent) voltages. If IRF is the only driving current, then IIF and IIM equal zero, and (5.20) reduces to (5.21), which includes only the first column of the resistance matrix. The parameter  of (5.20) and (5.21) is the determinant of the Y-matrix in (5.19). ( g 0  GIF )( g 0  GIM )  g12 VRF  V   1  g [ g  ( g  G )] 1 2 0 IM  IF    2  g  g ( g  G 1 2 0 IF ) VIM  

g1 ( g 2  g 0  GIM ) 2 ( g 0  GIF )( g 0  GIM )  g 2 g1 ( g 2  g 0  GRF )

2   I RF   g 2 ( g 0  GIF )  g1   g1 ( g 2  g 0  GRF )   I IF  2 ( g 0  GIF )( g 0  GRF )  g1   I IM 

(5.20) With the determinant  defined as: 2

2

  ( g 0  GIF )[( g 0  GRF )( g 0  GIM )  g 2 ]  g1 [2 g 2  ( g 0  GIM )  ( g 0  GRF )]

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Microwave Mixer Technology and Applications

VRF VIF  VIM

I RF 

g0  GIF g0  GIM   g12 g1 g 2  g 0  GIM  2 g1  g 2 g 0  GIF 

(5.21)

Conversion loss is defined in (5.22) in terms of transducer gain, which is the most common way of defining gain in microwave and RF circuits. LC 

PowerAvailableFromTheSource PowerDeliv eredToTheLoad

(5.22)

The following power definitions are required to evaluate the conversion loss. Equation (5.23) gives the available power in terms of signal current, IRF, and the signal source impedance, RRF. Equation (5.24) gives power delivered to the IF load in terms of IF current IIF and the IF port termination, RL. 2

R 1 I  2 PRF   RF  RRF  RF I RF 8 2 2  2

2

 V  1 VIF PIF   IF    2  RL 2 RL R R LC  10 log  RF IF  4

(5.23)

 I RF  V   IF 

(5.24)

2

  ,dB 

(5.25)

Noting that the ratio IRF/VIF is defined by the middle row of (5.21), the conversion loss in dB, Lc, is obtained: R R LC  10 log  RF IF  4 

 ( g 0  G IF )[( g 0  G RF )( g 0  G IM )  g 2 2 ]  g1 2 [2 g 2  ( g 0  G IM )  ( g 0  G RF )]    g1 [ g 2  ( g 0  G IM )]  

2

   

(5.26) The “Y” mixer conversion loss of (5.26) is valid for any termination at RF, IF, and image. It can be simplified in certain cases, which will be addressed next.

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255

5.3.2 Four Cases with Specific Port Terminations Various cases with different port terminations were examined in the classic linear analysis of Torrey and Whitmer [1] and later by Barber [2]. They are designated as L0 through L3. For the L0 case, the IF load is matched to the IF output impedance by (5.28), and the RF and IM ports are terminated in what is said to be the LO conductance given by (5.27). The historical significance of the L0 case is it was used for acceptance testing of the crystal rectifier [1].

GRF  GIM  g 0  g 2

(5.27)

2

GIF  g 0 

2 g1 g 0  g 2  GRF

(5.28)

Substituting these impedances into (5.26), results in conversion loss in (5.29). 2

L0 

2

4 g 0 ( g 0  g1 ) ( g 0  g 2 ) g1

(5.29)

2

The L1 case is defined for matched RF and IF ports with the image short circuited. The three port becomes a two port network, and the IF impedance is given by (5.30) [2]. 2

GIF  g 0 

g1 g 0  GRF

(5.30)

Minimum conversion loss is found by substituting (5.30) in for GIF in the conversion loss (5.26), and then taking the derivative with respect to GRF and setting it equal to zero. The resulting RF and IF impedances are given by (5.31), and conversion loss is given by (5.32).

g G RF  G IF  g 0 1   1  g0

g  L1   0   g1 

2

  

2

2  1  1   g1   g     0  

(5.31) 2

(5.32)

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Microwave Mixer Technology and Applications

The L2 case is the most widely used. It is defined for equal RF and IM port terminations, GRF = GIM, where GRF is optimized for minimum loss, given by (5.33). With this, optimum conversion loss and IF port termination, respectively, are given by (5.35) and (5.34) [2].

 GRF   g0

2

  GIM      g0

 G IF   g0

  

2

2

  g1        g0 

2

2   g 2  g 0  g g 1     0 2 2  2 g1   g 0  g1 

g  2 1  g  1  0  g 1 2 g0

    1    L2  1           

(5.33)

2

g g2  2 1 g0  g0  g  1  2  g0  

(5.34)

  

2

      

1/ 2

       

2

 g  1  2  g0    g1   g0

  

(5.35)

2

Case L3 is defined for optimal RF termination and open circuit image. This is a more complex case to analyze since the image port has image voltage developed at its terminal. The IF impedance as a function of RF impedance is defined by: 2

2

2

2

g G  g1 ( g 0  2 g 2 )  g 2 g 0 g1 GIF  g 0   1 RF g 0  GRF g 0 ( g 0  GRF )

(5.36)

The IF and RF optimum termination are given in (5.37) and (5.38), and the conversion loss in (5.39) [2].

 GIF   g0

  

2

g  1   1   g0   g 1 2 g0

2 2   g1  g  1  2   2  g0    g0  

(5.37)

Diode Mixer Theory

257

2

 G IF   g0

  

2

g  1   2   g0   2  g1  1     g0 

     1  g 2    g0  L3  1        1   g 1     g 0   

2  g2   g1  g2      1  1  2    g  g0   g 0    0 

 g    2 1    g0   2   g2   1   g 0      2

   

1/ 2

       

2

 g 1   1   g 0   g1   g0

  

   2

2

 1  g 2   g 0  

(5.38)

(5.39)

 g  1  2  g0  

Values for g0, g1, and g2 are derived later in this chapter and substituted into these equations; the conversion loss results are summarized in Table 5.3. 5.3.3 Optimal RF and IF Match with Arbitrary Image Terminations Equation (5.26) gives conversion loss for the “Y” mixer as a function of g0, g1, g2, GIM, and GIF. The RF and IF impedances are developed next. The derivations of conversion loss given by (5.26), and of RF input impedance are based on having only one source, IRF, at the RF input port. In contrast, obtaining the IF output impedance requires the source to be at the IF port. The top row of (5.21), which is Z11 equals VRF/IRF; the middle row is Z21, which equals VIF/IRF; and the middle column of (5.20) contains Z22, which equals VIF/IIF. From these we find the corresponding S parameters using (5.40) and (5.41) that relate Snn to Znn, and Snm to Znm [15]. They are different from the normal equations relating Z and S parameters because the impedance matrix here includes the terminating impedances [16]. Equations (5.42) and (5.43), respectively, give the reflection coefficient S11 at the RF input, and the RF input impedance ZRF. Equations (5.44) and (5.45), respectively, give S22 and ZIF. Appendix 5C gives the corresponding equations for the image port impedance, and conversion loss from RF to image that is used in the plots of Figure 5.13. Equation (5.26) gives Lc = 1 / |S21|, which can also be derived by substituting Z21 of (5.20) into (5.41).

S nm 

2Z nm  1; n  m; on diagonal Rn

(5.40)

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Microwave Mixer Technology and Applications

2Z nm

S nm 

S11 

Rn Rm

; off diagonal

(5.41)



2GRF ( g 0  GIM )( g 0  GIF )  g1

2



1 2 2 ( g 0  GIF ) ( g 0  GRF )( g 0  GIM )  g 2  g1 2 g 2  ( g 0  GIM )  ( g 0  GRF ) (5.42)



Z RF  RRF

Z RF 

g 02 GIM

1  S11 ; 1  S11



ZRF is RF port impedance

g 02  g 0 GIM  GIF   g12  GIM GIF  GIF   g 0 2 g12  g 22  GIM GIF  g12 2 g 2  GIM   g 22GIF  g 03





(5.43) S 22 



2GIF ( g 0  GIM )( g 0  GRF )  g 2

2



1 ( g 0  GIF ) ( g 0  GRF )( g 0  GIM )  g 2  g1 2 g 2  ( g 0  GIM )  ( g 0  GRF ) (5.44)



Z IF  RIF Z IF 

g 02 G IM

2

1  S 22 ; 1  S 22



2

ZIF is IF port impedance

g 02  g 0 G IM  G RF   g 22  G IM G RF  G RF   g 0 2 g12  g 22  G IM G RF  g12 2 g 2  G RF  G IM   g 03





(5.45) Referring to Figure 5.6, if the image port has a fixed termination that is not conjugately matched, then it is possible to conjugately match both the RF and IF ports, since the three port network effectively becomes a two port. (Actually, since this analysis has no reactance components, it might be more accurate here to say optimally matched instead of conjugately matched). By setting RF port impedance, ZRF, of (5.43) equal to the RF termination RRF = 1/GRF, and setting IF impedance, ZIF, of (5.45) equal to the IF termination RIF = 1/GIF, equations are derived that give the port impedances at which RF and IF are both optimally matched for any value of image termination. Equation (5.46) gives GRF,opt, which is the optimal RF port termination value, in terms of GIM, g0, g1, and g2. The value of GIF,opt is obtained by substituting GRF,opt into (5.47). The minimum conversion loss is obtained using these port termination values. Figures 5.13 and 5.14 plot

Diode Mixer Theory

259

calculated performance versus image termination, RIM, and LO power using GRF,opt and GIF,opt of (5.46) and (5.47). 2

G RF ,opt 

2

2

2

3

2

2

2

 ( g 0  g 0 G IM  g 2 )(2 g 0 g1  GIM g 0  g 0  g 0 g 2  2 g1 g 2  G IM g1 ) 2

2

( g 0  G IM )( g 0  G IM g 0  g1 )

(5.46) 2

G IF ,opt 

2

3

2

2

2

2

G RF ,opt ( g 0  G IM g 0  g1 )  g 0  g 0 G IM  g 0 (2 g1  g 2 )  g1 (2 g 2  G IM ) 2

2

G RF ,opt ( g 0  G IM )  g 0  g 2  G IM g 0

(5.47) While the foregoing analyses of the Y-mixer, truncated to include only the RF, image, and IF signals, gives approximate results, the closed form results give some valuable insights. First, (5.26) shows that conversion loss is a strong function of image termination, as expected. And (5.31) shows that for short circuit image, the RF and IF impedances equal each other, while (5.37) and (5.38) show RF and IF impedances diverge for open circuit image, with IF impedance being much larger than RF impedance. Equation (5.43) shows the RF port impedance is a function of the IF and image terminations. Similarly, (5.45) shows the IF port impedance is a function of the RF and image terminations. So for a fixed image termination, if the RF port termination varies it affects IF output impedance, and varying the IF port termination affects RF input impedance. Using this approach the optimal RF and IF port matches must be determined iteratively by trial and error. In contrast, (5.46) and (5.47) give the optimal RF and IF port matches directly for arbitrary image termination. In the following section values are obtained for the g0, g1, and g2 Fourier coefficients for three conductance waveforms that approximate the pumped Schottky diode: constant, linear, and exponential. Plots are then given for calculated conversion loss and port impedances that show the equations agree closely with harmonic balance simulations. 5.3.4 Pumped Nonlinear Conductance Figure 5.8 depicts three simplified I/V models of a resistive device. Values for g0, g1, and g2 are given based on (5.18). Model 1 assumes a linear relationship between voltage and current, resulting in a square wave for the pumped conductance. Model 2 assumes a quadratic I/V relationship resulting in a linear conductance. Model 3 assumes the conventional exponential I/V relation resulting in an exponential conductance. The conductance waveform, g(t), for the three models are adjusted to have equal g0 values, which is the DC conductance. The

260

Microwave Mixer Technology and Applications

resulting g1 and g2 values for models 1 – 3 are not equal; thus, conversion loss and optimal matching will vary among the three models. Given sinusoidal voltage excitation, the diode exponential I/V equation can be expanded into (5.50). The diode current with sinusoidal voltage excitation is:





I D  I S e x cos(LOt )  1 ; x 

VD VT

(5.48)

The exponential function with cosine in the exponent can be represented by a Fourier series with Bessel function coefficients given by (5.49), for which n is the harmonic and i is a dummy variable. This assumes the existence of a sinusoidal junction voltage, which in practice is not the case. The analysis assumes no parasitic resistance in series with the diode junction, which would otherwise distort the sinusoidal junction voltage by the voltage drop from the nonlinear current. However, regardless of these simplifying assumptions, the results agree with harmonic balance simulations for zero series resistance and yield useful insights reported in a later section. The addition of different values for series resistance are later included in harmonic balance simulations to see their effects on conversion loss and port impedances. 2i  n



e x cost  I 0 ( x)  2 I n ( x) cos(nt ) ; n 1

x    2 I n x       i  n !i! i 0

(5.49)

 2 I ( x)  2 I ( x) I D  I S I 0 ( x) 1  1 cos( LO t )  2 cos(2 LOt )  ... I 0 ( x) I 0 ( x)   (5.50) In(x) is a modified Bessel function of order n with argument x = VD/VT, and VT = ηq/kT ≈ 0.026 volts. VD is the voltage across the diode engendered by the LO voltage, VL, and equals LO voltage multiplied by the voltage divider formed by Rg and Rd of Figure 5.9. Rg is the internal resistance of the LO voltage generator, and Rd is the impedance offered by the diode to VD and is given by (5.53). VD and VL are related as follows:

Diode Mixer Theory

VD  VL

Rd Rg  Rd

261

(5.51)

The conductance values for model 3 of Figure 5.8 are given by (5.52). The numerical g values in Figure 5.8 for model 3 are obtained using (5.52) with VD = 0.4524 volts, VT = 0.026 volts, and Is = 3 x 10-10 amps. Conductance g(t) is given in Figure 5.8 in terms of (5.18), which gives the generalized Fourier expansion of the conductance waveform. V

dc I S VT  VD  g 0  e I 0   VT  VT 

(5.52a)

V

dc I S VT  VD  I ( x) g1  e I1    g 0 1 VT I 0 ( x)  VT 

(5.52b)

V

dc I S VT  VD  I ( x) g 2  e I 2    g 0 2 VT I 0 ( x)  VT 

(5.52c)

5.3.5 LO Impedance Consider a single diode in series with a sinusoidal LO voltage source, V L, and internal resistance, Rg, as in Figure 5.9. The diode can be represented by resistor RD, which is the resistance offered by the diode to the LO pumping voltage.

RG + VL

Figure 5.9

RD

LO voltage source in series with diode resistance.

RD equals the ratio of the sinusoidal LO induced voltage across the diode, VD, to the LO current, ID, through the diode. The current used in (5.53) is

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Microwave Mixer Technology and Applications

Model 1 Linear I = f(V) (dotted line) Constant g(V) (full line) square wave for g(t)

G(t) I, G

1 2  g (t )  0.08  cos  LO t  2  

V

g 0= 0.04, g1 = 0.08/π, g2 = 0 t

Model 2 Quadratic I = f(V2) (dotted line) Linear g(V) = di/dv= 2V (full line) Half sinusoidal for g(t)

2 1 1  g (t )  0.125  cos  LO t  cos 2 LO t  3  2  g0 = 1/8π, g1 = 1/32, g2 = 1/24π

t

G(t)

I, G V t

t

Model 3 Exponential I= f(eV) (dotted line) Exponential g(V) = di/dv = eV

g (t )  0.041  1.94 cos  LO t  1.8 cos 2 LOt 

g0 = 0.04, g1 = .039, g2 = .036

G(t)

I, G V

t

t Figure 5.8

Simple diode models, with the peak value of g(t) adjusted to give the same DC conductance in the three models.

Diode Mixer Theory

263

the Fourier component at the LO fundamental from (5.50), which gives the Fourier series for the nonlinear diode current. From this we get the LO input resistance, RD, to the diode, with x = VD/VT:

RD 

VD VD  I D 2 I S I1 x 

(5.53)

For low frequency operation, the diode junction and parasitic capacitances can be ignored. But for higher frequencies the capacitive reactance from diode junction and parasitic capacitances can be added using the zero voltage junction capacitance, Cj0, and estimates for package capacitance from EM simulation or data sheets. Power delivered to the diode, PD, is: 2

V  1 PD   D   VD I S I 1 ( x)  2  RD

(5.54)

As VL increases, PD also increases, causing RD to decrease. It is important to know how much available LO power, PAv,LO, is required to sustain PD, since mismatch occurs as RD varies. Equation (5.55) gives the required level of PAv,LO as a function of VD, RD, and Rg. 2

PAv ,LO

2 VD ( RD  Rg ) 2 ( RD  Rg ) 2 VL    PD 2 8Rg 4 RD Rg 8Rg RD

(5.55)

Table 5.1 gives calculated results for (5.53) through (5.55) as VD varies, with Rg = 50 ohms. When VD = 0.495 volts, RD = 49.6 ohms, giving the expected result that PAv,LO = PD. It is interesting to see that as RD goes to ~25 or ~100 ohms, PAv,LO is only 0.5 dB above PD; however, as RD drops to ~10 ohms, PAv,LO increases to 2.6 dB above PD, and as RD rises to 223 ohms, PAv,LO increases to 2.2 dB above PD. These results for PAv,LO and RD are in agreement with Figure 5.14 that gives calculated values, and Figure 5.18(a) that gives harmonic balance simulation results for a single diode operating as a Y-mixer, with 50 ohm terminations to the LO, RF, and IF, and diode series resistance Rs = 0. Referring to model 3 of Figure 5.8 and using (5.52), the diode voltage, VD, required to obtain the g values used in model 3 is Vd = 0.452 volts. The resulting diode impedance is RD = 223 ohms, and the power delivered to the load is PD = -3.4 dBm. The available LO power in 50 ohms required for this is PAv,LO =

264

Microwave Mixer Technology and Applications

-1.15 dBm. One would likely use a higher LO drive to a practical mixer, resulting in a lower value for RD. Table 5.1 Calculated LO Resistance of a Single Diode versus LO Power, PL

VD (volts pk) 0.540 0.514 0.495 0.475 0.452

PD (dBm) +11.6 +7.2 +3.9 +0.5 -3.4

RD (ohm) 10.0 24.3 49.6 101 223

PL(dBm) +14.2 +7.7 +3.9 +1.0 -1.2

Is = 3E-10 amps, VT = 0.026 volts.

5.3.6 Parasitic Losses The analysis so far assumes an ideal diode having no parasitic losses. The increase to conversion loss due to series resistance, Rs, junction capacitance, Cj0, and lead inductance, LD, has been approximated by including these diode parasitics in the RF input and IF output circuits as shown in Figures 5.10(a) and (b). This method of estimating parasitic losses is marginally useful as it does not include the effects on terminations to the image 2fL-fR, sum fL+fR, 3fL-fR, and other mixing products.

V1

LD

RS

V3

a) RF input circuit

V2 Ig

Rg

Cj0

RRF

V1

LD

RS

V3

V2 IIF

Figure 5.10

RIF

Cj0

b) IF output circuit

RL

Equivalent circuits at RF and IF frequencies. Is = 3E-10 amps, VT = 0.026 volts.

Diode Mixer Theory

265

Figure 5.10(a) represents the RF input circuit to the mixer. RG is the source impedance, and RRF is the diode RF input resistance. Figure 5.10(b) represents the IF output circuit. IIF is the IF current generated by the diode; RIF is the diode IF resistance; and RL is the IF load resistance. Insertion losses for both circuits are combined to estimate cumulative loss. Equation (5.56), derived in Appendix 5A, gives the total estimated parasitic loss.

LTotal,dB  10 logLRF LIF 

(5.56)

LRF

R 

2 2  Rs  RRF 1  RF LDC j 0   LD  RRF C j 0 Rg  Rs  RF

LIF

R 

 Rs  RIF 1  IF2 LDC j 0

g

L

 



2

2

4 RRF Rg



  L 2

D



 RIF C j 0 RL  Rs  IF2 2

4 RIF RL

In contrast to this analysis, an alternate analysis is presented in Appendix 5B using the conversion matrix with five mixing terms, which includes the effects of parasitic reactances on these terms, and agrees with harmonic balance simulations. The five terms in the numerical analysis are RF, IF, image, sum, and 3ωL-ωR; all others are short circuited. An example giving results for both analysis approaches is given for a mixer diode with image and sum terminations RIM = RΣ = 106 ohms (open), R3L-R = 10-6 ohms (short), and RF and IF terminations both equal 50 ohms. The diode pump voltage VD = 0.604 volts, and diode parameters VT = 0.026/,  = 1.16, IS = 9.5 10-14 amps. RF frequency, fRF = 11 GHz, LO frequency, fLO = 10 GHz, and IF frequency, fIF = 1 GHz. Conversion loss and port impedances are first calculated using the conversion matrix approach described in Appendix 5B. Equation (5.56) is then used to calculate the change in conversion loss due to the addition of parasitic reactances, using the values for |ZRF| and |ZIF| obtained from conversion matrix analysis. The increase in insertion loss, ΔL, is evaluated for Rs = 0, 5, and 10 ohms; Cj0 = 0 and 0.1 pF, and LD = 0 and 0.25 nH. Results are summarized in Table 5.2. The closest agreement in ΔL occurs for the addition of series resistance alone. The three different diode resistance parameters commonly discussed in manufacturer diode data sheets are described. The first is the junction resistance, Rj, given by (5B.5) in Appendix 5B. The second is the series resistance, Rs, caused by bulk resistance in the semiconductor. The third is dynamic resistance, RD = Rj + RS.

266

Microwave Mixer Technology and Applications

Table 5.2 Insertion Loss Including Parasitic Resistance and Capacitance

Parasitic Values Rs ohms 0 5 5 5 10 10 10

Cj0 pF 0 0 0.1 0.1 0 0.1 0.1

LD nH 0 0 0 0.25 0 0 0.25

5x5 Conversion Matrix Appendix 5B LTotal ΔL |ZRF| |ZIF| dB dB ohms ohms 2.92 0 44.5 139.4 4.23 1.31 60.3 166.5 4.22 1.29 57.28 135.8 6.63 3.44 153.5 224.8 5.36 2.44 76.1 192.5 5.38 2.46 71.2 152.9 7.42 4.49 155.8 222.5

Equation (5.56) LTotal ΔL dB dB 1.11 0 2.11 1.0 1.85 0.74 3.76 2.65 3.05 1.94 2.75 1.64 4.19 3.08

fRF = 11 GHz, FLO = 10 GHz, FIF = 1GHz.

5.3.7 Mixer Performance versus Conductance Waveform The conductance, g, values from the three models in Figure 5.8 are applied to the four conversion loss cases of (5.29), (5.32), (5.35), and (5.39), resulting in the conversion loss values given in Table 5.3. It is interesting to note that the exponential model provides lowest losses, which correlates with g1 being larger for model 3 than for models 1 and 2. Plots are generated for the Y-mixer squarewave and exponential diode cases of Figure 5.8 and Table 5.3. Figures 5.11 through 5.14 show variation in conversion loss and RF and IF port impedances as the image termination varies from a very low (short circuit), up through a very high resistance (open circuit). Figure 5.11 depicts the square wave case, and Figures 5.12 through 5.14 depict the exponential diode case. Table 5.3 Comparison of LC Values and Port Terminations for the Three Models in Figure 5.8.

L0 dB L1 dB L2 dB L3 dB

Square Wave 7.7 dB 8.9 dB 7.1 dB 5.6 dB

Quadratic 5.7 dB 6.3 dB 5.4 dB 4.1 dB

Exponential 4.0 dB 2.2 dB 3.9 dB 3.7 dB

RF term g0-g2 RRF Rmin RRF

IM term g0-g2 0 Rmin ∞

IF term RIF RIF RIF RIF

Figures 5.11 to 5.14 each have three plots for the RF and IF ports optimally terminated for open circuit image, broadband image, and short circuit image. Broadband image means the image and RF ports are terminated equally,

Diode Mixer Theory

267

not necessarily in 50 ohms. Optimally terminated means the RF and IF port terminations equal their respective port input impedances. The plots in Figures 5.11 and 5.12 are based on (5.26), (5.43), and (5.45). Figure 5.11 gives results for the square wave case with the g values of model 1 in figure 5.8: g0 = 0.04, g1 = 0.08/π, and g2 = 0 (no second harmonic). In Figure 5.11 (a), with open circuit image, the optimal RF and IF termination values, respectively, are 44.2 ohms and 74.2 ohms. These values were determined using equation (5.26) for conversion loss, setting image termination RIM to infinity (open circuit), then iteratively varying termination values RRF =1/GRF and RIF =1/GIF to make the IF input impedance ZIF = RIF and the RF input impedance ZRF = RRF. Figure 5.11(a) shows the minimum conversion loss is 5.6 dB for image termination, RIM, above 1000 ohms, effectively an open circuit. Minimum conversion loss occurring for open-circuit image is expected because the RF and IF port terminations were optimized for this. In contrast, Figure 5.11(c) gives results with RF and IF terminations optimal for short circuit image, with both RF and IF ports equaling 32.4 ohms, but unexpectedly conversion loss is still lowest when image is open circuited due to g2 = 0. Figure 5.11(b) gives results with RF and IF terminations optimal for broadband image. The optimal RF and IF port values, respectively, are 36.6 and 48.2 ohms. For all three image termination cases, the conversion loss was determined for fixed RRF and RIF while RIM was swept from small to large values and plotted in Figure 5.11. Notice G c increases with RIM due to g2 =0 for the square wave model-1 case. Figure 5.12 gives results for the exponential diode with the g values of model 3 in Figure 5.8: g0 = 0.04, g1 = 0.039, and g2 = 0.036. In Figure 5.12(a), with open circuit image, the optimal RF and IF termination values, respectively, are 326 ohms 1050 ohms, for which the minimum conversion loss is 3.71 dB at RIM = infinite. As image termination goes to a short circuit, conversion loss increases to about 5.8 dB. In Figure 5.12(b), with broadband image termination, the optimal RF and IF termination values, respectively, are 214 ohms and 374 ohms, for which conversion loss is 3.94 dB at RIM = RRF. Minimum conversion loss is 3.3 dB with image shorted, and maximum conversion loss is 4.62 dB with image open circuited. In Figure 5.12(c), with short circuit image, the optimal RF and IF termination values are both 103 ohms, for which the minimum conversion loss is 2.16 dB at RIM = short. As image goes to an open circuit, conversion loss increases to 8.16 dB. In contrast to Figures 5.11 and 5.12, Figures 5.13 and 5.14 give results for optimally terminated RF and IF ports. The RF and IF port values for Figures 5.13 and 5.14 agree with the optimal values of Figures 5.11 and 5.12. Figures 5.13 and 5.14 are based on (5.46) and (5.47), and give optimal RF-to-IF conversion gain, and optimal RF and IF port impedances. Figure 5.12 also gives optimal RF-to-image conversion gain.

268

Microwave Mixer Technology and Applications Square Wave - Opt RF & IF Terminations for Image = Open Circuit 0

80

0

80

2

70

4

60

6

50

8

40

R.IF

Gc

(a) Image open RF and IF terms opt for Image term = open ckt Opt RF term = 44.2 ohms Opt IF term = 74.2 ohms

R.RF

 10

30  10 3 110

0.01

0.1

1

3

10

100

3

30 5 110

4

110

110

4

R.IM

210

210

Square Wave - Opt RF & IF Terminations for Image = RF Term 0

80

2

70

4

60

6

50

8

40

Gc

(b) Image = RF impedance

RIF

RF and IF terms opt for Image term = RF term Opt RF term = 36.6 ohms Opt IF term = 48.2 ohms

RRF

 10 3 110

0.01

0.1

1

10

100

3

110

4

110

30 5 110

RIM

Square Wave - Opt RF & IF Terminations for Image = Short Circuit 0

80

2

70

4

60

6

50

8

40

(c) Image shorted

Gc

RIF RRF

 10 3 110

0.01

0.1

1

10

100

3

110

4

110

RF and IF terms opt for Image term = short ckt Opt RF term = 32.4 ohms Opt IF term = 32.4 ohms

30 5 110

RIM

Figure 5.11

Square wave Y-mixer calculated conversion gain, Gc, and RF and IF port impedances versus image termination, RIM. The values for RRF and RIF are optimized for image termination (a) open, (b) equal RRF, and (c) shorted. g0 = 0.04, g1 = 0.08/π, g2 = 0.

Diode Mixer Theory

269

Exponential - Opt RF & IF Terminations for Image = Open Circuit

(a) Image open

3

110

0

2

800

4

600

6

400

8

200

RF and IF terms opt for Image term = open ckt Opt RF term = 326 ohms Opt IF term = 1050 ohms RIF

Gc

RRF

 10 3 110

0.01

0.1

1

10

100

3

110

4

110

0 5 110

RIM

Exponential - Opt RF & IF Terminations for Image = RF termination

3

(b) Image = RF impedance

110

0

2

800

4

600

6

400

8

200

RF and IF terms opt for Image term = RF term Opt RF term = 214.2 ohms Opt IF term = 373.4 ohms RIF

Gc

RRF

 10 3 110

0.01

0.1

1

10

100

3

110

4

110

0 5 110

RIM

Exponential - Opt RF & IF Terminations for Image = Short Circuit

(c) Image shorted

3

110

0

2

800

4

600

6

400

8

200

Gc

RF and IF terms opt for Image term = short ckt Opt RF term = 102.8 ohms Opt IF term = 102.8 ohms RIF RRF

 10 3 110

0.01

0.1

1

10

100

3

110

4

110

0 5 110

RIM

Figure 5.12

Exponential Y-Mixer calculated conversion gain, Gc, and RF and IF port impedances versus image termination, RIM. The values for RRF and RIF are optimized for image termination (a) open, (b) equal RRF, and (c) shorted. g0 = 0.04, g1 = 0.039, g2 = 0.036, VD = 0.452, VT = 0.026, IS = 3 10-10.

270

Microwave Mixer Technology and Applications

Figure 5.13(a) uses the exponential diode g values of model 3 in Figure 5.8: g0 = 0.04, g1 = 0.039, and g2 = 0.036. As image termination varies from short to open circuit, optimal conversion loss varies from 2.1 dB to 1.6 dB, passing through the maximum loss of 3.4 dB at RIM = RRF ≈ 200 ohms. Also RF port impedance increases from 104 ohms to 668 ohms, and IF port impedance increases from 104 ohms to 2450 ohms. The conversion loss from RF-to-image is also plotted (at 10x), and peaks sharply at RIM = RRF where RF-to-IF conversion loss is also maximum. Since at this point RF-to-IF conversion loss is 3.4 dB, with approximately half the RF power converted to IF, the question arises as to where the other half of the RF power goes since it does not convert to image and there is no mismatch loss at RF and IF ports. An explanation for the lost RF energy is that it is converted to either DC or dissipated into high frequency components. Conversion to DC is shown by including the second order term for the RF signal in the analysis [15]. Figure 5.13(b) gives mixer performance with diode voltage, VD, equal to 0.4473 volts peak, IS = 3 10-10 amps, VT = 0.026/1.08 volts, giving g0 = 0.136, g1 = 0.132, and g2 = 0.122 using (5.52). As image termination varies from short to open circuit, optimal conversion loss drops from 2.1 dB to 1.5 dB, passing through the maximum loss of 3.4 dB at RIM = RRF = 65 ohms. Conversion loss from RF-to-image is also plotted (at 10x), and again displays a sharp peak at RIM=RRF where RF-to-IF conversion loss is also maximum. LO power is calculated using (5.55) with the diode voltage VD = 0.4524 that is used in Figure 5.13(a); also VD = 0.4473 is used in Figure 5.13(b), and the respective power levels are PL = -1.15 dBm and PL = +1.67 dBm. Saturation current IS = 3 10-10 for both, but η = 1 for Figure 5.11(a), and η = 1.08 for Figure 5.13(b). Comparing the Figures 5.13(a) and 5.13(b), RF-to-IF conversion loss values are almost identical, but the optimal RF and IF impedances are substantially higher for the lower LO drive level. Also the point at which RIM = RRF and where Lc is maximum shifts higher for the lower LO drive level. Figure 5.14 shows mixer performance with optimally terminated RF and IF, versus PL swept over -5 dBm to +15 dBm, for open circuit image, 50 ohm image, and short circuit image. In Figure 5.14(a) with open circuit image, as LO power increases, optimal conversion loss drops from 1.6 dB to 1.4 dB, RF optimal impedance drops from 1967 to 29 ohms, and IF optimal impedance drops from 7161 to 107 ohms. In Figure 5.14(b), for image terminated in 50 ohms, as LO power increases, optimal conversion loss begins at 2.6 dB, passes through a maximum of 3.4 dB at PAv,LO = 4.1 dBm, and then ends at 2.56 dB. Over the LO power range, RF optimal impedance drops from 397 to 15 ohms, and IF optimal impedance drops from 435 to 46 ohms. In Figure 5.14(c), for short circuit image, as LO power increases, optimal conversion loss drops from 2.2 dB to 1.9 dB, and RF and IF optimal impedance both drop from 328 to 3.8 ohms. In all three plots LO impedance decreases from 700 to 9 ohms. To calculate results in Figure 5.14, the LO power level, PAv,LO, and diode voltage, VD, are calculated and given in Table 5.1. Then a curve fitting process is employed to determine the polynomial

Diode Mixer Theory

271

in (5.57). For a given LO power, VD is then used in (5.52) to calculate the g values. The g values are used to obtain Lc, RRF, and RIF, respectively, using (5.26), (5.46), and (5.47). LO input impedance given by RD is calculated using (5.53). Equation (5.57) approximates VD as a function of PL, and agrees closely with the values given in Table 5.1. 2

3

4

VD  0.466  (0.01) PLO  (9.26 x104 ) PLO  (7.06 x105 ) PLO  (2.11x106 ) PLO (5.57) 5.3.8 Saleh Mixer Theory and Classification

The foregoing conversion matrix analysis has concerned the Y-mixer, for which all out of band mixing products have short circuit port terminations. Recalling the frequency notation ω±k = kωLO ± ωIF, the product is said to be of even or odd order, respectively, if k is an even or odd integer. Saleh also analyzed three other options: The Z-mixer having all out-of-band frequencies open-circuited; the Hmixer having all odd-order out-of-band frequencies open circuited and all evenorder ones short circuited; the G-mixer having all odd-order out-of-band frequencies short circuited and all even-order ones open circuited [3]. The phrase “out-of-band” refers to all frequencies other than the IF at ω0, the RF signal at ω1, and the image at ω-1. These classifications are summarized Table 5.4.

Even Order Open Short Short Open

Table 5.4 Mixer Classification Odd Order Designation Open Z-Mixer Open H-Mixer Short Y-Mixer Short G-Mixer

In the foregoing discussion for the Y-mixer, the analysis started with an admittance matrix that was inverted to obtain an impedance matrix, from which the conversion loss and port impedances were obtained. For the Z-mixer, the analysis begins with an impedance matrix that is inverted to obtain the admittance matrix, from which mixer performance is obtained. Since the G- and H-mixers have a combination of open- and short-circuited image terminations, the analysis is more complex. Saleh divides the voltage and current waveforms of the G- and H-mixers into even and odd components in the time domain, and uses superposition to obtain the solution. For the Z-, G-, and H-mixers, closed form approximations are obtained for conversion loss and port impedances that contain the Fourier coefficients of the switching waveform.

272

Microwave Mixer Technology and Applications 3

110

0

2

800

4

600

RRF RIF

Gc 6

400

8

200

 10 0.01

0.1

1

10

3

110

100

4

5

110

0 7 110

6

110

10ImL c

110

RIM

(a) 3

110

0

2

800

4

600

RRF RIF

Gc 6

400

8

200

 10 0.01

0.1

1

10

3

110

100

4

110

5

110

6

110

10ImL c

0 7 110

RIM

(b) Figure 5.13

Exponential Y-mixer performance versus image termination, RIM, with optimally terminated RF and IF ports. Plots are for RF-IF conversion gain, Gc; RF-Image conversion loss, ImLc; and RF and IF Port impedances. (a): g0 = 0.04, g1 = 0.039, g2 = 0.036, VD = 0.4524, η = 1.0, VT = 0.026, IS = 3 10-10. (b): g0 = 0.136, g1 = 0.132, g2 = 0.122, VD = 0.4473, η=1.08, VT = 0.024, IS = 3 10-10.

Diode Mixer Theory

273

Exp Diode - Image = Open Circ uit

4

0

110

1

110

(a) Image Open 3

RRF Gc  2

RIF

100

RLO

3

4 6

10

4

2

0

2

4

6

8

10

12

14

1 16

PL

Exp Diode - Image = 50 Ohms

4

0

110

1

110

(b) Image = 50 Ohm 3

RRF Gc  2

RIF

100

RLO

3

4 6

10

4

2

0

2

4

6

8

10

12

14

1 16

PL

Exp Diode - Image = Short Circuit

4

0

110

1

110

(c) Image Shorted

3

RRF Gc  2

100

RIF RLO

3

4 6

10

4

2

0

2

4

6

8

10

12

14

1 16

PL

Figure 5.14

Exponential Y-mixer performance versus LO power, PLO (dBm), with optimally terminated RF and IF ports. Plots are for RF-IF conversion gain, Gc; RF, IF, and LO port impedances. VT = 0.026, ISAT =3 10-10 amps.

274

Microwave Mixer Technology and Applications

5.3.8.1 Saleh Theory for Switching Model Saleh first analyzed the four mixer types using a switch in place of the diode, having “on” resistance, Rmin, greater than zero ohms; and, “off” resistance, R max, less than infinite ohms. His goals were to determine which of the four mixer types is best, and what the optimal switching waveform shapes are for the four types. After lengthy mathematical development the conclusion was the H- and G-mixer are superior to the Z- or Y-mixer types for the following reasons: 

Optimum pulse duty ratio for the Z- and Y- mixer depends on the values of Rmin and Rmax, in contrast, it is 50% for the H- or G-mixer types regardless of Rmin and Rmax.



Given Rmin>

-4

600

>

-7

p3

300

-8

p1 p2

200

-9

Image term = RF term Opt RF term = 214 ohms Opt IF term = 374 ohms

900 p4

ZIF (ohms) >>

-4

0

(b) Image = RF impedance

1000

RF: 11 GHz @ -10 dBm; RRF=214 ohms IF: RIF=374 ohms LO: 10 GHz @ -2.75 dBm; RLO=50 ohms Rs = 0 ohms

-3

-9

400 300

ZLO (ohms) >>

-10

-2

700

500

ZRF (ohms) >>

-8 -9

800

600

>

-3 -4

(a) Image open

1000

Ohms

-1

Single Diode Mixer RF: 11 GHz @ -10 dBm; RRF=326 ohms IF: RIF=1050 ohms LO: 10 GHz @ -2.75 dBm; RLO=50 ohms Rs = 0 ohms

Optimum RRF & RIF for RIM=Short Circuit

ZRF (ohms) >>

-10 .01

Figure 5.16

.1

1 10 100 1000 Image Termination, RIM (Ohm)

10000

Ohms

0

Ohms

280

100 0 100000

Exponential Y-mixer harmonic balance simulated performance versus image termination, RIM. Plots are for conversion gain, Gc, and RF, IF, and LO port impedances. Gc - blue; ZIF - orange; ZRF - cyan; ZLO – brown.

Diode Mixer Theory

Single Diode Mixer

-1

ZIF (ohms) >>

-2

450 400

ZLO (ohms) >>

0

-3

500

RF: 11 GHz @ -10 dBm; R=50 ohms LO: 10 GHz; R=50 ohms IF Term = 50 ohms Rim = 0,10,25,50,100,250,500,1k,5k ohms

5k

350

>

0

-10 -6

-4

-2

0

2

4

6

250 200

Ohms

0

Conversion gain - dB

281

150 100 50 0

8 10 12 Power (dBm)

14

16

18

20

22

24

(a)

Single Diode Mixer

0

ZIF (ohms) >>

-2 -3

400

>

0

450

350

5k

-4

300 p9 p8 p7 p6

-5

p5

-6

p4

0

-7

p3

-8 -9

0

5k

p36 p30 p35 p34 p2 p33 p32 p14 p13 p12 p11 p10 p31 p18 p17 p16 p15 p24 p25 p26 p27 p22 p23 p19 p20 p21

ZRF (ohms) >>

-10

250 200

Ohms

Conversion gain - dB

-1

500

RF: 11 GHz @ -10 dBm; R=50 ohms LO: 10 GHz; R=50 ohms IF Term = 100 ohms Rim = 0,10,25,50,100,250,500,1k,5k ohms

5k

150 100 50 0

-6

-4

-2

0

2

4

6

8 10 12 Power (dBm)

14

16

18

20

22

24

(b) Figure 5.17

Exponential Y-mixer harmonic balance simulated performance versus LO Power, PLO dBm, with diode Rs = 5 ohms. Image port termination, RIM varies over 0 to 5000 ohms, and RR F = RLO= 50 ohms. Plots are for conversion gain, Gc, and RF, IF, and LO port impedances. IF Termination: (a) RIF = 50; (b) RIF = 100. Gc - blue; ZIF - orange; ZRF cyan; ZLO – brown.

Microwave Mixer Technology and Applications

Single Diode Mixer

0 0

Conversion gain - dB

-1 -2 -3

Rs = 0,5,10 ohms

10

ZLO (ohms) >>

0

500

RF: 11 GHz @ -10 dBm; R=50 ohms LO: 10 GHz; R=50 ohms IF Term = 50 ohms Rim = 50 ohms

450 400

>

-7

100

0

0

-9

p12 p11 p6 p5 p9 p4 p8 p7

ZRF (ohms) >>

-10 -6

-4

-2

0

2

4

6

200 150

10

-8

300

8 10 12 Power (dBm)

14

16

Ohms

282

18

20

22

50 0

24

(a)

Single Diode Mixer 0

-1

Rs = 0,5,10 ohms

10

Conversion gain - dB

-2 -3

0

ZLO (ohms) >>

450 400

>

-10 -6

-4

-2

0

2

4

6

8 10 12 Power (dBm)

200 150

10

-8

300 250

p2 p3

ZIF (ohms) >>

-7

Figure 5.18

500

RF: 11 GHz @ -10 dBm; R=50 ohms LO: 10 GHz; R=50 ohms IF Term = 100 ohms Rim = 50 ohms

14

16

18

Ohms

0

20

22

50 0

24

(b) Exponential Y-mixer harmonic balance simulated performance versus LO Power, PLO (dBm), and diode Rs = 0, 5, 10 ohms. Plots are for conversion gain, Gc, and RF, IF, and LO Port Impedances. Port terminations: RIM = RRF = RLO =50 ohms, IF: (a) RIF =50; (b) RIF =100 ohms. Gc - blue; ZIF - orange; ZRF - cyan; ZLO – brown.

Diode Mixer Theory

Single Diode Mixer 0

-1

500

RF: 11 GHz @ -10 dBm; R=50 ohms LO: 10 GHz; R=50 ohms IF Term = 50 ohms Rim = 0 ohms

Rs = 0,5,10 ohms

10

ZLO (ohms) >>

-2

450 400

>

-10 -6

-4

-2

0

2

4

6

8 10 12 Power (dBm)

14

16

18

20

22

Ohms

0

Conversion gain - dB

283

0

24

(a)

Single Diode Mixer 0

Conversion gain - dB

-1

10

ZLO (ohms) >>

-2 -3

0

-4

500

RF: 11 GHz @ -10 dBm; R=50 ohms LO: 10 GHz; R=50 ohms IF Term = 100 ohms Rim = 0 ohms

Rs = 0,5,10 ohms

450 400 350

>

-8

100

10

-9

p6 p3 p12 p9 p5 p11 p8 p4 p7

0 ZIF (ohms) >>

-10 -6

-4

-2

0

2

4

6

8 10 12 Power (dBm)

14

16

18

20

22

50 0

24

(b) Figure 5.19

Exponential Y-mixer harmonic balance simulated performance versus LO Power, PLO (dBm), and diode Rs = 0, 5, 10 ohms. Plots are for conversion gain, Gc, and RF, IF, and LO port impedances. Port terminations: RIM =0, RRF = RLO =50 ohms, IF: (a) RIF =50; (b) RIF =100 ohms. Gc - blue; ZIF - orange; ZRF - cyan; ZLO – brown.

Microwave Mixer Technology and Applications

Single Diode Mixer

0 0 10

Conversion gain - dB

-1

10

0

500

RF: 11 GHz @ -10 dBm; R=50 ohms LO: 10 GHz; R=50 ohms IF Term = 50 ohms Rim = 1e6 ohms >

-6 -7

150

-8 10

-9 ZRF (ohms) >>

-10 -6

-4

-2

0

2

4

6

8 10 12 Power (dBm)

250 200

ZLO (ohms) >>

0

350

Ohms

284

14

16

18

20

22

p12 p11

100

p6 p5 p9 p8 p4 p7

50 0

24

(a)

Single Diode Mixer 0 10

Conversion gain - dB

-1

10

0

>

-6

300 250 200

ZLO (ohms) >>

-7

150

ZRF (ohms) >>

-8

10

0

-9

p12 p11 p6 p5 p9 p4 p8 p7

-10 -6

-4

-2

0

2

4

6

8 10 12 Power (dBm)

14

16

18

20

22

Ohms

0

100 50 0

24

(b) Figure 5.20

Exponential Y-mixer harmonic balance simulated performance versus LO Power, PLO (dBm), and diode Rs = 0, 5, 10 ohms. Plots are for conversion gain, Gc, and RF, IF, and LO port impedances. Port terminations: RIM =106, RRF = RLO =50 ohms, IF: (a) RIF =50; (b) RIF =100 ohms. Gc - blue; ZIF - orange; ZRF - cyan; ZLO – brown.

Diode Mixer Theory

Single Diode Mixer Rs = 5 ohms

-1

-10

-2 -3

500

RF: 11 GHz @ -10,-8,-6,-4,-2,0 dBm; R=50 ohms LO: 10 GHz; R=50 ohms IF Term = 50 ohms Rim = 50 ohms

450 400 350

ZLO (ohms) >> > 0

-8

100

0

p22 p20 p21 p23 p24 p12 p10 p11 p7 p8 p9 p14 p15 p16 p17 p18 p13

-9 ZRF (ohms) >>

-10 -6

-4

200

Ohms

0

Conversion gain - dB

285

-2

0

2

4

6

50 0

8 10 12 Power (dBm)

14

16

18

20

22

24

(a)

Single Diode Mixer Rs = 5 ohms

-1

-10

-2

Conversion gain - dB

-3

500

RF: 11 GHz @ -10,-8,-6,-4,-2,0 dBm; R=50 ohms LO: 10 GHz; R=50 ohms IF Term = 100 ohms Rim = 50 ohms

450 400 350

ZLO (ohms) >>

>

-8

200

Ohms

0

100

0

-9

p22 p20 p21 p23 p24 p10 p11 p7 p8 p9 p12 p14 p15 p16 p17 p18 p13

ZRF (ohms) >>

-10

50 0

-6

-4

-2

0

2

4

6

8 10 12 Power (dBm)

14

16

18

20

22

24

(b) Figure 5.21

Exponential Y-mixer harmonic balance simulated performance versus LO Power, PLO (dBm), and RF input power, PRF = -10 to 0 dBm. Diode series resistance, Rs = 5 ohms. Plots are for conversion gain, Gc, and RF, IF, and LO port impedances. Port terminations: RIM = RRF = RLO =50 ohms, IF: (a) RIF =50; (b) RIF =100 ohms. Gc blue; ZIF - orange; ZRF - cyan; ZLO – brown.

286

Microwave Mixer Technology and Applications

Single Diode Mixer

0 -1

-10

450 400

ZLO (ohms) >>

-3

350

>

-10

50 0

-6

-4

-2

0

2

4

6

8 10 12 Power (dBm)

14

16

18

20

22

24

(a)

Single Diode Mixer

0

RF: 11 GHz @ -10,-8,-6,-4,-2,0 dBm; R=50 ohms LO: 10 GHz; R=50 ohms IF Term = 100 ohms Rim = 0 ohms

Rs = 5 ohms

-1

-10 ZLO (ohms) >>

-3

-10

-4

450 400 350

>

0

-9

Ohms

Conversion gain - dB

-2

500

p12 p10 p11 p7 p8 p9 p24 p22 p23 p20 p21 p14 p15 p16 p17 p18 p13

ZIF (ohms) >>

-10

50 0

-6

-4

-2

0

2

4

6

8 10 12 Power (dBm)

14

16

18

20

22

24

(b) Figure 5.22

Exponential Y-mixer harmonic balance simulated performance versus LO Power, PLO (dBm), and RF input power, PRF = -10 to 0 dBm. Diode series resistance, R s = 5 ohms. Plots are for conversion gain, Gc, and RF, IF, and LO port impedances. Port terminations: RIM =0, RRF = RLO =50 ohms, IF: (a) RIF =50; (b) RIF =100 ohms. Gc blue; ZIF - orange; ZRF - cyan; ZLO – brown.

Diode Mixer Theory

Single Diode Mixer

0

500

RF: 11 GHz @ -10,-8,-6,-4,-2,0 dBm; R=50 ohms LO: 10 GHz; R=50 ohms IF Term = 50 ohms Rim = 1e6 ohms

Rs = 5 ohms

-1 -2 -3

450 400 350

>

-8

ZRF (ohms) >>

200 150

0

0

p20 p21 p22 p23 p24

-10

-9

0

-10 -6

-4

-2

0

p12 p10 p11 p7 p8 p9 p14 p15 p16 p17 p18 p13

ZRF (ohms) >>

2

4

250

Ohms

Conversion gain - dB

287

100 50 0

6

8 10 12 Power (dBm)

14

16

18

20

22

24

(a)

Single Diode Mixer

0 -1 -2

450 400

>

p8 p7 p10 p11 p9 p12 p14 p15 p16 p17 p18 p13

0

-10 -6

200

0 ZLO (ohms) >>

0

-9

ZRF (ohms) >>

-4

-2

250

Ohms

Conversion gain - dB

500

RF: 11 GHz @ -10,-8,-6,-4,-2,0 dBm; R=50 ohms LO: 10 GHz; R=50 ohms IF Term = 100 ohms Rim = 1e6 ohms

Rs = 5 ohms

100 50 0

0

2

4

6

8 10 12 Power (dBm)

14

16

18

20

22

24

(b) Exponential Y-mixer harmonic balance simulated performance versus LO Power, PLO (dBm), and RF input power, PRF = -10 to 0 dBm. Diode series resistance, R s = 5 ohms. Plots are for conversion gain, Gc, and RF, IF, and LO port impedances. Port terminations: RIM =106, RRF = RLO =50 ohms, IF: (a) RIF =50; (b) RIF =100 ohms. Gc - blue; ZIF - orange; ZRF - cyan; ZLO – brown.

288

Microwave Mixer Technology and Applications

Cases 3 and 4 show that IF impedance more than doubles while RF impedance is unchanged as 3L-R termination goes from short to open. Cases 3 and 5 show that RF impedance doubles and IF impedance changes very little as the L+R termination goes from short to open. And in general as the image, sum, and 3L-R terminations increase, the RF and IF port impedances increase almost proportionately. Table 5.5 Calculated Mixer Performance versus Port Termination Resistance at the Image 2L-R, Sum L+R, and 3L-R Frequencies.

Port Termination (ohms)

Calculated

Case

RF 11 GHz

IF 1 GHz

2L-R Image 9 GHz

L+R Sum 21GHz

3L-R 19GHz

Lc dB

|ZRF| Ohms

|ZIF| Ohms

1 2 3 4 5 6

50 100 100 100 100 147

50 100 100 100 100 287

50 100 short short short short

50 100 short short open short

50 100 short open short open

8.17 7.30 3.28 2.84 6.29 1.94

131 190 84 84 236 147

188 305 96 235 105 287

|Zxx| are port impedance magnitudes. LO port source resistance is 50 ohms, calculated LO input impedance, |ZLO|, equals 86 ohms, PLO = +4 dBm. Table 5.6(a) Harmonic Balance Simulation Results for LO Power Equal to +4, +5, and +6 dBm. ZRF and ZIF are the port impedances.

1 2 3 4 5 6

PLO = +4 dBm Lc |ZRF| dB ohms 8.37 137 7.31 172 3.20 113 4.01 116 6.07 152 2.93 172

|ZIF| ohms 188 299 118 231 143 269

PLO = +5 dBm Lc |ZRF| dB ohms 7.84 125 6.92 162 3.26 101 3.45 101 5.90 153 2.68 160

|ZIF| ohms 170 271 103 209 116 249

PLO = +6 dBm Lc |ZRF| dB ohms 7.48 117 6.64 155 3.47 91 3.14 91 5.69 155 2.63 151

|ZIF| ohms 157 248 91 192 99 229

Port terminations correspond to the six cases of Table 5.5; all other frequencies are short circuited.

Calculated results of Table 5.5 generally agree with simulated results of Table 5.6(a) for all six cases. The lowest conversion loss occurs for case 6, when the image 2L-R and sum L+R are both short circuited, the 3L-R product is open circuited, and RF and IF ports are optimally matched. Optimal match is obtained by iteratively adjusting RF and IF port terminations to equal the respective port impedances. For case 6, the calculated optimal RF port impedance and

Diode Mixer Theory

289

termination are 147 ohms; the calculated optimal IF port impedance and termination are 287 ohms; and calculated conversion loss is 1.94 dB. Harmonic balance simulated results obtained using these RF and IF port termination values with +4 dBm LO drive are given in Table 5.6(a). RF and IF port impedances, respectively, are |ZRF| = 172 ohms, |ZIF| = 269 ohms, and Lc = 2.93 dB. The simulations were carried out using the circuit of Figure 5.15 with all LO harmonics short circuited. The diode parameters appear in Appendix 5B. Table 5.6(b) Harmonic Balance Simulation Results for Case 6 with LO Power Equal +4, +5, and +6 dBm.

6

PLO = +4 dBm Lc |ZRF| |ZIF| dB ohms ohms 2.89 173 288

PLO = +5 dBm Lc |ZRF| |ZIF| dB ohms ohms 2.67 152 252

PLO = +6 dBm Lc |ZRF| |ZIF| dB ohms ohms 2.57 133 218

RF and IF ports are optimally matched at each LO power level; R IF=|ZIF|, and RRF=|ZRF|. Optimal terminations were obtained iteratively by simulation.

5.5 LARGE SIGNAL CONVERSION ANALYSIS As discussed earlier for the doubly balanced mixer of Figure 5.4, IM distortion is introduced by the RF signal affecting the phase of the conductance waveforms of the four diodes. In addition, IM distortion, gain compression, and cross modulation are introduced by the nonlinearity of the device I/V characteristic. IM distortion and gain compression reduce the upper end of the dynamic range; IM distortion is manifest as undesired mixing products that can obscure the desired IF and increase the effective system noise floor. Cross modulation causes a strong signal to be transferred to another signal, in particular, to the desired IF. All circuits display nonlinear behavior, the extent of which depends on drive level. If the RF signal level is low enough, it can be treated as a small signal, with its effects on the conductance waveform considered negligible. Small signal analysis results in the mixer output signals located above and below the LO harmonic frequencies as depicted in Figure 5.1. In contrast, if the RF input signal cannot be approximated as a small signal, then under large signal analysis there are multiple signals spaced about the LO harmonic frequencies as depicted in Figure 5.2. The multiple sidebands comprise RF harmonics that result from the RF signal being included in the nonlinear analysis. For a mixer with small signal RF input, a nonlinear analysis is first performed to determine the large-signal voltages and currents in the diode, from which can be determined the Fourier components of the conductance waveform. The coefficients are then used in a conversion matrix to perform the linear portion of the analysis. In contrast, for large signal RF input, the nonlinear analysis includes both RF and LO signals. Modern approaches to performing these analyses involve transient or harmonic balance approaches. These are widely

290

Microwave Mixer Technology and Applications

available in commercial circuit simulation software, and have been described by [12]. But simplified power series and switching models are still useful to understand performance limitations and trends. Two simplified approaches that give useful insight into mixer behavior given a large signal RF involve the power series expansion for the exponential diode, and the switching function approximation of the diode conductance. 5.5.1 Gain Compression 5.5.1.1 Exponential Model A popular analysis method in the literature for diode mixers is to apply a sinusoidal junction voltage to the exponential diode equation. Difficulties with the sinusoidal junction-voltage assumption are that the currents are very sensitive to the level of the junction voltage. Also, there is no saturation effect; that is, there is no point where the current components cease to change much with LO power. So, there is always some uncertainty as to what LO level to use in the calculation. 5.5.1.2 Switching Model Conversion gain compression of a single diode mixer can be represented using the switching model of the linear rectifier, which has very high off impedance and very low on impedance [6]. The applied voltage, V, is the sum of sinusoidal RF and LO voltages given by (5.64).

V  VL cos(Lt )  VR cos(Rt )

(5.64)

The switching function is defined so that the output voltage, V(t), of the mixer is proportional to input voltage V when V is positive, and V(t) is equal to zero when V is negative, as depicted in Figure 5.24. The RF signal level is much lower than the LO signal level. V 0

t

V(t) 0

Figure 5.24 Output voltage, V(t) in the switching model is proportional to input voltage V when V is positive. V(t) equals zero when input voltage V is negative.

Diode Mixer Theory

291

V(t) comprises harmonics of both LO and RF, and is represented by the double Fourier series in (5.65). The Fourier coefficients Amn and Bmn are solved by integrating over 0 to π for variables x and y, respectively, for LO and RF frequencies. When input voltage V passes through zero, the switch changes between on-off states. This threshold point is represented by (5.66), where input voltage equals zero, and the variable k is the ratio of RF to LO voltage. Since the output voltage is zero when input voltage V is negative, the integration range for x reduces to 0 to cos-1(-kcos(y)) radians, since output voltage is zero between cos-1(kcos(y)) and π. 



V (t )   [ Amn cos(n R t  m L t ) Bmn sin(n R t  m L t )] m0 n 0

(5.65)

cos( x)  k cos( y)  0 ;

k  VR / VL

(5.66)

The Fourier coefficients are computed by substituting in x = (ωLt) and y = (ωRt), then performing the double integration of (5.67) over the region bounded by (5.66). Bmn=0 due to orthogonality between sines and cosines.

Amn 

2V L



2





0

cos cos(ny )  0

1

(  k cos( y ))

[cos( x)  k cos( y )] cos(mx)dx dy  (5.67)

The details of how the integration is carried out are in the reference. The resulting Fourier coefficient A01 is the voltage component at the RF signal frequency [6]:

A01 

VR 2

(5.68)

The Fourier coefficient A11 is the voltage component at the IF:

A11 

4VL [(1  k 2 ) E (k )  (1  k 2 ) K (k )] 2 3 k

(5.69)

K(k) is the complete elliptic integral of the first kind, and E(k) is the complete elliptic integral of the second kind, given by (5.70).  /2

K (k )  

0

d 1  k 2 sin( )

(5.70a)

292

Microwave Mixer Technology and Applications  /2

E (k )  

0

1  k 2 sin( )d

(5.70b)

Conversion gain, GC, can be approximated by taking the ratio of A11/A01 in (5.71). GC is plotted in Figure 5.25 as a function of k = VR/VL.

A  GC  20 log 11   A01 

(5.71)

GC equals the theoretical value of 3.92 dB for k ≤ 0.1, where RF power is at least 20 dB below LO power, P LO. Gain compression at 1-dB occurs for k ≈ 0.876, corresponding to input RF power, P -1dB, being 1.15 dB below LO power. This result agrees with the harmonic balance simulation data in Figure 5.21 for a single-diode Y-mixer with RIM = 50 ohms with PLO = +1 dBm and PRF = 0 dBm. In contrast, the simulated results in Figure 5.22 for open circuited image have a wider spread, with RF input power at 1 dB gain compression, P-1dB ≈ (PLO – 3) dBm, likely due to unequal RF and IF impedances with open circuit image. 3

 3.5

4

Gc  4.5

5

 5.5

6 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

k

Figure 5.25

Conversion gain versus k = VR/VL for switched linear rectifier.

5.5.2 Intermodulation Accurate calculation of IM levels in mixers has long been an important goal, beginning with approximate power series and switching analyses in the 1930s, and progressing to the modern numerical approaches of the 1980s up to the present. The first report of accurate simulation of IM in a diode mixer including nonlinear capacitance was given in 1987 by Maas [11]. The analysis showed that

Diode Mixer Theory

293

terminating the image in a high-impedance inductive load reduced conversion loss as expected, but could also increase IM and noise levels. The analysis also showed that IM levels are minimized with high level LO, which moves the diode quickly between on and off states, minimizing time spent in the nonlinear region. This is facilitated by the diode having low junction capacitance and low series resistance. 5.5.2.1 Exponential Model A popular representation in the literature of IM generation in diode mixers is to use the exponential diode I/V equation with the assumption of a sinusoidal junction voltage. As stated previously for analyzing gain compression using the exponential model, the sinusoidal junction voltage assumption has serious limitations. And the method provides calculated IM levels that tend to be higher than measured results. Approaches have been developed for multiple input signals to a single diode [8] and for a doubly balanced diode mixer [10]. 5.5.2.2 Switching Model Given its simplicity, the switching model provides surprisingly good estimates for IM suppression. A version of it is used in two popular system simulators [9]. The Fourier series in (5.65), includes the RF and LO voltages for the switching model of a single linear rectifier. It is extended to include multiple RF signals by using a switching function, S(t), instead of using the limits of integration to effect switching as done in (5.67), [7]. Equation (5.72) gives the input voltage, V, comprising three sinusoids: the large signal LO voltage, VL, and small signal RF1 and RF2 voltages, VR1 and VR2. Equation (5.73) describes the output voltage, V(t), in terms of a triple Fourier series. Bmnp = 0 in (5.73) due to orthogonality between sines and cosines.

V  VL cos(Lt )  VR1 cos(R1t )  VR 2 cos(R 2t ) 



(5.72)



V (t )   [ Amnp cos(mL  nR1  pR 2 )t Bmnp sin(mL  nR1  pR 2 )t ] m 1 n 1 p 1

(5.73) 

S (t )   0



S (t )   0

sin(V )d

 sin(V )d



 

 2

 2

0

for V < 0

(5.74a)



for V > 0

(5.74b)

294

Amnp 

Microwave Mixer Technology and Applications

m n p 2 2

VL  1 n! p!

 m  n  p  1    V  n  V  p 1  2  R1   R 2    m  n  p  3   VL   VL    2 

( z  1)  z( z)  z( z  1)( z  1)... 1     2

 3    2 2

(5.75)

(5.76a) (5.76b)

(5.76c)

Equation (5.74) is the switching function, S(t), that is multiplied by input voltage, V of (5.72), to obtain the output voltage, V(t) of (5.73). The switching function comprises the signum function (definite integral) and an offset factor π/2, and equals 0 for input voltage, V < 0, and π for V > 0. Thus, output voltage, V(t), equals 0 for V < 0, and is proportional to V, for V > 0. The Fourier coefficients Amnp that give the voltage components, are developed in the reference and given by (5.75) that uses the gamma function defined by (5.76) [7]. Approximate conversion loss and IM levels given by (5.75) are summarized in Table 5.7, for VL = 0.495 volts, and VR1 = VR2 = 0.005 volts applied to a single diode mixer. The simplifying assumptions are made that input and output power, respectively, equal the voltage component, Amnp, squared and divided by the termination resistance. And that all source and termination resistances are equal. Again, the difference between single-tone and two-tone third order IM products is 9.5 dB as expected, and the same for the second-order IM products is 6.0 dB as expected. Furthermore, the absolute IM suppression levels are in line with typical measured values, as indicated by the positive input intercept IIP 3 and IIP2 values, respectively, at +12.9 dBm and +9.9 dBm. IIP 2 is normally higher than IIP 3 in doubly balanced mixers due to cancellation of second order products; however, IIP2 is lower than IIP3 for this single diode mixer model that has no such cancellation. Other interesting estimates are obtained from (5.75). The second order two tone input intercept point is found to be 6 dB above the LO power. This is obtained by equating |A110|= |A111|, and noting the result is (VR1/VL)2 = 4. Also, the third order two tone input intercept is found to be 9 dB above the LO power. This is obtained by equating A110 = A121, and noting the result is (VR1/VL)2 = 8 [18].

Diode Mixer Theory

295

Table 5.7 Calculated Conversion Gain and IM Levels for the Switching Diode

Description

Ratio

Conv Gain Two Tone 3rd Order Suppr. Single Tone 3rd Order Suppr. Difference IIP3 Two Tone 2nd Order Suppr. Single Tone 2nd Order Suppr. Difference IIP2

|A101| / VR1 |A121| / |A110| |A130| / |A110| |A121| / |A130| |A111| / |A110| |A120| / |A110| |A111| / |A120|

Value =20log(Ratio) -3.92 dB -97.9 dBc -107.4 dBc 9.54 dB +12.9 dBm -26.0 dBc -32.0 dBc 6.02 dB +9.9 dBm

The switching function analysis was extended to approximate IM levels for the doubly balanced diode mixer [9]. The analysis includes approximations for imperfections in baluns and diodes, and a nonzero diode threshold voltage given by (5.77). 

S (t )   0



S (t )   0

sinV  VF  d

 sinV  V F  d



 

 2

 2

0

for V < VF

(5.77a)



for V > VF

(5.77b)

Figure 5.27 depicts the threshold voltage at VF, where the diode switches between on and off states, depending on input voltage, V, given by (5.72) for a single RF input signal. The output voltage, Vout(t), is represented by the Fourier series of (5.78), and is proportional to V, for V ≥ VF, and equal to zero for V < VF. Referring to Figure 5.26 the RF and LO baluns, respectively, introduce the RF and LO input signals to the diodes differentially. Ideally the balun produces voltages that are equal in amplitude but 180 degrees different in phase at the two diode connection points. In contrast, the IF signal and ground connections, respectively, connect to the diodes in phase at the RF and LO balun diode connection points. Thus the RF and LO ports are ideally isolated from the IF port, with LO-to-IF isolation approximated as 20log(|1-α|) dB, and RF-to-IF isolation approximated as 20log(|1-β|) dB. The factors α and β provide a simplified means of quantifying the amplitude imbalance between the differential voltages at the diode connection points. Balun phase imbalance is not included. The effects of unequal voltages across the four diodes are also included in the

296

Microwave Mixer Technology and Applications

analysis using the factors δ2, δ3, δ4 that are defined by (5.85) as equaling the voltages across three of the diodes normalized to the first one.

Figure 5.26

Doubly balanced mixer with imbalance factors α and β.

Id

OFF VF Figure 5.27

ON Vd

Threshold voltage at VF for switching mixer.

Typical values can be assigned to the factors indicating imperfections in the mixer circuit as follows: α = β = 0.7; δ2= 0.85, δ3= 0.95, δ4= 1.05, and VF/VL = 0.1. Integer m refers to LO harmonics, while n refers to harmonics of RF. Positive values for m and n are used in (5.79) – (5.85), which provide the suppression results of Table 5.8 for single-tone IM products. It has been pointed out that (5.83) indicates IM suppression is equally affected by changes in LO or RF power [10]. It should be noted that the LO power is set at a level required to sustain the large signal switching waveform, and ΔP increases as RF signal level is reduced below this fixed LO level.

Diode Mixer Theory

Vout (t ) 





 

m  n  

Amn cos(m L  n R )t  Bmn sin(m L  n R )t (5.78)

S mn   m  1P  20 log Amn  dBc

Amn

297

(5.79a)

  m  n  1    2 1    1  B sin  m   sin  n    B cos m   cos n    ...   OO  EE      2   2  BIF n !  m  n  3  2   2   2           2 

m  n   2  VF ...   m  n  2  VL   2  

  m   n   m    cos   BEO cos   BOE sin     2  sin  2 2        

  n    2   

(5.79b) m = odd; n = odd:

BOO  1   4   ( 2   3 )  n  2   4   ( 2   3 )   ( 3   4 ) (5.80) m = even; n = even:

BEE  1   4   ( 3   2 )  n  4   2   ( 2   3 )   ( 3   4 ) (5.81) m = odd; n = even:

BOE  n   4   2   ( 2   3 )   ( 3   4 )

(5.82)

m = even; n = odd:

BEO  n  4   2   ( 3   2 )   ( 3   4 )

(5.83)

BIF  1   2   ( 3   4 )

(5.84)

2 

V2 V1

3 

V3 V1

4 

V4 V1

(5.85)

This analysis yields some interesting results. Equation (5.81) indicates that maximum suppression of IM products with even valued RF and LO harmonics is achieved if the four diode voltages are identical to each other

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Microwave Mixer Technology and Applications

regardless of the extent of imbalance in the baluns. This suggests that maximum suppression of these IM products is achieved using a monolithic diode ring or star quad. Additionally, for products having even LO harmonics and odd RF harmonics, and vice versa, the analysis indicates suppression is infinite if the threshold voltage, VF = 0. This result would likely not hold if phase imbalance had been included along with amplitude imbalance in the analysis. Table 5.8 Equations Approximating Suppression of Single Tone IM Products with LO Harmonics (m), and RF Harmonics (n)

m (LO) 1 1 1 2 2 2 3 3 3 4 4 5 5 6 6 7 7

n (RF) 1 2 3 1 2 3 1 2 3 1 2 1 3 1 2 1 3

Smn Suppression (dBc) 0 ΔP - 41 2 ΔP - 28 -35 ΔP - 39 2 ΔP - 44 -10 ΔP - 32 2 ΔP - 18 -35 ΔP - 39 -14 2 ΔP - 14 -35 ΔP - 39 -17 2 ΔP - 11

Calculated results are the same for ±m and ±n integer harmonic values.

5.5.2.3 Square Wave LO The analyses presented so far are for sinusoidal LO pumping. Square-wave LO pumping has also been analyzed and shown to be an important means of reducing IM levels. The effect of square-wave LO on the exponential diode is to minimize the time spent in the nonlinear region of the diode I/V curve. Similarly, with the switching model, a square wave LO reduces the ability of the RF signal to phase modulate the conduction waveform compared to an LO voltage with a finite transition time. A practical LO square-wave signal has finite rise and fall times, Ƭr, Tf, and it has been shown that IM levels are proportional to the product of Ƭ r and LO frequency. So for a given rise time, IM levels increase as LO frequency increases; and, as rise time approaches zero seconds, IM levels are said to

Diode Mixer Theory

299

approach zero [13]; however, given the fact that some nonlinearity always exists in the switching device one cannot expect perfect IM suppression. Some considerable improvement in single-tone and two-tone IM suppression has been achieved recently by commercially available diode mixers [19], with IIP3 levels in the range of +30 dBm to +50 dBm reported. 5.5.2.4 Definition of IM Order IM products comprise harmonics of both the large signal LO and the small signal RF inputs: fmnp = mfLO ± nfRF1 ± pfRF2; it is customary for the mixing order to be defined as |n| + |p| and not include the LO harmonic integer, m. This is because changes in the IM level equal the change in the small signal input power multiplied by the small signal harmonic integer associated with that input. The LO harmonic integer m is not included in the order because changes in IM level do not equal the change in LO power multiplied by m. For example the frequency of a two-tone-third-order IM product is given by mfLO ± 2fR1 ± fR2 or mfLO ± fR2 ± 2fR1, and the frequency of a two-tone-second-order IM product is given by mfLO ± fR1 ±f R2. The frequency of a single-tone-third-order IM product is given by mfLO ± 3fR, and the frequency of a single-tone-second-order IM product is given by mfLO ± 2fR. For the above m = 1 unless otherwise stated. 5.5.3 Mixer Classification The previous mixer classification proposed by Saleh relates to the various forms of input and output mixer terminations. Another classification relates to various diode configurations to improve IM suppression, [20]. Some of these circuit configurations are widely used today, but the proposed classification is more of historical interest and is described in Table 5.9. It is well known that adding one or more additional series diodes can dramatically improve IM suppression. For example, the diode ring quad of a doubly balanced mixer can be replaced with a diode octal ring having eight diodes. IM suppression increases because input RF voltage is shared across more diodes, so less RF signal voltage is available to phase modulate the switching point, assuming LO voltage remains the same across each diode. Of course, more LO power is required, and conversion loss increases due to the additional series resistance. The addition of a parallel RC in series with each diode also improves IM suppression, most likely by self biasing the diode so its conduction waveform is again less susceptible to phase modulation.

300

Microwave Mixer Technology and Applications Table 5.9 Classes of Mixers with LO Power Range

MIXER CLASS

CIRCUIT

LO POWER/DB MIXERS (dBm)

Class 1

+7 to +13

Class 2, type 1

+13 to +24

Class 2, type 2

+13 to +24

Class 3, type 1

+20 to +30

Class 3, type 2

+20 to +30

Class 3, type 3

+20 to +30

5.6 SUBHARMONIC MIXER For many mixer applications it is desirable to use an LO harmonic instead of the fundamental. Such mixers are commonly referred to as subharmonic mixers (SHM). The LO harmonic is generated by the mixer itself, with the LO harmonic sometimes referred to as the “virtual LO.” The SHM is more immune to DC offset at the output and to second order IM distortion than fundamental mixers, which is important when converting down to baseband. Also, the SHM is very useful at frequencies above 100 GHz and at THz, including radio astronomy, where fundamental LO sources are either cost prohibitive or unavailable. Most SHM applications use the second LO harmonic instead of higher harmonics, because conversion loss increases for higher LO harmonics. For some situations higher conversion loss is acceptable, allowing use of higher LO harmonics; one notable example is in spectrum analyzers. The remainder of this chapter discusses the SHM with the IF generated by the second LO harmonic, 2f LO. For this case the desired IF output is fIF = (2fLO-fRF) or (fRF -2fLO), with fLO ≈ fRF/2, and fIF 5 (7.17) provides results with 3% accuracy, and accuracy becomes 1% for x > 10.

In

n  x / 2 ( x) 

n

!

ex I n ( x)  2x

for x  0

(7.16)

 n2  1  2 x   

for x  5

(7.17)

The normalized Fourier coefficients are represented in the following plot, [3]. Notice they are a strong function of LO voltage for small amplitudes. Increasing the LO voltage above 5VT causes the normalized coefficients to saturate asymptotically and be less dependent on LO drive level.

2 I1 ( x) I 0 ( x)

1.6 1.2

2 I 3 ( x) I 0 ( x)

0.8 0.4 0

Figure 7.5

2 I 2 ( x) I 0 ( x)

Ratio

Bessel Function

2.0

2

4

6

8

10

12

14

16

x(VLO/VT)

Normalized Fourier components for sinusoidal excitation in BJTs. After [2].

7.1.2 Small Signal Impedance The small signal base impedance, Zbase, is calculated by applying a small RF voltage and calculating the resulting base current. The base current is defined by dividing the RF term from the collector current in (7.13) by the current gain, β. The base impedance is defined per (7.18), similar to the impedance of a linear amplifier, with transconductance controlled by the LO drive. The DC small signal collector impedance according to T-model is high for bipolar devices.

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Microwave Mixer Technology and Applications

1 I DCVRF cos( RF t )  VT vRF V  Z base   T  1 I DCVRF I DC g mQ  VT

iB 

(7.18)

7.1.3 Large Signal Impedance Conventionally, impedance is a linear concept where the relation between voltage and current is independent from the particular magnitude of the drive signal. In large signals both the voltage and current becomes distorted and are represented by Fourier series. The extension of the impedance concept to large signal is possible if it is constrained to the specific drive level and is defined by the ratio between voltage and current at the fundamental frequency of operation. One also has to determine conditions for the harmonics. In general they are considered shorted to ground. Therefore, the large signal base impedance at the LO frequency is defined by dividing the applied LO voltage (assumed sinusoidal in the equation) to the base current as given by (7.19).

 I DC  2 I1 ( x) 2 I ( x) cos  LO t  2 cos 2 LO t  ... 1    I 0 ( x) I 0 ( x)  V    D   (7.19) I B I DC  2 I1 ( x)  Gm VD  I 0 ( x) 

IB  Z LO

The conclusion from this equation is that LO impedance is equivalent to RF impedance with small signal transconductance replaced by its large signal value, Gm. At low frequency the diode impedance from the T-model can be given by similar values as those provided by Table 5.1. But due to transistor action the diode impedance appears at the base terminal multiplied by the current gain. Therefore, at low frequencies VD = VLO and there is nearly no power absorbed by the transistor. 7.1.4 Conversion Gain The schematic of the low frequency mixer represented in Figure 7.6 employs the device 2SC5006, which has fT = 4 GHz. Since operation below fT/10 can be

BJT Mixer Theory

439

considered low frequency, the mixer is set for down-converter operation with fRF = 130 MHz and fLO = 100 MHz. The quiescent operating point is set at I CQ = 2 mA, requiring a base voltage of 0.71 V. Both LO and RF signals are combined and then applied to the base as depicted in Figure 7.6. The IF frequency is extracted from the collector by the tank circuit, that presents a high impedance to IF signals and nearly zero impedance to all other frequency components. The IF currents flow to the load resistor, which determines the converted output IF voltage as in (7.20). The RF and image terminations at the collector are equal, and the IF load is not matched to the device output impedance. There are a couple of reasons to skip this matching, namely, the output impedance of bipolar transistors are high at low frequency which is the case of down conversion mixer; tentative to match will increase the voltage at the collector increasing the feedback through the capacitive divider to the base making the circuit potentially more unstable. In addition, if collector IF voltage becomes high, it has to be limited by the maximum available voltage swing in the circuit.

Vout  RLVRF

I DC I 1 ( x) cos LO   RF t VT I 0 ( x)

` +Vbias

+VCC L,C

RB VRF

(7.20)

Vout

RS RL

VLO Figure 7.6

RE Schematic of a BJT tuned mixer with LO and RF injected into the base.

Applying a LO peak voltage of 250 mV at 100 MHz, makes x  10 and the ratio I1(x)/I0(x) = 0.8, per the plot in Figure 7.5. The voltage conversion gain is equal to GCV = Vout/VRF = 6.15, for a load resistor equal to 100 ohm. The simulation of the same circuit with a full model using ADS gives a voltage conversion gain of 6.7. This is very reasonable accuracy for a hand calculation of a bipolar low frequency mixer. The conversion gain can also be normalized to the quiescent transconductance resulting in a function dependent on normalized quantities, VLO/VT. The conversion gain then follows the plot in Figure 7.5 for current I1/I0 (i.e., gain increases sharply with LO voltage and stabilizes for a value of x > 5).

440

Microwave Mixer Technology and Applications

GCV I ( x)  RL 1 g mQ I 0 ( x)

(7.21)

7.1.5 High Frequency and DC Effects The DC term in collector current expressed by (7.9) assumes a constant DC voltage applied to the junction. A more realistic result should include the base and emitter bias resistors, as they introduce DC feedback and affect the actual DC voltage applied to the junction. The junction voltage is therefore changed from VBE to VBE' according to the equation:

VBE '  VBE  VR

(7.22)

The bias circuit is represented by a voltage supply in series with a base bias resistor RB, in shunt with the base. Its value in general is large so that it does not affect the AC signals. The voltage VR is defined as follows:

R  VR  RE I E  RB I B  I C RE  B   VBE '  VBE  RI C

   RI C 

(7.23) (7.24)

The quiescent DC current is obtained by inserting the voltage from (7.22) into (7.9) and iteratively solving for DC current, resulting in:

I DCQ  I S e

VBE  RI DC VT

(7.25)

The introduction of LO drive voltage distorts the collector waveform, generating a DC term of its own. This effect is taken into account by considering the DC term of the Fourier expansion in (7.26). The feedback effect from RE needs to be accounted for in the amount of LO voltage applied to the junction; (i.e., VLO' = VLO - REIC). If the voltage developed in the emitter resistance is small compared to Vbe, then its effect can be ignored. On the other hand it has to be kept in mind that a minimum amount of RE is required for temperature stabilization in either BJTs or HBTs.

V '  I DC  I DCQ I 0  LO   VT 

(7.26)

BJT Mixer Theory

441

VLO '

With, I  V LO '   0   VT 

e 2

VT

V LO ' VT

This simplified analysis may not be adequate if the following conditions apply: (1) base capacitance has to be taken into account, requiring base current and consequently conjugate impedance matching for optimum performance; (2) the mixer is required to be matched to the generator, in which case feedback networks are inserted into the circuit, making it more difficult to determine voltages and currents without using a computer simulator; (3) emitter impedance is employed to stabilize the circuit, enhancing the amount of feedback. To address those conditions an improved approach to analyze bipolar mixers is described in the next section. 7.2 CONVERSION MATRIX It can be said the analysis method employed for the diode mixer is also valid for bipolar transistors, essentially by replacing the diode conductance with the transistor transconductance. As far as the nonlinear element behavior is concerned, that is correct, since the nonlinearities are described by similar equations. However, the nonlinear transistor elements are embedded in a larger number of parasitics, making it more difficult to treat analytically. An approach similar to the one applied to FETs reported in the literature, [4], is applied here, making use of a non-linear hybrid- type of model, represented in Figure 7.7. In this model, the base emitter capacitance, C, corresponds to the parallel combination of the junction capacitance, Cj, and diffusion capacitance, CD. Notice the base and collector physical ports are each terminated by four terminations at different frequencies, corresponding to 4 virtual ports. The top circuits are labeled LO for the input LO termination, ZGLO, and output LO termination, ZDLO. The other terminations correspond to the RF, image, and IF signals at the base and collector. All other frequency components are assumed to be shorted. The circuit is analyzed using the Z-matrix representation, (7.27), because there are more impedances in series in the circuit than in shunt and the resulting multi-frequency matrix is called the conversion matrix. Only the first two terms of Fourier series for gm(t) are necessary to simulate image, RF and intermediate frequencies, i.e. gm0 and gm1, and are include in the conversion matrix for the circuit. The conversion matrix does not take into account the effect of feedback capacitance, Cµ.

442

Microwave Mixer Technology and Applications

ZGLO

FDLO

FG

ELO

LO

Cµ

1=RF

1=RF

RB

F1

Z1

F4

ERF

Rj

Figure 7.7

2=IM Z5

F5

gm(t)Vb

RE

3=IF Z3

M)

C F2

Z4

R0

Vb

2=IM Z2

ZDLO

3=IF

F6

F3

Z6

Representation of a transistor mixer model with ports terminated at RF, IF, IM and LO frequencies. After [4].

 E1*   Z11*  Z1*    0 0   0   0   * Z 0    41 0   0    * 0   Z 61

0 Z 22  Z 2 0

0 0 Z 33  Z 3

Z14 0 0

0 Z 52 Z 62

Z 43 Z 53 Z 63

Z 44  Z 4 0 0

*

0 Z 25 0 *

*   I1      I 2  (7.27) I3   *  0 I 4    0 I5  Z 66  Z 6   I 6 

0 0 Z 36

0 Z 55  Z 5 0

The matrix components are defined by the following equations. The termination impedances at each port are denoted Zk with k = 1, 2, ...6.

Z11  Z 22  Z 33  rB  RE 

Z 44  Z55  Z66  RE  R0 Z14  Z 25  Z36  RE

jR j 1  jk C R j

with k = 1,2,3

(7.28a) (7.28b) (7.28c)

BJT Mixer Theory

Z11  Z 22  Z33  RB  RE 

Z 41  RE  Z 52  RE 

g m 0 R j R0 1 j1C R j g m 0 R j R0

1 j2C R j

Z 63  RE  j

g m 0 R j R0 1 j3C R j

g m 0 R0 j1C g R Z 63  RE  m 0 0 j3C

Z 41  RE 

443

1 jk C

(7.28d)

7.28e)

Z 61  

Z 62  

(7.28g)

g m1 R j R0

g m1R j R0

(7.28h)

1 j1C R j

(7.28i) Z 43  Z 53  

Z 52  RE 

(7.28k)

(7.28f)

1 j1C R j

g m1 R j R0 1 j3C R j

g m 0 R0 j2C

(7.28j)

(7.28m) (7.28n)

Once the conversion matrix is established, the available conversion gain can be solved by computer simulation using (7.29). Simulation will also provide information about the effect of terminations on conversion gain and port impedance at the frequencies of interest.

I 6 ReZ 6  2

GC 

2

E1 ReZ1  4

I  4 R g RL 6 E1

2

(7.29)

An analytical solution for the conversion gain is possible by calculating the ratio (I6/E1), in the manner proposed by reference [5]. It consists in calculating the ratio of two determinants, /z. The first determinant, , is obtained by deleting the first row and sixth column of (7.27). And the second, z, is the determinant from (7.27). *

*

I6 (Z 22  Z 2 )(Z33  Z3 )(Z 44  Z 4 )(Z55  Z5 )Z 61  * * E1 (Z11  Z1 )(Z 22  Z 2 )(Z33  Z3 )(Z 44*  Z 4* )(Z55  Z5 )(Z 66  Z 6 )  Z 41Z52Z 63Z14Z 25Z36

I6 Z 61  * * E1 ( Z11  Z1 )(Z 66  Z 6 )  

(7.30)

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Microwave Mixer Technology and Applications



Z14Z 25Z 36Z 41Z 52Z 63 * * ( Z 22  Z 2 )(Z 33  Z 3 )(Z 44  Z 4 )(Z 55  Z 5 )

(7.31)

In down conversion mode, Z1 and Z6 are matched and Z4, Z5 are nearly zero with the purpose of improving the trans-conductance modulation. The reverse and forward transfer parameters Z25, Z36, Z52, Z63 are low due to the low impedance attached to the collector at these frequencies, making parameter  small and able to be disregarded. In up conversion mode, Z 1, Z4, Z5 are matched, while Z6 is shorted. Z2 corresponds to the image frequency, which is either close to the RF impedance or filtered. The reverse transfer parameters are still low in value but the forward transfer parameters are not negligible in this condition. So,  is not zero but still is smaller than the first term of denominator of (7.30), and a small error occurs if it is disregarded. The error obviously depends on circuit parameters and frequency of operation. Thus the following approximation can be made:



g m1R0 j1C j

I6  E1 ( Z11*  Z1* )( Z 66  Z 6 )

(7.32)

The output resistance R0 is included in the numerator of (7.32), and in the denominator since it is included in the expression Z66 = RE + R0. This resistance is usually high in small size bipolar devices, and if it is much larger than load impedance it is not practical to match. In this case it can be neglected, so the available conversion gain can be defined as: 2

R g RL  2g  R GC   m1   L * * 2 Rg  1C  ( Z11  Z1 )

 g m1     1C 

2

(7.33)

7.2.1 Transconductance modulation The small signal components of the collector current can be described by multiplying the time varying transconductance with the input small signal voltage, as in (7.34), with gm(t) defined in (7.35). The time varying transconductance is obtained from the derivative of current with respect to base voltage, evaluated over the LO base voltage waveform. The term transconductance mixer is sometimes found in the literature to describe the same modulation effect. To obtain gm(t) from I/V characteristics it is first necessary to obtain gm(VBE). In this case it was approximated in Figure 7.8 by a bi-linear relation, where gm = 0 for voltage below VBEQ and is proportional to VBE for voltages above. The best bias point for mixer operation is the one that maximizes the transconductance

BJT Mixer Theory

modulation, (i.e. Gm / VBE V

be V BE

445

). The best bias point is therefore at VBEQ where

the Fourier coefficients of rectified sinusoidal LO voltage are maximum.

iC (t )  g m (t )vbe (t )

(7.34)

g m (t )  g m0  g m1 cos LOt  g m2 cos 2LOt  ...

(7.35)

The output waveform is symmetric around t = 0 so that the Fourier series can be of the cosine form. The coefficients for this waveform are standard and described by (7.36a, c). The parameter to be determined in this case is the gmax function of LO drive level.

g m0 

g max

(7.36a)



g max 2 2 g max  3

g m1 

(7.36b)

gm2

(7.36c)

gm(VBE)

gm(t)

gma x

VBE

VBE

Q

t=0

t

t= 0

t Figure 7.8

Representation of gm(t) assuming a linear gm(Vbe), derivated form I, V plot.

Application of harmonic balance simulation to determine the large and small signal coefficients can be carried with the circuit shown in Figure 7.9. In this schematic, the collector is short circuited, the base is biased with voltage source VB and resistor RB, and the source is decoupled with base capacitor CB. The simulated coefficients are obtained by taking the ratio between collector currents and applied base voltage in (7.37). The large signal impedance is obtained from same equations but evaluated at the LO frequency only.

446

Microwave Mixer Technology and Applications

iC ( RF ) vRF i (  LO ) g m1  C RF vRF i (2LO  RF ) gm2  C vRF g m0 

(7.37a) (7.37b) (7.37c)

IC

CB IB

VRF

RB

VLO

+VBB

Figure 7.9

+VCC

RE

Circuit schematic to determine Fourier coefficients from an electrical simulator.

Employing the circuit of Figure 7.9 and use the parameters for the bipolar device type 2SC5006 by NEC, (7.38) was applied to determine the transconductance Fourier coefficients, for different values of base resistor, bias, and LO drive voltages. The results are in table 7.1. By replacing the small signal voltage VRF by VLO in (7.38), one obtains the large signal coefficients described in Table 7.2. Table 7.1 Small Signal Transconductance Fourier Coefficients Function of LO Drive and Bias Resistor RB

VLO (mV) 125

250

500

RB 170 340 680 170 340 680 170 340 680

gm0 (mS) 105 96 84 127 117 103 145 133 116

gm1(mS) 74 70 63 94 89 80 107 102 95

gm2 (mS) IDC (mA) 0.018 7 0.02 6 0.022 5 0.024 15 0.03 12 0.034 9 0.025 33 0.033 27 0.042 20

BJT Mixer Theory

447

The parameter of importance in the conversion is gm1, which is responsible for the conversion process for fundamental mixers and therefore should be maximized. It should also be noted that gm1 can be increased by external bias or by applying a larger LO voltage. The peak value of 0.5 V corresponds to +4 dBm considering a matched 50 ohm generator. The dependency of gm1 on equivalent base resistance, RB, suggests the option of employing an active base bias capable of providing low source impedance. The large signal parameter is useful in the matching of large signal LO generator. Table 7.2 Large Signal Transconductance Fourier Coefficients as a Function of LO drive and Bias Resistor RB

VLO (mV) 125

250

500

RB 170 340 680 170 340 680 170 340 680

GM0 (mS) GM1(mS) 87 88 64 77 46 62 78 102 59 86 43 68 77 114 60 95 44 73

GM2 (mS) 38 35 31 52 47 40 60 54 45

IDC (mA) 7 6 5 15 12 9 33 27 20

7.2.2 Nonlinear Capacitance The nonlinear capacitors on bipolar devices have already been discussed in Chapter 3 and the equations for the simplified T- model are reproduced in (7.38), (7.39). The capacitances are voltage dependent, set by external bias and large signal drive level. Nonlinear reactances are not efficient at frequency conversion, except in the special case of varactors. However they affect mixer impedance which in turn affect mixer performance. In general their average value is sufficient to determine its impedance contribution. The base capacitance, C , is given by, (7.38) composed of two terms: the first is the diffusion capacitance expressed as the product = gmQ, where  is the transit time of carriers in the base; the second is the junction capacitance relevant when the junction is reverse biased. The feedback capacitance Cµ is the capacitance of the collector base reverse biased junction and Vbi is the built in voltage.

 V  dI C   C  C j 1  be  dV j  Vbi 

0.5

 V   g m  C j 1  be   Vbi 

0.5

(7.38)

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Microwave Mixer Technology and Applications

 V  C  C j 1  bc   Vbi 

0.5

(7.39)

7.2.3 Base and Collector Impedance The base and collector impedances can be determined directly from matrix (7.27) replacing gm(t) by gm0 and applying the corresponding terminations as follows: for base impedance calculations at RF, LO and image frequencies, the collector is terminated with a low impedance at those frequencies, and with a load resistor, R L at the IF. For collector impedance calculations at IF frequency, the base is terminated either with 50 ohm, an open, or a short circuit to ground. An open circuit is a better termination as it minimizes effects from negative feedback. Analytical expressions can be obtained by reducing the schematic of Figure 7.7 to the schematic of Figure 7.10. The base RF impedance in terms of device parameters, load impedance, and source impedance is in (7.40). In the particular case where the load impedance is low, and the base emitter reactance is much lower than the junction resistance the impedance is described by (7.41). The collector termination can make use of L,C elements but a better choice is an open stub measuring quarter wave length at the LO frequency. It provides a near zero impedance at that frequency and a low impedance at RF and IM if the IF frequency is much lower than either frequency. At IF frequency the open stub reactance is small in value and does not interfere with power transfer to the load. Cµ

rB

C

B C

Rj

gm0Vb

RE E Figure 7.10

gd

E

Schematic for determining in/out device impedance.

The low frequency base impedance is obtained by making   0, resulting in ZB  rB+Rj + RE(1+gmRj)  rB+ Rj +RE( +1). This expression is equivalent to (7.18) when RE = 0. Equation (7.42) gives the output impedance at the IF frequency with the base impedance terminated in the generator impedance,

BJT Mixer Theory

449

Zg. The term ZE represents either an emitter resistance, RE, or an inductance, jLE, but not both simultaneously. The collector impedance at other frequencies is not relevant since it is nearly shorted in normal operation.

  1 1 (1  jZ L C )  g m 0    jC  ZE Rj   (7.40) Z B  RB  Z E 1  1  jC  jC Z L   jC  g m 0 (1  Z E ) Rj  R j 

Z B  RB  Z E  1 Z IF 

g mQ Z E 1  jC jC

C C

 1  1    gm  Z   g Rj 



(7.41)

1 1  Zg Rj  1  1 jC     gm  Z   g Rj 

(7.42)

The collector impedance at IF frequency can be approximated by the feedback capacitance, Cµ, if the termination between base and ground is a short circuit. 7.3 MIXER PROPERTIES 7.3.1 LO Power Ideally the base impedance should be conjugately matched at RF and LO frequencies for best efficiency. However, it is not possible to achieve a simultaneous match at both. The general approach is to match the device at RF and accept the mismatch at the LO. This mismatch reduces the LO voltage appearing across the internal base-to-emitter node that controls the current, reducing transconductance modulation and consequently conversion gain. It is simple, though, to increase the LO drive power to recover the conversion performance. Once the desired LO voltage is known for a given application, the trans-conductance can be linearized at this LO voltage and frequency. The large signal LO impedance is determined by replacing in (7.40) the small signal gm of (7.1) with the large signal trans-conductance, Gm of (7.2). The large signal base

450

Microwave Mixer Technology and Applications

impedance can then be determined allowing calculation of drive LO power to the mixer according to the equation: 2

1 VLO PLO  2 Re( Z11)

(7.43)

7.3.2 Noise Figure The mechanism of noise generation in a mixer is different than that of an amplifier because the device parameters vary in time in response to the LO signal. An approach, [6], was proposed to use the time averaged values for the noise parameters using a similar equation developed for amplifiers by making the following substitutions: ω  ωRF, gm  gm1 and Cbe  Cbe,av, where the suffix av stands for average value. Essentially three noise sources exist in a bipolar device: flicker noise, shot noise, and thermal noise. The first is inversely proportional to frequency so it is not generally relevant at high operating frequencies. The shot noise is modeled by current sources proportional to collector current and is connected in parallel with the collector to emitter signal generator. The thermal noise is associated with any lossy element in the circuit, and is represented as a current generator for any parallel resistor, and as a voltage generator for any series resistor. A dedicated computer program is required to enter the noise sources and calculate the noise figure, taking into account fundamental and harmonics of the LO generator. The noise figure for a bipolar mixer in terms of the device parameters can be estimated from (7.44), where, gm1 corresponds to the conversion trans-conductance, and the effect of frequency is normalized to fT. At high frequencies the limitation becomes the reactance C, included in the equation through the parameter fT = gm/(2C).

g ( R  RB  RE ) 2 R  RE 1 F  1 B   m1 S RS 2 RS g m1 2 RS

1  f       f T

  

2

  

(7.44)

With Rs = Real part of generator impedance, ZS For the noise factor of (7.44), it is not practical to include effects from the source impedance. A better approach for circuit simulation is to consider the noise factor of (7.45).

NF  NFmin 



Rn YS  YSopt GS



2

(7.45)

BJT Mixer Theory

451

Where YS  GS  jBS is the source admittance connected to the base

Rn is the equivalent noise resistance NFmin is the minimum noise figure The minimum noise figure is obtained from the derivative of (7.45) relative to the source impedance equated to zero, (i.e., dF / dZ S

 0 ). The

definitions [7] of these parameters for a SiGe HBT are given by the set (7.46). In spite of differences in technology, the noise sources for the model used to derive the expressions are similar to the ones employed in silicon devices.

NFmin

 1  f 2   1   2 g m1 RB          f T   1

g m1 1 C   2 Rn  2 g m1 Rn 2

GSopt 

BSopt  

C 2

 1 1   2 g m1 Rn

  

2 g m1 Rn 1 Rn  RB  2 g m1

(7.46a)

(7.46b)

(7.46c) (7.46d)

7.3.3 Linearity While single ended BJT mixers provide good noise figure results, the conversion process is not actually linear, there is always a certain degree of distortion even at low voltage levels. The intermodulation products in particular of third order are more harmful because they are translated in frequency and show up within the IF band. For intermodulation analysis the nonlinearity is considered mild and Taylor series expansion is usually applied to the transconductance as shown in (7.47). The problem in mixers is the coefficients are a function of time due to the effect of the LO signal, and thus are more complex to determine compared with amplifiers. The terms gm(t), gm2(t), and gm3(t) are obtained from the first, second and third order derivatives of the IC(Vbe) evaluated at the quiescent point with LO superimposed.

I C (t )  g m (t )VRF (t )  g m2 (t )VRF (t ) 2  g m3 (t )VRF (t )3

(7.47)

The analysis of bipolar linearity for single ended mixers is rare in the literature since efforts have been dedicated to the differential mixer approach,

452

Microwave Mixer Technology and Applications

which provides a great advantage in reduction of distortion. Detailed information on the gmn(t), n =1,2,3, is demonstrated for FETs in Chapter 9 and can be extended to bipolars. One method to improve linearity is similar to amplifier linearization, namely, inserting an emitter resistor, which is briefly addressed next. To a first order approximation, the non-linear effects can be lumped into a dynamic emitter resistance, re, which is given by (7.48). Applying a voltage signal at the transistor base, and employing a T-model for the transistor, it is easily concluded that the voltage will appear at the terminals of re, assuming rB is of much smaller value. Note the emitter resistance, re, in the model of Figure 7.3 corresponds to the differential diode resistance. This voltage will develop a non-linear current IE.

re 

VBE V  BEVBE IE I S e VT

(7.48)

Inserting an external emitter resistance, RE, the base voltage is applied to (re + RE). The input voltage is now shared between a linear and a non-linear resistor. The higher the RE, the lower the portion of voltage dropped across the nonlinear resistance, so the circuit becomes more linear at the expense of gain reduction. If an external base resistance, RB, is added, it will also linearize the circuit. Therefore, constant current drive and linearization by an external resistor fundamentally produce the same effect; they both minimize the internal Vbe swing on the nonlinear element compared with the applied signal. However, these corrections must be traded-off with gain reduction to obtain the optimal result. This effect, discussed for the common emitter circuit is equally applicable to the common base. This approach works well at low frequencies where gain is high but not so well at high frequencies due to gain reduction. An alternative at high frequencies is to use inductive feedback instead of resistive, preserving the DC circuit conditions. 7.3.4 Stability A common problem when matching impedance in active devices is the possibility of unstable operation, which is a function of device terminations and bias conditions. In general, oscillations in active mixers are of linear nature, generated by positive feedback in the circuit. Therefore, S-parameter theory employed in amplifier stability analysis can equally be applied with some modifications. The mixer is considered as a two port network, RF input and IF output and all other terminations are considered part of the two port network and properly terminated. For instance, a low impedance at RF, image and LO frequencies is connected to the collector. Using stability circles, one can check the potential stability at one port as the terminating impedance at the other port varies. The first condition to

BJT Mixer Theory

453

evaluate is stability as a function of bias variation around the bias operating point. The second stability condition is to check for negative real impedance observed on small signal S11 from a LO pumped device. Another test is to increase the LO drive from a low level up to a large signal level and verify the magnitude of S 11 over the desired frequency range is always less than one. An additional test is to verify conversion gain stability. The conversion gain should track the LO input drive level. Any erratic behavior or gain jump, it is a symptom of instability. A better tool to verify stability is the time domain check with the application of a small pulse to the circuit. If oscillations build up, then the circuit is unstable and the frequency where the instability occurs can be ascertained from the output spectrum. If the transients impressed in the circuit decay, then the circuit is stable. This verification works well with circuits modeled with lumped elements. At higher frequencies the distributed nature of elements must be considered, and time domain analysis may not always work. 7.4 DESIGN STUDY: CDMA DOWN – CONVERTER The design target is to convert the 869 – 894 CDMA band to an IF of 85 MHz using high side LO. The 2SC5006 device was used, and the design parameters were determined using the circuit of Figure 7.11. The device parameters are found in Appendix 7A for the Gummel–Poon model. S-parameter blocks were employed to artificially terminate the circuit with desired impedances, and the Gummel Poon device model was used in the ADS simulator. The base bias is delivered by the resistive divider Rb1, Rb2, selected to deliver a quiescent current of 2.0 mA into a 5 ohm emitter resistor. Initially, the output S-parameter block, which acts as a low pass filter with short circuit impedance at the collector, was set to pass IF signals and block all higher frequency signals. +VCC Lbias [S]out

Rb1 [S]in Vin Rb2

Figure 7.11

RE

Mixer initial schematic with virtual input and output matching network.

Vout

454

Microwave Mixer Technology and Applications

The input S-parameter block was set as a through line for LO and RF signals, and image and IF terminations were varied. Simulation results for conversion gain are summarized in Table 7.3. It is interesting to note that an open circuit IF impedance at the base degrades conversion gain regardless of image termination. This is due to the fact that an open circuit condition enhances internal device negative feedback. If the IF termination is 50 ohm or a short circuit, then there is less effect from image termination on the conversion gain. Table 7.3 Effect of IM and IF Termination at the Base

ZIM 50 open open 50 short open short open 50

ZIF 50 open 50 open short short open 50 short

Gc dB 9.6 3.37 8.2 3.8 10.2 8.9 3.47 8.37 9.38

The test circuit was then extended to the mixer circuit of Figure 7.12. The collector S-parameter block was replaced by an open circuit stub tuned at the LO frequency, and an external load of 100 Ohm. An initial stability analysis showed instability occurring in the low MHz range of frequencies, which can be suppressed by a parallel RC resistor combination in series with the base. +VCC /4

Lbias Cst

Rst Cp

Figure 7.12

LS Rb2

Vout

2

Rb1 Vin

Ls

RE

Mixer final schematic with bias network Rb1, Rb2.

Cp2

RL

BJT Mixer Theory

455

The base impedance at LO and RF frequencies were determined by applying two voltage generators at the Vin port and measuring the base current at the corresponding frequency. The results were: ZRF = 12.38 - j35.8  and ZLO = 7.9 - j33.7 . Approximate results from (7.41), for the desired bias and values from Tables 7.1 and 7.2, provide the following impedance values ZRF = 9.37 - j35  and ZLO = 9.37 - j30. A simple LsCp network was applied to match the gate to the RF generator impedance. The collector impedance at IF, with the base terminated in 50  per (7.43), is equal to ZIF = 158 - j663 , which is approximately equal to the simulated small signal impedance, Z IF = 360 - j890 . A similar network was applied at the output side to transform the external load impedance of 50  to 100  at the collector terminal. This network helps to short higher harmonics and other high frequency mixing products. The component values are given in Table 7.4. The LO generator voltage was set to 0.25 V at the device base. The level of LO power to achieve this voltage is + 2 dBm. Table 7.4 Circuit Element Values

Element LS CP Rb1 Rb2

Value Element Value 10.36 nH LS2 60 nH 5.45pF CP2 13.8 pF 4500  Rst 100  800  Cst 10 pF

Impedance matching at RF and LO frequencies is depicted in the Smith charts of Figure 7.13. The first plot on the left shows impedance optimized for RF match and with LO mismatched with a VSWR of 3:1, giving gain of 15 dB. It is intuitive that matching the base impedance at LO frequency will result in a mismatch at RF frequency due to the frequency difference and large signal effects.

SZLO SZRF

RF LO

RFf req (850000000.000 to 950000000.000)

Figure 7.13

SZLO SZRF

RF RFfreq= 8.950E8 SZRF=0.039 / 165.059 impedance = 46.375 + j0.930 RF

LO

LO RFfreq= 8.950E8 SZLO=0.541 / -27.711 impedance = Z0 * (2.113 - j1.503)

RFfreq (850000000.000 to 950000000.000)

(a) RF matched (b) Best tradeoff for gain vs. match RF and LO base impedance represented in the Smith Chart.

RF RFfreq= 8.950E8 SZRF=0.233 / 171.930 impedance = 31.205 + j2.157 LO RFfreq= 8.950E8 SZLO=0.375 / 5.615 impedance = 108.996 + j9.307

456

Microwave Mixer Technology and Applications

A better trade-off between impedance and gain is possible by slightly mismatching the RF and improving LO match. The plot on the right shows RF matched with VSWR of 1.6 and LO VSWR of 2.2. Improving LO match results in more LO signal power into the device and better gain performance, at the expense of slightly degraded RF mismatch. This tradeoff increased conversion gain from 15 to 17 dB. The conversion gain is in Figure 7.14 for RF frequency swept from 850 to 950 MHz, with the LO frequency fixed at 1 GHz. 25 23 21

Gainc

19 17 15 13 11 9 7 5 8.0E8

8.2E8

8.4E8

8.6E8

8.8E8

9.0E8

9.2E8

9.4E8

9.6E8

RFfreq

Figure 7.14

Conversion gain for near matched RF and LO impedance at Vcc = 6 V.

L_StabCircle1

S_StabCircle3

The stability analysis of the final circuit revealed stable operation for bias supply ranging from 1 to 10 mA over the 100 MHz to 12 GHz frequency range. The source and load stability circles within this range are depicted in Figure 7.15. A transient analysis was also performed for this circuit and it showed a decaying transient within a few nS.

indep(S_StabCircle3) (0.000 to 51.000)

Figure 7.15

indep(L_StabCircle1) (0.000 to 51.000)

(a) Source stability circles (b) Load stability circles Stability circles for input and output mixer circuits.

BJT Mixer Theory

457

7.5 CASCODE APPROACH The cascode amplifier is known to provide higher gain compared to using a single stage due to higher output impedance and higher reverse isolation. In addition, the cascode topology offers higher frequency of operation due to a reduction in the equivalent input capacitance. The effective input capacitance of a single common emitter amplifier equals the base emitter capacitance, Cπ, plus of Cµ multiplied by gmRc after Miller’s theorem:

Ceq  C  (1  g m Rc )C

(7.49)

In contrast, the effective shunt capacitance appearing at the input of a cascode amplifier is lower, being a function of total voltage gain, A V. If the devices are similar, then AV is given by the ratio of the trans-conductance of each device, Av = – gm2/gm1 ≈ -1, therefore:

Ceqcascode  C  (1  Av )C  C  2C

(7.50)

Since the input capacitance is smaller, the circuit will operate at higher frequencies. It can also be shown that output capacitance of a cascode is given by Cµ of the common base device, minimizing internal feedback from the collector of transistor-2 to the base of transistor-1. In the conventional cascode mixer, represented in Figure 7.16, the RF signal is applied to device Q1 operating as a linear RF amplifier/mixer. The LO is applied to Q2 operating as an emitter follower that switches the collector voltage of Q1 thereby modulating its transconductance. +VCC

RLC Vout

RLO Q2 VLO RRF

Ie Q1

VRF

Figure 7.16 Cascode down-converter configuration with tank circuit at the drain.

458

Microwave Mixer Technology and Applications

Device Q2 is also switched and approximately operates as a common base IF amplifier. Under these conditions, the amplifier benefits of a cascode configuration are not known to be applicable to a cascode mixer. The real benefit here is the natural isolation between the base 1 from common emitter and base 2 of common base device allowing adding both LO and RF signals, eliminating the need for other components to carry out this task. The transconductance modulation is illustrated in Figure 7.17 where a 100 MHz voltage applied to the base of Q2 (black) modulates the current (red) at the same frequency, but with a shift in phase between the two. Thus, the LO voltage modulates the current at the LO frequency, and that current is multiplied by the RF current component from Q1, providing the mixer function. 5

3.5 3.0 2.5

3

2.0 2 1.5 1

ts(VLO), V

ts(IC.i), mA

4

1.0

0

0.5

-1

0.0 0

2

4

6

8

10

12

14

16

18

20

time, nsec

Figure 7.17 Transconductance modulation observed on the collector current.

7.5.1 Design Study: Cascode mixer The same frequency and device of the design study 7.4 is employed here. A more detailed schematic of a cascode mixer is in Figure 7.18. The Q1 collector voltage is biased near the saturation voltage, and current for the cascode Q 1, Q2 is set by the current mirror device Q3. A low impedance at the LO, RF and sum frequencies at the collector of Q2 is obtained from a quarter wave long open stub at LO frequency. The level of second harmonic voltage is high at the collector, so to reduce it an open circuit stub that is a quarter wave long at the second harmonic was also applied at the collector. To complete the output circuit a low pass filter Lout, Cout is employed to deliver a clean IF signal to the load. One problem in this type of design is how to guarantee circuit stability which can be generated from one of the following circuit locations: the emitter impedance of transistor Q1 can generate negative resistance at the base if the parasitic capacitance to ground dominates the impedance, according to (7.41); the transistor Q2 is a common base device with

BJT Mixer Theory

459

collector grounded at the frequency of operation. One can easily demonstrate with a simple equivalent circuit for the transistor that emitter impedance can be determined from (7.51). If the element Rst represents a resistance, then the impedance is positive. However, if it represents a reactive inductance, then the emitter impedance can become negative. In addition to these two locations, the circuit layout can also introduce positive feedback not represented in the schematic. After addressing these causes of instability, addition of a negative feedback circuit, Rf, Cf can also help stabilize the circuit.

Z E ,Q 2 

1  jCRst g m  jC

(7.51)

Lbias

+VCC



3

1



Rst Q3

VIF

Cout

Q2 Rf

VLO

C Cp2

2

Lout

LS2

f

IE Q1 LE

VRF Cp1

LS1

RE

Figure 7.18 Cascode down-converter configuration employing similar RF and LO matching circuits.

The base matching at Q1 and Q2 are of similar topologies. The base impedance of each was determined by applying a large signal LO voltage generator to the base of Q2 and a small signal voltage to the base of Q1 and monitoring the base current of both. The Q2 base impedance can be estimated from (7.41) by making ZE = ZE,Q2 the collector impedance of Q1. On the other hand resistor Rst shunts the base impedance of Q2, contributing to the impedance matching at the cost of gain. A trade off analysis is required to optimize this resistor for matching, stability, and gain. The component values for the RF frequency band of 864 to 869 MHz and LO at 1 GHz are in Table 7.5. The mixer was biased at 5V and 2 mA, and the output load was set to 100 ohm. The conversion gain is illustrated in Figure 7.19 for two conditions. The first shows

460

Microwave Mixer Technology and Applications

the designed conversion gain which is relatively constant at 15 dB over the RF frequency range of 780 to 940 MHz, and LO power of 4 dBm. The circuit was simulated to be stable for an LO range from 1 to 7 dBm. The simulation assumes a Q-factor of 6 for the on-chip inductors. The second condition applies to conversion gain under the conditions provided by a recent publication on a BICMOS technology [8]. The matching inductors were replaced by the models from that publication. The device bias was made equal to the one from the paper, (i.e., 2V and 1.7 mA). And the bias inductor, Lbias was replaced by a 300  resistor. The resulting conversion loss, depicted by the second plot, shows a 10 dB degradation compared with the initial ideal design. This agrees with the performance reported by [6] is about 5.9 dB conversion gain at the RF frequency of 1.9 GHz, and LO drive of 0 dBm. The reported noise figure is equal to 15 dB. Table 7.5 Circuit Element Values

dBm(SPAR_CAS..Vload[::,1])-dBm(SPAR_CAS..Vin[::,4]) dBm(Vload[::,1])-dBm(Vin[::,4])

Element LS1 CP Lbias 1

Figure 7.19

Value Element 7.8 nH LS2 8.8 pF CP2 300 nH Rst 96 Cst

Value

3

40

2

47

18 nH 4.5 pF 230  12 pF

20 18 16 14 12 10 8 6 4 2 0 7.8E8

8.0E8

8.2E8

8.4E8

8.6E8

8.8E8

9.0E8

9.2E8

9.4E8

RFfreq

Conversion gain performance for GaAs technology (top curve), MOS technology (bottom curve).

BJT Mixer Theory

461

7.6 SINGLY BALANCED MIXER A singly balanced transistor mixer could be designed in the same approach as a diode mixer, with a balun to insert LO signal and a power divider to insert the RF signal into say the base of transistors. And this was effectively common when Silicon technology became available in the early 1960s. With the maturity of IC technology the trend moved to integrate circuits for lower cost production and the application of a differential amplifier circuit as a mixer became a better alternative. Usually the differential devices are driven by the LO signal operating as switches and the current source device operates as a linear RF amplifier. Therefore, instead of selecting a device when starting a design, one starts by considering which technology is best suited for the design at hand. One of the key parameters to be selected is the process fT, which is inversely related to the parasitic capacitances of the device. Usually fT is selected to be much higher than the maximum target frequency of operation. However, high fT has a higher cost that must be traded off with process performance. The majority of differential mixers, either singly- or doubly-balanced, do not use inductors for compensating capacitance degradation, which in turn limits high frequency performance. If image or sideband rejection is desired, then filters or phasing circuits must be applied. In the differential schematic of Figure 7.20, the LO voltage applied at the bases of the differential pair, Q1, Q2 are of equal amplitude and 180º opposed in phase. This makes the emitters a virtual ground to the LO, and isolates the LO from the current source created by Q3.

+VCC RL (-)VIF/2

RL IM

IN

Q1

(+)VIF/2

Q2

(+)VLO/2

(-)VLO/2 IE

VRF Figure 7.20

Q3

Differential amplifier operating as a single balanced mixer.

462

Microwave Mixer Technology and Applications

In a differential amplifier, the largest contributor to distortion is the base nonlinearity of the top devices, but in mixers they operate as switches which theoretically are linear. They do however introduce nonlinear effects during the transition time but the major source of nonlinearity is the lower device acting as an RF amplifier. If the output signals at RLare combined differentially, then the RF signal is cancelled, and the IF and LO signals add constructively. In downconverter mode the LO can be suppressed by a lowpass filter comprising a capacitor connected in shunt with the differential output, resulting in a clean IF signal. 7.6.1 Analysis of Mixing Effects A detailed analysis of the conversion process can be obtained by calculating the currents IM and IN in the circuit of Figure 7.20. The emitter current of Q1 is related to base emitter voltage by the expression below. In general, collector current is approximately equal to the emitter current, IM = IEQ1, where  ≈ 1 and IS is the saturation current, therefore:

I EQ1  I S e

VLO 2VT

IM  ISe

VLO 2VT

IN  ISe



VLO 2VT

From this result, the ratio between IM and IN is given by the exponential relation (7.52) and the current IE, usually called tail current, by (7.53). VLO

IM  e VT IN IE  IM  IN

(7.52) (7.53)

Expressing IM and IN in terms of IE and replacing the term VLO/VT by , one obtains (7.54a,b).

IM  IE  IN  IE  IM e

IE IN  1  e 

(7.54a)

VLO VT

and

IM

I E e   1  e 

(7.54b)

The differential output current is equal to id = (IN - IM) resulting in (7.55).

BJT Mixer Theory

 1  e  I d  I E    e e

  

463

(7.55)

Multiplying and dividing the equation by e/2 the equation for differential current is given by (7.56), a familiar relationship between output current and input voltage for a differential amplifier.

V I d  I E tanh LO  2VT

  

(7.56)

The current in each collector is found from (7.54) and (7.55), resulting in (7.57a,b).

      V  I   E 1  tanh LO   2   2VT  

IM 

IN

IE 2

 V 1  tanh LO  2V   T 

(7.57a)

(7.57b)

If a small signal RF voltage is applied to Q3, then the emitter current for the differential pair, IE, is obtained by adding the DC collector current of Q 3 to the amplified RF current. Employing the hybrid  model, the RF current is approximated by gmVRF.

I E  I DC  g mVRF cos RF t 

(7.58)

Two conditions exist for the hyperbolic tangent in (7.56): a. small signal where the hyperbolic tangent can be replaced by a McLauren expansion, ( i.e. tanh(x) = x); b. large signal where the full hyperbolic tangent is applied. 7.6.1.1 Small Signal LO If the circuit operates as a mixer, then (7.58) can be directly applied to (7.56). The LO voltage is also approximated by MacLauren expansion and the output current is defined by (7.59a). The output voltage is determined from the voltage developed differentially across output load resistor RL.

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Microwave Mixer Technology and Applications

V  I out  I DC  g mVRF cos  RF t  LO cos  LO t   2VT 

I out  I DC

Vout  I DC

(7.59a)

V  VLO cos  LO t  g mVRF cos  RF t  LO cos  LO t  2VT  2VT 

(7.59b)

RLVLO V V cos  LO t  g m RL RF LO cos RF   LO t 2VT 4VT

(7.59c)

7.6.1.2 Large Signal LO If the LO voltage is large, then one has to consider the behavior of the hyperbolic tangent in (7.57), [9]. This complex function can be visualized in the plots of Figure 7.21, which shows complete current switching occurs when the magnitude of sinusoidal voltage applied to the base is greater than 5 times VT. IM/IE, IN/IE

IM/IE, IN/IE 1.0

Ip/2 t

Vin Vin -8VT

-VL=2VT

0

VL=2VT 8VT

Vin

t Figure 7.21

Collector current versus input voltage. Solid line for x =1VT; dotted line for x = 5VT; traced line for x >> 5VT.

BJT Mixer Theory

465

In the circuit of Figure 7.20, half of VCC is applied to the collector of the RF amplifier and half to the differential amplifier. The maximum current of Q1, Q2 is approximately given by IQ1 = (VCC/2-Vsat)/RL which is approximately twice the peak current. The center of the plot in Figure 7.21 corresponds to the quiescent bias current IQ1  Ip/2 = IE/2. Replacing the hyperbolic tangent by a Fourier series representing a square wave, the currents IM, IN, becomes:

IM 

IN 

IE 2

IE 2

 V 1  tanh LO  2V   T 

 V 1  tanh LO  2V   T 

 I E   4      2 1    cos  LO t     

 I E   4        1  cos  t      LO   2     

(7.60a)

(7.60b)

The differential current is therefore expressed by (7.61). Substituting IE by (7.58), the output current is then given by (7.62).

4  I out  ( I M  I N )  I E  cos LO t    I out  I out 

4 I DC



4 I DC



cos  LO t  g mVRF

cos  LO t 

2



4



(7.61)

cos  RF t cos  LO t

g mVRF cos LO   RF t  cos LO   RF t  (7.62)

Notice the conversion process is independent on the LO voltage as long as it is much larger than VT, but is dependent on bias. The voltage developed by Iout on a differential output load resistance RL is given by (7.63). If the emitter resistor is considered, then output voltage is given by the next equation. An approximation valid in general is to consider gm(RE + re) > 1 giving Vout expressed in (7.64).

Vout  Vout 

2



g m RLVRF cos LO   RF t

g m RLVRF cos LO  RF t  1  g m (re  RE ) 2

(7.63)

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Microwave Mixer Technology and Applications

Vout 

2

RLVRF cos  LO   RF t  (re  RE )

(7.64)

High side LO mixing refers to the situation where the LO frequency is higher than RF, and down conversion refers to the IF equal to the difference frequency term (LO - RF). In general power is lost in the unwanted sum term (LO + RF), whose conversion loss is approximated by the factor 2/. In low side LO mixing, the LO frequency is lower than RF frequency and the IF remains the difference. The conversion trans-conductance assuming RE = 0 and gmre 2 the peak value is equal to IDC defining the magnitude of the square waveform. For the differential configuration, 2I DC = IE, so the emitter current from each device under square wave drive becomes:

IE  4 4  cos 3 LO t  1  cos  LO t  2   3  I  4 4  I N  E 1  cos LO t     cos 3 LO t    2   3  IM 

(7.70)

(7.71)

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Microwave Mixer Technology and Applications

The differential current description assumes LO voltage is applied equally to both bases, and total emitter current is equal to the current source current. The differential Iout is therefore the same as (7.57) for IM and IN, meaning that conversion efficiency of a differential amplifier is the same for sinusoidal or square wave drive. This conclusion does not take into account the transition on-off time for square wave which is lower than sinusoidal. The longer transition time of sinusoidal drive means the RF signal can modulate the switch conductance resulting in greater distortion. Therefore, LO voltage rise time affects intermodulation, so whatever is faster provides better intermodulation performance. In practice, the square wave drive makes sense if the switching device capacitance does not affect the rise time, otherwise the drive waveform will have no effect on distortion. 7.6.2 Impedance Matching For the differential configuration two modes of operation exist: differential mode and common mode. The normal mode of operation for Q 1, Q2 of Figure 7.20 is the differential mode; the common mode is of no interest, except for the stability analysis. Transistor Q3 operates in single ended mode. 7.6.2.1 Differential Mode The LO impedance for Q1, Q2 is obtained by considering one half of the differential circuit since the emitter is a virtual ground relative to the differential LO drive. The impedance is therefore determined from a single switching device with the collector terminated into load RL, and with the same bias as in the differential configuration. Equation (7.41) can be applied to make an estimation of input impedance, with gm replaced by gm0. The reactive elements in the model are also calculated at the bias point. The resulting base impedance (at RF or LO) is doubled to obtain the total differential mode impedance. The IF impedance is differential and determined in a similar way. If the LO generator delivers a square wave to the base, then the impedance matching process does not follow the same approach as for a sinusoidal drive. It is important to maintain the square wave shape so reactive elements must be used with care to avoid distorting the square wave. If the LO is close to the mixer, transmission lines are minimum and the circuit connections can be considered lumped. If the device reactances are low enough, then instead of conjugate matching it is more straightforward to simply ensure the mixer base impedance is higher than the generator output impedance, so the mixer does not load the generator. If the LO is physically far from the mixer, or external to the die, the best interface is a uniform transmission line whose characteristic

BJT Mixer Theory

469

impedance equals the generator impedance, and terminate the line with a matched load. 7.6.2.2 Common Mode In this case half the differential circuit is again considered, g m is replaced by gm0, and the emitter resistance of Q1 is replaced by half the collector impedance of Q3. The resulting impedance is divided by 2, as both sides of the differential circuit are connected in parallel. 7.6.3 Conversion Gain According to (7.41), the RF input impedance at low frequencies is relatively high compared to 50 Ohm, depending on the device size and bias. Therefore the generator is only lightly loaded, and there is no need to conjugate match the RF port. In this condition it is more practical to calculate the voltage conversion gain, GCV, (7.72). If an external emitter resistor is added, then the second equation is applicable

2 RL  g m  2    g m RL for gmre 1  L    1  g m (re  RE )   (re  RE )

Gcv  GCV

When the frequency of operation is high the input RF current is no longer negligible, then power conversion gain should be used. Assuming conjugate match and the fact the differential pair is switching in a square wave manner, the conversion gain can be described by (7.73).

 R  2 1  GC  L  Rin   RF C (re  RE ) 

2

(7.73)

A single balanced differential mixer can provide lower noise figure than double balanced or other architectures because fewer sources of noise are participating in the frequency translation. The effect of capacitive emitter impedance was discussed in terms of the circuit stability. If it is made inductive, then there is some gain reduction, but stability, linearity and matching are improved.

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Microwave Mixer Technology and Applications

7.6.4 Linearity in Singly Balanced Mixer The distortion in a differential mixer is generated by the RF amplifier while the mixer switch performs the frequency translation with comparatively small amount of distortion. Any nonlinearity generated at this level will be transferred to the signal during the conversion process. For inter-modulation analysis the nonlinearity is considered mild and the Taylor series expansion is usually applied to the trans-conductance as shown in (7.74), [11]. The first term corresponds to the linear term, ic = gmVRF. 2 3 V 1  VRF  1  VRF   RF      ic  I C    2  VT  6  VT    VT 

(7.74)

The second order inter-modulation product is given by the ratio of second order distortion with fundamental component, (7.75). The third order is given by relating third order level with fundamental, (7.76). In Chapter 2 the intercept point was defined as the point where the IF power and intermodulation power, both as a function of RF input power, equal each other. A similar definition has been demonstrated in [11] using voltages instead of power. The former is more often employed at microwave and higher frequencies where it is easier to measure power while the latter is employed in lower frequencies. Therefore, the second order intercept point is described by (7.77) and the third order by (7.78). The effect of the emitter resistor is obtained by multiplying V RF/VT by the factor 1/(1+gmRE). The input intercept point is obtained by applying the relation IP 2 = Vin/IM2 and IP3 = Vin/IM3.

IM 2 

1 V RF 2 VT

1 V IM 3   RF 8  VT IP2 

IP3 

(7.75)

  

2

VRF  2VT IM 2

VRF IM 3

 2 2VT

(7.76) (7.77) (7.78)

The definition of distortion in terms of device parameters is available in a classical paper on bipolar amplifier linearity, [12]. The paper assumes the gain of a common emitter amplifier is given by (7.79), showing a predominance of

BJT Mixer Theory

471

collector to base capacitance, CJC. Therefore the analysis assumes CJE to be constant in the forward region and concentrates on the effect of feedback capacitance nonlinearity. The source, RS and Load RL are assumed to be resistive and the subscript Q designates quiescent values for voltages and currents.

S 21 

2 RS RS  RB

1 C JEQ   j  1   C JCQ R L  gm  

(7.79)

The conclusions for intermodulation are given by (7.81) and (7.82) referred to the output. The diffusion capacitance, CD, is considered constant and replaced by its equivalent, (7.80), with 1 representing the transit time of carriers in the base. The collector to emitter capacitance is represented by C0. The factor 1/n is the exponent used to describe the relationship between capacitance, C JC, and voltage.

C D  g m 1 

IQ VT

1

C JEQVT 2

IM2 

(7.80)

 C jCQ R L

Vout R L I Q C JEQ VT 2 1   C JCQ R L IQ C JEQ VT

1  1 1 1   2n  n  VCEQ 2  Vout  R L I Q   C JEQ VT  2  1   C0 RL IQ 2

IM3

1 1 n VCEQ

2

2

(7.81)

 C0 RL

(7.82)

The distortion is expressed in terms of Vout, the peak value of output signal voltage across RL, at the IF frequency. The equations shows distortion, to a first order approximation, is frequency independent. Also note the only mixing effect assumed is the conversion loss represented by the 1/ term. The equations show a linear relationship to signal level at the load, Vout, for second order products; and a squared relationship for the third order. In both it is seen that intermodulation levels decrease with higher bias voltage and current. The equations

472

Microwave Mixer Technology and Applications

were developed for operation at high frequencies, therefore above f T/. The equations do not take Kirk effect into account which may provide an optimistic distortion at high currents. 7.6.5 Noise in Singly Balanced Mixers The treatment of noise figure in a differential mixer is more straightforward compared to a single stage or cascode because the noise is assumed to originate mainly from the RF amplifier. This assumption is correct as long as the switches are driven by a large signal LO that makes the switching nearly square wave. Under this assumption the RF amplifier noise is transferred to the IF band. The instantaneous switching process increase the input referred noise contribution from the RF amplifier by a factor of (/2)2 or 3.9 dB, [13]. As shown in Figure 7.23, the noise sidebands about the LO convert in frequency down to IF band. Power

IF

RF LO IM

3LO

5LO

Frequency Figure 7.23

Noise contribution from the harmonics of LO. After [13].

The noise figure of a bipolar device in terms of the device parameters from (7.44) is reproduced next, where gm1(t) becomes gm. 2 r  RE g ( R  r  RE ) 2 1   F    1  b   m S b RS 2 RS g m 2 RS 2  

 1  f  2        (7. 83)    f T   

The four noise parameters NFmin, Rn, GS, and BS defined by the set (7.45) are equally applicable with NFmin multiplied by the conversion loss. A larger RF device will reduce the base resistance and emitter access contribution to lower noise figure. Also, based on Friss equation for cascaded noise figure, the RF gain should be maximized to minimize the contribution of noise figure from the switching quads. During design those parameters need to be traded off with linearity.

BJT Mixer Theory

473

7.7 SINGLY BALANCED SUBHARMONIC The differential amplifier circuit was also considered for subharmonic mixers at about the same time fundamental mixers were investigated. A subharmonic differential mixer proposed in a patent by Motorola, [14], is represented in Figure 7.24. The LO and RF signals are applied to the base of Q1 and the IF signal is extracted from the collector of Q2. In order to understand the operating principle of this circuit, let’s calculate the trans-conductance that relates the collector current of Q2 to the base voltage of Q1. The resulting expression, (7.84) has a bell shaped characteristic around zero volts for g m(Vbe), therefore its trans-conductance is symmetrical for positive and negative values of input voltage. A linearized version of this function is the simple triangular shape shown in Figure 7.25, which adequately represents the application of a dynamic LO voltage at the base of transistor Q1. Icc C C +Vcc αIE2 αIE1

Rb B

Rb

IE1

IE2

Q2

Q1

IE

B

IE

RE

-VEE (a) Figure 7.24

(b)

Differential amplifier as subharmonic mixer (a) schematic, (b) equivalent circuit. qV

di qI e KT gm  E2  T qV 2 dV KT   1  e KT   

(7.84)

In this case V = Vbe2 – Vbe1 = Vbe1 since Vbe2 = 0. Applying a LO voltage lower than V0 (100 mV), the resulting modulation for g m(t) is a DC term plus a sinusoidal term that is twice the frequency of the applied LO voltage. If the LO voltage is much greater than V0, the modulated gm(t) has its negative peaks clipped, turning the waveform into a sequence of half rectified peaks. The Fourier components are still predominantly composed of even harmonics. The equation for gm therefore is defined for two conditions depending on LO voltage.

g m (t )  g m0  g m2 cos 2 LO t for VLO < V0

(7.85a)

474

Microwave Mixer Technology and Applications

g m (t )  gm0  gm2 cos 2 LO t  gm4 cos 4 LO t  ... for VLO > V0

(7.85b)

If a small RF signal is added to the LO signal before application to the base of Q1, then the current at Q2 is approximated by the product as follows:

I C (t )  g m (t )VRF cos  RF t

(7.86a)

I C (t )  ( gm0  gm2 cos 2LOt ) cos RF t

(7.86b)

I C (t )  gm0VRF cos RF t 

VRF gm2 cos(2LO  RF )t  cos(2LO  RF )t  2 (7.87)

gm(V)

-V0

gm(t)

V0

VLOV0

t Figure 7.25

t

Idealized transconductance waveforms.

An additional approach proposed by the inventors is to apply the RF signal at the base of Q1 and the LO signal at the base of Q 2. This does not change the trans-conductance function of the drive voltage, and it is a good way to isolate the RF signal source from the LO source. 10

IF freq=200.0MHz dBm(Vload)=-17.536

RF freq=1.800GHz dBm(Vload)=-9.900 LO2 freq=2.000GHz dBm(Vload)=-16.094

Figure 7.26

0 -5

dBm(Vload)

LO freq=1.000GHz dBm(Vload)=1.668

LO

5

RF

-10

LO2

IF

-15 -20 -25 -30 -35 -40 -45 -50 0.0

0.2 0.4 0.6 0.8

1.0 1.2 1.4 1.6

1.8 2.0 2.2 2.4

freq, GHz

2.6 2.8 3.0

Spectrum of differential harmonic mixer, illustrating besides the LO, RF and IF components, the spurious generated.

BJT Mixer Theory

475

The circuit from Figure 7.24 was analyzed using ADS with the RF frequency at 1.8 GHz and LO at 1 GHz. The input circuit was matched by a simple series L, parallel C circuit, resulting in conversion gain of 5 dB, with devices biased at 4 mA each. The spectrum of the simulation is in Figure 7.26. 7.8 DOUBLY BALANCED MIXER The analysis of a single balanced differential mixer can directly be applied to a double balanced differential, since the double balanced topology can also be thought of a parallel “push-pull” combination of two single balanced mixers. When doing this combination, the collectors are cross coupled eliminating the LO and RF voltages at the sum points, while adding the IF currents. A direct consequence of parallel blocks is that conversion gain remains unchanged if the circuits are well balanced. Compared with singly balanced, the doubly balanced configuration has higher port-to-port isolation between RF, LO and IF signals and it generates lower spurious levels. The reason for double balanced configuration is a higher port-to-port isolation between RF, LO and IF signals, generating less spurious and suppression of both RF and LO signals. One of the first reports of using two differential amplifiers with cross-coupled collectors appeared in the patent for a phase detector [15].

+VCC RL

IV

RL

IW

(+)V0/2

(-)V0/2 IR Q3

IU IT

IO

Q5

Q4

(+)VLO/2 IM

(-)VLO/2

Q1

Q6 (+)VLO/2

IN Q2 (-)VRF/2

(+)VRF/2 ZE

Figure 7.27

IE

Differential amplifiers comprising a doubly balanced mixer.

476

Microwave Mixer Technology and Applications

The objective of the inventor was to eliminate transformers by making use of the 180° phase shift provided by differential circuits. The circuit schematic was originally built with discrete components and later was implemented into IC technology in the form shown Figure 7.27 after the patent [16]. The proposed inventions are capable of providing an output voltage proportional to the product of two input voltages with the same or different frequencies. They operate as a mixer, when two signals with different frequencies are supplied to the circuit, the desired mixing products appear at the output. If two input voltages with equal frequencies are applied, then the output voltage is proportional to the phase difference between the signals. The multiplier function is understood by considering the currents in the circuit as follows. The differential output current for this circuit is obtained from the sum of the currents in Figure 7.27. Assuming the relation between IR, IO and IU, IT is given by (7.88), the set (7.89a,d), defines each current component as a function of IM, IN, respectively and as function of the LO voltage VLO.

I out  I R  IU  ( I O  I T ) IM IR  V

(7.88) (7.89a)

LO

1  e VT

IU  IO 

INe

(7.89b)

VLO VT

1 e IM 1 e

IT 

VLO VT

INe

(7.89c)

VLO VT

VLO VT

1 e

(7.89d)

VLO VT

Noting that IV is the sum of IR and IU, and subsequently replacing the terms IM, IN, the output is given by (7.90a). A similar argument is valid for current IW in (7.90b).

IV  I R  IU 

IM 1 e

VLO VT



INe

VLO VT

1 e

VLO VT

(7.90a)

BJT Mixer Theory

IM

IW  I O  I T 

1 e

IV  I E

1 e VLO  1  e VT  

VLO VT

VLO VT

e



INe

477

VLO VT

1 e

(7.90b)

VLO VT

VRF VT

VRF  1  e VT  

(7.91)

   

Adding IE/2 and -IE/2 to (7.91), then multiplying and dividing the equation by exp(-VLO/2VT) and exp(-VRF/2VT), gives the following equations for IV and IW:

IV 

IE 2

  VLO 1  tanh  2VT 

 V  tanh RF   2VT

  

(7.92a)

IW 

IE 2

  V LO 1  tanh  2VT 

 V  tanh RF   2VT

  

(7.92b)

For mixer operation, the RF is small signal, so the hyperbolic tangent can be replaced by the first term of the MacLauren expansion, tanh(x)  x. The LO can be assumed to have square wave shape, and is replaced by the first term of the Fourier series, tanh V LO   2 . After the substitutions, (7.92) becomes:  2V    T

IV 

IE 2

 VRF  4 cos RF t  cos  LO t  1  VT   

(7.93b)

IW 

I E I EVRF 4  cos RF t  cos LO t  2 2VT 

(7.93b)

The differential converted current is obtained from the difference between the two currents:

478

Microwave Mixer Technology and Applications

I out 

2 I EVRF cos RF   LO t  VT

(7.94)

If the circuit is weakly nonlinear in terms of the RF voltage magnitude, the voltages and currents are considered to be of mild non-linearity and the hyperbolic tangent becomes, tanh( x)  x 

x3 . Applying two equal tones at the 3

RF frequencies 1 and 2, the third order distortion component is given by (7.95), and the argument x becomes defined by the sum of both tones. An illustration of the relative magnitude of the distortion components along with the fundamental is in Figure 7.28.

x

VRF (cos 1t  cos  2 t )  a(cos 1t  cos  2 t ) 2VT

I out 2a 3a 3 cos22  1 t  cos21  2 t cos  LOt  (cos 1t  cos 2t ) cos  LO t  IE  24 (7.95) 2a/π 3

3a /(48π)

 LO-1 LO-2 LO - ( 21 - 2) Figure 7.28

LO - ( 22 - 1)

IF spectrum for two tones after (7.95).

Another alternative to analyze the mixer operation is to consider the differential output current to be defined as a function of currents I M, IN, by rearranging (7.89) as shown in (7.96).

I out  I R  I O  ( I T  IU ) V I out  I M  I N  tanh LO  2VT

(7.96)

  

(7.97)

Where IM, IN, delivered by the collector currents of devices Q1 and Q2 are given by (7.98)a,b, defined in terms of gm, VRF, and RE.

BJT Mixer Theory

479

IM 

 gm 1  I E  VRF cos  RF t  2 1  g m (re  RE ) 

(7.98a)

IN 

 gm 1  I E  VRF cos  RF t  2 1  g m (re  RE ) 

(7.98b)

Substituting (7.98) into (7.97), and replacing the hyperbolic tangent by its square wave fundamental coefficient, one obtains the small signal mixer output differential current, (7.99). The converted voltage is calculated from the flow of Iout current into the output load, RL. The voltage conversion gain can easily be obtained from (7.100). If there is conjugate match at the input, then the internal base voltage is no longer equal to generator voltage, V RF and the effect of input base match has to be included in the equation. At lower frequencies the voltage gain is sufficient, and if gmRE >> 1 it becomes essentially the ratio of RL to RE.

I out 

Vout 

2g m 2  VRF cos  RF t cos  LO t 1  g m (re  RE )   

(7.99)

RL g mVRF cosLO  RF t  cosLO  RF t   1  g m (re  RE ) 2

(7.100)

GCV

RL g m 2 RL    1  g m (re  RE )  RE 2

(7.101)

The gain expression is useful for an initial design when doing hand estimations for the circuit. Its main approximation is the assumption of square wave form for the LO voltage. Comparing results for double balanced with single balanced, it is observed that both have similar output voltage, but the LO term is suppressed by circuit phasing at each collector for double balanced. A simple low frequency transformer is sufficient to recover the IF signal. The RF section is differential, so the impedance can be calculated from a single stage and doubled to provide the differential impedance. 7.8.1 Gilbert Multiplier The work of Barrie Gilbert, [17, 18], was initially intended for wideband amplifier operation, but it resulted in a current multiplier circuit that, like all multiplying devices is applicable to frequency conversion as well. The initial application of

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Microwave Mixer Technology and Applications

the differential topology was intended for voltage multiplication, which is constrained to small voltages to avoid distortion. Gilbert, [13], proposed the use of a differential topology for multiplying input and output currents, avoiding the voltage limitation and therefore improving linearity. The conversion of the applied voltage to current is provided by external circuits. The key contribution of his invention is depicted in Figure 7.29, showing a differential amplifier with the addition of current mirrors using bipolar devices connected as diodes. I1

Ia

VQ VO

I2

IE

Ib

VR VS

ID

Figure 7.29

Differential current amplifier.

This arrangement transforms the applied current into voltage following a logarithmic law, which when applied to the bases of the differential devices, generates current having an anti-logarithmic characteristic. The resulting relation between the collector and base current turns out to be linear. In this circuit the following relation between the currents are valid, assuming the devices are maintained active by biasing the emitter-base junction much higher than the thermal voltage. The dc current gain is also assumed to be high so that collector and emitter current are approximately equal.

I D  Ia  Ib I E  I1  I 2 In the original work, Ia was defined as xID and Ib as (1-x)ID. and x varies between 0  1, which means they are complimentary. The parameter x was defined as the modulation factor for the current ID. The voltages between the baseemitter junction of the various transistors are denoted as V O, VQ, VR, and VS as depicted in Figure 7.28.

BJT Mixer Theory

KT  I a  ln   (7.102a), q  I S  KT  I E  I1   (7.102c), VR  ln  q  I S  VO 

481

KT  I1  ln   (7.102b), q  I S  KT  I b  VS  ln   (7.102d) q  I S 

VQ 

Note that since all devices share the same chip, they have similar parameters, in particular the saturation current. The loop equation applied to the circuit provides:

VO  VQ  VR  VS Replacing the terms, eliminating the common factor I S, and performing some algebra, one obtains the following collector currents as a product of the currents IaIe, IbIe, normalized to the total current ID.

I1 

Ia I e (7.103a), ID

I2 

Ib I e (7.103b) ID

One can notice the ratio of emitter currents from the differential amplifier is the same as the ratio of the current in the mirrors. Replacing Ia  [xID] and Ib  [(1-x)ID] it is found that the two ratios are independent of the absolute value of ID. In order to validate the multiplication between the currents, it is essential that the normalization factor ID be maintained constant, which is accomplished if the input currents are derived from another differential circuit. The multiplier circuit requires the currents Ia, Ib, IE to be positive to maintain the devices active, which means operation in the first quadrant where collector current and base-to-emitter voltage are both positive. However, in a differential amplifier, two quadrant operation is possible if the devices are biased at the center of the transfer characteristic, so that output net current is zero for a zero input current. This extended operation is accomplished by defining a variable Ii(t) as follows:

I i (t )  2I a ' I D  I D  2I b ' and the new prime currents are defined by:

I a ' (t ) 

Ia  ID I  ID , I b ' (t )  b 2 2

(7.104)

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Microwave Mixer Technology and Applications

The collector currents Ia’(t) and Ib’(t) will swing between +/- Ia, Ib around ID. Combining (7.104) and (7.103) we obtain (7.105a,b) that defines the new I1, I2:

I1 

IE 2

 I  I  I  1  i  (7.105a), I 2  E 1  i  (7.105b) 2  ID   ID 

The currents represented by previous equations describe the performance of a single balanced circuit in two quadrants. A four quadrant multiplier is obtained by paralleling an additional differential amplifier covering the other two quadrants. This option was proposed in the same patent by doubling the previous circuit and cross coupling the collectors, as illustrated in Figure 7.30. The input signal is the pair of currents xIB, (1-x)IB and yIE, (1-y)IE. Vo = RLIout (1-x)IB

xIB Q2

IC2

IC5 I C6

IC3

Q1

Q3 yIE

Figure 7.30

Q6

Q4

Q5 (1-y)IE

Four-quadrant current multiplier.

The output collector currents in this circuit are given by the following set of equations:

I C 2  xyI E I C 3  (1  x) yI E

(7.106a)

I C 5  x(1  y) I E I C 6  (1  x)(1  y) I E

(7.106c)

(7.106b)

(7.106d)

The value allowed for y is the same as for x, and varies between 0 and 1. The differential output current is given by the following equation, and the differential output voltage is developed across the resistor, R L:

BJT Mixer Theory

483

I out  I C 6  I C 2  I C 3  I C 5

(7.107)

Vout  I outRC

(7.108)

The multiplication effect can be understood as follows. The output current is normalized to the emitter current, resulting in the function Z:

Z

I out  1  2 y  2 x  4 xy IE

(7.109)

The polynomial can be factored as follows:

Z  (2 x  1)(2 y  1)

(7.110)

The parameters x, y are proportional to the input signals xI B, yIE respectively. The input signals can also be described by (7.111). If the devices are biased such that x = y = 0.5 then the output voltage is balanced and equal to 0.

X  2x 1

(7.111)

y  2Y  1

(7.112)

The output differential current can be defined by (7.113), as the product of two input currents. This redefinition of variables results in input current amplitude ranging from -1 to +1.

Z  XY

(7.113)

In addition to the work on the differential current multiplier, Gilbert also realizes in 1967 [19] that the switching quad performs the same tasks as a diode ring mixer. Since then the circuit topology became known as Gilbert mixer, or Gilbert cell. 7.8.2 Linearity and Noise Figure Similar assumptions made for the singly balanced mixer is applied here, i.e., the switches are considered ideal and do not introduce distortion in the conversion process, and the differential RF amplifier or transconductor does. The amplification is assumed linear as long as the amplitude of the input signal is smaller than the maximum input voltage, V L, determined from the transfer

484

Microwave Mixer Technology and Applications

characteristic in Figure 7.22, valid for a differential amplifier circuit. Above this point there will be compression and clipping of the signal, generating distortion. The differential output current expressed in terms of output differential current is given by (7.114), [11]. Expanding the hyperbolic tangent after Taylor the coefficients a1, a2, a3 are determined. In a perfect differential amplifier a 2 = 0. Therefore, the input referred voltage intercept point for the third order intermodulation is provided by (7.115).

V I dout  tanh di 2I  2VT

   a1vdi  a2vdi 2  a3vdi3  ...  1 1 a1  ; a2  0 ; a3  3 2VT 32VT  Vip 3 

4 a1  4VT 3 a3

(7.114)

(7.115)

Small differences in each half of a differential amplifier caused by imperfect layout symmetry, imperfections in the epitaxial layers, and other small imperfections give rise to mismatches in circuit parameters, resulting in an offset voltage, Vos. If there is mismatch, the second order input referred voltage intercept point is given by (7.116). If there is an emitter degeneration resistor, then the distortion improves accordingly, predicted by (7.117) and (7.118).

Vip 2

2  2VT  

Vip 2

2  2VT   (1  g

(7.116)

Vos

Vos

m

RE )3

Vip3  4VT (1  g m RE )3 2

(7.117) (7.118)

In reality assuming the Gilbert cells do not introduce non-linearities is an approximation. The effect on non-linearity originating from this differential cell was analyzed by Meyer, [20], where he combined analytical and computer simulation means. The inter-modulation expressed as the ratio of amplitude of third order current component and fundamental component, referred to the output is in (7.119). In this function the input signal current Is was normalized to the quiescent current and the LO voltage normalized to the thermal voltage.

BJT Mixer Theory

I VIM 3   K I  Q With

485

2

  VLO   f2   f ( A)  f 3 ( B)  V  3   T  IQ A  0 1rb VT V B  0C JE T IQ

(7.119)

I K  I RF cos RF 1t  cos RF 2t 

The first conclusion from the equation is that the distortion level increases as the square of signal level, the same as for an amplifier as described in (7.82). The effects of other parameters are shown in Figures 7.31 and 7.32, where the normalized input signal is fixed at IK/IQ =0.2.

IM3 - dBc

-60

Figure 7.31

0

1 0

2 0

3 0

4 0

VLO/VT

-70 -80

--- 01rbIQ/VT = 0.6 .... 01rbIQ/VT = 0.3

IM3 function of LO drive level. After [20].

Figure 7.31 shows the VIM3 level improves slightly with higher LO drive, confirming that distortion is mainly a function of the RF stage. The dependency observed is a function of the switching waveform, which approximates the ideal square wave at larger drive (dashed line in the figure), with a consequently higher value of conversion trans-conductance. For this plot CJE = 0. In the plot of Figure 7.32(a), CJE is still disregarded, the LO voltage is fixed at 500 mV and the ratio of signal current to bias current is also fixed. Therefore, increasing any of the parameters bias, LO frequency, base resistance or transit time causes the IM 3 level to increase. Those parameters are defined by functions A, B for (7.119). If rb is made equal to zero, then IM3 drops to -100 dB. In Figure 7.32(b), the ratio of signal to bias, and LO drive level are held constant, while CJE is varied. With rb = 0 the simulation result indicates a large dependency on base capacitance, C JE.

486

0.01

LO1rbIQ/V 0.1 T 1

-3

-40

-70

10

IM3 - dB

IM3 - dB

-60

Microwave Mixer Technology and Applications

LOCJEVT/IQ -1 -2 10 10

-60

CJE = 0 IS/IQ =0.2 VLO= 500 mV

-80

rb = 0 IS/IQ =0.2 VLO= 500 mV

-80

(a) IM3 function of ωLO1rbIQ/VT (b) IM3 versus LOCJEVT/IQ Figure 7.32 VIM3 level versus frequency, bias, and device parameters. After [20].

The noise mechanism in a doubly balanced mixer using differential amplifier configuration is similar to singly balanced. In [13] it was pointed out that theoretically both singly and doubly balanced mixers have similar noise figure, but doubly balanced are more complex to layout and have more noise contributors resulting in higher noise figure. 7.8.3 Linearization of Mixers Built with Differential Amplifier The linearization process largely consists of correcting the nonlinearities of the RF amplifier. A simple method to improve linearity is by emitter degeneration, which means adding a resistor to the emitter, transforming the exponential Ice(Vbe) relation to be more linear at the cost of lower gain. Vout 1.0 RE

Vin=VLO -8VT Figure 7.33

-2VT 0 2VT

8VT

Effect of emitter resistance on the amplifier transfer characteristic.

The output differential voltage is given by the difference of output currents 7.57a, 757b on a load RL represented in Figure 7.33 by the solid line. Adding an emitter resistor it can be proven the transconductance changes from

BJT Mixer Theory

487

IE/2VT to IE/(2VT+REIE), resulting in lower gain. However the input voltage swing limited to 2VT is now increased to 8VT, so that the maximum allowable input differential input voltage increases and the circuit becomes more linear [21]. The linearization method represented in Figure 7.34, employs an emitter resistor connected differentially from emitter to emitter. The differential connection of RE allows the level of degeneration to be adjusted without affecting DC bias.

+VCC RBB

IC2

IC1

R1 Q1

RE

R2

Vin(t)

Q2

IDC/2

IDC/2

IE

IE

RB Q3

IE

Q4

Q5

-VEE Figure 7.34

Linear RF transconductor amplifier.

The collector of Q1, Q2 are attached to the top quad devices not shown in the figure, which are driven by the LO voltage. The bias current is determined by the resistor RB, which delivers the base current for Q4, Q5, plus emitter current for the mirror Q3, and RBB which also provides base current to Q1, Q2. Since the currents are equal in each device, the currents in RB and RBB are given by (7.120a) and (7.120b) respectively, where  is the ratio of collector to emitter current. The voltage on RB is given by the difference between –VEE and the Q3 transistor voltage, VCE.

I RB  (3  2 ) I E (7.120a);

I RBB  (5  4 ) I E (7.120b)

I DC  (VCC  VEE  VCE ) (V  VCE )  I E   EE 2 RBB (5  4 )  RB (3  2 ) RBB  RB

(7.121)

The current in each collector is given by the sum of DC current and the signal current generated by the applied voltage, Vin. Using the T-model described in Chapter 3 to represent the bipolar devices, this voltage is developed around the RE terminals. For differential signals, half of this resistor is on each side of the

488

Microwave Mixer Technology and Applications

circuit. If one considers the applied peak to peak voltage the following equations describe IC1 and IC2.

I DC Vin (t )  2 RE I V (t )  DC  in 2 RE

I C1  i E1 

(7.122)

I C 2  i E 2

(7.123)

An alternative to RF linearization proposed by Gilbert is the use of the Micromixer, [23], represented in the next figure. The traditional differential RF amplifier was replaced by a class AB amplifier while the mixer quad is conventional. The RF stage comprises a current mirror and a common base transistor formed by Q1, Q2, Q3 and CD. The RF signal, Vgen, drives the input stage through the current mirror and the emitter of the common base transistor, Q 1. On the positive RF cycle the current flows mainly through the common emitter device, Q3, while the common base is non conducting. Therefore a half sinusoid current IC3 is generated. Alternately, when the RF signal is negative the common base device conducts and the common emitter is not operational. The current I C1 is then a half sinusoid on the negative side and of opposite phase compared to I C3. The mixer is driven by the differential current, I C1 – IC3, where each current is highly non linear but the difference is linear.

ZS

CC

IC

IC

1

3

IB

Q1 Q3

Vgen

VR F

Q2 IC

QZ1

CD

QZ2

2

Figure 7.35

Micromixer schematic. After [23].

Another linearization method reported by Gilbert is the technique called multi-tanh, [24]. In this method two or more differential pairs are paralleled with different emitter areas and turn on voltage. The effect is to extend the input voltage capacity, and therefore improve linearity of the equivalent transconductance. An example of three differential pairs is in Figure 7.36, where the emitter areas of Q1a and Q2a are equal and emitter areas of Q1c and Q2b are set A

BJT Mixer Theory

489

times larger than the area of Q1b and Q2c. The bias current for the center pair is K times the bias of the outer pairs.

Iou Q1c Vin

Q2c

IE Figure 7.36

t

Q1b

Q1a Q2b

IE

Q2a

KIE

Multi-tanh schematic. After [24].

The current sources are controlled by a single current mirror. The transconductance of a differential pair is obtained by taking the derivative of output current with respect to input voltage, as in (7.124). The composite transconductance in (7.125) obtained from reference [25], is shown to be set at different input voltages adding an offset voltage dependent on the ratio of device area.

V  I out  I E tanh in   2VT  V dI I g m (Vin )  out  E sec h 2  in dVin 2VT  2VT

  (7.124)  V   V  Vos   V  Vos  I    sec h 2  in  g m (Vin )  E  K sec h 2  in   sec h 2  in 2VT   2VT   2VT   2VT  (7.125) With

Vos  VT ln( A)

For small signal input, the transconductance of the composite structure is given by (7.125). The parameter values must be properly calculated to guarantee a constant gm. By differentiating (7.125) and equating to zero, the optimum values for the triplet provided in the literature are A = 13 and K = 0.75. An example of use of these parameters, applied to SiGe technology is in Figure 7.37, [26], where

490

Microwave Mixer Technology and Applications

gm is approximately constant from - 0.075 to + 0.075V. The transconductance for each pair and for a single differential pair are also plotted for comparison. The plot shows the composite gm is reduced to more than half the value of a single differential pair. This is agreement with (7.126) that shows gm0 is divided by 2.15, using the above values for A and K.

g m (Vin )  g m 0

1 2 K

 8A   K  2  1  A 

(7.126)

The multi-tanh circuit provides improvement in both noise figure and linearity: V1dB increases by 10 dB and input noise power increases by 3.3 dB, for a net increase in dynamic range of roughly 7 dB. The V 1dB is the voltage for 1 dB gain compression, where 0dBV corresponds to +10 dBm in 50. The feedforward technique, [25], the classical linearization method applied in amplifiers, has also been adapted for mixer application by combining two signal paths with proper phase to cancel distortion.

Figure 7.37

Composite gm compared to a single differential pair at the same total current. After [26].

7.9 DESIGN STUDY: WiFi 2.45 GHz GILBERT MIXER For this exercise the mixer is assumed to operate from 2.4 GHz to 2.5 GHz with an IF of 100 MHz. The device selected for this example is the HP AT305, which has fT = 10 GHz at 1 mA, whose parameters are in Appendix 7B. 7.9.1 Selecting Bias and LO Power Let's start with a singly balanced circuit that is half the full Gilbert cell circuit to determine bias voltage levels. The supply voltage is split, as shown in Figure 7.38,

BJT Mixer Theory

491

across current source Q4; RF amplifier, Q3; switching devices, Q1, Q2; and output load RL. The voltage values indicated are typical for a 5V supply voltage and current bias of IE = 2 mA, allowing a voltage swing of 1.5 Vpp at the LO frequency and nearly the same IF and RF voltage swings. A base peak LO voltage of 5VT = 125 mV at each device is sufficient to switch the current on-off. That corresponds to a power of –10 dBm into 50 Ohm. In order to improve square wave shape of the switching waveform, the drive level was increased to 250 mV. Device Q3, Q4 carries twice the current of either Q1 or Q2 and should be designed with twice the emitter area. 5.0 V

+VCC RL

RL DC Voltages

3.5 V

I1

I2

2.7 V

Q1

Q2

2.0 V

IE

1.1 V

Q3 1V .8 V

Figure 7.38

Q4

Selection of DC voltages in a differential configuration.

7.9.1.1 Impedance Considerations The equivalent impedance at the fundamental frequency, in differential mode can be obtained from a single ended grounded emitter transistor. The collector was also assumed grounded at the LO frequency since in doubly balanced operation the collectors are virtually shorted. Transistor Q4 was bypassed by a capacitor connected between collector and ground. The impedance was determined by applying three different LO voltages to the base from 100 MHz to 10 GHz and reading the resulting current at the fundamental frequency. The results are displayed in Figure 7.39(a) as follows: (1) small signal; (2) VLO = 0.125 V; (3) VLO = 0.25V in the direction of the arrow.

492

Microwave Mixer Technology and Applications

Figure 7.39

M1

M

SZin1 SZin

SZin2 SZin1 SZin SZin3

The impedance indicated by marker M1 in the figure corresponds to 2.4 M GHz. The differential impedance is doubled and indicated by marker M. This indep(M)=0 SZin3=0.747 / -11.111 impedance allows high impedance assumption at the LO port so that impedance impedance = Z0 * (4.800 - j3.120) was not conjugated matched. In this M3 case a low impedance voltage source is LOfreq=2.400E9 M1 / -4.591 sufficient common mode impedance requires the LOfreq= 2.400E9 to drive the switches. TheSZin1=0.965 impedance = Z0 * (9.292 - j20.804) SZin2=0.560 / -23.444 calculation of- Q collector impedance, to serve as an emitter impedance of Q 1. impedance = 119.953 j77.811 3

M3

LOfreq (100000000.000 to 10000000000.000) a) Differential b) Common mode Large signal base impedance for common and differential mode.

(0.000 to 0.000) LOfreq (100000000.000 to 10000000000.000)

SZin

The net common mode impedance provided by Figure 7.39(b) corresponds to Q1, Q2 in parallel and with the Q3 device connected to the emitter. M LOfreq= 2.400E9 the impedance trace is outside the Smith chart The small signal simulation shows SZin=0.448 / -81.344 indicating unstable circuit.impedance = Z0 * (0.750 - j0.831)

M

LOfreq (100000000.000 to 10000000000.000)

Figure 7.40

Small signal RF port impedance.

A feedback resistor in series with a 1 pF decoupling capacitor was placed between the collector and base of Q1 and also of Q2, which moved the trace inside the chart as indicated by the black line with marker M3 at 2.4 GHz. The RF device is twice the size of switch devices and carries twice the current. The small signal impedance at the RF port from 100 MHz to 10 GHz is indicated in Figure 7.40.

BJT Mixer Theory

493

The marker is at 2.4 GHz, where the RF impedance is equal to ZRF = 37 – j40 Ω, which can be matched by simple series L parallel C elements. 7.9.1.2 Collector Waveforms

5

5

4

4

3

3

ts(Vc1), V ts(Vc), V

ts(Vc1), V ts(Vc), V

An ideal transformer was employed to drive the differential mixer. The collector waveforms in Figure 7.41(a) have a square wave shape when driven by a sinusoidal generator at 1 GHz. The effect of frequency is illustrated in Figure 7.41(b) where the frequency is at 2.4 GHz. The plots are for device Q1 and a similar waveform phase shifted by 180 is obtained for Q2. The required LO voltage was assumed to be 250 mV for a single device. Since there are two devices in differential mode, the drive generator voltage was doubled to 500 mV. This waveform distorts above 1 GHz, becoming a half wave sinusoid at 2.5 GHz and a distorted sinusoidal a 5 GHz due to effect of device capacitances.

2

2

1

1

0

0 0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

time, nsec

Figure 7.41

1.8

2.0

0

100

200

300

400

500

600

700

800

900

time, psec

a) fLO = 1 GHz b) fLO= 2.4 GHz Collector waveforms with base driven by a sinusoidal signal.

7.9.1.3 Doubly Balanced Circuit Before building the Gilbert cell it is advisable to perform a large signal analysis on the singly balanced differential mixer, with LO signal only, to make sure the LO voltage at the collector swings within the expected voltage levels. The Gilbert cell in Figure 7.42 comprises devices Q1, Q2, Q5, Q6 and the RF amplifier the devices Q3, Q7. The linearization resistor R was set to 100 Ohms. The output signals are buffered by devices Q9, Q10 and are converted from balanced to single ended by an active 180 combiner Q14, Q15. Bias for the active combiner is not shown in the figure. The LO and RF signals are coupled by means of transformers to balance the unbalanced input signals. RF transformers using wire and ferrite material operating in excess of 3 GHz are commercially available. And if higher operating frequency is desired, a transmission line balun can be used, either surface mounted or printed directly onto the circuit board. The RF amplifier in the simulation was matched with simple series L parallel C network not shown in the figure. An

494

Microwave Mixer Technology and Applications

external low pass frequency filter was added at the IF output to remove high residual frequency components, also not shown.

+VCC Rb

RL

RL

T2

Q1 Q2

VLO

Rb

Rc

Q9

Q10

Q5 Q6

C +VCC

Q3

VRF

Q7

Q14

C

Rb 3 Q8

Q4

T1 Figure 7.42

R

Q15

Q12

Q11

VIF

Q 13

Circuit with external transformers.

The conversion gain from 2.2 to 2.8 GHz with an IF of 100 MHz is in Figure 7.43(a), showing 23 dB within the desired band of 2.4 to 2.5 GHz. The LO frequency is at the low side and drive power is + 3 dBm. The figure on the right shows an IF output voltage of 400 mV after filtering, corresponding to + 2 dBm. 25

600

24

m7

23

m8

400

22

20 19

200

m7 LOfreq=2.400E9 gain2=22.957

m8 LOfreq=2.500E9 gain2=22.992

18 17

ts(Vif), mV

gain2

21

0 -200 -400

16 15 2.2E9

-600 2.3E9

2.4E9

2.5E9

LOfreq

2.6E9

2.7E9

2.8E9

0

2

4

6

8

10

12

14

16

18

20

time, nsec

(a) Conversion gain (b) IF voltage waverform. Figure 7.43 Gilbert mixer performance in terms of conversion gain and IF voltage waveform.

These results do not include inductor parasitic losses in Silicon technology, where Q factor can approach unity. An additional improvement to this circuit would be to replace the resistive bias network with an active circuit containing a bandgap stable reference, such as the one shown in Figure 7.44. The bandgap comprises transistors Q3, Q4 and resistors R4, R5. The other resistors are simple voltage dividers that are buffered by transistors Q 1, Q2, Q3 and useful to bias the base of Gilbert cell devices.

BJT Mixer Theory

495

+VCC

R1 Q1 Vb R2

1

Q2 Vb

R3

2

Q3 Q4 R4 Figure 7.44

Vb3

R 5

GND

Gilbert cell biasing circuit.

7.10 DIFFERENTIAL TRIPLE LEVEL The work by Gilbert on the current multiplier was extended to a triple multiplier by Choma, [27], making the output voltage the product of three input voltages, Figure 7.45. His proposal was to stack one more cross coupled cell, resulting in an additional level of differential amplifier. A later patent [28], was applied to the use of the triple level concept for a stacked mixer. The proposal is to use the second level as a first mixer similar to conventional Gilbert cell. The output of the first mixer is applied to an additional mixer quad constituting a dual conversion system. To gain more insight into this type of structure, let us assume a fundamental LO and RF are applied, respectively, to the first (lowest) and second (middle) levels, represented by simplified (7.127) to (7.130), which are related to the currents in the schematic of Figure 7.45. As for the device modulation, only the first Fourier coefficient, gm1 = 2/, is considered.

  V I C1   I T  RF cos(RF t ) g m1 cos(LOt ) RE     V I C 3   I T  RF cos(RF t )  g m1 cos(LOt ) RE     V I C 2   I T  RF cos(RF t )  g m1 cos(LOt ) RE  

(7.127)

(7.128)

(7.129)

496

Microwave Mixer Technology and Applications

  V I C 4   I T  RF cos RF t g m1 cos LOt  RE  

RL

(7.130)

+VCC

RL

Vo=VXVYVZ

IV

IW IC5 IC6

Q5

Q7

Q6

IC7

IC8

Q8

VZ IC1

Q1

IC2

Q2 Q 3

IC3 I C4

Q4

VY Q9

IT-IX/2

IT+IX/2

Q10

VX RE

RE 2IT -VEE

Figure 7.45

Triple multiplier topology. After [27].

Adding the first two currents together, it is found the fundamental LO is cancelled and conversion to the first IF is available at the collector of devices Q1,Q3. The same is applied to the last two currents, and the IF is available at the collectors of Q2,Q4.

I C1  I C 3  2 g m1

VRF V cos LOt cos  RF t  g m1 RF cos LO   RF t (7.131) RE RE

BJT Mixer Theory

I C 2  I C 4  2 g m1

497

VRF V cos  LO t cos  RF t   g m1 RF cos LO   RF t (7.132) RE RE

On the third level, the signal is mixed at the left with a second LO, LOI with reference phase equal to zero. The modulation of the top quad is assumed to generate a similar conversion gain, gm2 = 2/.

 1 1 V I C 5  ( I C1  I C 3 )   g m1 RF cosLO  RF t  g m 2 cos LOI t (7.133) 2 2 RE   V 1 1 I C 7  ( I C 2  I C 4 )    g m1 RF cos LO   RF t g m 2 cos LOI t    (7.134) 2 2 RE  Adding both IC5 and IC7, one obtains IV in (7.135) and by similar development, IW, that is equal to the sum of IC6 and IC8, (7.136). Both currents develop an IF voltage on load R L, which operate differentially. Notice that if we make ωLOI = (ωLO – ωRF) then the resulting frequency is directly at base band.

IV 

1 V g m 2 g m1 RF cos LOI t cosLO  RF t 2 RE

(7.135)

1 V IW   g m 2 g m1 RF cos LOI t cosLO  RF t 2 RE

(7.136)

While there is not much to gain by using this topology directly as two stacked mixers, the patent reported that by adding one more quad at the third level in parallel with the previous quad, it is possible to generate an I, Q mixer, provided the top quads are fed with quadrature LO. The concept is illustrated in the block diagram of Figure 7.46 illustrating an AGC to control the gain level of an RF amplifier. The circuit schematic in Figure 7.47 shows similar equations are developed for the quad on the right. Therefore, currents IC 9 and IC11 are in (7.137) and (7.138).

 V 1 1 I C 9  ( I C1  I C 3 )   g m1 RF cos LO   RF t  g m 2 sin  LOQ t (7.137) 2 2 RE   V 1 1 I C11  ( I C 2  I C 4 )    g m1 RF cos LO   RF t  g m 2 sin  LOQ t    (7.138) 2 2 RE 





498

Figure 7.46

Microwave Mixer Technology and Applications

Block diagram of a dual conversion system.

RL

RL IV

IW

IC5 IC6

+VCC

RL

RL

IX IC9 IC10

IC7 IC8

IY IC11 IC12 127

LOI - 

LOQ- +90 IC1 IC2

IC3 IC4

LO_1 RFin

IT-IX/2

RE

IT+IX/2

RE 2IT

-VEE Figure 7.47

Schematic of inntegrated stacked mixer.

BJT Mixer Theory

499

The currents IX, IY from (7.139), (7.140) have same magnitude as IV, IW but are in quadrature. The converted signal is again at baseband if the second LO is given by ωLOI = (ωLO – ωRF).

IX 

1 V g m 2 g m1 RF sin LOQt cosLO  RF t 2 RE

1 V IY   g m 2 g m1 RF sin LOQt cosLO  RF t 2 RE

(7.139)

(7.140)

In the original patent, this circuit was developed for a RF frequency located at 930 MHz and the LO is located at 744 MHz. The second LO is applied at 186 MHz, the same frequency as first IF, so the output of the second conversion is at baseband. Notice the second LO voltage requires splitting the phase into 4 different phases: 0º, 90º, 180º, 270º. The inventor suggests an addition application where the LO frequency of left quad is different than the frequency of the right quad resulting in two different IFs for the same RF input. An alternative to obtain the differential quadrature signals for LO 2 is with the use of a polyphase filter such as the one indicated in Figure 7.48. The design condition is RC = 1, therefore, if R = 100  the capacitor at 186 MHz is equal to 0.855 pF. Harmonic balance simulation of the circuit shows the input signal amplitude is divided by two at each port, and the relative output phase angles are as shown in the figure. R V2 = V0

C R V1

V3 = V- 90

C R

V4 = V180

C R C

Figure 7.48

LO differential quadrature generator.

V5 = V90

500

Microwave Mixer Technology and Applications

Those are values calculated for an open circuit, which is a close approximation if the load is equal to 1000 . Lower load resistor values start degrading the phase relationship, so some kind of buffering is required. 7.11 DOUBLY BALANCED SUBHARMONIC The need for subharmonic mixers in direct conversion receivers resulted in several innovations in circuit techniques that were used in conjunction with the quad cell. Two main approaches became standard: (a) QLT – quadrature on top, where the quad is transformed to operate subharmonically; (b) DRT – Differential on top, where the LO frequency is first frequency doubled and then fed to a conventional quad. 7.11.1 QLT – Quadrature LO on Top In this topology, [29], each of the four LO switches in the quad are replaced by a pair of transistors that are parallel connected at emitter and collector, with the bases driven by an LO voltage in counter phase to the other. The paralleled transistors generate voltage at the second harmonic and suppress the fundamental frequency components, resulting in the mixing with one half the LO frequency. The second harmonic is generated by the application of a fundamental sinusoidal signal at the base of switching devices. RL

RL

IF+ IF-

LO+

Q3 Q4

LO-

LO90+

RF+

Q7 Q8

Q5 Q6 LO-

Q9 Q10

LO90+

Q1

Q2

RE Figure 7.49

LO-

QLT subharmonic topology. Ater [29].

RF-

LO+

BJT Mixer Theory

501

It has been shown, [30], that a square wave drive is not very efficient as a subharmonic mixer. The schematic in the figure shows the two pairs are driven 90° apart in phase, with the phase difference becoming 180 at the second harmonic. The current provided by the RF devices Q1 and Q3 is described by (7.141). The DC term was dropped since it will be suppressed by the differential output volage.

I Q1  I Q 2 

VRF (cos  RF t  cos 2 RF t ) 2 RE

(7.141)

The current provided by device Q3 is a half sinusoid and is multiplied by the current from the RF device. The same is true for device Q 4 with LO phase shifted by 180º. The current subscripts take the number from the devices in Figure 7.48.

 V I 3  I Q1 1  tanh LO  2VT   V I 4  I Q1 1  tanh LO  2VT 

    I Q1 1  g m1 cos( LO t )  g m 2 cos(2 LO t ) (7.142)       I Q1 1  g m1 cos LO   t  g m 2 cos(2 LO  2 )t    (7.143)

Equation (7.144) shows the combined output from devices Q3,Q4 cancels the mixing products with fundamental and odd LO harmonics, while those with even LO harmonics add constructively. The analysis is extended to combine the currents from the pairs Q7,Q8 and Q3,Q4, since they are connected together.

V  I 34  2(1  g m 2 cos 2 LO t )  RF cos  RF t  cos 2 RF t   2 RE 

(7.144)

V  I 78  2(1  g m 2 cos 2 LO t )  RF cos  RF t  cos 2 RF t   2 RE 

(7.145)

The result of adding I34 and I78 is the cancellation of mixing products with the RF second harmonic, leaving the desired product with second LO harmonic and fundamental RF, as shown in (7.146). The fundamental RF component is doubled at the output terminal but it is suppressed when taking the output differentially.

502

Microwave Mixer Technology and Applications

I IF   2 g m 2

VRF cos 2 LO t cos  RF t RE

(7.146)

The factor gm2 represents the second harmonic trans-conductance coefficient for a half sinusoidal waveform, which is equal to 2/(3). The IF voltage considering downconverter application is provided by (7.147) and the voltage conversion gain by (7.148).

VRF RL cos2 LO   RF t  RE 2 RL  3 RE

VIF  gm2

(7.147)

GCV

(7.148)

It is important to note that in this configuration the LO is required to provide four phase angles: 0º, 90º, 180º and 270º. Such an LO can be generated with a polyphase filter, using hybrid couplers, or from active circuits such as the one from next subsection. The alternative reported in this reference uses a ring oscillator that provides two balanced outputs in quadrature. The down-converter architecture using this subharmonic mixer in combination with ring oscillator is shown in Figure 7.50.

Figure 7.50

Architecture of a subharmonic down-converter mixer. From [29].

7.11.2 DRT – Differential RF on Top This proposed topology, [31], reverses the traditional RF and LO ports, used in fundamental mixers, observed in the schematic contained in Figure 7.51. The switching devices are at the bottom and are replaced by a pair of paralleled devices. The applied LO is frequency doubled and then applied to the top quad for mixing with the RF. The RF is applied to the top quad and the converted signal is collected differentially from the load resistor terminals.

BJT Mixer Theory

503

+VCC R

R

Vout

1

2

(+)VRF/2

(+)VRF/2

(-)VRF/2 I1

I2 VLO/2270

VLO/2180 VLO/290

VLO/290 IE

Figure 7.51

DRT subharmonic topology. After [31].

In conventional mixers, the current I1 and I2 is 180º out of phase, which is obtained here by driving the second doubler at 90º offset compared to the first pair. The resulting second harmonic will be at the desired 180º phase. The generation of LO signals was obtained by first balancing a single ended signal then they are processed by a simple polyphase filter represented in Figure 7.52, The filter in this case is a simple high pass, low pass network applied to each side of the balanced signal to generated the requires four phases in quadrature. The required phase differences generated by the polyphase filter are maintained over a wider frequency range than the amplitude variation. Fortunately, conversion gain is not much affected as long as it is larger than the minimum required to switch the base emitter voltage. The differential amplifiers are used to generate the balanced signal and as buffer amplifiers after the filter to compensate for losses. In this particular design described in the reference a 0.6 nH inductance is inserted in series with emitter of RF quad to improve matching with minimum effect on noise figure.

504

Microwave Mixer Technology and Applications

VLO0 VLO180

VLO VLO90 Polyphase filter

Figure 7.52

VLO270

Quadri-phase LO generation. After [31].

7.12 SUBHARMONIC TRIPLE LEVEL The triple level SHM, [26], is similar to the stacked mixer described earlier, with a single LO frequency applied to two levels in quadrature with each other. Instead of a dual-conversion system, the authors decided to call this a successive mixer system. Each of the dual level mixers operate at a frequency half that of a fundamental mixer. The schematic of the mixer is in Figure 7.53, and the current equations for half RF amplifier, IC, and half of first level mixer, I A, IB are defined by (7.150), (7.151). The current subscripts are defined as defined in the figure.

I C  I DC 

VRF cos RF t  RE

(7.149)

  V I A  g m1 cos LO t  g m 2 cos 2LO t  I DC  RF cos RF t  RE  

(7.150)

  V I B   g m1 cos  LO t  g m 2 cos 2 LO t  I DC  RF cos  RF t  RE  

(7.151)

BJT Mixer Theory

505

The sum of currents flowing into Q1 and Q3 is given by (7.152) that shows cancellation of the fundamental LO frequency and constructive addition of the fundamental mixing between LO and RF. The LO second harmonic is removed if the LO drive waveform is square wave, in which case g m2 = 0; it is also eliminated at the differential output by cancellation. Similarly, the sum of current flowing into Q2, Q4 is given by (7.153).

VRF cos LO t cos RF t RE V I E  I F  2 I DC g m 2 cos 2LO t  2 g m1 RF cos LO t cos RF t RE I A  I B  2 I DC g m 2 cos 2LO t  2 g m1

RL

IA’ IF’

Q5

Q7

Q6

Q8

IB'’ IE’

VLOQ IA

Q1

IF

Q2 Q3

IB

Q4

IE

VLOI Q9

IC

ID

RE

RE

VRF

2IDC -VEE Figure 7.53

Subharmonic triple level mixer. After [26].

Q10

(7.153)

+VCC

RL

Vo=VXVYVZ

(7.152)

506

Microwave Mixer Technology and Applications

The LO feeding the top quad is in quadrature with respect to the first quad, so the signal provided by the first mixing is multiplied by an LO voltage whose phase is shifted by /2. The resulting current in the devices Q5, Q7 is the current IAB multiplied by the LO switching voltage, g m1sin(LOt), depicted in (7.154). The LO voltage for the other half, Q6, Q8, is shifted in phase by 270. The resulting current for devices Q6, Q8 is in (7.155). In the expressions I AB = IA + IB and IEF = IE + IF and currents with (') corresponds to current on the third level where the second harmonic of LO is generated.

I AB g m1 sin( LO t )  2 g m1

2

VRF cos( LO t ) cos( RF t ) sin( LO t ) RE

VRF (7.154) cos(2 LO t ) cos( RF t ) RE 2 V I EF g m1 sin( LO t   )  2 g m1 RF cos( LO t ) cos( RF t ) sin( LO t   ) RE 2 V (7.155) I B '   g m1 RF cos(2 LO t ) cos( RF t ) RE I A '  g m1

2

Adding the currents from (7.154), (7.155) the positive side of the converted IF current is given by (7.156). The negative side is developed similarly for the currents IE', IF ' and is given in (7.157).

VRF cos 2LO t cos RF t RE 2 V I IF   2 g m1 RF cos 2LO t cos RF t RE I IF   2 g m1

2

(7.156) (7.157)

The switching for the devices in second and third level is clearly described by waveforms in Figure 7.54. Each mixer is switched by a square wave shifted 90 º in phase relative to the other switch. The product of the two f LO signals result in an equivalent 2fLO signal. Figure 7.55 illustrates the application of two subharmonic mixers in a direct conversion receiver to convert a differential RF input signal to differential baseband, obtained from reference [26]. The differential quadrature signals have a 45 phase difference between each port at the fundamental frequency. Generation of the second harmonic is accomplished by multiplying the frequency and phase by two, creating the differential quadrature at the second harmonic. The poly phase filter of Figure 7.55 can be used for this purpose by making R = 100  and

BJT Mixer Theory

507

C = 0.939 pF. The topology is able to provide 0, -45, 180 and 135 with the same voltage magnitude at each port.

S1(t) t S2(t) t S(t)=S1(t)S2(t) t Figure 7.54

Equivalent switching waveform.

Figure 7.55

Subharmonic triple level mixer. From [26].

A parameter of importance in direct receivers, which is described in chapter II, is the second order intercept (IP 2). The two-tone IP2 measures rejection of intermodulation (IM) appearing at DC and baseband, and its frequency equals the difference in frequency between two closely spaced signals at the RF input.

508

Microwave Mixer Technology and Applications

Normally one signal is the desired one, and the other is an interferer. Ideally, the second order intercept point of the mixer is very high. The generation of IM levels is reduced by reducing nonlinearity, and further reduced in balanced circuits by cancellation that requires excellent circuit symmetry. Unfortunately, imperfections in lithography and asymmetry in circuit layout cause IP2 values that can fall short of the system requirements. An expression for sensitivity of input IP2 from the same reference [26], is given in (7.158). It is derived in terms of voltage referenced at the input, and differences in load resistors R L, bias current, IQ, and duty cycle for each switch. The effect of non zero output conductance is assumed to be the inverse of Early voltage, V A, a transistor parameter defined in Chapter 2.

IIP2  

1  1 RL d 2(1  2d ) I Q 2   d VT IQ V A RL

  

(7.158)

A typical example from the reference assumes a mismatch of all components of 1%, an Early voltage of 199 V, and a duty cycle of 0.49. This results in IIP2 equal to 38 dBm in a 50 ohm system, which is not very high.

7.13 SUMMARY The basic principles of active BJT mixers were described. Similarities with diode mixers were noted due to the similar exponential relation between current and voltage. The mixing process was discussed in detail sufficient for the reader to explore specific topics further using the references cited. The most common topologies were addressed, including the single ended mixer with base LO injection, the cascode mixer, and the differential singly- and doubly-balanced mixers. All these circuits can be built as either discrete or integrated circuits. Subharmonic mixers were also discussed and shown to have similar advantages to diode APDP SHMs, which respectively refer to anti-parallel diode pair and subharmonic mixers.

BJT Mixer Theory

509

REFERENCES [1] C. L. Searly, A, R. Boothroyd, E. Angelo, Jr., P. E. Gray, and D. O. Pederson, Elementary Circuit Properties of Transistors, John Wiley, 1964, pp. 81. [2] K. K. Clarke and D. T. Hess, Communication Circuits: Analysis and Design, Addison Wesley Publishing Company, 1971, pp. 637. [3] K. K. Clarke and D. T. Hes, Communication Circuits: Analysis and Design, Addison Wesley Publishing Company, 1971, pp. 109. [4] B. A. Xavier and C. A. Aitchinson, “Simulation and Modelling of a Hetero junction Bipolar Transistor Mixer,” 1992 IEEE MTT International Microwave Symposium Digest, pp. 333-336. [5] R. A. Pucel, D. Masse, and R. Bera, “Performance of a GaAs MESFET Mixers at X Band,” IEEE Transactions on Microwave Theory and Techniques, Volume MTT 24, No. 6, June 1976, pp. 351-360. [6] B. A. Xavier and C. A. Aitchinson, “The Measured and Predicted Noise Figure of a GaAs Heterojunction Bipolar Transistor Mixer,” 1997 RFIC Symposium, pp. 135-138. [7] Guofu Niu, “Noise in SiGe HBT RF Technology" Physics, Modeling and Circuit Implications,” Proceedings of the IEEE, Volume 93, No. 9, September 2005, pp. 1583-1597. [8] N. Suematsu, M. Ono, S. Kubo, H. Sato, Y. Iyama, and O. Ishida, “L-Band Internally Matched Front-End Si-MMIC,” 26th European Microwave Conference, September 1996, pp. 37-40. [9] K. K. Clarke and D. T. Hess, Communication Circuits: Analysis and Design, Addison Wesley Publishing Company, 1971, pp. 117. [10] K. K. Clarke and D. T. Hess, Communication Circuits: Analysis and Design, Addison Wesley Publishing Company, 1971, pp. 105. [11] Willy Sansen, “Distortion in Elementary Transistor Circuits,” IEEE Tranactions on Circuits and Systems-II: Analog and Digital Signal Processing, Volume 46, No. 3 March 1999, pp. 315-325. [12] Howard E. Abraham and Robert G. Meyer, “Transistor Design for Low Distortion at High Frequencies,” IEEE Transactions on Electron Devices, Volume ED 23, No. 12, December 1976, pp. 1290-1297 [13] Keng Leong Fong and Robert G. Meyer, “Monolithic RF Active Mixer Design,” IEEE Transactions on Circuits and Systems II Analog and Digital Signal Processing, Volume 46, No. 3, March 1999, pp. 231-239. [14] E. Thompson, “Integrated Harmonic Mixer Circuit Including an Emitter Coupled Differential Amplifier,” US Patent 3,491,301, issued January 20, 1970. [15] H. E. Jones, “Dual Output Synchronous Detector Utilizing Transistorized Differential Amplifiers,” US Patent 3,241,078, issued March 15, 1966. [16] R. R. Sinusas, “Double-Balanced Modulator Circuit Readily Adaptable to Integrated Circuit Fabrication,” US Patent 3,550,040, issued December 22, 1970. [17] Barrie Gilbert, “A new Wideband Amplifier Technique,” US Patent 3,689,752, issued September 5, 1972.

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Microwave Mixer Technology and Applications

[18] Barrie Gilbert, “A Precise Four-Quadrant Multiplier with Sub-nanosecond Response,” IEEE Journal of Solid State Circuits, Volume 3, No. 4, December 1968, pp. 365-373. [19] Barry Gilbert and Rupert Baines, “Fundamentals of Active Mixers,” Applied Microwave & Wireless, Winter 1995, pp. 10-27. [20] Robert G. Meyer, “Inter-modulation in High-Frequency Bipolar Transistor Integrated Circuit Mixers,” IEEE Journal of Solid State Circuits, Volume 21, No. 4, August 1986, pp. 534-537. [21] P. R. Gray, P. J. Hurst, S. H. Lewis, and R. G. Meyer, Analysis and Design of Analog Integrated Circuits, John Wiley, 2001, pp. 218. [22] K. K. Clarke and D. T. Hess, Communication Circuits: Analysis and Design, Addison Wesley Publishing Company, 1971, pp. 371. [23] Barrie Gilbert, “The MICROMIXER: A Highly Linear Variant of the Gilbert Mixer Using a Bisymmetric Class-AB Input Stage,” IEEE Journal of Solid State Circuits, Volume 32, No. 9, September 1997, pp. 1412-1423. [24] Barrie Gilbert, “The Multi-tanh Principle: A Tutorial Overview,” IEEE Journal of Solid State Circuits, Volume 33, January 1998, pp. 2-17. [25] ST Lim and John R. Long, “A Low Voltage Broadband FeedforwardLinearized BJT Mixer,” IEEE Journal of Solid State Circuits, Volume 41, No. 9, September 2006, pp. 2177-2187. [26] L. Sheng, J. C. Jensen and L. E. Larson , “A Wide Bandwidth Si/SiGe HBT Direct Conversion Sub-Harmonic Mixer/Downconverter,” IEEE Journal of Solid State Circuits, Volume 35, No. 9 September 2000, pp. 1329-1337. [27] J. Choma, Jr., “A Three Level Broad Banded Monolithic Analog Multiplier,” IEEE Journal of Solid State Circuits, Volume 16, No. 4, August 1981, pp. 392-399. [28] W. J. Grandfield, “Stacked Double Balanced Mixer Circuit,” US Patent 5,448,772, issued September 5, 1995. [29] R. M. Kodkani and L. E. Larson, “An Integrated 50 GHz SiGe SubHarmonic Mixer/Downconverter with a Quadrature Ring VCO,” 2007 Topical Meeting on Silicon Monolithic Integrated Circuits in RF Systems, January 2007, pp. 223 -226. [30] M. Goldfarb, E. Balboni, and J. Cavey, “Even Harmonic Double -Balanced Active Mixer for Use in Direct Conversion Receivers,” IEEE Journal of Solid State Circuits, Volume 38, No. 9, October 2003, pp. 1762-1766. [31] K. Nimmagadda and G. M. Rebeiz, “A 1.9 GHz Double-Balanced Subharmonic Mixer for Direct Conversion Receivers,” 2001 IEEE Radio Frequency Integrated Circuits Symposium, Phoenix, 2001, pp. 253-256.

BJT Mixer Theory

511

APPENDIX 7A 2SC5006 GUMMEL – POON PARAMETERS (Device made by NEC)

Parameter

Value 6.16E-16

Parameter

Value

Parameter

IRB 7.56E-04 Ne 114 RC 5 VTF 0.98 CJE 2.80E-12 ITF 50 VJE 0.954 PTF 1.50E-03 MJE 4.08E-01 TR ISE 3.82E-13 CJC 1.00E-12 EG BR 14.4 VJC 6.67E-01 XTB NR 0.991 MJC 0.408 XTI VAR 2.4 XCJC 0.8 KF IKR 0.32 CJS 0 AF ISC 0 VJS 0.75 fT NC 1 MJS 0 nf 1GHz RE 0.4 FC 0.5 nf 2GHz RB 4.37 TF 2.00E-11 Gain 1 GHz RBM 2.33 XTF 6 Gain 2 GHz BIAS: VCE = 2.5 to 5.5 V; IC = 1 to 8 mA PT = 150 mW IS BF NF VAF IKF

Value

2.19 0.668 0.7 40 0 1.11 0 3 0 1 4.50E+09

1.4 dB 2.2 dB 12.5 dB 6.5 dB

512

Microwave Mixer Technology and Applications

APPENDIX 7B AT305 SPICE PARAMETERS (Device originally made by Hewlett Packard, now by AVAGO)

AT305 Parameter

Value

Parameter

IS BF NF VAF IKF

7.80E-17

ISE

2.40E-13

BR NR VAR IKR ISC NC RE RB RBM

14.4 0.991

IRB RC CJE VJE MJE CJC VJC MJC XCJC CJS VJS MJS FC TF XTF

100 1.03 20 6.30E-03

2.44 38.49

Value

61.57 1.10E-13

1.01 6.00E-01

5.10E-14 7.60E-01

0.52 19 7.00E-14 0.75 0.75 0.5

Parameter

Value

Ne VTF ITF PTF TR EG XTB XTI KF AF fT nf 1GHz

2.5 6

1.20E-11 Gain 1 GHz

4

1.40E-02

25

1.818

1.00E+10

1.1 dB 14 dB

Chapter 8 Bipolar Junction Transistor Applications In the 1960s Silicon bipolar transistors started replacing vacuum tubes in various applications, in particular AM, FM and TV broadcast receivers. By the end of the decade television receivers had transitioned from tube to solid state technology, with the exception of the high voltage generation that remained dependent on tubes. Many innovations in circuit topologies occurred, to improve selectivity, sensitivity, power consumption and cost. An early innovation included combining a few transistors and capacitors onto a single monolithic circuit, surrounded by discrete components to complete the amplifier, oscillator and mixer functions. One of the critical challenges was to provide on chip inductors or alternate circuit techniques to replace or emulate such components in silicon IC technology. In the early 1990s bipolar technology started moving into areas that had been dominated by III-V compound FETs for many years. Most notably was the advancement in Hetero-junction devices, in particular the Aluminum, InGaP HBT and SiGe HBT. These extended the performance advantages of bipolar technology into the microwave and millimeter-wave frequencies previously dominated by FETs. In the case of SiGe devices the transition allowed easy integration with MOS technology, allowing the inclusion of digital and analog functions, and the subsequent evolution of subsystems onto a single chip. This chapter provides a summary of the most important work in the area of Si-BJT applied to mixers, selected from various patents from the early 1960s up to recently published work. 8.1 SINGLE ENDED Most of the single ended BJT mixer applications were developed for TV and AMFM radio receivers. The initial applications largely consisted of replacing crystal diodes and tubes following classical mixer topologies. 8.1.1 Regenerative Feedback In 1963 Barditch [1] et al. published a patent for a monolithic mixer circuit with regenerative IF feedback that obviated the need for the inductor. The schematic is 513

514

Microwave Mixer Technology and Applications

depicted in Figure 8.1, comprising a grounded emitter transistor driving an emitter follower with regenerative feedback at the IF frequency. The large signal local oscillator and small signal inputs are both applied to the base of the emitter follower, generating mixing products at the emitter node 27. The RC filter 29 shifts the phase of the IF by 180 degrees and is fed back to the base of the inverter stage. Thus a highly selective regenerative feedback is obtained without the use of a tuning inductor. A few circuit variations are given in the patent, including one with the phase shift network placed between the two transistors, and another with the LO fed at the emitter of the first transistor.

Figure 8.1

Monolithic semiconductor mixer with positive feedback.

8.1.2 Autodyne Approach In the early days of portable transistorized AM radios, a single transistor was used in an “autodyne” or SOM (self oscillating mixer) configuration to generate the LO and provide the mixing function. A problem encountered with the autodyne circuit was detuning and reduced selectivity of the IF tuned circuit, caused by varying reactance in the mixer transistor, which in turn was caused by varying RF signal strength. The mixer of Figure 8.2 was proposed to minimize this problem by using a low impedance transformer, T 56, to couple the IF signal from the mixer to the IF tuned circuit [2]. The Autodyne function is performed by transistor Q 44, with tank circuit composed of capacitors C52, C54, slug tuned transformer with primary and secondary inductances L50, L51, and inductor L56. The RF signal is amplified by transistor Q12 and applied to the base of Q44 mixing with the LO signal. In order to minimize the detuning problem caused by variations in the collector impedance with signal strength, the collector impedance of Q 44 is minimized by making the L56 reactance low and loosely coupled to the tuned IF transformer. The LO frequency tuning is performed by C52.

Bipolar Junction Transistor Applications

Figure 8.2

515

Mixer circuit for autodyne receiver with improved IF coupling.

8.1.3 Cascode Topology The cascode topology was a common feature of many early television mixers using bipolar transistors due to its higher frequency response and higher gain. One such mixer is shown in Figure 8.3, which was introduced in 1972 for use in a television receiver [3]. The LO and RF are coupled by capacitors 20 and 22 into the base of the common emitter transistor, Q24 where mixing is performed. The generated IF current is then amplified by Q26.

Figure 8.3

Cascode mixer circuit with base LO injection.

A low impedance for RF and LO signals at the Q 26 collector is provided by C48, which is also part of IF  type impedance transformer, L44, C50. Since the cascode configuration has high output impedance, the output transformer has high Q, improving rejection of unwanted signals. An IC version of the cascode mixer is in Figure 8.4 also designed for a television receiver in 1976 [4]. The contribution here is the use of an emitter follower at the mixer-IF amplifier,

516

Microwave Mixer Technology and Applications

buffering the mixer from IF load reactance, eliminating the need for high Q, high value inductors.

Figure 8.4

Monolithic cascode mixer with emitter follower circuit at the output for buffering load effects.

8.1.4 Mixer Linearization A simplified block diagram of a TV receiver is given in Figure 8.5, comprising a mixer capable of converting high level signals. The objective of the mixer disclosed in 1989 [5], was to minimize cross modulation between TV channel carriers that otherwise degraded reception. In the mixer circuit of Figure 8.6, the LO and RF signals are applied to the base. The signal voltage is imposed across the base-emitter junction and the Schottky diode. The LO modulates the transistor trans-conductance and diode conductance simultaneously, both generating mixing products. Therefore the diode acts as an extended mixer improving power performance and consequently reducing intermodulation levels in the desired IF output. At the collector the L224, C238 circuit is tuned to the IF frequency rejecting high level LO and RF signals. The Rstab resistor improves circuit stability. Improved IM rejection is obtained by applying lower voltage to each non linear element while maintaining the same signal current, a feature used in cascaded diode mixers that is known to improve linearity. The authors reported a reduction of 13 dB in IM levels. The contribution to extending the operating level by the diode is controlled by the amount of coupling between the diode and the transistor, and the DC bias level applied to the diode. The frequency coverage of the mixer is sufficient for TV applications that operate over a wide frequency range in the VHF and UHF bands.

Bipolar Junction Transistor Applications

Figure 8.5

RF Amplifier and Filter

Extended Square Law Mixer

Source RF signals

Local Oscillator

517

Signal Processing Unit

IF Filters

Sound and Vision Reproducing unit

Simplified TV receiver block diagram. R222

R232

+VCC C236

L224 R234 Rstab

C238

Output

Input R218

Figure 8.6

+Vd

Diode added to emitter circuit reduces intermodulation.

8.2 PARALLEL COMBINED MIXERS Early in their development, single ended BJT mixers were combined using RF transformers, which were a mature technology having been developed for tube circuits. This approach was applied in many designs and configurations and still finds application today in many circuits. Transformers were later replaced by a topology successfully applied in solid state audio power amplifiers, namely, complimentary pair npn and pnp transistors. 8.2.1 Transformer Coupled Some of the early work done with balanced BJT mixers is described in a patent disclosed in 1959, [6], for mixer topologies with npn and pnp transistors. In the topology of Figure 8.7(a), the devices are connected in common base; and in

518

Microwave Mixer Technology and Applications

8.7(b), they are connected in common emitter. The operating principle of both is the same, but have different input and output impedance levels. Note that device bias is not included in the figure.

17

30 24

Figure 8.7

17

35

30 24

35

(a) (b) Balanced configurations for bipolar mixers.

Assuming up-converter operation, the LO may be injected at terminal 17 and coupled to the emitters in Figure 8.7(a), causing the collector current to be modulated. The devices are driven by the LO in opposing phase through the transformer, but due to the opposite device polarities, pnp and npn, both devices conduct at the same time. The collector voltage waveforms have the same phase but different DC offset levels; the npn-transistor is biased positively and the pnptransistor is biased negatively. After proper DC decoupling, the waveforms are applied to the transformers having output ports 30 and 35. The low frequency RF signal is applied to terminal 24, and coupled in phase to both emitters. The RF signal is therefore combined with the LO and the two mix together by the transistor non-linearity. The up-converted sidebands appear at each collector with opposite phase, as the phase of the L-R sideband is the difference between LO and RF phase angles, while the phase of the L+R sideband is their sum. The voltages from both collectors combine in the output transformers. The LO fundamental and harmonics, and mixing products involving even RF harmonics appear at terminal 30, and mixing products with odd RF harmonics, including the upconverted IF sidebands, appear at terminal 35. A different topology using pnp and npn devices with their collectors tied together is depicted in Figure 8.8. The LO is applied to terminal 67, and couples in phase to the bases of both transistors through transformer T 45. LO harmonics are generated by the transistors, with even ones cancelling at the collector connection due to the reverse polarities of the transistors. The low frequency RF signal is applied at terminal 39. It couples in counter phase to the base of each device through transformer T45, generating RF harmonics with the same polarity since the two transistor polarities are opposite, pnp and npn. The result is the IF sidebands at each collector are in phase and add constructively before being filtered by the output tuned circuit. If the RF and LO inputs are reversed, then the LO fundamental and harmonics, plus mixing product involving odd RF harmonics including the desired IF sidebands are produced. If the RF and LO are both applied at terminal 67, then the RF and LO fundamental and harmonics appear amplified at the output, and the desired IF sidebands are cancelled.

Bipolar Junction Transistor Applications

519

39

T45

67 +V

Figure 8.8

-V

Alternative topology for a balanced mixer with phasing introdued by transformer.

A second work following similar principles, patented in 1969 [7], comprises a push-pull modulator using two npn devices. The input and output transformers shown in Figure 8.9 have center taps to apply DC bias to the circuit. The RF signal is applied in phase to the bases of Q 1 and Q2 through the center tap of transformer Tr1, and cancels at the IF output node 2 by the common mode rejection of transformer Tr2. The LO is applied to terminal 1, and the resulting IF currents appear at the collectors of Q 1 and Q2 in counter phase. The IF adds constructively in the output transformer and is available at terminal 2, along with mixing products with odd LO harmonics; those with even LO harmonics cancel. The main contribution of this patent was the use of feedback resistors R 1 and R2 which play a regulating effect with the application of large signal RF and class AB operation. Q1 R3

R1 2

R4

1 R2

Tr1

3 Figure 8.9

Tr2

Q2

+V

4

Active push-pull modulator.

The increase in the RF signal amplitude generates a DC current due to rectification, which flows in the resistors R1 and R2, decreasing the base-to-emitter voltage since the external DC voltage is constant. The device gain is thus automatically reduced, maintaining the output level.

520

Microwave Mixer Technology and Applications

8.2.2 Differential Pair with Transformer A differential SBM using transformers was proposed in 1972 [8], using four npn transistors, as shown in Figure 8.10, with LO square wave shaping. In this approach the RF signal, E1, is applied to the primary winding of transformer T r1. The signal on the secondary winding is directly connected to the bases of T 1, T2. The LO is applied at terminal T and coupled to the bases of T 3, T4 forming differential pairs with T1, T2, respectively, through resistors R1 and R2. The bias circuits are designed for quiescent operation with each device sharing equal current with its pair when no LO is applied. When the polarity of the applied LO voltage is positive, T3 and T4 are highly conductive and pass the majority of total emitter current, thus switching off devices T 1, T2. As soon as the LO voltage becomes negative, the total emitter current becomes dominated by T 1, T2. Therefore, a low alternating LO voltage is sufficient to rapidly switch T 1, T2 in a square wave manner.

Figure 8.10

Differential pair with transformer for mixer and LO waveform generation.

The IF signals from the transistor pairs have opposite phase, and are combined differentially by transformer Tr2, to become available at terminal A1. Since the devices are switched in-phase, the LO voltage is cancelled at both E 1 and A1 ports. If the output transformer has a broadband frequency response, a push-pull amplifier is formed by T 1, T2 for the RF signal. The mixer provides an

Bipolar Junction Transistor Applications

521

additional A3 port that can be used to provide square wave LO signal to other mixers. 8.2.3 Active Modulator with Ratrace As operating frequencies increase above UHF into the microwave range, the performance of RF transformers degrade in comparison to distributed hybrids due to their semi lumped composition. The rat-race hybrid fabricated using microstrip line was applied to a singly balanced bipolar mixer patented in 1978 [9]. The circuit in Figure 8.11 is a modified version of that patent, originally designed for an FM radio operating near 800 MHz, with IF output at 45 MHz, and instantaneous bandwidth of 20 MHz. The LO signal is applied at the differential port, and the RF at the common mode port. The converted IF signals at the collectors are filtered by L1, C3 and L2, C4 to remove high frequency components and to apply low impedance at the collector. IC1 /4 L1 Q1 C3 IF /4 C5 R1 C2 R3 IO /2 RF R2 C6 /4 /4 C1

R4

LO Figure 8.11

C4

Q2 IC2

L2

R6

R7 +V

C7

R5

Schematic of active mixer employing a ratrace hybrid for RF and LO.

To analyze the frequency components at the output, the collector current is expressed by (8.1) where the transconductance coefficients are from the Taylor expansion at the bias point, where n is the number of coefficients under consideration. 2

3

I c  g m0VBE  g m1Vbe  g m 2Vbe  g m3Vbe  ...  g mnVbe

n

(8.1)

The voltages applied to the transistor bases are given by the sum of RF and LO voltages, where for this ωO is the LO frequency per the reference:

522

Microwave Mixer Technology and Applications

vR (t )  VR cos  R t vO (t )  VO cos O t Vb1  v R (t )  vO (t ) Vb 2  v R (t )  vO (t )

(8.2) (8.3) (8.4) (8.5)

The collector current is obtained by substituting the base voltages into (8.1). The first term in the equation corresponds to the DC as determined by application of bias to the circuit. After LO injection, all even order g m coefficients contribute to the DC term. The second term corresponds to amplification of signals applied to the base that are filtered by the LC elements connected to the collector. The mixing currents of interest are given by (8.6) and (8.7).

I c1  g m 2 v R (t )  vO (t )  g m 4 (v R (t )  vO (t )) 4  ... 2

(8.6)

I c1  g m 2 v R (t )  vO (t )  g m 4 (v R (t )  vO (t )) 4  ... 2

(8.7)

The IF current, I0, in the secondary of the output transformer will be proportional to the difference of the above collector currents.





I 0  I C1  I C 2  g m 2 v R (t )  vO (t )   (v R (t )  vO (t )) 2 



2



g m 4 v R (t )  vO (t )   (v R (t )  vO (t )) 4  ... 4



3  2 2 I 0  2VRVO  g m 2  g m 4 V R  VO 2 

 cos( 

R

  O )t

(8.8)

(8.9)

Mixing products involving even LO harmonics do not appear in the output current IO because they cancel in the subtraction of I C1-IC2. This is beneficial, for example, in suppression of mixing products involving the second LO harmonic. The patent reference is concerned with suppression of an intermodulation (IM) product caused by an undesired input signal, ω R’:

2 R '2O   IF   R  O

R ' 

 R  O 2

(8.10) (8.11)

The undesired ωR’ input signal mixes with the fundamental LO to generate an output at half the desired IF. The mixer also generates the second

Bipolar Junction Transistor Applications

523

harmonic of this “half IF”, which interferes with the desired IF that resulted from the desired RF signal, ωR, mixing with the fundamental LO. An objective of the patent was to suppress this half IF signal. For example, if the undesired input signal is at 857.5 MHz it will mix with the LO at 835 MHz and convert down to 22.5 MHz. The second harmonic of this is at the desired IF at 45 MHz. The desired RF frequency is at 880 MHz. Since mixing products involving even LO harmonics are suppressed by this mixer, the circuit has immunity to this interference. The authors claim the rejection of the half IF spurious to be on the order of 70 dB. 8.2.4 Complimentary Devices The block diagram of Figure 8.12 describes an invention that was proposed for TV receiver applications. The signals from the antenna are diplexed into two outputs, one at UHF (300 to 900 MHz) and the other at VHF (50 to 300 MHz). In VHF operation the oscillator 87 is applied to mixer 88, down-converting the VHF band to a fixed IF. In UHF operation, oscillator 87 is switched off, and oscillator 85 is switched on. The IF signal is applied to the tuner amp and to the mixer/amp block, both operating as high gain IF amplifiers. Therefore, block 88 has to perform both as a mixer and as an IF amplifier, without distorting the AM information contained in the RF carrier. Those requirements drove the development of this invention disclosed in 1978, [10], which makes use of complimentary devices, minimizing the number of transformers required for the mixer operation.

Figure 8.12

Front end block diagram of a TV receiver.

The schematic for this circuit is in Figure 8.13. The circuit utilizes two bipolar devices, one pnp and one npn as mixing elements, with similar DC and RF characteristics. The RF and LO signals are both applied at the base of each device, and since they are complimentary, the LO signal at the collectors are in counter phase and therefore rejected. The IF current at the collector of device Q16 flows

524

Microwave Mixer Technology and Applications

through the primary transformer T 60 and into the collector of device Q20. The series combination of capacitors C48 and C50, are in parallel with the primary winding of the output transformer that is tuned to the IF frequencies, and which presents a low impedance to ground for LO, RF and harmonics. The interesting aspect of this invention is its ability to operate in two modes: in near class AB mode where each device has its transconductance modulated by the large LO drive; and, in deep class B where each device acts as a sampler.

Figure 8.13

Complimentary NPN, PNP, pair mixer.

The operation in transconductance mode is similar to the modulated conductance of a Schottky diode. The operation as a sampler requires further explanation. Consider the plots of Figure 8.14 that contains two DC levels, 69 and 73, respectively, corresponding to the npn and pnp conduction levels. It also contains the LO sinusoidal waveform, 68, and another sinusoid, 70, representing the RF modulation signal. The DC level 73 in reality is higher than 69, but to simplify the explanation this arrangement is more clear. Level 69 represents the threshold of conduction for the npn device, which functions as a switch that is turned on by the tips of LO waveform above level 69. Therefore, samples of waveform 70 are applied to the primary of the transformer at terminal 42. The current pulses are filtered by the tuned circuit to provide IF waveform 72. Level 73 is the conduction threshold for the pnp device. The negative portion of waveform 68 below level 73, makes the pnp device to conduct at alternate half cycles of the LO signal to produce pulses 74, which are filtered to generate waveform 75. The balanced IF signal generated is converted to single ended by the IF transformer. This circuit presents several advantages compared with single ended BJT mixers, one of which is the minimization of driving source distortion. During dynamic drive by the LO signal the impedance is modulated, affecting the driving waveform. In this arrangement, when the LO level is low, one device is turned off and the other conducts.

Bipolar Junction Transistor Applications

Figure 8.14

525

Waveforms for the complimentary pair circuit detailing the sampling process.

Impedance

As the signal increases, the off device starts to conduct, while the other decreases its conductivity. When the LO reaches the zero crossing, both devices are ideally showing the same conductivity. The LO progresses until maximum voltage is applied, reversing the conductance compared to the initial state. The resulting impedance, depicted in Figure 8.15, is essentially constant for the LO and RF drive, minimizing distortion to these sources.

Z of 20 Z of 16 T1

Figure 8.15

T2

T3

time

LO and RF impedance as a function of time.

Another advantage of this circuit is the minimization of nonlinearities at the base-emitter junction of the transistors. Those signals are fed back by the collector to base capacitance, C78 and C80, and are in counter phase to the incoming signal, reducing cross modulations. The requirement to operate both as a mixer or amplifier is realized by simply switching off half of the circuit and using

526

Microwave Mixer Technology and Applications

the remaining circuit as an IF amplifier. Another embodiment of the invention is depicted in Figure 8.16, where the RF is applied to the base of each device and the LO to the emitter. This alternative arrangement offers a reduction of LO leakage at the RF input port.

Figure 8.16

Alternative embodiment of the invention.

8.2.5.1 Complimentary Pair with Transformers/Inductors A recent work which discloses the use of lumped inductors, transformers and complimentary devices in four different arrangements was introduced in 1999, [11]. In the first topology, shown in Figure 8.17, the LO is applied to the base of two devices with emitters tied together.

Figure 8.17

Complimentary devices with output from paralleled collectors.

The RF source is connected at the emitters, and the IF currents at both collectors are in phase. They are combined by capacitor C4, which resonates with the series connection of Z7 and Z8 at IF. At LO and RF frequencies Z7, and Z8 should present a low impedance to ground and high impedance at IF. The

Bipolar Junction Transistor Applications

527

impedances Z1 and Z2 should match the floating LO impedance to the series combination of T1 and T2. The floating generator would require a transformer or another type of floating circuit. The same applies to the floating DC supply. The RF and LO signals at the output terminal are opposed in phase.

Figure 8.18

Complimentary pair with collectors transformer coupled.

The next proposal in Figure 8.18, uses transformers to couple the RF signal to the emitter, and the base circuit is similar to that of the previous circuit. The IF currents are extracted from the collector of each device by means of a transformer. In the example shown in Figure 8.19, the collectors are tied together and the LO signal is still applied to the base of each device.

Figure 8.19

Complimentary pair with emitter transformer coupled.

The LO impedance is matched by impedance Z5 and impedances Z3, Z4 simply decoupled DC power supplies DCS1, DCS2. The RF is coupled to the emitters via a transformer and the IF is extracted from the collector shunted by an

528

Microwave Mixer Technology and Applications

inductor with impedance Z6. The circuit in Figure 8.20 uses the base approach of the previous figure and emitter connection from Figure 8.18.

Figure 8.20

Balanced configuration with collectors tied together.

8.2.2 Doubly Balanced Transformer Coupled The application of bipolar transistors as switches is the object of a couple of patents, one of which is in reference [12]. The collectors and emitters of four transistors are connected together as shown in Figure 8.21 in a “star” configuration. The transistors are connected to the source and load with transformers. Notice that transistors Q3 and Q4 are in series with the load, while Q1 and Q2 are in parallel. If the LO signal is applied to terminal S2, then the positive cycle switches off Q1 and Q2 and switches on Q3 and Q4 , with the reverse occurring when the LO is negative. The LO signal is in counter phase at the primary winding of transformer U3 and thus cancels at the load at the output of U3. The RF signal is applied at terminal S1, and is modulated by the devices switching on/off. The resulting IF sideband signals are in counter phase in the top circuit (Q1 and Q2) relative to those in the bottom circuit (Q3 and Q4), and thus add constructively at transformer U3. If the bandwidth of transformer U3 is wide enough, RF will also add constructively and appear at the output. If the LO carrier is applied to S1, then the top devices (Q1 and Q3) are fed in counter phase with respect to the bottom devices (Q2 and Q4). Therefore, a positive LO cycle at the primary winding of U1 switches Q1 off and switches Q3 on. At the same time, Q2 is switched on and Q4 is switched off. The RF signal is applied to S2, and is cancelled by phasing at the output transformer U 3. The phase of the RF signal appearing at the output transformer U 3 is reversed at the LO rate, by Q3 and Q4 alternately switching on and off, resulting in only mixing products

Bipolar Junction Transistor Applications

529

appearing with odd LO and odd RF harmonics, which include the desired IF sidebands.

Q1

U1

U3 Q3

S1

U2

V + -

S2

Figure 8.21

Q2

S1  S2

Q4

Signal switching with 4 transistors.

Another mixer employing a transformer, [13], is depicted in Figure 8.22. It has an interesting arrangement of four transistors resembling a triply balanced diode mixer. The circuit has input and output transformer T 1, T2, respectively, and two pairs of devices, TR1, TR2 and TR3, TR4. Mixing is achieved by commutating the RF phase by 180 degrees at the LO frequency, as the transistor pairs alternately switch on and off. The four transistors can be thought of as common base amplifiers. Linearity is improved by adding a resistor, R1 to R4, in series with each emitter.

Figure 8.22

Transformer coupled differential mixer.

530

Microwave Mixer Technology and Applications

The LO signal is applied to the bases of the four transistors via transformer T3, causing them to switch on/off as their gain is modulated. Linearity can be improved by increasing the values of resistors R 1, R4; however, this limits the conversion gain of the mixer due to power lost in the resistors. The load impedance could be increased to recover the losses as long as it is not overcome by the non-linear collector to base capacitance. The RF signal is applied to the common base amplifiers TR5, TR6 via transformer T1. Resistors R9 and R10 are used for linearity improvement. A doubly balanced mixer using transformers and operating in passive mode was described in a 1982 patent [14], Figure 8.23, where RF signals are applied at terminal 3, LO at terminal 29 and IF extracted from terminal 20. The emitters and collectors are DC connected to ground, with device conduction controlled by the LO signal applied to the bases. A positive LO polarity causes the collector-emitter paths for transistors 7 and 10 to conduct, and transistors 8 and 11 to be biased off; the reverse occurs for the negative LO cycle. The result is to commutate the phase of the RF signal appearing at the IF output by 180 degrees at the LO frequency. A self bias resistor is also proposed for higher RF level operation, making use of rectified LO current from the base emitter junction to generate a base bias voltage on resistor R30, which is held during each cycle by the capacitor C31. The additional self bias circuit is represented in part b of the figure. A small external bias can also be introduced at this point to improve performance at lower RF operation.

(a) Schematic of passive mixer Figure 8.23

Passive and active mode bipolar ring mixer.

(b) Self bias and external bias

Bipolar Junction Transistor Applications

531

8.3 INTEGRATED CIRCUIT TOPOLOGIES Singly balanced mixers migrated from solely using discrete devices to being realized as integrated circuits, due to the ease of integrating differential topologies, the high level of circuit balance obtained from nearly identical devices, high isolation between ports, and the ability to provide gain with reasonable linearity. In the early stages of IC technology, the die contained just a few elements, which were packaged and inserted in between external components. The level of integration increased over the years and today fully integrated mixers are available. In addition, they have become integrated with other functions on a single die with minimal cost increase. The most common additional function is the Local Oscillator. But the discrete hybrid approach is still applicable today when special features or performance levels cannot be provided by a fully integrated design. 8.3.1 Self Oscillating Mixer A self oscillating mixer, SOM, is depicted in Figure 8.24, [15]. It combines three circuit functions using differential amplifier topology. The oscillator function is performed by transistor Q2, where the signal is fed back from collector to base, introducing an additional 180 phase shift in the voltage, which is a necessary condition for oscillation. The oscillation frequency is determined by the tank circuit, L0, C0 plus the transistor parasitics. The RF amplifier function is performed by Q1, whose input impedance is matched to the external source by a tuned transformer. And the mixing function is performed by Q 3. The RF frequency is selected by tuning L1, C1. Capacitors C0, C1 are varactors that are tuned such that the IF frequency is constant.

+VC C0

In

Out

C3 Q3

Q2

L0

L3

+Vb1 L1

Q1

C1 +Vb2 Figure 8.24

Integrated self oscillating mixer using a diffierential configuration.

532

Microwave Mixer Technology and Applications

5.3

150

5.2

100

5.1

50

Vload, mV

Vc, V

The inventors assumed the LO current generated by the oscillator Q2 along with RF current from Q1 is applied to the emitter of mixer Q3. The tuned output transformer selects the converted IF signal and rejects all high frequency components. The devices comprise a classical differential topology in DC mode but the mixer operates single ended.The reported performance in the FM range (80 to 120 MHz) shows 30dB of conversion gain, input impedance equal to 100 ohms, and load impedance of 10 KOhm. The bias current is only 2 mA. A simulation exercise was carried out for this circuit using device 2SC5002.

5.0

0

4.9

-50

4.8

-100 -150

4.7 60

63

66

69

72

75

78

81

time, nsec

(a)

Figure 8.25

84

87

90

140

146

152

158

164

170

176

182

188

194

200

time, nsec

(b)

(a). Voltage waveform at Q3 collector; (b). Voltage waveform at load.

The circuit was tuned to oscillate at 1 GHz with input signal frequency set to 900 MHz. The circuit provided 5 dB gain into an output load of 50 ohms at a current of 12 mA. The ADS transient engine was used to determine the waveforms. Notice at collector of Q3 the LO frequency adds to the converted IF frequency. 8.3.2 Linearized Modulator The linearization of the IF amplifier used to drive the differential modulators was proposed as a means to achieve low distortion in this patent, [16]. The inventors applied feedback to the base of Q2, as shown in Figure 8.26, by sampling the emitter voltage to the op amp input, maintaining a linear correlation between output and input voltage. The inventors claim the circuit is capable of 100% AM modulation. The current source provides a constant bias current for the differential amplifier, contributing to maintain a constant gain, as long as the carrier amplitude is constant. If the modulating signal exceeds a certain level, the bias current is reduced, reducing the differential amplifier gain and maintaining the output level constant with minimum distortion. If the modulation is set to operate at this limiting condition, the carrier signal does not need to be constant to maintain a constant output.

Bipolar Junction Transistor Applications

533

Figure 8.26 AM modulator linearized by feedback Figure 8.26

AM modulator linearized by feedback.

An improved version of this principle was patented by Daniel Talbot, [17]. He built a multiplier circuit employing two operational amplifiers, one for each phase. The circuit schematic is represented in Figure 8.27, which shows the multiplier input “X” where the LO signal is applied and input “Y” where the small signal is applied. In this circuit, the resistors R44, R46 and capacitors C52, C54 stabilize the op amp. The RC roll off is below the op-amp cut-off frequency. The base-emitter junction resistor is in the feedback loop of the op amp. Its function is the same as previously, namely, to minimize distortion in the voltage to current conversion. The resistor R38 can be assumed to be two equal resistors in series with a virtual ground at the connecting point, therefore it is also part of the feedback loop. Since it directly affects noise figure and inversely linearity, its value is set to optimize that trade off. A simple linearity test carried out by the inventor is worthy of note. A 4.5 MHz square wave signal was applied to “X” input. The R38 was set to be 1.8 K ohms the DC current for devices Q 30, Q34 set to 3 mA by adjusting current sources 32, 36. A DC voltage of 1 volt was applied to one “Y” terminal, while a 1 KHz audio signal with 2 V peak to peak was applied to the other. This resulted in 1 KHz side bands around 4.5 MHz, each 6 dB down (AM with carrier) the carrier at 100% modulation. The 2 KHz distortion sidebands were lower than -60 dBc.

534

Figure 8.27

Microwave Mixer Technology and Applications

Alternative modulator linearization with op amp.

8.3.3 Frequency Converter A frequency converter comprising a local oscillator and a doubly balanced type of mixer is reported in a patent from 1977, [18]. The schematic of the IC containing the converter is in Figure 8.28. The LO is generated by differential pair Q2, Q3, with a RC feedback coupling the collector of Q3 to the base of Q2. The RC is connected externally, so that frequency can be customized. A resonant parallel LC circuit is also connected to collector of Q3, which fine tune selects the frequency of operation. The signal at the base of Q2 is opposed in phase to the base of Q3 due to the differential configuration, and it is coupled to the bases of Q 8, Q9. Note the LO signals flowing through resistors R4, R5 are cancelled at their connection with R6, keeping LO current from flowing through the bias circuit. The LO in this configuration is applied to the differential pair Q8, Q9, whose collector currents modulate the quad BJT ring Q4, Q5, Q6, Q7 at the LO frequency. As is the case with most mixer circuits, the RF and LO ports may be reversed in this circuit resulting in the mixer operating similarly. In this case the RF signal is applied to the quad Q4, Q5, Q6, Q7. Since the RF signals are isolated from the signals at the trans-conductor, the mixing still takes place in the quad. The modulated transconductance of the RF amplifier results in a lower value of transconductance compared to the conventional port connection; assuming square wave modulation the gain is reduced to lower than half its original value. The output signal is extracted in single ended form from terminal t7 and then through a tuned transformer circuit. The IF filter in the figure illustrates the use of a high Q crystal resonator to recover the desired signal and reject all others.

Bipolar Junction Transistor Applications

Figure 8.28

535

Frequency converter schematic. Dotted line indicates the IC die.

A more recent converter was disclosed [19] in 1981 in a patent for a differential monolithic converter. The die contains mixer and LO functions using resistors and capacitors, with minimum external components. The initial proposal uses the topology shown in Figure 8.29(a). The dashed line denotes the silicon die area. The RF signal is applied at the base of device Q 6 with the collector current linearized by emitter resistor R8. The Q6 collector signal is applied to the differential pair, devices Q7, Q8. The mixing device Q8, is switched on-off by the LO voltage, while transistor Q7, whose base is AC grounded, is switched by the emitter current from Q8. But the mixing currents originating from Q7 are not in the load path and are not used. Therefore, this topology is not a true differential pair, it approximates a cascode mixer. Device Q7 however is important to control variation with temperature and voltage bias for device Q6. The large voltage applied on Q8 changes its DC level to accommodate the nonlinear collector-emitter waveform affecting the bias current of Q7. This DC level shift is sensed by the base current of Q9 and delivered to bias the base of Q6 through a resistive divider in a negative feedback manner, maintaining a constant DC current during the mixing process thus stabilizing conversion gain. Resistor R14 is preferably made of the same material as the device base, so that its value tracks the DC gain of Q 6, making the design more robust for manufacturing. This circuit also provides temperature compensation, using the drop of the base emitter junction voltage from Q9 when temperature increases. Lower Q9Vbe voltage results in increased base bias voltage for Q6, maintaining the collector current constant. The mixing products generated

536

Microwave Mixer Technology and Applications

at the collector of Q8, are delivered to an external tuned transformer that filters out the desired IF output.

Figure 8.29

(a) Cascode (b) Differential Integration of mixer and LO function in the same die.

The oscillator is composed of an amplifier Q10, which is connected to an external tank circuit. The collector voltage is sampled back to Q10 through a resistive divider and buffer device Q11, which also acts as a power divider, delivering the LO signal to the base of mixing device, Q 8. The resistor R15 controls the bias for the devices Q8, Q10, Q11 and also controls the loop gain for the oscillator. Bias is adjusted so that dynamic LO voltage switches Q 8 from nearly off to full conduction. Therefore, the collector to emitter junction impedance is close to an ideal switch providing linear mixing operation. In order to improve conversion gain, a second embodiment of the patent was proposed by the inventor, by connecting the collector of Q 7 to the tank circuit, shown in Figure 8.29(b). The LO signal is applied to the base of Q 8, which also receives RF signal from the emitter, generating IF signal at the collector. The LO signal is also applied to the emitter of Q7, along with the RF signal, due to the high collector impedance of Q6. IF signal is available at the collector of both devices which are delivered to the output load. The inventor also modified the bias, inserting an emitter follower to provide base bias for Q 8. That limits the amount of positive peak voltage on the base, but protects the device from a direct connection from collector to base.

Bipolar Junction Transistor Applications

537

8.3.4 Differential Active Balun and Mixer An IC singly balanced mixer using differential amplifiers and an external IF output transformer is shown in Figure 8.30, [20]. The main contribution reported in this patent is the use of an active balun to drive the mixer devices. The RF is applied at node 3 to the emitters of transistors 12, 13 that are the mixing devices. The single ended LO is applied at node 23 to the differential transistor pair 24, 25 that provide the balanced LO drive to the mixing pair via coupling capacitors 9,10. The differential outputs at the collectors of transistors 12, 13 couple to the single ended IF output at node 21 through transformer 19.

Figure 8.30

Differential amplifier mixer with active LO balun.

The two problems this circuit addresses are obtaining low noise and high linearity simultaneously. The solution is in two parts, first the RF is applied directly to the emitters of the mixing pair without an RF amplifier that would degrade linearity. Secondly, noise at the IF frequency range generated by the active LO balun is suppressed by the high pass RC filter comprising the collector bias resistors 27, 28, base bias resistors 15, 16, and coupling capacitors 9, 10. Also, the values of resistors 15, 16 are minimized to minimize their noise contributions. 8.3.5 Darlington Dielectric Resonator TVRO Converter An interesting mixer, disclosed in 1984, [21], is used in a TVRO block converter and shown in Figure 8.31. The converter translates the 3.7 to 4.2 GHz band to a first IF located around 1 GHZ, which is then converted to a second IF. Since this

538

Microwave Mixer Technology and Applications

is a consumer application, low cost and high performance are important, and the design approach supports these goals. The LO is frequency stabilized by means of a dielectric resonator coupled in the feedback loop of a Darlington MMIC amplifier gain block. The schematic is given in Figure 8.31(a), depicting the Darlington MMIC, the dielectric resonator represented by a tank circuit, and the coupling between the resonator and microstrip lines represented by transformers T1 and T2. The transmission line dimensions and the transformer coupling are calculated to meet the oscillation conditions. The RF and LO signals are applied to the base of Q1 and mixed. The IF output is amplified and delivered to the output of Q2. Out

DR / 4

12 T2

T1

10 8 6

Q1

Q2

3.6

In

3.8 4.0 Frequency - GHz

4.2

MMIC

Figure 8.31

(a) Schematic (b) Measured conversion gain Prototype Darlington dielectric resonator mixer. After [21].

A prototype circuit was built by the authors using a PC board substrate. Conversion gain for the mixer is shown in part b of Figure 8.31. The conversion gain has positive slope, ranging from 8 to 10 dB over the 3.7 to 4.2 GHz band. 8.3.6 Doubly Differential Transformer Coupled Transformers are combined with differential amplifiers to realize a mixer having an improved tradeoff between noise figure and linearity, [22]. The schematic in Figure 8.32 represents the mixer as an up-converter, with IF input and RF output. A few circuit topology versions are given that minimize the effect of nonlinear emitter resistance in the IF input amplifiers. The circuit comprises two differential pairs that are each driven by a common base IF amplifier.

Bipolar Junction Transistor Applications

539

Transformers are used to distribute the collector currents of the four differential pair devices, and also to feed signal back to the IF input. The LO and IF inputs to the mixer are differential, requiring additional transformers not shown in Figure 8.32 to interface with single ended sources.

Figure 8.32

Differential circuit coupled with RF transformer.

The IF input signals are applied in counter phase at the emitters 113, 114 of devices 103, 106 operating in common base, as defined by (8.12).

I113  I Q 

A cos( IF t ) (a) Rin

I 114  I Q 

A cos( IF t ) (b) Rin

(8.12)

Where A = voltage amplitude of IF signal IQ = quiescent bias current Rin = input resistance Currents I113 and I114, respectively, are applied to differential pair 101, 102 on the left half side and differential pair 104, 105 on the right half side. The two pairs together comprise a ring doubly balanced mixer. The positive LO signal is applied at the bases of 101, 105, and the negative LO signal is applied at the bases of 102, 104. The corresponding currents at the collector of each device in the ring are represented in (8.13) with the proper LO phase. The equation assumes the quad is switched by a unit voltage.

540

Microwave Mixer Technology and Applications

IC  I E

1  cos  LO t 2

(8.13)

Combining (8.12) and (8.13), the RF, IF, and LO currents are derived for each collector, and given by (8.14). I116  IQ

1  cos(LOt ) cos(RF t ) A cos(RF  LO )t  cos(RF  LO )t  A  2 Rin 4 Rin

(8.14a)

I115  IQ

1  cos(LOt ) cos(RF t ) A cos(RF  LO )t  cos(RF  LO )t  A  2 Rin 4 Rin

(8.14b)

I117  IQ

1  cos(LOt ) cos(RF t ) A cos(RF  LO )t  cos(RF  LO )t  A  2 Rin 4 Rin

(8.14c)

I118  IQ

1  cos(LOt ) cos(RF t ) A cos(RF  LO )t  cos(RF  LO )t  A  2 Rin 4 Rin

(8.14d)

The common mode collector currents, represented by (8.15) and (8.16), appear at the center tap of transformers 111, 112 and are extracted and terminated into an IF load R11 through transformers 107, 108. The secondary windings of transformers 111, 112 feed part of the amplified common mode signal back to the IF amplifier input, helping to correct the linearity of amplifier.

A cos( RF t ) Rin A cos( RF t ) I 120  I 117  I 118  I Q  2 Rin

I 119  I 115  I 116  I Q  2

(8.15) (8.16)

The differential mode currents appearing at the secondary of transformer 111 is given by (8.17), and at the secondary of transformer 112 by (8.18). A [cos(RF  LO )t  cos(RF  LO )t ] (8.17) 2 Rin A  ( I117  I118)   K cos(LOt )  [cos(RF  LO )t  cos(RF  LO )t ] (8.18) 2 Rin

I111  ( I115  I116)   K cos(LOt ) 

I112

The current in the load is given by I121 = I111 + I112, so the LO is suppressed and the up-converted RF currents add constructively resulting in (8.19).

Bipolar Junction Transistor Applications

I121  

A cos(RF  LO )t  cos(RF  LO )t  Rin

541

(8.19)

The specific contribution of this invention is the proposal of a set of linearized IF amplifiers to replace the amplifiers 103, 106 represented in Figure 8.32. The first proposal is the application of an “augmented” common base amplifier shown in Figure 8.33. An inverting amplifier is connected between the emitter and base of device 305. The base voltage is given by (8.20), and the internal VBE voltage is given by (8.21).

Figure 8.33

Augmented op- amp common base amplifier.

V307   AV V304

(8.20)

VBE  V307  V304   AV V304  V304  V304 (1  AV )

(8.21)

The output impedance of the inverting amplifier is low so for practical purposes, the base of the circuit is still grounded. The input impedance of the inverting amplifier is high and can be neglected. The effective input resistance, R in is given by (8.22), with VT =kT/q.

Rin  RE  re  RE 

V304  RE  IE

VBE (1  AV ) I s e

VBE VT

(8.22)

The second term of (8.22) represents the device emitter resistance, re, which tends to zero at high current, turning the emitter into a virtual ground. The input impedance is then equal to RE. An alternative approach to decrease the effect of nonlinear re is shown in Figure 8.34, where the op-amp is replaced by a common emitter stage. The total current entering the node connecting base of 506 and emitter of 504 is given by (8.23). Current IE1 flows into the emitter 504, IB2 flows into the base 506, and IB1 flows out of the base 505 and flows into the collector 507.

542

Microwave Mixer Technology and Applications

IE’ 503

IE1 504

RE IB2 506

501 Figure 8.34

IB1

505 RL

507

Augmented amplifier with bipolar transistor.

qvBE   I  I  I E '  I E1  I B 2  I B1 1  1  B1   I s 2e KT 1  1  B1  2  2   

(8.23)

Replacing this current into (8.22), there results:

Rin  RE 

V504  RE  IE '

V504 ( 1  1 

1

2

)I s2e

(

qVBE ) kT

(8.24)

The DC current gains, β1 and β2, respectively, apply to base currents 505 and 506. Equation (8.24) shows the non-linear resistance, re, of device 505 is reduced by its DC gain, making the effective impedance equal to R E. An approach to decrease noise figure is to replace the bipolar or op amp device driving the base, by a transformer in the manner of Figure 8.35. In this figure, the voltage relations for a transformer, provides the following relation between base-emitter voltage and emitter to ground voltage:

VBE  V807  V804  LV804  V804  V804(1  L) Where, L represents the transformer turn ratio

IE’ -LIB

Figure 8.35

IE IB

Augmented amplifier with transformer.

(8.25)

Bipolar Junction Transistor Applications

543

Similarly, the current relations at the emitter, are composed of three currents, IE‘ from the generator, IE from the device emitter and a current to the primary of transformer given by I = -LIB, where IB flows out of the base and into the secondary of the transformer.

I E '  I E  LI B  I E  L

IE



 I E (1 

L



)

(8.26)

Replacing the emitter current by its exponential relation to current and calculating the equivalent dynamic emitter resistance, one obtains (8.27).

I E  I se I E ' I se re ' 

qVEB kT

 I se

qV804(1 L ) kT

V804  IE '

qV804(1 L ) kT

L

(1 



)

V804 (1 

L



)I s e

(8.27)

q (1 L )V804 kT

(8.28)



VEB (1  L)(1 

L



)I s e

qVEB kT

(8.29)

The equation demonstrates the transformer reduces the effective nonlinear resistance, inversely proportional to the number of turns. Any of the linearized amplifiers proposed can be used to replace the common base IF amplifier of Figure 8.32 with improvements in linearity of the mixer. 8.4 DOUBLY BALANCED In 1966 a patent for a phase modulator was proposed by Jones, [23], as a means to avoid the use of RF transformers. The circuit topology became known as the Gilbert cell based the subsequent work Gilbert did on linear analog multipliers. Most mixer designs using this circuit are not linear multipliers between two input signals, because one of the two inputs, VB of Figure 8.36, is switched on-off, while maintaining linearity to the other input voltage, VA, which is sufficient in mixer applications. An extended application of Jones’ topology as a doubly balanced demodulator was reported by Sinusas in patent [24], using dual differential amplifier circuits.

544

Microwave Mixer Technology and Applications

86

IV

30

28 IR

+V

IW

96

40

38

IS

IU

IT

VB

IN

IM IB1

48

VE

50

IB2

VA

-V Figure 8.36

I = constant

Dual differential amplifier modulator.

The time and frequency domain waveforms are displayed in Figure 8.37, where the voltage VB is supplied by a square wave generator to switching devices 28, 38 with positive polarity, and to devices 30, 40 with negative polarity. The RF modulating signal, VA, in this example is triangular, and applied in counter phase to the bases of devices 48, 50. The signals V A and VB are interchangeable and their waveforms can be arbitrary. The modulating signal is linearly amplified by the devices 48, 50, and the instantaneous current at the collector of device 48 is equal to the collector current 50 shifted by 180. With no signals applied to the circuit, DC current flows from the positive voltage terminal and splits equally between the two differential amplifiers, and then is joined by the emitter resistor before sinking into the negative source. A well known property of the differential amplifier is the application of the voltage from Figure 8.37(a) to VB of Figure 8.36, which causes devices 28, 38 to conduct and take over the current from devices 30, 40 effectively shutting devices 30, 40 off, and vice versa as the square wave changes polarity. VB is considered as the LO. When devices 28,38 conduct, the current IM in collector 48 will correspond to the current IR in transistor 28, and the current IN in transistor 50 to the current IT in transistor 38. In the next half LO cycle, the inverse process takes place and the currents are supplied by devices 30 and 40. The triangular modulating signal of Figure 8.37(b) applied to node VA in Figure 8.36, is initially negative going and causes device 48 to conduct less and

Bipolar Junction Transistor Applications

545

device 50 to conduct more compared to quiescent conditions, represented by the horizontal line at zero. The currents at the collectors of 48, 50 are no longer equal. LO a

+V 0 -V

b

+V 0 -V

c

+I 0 -I

d

+I 0 -I

e

0

f3LO

f5LO

RF 0

fRF

Current Lead IM

Current Lead IN

+V 0

Figure 8.37

fLO

RFxLO

0 Time Domain Waveforms in the differential circuit.

fLO±fRF 3fLO±fRF 5fLO±fRF Frequency Domain

The current from the linear amplifier circulates directly to loads 86,96 during the first half LO cycle through devices 28, 38 respectively. Devices 28 and 38, respectively, deliver voltages inversely proportional to the current in Figure 8.37(c) and 8.37(d), developed across load resistors 86 and 96. Taking the difference between the voltages across load resistors 86 and 96 results in the voltage shown in Figure 8.37(e), which changes between positive and negative at the LO frequency, as currents IM and IN are steered between devices 28,38 and 30,40. It is important to observe that the output voltage is a direct result of the switching action and there is no nonlinear effect of the devices required by the process. The output waveform contains only the side bands, with the LO and carrier frequency components suppressed. The corresponding frequency domain of the waveforms are shown in the right side of Figure 8.37. In addition to functioning as a mixer, it also works as a phase detector. The voltage multiplier analysis was discussed in Chapter 7, resulting in the equation describing the multiplication of two input voltages, VA and VB, being given by the product of two hyperbolic tangents. Applying the small signal hyperbolic tangent approximation, the collector current, IW, of device 96 is given by (8.31).

546

Microwave Mixer Technology and Applications

IW 

IE 2

 VA V  tanh B  1  tanh 2 V 2 VT  T 

tanh x  x  IW 

(8.30)

x3 2 5  x 3 15

I  V AVB 1  1   1  2  4VT 2 3  2VT 

4    V AVB (V A 2  VB 2 )  ...   





(8.31)

The operation of this circuit as a phase detector can be understood by considering VA, VB as two sinusoidal small signals of equal amplitude and frequency, and offset in phase by, (t). The differential output voltage across load resistors 86,96 according to (8.32) is given by:

Vout  K cos[( A  B )t   (t )]  K cos[ (t )]A B

(8.32)

The output voltage, Vout, for phase offset (t) = nπ/2 is zero for n=1,3,5…; and maximum at ±K volts for n = 0,2,4,… The differential voltage in this detector therefore goes to maximum for in phase, and minimum for 180 degrees out of phase. If linear voltage versus phase is desired with voltage equal zero when in phase, (t)=0, then a /2 phase shift is added to one of the signals, making (8.32) change from cosine to sine. If the signals are small, sinx  x and the output voltage is linearly proportional to the phase difference in radians. Accordingly, the plot in Figure 8.38 shows zero volts for in-phase signals. As the phase difference increases, the voltage change is approximately linear for a relatively wide range, until the deviation from linear becomes significant and the sine of angle has to be included. For large phase differences the circuit still detects the difference, but is no longer linear with the phase difference.

Vout '  K sin[ (t )]A B

(8.33)

VA

V

(t)

A

(t) Figure 8.38

Phase detector operation.

Bipolar Junction Transistor Applications

547

8.4.1 Gilbert Multiplier The next major contribution to BJT mixer technology was an analog multiplier patented in 1970 by Barrie Gilbert, [25]. This was already introduced in Chapter 7. His multiplier is based on current multiplication and includes a voltage to current converter, supporting the implementation of RF and LO sources as voltage generators. A general diagram proposed in his patent is depicted in Figure 8.39. The input signal M is applied to a “linearized” differential amplifier, 70, and the input signal N to another differential amplifier, 90. The linearization of amplifier 90 is obtained by applying the output collector currents from terminals 91, 92 to the emitters of devices 98, 96, whose bases are grounded and collectors connected together at the voltage rail.

Figure 8.39

Four quadrant linear multiplier.

Devices 96, 98 are essentially diodes, generating an emitter voltage that is the logarithm of current. This current is applied to the base of devices 72, 74 and 76, 78, generating collector current in each that is exponentially related to the base voltage. Therefore, this logarithm conversion process results in a linear relation between collector current of devices 72, 74 and 76, 78 and the voltage applied at terminal N. It is interesting to note that the multiplication is achieved by means of the transistor non-linear characteristic, the multiplication itself is achieved largely distortion free and operates a over wide frequency range. An example of the linearization of a differential amplifier was described earlier that makes use of emitter degeneration. Gilbert proposed a different type of amplifier for this purpose, whose cell is shown in Figure 8.40. In this circuit, the input devices labeled 18’ and 20’, respectively, have their collectors cross connected to the collectors of devices 12’ and 10’. The output currents then become X(ID+IE) and (1-X)(ID+IE). The current gain, defined by the ratio of output current, X(ID+IE) to the input current XID is given by (8.34). Notice that if IE =0

548

Microwave Mixer Technology and Applications

the gain is equal to common base gain,   1. With this arrangement the gain is boosted by the ratio IE/ID and the linear relation between output and input current makes the process a linear one. To be valid these relations requires: small signal currents, high DC gain, i.e. qV/kT >>1, and small series resistances.

 I  Gain  1  E   ID 

(8.34) X(ID + IE)

(1-X)(ID + IE)

18'

20' 12'

10' XID Figure 8.40

IE

(1-X)ID

Cross coupling of collectors to improve gain.

With the purpose of maintaining the gain low to avoid excessive current ratios and a larger dependency on , the inventor proposed cascading or “stacking” stages of this amplifier, as displayed in Figure 8.41. Note the output current of one stage becomes the input current to the next. In this particular example one of voltage inputs was grounded, transforming the differential input into single ended.

Figure 8.41

Cascaded current amplifier stages connected in series.

Bipolar Junction Transistor Applications

549

The gain is determined from the value of current I 1 and the ratio of each current I2, I3, I4 to I1. The output signal is extracted differentially from load resistors 34, 36.The first stage, devices 40, 42 have emitter degeneration caused by resistors 48, 50 improving linearity and dynamic range of the stage. If the available rail voltage is low, the author proposed using the paralleled version of this amplifier represented in the Figure 8.42. Here the input devices 10” and 12” are cascaded and the collectors of each device are cross-crossed paralleled to terminals 30 and 32. The signals add in phase at the output terminals and the gain control shows the same relations between the emitter currents.

Figure 8.42

Cascaded current amplifier stages connected in parallel.

8.4.2 Micromixer - Matched Input Impedance A circuit that improves impedance matching and biasing was also patented by Barrie Gilbert, [26], which he called a “micromixer” [27]. The schematic, shown in Figure 8.43, contains three sections: the mixer, block 24; the RF input section, block 26; and the bias section, block 28. The mixer section is conventional, with RF input at 38, 40; LO input at 30, 32; and IF output at 34, 36. The RF section contains transistor Q11 operating in common base and a current mirror Q 12-Q13. The RF input current applied to terminal 44 is divided into emitter current in Q 11 and the current in the mirror, Q12. These currents are non-linear as previously demonstrated in Chapter 7, but the differential current applied to the mixer, given by the difference in currents, (I1 - I2 = IRF) is linear relative to input RF voltage.

550

Figure 8.43

Microwave Mixer Technology and Applications

Matched RF input impedance.

The input impedance of this circuit is given by the parallel combination of the impedance of common base device, Q 11, and the impedance of current mirror, Q12, paralleled with common emitter device Q13, impedance. The common base input impedance is approximately given by the dynamic impedance of emitter base junction, re. The common emitter impedance is at least one order of magnitude higher than the impedance of the mirror, Q 12, and can be disregarded in a first order approximation. Therefore, the impedance of both paths are substantially the same and the net input impedance is given by their parallel combination. The small signal input impedance is therefore given by (8.35), where IZ is the quiescent current of I1, I2.

Rin 

VT 25mV   50 2 I Z 2 * 0.258mA

(8.35)

The matched condition is obtained for a specific bias current and is very sensitive to input drive. The author showed previously [27] that R in mismatch occurs due to a very low input impedance under large signal drive. The optimum impedance is obtained by adding resistive attenuation at the input, as shown in Figure 8.44(a), allowing a larger bias current and less impedance variation versus signal drive level. Simulations showed that minimum impedance variation is obtained for attenuator resistor values between 50 and 100 ohms. But for better linearity a lower value is preferred, on the order of R p = 33.3 ohms, where input impedance changes from 43 to 50 ohm. An alternative means to improve input impedance is in Figure 8.44(b), with device Q21 added to the circuit, operating in

Bipolar Junction Transistor Applications

551

cascode with Q13. The added device shields the RF input device Q 13 from variations in the supply voltage, and from spurious generated by the quad mixer. 42

38 Q11

VBIAS

VBIA Rp

38

40 Q21

Q11 Rp

S

44 RF

42

40

44 Rp

RF

Rp

Q13

Q13

GND (a) Input impedance padding Figure 8.44 Matched input impedance options.

GND (b) Addition of cascode stage Q21

The bias circuit section of Figure 8.43, block 28, contains a current source applied to two diode-connected transistors, Q14, Q15. The current source IQ provides current to the diodes establishing a bias voltage VBIAS, and a current to bias Q11 and Q12. Over temperature, the voltage drop of Q 11, Q12 is tracked by VBIAS resulting in a constant base current to Q11, constant collector current to diode connected Q12, and constant collector current Q13. The emitter area of the bias diodes can be scaled in accordance with the current required by the devices being biased, establishing the desired ratio between I Q and ID. The capacitor C1 connected across the biasing diodes guarantees a low impedance to RF. If lower frequency operation is desired, an off chip capacitor should be added. 8.4.3 Feedforward Linearization in Mixers Another alternative to minimize mixer distortion is to make use of feedforward concepts: distortion produced by a main and second error amplifier are designed to be equal in magnitude and opposed in phase so that they cancel when summed. In this regard, the feedforward amplifier of Figure 8.45 is an example that can be used as the RF amplifier in a Gilbert mixer, [28]. In this configuration the distortion voltages on Q1, Q2 are similar to the distortion voltages on Q3, Q4. This signal is detected by Q5, Q6 and fed forward in opposed phase at the collectors of Q3, Q4. It can be shown, [29], that the differential output current of this amplifier is given by (8.36). The parameter VBE is the distortion differential voltage generated by Q1, Q2 and VBE' is the distortion generated by Q3, Q4. If transistor transconductance, gmE is made equal to 1/RA then distortion is suppressed.

552

Microwave Mixer Technology and Applications

Figure 8.45

CasComp transcondutor. From [28].

I out 

Vin VBE   VBE ' g mE RA RA

(8.36)

The second source of distortion in the Gilbert mixer is the switching quad, usually considered linear in most designs. In principle, the same technique can be applied to minimize distortion introduced by the switching quad. However simulations has shown it is not practical, one reason is the voltages VBE and VBE' in the (8.36) are not at the same frequency, and besides there is distortion introduced by the LO waveform. The proposal from the reference to circumvent this problem is to parallel two mixers as depicted in Figure 8.46. The fundamental components add in phase and the intermodulation are in counter phase and can be suppressed. The resistors RA, RB, are used to adjust the magnitude of the intermodulation products. The authors found an improvement of more than 7 dB in IIP3 compared with a single ended mixer, and it improves further at lower frequencies. The concept was confirmed by fabrication of the circuit on a 0.2 um SiGe HBT technology.

Bipolar Junction Transistor Applications

Figure 8.46

553

Two mixers in parallel improve IIP3 by cancellation. From [28].

8.4.4 Linear Low Noise Differential mixers in general have good conversion gain, wide frequency coverage, good LO suppression, and high linearity. However, those characteristics are obtained at the cost of high noise figure, degraded by the emitter degeneration resistor, RE, used in the RF amplifier. In this proposal, the RF amplifier is tuned for a specific frequency range, trading off bandwidth for noise, [30]. The circuit of this mixer is depicted in Figure 8.47. Two improvements are made to reduce noise figure compared to a conventional design: the degeneration resistor, R E, is replaced by an inductance LE; and a tuning inductance L1 is added in series with the base of transistor Q5 in the RF differential amplifier. Note that the RF amplifier is also a single ended to differential converter. Assuming the presence of resistor, RE, the RF input impedance is given in Chapter 7 and reproduced in (8.37). The equations are valid for frequencies below T/0, where T is the transition frequency and 0 is the transistor DC current gain.

Z in ( j )  (2rb  RE ) 

1  1   RE T   j  C / 2 

(8.37)

Replacing RE by an equivalent series RI-LE, representing the inductance and its parasitic resistance, the input impedance is modified to:

554

Microwave Mixer Technology and Applications

Z in ( j )  (2rb  RI  T LE ) 

Figure 8.47

 1  1  RI T    2 [ L1  LE ]  (8.38) j  C / 2 

Low noise mixer using inductor emitter degeneration.

The remaining parts of the circuit are similar to a conventional design. The capacitors C4, C5, C6 and C7 were added to minimize the effect of parasitic inductances from the transistor package. The inventors reported in their patent a noise figure of 10 dB for the mixer, instead of the usual 15 to 16 dB. 8.4.5 The Tree Mixer A variation of the differential BJT mixer, called a tree mixer by the inventors [31], is depicted in Figure 8.48. It consists of a first differential pair Q 11, Q12 having a current source I at the emitters, and two differential inputs I/P1+ and I/P1– at the bases. The second differential amplifier actually contains two differential pairs with tied emitters Q15, Q16 and Q17, Q18; and cross coupled collectors, Q15, Q18 and Q16, Q17. The inputs to the bases of this differential amplifier are denoted I/P 2+ and I/P2– and connect, respectively, to Q15, Q17 and Q16, Q18. The differential output is taken from load resistors R19, R20.

Bipolar Junction Transistor Applications

Figure 8.48

555

The highly linear tree mixer.

The principle of operation consists in adding an additional current source coupled to the collectors of the first differential amplifier through isolating impedances Z23 and Z24. Current source I1 adds with the small signal collector currents in Q11 and Q12, equaling the source I so that the first differential amplifier operates at a high current minimizing distortion compared with a conventional approach. The second differential amplifier operates with the current difference (I 1 - I), and therefore at a much lower current. So it generates lower noise, and reduces the headroom constraints in relation to the power supply voltage due to the voltage drop across R19, R20. The effective trans-conductance for the first differential amplifier is given by 1/(2RE) where RE = R13 = R14. The LO applied to the second differential amplifier acts to multiply the currents from Q 11, Q12 alternately by +1 and –1. The conversion gain, assuming output voltage is developed at the resistors R19 = R20 = RC, is equal to (8.39). By way of illustration the current, I, can be made equal to 30 mA and I 1 = 24 mA, so the bias current for the second differential amplifier is 6 mA.

GCV 

2 RC  RE

(8.39)

Another alternative proposed in the invention that is interesting for the case of low VCC is represented in Figure 8.49. The differential RF amplifiers use pnp devices and an emitter resistance to improve linearity. The total current passing through devices 11, 12 is greater than the total current in devices 15 to 18. The current in devices 11,12 is given by I/2 and the current in devices 15 to 18 can be selected to be less than (2I2-I).

556

Figure 8.49

Microwave Mixer Technology and Applications

The low voltage version of the tree mixer.

8.4.6 Chroma Modulator A differential multiplier was invented for the modulation of video chroma signals from fs = 100 Hz to 1100 KHz, to a fixed carrier at fc = 4.58 MHz, [32]. The circuit of a conventional multiplier is shown in Figure 8.50(a). This is not a linear multiplier, since the collector current of Q1,..,Q4 are not linearly related to the ec voltage. For mixer applications this linearity is not required, because the purpose is frequency translation of information contained in the signal e s to the output sideband signals e1, e2. One problem the inventors had with the conventional approach is the leakage of carrier signal at the output that cannot be eliminated by filtering. The inventors identified the problem as unbalance caused by nonsimilar source impedances for the devices Q1, Q4, equal to the parallel combination of R11 and the carrier source impedance, while the Q 2, Q3 impedance is given solely by R12. This impedance imbalance results in different voltages at the base-collector and different Miller effects. In the proposal, shown in Figure 8.50(b), devices Q11 and Q12 are added with bias device Q13.The base of Q11, Q12 is now connected to a low impedance given by the emitter of Q 13, forcing the load impedance for devices Q1,..,Q4 to be low as well. So, the unbalancing effect becomes less important and is minimized in the output signal taken from the collector of device Q12. Notice the single ended output is obtained by connecting only one side of the differential circuit to the external load, so the circuit is not fully doubly balanced.

Bipolar Junction Transistor Applications

Figure 8.50

(a) Conventional Analog multiplier circuits.

557

(b) Improved

8.4.7 Front End Mixer A balanced modulator (or down-converter mixer) built with differential amplifiers was disclosed in 1982, [33], and used in a front end receiver depicted in Figure 8.51. The important parameters for a mixer in this configuration are the conversion conductance, noise figure, and maximum input signal level. Those parameters are controlled by a DC feedback loop from the detector output to the balanced modulator. The DC level is proportional to the input signal, and is used to control bias to the modulator and the amount of LO injection, to maintain the modulator gain within the optimum range of operation. The schematic for the balanced modulator is depicted in Figure 8.51. The modulator consists of ring Q17, Q18, Q19 and Q20 plus RF amplifier devices Q15, Q16. The current source for this stack of amplifiers is obtained from transistors Q 13, Q14. The bias of this current source is controlled by the by the external DC level generated by the detector and applied to current mirror Q21. The LO signal injection is applied to an input variable attenuator controlled by the channel resistance of the FET, Q 9. The single ended LO voltage is transformed into differential by the differential amplifier Q10, Q11 before being applied to the base of the mixing transistors.

558

Microwave Mixer Technology and Applications

Figure 8.51

Mixer for front end receiver.

The conversion trans-conductance is a function of R14 and bias, GC = k(1/R14, IQ) and the maximum acceptable input signal is given by V in,max = R14IQ. The optimal receiver sensitivity is obtained by setting a high conversion gain and low noise figure when the DC detected control voltage is low. The RF amplifier noise figure is lower when operating at a lower bias current and mixer gain is maximum with higher LO drive. With an increase of signal level, the detected voltage increases and so does the bias current for the RF amplifier. +VCC IFout LOin n

Detect Input GND RFin Figure 8.52

Front-end receiver with automatic level control for constant conversion gain.

That degrades noise figure but signal level is higher so S/N ratio is not affected. In the specific case of this invention, the DC control voltage is also applied to the gate of the FET decreasing the amount of LO power applied to the mixer. The authors found the circuit to perform with lower distortion in this condition. However, notice that most mixers show lower cross and intermodulation with higher LO power drive. The circuit can be corrected easily by inverting the gate voltage controlling the attenuator.

Bipolar Junction Transistor Applications

559

8.4.8 Amplifier Bypass A patent was proposed to extend the dynamic range of a differential pair mixer using the amplifier bypass technique [34]. As RF input signal level increases into the LNA stage, at a certain threshold level the RF is bypassed around the LNA, reducing the level of distortion. The LNA shown in Figure 8.53 comprises an input differential amplifier connected in cascode. The circuit also includes a second differential amplifier that provides the RF input to the doubly balanced mixer quad. Additional functions are shown as blocks, including the LO and AGC. Initially let’s assume transistors T 12, T13, and resistors R7, R8 are not in the circuit. The RF amplifier consists of devices Q1 to Q4, biased by a voltage supply in series with a resistor. The base of Q2 is RF grounded through capacitor C 1 to generate the single ended to differential conversion. The differential signal from the RF amplifier connects to the bases of differential amplifier Q5, Q6, which is linearized by resistors R3, R4. The Q5, Q6 collectors connect to the emitters of mixer quad Q8 to Q11. The collectors of Q8 and Q10 are RF grounded to deliver a single ended output from collectors of Q9 and Q11. The AGC block detects the output signal level and controls the bias of Q 1, Q2 and Q5, Q6.

+VCC

Q8

IF

Q9 Q10

LO

Q11

R5

R6 Q5

R3

R4

Q6

Q7 Q13

Q12 R7

Q3 Q1

Vr2

Q4 Q2

RF R1 Figure 8.53

RF

AG C

R

R2

8

Vr1

C1

Frequency converter with RF amp bypass.

Therefore, with an increase of signal level, the bias decreases reducing the gain of the two differential stages, thus maintaining the output level constant.

560

Microwave Mixer Technology and Applications

Let us now introduce the components Q12, Q13, R7 and R8. Under normal operating conditions, the resistors R7 and R8 are calculated such that the devices Q12 and Q13 are cut off. With an increase of signal level, voltage Vr1 decreases by AGC action making the devices enter into conduction. At this point part of the input signal starts to be bypassed directly to the mixer, avoiding distortion in the amplifiers. The resistors R7, R8 are noise sources, which are almost negligible when the devices Q12, Q13 are cut off. At large signal levels the resistors increase the amplifier noise level, but the overall signal to noise ratio is not degraded since the signal is larger.

8.4.9 Balanced/Unbalanced Gilbert Cell Mixers A means of providing a single ended output using an active circuit was proposed in patent [35]. The circuit consists of a differential pair with a current mirror load, as shown in Figure 8.54. The current mirror comprises two identical pnp transistors Tr1 and Tr2, where Tr1 is connected as a diode, and the base of both devices connect together. Assuming R3, R5 are not in the circuit, then IC1, IC3, IC4 equal each other as in (8.40). The result in this arrangement is the the input current IC1 is impressed at the output load, with output current described by (8.41).

IC3 IC1

Figure 8.54

IC4

Iout

IC2

Unbalanced output converter with current mirror.

A better description of this mechanism can be found if one considers a V applied at the base of Tr7, resulting in IC1 = IT/2+I. Applying the same V to

Bipolar Junction Transistor Applications

561

device Tr8, the current IC2 becomes IT/2-I. Applying (8.41), the output current is given by Iout = 2I, demonstrating that the output current fully recovers the signal without any loss.

I C 3  I C 4  I C1

(8.40)

I out  I C 4  I C 2  I C1  I C 2

(8.41)

A more exact development of the relationship between the output signal and the RF input voltage becomes too complex, since the mirror load is at the third level of the circuit. The resistors R1, R2 are added to balance any base/emitter voltage offset due to the mismatch between devices, and mismatched temperature coefficients including Vbe/T. R1, R2 are also used to limit the gain introduced by the current mirror, which is an active load for the mixer. If signal level is high a small distortion is introduced, as shown in Figure 8.55. The resistors R3, R4 are used to match the output impedance to the filter impedance.

Figure 8.55

Distortion introduced at high signal level.

8.4.9.1 Unbalanced RF Input The proposed circuit, [36], uses an active common base, common emitter circuit to balance an input signal. The schematic is split into two sections: the RF and the mixer. The modified RF section is illustrated in Figure 8.56(a), and the mixer section is in Figure 8.56(b). The single ended input AI is at node NC, which connects to the emitter of transistor Q101 by resistor R111, and to the base of transistor Q102. Transistor Q101 is arranged in common base, with the base connection to ground via Schottky diodes D241, D242, and D243 plus impedance R244. Transistor Q102 is common emitter with

562

Microwave Mixer Technology and Applications

impedance R112 connecting the emitter to ground. The collectors of Q 101 and Q102 constitute the differential output RF amplifier that drives the mixer section. The common base transistor Q101, is a noninverting current follower. The input impedance to this stage is given by R111 in series with the emitter impedance of Q101, which is a function of base to ground impedance. The common emitter transistor Q102, is an inverting trans-admittance stage with input impedance higher than the common base stage Q101. Its transadmittance gain, or voltage to current conversion factor, is approximately equal to the inverse of input impedance. The RF amplifier input impedance is approximately given by the impedance of the common base device. The input current applied to AI will substantially flow to the emitter of Q101 and to the symmetrical output S01. The signal voltage at node NC is approximately equal to the product of the input current and the input impedance of the current follower. This signal voltage is simultaneously applied to the transadmittance amplifier. The symmetry between the signal currents at the collectors of Q101 and Q102 is obtained if the product of trans-impedance gain and the input impedance of the current follower approximates unity gain. This is approximately true if the bias of both Q101 and Q102 are similar and the impedances R111 and R112 are also close to being equal. Some asymmetry may result from the base to ground impedance of Q101. This asymmetry may be compensated by decrementing the magnitude of the impedance R111 with respect to that of impedance R112.

Figure 8.56

(a) (a) Modified RF section; (b) mixer section.

(b)

The linearity of this stage is demonstrated by studying a few factors, starting with the sum of base voltages which is constant and equal to V c. An input signal current applied at AI flows to the emitter of Q101 modulating its base-emitter junction voltage. This current to voltage conversion is logarithmic. The collector

Bipolar Junction Transistor Applications

563

current of Q101 substantially follows the input RF current, which is a property of the common base topology. The base-emitter junction of Q102 is modulated inversely and also non-linearly, since the sum of base voltages of both devices is approximately constant. The collector current of Q102 is linear for a logarithmic base emitter voltage. Therefore, the transfer of an input current from terminal A I to SO1 and SO2 is substantially linear as long as the drive level is below a certain level. The presence of impedances R111 and R112 improves linearity since they increase the amount of input voltage that can be applied to the circuit. The stability of output signal currents at SO1 and SO2 over wide temperature ranges is obtained from the bias circuitry. The bias control consists of two current sensing transistors, Q 201 and Q202. The collector current of each device is scaled from Q 101 and Q102, respectively, and impedances R211, R212 also scaled from R111 and R112. The collector of Q201 is biased by current source Q221 that mirrors the same current to device Q222 with opposite phase. It is therefore subtracted from the collector current of Q202. Therefore the difference between both collector currents of the sensing devices feeds the base of device Q240. The voltage Vc biases Q101 and Q102. In steady state operation the bias current of Q101 and Q102 are substantially equal. The bias control circuit is an integrating control loop that maintains zero error; which equates to a zero difference current. Any difference is updated into control voltage VC and base bias of Q202 through voltage divider R211, R120. If there is a drop in voltage over R112, then the same drop occurs on R120, resulting in an increase in base voltage of Q240 and consequently an increase in VC. The mixer section is a conventional ring mixer topology, where the RF signals are applied from SO1 and SO2 to terminals 3, 4, and the output signals are extracted from the load resistors R5 and R6. The LO signal is coupled by means of an external balun with positive polarity to the bases of devices T1, T4, and with negative polarity to T2, T3. 8.4.10 Linearization with Transconductance Amplifier This circuit replaces the usual RF amplifier in a differential mixer with one having a high degree of linearity, [37]. The mixer quad comprises devices Q306, Q308, Q310, Q312, which are biased through resistors R302 and R304 from a positive supply voltage. The trans-conductor for the positive RF input comprises devices Q318, Q320 and Q314, plus the current sources I315 and I330 shown in Figure 8.57. The object of this circuit is to transform the input voltage into a linear collector current. This process can be analyzed by first simplifying the transconductor circuit. Since the circuit is differential, it can be split in two independent and equal circuits. The connection between the two parts is where resistors R 322 and R324 tie together, which is a virtual ground. The two resistors can be separated and the newly open ends can be grounded, with the resistors replaced by RE as in Figure 8.57.

564

Figure 8.57

Microwave Mixer Technology and Applications

Balanced mixer with linearized transconductor.

Assuming there is no feedback in the circuit, and applying a RF signal voltage at Vin, the resulting collector current is exponentially related to Vin. The base voltage of Q320 swings in an inverse exponential relation to input voltage V in. This distortion is cancelled in device Q314 by the exponential voltage current relationship, making the output current IC a linear function of Vin. Closing the feedback from the Q314 emitter to the Q320 base improves the linearization, and the addition of RE causes the amplifier gain to become proportional to R E.

+VC C

IC Q314

Q318 Q320 Vin

Figure 8.58

Simplified transconductor circuit.

RE

Bipolar Junction Transistor Applications

565

The linear relationship between collector current Ic and input voltage Vin is demonstrated by (8.42):

IC 

VRE Vin  Vbe1  Vbe2 Vin   RE RE RE

(8.42)

In conventional mixers RE improves linearity but also degrades noise figure. However, in this approach the negative effect of R E on the input impedance is buffered by device Q320. Therefore, the circuit offers a better tradeoff between noise figure and linearity. 8.4.11 Low Voltage Operation Portable equipment like cell phones, require low bias voltages, usually in the order of 3.0 V. This trend to require operation with lower rail voltages is driven by battery technology. The inventors of this patent, [38], developed the circuit in Figure 8.59 with reduced rail voltage in mind. The core cell is conventional, but the trans-conductor has different features. The RF signal is split into two parts, one feeding the base of Q11 via C15 and the other feeds the emitter of Q12, via R14.

Figure 8.59

Low voltage operation.

566

Microwave Mixer Technology and Applications

The impedance adjustment between both devices is carried out by R51. The bias for Q12 comes from R21 that sets the emitter current. In order to allow Q11 to be biased at the same current, an additional voltage source is created by Q 24 that biases Q11. Note the potential V2 is lower than V1, which means the voltage on Q11 is lower than Q12, but the same amount of current flows in both devices if the resistors are properly adjusted. Since low value resistors are used to bias Q 11, Q12, it can operate properly using low supply voltage. 8.4.12 Cross Coupled Even Order The circuit of Figure 8.60 is referred to as an even order mixer since it rejects all odd order components at the output, [39]. The circuit schematic shows the LO V1 is applied to the base of Q1 and emitter of Q2. The signal V2 is applied at the base of Q2 and emitter of Q1. If the amplitudes are properly balanced those signals cancel at the output node V0. The output voltage is collected on the resistor RC and is given by (8.43).

V0   RC ( I C1  I C 2 )

I C1  e IC2  e

VBE V1 V2 VT

VBE V1 V2 VT

 V  V2   V0  2 I E RC cosh 1  VT 

(8.43) (8.44a) (8.44b) (8.44c)

By approximating the argument of the hyperbolic cosine as a small signal, the output voltage is shown to have only even order terms.

 V  V 2 V  V 4 V  V 6  V0  2 I E RC 1  1 22  1 42  1 62  ... 2!VT 4!VT 6!VT  

(8.45)

This equation shows if the two signals are at different frequencies, the square term generates frequency translation. If they are at the same frequency it can operate as a frequency doubler. As a matter of fact it also shows potential for subharmonic application which appeared in the patent described at the end of chapter. The application of two signals, V1 = 1200 MHz at -20 dBm and V2 = 900 MHz at -30 dBm resulted in the following output spectrum:

Bipolar Junction Transistor Applications

Figure 8.60

Cross coupled even order type.

Table 8.1 Spectrum of Output Voltage V0

Term f1-f2 2(f1-f2) 2f2 f1+f2 2f1 3f1-f2 2(f1+f2) 3f1+f2 4f1

Order

Amplitude (dB)

2 4 2 2 2 4 4 4 4

0 -31.2 -14.7 0 3.6 -24.9 -31.2 -24.9 -26.9

567

568

Microwave Mixer Technology and Applications

8.4.13 RF Section with Feedback Typical specifications for a cell phone down-converting mixer are conversion gain of 10 dB, input 1-dB gain compression point of –10 dBm, and noise figure at less than 9 dB. The conversion gain of a typical mixer can be higher than 20 dB, which may be undesirable if the signal levels at the output are above the compression point. A design strategy that sets the maximum input signal power equal to the input compression point for a high conversion gain mixer will have a small margin for thermal noise at the IF. The objective of this patent, [40], is to improve linearity without sacrificing noise figure by using reactive feedback networks. Essentially the proposal is to add a parallel RC feedback network between the collector and base of RF amplifier, associated with an inductive series feedback connected to the emitters of same amplifier, as in Figure 8.61. OUT2

IN1

OUT1

D1

D2

IN2 Rb

Cb

Cb

Q2

Q1 IN3

Rb

Lb

Lb

IN4 COMMON Figure 8.61

Reactive feedback mixer.

8.4.13.1 Alternative Feedback In this proposal, [41], an additional trans-conductance stage is added in parallel with the main stage, Figure 8.62. The assumption here is that nonlinearities at the collector of both Q44 and Q46 contain similar types of distortion, so that the extra stage is used to feedback the distortion voltage, to add with the incoming RF signal. Since the base of both devices are in parallel they receive the same feedback corrected signal. In addition, the emitters of Q 44 and Q46 are tied to a

Bipolar Junction Transistor Applications

569

series feedback inductance, which creates feedback without degrading noise figure. The load resistor R54 for the feedback transconductor controls the amount of feedback along with impedance R56. Since two transistors are in parallel, there will be some noise increase compared to a conventional single gain stage, but the noise degradation is still lower than for other linearization approaches. Additionally, the trans-conductors are designed with large devices to minimize active base resistance and consequently also minimize noise figure. +V R54 LO

IF Q1

Q2 LO

R56 RF

Figure 8.62

Q44

Q46

Schematic of feedback mixer.

8.4.14 RF Transformer Coupled In an attempt to improve linearity, a transformer connection was proposed [42], to replace the RF amplifier used in conventional differential or Gilbert cell mixers. The circuit schematic is represented in Figure 8.63, where two resistors have been added to the RF path, and also serve as bias for the differential mixer ring. The LO source impedance driving the bipolar quad ring is low, so the devices can be considered as common base for the RF signals applied to the emitters. The emitter impedance of a common base device is the inverse of its transconductance, which is degraded by the emitter resistors RE. The voltage gain is given approximately by (8.46):

GCV 

g m 0 Rc R  c 1  g m 0 RE RE

(8.46)

In this circuit, the input impedance of common base can be controlled by the amount of bias current while the impedance of the RF generator is controlled by the transformation ratio.

570

Microwave Mixer Technology and Applications VDD

RC

RC IF

LO RE

RE

Figure 8.63

RFIN RF transformer coupled differential mixer.

The circuit Figure 8.64 adds an RF gain stage between the transformer and the mixer quad, [43]. The RF gain stage functions as a voltage-current converter, and comprises a common base differential amplifier, in contrast to the usual common emitter circuit used for this function. The benefit of the common base topology is it has better linearity and also better signal-to-noise ratio. For optimum noise and linearity the ratio of the input impedance of the common base circuit to the transformed source impedance can be adjusted.

R

R

VIF

1

T1

2

T2

T5

T5

T6

T4

VLO

-VEE

-VEE VRF

Figure 8.64

RF transformer coupled with common base.

Bipolar Junction Transistor Applications

571

The source impedance is controlled by the turns ratio of the transformer, and the common base input impedance is controlled by the varying V EE. Increased linearity results from increased transformed source impedance relative to common base input impedance. 8.4.15 High Dynamic Range The proposal described in patent [44], is to bias the transistors like a class AB amplifier, as a means of improving the dynamic range of differential mixers. The schematic of the RF section is in Figure 8.65(a), where the differential RF signal is applied between RF1 and RF2, and the amplified signals OUT 1 and OUT2 are applied to a conventional ring mixer not shown in the figure. RF inputs RF 1 and RF2 each drive a gain stage. Each gain stage consists of two parallel transistors of different size, so that one has a higher trans-conductance than the other and an inversely higher emitter resistor value. One is biased class A and the other biased class AB. The device biased in class A operates normally under small signal conditions, exhibiting trans-conductance compression when the input signal increases above a certain level. The device biased in class AB exhibits class A operation at small signal conditions, but its trans-conductance increases with drive level. Combining both devices in parallel, the total trans-conductance at larger signal levels is kept essentially constant for a wider range of input signals, as depicted in Figure 8.65(b). Out1

61

ID

62

N12 N21

N11 70

Out2

RE11 RE12

N22 RE21 RE22

+VCC

RB71

RB1

RF1

RF2

g RB2

g0=g1+g2

g2 RE71

g1 V

Figure 8.65

(a) Schematic (b) Combined transconductance Class AB mixer using two transistors of different size to obtain constant gain over a wider input power range.

The supply voltage circuit for the base of the RF amplifier is given by circuit 70, containing a configuration that is independent of transistor DC current

572

Microwave Mixer Technology and Applications

gain . Additionally, making the devices RE71 and RB71 proportional, respectively, to RB and RE, then the currents in 61 and 62 track the current in voltage supply circuit 70. Two options are proposed to achieve this. First, the transistor ratios are the same; i.e., the devices N11, N12, and N21, N22 have the same area and transconductance, as set by the ratio of resistors R B1/RE11, RB1/RE12 for branch 61, and RB2/RE21, RB2/RE22 for branch 62. Second, the transistor ratios are different and resistor bias can be the same. 8.5 IMAGE REJECT The amount of image rejection required in a radio receiver is dependent upon the system specification and the IF frequency. In many communication systems the requirement is on the order of 80 dB or more, which can only be achieved by filters. Since filters are large and have limited Q in monolithic form, achieving high levels of image rejection in IC has proved difficult. However, in applications where less rejection is required, a monolithic approach becomes feasible. This is possible where a simplified filter can be integrated, or in the case of direct receivers or quasi-zero-IF receivers polyphase RC filters are relatively easy to integrate. These provide the accurate phase shifting required for image rejection obtained by phase cancellation instead of filtering, or in addition to it. 8.5.1 Voltage Tuned Image Filter A proposed approach uses a tunable notch filter between LNA and mixer to reject the image in a classical Super-heterodyne system, [45]. A circuit designed for Silicon technology operating at 1.9 GHz offered rejection on the order of 50 dB. The circuit schematic of the proposed receiver is in Figure 8.66, which comprises an LNA stage at the input composed of transistor Q1, emitter degeneration inductance Le1, and collector resonant circuit L1, C1 tuned at 1.9 GHz. A large device was used, 80x0.5 µm2 to decrease base resistance and improve noise figure. The image reject filter consists of transistors Q2, Q4 forming a cascode stage, and a notch filter Q3, L3, CR connected at the floating terminal. The resonance capacitance CR represents the varactor plus other associated capacitances. The resonance frequency is calculated using the equation and is tuned to notch the image frequency.

f notch 

1 (C 3  C )C R 2 L3 (C 3  C )  C R

(8.47)

Bipolar Junction Transistor Applications

Figure 8.66

573

Superheterodyne receiver. From [44].

At the resonance frequency the filter creates a zero impedance at the emitter of Q4, making the base impedance of Q 3 have a negative resistance component. The resonant Q factor is increased by the negative resistance, allowing a deep notch in the image. The mixer itself is a conventional Gilbert cell with single ended input and differential IF output. Applying a -40 dBm RF signal at 1.9 GHz, the IF output power at 300 MHz is on the order of -6.5 dBm. The measured notch frequency tuning range was 230 MHz, from 2.40 to 2.63 GHz. The unit conversion gain is equal to 33.5 dB, the noise figure is 4.9 dB, and the IIP3 is -28 dBm. 8.5.2 Conventional Image Rejection Mixers An interesting publication, [46], describes an integrated IRM comprising an LNA and two mixers on the same SiGe chip, with the IF quadrature elements off chip, whose block diagram is in Figure 8.67. To achieve the required 40 dB image rejection, both IF outputs must track each other within 0.2 dB amplitude and 1 phase. This requires the mixer circuits to have a very symmetrical layout. The LNA and mixers are also located closely together by using planar trifilar transformers to couple the RF amplifier Q1, Q2 to the mixers. These transformers offer reduced size over other implementations. The LNA had its impedance matched for optimum noise figure by properly dimensioning the emitter inductance. The resulting circuit, which operates over 5.1 to 5.8 GHz, is shown in Figure 8.68. Bias decoupling using resonant circuits allowed the circuit to operate with a very low Vcc rail voltage. The current for the Gilbert cell transistors was selected to provide a good match to the emitter, which is the load for the RF

574

Microwave Mixer Technology and Applications

amplifiers. The emitter impedance of the switching quad is proportional to the transconductance of the bipolar devices, so bias was selected to provide the desired load impedance to the RF amplifier. The mixer provided a conversion gain of 14 dB, noise figure equal to 7 dB and image rejection in the order of 35 dB.

90º RF

90º

+

LNA 0º 0º LO

90º Hybrid

Figure 8.67

Conventional image rejection architecture. After [45].

Figure 8.68

Schematic of down-converter. From [45].

IF

Bipolar Junction Transistor Applications

575

8.5.3 Quasi-Zero IF Image Reject In direct conversion receivers the demodulated signal spectrum remains at low frequencies, starting at DC. This reduces the component cost compared to double conversion due to the reduced parts count and removal of the IF filter. As a result much attention has been paid to direct receivers for the cell phone market, which is extremely cost sensitive. Some tradeoffs exist, including DC offsets at the IF output caused by circuit imbalance, and the flicker noise corner frequency prevalent in CMOS. DC offsets are particularly difficult if they vary with time because they appear as part of the desired signal, corrupting its integrity. The DC and flicker noise problems can be avoided by slightly increasing the IF frequency. The use of a quasi-zero IF, in which the IF is low but still relatively far from DC, allows use of reasonably sized filters. Image rejection is required by both zero-IF and quasi-zero IF receivers, but only about 15 dB rejection is required by zero-IF since the image level is known because it is the lower half of the channel. In contrast, in quasi-zero IF, the image is separated from the desired signal by twice the IF, and can have a much higher level depending on whatever signals are present, requiring image rejection up to 60 dB or so. Image rejection using filters is difficult due to the very high resonator Q to achieve adequate selectivity, thus image rejection using quadrature mixing is the preferred approach. The image rejection mixer block diagram shown in Figure 8.67 uses two mixers. Its image rejection is limited to amplitude and phase imbalance to the first order. In contrast, if four mixers are used with differentially connected quadrature circuits, then higher image rejection will be achieved for the same circuit balance, as image rejection is limited by phase imbalance squared (second order). It is still limited to the first order by amplitude imbalance. An alternative to the conventional IRM topology is to use polyphase filters, [47], which have the capability of discriminating between a positive and negative sequence of voltage vectors. If the signal and image are up-converted by two mixers whose LO’s are in quadrature, then the two IF output signals are described by the equations: 1 V01  sin[( LO   IF )t ] sin  LO t   cos  IF t 2 1 V02  sin[( LO   IF )t ] cos  LO t   sin  IF t 2

(8.48) (8.49)

The two IF outputs for the desired signal are cos() and sin()=cos(+90) and the two IF outputs for the undesired image are cos() and -sin()=cos(-90). The two vector pairs are applied to the polyphase filter of Figure 8.69, and seen to add constructively for the desired IF (+,+), and cancel for the undesired image (+,-). In Figure 8.69(a), the cos() signal is applied at the top, and becomes cos(+45) at the output. The cos(+90) signal is applied at

576

Microwave Mixer Technology and Applications

the bottom, and also becomes cos(+45) at the output, so both signals add constructively. In contrast, in Figure 8.69(b), the cos() signal is applied at the top and again becomes cos(+45) at the output. But the cos(-90) signal is applied at the bottom and becomes cos(-135°) at the output, so the two signals cancel. Therefore, the circuit is capable of suppressing the upconverted image. V0

V90 

V0

V0+45

V0+45

V-90

V-90 - 45= V -135

V90-45 =2V45 Figure 8.69

(a) (+,+) addition Simplified two cells of a polyphase filter.

=V45+V-135 =0 (b) (+,-) cancellation

If Gilbert cell mixers are used in a four-mixer IRM, then eight polyphase cells are required: four for each mixer, and twice that for differential connection. The bandwidth of effective image suppression is narrow for the single-stage polyphase filter of Figure 8.69. To increase bandwidth, more stages are added in cascade as shown in Figure 8.70, [48].

Figure 8.70

Cascade of polyphase filters for two of the four Gilbert mixers.

A quasi-zero IF mixer designed after this principle is shown in Figure 8.71, using four CMOS FET ring mixers connected differentially using polyphase filters for the LO, RF, and IF circuits [47].

Bipolar Junction Transistor Applications

Figure 8.71

577

Block diagram of the polyphase IRM. From [46].

Due to the reduced sensitivity to phase imbalance, the mixer achieves average image rejection of 58 dB over a 10 MHz frequency range near DC, as shown in Figure 8.72. This represents a very wide percentage bandwidth at the IF output, and is obtained using a five-stage polyphase filter.

Figure 8.72

Performance of the polyphase IRM, comparing simulation with measured results on more than 5 samples. From [46].

578

Microwave Mixer Technology and Applications

8.5.4 Note on Polyphase Filters The polyphase filter can be understood by analyzing an I,Q generator using a high pass and low pass filter as depicted in Figure 8.73. The transfer function is represented in terms of the Laplace transform, s, for the path IN-to-I by (8.50), and for IN-to-Q by (8.51).

H I ( s) 

I ( s) 1  IN ( s) 1  sRC

(8.50)

H Q ( s) 

Q( s ) sRC  IN ( s) 1  sRC

(8.51)

Figure 8.73

I,Q generator with high pass, low pass filters.

100

HQ

HI

10-1

-2

10

102

Figure 8.74

104 Frequency (Hz)

106

Phase of HI(s) and HQ(s)

Amplitude of HI(s), HQ(s)

The magnitude of the responses are shown in Figure 8.74(a) for the low pass and high pass paths. The phase responses are in Figure 8.74(b), negative for the low pass and positive for the high pass. At the 3 dB point (0.707xVoltage) the two signals have equal magnitude with 3dB loss relative to input at IN, and the phase is -45C and +45C respectively.

(a) Magnitude Frequency response of an I,Q generator

100 50 0

HI

HQ

-50 -100

102

104 Frequency (Hz)

106

(b) Phase

In the reverse the path, the transfer function becomes (8.52). A simple poly-phase filter is represented in Figure 8.75, comprising two instances of the circuit of Figure 8.73 connected back-to-back.

Bipolar Junction Transistor Applications

I i ( s) 

QsRC I  1  sRC 1  sRC

(8.52)

C

R

Ii C

I Q

Figure 8.75

QBi

R

QB

R

Qi

579

IB C

C R

IBi

Two high-pass low-pass circuits back-to-back.

If the two pairs of quadrature input signals, I i, Qi and IBi, QBi are 180 apart in phase (differential), then the connection points of the two pairs are virtual grounds to each other, and (8.52) applies. The output currents in terms of applied currents are given by (8.53a, b).

I ( s) 

Ii Q sRC  Bi 1  sRC 1  sRC

I B ( s) 

I Bi Q sRC  i 1  sRC 1  sRC

Qi I sRC (b)  i 1  sRC 1  sRC

(a)

Q( s ) 

(c)

QB ( s ) 

QBi I sRC (d)  Bi 1  sRC 1  sRC (8.53)

The filter is analyzed with the input signals given by the set of equations described below and the output current given by (8.54). Taking only the frequency effects the Laplace variable becomes, 0 = 1/RC.

I i (t )  cos(t ) Qi (t )  cos(t  90) I Bi (t )   cos(t ) QBi (t )  cos(t  270) I (t ) 

1 1 cos(t  45)  cos(t  270  45)  2 cos(t  45) (8.54) 2 2

580

Microwave Mixer Technology and Applications

Take now the case of negative frequencies, which results after the conversion process. The output current is now given by (8.55).

I i (t )  cos(t )

Qi (t )  cos(t  90)

I Bi (t )  cos(t )

QBi (t )  cos(t  270) 1 1 I (t )  cos(t  45)  cos(t  270  45)  0 2 2

(8.55)

Polyphase filters distinguish positive from negative frequencies resulting from signal and image down conversion, at the frequency where the equality 0 = 1/RC holds. The rejection properties for the single stage poly phase circuit is in Figure 8.76(a). The effect of more RC cells tuned at different frequencies results in larger bandwidth rejection, depicted in Figure 8.76(b). Positive Frequency 100

0

Amplitude

Amplitude

10

10-2

10-2 Negative Frequency

103

4

10

105

Frequency (Hz)

Figure 8.76

(a) Single-stage response Polyphase filter responses.

0

frequency

1=1/R1C1 2=1/R2C2 3=1/R3C3 (b) Three-stage response

8.6 SUBHARMONIC TOPOLOGIES BJT and HBT devices compete with the antiparallel diode pair (APDP) in subharmonic mixers. The advantage with the use of active devices is lower conversion loss or even conversion gain. Examples of subharmonic mixers using GaAs, SiGe HBTs and conventional Si BJT follow.

Bipolar Junction Transistor Applications

581

8.6.1 Complimentary Approach A simple circuit proposed in reference [49], emulates the properties of the APDP for subharmonic conversion using bipolar transistors. The topology of the mixer is depicted in Figure 8.77. It contains a diplexer to combine the LO and RF signals to feed a pair of complimentary NPN-PNP transistors. The emulation is possible because the base emitter junction of these devices have similar properties to simple diodes. The currents I1 and I2 contain the same harmonic components as diode mixers, except that instead of conductance modulation, transconductance modulation is employed. Subharmonic conversion is feasible because g m is modulated at twice the LO frequency, as depicted in Figure 8.78. Depending on matching and harmonic terminations in the circuit, conversion gain can be obtained. C i = i1 +i2 IF

LO

i1

LO

i2 IF

RF RF Figure 8.77

Complimentary transistor subharmonic mixer. i

gm = gm1+gm2 V T

t

T

Figure 8.78

Complimentary transistors transconductance modulation.

In analytical terms the transconductance of both devices is given by (8.56). Introducing the LO modulating voltage in the equation as V=VLOcos(LOt) one obtains (8.57). The Fourier expansion in the next equation shows there is no fundamental LO component in the transconductance.

gm  gm1  gm 2  I S (eV  eV )

(8.56)

582

Microwave Mixer Technology and Applications

g m  2I S cosh(VLO cos  LO t )

(8.57)

g m  2I S I 0 (VLO )  2I 2 (VLO ) cos2 LO t   2I 4 (VLO ) cos(4 LO t )  ... (8.58) A detailed circuit schematic of the IC is represented in Figure 8.79, where the input signal is fed to a pair of emitter followers, Q 1, Q2 that provide bias and buffering for the mixing stage, Q3, Q4. After mixing, another emitter follower is employed for level shifting, and the output stage combines the currents from both collectors. DC bias and Emitter follower

Level Shifter

R3 R2 RFin

HPF

Q5 Q1

Q2

Q3 Q4

LPF LOin

3

R5

R6 Q6

R1

Q7 R7

Q8

IFout

R8

R4 Core Cell Figure 8.79

Current Combining

Schematic of complimentary pair subharmonic mixer.

The plots in Figure 8.80(a) show down conversion performance, for a fixed IF at 7 MHz, RF swept from 500 MHz to 5 GHz, and LO swept at half-RF frequency plus IF. LO power is +5 dBm and RF power is -10 dBm. The result was obtained with no filters attached nor impedance matching. Down-conversion performance is flat within a couple of dB over 1 to 4 GHz and peaks positively at 2 GHz with 2 dB gain. The up-conversion gain is negative, but still comparable to conventional mixing with -7 to -9 dB within the same band. The fundamental conversion is in the same plot at under -30 dBc suppressed. The plot in Figure 8.80(b) shows the 2XLO-RF isolation below -60 dBc at the IF port, and below -120 dBc at the RF port. The fundamental LO leaks with almost no loss at both RF and IF ports.

Bipolar Junction Transistor Applications

(a) Conversion Gain Figure 8.80 Experimental results for IC SHM.

583

(b) Isolation

An alternative approach has been proposed for subharmonic conversion, [50], composed of: 1. frequency doubler stage to generate second harmonic of LO; 2. RF amplifier for low noise performance; 3. addition of both signals and subsequent application to a single ended base mixer. The mixer in this case operates at its fundamental frequency which is twice the LO frequency. 8.6.2 Push-Push Self-Oscillating Mixer Another interesting alternative approach is the use of a push-push oscillator-mixer configuration patented in 1989, [51] and displayed in Figure 8.81. The oscillator makes use of the fact that capacitive emitter impedance can generate negative resistance. The capacitor C in differential mode terminates the emitter capacitively and a resonator circuit R is inserted between the bases of both devices. The signal to be converted is applied to a buffer device T 3 and to the base of current sources T4, T5. The collector current of those devices becomes the emitter current of T 1, T2, generating mixing products in each device. Since the signal is applied in common mode, it does not "see" the capacitor C. The following mixing products are available at node N: -Fundamental LO signal and fundamental conversion are suppressed due to phasing. -Second LO harmonic adds constructively. -Harmonic converted signals are in phase and add constructively. -Fundamental LO can be tapped from the emitter of either T 1 or T2 and delivered to a prescaler for frequency control or other application. The patent was developed on silicon 3 µm emitter geometry for applications on Direct TV operating at C-band (3.7 - 4.2 GHz). The LO fundamental frequency is at 2 GHz and the converted band is from 950 to 1750 MHz. The circuit required +5V and – 2V biased with 6 mA.

584

Figure 8.81

Microwave Mixer Technology and Applications

Push-push oscillator-mixer.

8.6.3 Cross Coupled mmWave Mixer The exploitation of subharmonic mixing using SiGe devices operating at 60 GHz is reported in [52]. The circuit functions by cross coupling two devices to operate as a subharmonic mixer, depicted in Figure 8.82. The circuit is differential in DC but at mmWave the collectors are both shorted to ground via the open stub. The LO signal at half the frequency is applied to the base of Q 2 and to the emitter of Q1 via coupling capacitor C4. The circuit used for RF impedance matching, C1,T1, can be made to short the base of Q1 at the LO frequency.

Figure 8.82

Schematic of a SiGe Subharmonic mixer. From [51].

Bipolar Junction Transistor Applications

585

Therefore, the first device is common emitter and the second is common base at this frequency, generating harmonics in antiphase at the respective collector currents. At RF, at double the LO frequency, Q 1 has its emitter shorted to ground by LO impedance matching circuit C4, T2. The RF signal is applied in common emitter to transistor Q1 and common base to transistor Q2. The differential IF signal is extracted from the collector of each device. Measured conversion loss for fIF=fRF–2fLO with fIF=500 MHz and LO power at 0 dBm is given in Figure 8.83. Figure 8.83(a) is for fRF swept over 55 to 65 GHz, showing conversion loss between 5 and 7.8 dB. Figure 8.83(b) shows IF bandwidth with fRF swept over 51.5 to 57.5 GHz, fLO fixed at 27.25 GHz, and IF varying between 0 and 3 GHz. The IF bandwidth is roughly +/- 1 GHz for a 1 dB loss variation. 0

Conversion Gain (dB)

Conversion Gain (dB)

0 -3 -6 -9 -12

55 Figure 8.83

-3 -6 -9 -12

55 60 65 GHz -3 -2 -1 0 1 2 3 GHz (a)Versus RF frequency. (b) Versus IF frequency. Measured conversion gain of the mmWave subharmonic SiGe mixer. From [51].

8.7 SUMMARY This chapter provided a brief summary of selected patents on the application of bipolar devices to mixer functions. The presentation is more or less chronological, with an attempt made to cover the important aspects of this technology. For example, while most designers avoid using transformers, there are instances where it still is the best choice. The transition from discrete components to integration into a monolithic substrate followed the same motivation that motivated Fairchild to develop the IC technology, cost reduction and in many instances improved performance. An insight into those innovations are covered in this chapter with examples of multifunction in a single die, containing mixer, RF amplifier and local oscillator. The chapter closes with examples of image reject and subharmonic converters where the conventional functions of transformers and couplers are replaced by new circuit approaches.

586

Microwave Mixer Technology and Applications

REFERENCES [1] Irving F. Barditch et al, “Monolithic Semiconductor Mixer Apparatus with Positive Feedback,” US Patent 3,107,331, issued October 15, 1963. [2] Bernhard Birkenes, “Mixer Circuit for Autodyne Receiver in Which Untuned Coil Couples Signal to Intermediate Frequency Transformer,” US Patent 3,165,700, issued January 12, 1965. [3] David John Carson, “Mixer Circuit,” US Patent 3,694,756, issued September 26, 1972. [4] Seishi Watanabe and Takashi Yoshikawa, “High Frequency Amplifier with Frequency Conversion,” US Patent 3,949,306, issued April 6, 1976. [5] Max W. Muterspaugh, “Transistor Mixer,” US Patent 4,850,039, issued July 18, 1989. [6] Jakob Zawels, “Non-Linear Semi-Conductor Signal Translating Circuits,” US Patent 2,890,418, issued June 9, 1959. [7] Heinz Georg Karl and Borse Mais Ingmar Roos, “Modulator Employing a Pair of Transistors Biased for Class AB Operation,” US Patent 3,469,212, issued September 23, 1969. [8] Erwin Glock, “Active Balanced Modulator Circuit,” US Patent 3,636,478, issued January 18, 1972. [9] F. J. Cerny, Jr., G. D. Helm, and R. G. Wesoloski, “Balanced Active Mixer Circuit,” US Patent 4,193,036, issued November 3, 1980. [10] William J. Howell, “Balanced Mixer Using Complimentary Devices,” US Patent 4,080,573, issued March 21, 1978. [11] Heinz Rinderie and Hans Sapotta, “Additive HF Mixer,” US Patent 5,732,344, issued March 24, 1998. [12] Reginhard Pospischil “Double Balanced Modulator,” US Patent 3,124,767, issued February 27, 1962. [13] Toshiaki Sudoh, “Double-Balanced Modulators of the Current Switching Type,” US Patent 3,614,668, issued October 19, 1971. [14] Albert Boubouleix, “Broad-Band Passive Ring Mixer,” US Patent 4,317,230, issued February 23, 1982. [15] Tokinor Kozawa, Kokubunji-Shi, “Frequency Converter Circuit,” US Patent 3,555,303, issued January 12, 1971. [16] Thomas K. Lisle, Jr. and Joseph Romanchak, “Amplitude Modulator Having a Transistor Controlled Bias Current,” US Patent 3,866,148, issued February 11, 1975. [17] Daniel B. Talbot, “Amplitude Modulator Having Substantially Zero Modulation Distortion,” US Patent 4,485,359, issued November 27, 1984. [18] Mitsuo Ohsawa and Wataru Yamanati, “Double Balanced Frequency Converter,” US Patent 4,058,771, issued November 15, 1977. [19] Max E. Malchow, “Frequency Converter, as for First Detector of Heterodyne Radio Receiver,” US Patent 4,253,196, issued February 24, 1981.

Bipolar Junction Transistor Applications

587

[20] Kenzo Tanabe, Junji Suzuki, and Masashi Kanno, “Transistor Balanced Mixer,” US Patent 4,461,042, issued July 17, 1984. [21] Issy Kipnis and Amarpal S. Khanna, “Computer Simulation Models Performance of RR Converters,” Microwaves & RF, May 1989, pp. 183-192. [22] Christopher Trask, “Low Distortion Lossless Feedback Double Balanced Active Mixer Using Linearity Augmentation,” US Patent 6,242,964B1, issued June 5, 2001. [23] Howard E. Jones, “Dual Output Synchronous Detector Utilizing Transistorized Differential Amplifiers,” US Patent 3,241,078, issued March 15, 1966. [24] Robert R. Sinusas, “Double-Balanced Modulator Circuit Readily Adaptable to Integrated Circuit Fabrication,” US Patent 3,550,040, issued May 31, 1968. [25] Barrie Gilbert, “Four Quadrant Multiplier Circuit,” US Patent 3,689,752, issued April 13, 1970. [26] Barrie Gilbert, “Double Balanced RF Mixer with Predetermined Input Impedance,” US Patent 5,826,182, issued October 20, 1998. [27] Barrie Gilbert, “The MICROMIXER: A Highly Linear Variant of the Gilbert Mixer Using a Bisymmetric Class-AB Input Stage,” IEEE Journal of Solid State Circuits, Volume 32, No. 9, September 1997, pp. 1412-1423. [28] Su-Tarn and John R. Long, “A Low Voltage Broadband FeedforwardLinearized BJT Mixer,” IEEE Journal of Solid State Circuits, Volume 41, No. 9, September 2006, pp. 2177-2187. [29] Patrick A. Quinn, “A Cascode Amplifier Nonlinearity Correction Technique,” IEEE International Journal of Solid State Circuits Conference, February 1981, pp. 188. [30] Nhat M. Nguyen, “Low Noise Active Mixer,” US Patent 5,379,457, issued January 3, 1995. [31] G. Khoury, R. D. Beards, and J. J. Nisbet, “Linear Low Noise Mixer,” US Patent 5,532,637, issued June 2, 1996. [32] Akira Shibata and Yoshizumi Watatan, “Integrated Circuit for Frequency Conversion,” US Patent 4,216,431, issued August 5, 1980. [33] K. Tanabe, J. Suzuki, and M. Kanno, “Balanced Modulator,” US Patent 4,344,188, issued August 10, 1982. [34] Rinaldo Graziadei and Michelangelo Lorusso, “Transistor Mixer and Amplifier Input Stage,” US Patent 4,480,337, issued October 30, 1984. [35] Joan Nyquist, “Circuit for Making a Differential Output Single-Ended,” US Patent 5,001,372, issued March 19, 1991. [36] Gilles Chevallier and Eduard F. Stikvoort, “Transformer Circuit, Double Balanced Mixer,” US Patent 5,825,231, issued October 20, 1998. [37] Joseph P. Heck, “Balanced Mixer Circuit with Improved Linearity,” US Patent 5,548,840, issued August 20, 1996. [38] Nader Fayyaz, “Low Voltage Mixer,” US Patent 6,335,651B1, issued January 1, 2002.

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Microwave Mixer Technology and Applications

[39] David E. Bien, “Even Order Term Mixer,” US Patent 5,465,414, issued November 7, 1995. [40] Roger Branson, “Mixer Using Four Quadrant Multiplier with Reactive Feedback Elements,” US Patent 6,255,889B1, issued July 3, 2001. [41] J. B. Groe, “Active Radio Frequency Mixer Circuit with Feedback,” US Patent 6,205,325B1, issued March 20, 2001. [42] Russel D. Wyze, “Active Commutated Double Balanced Mixer,” US Patent 6,230,001B1, issued May 8, 2001. [43] A. Jongepier, W. Kasperkovitz, “Double-Balanced Mixer Circuit,” US Patent 4,636,663, issued January 13, 1987. [44] Didier Belot and Pascal Persechini, “Class AB Differential Mixer,” US Patent 6,882,194B2, issued April 19, 2005. [45] J. Macedo, M. Copeland, and P. Schvan, “A 1.9 GHz Silicon Receiver with On-chip Image Filtering,” IEEE 1997 Custom Integrated Circuits Conference, pp. 181-184. [46] John R. Long, “A Low Voltage 5.1-5.8 GHz Image Reject Downconverter RF IC,” IEEE Journal of Solid State Circuits, Volume 35, No. 9, September 2000, pp. 1320-1328. [47] Behbahani, Kishigami, Leete, Abidi, “CMOS Mixers and Polyphase Filters for Large Image Rejection,” IEEE Journal of Solid State Circuits, Volume 36, No. 6, June 2001, pp. 873-887. [48] P. J. Mole, H. V. Vliet, and C. Babla, “Image Rejection Mixer Circuit and Method for Image Rejection,” US Patent 6,226,509B1, issued May 1, 2001. [49] Kevin Kobayashi, “Complimentary Bipolar Harmonic Mixer,” US Patent 6,901,249, issued May 31, 2005. [50] T. K. Johansen, J. Vidkjaer, A. Konczykowska, M. Riet, F. Jorge, and T. Djushuus, “A High Conversion-gain Q-Band InP DHBT Sub-Harmonic Mixer Using LO Frequency Doubler,” IEEE Transactions on Microwave Theory and Techniques, Volume MTT 56, No. 3, March 2008, pp. 613-614. [51] N. P. Cowley, R. J.Lawton, and T. D. S. McClelland, “Frequency Doubling Oscillator and Mixer Circuit,” US Patent 4,810,976, issued March 7, 1989. [52] B. G. Perumana, et al, “A SiGe Sub-Harmonic Mixer for Millimeter Wave Applications,” Proceedings of the Second European Microwave Integrated Circuits Conference, Munich, Germany, October 2007, pp. 80-83.

Chapter 9 FET Mixer Theory MESFETs became widely accepted and implemented into microwave/mmWave mixers due to their superior high frequency performance compared with silicon bipolar devices. However, in recent years this distinction has diminished with the advent of HBTs in either SiGe or GaAs technologies. But active MESFETs and PHEMTs in GaAs technology are still widely used at high frequencies due to their simpler circuitry. With the recent advances in CMOS technology to obtain very small gate dimensions, the fT for CMOS FETs has become comparable to their GaAs counterparts, resulting in a drastic change in technologies available for mixers. Today CMOS mixers are the preferred choice for high volume applications, and their design is similar to any other type of FET. As a matter of fact, their I/V relationships are similar, as confirmed by (9.1a) for MOSFETs and (9.1b) for GaAs FETs.

W Vgs  Vth 2 L 2 I ds   Vgs  Vth  I ds  B

(9.1a) (9.1b)

With: B = µnCox (mobility multiplied by oxide capacitance per unit area), =IDSS/Vth2 (Current at zero gate bias divided by squared threshold voltage) The dimensions W and L, respectively, correspond to channel width and length, and Vth to threshold voltage, the gate voltage above which drain current starts to flow. The difference from the circuit point of view is in the values of the constants and the fact that MOS FETs for RF applications are usually enhancement devices, and GaAs FETs used in industry are both enhancement and depletion type devices. One of the drawbacks of MOS compared to GaAs MESFETs is its lower breakdown voltage for the same gate length, which limits the operational power level. Any of the FET devices can also be operated in passive mode, allowing development of mixers with performance not attainable 589

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Microwave Mixer Technology and Applications

either by diodes or bipolars. Passive operation is preferred in applications requiring low noise figure, high linearity, and circuit simplicity. A simple examination of a depletion mode FET static I/V characteristic allows one to conclude that in spite of its widespread usage for linear amplification, it is a very non-linear device. Nonlinearity exists for drain current versus voltage in two regions: at the onset of current saturation where current is more a function of VGS, ID(VGS); and in the pinch-off area, where current is more a function of VDS, ID(VDS). Coincidently these are the areas where a device can be biased and driven by a large signal voltage, generating the maximum variation (derivative) of the parameter responsible for the frequency conversion, either conductance or transconductance. Depending on the topology employed, one parameter can dominate the other, and in most cases both participate in the conversion process. The options for generating distortion in the drain current for active mixer design are similar to those for bipolars, depending on where the LO is injected, as depicted in Figure 9.1. ID

IG

RIF

RIF

VLO IS

RIF

VRF VLO

VRF (a) Figure 9.1

ID

ID

VRF

VLO

(b) (c) FET mixer classification: (a) gate injection; (b) source injection; (c) drain injection.

LO injection at the gate modulates the transconductance, clipping the drain current to generate harmonics. LO injection at the drain generates harmonics by modulating both transconductance and output conductance. The disadvantage of this approach compared to gate injection is the larger LO power required to modulate the current. LO injection at the source has the same effect as drain injection but the circuit design is more complex due to instabilities that can occur when reactive elements are introduced between source and ground. A special case of LO gate injection is the resistive mixer, with a configuration similar to drain injection with RF and LO generators swapped. The LO applied to the gate modulates the device channel resistance and mixing is generated by application of RF signal voltage at the drain. The mixer action can be achieved either by a linear multiplier or by first combining signals and then applying them to a non-linear element. The former approach requires a circuit with a high degree of complexity, while the latter can be obtained with the simple circuits of Figure 9.1. The calculation of the spectral components of the drain current from a FET for a mixer circuit is complex, requiring development of multiple Fourier integrals, one for each input signal including the LO. This task can only be handled by computer analysis.

FET Mixer Theory

591

Fortunately, the conventional “small-large signal” approximation where the signals to be converted are of much lower amplitude compared to the local oscillator component is applicable in linear mixing. This condition allows simplification of multiple Fourier integrals to a single one at the LO frequency. In this chapter the basic concepts of frequency conversion for the various FET configurations are discussed.

9.1 GATE LO INJECTION The circuit schematic is represented in Figure 9.2, with RF and LO applied at the gate, and IF extracted at the drain. It is assumed the RF and image have similar terminations, which is true for a down-converter with low IF frequency. The IF is shorted to ground at the gate by the bias decoupling stub, 3. The phase angles of the transmission lines in the figure are defined at the RF frequency. Note the RF and LO generators share the same impedance, which is convenient for circuit simulation. A typical means to combine RF and LO is with a directional coupler, where RF uses the direct port and LO uses the coupled port. The gate matching circuit consists of an open stub, m2, and a series line, m1, designed for best energy transfer at the RF frequency. If the IF frequency is much lower than RF, then the LO generator impedance match is close to that of the RF.

4 = 90 m1

ZS

3 =90

VRF VLO Figure 9.2

m2

+

VGS

Lm

Lbias

+ -

RL VDS

Cm

Schematic for a gate mixer.

On the drain side, an open stub, 4, which is a quarter wave long at the LO frequency shorts the LO voltage to ground. This stub also causes the impedances at the RF and image frequencies to be low, which minimizes the effect of output conduction non-linearity and simplifies the analysis of gate mixers. At IF the open-circuit drain stub is a small reactance, and drain bias is introduced by a large reactance inductor, Lbias. The IF output impedance usually ranges from 75 to 100 ohms. The Lm, Cm elements provide the proper impedance match to the external load RL, and also acts as a low-pass filter to reject undesirable signals.

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Microwave Mixer Technology and Applications

9.1.1 The Quadratic Mixer This is a popular representation for mixer study due to its simplicity and capability to describe the conversion effects in closed form equations. The quadratic I/V description in (9.3a) is found in Curtice's paper, [1] for the case n = 2, and is valid for a gate voltage that is less than VP and equal to zero volts or higher. The total gate voltage is given by (9.2) where the LO and RF are added to the DC supply voltage. Note it is also assumed RF and LO voltages have the same phase. In some engineering texts the drain equation is described by (9.3b), which essentially the same as Curtice, if  is replaced by its definition, IDSS/Vth2 and dropping the built in Vbi voltage from the equation defining the threshold voltage Vth = Vp + Vbi. The threshold voltage is measured between the gate and source and determines the voltage at which drain current starts to flow.

Vgs  VGS  VRF cos(RF t )  VLO cos(LOt )

(9.2)

I ds   Vgs  Vth  (1  VDS ) tanh(Vds ) , if Vgs < Vth

(9.3a)

2

 V I d  I DSS 1  GS VP 

n

  (1  V DS ) tanh(V DS ) , if Vgs < Vp 

(9.3b)

I ds  0 , if Vgs > Vth Where,  = DC output conductance of device;  = fitting parameter defining the drain voltage above which the drain current becomes saturated. Substituting (9.2) into (9.3b), with n = 2, and developing the terms, the drain current components can be defined as follows:

 V I ds (t )  I DSS 1  GS  V P

2

2

  V RF  V     cos 2 ( RF t )   LO   VP   VP

2

  cos 2 ( LO t )  .. 

 V V  V V  21  GS  RF cos(RF t )  21  GS  LO cos(LOt )  ...  VP  VP  VP  VP  V V  2 LO RF cos(RF t ) cos(LOt )(1  VDS ) tanh(VDS ) VP VP 

(9.4)

FET Mixer Theory

593

The operating point is selected to be at the middle of the gate transfer characteristic, ( i.e. VGS = - VP/2, IDS =IDSS/4), and drain voltage is in the middle of the saturation and breakdown voltages, VDS = (VB -Vsat)/2. For maximum LO swing at the gate, the LO generator voltage is set at V LO = VP/2. The LO voltage has to take into account the RF voltage that adds up at the gate. If it is assumed the RF voltage is of small amplitude then total gate voltage is approximately given by the LO voltage. The output conductance coefficient is defined by  = 1/Rd0IDSS. An additional assumption is made for the drain voltage swing: it is assumed that total voltage will be larger than the saturation voltage and lower than the breakdown voltage. Under these conditions the hyperbolic tangent becomes equal to unity. Including these assumptions, (9.4) reduces to:

 1  2  V I ds (t )  I DSS     RF  2   VP

2

 V 1  cos 2 ( RF t )    cos 2 ( LOt )  RF cos( RF t ).. VP 2  2

 1 V 1  cos(LOt )  RF cos(RF t ) cos(LOt ) (1  VDS ) 2 VP Rd 0 I DSS 

(9.5)

Figure 9.3

3fLO- fRF

f-1

f+1

f-2

fLO+ fRF

fRF

fIF

2fLO

2fLO- fRF

0

fRF -fLO

Ids(f)

fLO

The spectrum for the current is depicted in Figure 9.3, similarly to Figure 5.1. Only the positive frequency components were considered, i.e. f = ± nfRF + mfLO and m, n = 0, 1, 2, 3. Notice that applying fLO and fRF there are two main signals generated, fRF - fL0 and fRF + fLO. The DC component comprises the applied DC and the DC generated from detection. The dotted line components are not in (9.5) and should not be present in an ideal unilateral model; however, they are present in real circuits.

f+2

Spectrum of drain current in a quadratic mixer.

Retaining only the terms containing the mixing products, the drain current reduces to two components as follows:

594

Microwave Mixer Technology and Applications

I ds (t )  I DSS

VRF cos(RF  LO )t  cos(RF  LO )t (1  1 VDS ) 2VP Rd 0 I DSS

(9.6)

The IF current circulates in the output load, generating a drain voltage, VDS = - RLIds(t), which can be inserted into (9.6). The down-converted signal is given by the frequency difference, and the up- converted by the frequency sum. The undesired sum signal is filtered out, with the drain voltage shorted to ground at that frequency. Given these conditions, (9.6) becomes:

VRF cos(RF  LO )t 2VP I ds (t )  R V 1  L RF cos(RF  LO )t Rd 0 2VP I DSS

(9.7)

The second term in the denominator of this equation will be relevant when Rd0 Vdsat, and the gate is switched between the on and off states. However, analytical problems appear when one tries to apply (9.3) to switched operation, due to the discontinuity of gate voltage for part of the LO period. The analytical solution for the drain current requires more sophisticated models. For example, the current can be obtained from a multiple Fourier series, one for each of the input frequencies, with no restriction on signal amplitude. Such a complex approach is not necessary if only one voltage is considered large (i.e., the LO voltage). And the signals to be converted are small (i.e., the RF voltage). Under this condition, two alternative approaches exist to find the solution: use a Fourier series to represent the gate voltage, or apply a modulating function. An example of modeling based on the Fourier series is reported in [2], where the ID, VG is defined by a sum of sine and cosine functions. Overall the majority of solutions found in the literature make use of the same model given in (9.3), with the IF current defined by the product of a modulating function determined by the large signal, and the small signal RF voltage. This is discussed further in the next section. Equation (9.11) also allows comparison between LO power requirements for different technologies. For example, a technology offering lower capacitance and lower series resistance requires lower power. The LO voltage is another important parameter. The LO signal must cause the gate voltage to swing between the threshold voltage, Vth and the forward conduction voltage, Vf. As this difference between threshold and conduction decreases, the required LO drive also decreases. Hence a depletion device with Vth = -0.5V requires less LO power than a device with Vth = -1V. Enhancement mode FETs are particularly attractive since the threshold voltage is a few tenths of a volt on the positive side.

9.1.2 Conversion Matrix Analysis This method was introduced in Chapter 5 for a nonlinear conductance and is extended here to FET transistors. The concept of large-small signal was described

596

Microwave Mixer Technology and Applications

in detail in [3]. It starts by considering a Taylor expansion applied to the I/V relation for the device, I = f(V). The expansion comprises the sum of a large signal, VLO and a small signal component, v. The current is then expressed by the equation: I (VLO  v)  I (VLO ) 

d 1 d2 1 d3 I (V ) v I (V ) v2  I (V ) v 3  ... 2 3 dV 2 6 dV dV V VLO V V V V LO

LO

(9.12) For small signal operation, voltage v 7 dBm and VG = - 1.6 V, and the best IM2 result is at VG = -1.1V. Designing the mixer with a low impedance at the drain for the LO signal voltage minimizes leakage from gate to channel.

4

- 40

6

- 50

8

- 60

10

- 70

PIM3 - dBm

PIM2 - dBm

-1.6V PIM2

- 30 - 40

-1.1V

- 50 -1.1V

fRF - GHz 10.0 10.1 10.2 10.3 10.4 10.5 1.2

1.3 1.4

1.5

1.6 1.7

- 60

- 80

-1.6V

PIM3

- 90

5

fIF - GHz

PLO - dBm

10

(a) Mixer conversion loss (b) Mixer IM2 and IM3 versus LO power Figure 10.3 Mixer performance for PLO = 10 dBm; VG = - 2.1 V and VD = 0 V.

An improvement in this basic circuit that minimizes LO leakage into the channel is found in the patent [6]. The improvement is obtained by adding an isolation reactance, 16, comprising a series capacitor and transmission line that parallel resonates the drain-gate capacitance at the LO frequency. The reduced LO leakage into the drain improves linearity at high LO drive. The mixer, whose schematic is in Figure 10.4, was built on GaAs material, and operates over 32 to 36 GHz.

2 pF

L - 400 W - 45

L - 167 W - 34

RF 16

RIF

1 pF

L – 641 W-5

L - 173 W-5

L - 150 W - 72

2 pF

LO Figure 10.4

L - 150 W - 72

MMIC MESFET with tuned Cgd.

-Vg

L - 150 W - 72

Passive FET Applications

691

The source is terminated by a low pass filter that grounds the port for RF and LO frequencies, and IF is extracted from that port. The RF signal is injected to the drain and filtered from IF through the LC circuit. The LO is injected to the gate, which is biased through a low pass circuit. Over 32 to 36 GHz, with P LO = 15 dBm and VG = -1.9V, the best RF to LO isolation ranged from 15 to 19 dB. The gate drain capacitor is resonated with an inductive line connected in series with a DC blocking capacitor, between the drain and gate. 10.1.4 MESFET Self Biased Mixer The next selected patent, [7], employs a similar L1C3 resonant circuit to improve isolation between gate and drain, and in addition, it introduces a clamp circuit to generate the gate bias from the LO signal, eliminating any external supply bias. L1

C3

C2

C1

C4

L2

RF

LO M1

L4 R1

Figure 10.5

D1

L3 IF

C5

A MESFET single ended resistive mixer wits self biased gate.

The gate circuit comprises a series LC circuit and a clamp diode that adds capacitance to the circuit. The LO generator is matched to the circuit by a parallel reactance, L4 and series reactance L2,C2. When matched, the gate source capacitance is tuned to maximize the LO amplitude. The clamp circuit prevents negative half-cycles from reaching the mixing device, and also avoids reversebreakdown of the gate source junction. The clamping voltage is approximately one diode drop, about 0.7V below ground. The capacitor C 2 and the gate Schottky barrier diode provide a self-bias circuit, which charges the capacitor to the LO peak voltage. The diplexer connected at the drain separates the RF and IF signals. The LO voltage impressed at the gate must swing between pinch-off (-1 V for a typical 0.25 m gate FET) to a few tenths of positive gate voltage (0.5V for the same device). A variation on this gate self bias circuit is found in [8] that adds a parallel RC in series with each gate for LO pulse shaping. It is said to reduce the LO rise and fall times, and thus increase linearity.

692

Microwave Mixer Technology and Applications

Another interesting improvement for single ended mixers was reported in [9], which makes a statistical study of drain terminations at RF and IF frequencies. The study was applied to a 300 µm gate GaAs FET with termination impedances ranging from 0 to 500 ohms for the real part, and -500 to 500 ohms for the reactive part. The conclusions revealed that besides the RF, IF and image termination, the sum fLO + fRF is equally important. This frequency is located near the second LO harmonic frequency in down converter mixers with small IF relative to LO frequency. The optimal embedding terminations for gain are: resistive finite impedance for RF and IF frequencies, high impedance (open) for the sum and image impedances. 10.1.5 CMOS Technology A CMOS version [10], depicted in Figure 10.6, was developed by IMEC on 90 nm CMOS technology with fT = 150 GHz for 60 GHz applications. The LO signal is applied to a device measuring to 2X20 µm2, with transmission line circuits for impedance matching and gate bias insertion. The RF is applied directly to the drain with a coupled line to provide isolation to DC and IF. The reported performance includes 11.6 dB conversion loss at +4 dBm LO drive and - 0.45 V gate bias. Conversion gain drops by 1 dB when the IF increases up to 6 GHz. At this frequency the P1dB is +3 dBm and the IIP3 is +16.5 dBm. The IIP3 improved 2 to 3 dB by applying a low drain bias on the order of 0.1 volts. Vg RF filter

RFin LOin

Matching network

IF filter

IFout

(a) Circuit schematic. After [10]. (b) Photo of chip. From [10]. Figure 10.6 A CMOS single ended mixer for 60 GHz operation.

10.1.5.1 CMOS Series Configuration In another application, [11], the FET is connected in series instead of shunt to ground, to build a resistive mixer to operate within the 20 to 30 GHz frequency

Passive FET Applications

693

range. The circuit was designed on 90 nm SOI CMOS technology by IBM. The circuit schematic in Figure 10.7 includes filters with inductance connected to ground, which is claimed by the author to provide higher Q, a premium parameter in CMOS. The circuit also provides a convenient DC ground, maintaining drain and source at zero volts. RF (fRF= 27 GHz)

Wg = 64 µm

L1

C2

C1 R Vg=0.45V C3 Figure 10.7

IF (fIF = 2.5 GHz)

L2

L3 LO (fLO = 24.5 GHz)

L1 = 180 pH L2 = 1.4 nH L3 = 0.35 nH C1 = 210 fF C2 = 4 pF C3 = 5 pF R = 300 

A CMOS series connected single ended mixer.

The author addresses the theory of conversion loss by first examining how the drain resistance is modulated by the LO voltage in this technology, resulting in the waveforms of Figure 10.8. Conversion loss is evaluated using the first two terms of the Fourier series for the resistance waveform, R ds0 and Rds1. These are expressed in terms of the duty cycle,  of the waveform.

Rds1 sin( )  Rds0  r (t )  Rds0  2Rds1 cos( LO t )  ...



Lconv 

available  input  power 1  1   2  output  power 1 1  2

(10.1a) (10.1b) (10.2)

Conversion loss expressed in terms of on and off resistances, according to Saleh is given by (10.2). The author assumed a duty cycle of 35% based on Figure 10.8, which gives a conversion loss of 5.9 dB. Applying the simplified wave theory where a square wave modulation is assumed, the minimum resistance of 10 Ω introduces a conversion loss of 5.5 dB. Adding the filter and other losses the total predicted conversion loss is in the order of 10 dB. The measured conversion loss ranges from 9.5 to 12 dB within the 26.5 – 30 GHz band,

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Microwave Mixer Technology and Applications

depicted on Figure 10.9(a). The conversion loss versus LO drive power for an input signal at 27 GHz and IF output at 2.5 GHz is in Figure 10.9(b). Additionally, measured IP3 is + 16 dBm, and noise figure is 11 dB, at +5 dBm LO power.

Figure 10.8

Nonlinear characteristic of Rds as a function of LO drive. From [11].

Figure 10.9

(a) Lconv versus fRF (b) Lconv versus LO power Conversion loss of a CMOS series connected single ended mixer. From [11].

10.1.5.2 CMOS Subharmonic X2 A CMOS resistive mixer operating at 28 GHz was connected to a LO frequency doubler as an alternative to using a single ended subharmonic mixer [12]. The doubler is built with a bridge comprising two each N and P CMOS devices driven from a differential LO source. The bridge circuit suppresses the LO fundamental but generates and applies the LO second harmonic to the resistive N-channel

Passive FET Applications

695

mixer. With RF input covering 25 to 29 GHz and IF output at 1.5 GHz, measured conversion loss is 11 to 12 dB, noise figure is 12 dB, and P 1dB is - 2.7 dBm.

Figure 10.10 A CMOS mixer with a LO frequency doubler. From [12].

10.1.6 mHEMT Subharmonic X4 A times four subharmonic mixer built with mHEMT technology for operation at 60 GHz was reported in [13]. The paper highlights contributions in the understanding of channel conductance in subharmonic mixers, and a comparison is given for performance as a fundamental, x2, and x4 mixer. The modulation of the device channel conductance for a fundamental mixer, f LO, a times two subharmonic, fLO/2 and a times four subharmonic mixer, fLO/4, are depicted in Figures 10.11(a) – (c). These theoretical plots show the channel is switched once per LO cycle in a fundamental mixer, twice in a times two, and four times in a times four.

(a)

(c)

(b)

Figure 10.11 Channel conductance modulation for x1, x2, and x4 LO harmonics. From [13].

696

Microwave Mixer Technology and Applications

The author points out the conduction waveforms are ideal, but in real circuits the nonzero “on” resistance and device capacitance slow the switching of channel conductance at mmWave frequencies, causing simultaneous conduction of FETs that in turn degrades the conversion loss. The circuit schematic for the proposed X4 subharmonic mixer is in Figure 10.12, illustrating the four FET switches, the RF/IF network, and the LO feed network. The drain circuit is simple, consisting of an RF/IF diplexer with the RF filter composed of an edge coupled line, and the IF low pass filter comprising a transmission line with two open circuit shunt stubs. The LO circuit on the other hand is quite complex. It includes one section of impedance matching, an LC quadrature splitter network and a lattice balun.

Figure 10.12 Schematic of X4 mixer. From [13].

Figure 10.13 compares the performance of different 60 GHz mixer types: single ended resistive mixer, (SRM); single ended resistive mixer with a four times lower LO pump frequency, (X4SRM); times 2 subharmonically pumped resistive mixer, (X2SHPRM); and times 4 subharmonically pumped mixer, (X4SHPRM). All four designs use 2X50 µm2 mHEMT devices having fT equal to 120 GHz. The IF is 200 MHz, and the best bias voltage for each design, respectively, was -0.75V, -1.15V, -1.0 and -1.8V. The fundamental mixer has the best conversion loss as expected, but conversion loss of the times two mixer approaches it at LO power above 10 dBm. The X4 version has conversion loss around 15 dB for LO power ranging from 5 to 15 dBm.

Passive FET Applications

697

Figure 10.13 Performance comparison between four mixer types. From [13].

10.2 FLOATING APPROACH For this class of mixers the switching devices have no common ground connection, and instead “float” above ground potential. The advantage is simpler and more compact circuits. For example, a singly balanced design usually comprises two FET devices with grounded sources, with RF current at each drain in counter phase due to transformer action. In the floating approach a single FET device is used with the drain and source currents in counter phase with each other. This way a singly balanced mixer can be built with one device and a balun. Floating topologies fit well with resistive mixers, but are not a good option for active mixers due to the difficulty in controlling circuit stability. 10.2.1 The MESFET Sandbox A generalization of floating mixer circuits appeared in the patent literature, [14], advocating the application of this topology for various circuit functions to make a better use of the symmetrical I/V properties of FET devices. The proposal was made for MESFETs but it equally applies to other FET technologies. The proposal represented in Figure 10.14 is a kind of electronic sandbox whose functions are described next. 10.2.1.1 Singly Balanced Mixer The circuit was simulated in ADS employing the NE673 MESFET device. The parameters were set to L1 = 0, C1 = 1 pF, L2 = 20 nH, C2 =100 pF, and the

698

Microwave Mixer Technology and Applications

transformer ratio is 1.2. In addition the LO and RF were impedance matched with a series line and an open stub. With the RF signal ranging from 9.95 GHz to 10.05 GHz, and a fixed LO frequency of 10.1 GHz, the performance results are given in Figure 10.15. The LO drive power is equal to + 13 dBm and the gate is biased at 1.2 V. P1 = LO input P2 = Vg bias at pinch off P5 = IF out P7 = RF source Switches as drawn = on position

S2

S3 S5

P1 P2

S1

S4

P9 P8 P7

L1 C1 C2 L2

P6 P5 P4 P3

Figure 10.14 Typical sandbox circuit with a floating FET.

0

0

-1

-6

-2

-12

-3

-18

-4

-24

-5

-30

-6

-36

-7

-42

-8

-48

-9

-54

-10 9.950E9

-60 9.970E9

9.990E9

1.001E10

RFfreq

Figure 10.15 Single balanced floating mixer conversion gain.

1.003E10

1.005E10

LR_dBc

Gc_dB

The top plot indicates the conversion loss (< 6 dB) referenced to the left axis and the bottom indicates the L - R isolation, corresponding to better than -54 dBc on the right axis.

Passive FET Applications

10.2.1.2 Amplifier The fact that the source is now floating gives rise to instability. This problem is solved by shunting the gate with a 75 ohm resistor. The bias inductors are 20 nH each and the input/output matching circuit is similar to the one applied in the mixer. The resulting 5 dB gain is flat within the band 9.95 to 10.5 GHz, which is about 6 dB lower gain compared with conventional source grounded operation. P1 = RFin P5 = -0 V

P6 = RFout P9 = + 3V

P2 = Vg bias linear Switches all off

10.2.1.3 Frequency Tripler Operation as a frequency tripler under high LO drive causes the source-gate and drain-gate voltages to alternate between positive and negative polarities. This modulates the channel conductance, causing it to have a square wave shape with a strong third harmonic component. In this passive multiplier, the LO drive power is equal to 13 dBm and the results are in Figure 10.16. The generated third harmonic is - 5 dBm and the second harmonic rejection is more than 30 dB. The fundamental output level is about 0 dBm and can be minimized with a simple filter. P7 = fLO P2 = -1.3V P5 = 3fLO Switches all off

0

0GHz =-0.505

.

m4

m5

-10 Spectrum

0GHz =-5.209

699

-20 -30 -40 0.000

10.0G

20.0G

30.0G freq, Hz

Figure 10.16

Spectrum of the floating frequency tripler.

40.0G

50.0G

60.0G

700

Microwave Mixer Technology and Applications

10.2.1.4 Voltage Multiplier The circuit can be operated as a voltage multiplier at low voltages, where the applied drain voltage is multiplied by the variable channel resistor. This fourth application proposed by the reference patent is obtained by making the following settings: P1 = Vg () P2 = Vg bias at linear P6 = Vd () P4 - P8 = Vab =K Vg()Vd() Switches all off Further development of this work is found in [15], where the output balun, required to cover IF and RF frequency ranges, is replaced by a modified balun operating at RF frequencies only. The circuit inserted in the center tap of the transformer’s primary winding allows extraction of IF without the use of an IF balun. The external gate voltage supply can be suppressed by connecting a resistor to ground. The LO peak voltage will be rectified through the drain ground connection, developing a self-biasing voltage on the resistor. RFin LOin

-Vgs IFout Figure 10.17 Floating mixer with modified IF circuitry.

10.2.2 Basic Vice Floating Circuit A series of inventions proposed by Michael W. Vice have a common basic diagram as illustrated in Figure 10.18. This topology has two devices connected in series, with the gates and sources, respectively, tied together. The LO is applied across the gate and source, while the RF signal is applied across the series connected FETs. The following analysis demonstrates how this topology minimizes inter-modulation distortion. It is known that LO leaks into the channel through the gate-drain and gate-source capacitances, which are undesired effects degrading linearity. The voltages in the FET are given by the following equations, and the control voltage is applied across the gate-source terminals.

Passive FET Applications

701

Vgd1  Vgs1  Vsd1

(10.3)

Vgd2  Vgs2  Vsd2

(10.4)

Load D1

Vgd1

Vcontrol

G1

Vsd1

Vgs1

S1

Vgs2

S2

G2

Small signal

Vsd2

Vgd2

D2

Figure 10.18 Basic Vice back-to-back connection of two FETs.

However, symmetry requires that V sd1 = -Vsd2, so that a voltage perturbation ∆Vgs equates to ∆Vgd. Therefore, a voltage perturbation applied to the gate can be approximated by the next equation, which gives a linear relation between dynamic resistance, ΔRds, and gate to drain voltage, ΔVgd.

Rds  cVgd

(10.5)

Let’s now assume the channel resistance variation ∆Rds1ds2 causes intermodulation. Minimum inter-modulation is obtained if this variation tends to zero, hence: (10.6) Rds ds  c(Vg d  Vg d )  c(Vs d  Vs d )  0 1

2

1 1

2 2

1 1

2

2

The effect of LO leakage in the channel is suppressed for the signal extracted to the load, resulting in the invention of Figure 10.18, [16], in which the drain/source are floated. Additionally, the circuit variation of Figure 10.19 has the gates floated by means of a transformer. The elements L1, C1, TL, C3 comprise a diplexer to separate RF and IF signals, and provide impedance matching. The TL transmission line element is a quarter wave long at the RF frequency. The element L2, helps improve impedance matching at the LO frequency. The power gain and compression simulation for the RF band over 10.025 to 10.175 GHz with LO fixed at 10 GHz is represented in Figure 10.20. The NE67300 device Root model available in ADS is used in the simulations.

702

Microwave Mixer Technology and Applications

C1 VG

Lbias M1

LOpor t

T = 1:1

Figure 10.19

L2

RFport

L1

C4

TL

C3

IFport

C2

M2 m3 ind Delta=-29.500 L2 dep Delta=1.090 Delta Mode ON m2 RFpower=15.500 Floating gate mixer and matching elements. Lc_dB=-7.151

L1 = 0.33 nH C1 = 0.5 pF TL = 50 /82 C3 = 0.2 pF L2 = 0.4 nH

0

10

-1 0

-2

-10

-4 -5

-20

m3

-6

m2

-7

Pout_dBm

Lc_dB

-3

-30

-8

-40

-9 -10

-50 -40

-30

-20

-10

0

10

20

RFpower

Figure 10.20 Power performance at 10 GHz; PLO = + 10 dBm.

The circuit provided a conversion gain of 6.25 dB, output P 1dB in excess of + 15 dBm, and L-to-R isolation of 28 dB, for a LO drive level of + 13 dBm. The fact the P1dB is 2.5 dB higher than the LO drive gives an indication of the high linearity achieved with this approach. C1

C3

M1

L2

L1

RFpor t

IFport

LOport

M2

M3 M4 Figure 10.21 Circuit with better LO impedance match.

C4

C2

Passive FET Applications

703

The author also proposed adding the devices M3, M4 to maintain the same capacitance seen by the LO generator in either the positive or negative halves of the LO cycle. The added symmetry improves LO matching but is not essential to the mixer linearity performance. 10.2.2.1 MOS Technology The use of transformers for impedance matching and circuit balance in a FET MOS mixer is addressed in this patent [17]. The singly balanced version in Figure 10.22 has the drain and source connected to two ends of the primary transformer windings, 81. The other two ends of the primary windings are connected to voltage source, V3. Voltages on windings 84 and 86 are opposed in phase. The source V2 is applied to a secondary winding that magnetically couples signals to the primary in counter phase. Source V1 is applied across the virtual ground of V3 and the gate of the FET, which is approximately in the center of transformer primary winding, 81. Notice the sources V2 and V3 are balanced so they are orthogonal to V1, but they are not isolated from each other. If the impedances presented to the FET are low, then the voltages developed will be low, and less distortion will be generated. In this circuit the signal voltages V 2 and V3 can represent RF, IF or LO sources or loads. V1 can represent an input source only, either LO or RF.

84 V3

V2

81

86

VDC

V1

Figure 10.22 Simplified singly balanced version,

The circuit of Figure 10.22 is capable of performing the multiplication function, as shown in the following analysis. The drain current is first defined for the device with ungrounded source:

704

Microwave Mixer Technology and Applications

ID 

V V Z eff Ci [(VG  VT  D )VD  (VG  VT  S )VS ]  otherterms L 2 2

(10.7)

This can be approximated as: ID 

Z Z eff Ci [(VG  VT )VD  (VG  VT )VS ]  eff Ci (VG  VT )(VD  VS ) L L

(10.8)

The circuit is approximately symmetric with equal drain and source voltages that are opposed in phase. The drain and source voltages can be referenced to a midpoint between source and drain, V R/2.

VR 2  VR VS  2 VD 

(10.9) (10.10)

Substituting (10.9) and (10.10) into (10.8), the drain current becomes:

ID 

Z eff Ci [(VG  VT )VR ] L

(10.11)

In the floating circuit, there will be currents circulating between drain and source, and currents circulating between drain and source to ground. Therefore, the circuit can be considered as two FETs in parallel driven differentially, with the gate voltage comprising even and odd components. VG = VGE +/- VGO

(10.12)

The common (even) mode drain current becomes: I DE  2

Z eff Ci [(VGE  VT )VR ] L

(10.13)

For differential (odd) mode the VGE term is equal at the source and drain, so the differential current becomes:

I DO  2

Z eff CiVGOVR L

(10.14)

Passive FET Applications

705

A perfect multiplication function is obtained for the differential mode. If the substrate is not grounded, then the voltages need to be subtracted by the substrate voltage, VB, and similar conclusions are obtained. A doubly balanced configuration obtained from the singly balanced is shown in Figure 10.23. Windings 64 and 66 have center taps, across which a balanced signal V3 is applied. One end of winding 66 is connected to an end of winding 64 through the drain-source of FET 72, and a similar condition is found for FET 70. The signal source V3 is impressed in phase to the drains and in counter phase to the sources due to the cross connected transformer terminals. The FETs are connected in series at DC. Another signal source, V2 is magnetically coupled in phase to the sources and in counter phase to the drains. Bias can be applied from the center tap to a virtual terminal within source V1. A time domain analysis shows that when both device conductances are similar, the signal impressed by V 2 is cancelled at V3. If FET 70 conducts, it provides a low resistance connection from the top of winding 64 to the bottom of winding 66. The opposite occurs when FET 72 conducts. The impressed voltage V2 appears at V3 with a magnitude depending on the device conductance, and phase that shifts 180° as the polarity of signal V 1 varies. IF signals generated by the mixing of V2 and V1 are in counter phase at each device and therefore combine at the output V3. Similar to the singly balanced case, the signal at V1 is always an excitation signal, while V 2 and V3 can be either an excitation or a load. V1

70

VDC

72

62

60 V3

V2 64

66

Figure 10.23 Doubly balanced floating mixer.

An application example of the mixer is shown in Figure 10.24, where the RF signal is applied at node 104 to the gate through transformer 102, and the LO is applied at node 105 to the drain and source through transformer 60A. The circuit operates as a multiplier for small signals per (10.14), and as a mixer if a large signal LO signal switches the FET channels on-off. The IF signal is

706

Microwave Mixer Technology and Applications

extracted from the center taps of transformer 60A and fed to a common gate differential pair presenting a low impedance to the mixer. The single ended output is obtained by taking the signal from one side of the differential amplifier.

Figure 10.24 Mixer including LO and IF couplings.

10.2.2.2 Single Transformer An alternative to the previous work was proposed in a patent referenced in [18], where the IF balun is eliminated. In this proposal, illustrated in Figure 10.25, the gates of both devices are connected together, so the LO signal is in phase at each drain. The source of the bottom device is RF, LO and DC grounded while the source of the top FET is grounded only at RF and LO by means of a capacitor. The RF is applied to the drain by means of a transformer balun, so the RF signals at the drains are in counter phase to each other. The IF currents at the two drains are opposed in phase; (e.g., leaving FET2 and entering FET1), but combine in phase at the source of FET 1. RF and LO signals are both grounded at the sources. A similar description applies to the up converter case, where the generated RF voltages are differential and applied to output load. The LO signals are common and suppressed by the differential transformer at the RF output. The LO is coupled to the gates via a capacitor C1, which allows for passive self-biasing of the FETs if a resistor is added to ground. Capacitors C3 and C4 in the original invention are used for fine adjustment to balance the signals, and capacitor C 6 is used to improve symmetry of the RF signals.

Passive FET Applications

707

IF D1 G1

C3

VRF LO

S1 S2

C1 G2

VRF

RF C6

C2

C5 C4

D2 Figure 10.25 Elimination of the IF balun from the FET mixer.

10.3 SINGLY BALANCED A balanced resistive mixer usually refers to a pair of single ended mixers combined with a 180º transformer coupled to the gate of each mixer. This suppresses even harmonics of LO and provides high L–R isolation. In some cases a 90º hybrid is used instead, causing suppression of both LO and RF, but this also gives inferior L–R isolation. The following section first discusses mixers using discrete devices, and the second section discusses MMIC implementations. 10.3.1 Hybrid Topologies 10.3.1.1 JFET Technology The balanced mixer patent, [3], mentioned at the beginning of this chapter records one of the first instances using the channel resistance as a mixer. In this proposal, two JFET devices are mounted in a push-pull type of configuration using transformers to generate the LO in counter phase with the IF signals added in counter phase at the output. The schematic shown in Figure 10.26 depicts its application as a receiver. The ideal IF output includes only odd LO harmonics as shown by (10.15).

I1   gmnVRFm e j ( nLOt  mRF t )  I mn

I 2   gmnVRFm e j ( nLOt  mRF t )e jn  I mn ( 1)n

I out  I nm[1  ( 1) n ] = 0 for even n; ≠ 0 for n odd

(10.15)

708

Microwave Mixer Technology and Applications

Thus fundamental mixing occurs and products with LO second harmonics are suppressed. The odd harmonics add constructively, but inclusion of resonant circuits further suppresses the additional undesired components. The RF signal incident to the antenna is coupled to the FET drain by the tuned transformer. Due to the balancing of IF and LO signals, the point of insertion of the RF signal is a virtual ground for the LO and IF, creating L–R and I–R isolation. The LO signal is applied to the gate through transformer T 1, which is part of the gate bias circuit. The DC bias is usually set close to the pinch off voltage. IF out RF in

I1

I2

LO in Figure 10.26 Singly balanced receiver with JFET.

It is assumed both devices have matched DC and RF characteristics. Small imbalances can be compensated by adjusting capacitors 38 and 40. The IF signals generated at the drains are combined at transformer T 2 and delivered to the IF amplifier. The transformer T 2 is tuned at the RF frequency, so ideally there is no loss of RF energy. Besides the L-R isolation provided by the circuit topology, the LO voltage modulates the channel resistance and there is no LO current in the channel, except for a low leakage level due to parasitic capacitances. The mixer noise figure is low, equivalent to the conversion loss of a diode mixer, which is around 6 dB. Consequently this circuit can be directly connected to the antenna circuit. It also provides a very wide dynamic range without any significant distortion.

Passive FET Applications

709

10.3.1.2 DSB Modulator The same concept from the previous patent was later employed in a balanced configuration proposed to up-convert an audio input signal to a DSB output signal with high carrier suppression. The schematic of the invention, [19], is shown in Figure 10.27. Specifically, the modulator up-converts a base-band audio signal to a 450 KHz IF carrier with low inter-modulation distortion (< - 40 dBc) and high carrier rejection (> than 40 dBc) with minimum adjustments. Since discrete components were used for this invention, the FETs must have matching characteristics to achieve the required performance. The FET’s are in “push-push” for the carrier signal. That means the input carrier or LO signal is split into two counter phase signals through transformer 19 and applied to the control electrode gates of FET’s 13 and 22. The output LO signals are cancelled by tying the drains together. The source of FET 22 is grounded and the source of FET 13 is AC grounded at the LO frequency through capacitor 15. The 450 KHz LO voltage at the transformer secondary output is approximately equal to 500 mV peak-to-peak. This peak gate voltage is below the junction voltage of silicon, so the forward conduction at the gates is negligible. The FET’s are in series configuration for the audio signal applied to the source of FET13. The audio current is first applied to the channel for FET 13 and then to the channel for FET22, where it mixes with the carrier signal. The DC current in the channel flows to ground through resistor 16. It is assumed the Utilizing Equipment, or next circuit, has a filter to block the audio signal. The carrier is applied in counter phase to each FET while the audio signal is applied in phase, so that (10.15) is equally valid, therefore, the up-converted currents will be in phase. The two sideband currents generated around the 455 KHz LO frequency flow to the next stage.

Figure 10.27 Schematic of a JFET balanced modulator.

710

Microwave Mixer Technology and Applications

10.3.1.3 Square Wave Drive in MOSFET Technology The noise contained in the LO spectrum is transferred to IF in the conversion process. The process is rather complicated and the end result is an increase in the effective receiver noise figure. In some cases the oscillator can be filtered, but the best approach is to use a balanced mixer that will attenuate this effect proportionally to the degree of balancing. However, in wide band mixers the problem is aggravated by the presence of LO harmonics and their associated noise. In particular the effects of FM on noise from harmonics are more pronounced. The mixer can reject even mode effects but is useless for odd mode, which is the case of noise contained in the third harmonic of LOs. The objective of this invention, [20], is to provide a linear balanced mixer with a special LO waveform that reduces the third harmonic effect. VA +



VRFi

Vout

-

n

VB Figure 10.28 Ideal swith model of a balanced mixer.

The balanced mixer is modeled after Figure 10.28, where the individual mixer is represented by ideal switches, A and B. They are driven in counter phase generating IF currents that are also in counter phase and require a differential circuit for combining constructively at the final IF signal.

Figure 10.29 Time domain depiction of voltage waveforms VA and VB.

Passive FET Applications

711

Each of the digital waveforms V A and VB shown in Figure 10.29, are periodic with a period T, equal to the reciprocal of the oscillator frequency, f. The oscillator waveforms are shifted by one half of the oscillator period, so they are 180 out of phase with each other. The duration of each pulse is equal to 1/3 of the total oscillator period T. For a 50% duty cycle, the sinc function reveals the presence of fundamental and third harmonic, while the second harmonic is suppressed, waveform 40 in Figure 10.30. The duty cycle for the waveforms in Figure 10.29 are not equal, so, according to the sinc function (10.16), the third harmonic is suppressed. Notice the resulting waveform 30 contains, besides the fundamental a small content of second harmonic.

X( f ) 

T0 nT 1 n sin c( 0 )  sin c( ) T T 3 3

(10.16)

n = 1, 2, 3…

Figure 10.30 Spectrum of IF signal when LO voltage waveform of Figure 10.29 is used.

But the second harmonic will be eliminated by the balanced circuit. The effective output waveform is the one represented by the dotted line, waveform 50 in Figure 10.30. The full circuit is in Figure 10.31 and illustrates one way of digitally generating the waveforms VA and VB using an oscillator frequency 6 times higher than the mixer LO. The circuit makes use of a synchronous 4-bit counter, type D flip-flops, and NAND gates.

–

Figure 10.31 Schematic of linear mixer with modified LO waveform.

712

Microwave Mixer Technology and Applications

10.3.1.4 MESFET Technology The gate/drain capacitance of FETs used in resistive mixers is large compared to the capacitance of a normally biased device, where it is reverse biased across gatedrain. This reduces the R-L isolation and allows LO signal applied to the gate to leak into the channel, and RF signal from the channel to leak into the gate, both degrading inter-modulation performance. A simple solution was disclosed in 1990 [21], where the LO voltage at the drain, and RF voltage at the gate are shorted.

Figure 10.32 Singly balanced resistive MESFET mixer.

10

-30

8

IM3 Level - dBm

Conversion Loss - dB

Conversion Loss - dB

The short circuits can be implemented by filters or signal balancing. The circuit above depicts the topology of a singly balanced MESFET mixer, where the RF is applied in phase to the drain of the balanced devices by means of a quarter wave long impedance transformer. The transformer functions as a RF power divider that must be interrupted at IF frequencies to avoid shorting the IF currents, which are 180 out of phase. Also, a small value capacitance shorts both FETs at the RF and LO frequencies, while providing a largely open circuit for IF signals. The LO is injected to the gates by means of a half wave 50 ohm microstrip balun between the two gates. Therefore, the LO voltages at the gates are 180 out of phase with each other.

6 4 2

10.0

10.2 10.4

10.6

Frequency - GHz

10.8 11.0

-50 -70 -90 15

10

15

20

LO Power - dBm

(a) Conversion Loss vs. frequency (b) IM3 vs. LO power at Vg = -1.1V Figure 10.33 Conversion loss as a function of frequency and linearity as a function of LO power.

Passive FET Applications

713

The LO signals leaking to the drain are also 180 out of phase, and are shorted by means of capacitor 12. The RF signals leak in phase to the gates, and are cancelled by the common mode rejection of the balun. The IF signals available at X and Y, in counter phase from each drain, are combined by means of a balun transformer, depicted in the left of Figure 10.32. Simple low pass transmission line filters are employed to separate IF from RF and LO signals at the drain. The circuit was tested at an RF frequency of 10 GHz with IF from 10 to 300 MHz. The conversion loss as a function of RF with IF fixed at 60 MHz is shown in Figure 10.33(a). Effects of LO power and DC bias can be found in Figure 10.33(b), showing inter-modulation levels for an RF input drive of - 3.0 dBm. The best result at 18 dBm LO drive is obtained for a gate bias of -1.1 V. 10.3.1.5 3-dB Hybrid Coupler The advantages of 90 couplers can also be realized in singly balanced mixers, [22], shown in Figure 10.34, depicting two couplers, one connected to the gates and the other to the drains. The RF and LO ports have good impedance match, with reflected energy dissipated into the 50  isolated ports. Both LO and RF are in quadrature in the mixing device and the IF is dependent on the circuit operation, (i.e., IF currents for a down-converter have different phases than for an up-converter). For example, let’s consider the down-converter case. The signals at the FET channels are given by the following relations:

Figure 10.34 Down-converter with 90 3 dB couplers.

VA (t )  ALO cos( LO t )VRF cos( RF t )   VB (t )  ALO cos( LO t  )VRF cos( RF t  ) 2 2

(10.17) (10.18)

714

Microwave Mixer Technology and Applications

Developing the equations one obtains the following IF currents at the drains of FETA and FETB:

k k cos( RF   LO )t  cos( RF   LO )t 2 2 k k I B (t )  cos[( RF   LO )t ]  cos[( RF   LO )t   ] 2 2 I A (t ) 

(10.19) (10.20)

Where k is a constant representing the amplitudes of mixed signals. It is obvious from the equations that down-converted signals are in phase and can be tied as represented in the figure, while the up-converted signals cancel. A simple LC low pass circuit at the connection is enough to reject undesired signals at the IF output. These equations indicate that for the up-converting case, the drains cannot be connected together at the IF side. Let’s assume (10.19) nand (10.20) are the same for the up-converter case, with the RF index replaced by the IF index. Let’s also assume IF current IB(t) is applied with a 180 hybrid.

I B (t ) 

k  k 3 cos[( LO   IF )t  ]  cos[( LO   IF )t  ] 2 2 2 2

(10.21)

At the output, current IA does not have any phase shift, while current IB is shifted by 90. Rewriting the equations at the coupler output port, including coupler phase shift, the new set of equations are (10.22) and (10.23). It is now evident the signals are in counter phase for the difference frequency, and have the same phase for the sum frequency.

I A (t ) 

k k cos(LO  IF )t  cos(LO  IF )t 2 2

I B (t ) 

k k cos[( LO   IF )t   ]  cos[( LO   IF )t  2 ] (10.23) 2 2

(10.22)

Therefore in the up-converter case the IF currents at the drain have to be split by a 180 hybrid. An additional solution is to extract IF from the drain of FETA and from the source of FET B that is grounded by capacitors. This is an attractive solution for lower frequencies where grounding is less critical and an IF balun can be eliminated.

Passive FET Applications

715

10.3.1.6 Bifilar MOS Technology The next patent, [23], relates to application of MOSFET devices to bifilar transmission line transformers to build high intercept mixers. The transformer used to apply LO power to the gates of devices Q1, Q2 is a Guanella type 4:1 balun transformer described in Chapter 4. This causes LO voltage at the gates to be equal in amplitude and opposed in phase. The bifilar transformers T 1, T2, T3 connected to the drain comprise a different type of balun. The applied RF signal is balanced by transformer T 1 , with the balanced output voltages are subsequently phase inverted by T2, T4 before being applied to the drain. The RF signals are applied to the drains in counter phase, and are multiplied by the channel resistance switched by the LO signal also in counter phase. Consequently the IF signals at each drain are in phase with each other and add constructively at the IF terminal. The IF signal is blocked at the RF terminal due to the high impedance that the differential transformer T1 offers to the common mode IF. At the input a parallel inductance was added to resonate the FET capacitance. The IF terminal is DC connected to the FETs resulting in IF response down to DC, an important feature in direct conversion receivers.

Figure 10.35 Schematic of bifilar CMOS single balanced mixer.

10.3.1.7 Trifilar Transformer Resistive mixers can operate at very high power levels and wide bandwidths if the LO, RF and IF signals are coupled properly, [24]. This invention used a bifilar balun transformer connected to a trifilar balun transformer to apply high level RF signal to the FET switches. Both transformers are formed of wire wrapped on a small ferrite core, and comprise a hybrid circuit as discussed in Chapter 4. The schematic of Figure 10.36(a) depicts the balanced LO source feeding the FET gates in counter phase. The blocking capacitors 52 and 53, with resistors 34, 50 and 51, provide self bias for the FETs. The RF signal is applied to the trifilar transformer that provides two counter phase outputs to the FET drains. The IF signal is extracted from the center tap of the trifilar. The effect of switching the

716

Microwave Mixer Technology and Applications

FET channel resistance is displayed in Figure 10.36(b). Waveform A is at the secondary of transformer, so it is floating around +/- VRF. Waveform B indicates the LO switching applied in counter phase to the gate of each FET. Waveform C indicates the resultant waveform available at the IF output terminal of the mixer. This waveform includes the effect that switching has on the incoming RF signal. Simple filtering can be used to eliminate high frequency components. A small drain bias voltage can be applied to optimize the tradeoffs between distortion, conversion loss, and undesired leakage of LO into the RF port.

(a) Schematic. (b) RF, LO, and IF voltage waveforms. Figure 10.36 High level balanced resistive mixer.

10.3.2 MMIC Topologies 10.3.2.1 MESFET Technology A resistive mixer built in a MMIC technology using a RF amplifier appeared in 1998, [25]. Figure 10.37(a) reveals the basic cell for this mixer consisting of a RF amplifier loaded with a current source, delivering signal to the mixer device, M 2. The source of M2 is AC grounded by means of a capacitor that needs to be large enough to cover the low end of the IF range. The LO signal applied to the gate of M2 does not modulate the bias of the RF amplifier, but acts as a switched load. Making a few assumptions for this circuit, a simplified circuit, depicted in part (b) of the figure can be applied to this type of mixer topology: (1) The loading of node 1 is due only to M1, M2, and M3. Subsequent circuits are considered high impedance.

Passive FET Applications

717

(2) The mixer impedance is matched to the RF and IF impedances, which are equal and resistive. (3) The switching mixer does not change the bias point for M 1, M3, so gm remains approximately constant. VDD LO

RF

M3 1 M1

RF

R3

VGS

M2 C

CGS

IF gmVGS

gd

RL

(a) Schematic (b) Equivalent circuit Figure 10.37 Core cell mixer. There is no bias applied to the mixing FET.

The load impedance for the RF amplifier is given by the parallel combination of M2 switched drain-source resistance and M1, M3 drain resistances. The switching of M2 expressed in terms of drain source resistance, and the LO voltage applied to the gate, is given by (9.49) and reproduced as (10.24). The values for Rd0 and Rd1 are defined in (8.49a) and (8.49b).

Rd 2 (t ) 

1  Rd 20  Rd 21 cos  LOt  ... g d (t )  g L

(10.24)

The IF voltage available at the drain can be calculated from the (9.53) on Chapter 9 assuming gm is constant. The result in (10.25) assumes RL is given by the parallel combination of M1, M3 resistances and the waveform for gd(t) is a square wave. The effects of device capacitances are not included, so the equation is valid at lower frequencies. The RON value is in the order of 10 ohms for FETs with gate geometry equal to 0.5x100 m2. The ROFF value is high for small devices, but in general it is shunted by parasitic capacitances.

VdIF g m RL  VGS 

 Rds2 RON     cos( LO   RF )t R  R R L ON  RL   ds2

(10.25)

A value of ROFF between 200 to 300 ohms is usual for GaAs FETs. Let us select RL to be Rds2/2 = 100 ohms, gm = 40 mS, which gives a voltage conversion gain of 0.85. At high frequencies power conversion gain is more useful, obtained from IF power at the load and available power from the source. The following equation assumes a conjugately matched RF impedance.

718

Microwave Mixer Technology and Applications 2

g m RL PIF  Pavs (C gs ) 2 Rin

 Rds2     Rds2  RL 

2

(10.26)

This simplified approach is useful to find first order effects of terminations and to size the FETs in the circuit. The inventor provided balanced LO signals using a differential amplifier. VDD IF+

M5

M7

RF

C1

M3 R1

M6

M8

RF

M4

R1’

R2 VG1 R2’

-VG2

VDD

C2 R4

M1

LO C5 -VG2

R5 -VSS

IF-

-VG2

C2’ R4’ M2 C5’

R5’ -VG2

Figure 10.38 Schematic of full differential mixer.

The overall MMIC circuit is in Figure 10.38, which depicts the ground bypass capacitor shared by both mixers. Both FET mixers M7, M8 are DC connected, but since they are at the same potential no DC current flows from drain to source. 10.3.2.2 PHEMT Technology A couple of PHEMT resistive mixers are described next. The first comes from a publication, [26], where the synthesis method introduced in Chapter 4 is applied to the design of subharmonic mixers. In this publication, instead of using the Sparameter approach, they used the Y-parameter approach. The two-sets of quasilinear simulations are performed at a given LO level, one in up-convert mode to obtain Y11 and Y21 and down-convert mode to obtain Y12 and Y22 for the two port

Passive FET Applications

719

network of Figure 10.39. The Y-matrix representing the network is described in (10.27).

Figure 10.39

Two port representation of quasi-linear mixer. From [26].

 I1IF  Y11 Y12  V1IF   I   Y Y  V   2 RF   21 22   2 RF 

(10.27)

From this matrix, the conversion loss is obtained from (10.28), and the IF and RF admittance for conjugate matched operation are obtained from (10.29) and (10.30). Lc21 



2 Y12

2

A B

(10.28)



2

A  2G11G22  Re(Y12Y21 )  Y12Y21

B  2G11G22  Re(Y12Y21 )  Y12Y21 GIF 

1 2G11

2G

G22  Y12Y21 cos    Y12Y21 2

2

11

G12 B21  G21B12  B22 2G11 1 2G11G22  Y12Y21 cos  2  Y12Y21 2  2G22

(10.29a)

BIF 

(10.29b)

GRF

(10.30a)

G12 B21  G21B12  B11 2G22   Phase(Y12Y21 ) BRF 

(10.30b)

To demonstrate the theory, a subharmonic mixer operating at 43 GHz was reported after the topology in Figure 10.40(a), containing a Lange coupler at the input and an IF balun at the drain side.

720

Microwave Mixer Technology and Applications

(a) Block diagram (b) Load filter Figure 10.40 Quadrature/balanced FET mixer. From [26].

The signals are applied in phase to the drain, by means of a load filter displayed in Figure 10.40(b). It includes a low-pass filter with two open stubs at the fundamental LO for high rejection of signals at this frequency. The authors reported a conversion loss of 12 to 15 dB within the 2 to 5 GHz IF band. The P 1dB is equal to + 15 dBm, and the 2LO-RF rejection in the linear operation is equal to 20 dBc. Another application of a subharmonic mixer in pHEMT technology for operation at 28GHz employing a discrete balun is found in the literature, [27]. The schematic of the circuit is in Figure 10.41, showing the balun is closely related to the Marchand balun equivalent circuit, where the inductances represent the transmission lines and the capacitances the coupling between the lines. The balun reactances are equal to j70.7  at the center frequency, for equal 50  impedances.

Figure 10.41 Circuit Schematic for balun/balanced FET SHM. From [27].

Passive FET Applications

721

Therefore the generator impedance for each gate is half the value. The RF signal is applied in phase to the devices, providing in-phase IF currents at the drain that add constructively at this frequency with no need for an IF balun. The conversion loss as a function of RF frequency is in Figure 10.42(a) and the isolation in 10.42(b). 60 50

14

Isolation - dB

Conversion Loss - dB

16

12

10 26

27

28

29

30

40

L-R L-I

30 20 12.4

Frequency - GHz

12.8

13.2

13.6

14.0

LO Frequency - GHz

(a) Conversion loss Figure 10.42 Conversion loss for IF = 1 GHz. From [27].

(b) Isolation L-R and L-I

Additional MMIC subharmonic mixer examples, [28], [29], using 0.25 µm PHEMT are represented in Figures 10.43 and 10.44. In the former two, Lange couplers comprise baluns using an area of 0.8X1.4 mm2. The reported performance for this topology is 9.5 dB conversion loss from 34 to 40 GHz with 5 dBm of LO drive. The latter is a quadrature mixer designed for a direct conversion receiver operating between 15 and 20 GHz with an IF of 200 MHz and LO power of +13 dBm. The fundamental LO is split in-phase and then applied to the I and Q channels, respectively, with phases indicated in the figure. Due to push-pull operation even order terms at the drain are minimized, reducing LO leakage to the antenna and preventing DC offset problems that are critical in direct receivers. The Lange couplers were meandered to minimize area, and the whole circuit fits into an area of 2.1X2.9 mm2. RF

RF filter

Vg

IF IF filter Figure 10.43

Singly balanced subharmonic mixer. After [28].

LO

722

Microwave Mixer Technology and Applications

I

0

RF 0

LO

0

180 0

Q

180 Figure 10.44 Quadrature sub harmonic mixer. After [29].

The mixers were built on 0.25 µm PHEMT technology, and best conversion loss was 14 dB at 200 MHz IF. The plot below shows the converter is efficient up to nearly 500 MHz. The LO-RF isolation is better than 30 dB over the 15 to 20 GHz band.

Conversion Gain dB

-10 -20

-30 102

Frequency - MHz

103

Figure 10.45 Performance of quadrature subharmonic mixer.

10.3.2.3 InP Technology An example application of this technology is found in the realization of an IRM operating in the 92 - 96 GHz band, [30]. The topology in Figure 10.46 shows the use of two Lange couplers, one for the IF signal at 1.5 GHz and one for the RF signal at 94 GHz. The LO is injected by a Wilkinson divider and the separation of IF, RF signals is performed by simple high pass, low pass filters. A contribution from this paper is the discussion relating technology parameters and mixer performance. The paper shows the high mobility offered by InP FETs that allows a much lower transition time between ON-OFF states, which is demonstrated next. The drain current of a PHEMT can be defined as a function of device geometry and technology parameters expressed by (10.31), valid for low drain voltages.

Passive FET Applications

W  I DS   q nVDS nch L IF1

723

(10.31)

RF

LO

USB

IF2 Figure 10.46 Image reject mixer schematic. After [30].

The current is directly proportional to the electron mobility, µn and to the sub-threshold channel electron density at the source, nch. The channel conductance is defined by the derivative of current with respect to drain voltage, according to (10.32). Gch 

dI DS  W    q n nch dVDS  L 

(10.32)

The subthreshold density can be expressed as a function of gate voltage, Vg, per (10.33). The capacitance per unit area, C g is determined from gate metal area and distance from GaAs channel, d, according to (10.34).

qnch  Cg (V gVth )  C g  WL d

(10.33) (10.34)

The next expression demonstrates how fast the conductance changes with gate voltage, expressed in terms of mobility and capacitance. Therefore, the device with higher mobility and lower capacitance will "transit" faster.

dGch  W  dVG  L

  n C G 

(10.35)

The ON impedance at the FET’s drain is given by (10.36), which considers the access series resistances, Rs, Rd and the channel conductance Rds.

724

Microwave Mixer Technology and Applications

The channel resistance is device size dependent and is in the same order of magnitude as Rs and Rd. The channel resistance of an FET is determined by R ds = L/(sW)= L/(qµnns), with ns being the free electron sheet density. The OFF resistance is a high value for the real part compared to a standard 50 . But the reactive part will shunt the resistive part and will limit the frequency of operation.

Z ON  Rs  Rd  Rds 1 ZOFF  Rd  jCd

(10.36) (10.37)

The drain capacitance in (10.37), consists of the drain gate capacitance in series with a parallel combination of generator impedance and gate source capacitances. For an unbiased FET, Cgs is equal to Cdg, so a reasonable approximation is to consider Cd = Cgs/2. Therefore, the theory for resistive mixers in Chapter 8 is not quite valid at high frequencies when the open circuit assumption is no longer valid. A coefficient , relating ON and OFF impedances, was defined by (10.38), expressing this ratio in terms of device technology.

 

Z ON L2  Z OFF q n ns d

(10.38)

The conversion loss for this condition is proposed in (10.39), defined by Saleh. The conclusion is that a minimum conversion loss is obtained when  is minimum, achieved with large channel conductance, (i.e., large µn, ns and small gate capacitance). 2   1    LC  1  2 1  1          2

(10.39)

The output conductance and gate capacitance for a 0.1x20 µm2 device as a function of gate bias is shown in Figure 10.47(a, b). From these parameters, one can determine  and estimate LC, to be on the order of 3.2 dB at 90 GHz. Better accuracy was obtained from a nonlinear model based on the equation for output conductance, (10.40) and capacitance, (10.41). Both expressions are similar, where the coefficients g0(C0), g1(C1), g2(C2) and C3 are determined by curve fitting procedures.

g ds (Vg )  g 0 tanh[(g1Vg  g 2 )  1]

(10.40)

Passive FET Applications

Cgs (Vg )  C0 tanh[(C1Vg  C2 )  C3 ]

725

(10.41)

.

(a) Output conductance (b) Performance results Figure 10.47 Capacitance and device conductance versus VG. From [30].

The measure and simulated performance is in Figure 10.48, showing a low dependency on gate bias for a high LO drive power. The measured conversion loss is 9 dB and image rejection is in the order of 20 dB from 92 – 95 GHz.

(a) Conversion loss vs. LO power and Vg (b) Conversion loss and image rejection vs. LO power Figure 10.48 Mixer performance at 90 GHz. From [30].

10.4 DOUBLY BALANCED Doubly balanced mixers comprise four switching devices connected using two baluns, respectively, for the LO and RF. The use of two baluns creates an orthogonality between the RF and LO by circuit balance. This provides high

726

Microwave Mixer Technology and Applications

isolation between LO, RF, and IF ports, and cancellation of the even LO and RF harmonics. In some doubly balanced configurations, the IF outputs are combined using a third balun allowing the mixing elements to be completely floating. 10.4.1 Hybrid Technologies 10.4.1.1 MESFET_ MIC_Technology This circuit uses cold (unbiased) FETs in a ring configuration built with discrete devices, [31]. The circuit schematic contained in Figure 10.49 shows the gates of opposing FETs are connected together, and the LO signal is applied to these two pairs of gates. Similarly, the drain electrodes of adjacent FETs are connected together and the RF signals are applied to opposing drain contacts. The LO and RF signals are applied to baluns 16 and 14, respectively. In this work, the LO and RF baluns are also employed as impedance transformers from 50 ohms to 100 ohms. The IF signals are also generated in balanced form and can be extracted by another balun, or can be unbalanced as represented in the figure. The mixer was constructed to down-convert signals from 2 to 12 GHz within an IF band from 10 MHz to 2GHz.

Figure 10.49 Ring mixer with discrete FETs.

The invention takes advantage of the ring topology where the RF connections are a virtual short to the LO connections and vice versa. The circuit was built with packaged devices, type NE900089 by NEC, which requires special care on assembling to minimize parasitics. Figure 10.50 illustrates the proposed assembly, where advantage is taken of the fact that both sources can be tied together for each pair of devices. The structure is based on a diode mixer, consisting of a LO balun in orthogonal planes compared to the RF balun. Both baluns have a taper in the ground plane from the unbalanced port where signal is injected, to the point where a horse-shoe shaped balun is derived. On the LO side one arm of the balun is twisted and connected to the gate

Passive FET Applications

727

junctions, while the other arm is connected directly to the gates. The twist was necessary to maintain the phase angles at both gate junctions the same.

Figure 10.50

Ring FET mixer with packaged devices.

A similar topology is applied to the RF signal, but instead of twisting the transmission line, a jumper is inserted in one of the arms to bring the balanced signal to the drain connections. An illustration of the FETs details are in Figure 10.51. Note the twisting of the gate connections on the top. The sources are on the plane of paper and are either connected as indicated in the figure or connected to a bifilar balun for signal extraction.

Figure 10.51 Cross view (3) of assembling of packaged FETs.

A self bias for the gate or external supply voltage is required and not shown in the figure. The results for conversion loss indicated in Figure 10.52 shows a value ranging from 7 to 9 dB within the RF band of 2 to 12 GHz. The VSWR is below 3:1 for RF frequencies below 10 GHz and rises to 6:1 over 10 to 12 GHz. Input third order intercept (IP3) is given by curve 60 for LO at +20 dBm, showing input IP3 ranging from 25 to 30 dBm.

728

Microwave Mixer Technology and Applications 60 ’

30 20 15

60

Input IP3 - dBm Conversion Loss dB

9 7 5

VSWR

3 2 1

0

2

4

6

8

10 12

RF frequency - GHz

Figure 10.52 Performance results of FET mixer.

Curve 60’ shows that increasing LO drive to +23 dBm slightly improves IP3. The same IP3 performance as 60’ was achieved with LO power at +19 dBm by adding a small DC bias to the drain. 10.4.1.2 MESFET Vice Mixer The patent from Michael W. Vice, [32], depicted in Figure 10.53, is similar in almost all aspects to the one described in the active circuits section, (Chapter 11, [24]), except that no external bias is applied. It is also a balanced version of the mixer of Figure 10.18.

Figure 10.53 Floating DBM with self-generation of gate bias.

The resistors R3, R4 and the capacitors C1, C2 act with rectified gate current to provide a small forward self-bias to the FET. An interesting contribution is the use of a trifilar transformer to combine RF and IF signals

Passive FET Applications

729

whose equivalent circuit is in Figure 10.54. The inventor assigned the name reflection transformer to this component. The FETs switch the RF signal so that it appears at the IF port with reversing phase with each LO cycle, which is equivalent to mathematical multiplication of the RF and LO signals.

Figure 10.54 Transmission line circuit for a trifilar transformer.

The construction details of this mixer are in Figure 10.55, where the trifilar transformer is identified as item 20 assembled within a TO-8 type of package. The equivalent transmission lines T1, T2 have a combined impedance of Z0/2, which add in series with similar impedance of T 4,T5, matching the RF impedance to 50 ohms.

Figure 10.55 Construction details, the trifilar balun is item 20.

On the other hand T1, T2 are in series with respect to IF port, making a total impedance of 2Z0, which are paralleled with T4, T5, given a matched Z0 impedance for the IF port as well.

730

Microwave Mixer Technology and Applications

10.4.1.3 H-Mode Mixer As with the diode, the doubly balanced FET mixer can be realized as a star as well as a ring. One configuration, known as the H-mode FET mixer, uses four FETs, with the four sources grounded, and the four drains connected through a hybrid-Tee to the RF and IF ports. It is said that grounding the sources improves linearity by reducing the RF signal voltage appearing at gate, thus reducing the ability of the RF signal to perturb the switching waveform that ideally is controlled by the LO at the gates. Input intercept point levels in excess of +50 dBm at HF frequencies have been reported using this type of circuit, [47]. 10.4.2 MMIC Technologies 10.4.2.1 MESFET Technology An integrated version of a ring FET mixer with an active balun is found in this design of a full converter on a single die, [33]. In reference to Figure 10.56(a) the LO input power is applied to a pair of amplifiers driving the LO ports 1,2 from the quad mixer. The RF signals are applied to another balun connected to ports 3,4 of the quad. The IF signals are delivered to ports 3,4.

Figure 10.56

(a) Block diagram MMIC MESFET quad converter.

(a) Power Splitter Figure 10.57 Active power splitter and combiner.

(b) Quad FET mixer

(b) Power Combiners

Passive FET Applications

731

The FET quad comprising the mixer core is in part (b) of the figure showing the MESFET connections. The quad mixer employed was built with device sizes capable of exhibiting IP3 > +30 dBm. The baluns used in blocks 11 and 15 were obtained by employing an integrated active power splitter followed by all-pass networks capable of rotating the phase +90 and -90 on each port. The applied power at terminal 40 is equally applied to each of the four gates due to circuit symmetry. Between each gate there are two drains that independently apply power to the output load. One grounded via exists to ground the sources. The gate and drains are biased by means of resistors, and resistive feedback networks are applied from each output to the input to improve stability and impedance matching. An active combiner was also employed to add the IF signals and is represented in Figure 10.57(b). The combiner is a common gate topology, with two sources where both signals are applied in phase. The drain current contains the sum of both currents and signal power is extracted from this point. Current sources are applied to each source and drain for proper device biasing. Transmission line 75 represents matching added to recover gain and power performance. All these functional blocks were incorporated onto integrated circuit chips on GaAs substrates to operate within the 8.5 to 10.5 GHz band for the RF and LO signals. The IF band is located within the frequency range of 0.5 to 2.0 GHz. The obtained conversion gain is equal to 8 dB and the IP 3 is better than 30 dBm. The authors proposed an alternative to the active balun and power combiner, using a distributed approach, Figure 10.58. The basic active cell consists of a common gate and a common source device, whose low frequency Sparameters, respectively, were defined in Chapter 4 by (4.57) and (4.58). S21 is the gain from input to CS drain, and S31 is the gain from input to CG drain. The CS and CG output loads both equal g, and the input termination is g1.

S 21 

S 31 

 2 g m1 gg1

g g1  g m 2  g ds   g1 g ds  2 jC gs g  g ds  2g m 2  g ds  gg1

g g1  g m 2  g ds   g1 g ds  2 jC gs g  g ds 

(10.42)

(10.43)

These equations were developed under the assumptions that g m > gd and Cgs, is the only capacitance of importance in the circuit. Under these circumstances, the phases at each drain are different by 180 and voltage magnitudes are the same. With a significant increase in frequency those conditions are no longer valid but can be computer optimized to maintain circuit balance at higher frequencies. In this regard the distributed topology helps by absorbing the input and output capacitances into their artificial transmission lines. The top line

732

Microwave Mixer Technology and Applications

collects signals from the common gate device, so are in phase with the input signal or 0. The bottom transmission line collects the signals from the common source device, phase shifted by 180. The low pass nature of the structure minimizes high frequency harmonics. The schematic of the balun/divider in Figure 10.58 can easily be transformed into a balun/combiner by reversing the balanced output to the input of inductors 30, 39 and by moving the single ended input to the output of inductor 29.

Figure 10.58 Active distributed MESFET balun.

The previous input at inductor 24 is terminated by the line characteristic impedance. The alternative distributed in-phase power combiner shown in Figure 10.59 also contains three transmission lines, the distinction here is the input lines are connected to ports 1 and 2 and the output line is connected to the drain. The design of those structures follows the same approach employed in the design of distributed amplifiers.

Figure 10.59 Active distributed MESFET power combiner.

Passive FET Applications

733

10.4.2.2 PHEMT Technology A self oscillating mixer was proposed [34], to eliminate one of the baluns required by conventional doubly balanced topologies. This is achieved by designing a balanced oscillator directly connected to a resistive quad FET, as indicated in the block diagram, Figure 10.60. It was designed on 0.25 µm PHEMT technology to operate at 25 GHz.

Figure 10.60 Converter block diagram. From [34].

The oscillator is of the push-push type, where the anti-phase operation is locked by a transmission line connecting both gates in the manner indicated in Figure 10.61(a). Its length is calculated to invert the gate impedance with respect to the other gate creating one of the conditions for oscillation, (i.e., the sum of reactances equals to zero). The second condition is the generation of negative resistance, obtained by attaching a capacitor to the source. The quad FET is represented each by a mixer block in the figure, and the overall performance achieved by the mixer circuit is an average conversion loss of 12 dB, with similar performance in the upper and lower side bands. -10

G1-2G

G2

G1

G1-G+jB Figure 10.61

G1-G-jB

Conversion Loss - dB

Transmission Line

-12 -14

-1 -.8 -.6 -.4 -.2 0 .2 .4 .6 .8 1

IF Frequency (GHz)

(a) Oscillator topology (b) Conversion loss performance Oscillator detail and conversion performance. From [34].

734

Microwave Mixer Technology and Applications

10.4.2.3 CMOS Technology Circuit-1 A direct conversion mixer is described that up- or down-converts RF/IF signals using a non-conventional approach where the RF and LO signals are not at the same frequency. It uses a dual conversion approach and orthogonal mixing [35], removing the self mixing of LO that occurs in conventional mixing. In order to fulfill the goal of direct conversion, only one VCO source can be used, and the mixing and harmonic frequencies generated must be selected so there are no carrier components in the converted signal. The schematic represented in Figure 10.62 shows the application of this mixing principle using a Gilbert mixer cell in a nonconventional form. The RF signal is converted by the first mixer, using the usual RF amplifier from a Gilbert cell. The second conversion is performed by the ring mixer, which is the usual LO mixer cell. All mixers are resistive, the LO drive is applied to the gate, and therefore orthogonal to the converted signals. In order to use a single VCO, the frequency of the first and second mixers is generated by frequency dividers or multipliers. The output signals are balanced and applied to differential amplifiers acting as detectors by rejecting the RF through the use of its common mode rejection. An additional op amp is required to combine the two differential outputs.

fRF

0 180

f1

0 180

f2

Figure 10.62 Diagram of direct conversion circuit with two LO frequencies.

The frequency selection for f1 and f2 includes the fact that the final mixer uses an equivalent fLO equal to fRF. The output equivalent fLO is equal to the sum of two signals at f1 and f2, (i.e., f1 + f2). In resistive mixers the frequencies are related to the time varying rate of channel resistance at the f1 + f2 frequency. An

Passive FET Applications

735

example set of frequencies is shown in the Table 10.1, where in one case a 3 GHz VCO is divided by 3 to generate the first LO at 1.0 GHz and divided by 2 to generate the second LO at 1.5 GHz. The second option uses a generator at 1.66 GHz and a divide by two. Note that both approaches generate the equivalent LO of 2.5 GHz used in a conventional direct conversion mixer. Since the two LO signals are orthogonal in the mixing process they do not appear in the spectrum of the converted signal. Table 10.1 Selection of Frequencies for Direct Conversion fVCO Operation Operation (1st step) f1 (GHz) (2nd step) f2 (GHz) Example (GHz) I 3 divide by 3 1 divide by 2 1.5 II 1.666 divide by 2 0.833 1:01 1.666

V1 2 1

f1

0

V2

2

f2

1

Vmix 0

2 1 fmix 0 1Time - nsec 2 0 0.5 1.0 1.5 2.0 2.5 3.0 Figure 10.63 Control voltages at frequencies f1 and f2, and FET channel modulation at fmix.

In another example, the simulated signals in a mixer using MOS technology are shown in Figure 10.63 in terms of normalized voltages. The top sinusoidal signal is at 800 MHz and the middle is at 1600 MHz. The f mix is a function representing the FET channel conductance variation showing a dominant component as (f1 + f2) in the time domain. The spectrum at the output of the first mixer is in Figure 10.64(a), showing the input signal at 2.4 GHz converted to 1.6 GHz and 3.2 GHz. Since this is a resistive mixer, no voltage or current signals exist in the channel at the LO frequency except leakage. The input signal is present at each drain and fed into the second mixer along with the converted signals. The output spectrum of second mixer is depicted in Figure 10.64(b), where the input signals are rejected and only the converted signals are present. Only residual signals are present at 0.8 GHz, 1.6 GHz, and 3.2 GHz; only the generated difference and sum for first and second converted side bands exist. The

736

Microwave Mixer Technology and Applications

Amplitude - dBm

signals (f2 - f1) appearing at (1.6 - 0.8 = 0.8) GHz and (3.2 - 0.8 = 2.4) GHz are the result of mixing but not self mixing. Thus, the degrading effect of self LO mixing which causes a variable DC offset at baseband is greatly reduced. The pulling on the VCO that can occur when strong RF signals are present or when TX, RX isolation is not high enough in a transceiver operating at the same frequency is also minimized. 0 -20

(a)

-40 -60 -80 -100 0

0.8

1.6

2.4

3.2

4.0

4.8

5.6

6.4

Amplitude - dBm

Frequency - GHz 0 -20

(b)

-40 -60 -80 60 -100 0

0.8

1.6

2.4

3.2

4.0

4.8

5.6

6.4

Frequency - GHz Figure 10.64 Output spectrum: (a) first mixer with LO at 800 MHz, RF at 2.4 GHz, and converted signals at 1.6 and 3.2 GHz.; (b) the second mixer operates at 1.6 GHz. The input signal at 2.4 GHz is suppressed at the output. The outputs of the first mixer represented by transparent bars are also suppressed at the output of the second mixer.

This concept was extended to convert signals to a complex baseband signal where I and Q signals are required, which is obtained by paralleling two mixers and phasing their LO’s as indicated in the diagram of Figure 10.65. The first LO signal is split into 0 and 180 by conventional active or passive circuits. The second LO requires differential quadrature signals, obtained by the simple polyphase filter indicated in Figure 10.66. To understand its operation let's first consider that any input signal at the left appears at the right with a phase (+45, 45), plus the phase of the applied signal. By adding the phases of the input signals, it is seen that the output consists of four signals with phases 0, 90, 180, 270. The same approach applied to the down-converter can be applied to upconverters. The fact that signals are in quadrature at the output means the image signals, including (f2 - f1) can be eliminated. Application of this mixer in a

Passive FET Applications

737

transceiver configuration as proposed by the inventors is shown in Figure 10.67. A single VCO generator system is used for both up and down conversion.

I

fRF

0 180 0 180 f1 f2 0 180 90 270

Q

Figure 10.65 Conversion from RF to differential baseband in complex I – Q signal format.

V0

V’(180+45) + V’-45 = V’’ -90 V’+45 + V’-45 = V’’ 0 V’+45 + V’(180-45) = V’’ 90

V180

V’(180+45) + V’(180-45) = V’’ 180

Figure 10.66 Generation of differential quadrature signals. There is a -45° shift through each resistor, and a +45° shift through each capacitor.

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Microwave Mixer Technology and Applications

Figure 10.67 Transceiver for 2400 MHz. The balanced VCO at 1.6 GHz is applied to the first set of switches. The second LO at 0.8 GHz is digitally generated and applied to the second switch set.

10.4.2.4 CMOS Technology Circuit 2 A high IIP2 mixer for direct conversion receivers uses a RF amplifier, followed by a resistive mixer and an additional differential amplifier, shown in Figure 10.68 [36]. A high performance system requires low noise and high IIP3 from the RF amplifier and a high IIP2 from the mixer. The mixer is a resistive quad FET type, built with CMOS FETs and driven by a square wave LO.

Figure 10.68 A low distortion front end receiver.

Passive FET Applications

739

An example of a buffer amplifier is depicted in Figure 10.69, illustrating a cascode differential amplifier with input impedance determined by resistor R CM. The input RF amplifier uses capacitive feedback to improve IIP 3 linearity. The RF output uses inductors to supply voltage to the drain instead of resistors, allowing a larger voltage swing at the output and consequently improves third order linearity. If two tones, RF2 and RF1, are applied to the amplifier, it generates low frequency second order distortion at RF2 - RF1, which is filtered by C1, C2.

Vout-

Vout+

Vout Vbias

-

Vbias

Vin+ -

+

Vin

Vin RCM

RCM VCM

Ibias

VCM

Figure 10.69 Cascode RF differential amplifier.

The IIP2 performance for a conventional Gilbert mixer is a function of symmetry in the differential topology. The author made a detailed study of asymmetry on second order distortion. The study started with the definition of  and Vth for a MOS device and equated the asymmetries as: W L 1    

and

 2    

and

  Cox

V1th  Vth  Vth V2th  Vth  Vth

With a simple FET model the authors analyzed the effect of small differences in the RF voltages, linearizing resistors, feedback resistor,  and Vth on the IIP2 performance. The conclusion for a differential amplifier with minimum mismatch at the load and factor , (i.e. RL =  are near 0) making IIP2 very high in (10.44). If the mismatch from other circuit parameters are taken into account then (10.45) applies and it can be seen that IIP2 degrades quickly with parameter mismatches.

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Microwave Mixer Technology and Applications

IIP2 (V ) 

2(Vgs  Vth  VRF ) 



IIP2 (V ) 



(10.44)

RL RL

2(Vgs  Vth  VDS )

1  Rs  (VGS

1   2  Vth  VDS ) 

R f Rs  (VGS  Vth  VDS  Rs 1  1  Rs  (VGS  Vth  VDS ) R f 1  Rs  (VGS  Vth  VDS )2 Rs

(10.45)

The authors showed that adding Rs in between LNA and mixer, according to (10.45) increases IIP2. The inventor claims improvements in IIP 2 as high as 20 dB or more depending on how tight the technology parameters are for RS in a given foundry. 10.4.2.5 CMOS Technology P and N Gates The block diagram in Figure 10.70 is for a communication system used in notebook computers, cordless telephone or other applications. If applied to meet the IEEE 802.11 standards at 2.45 or 5.0 GHz it must provide two-way wireless transfer with speeds up to 54 Mbps with high linearity.

Figure 10.70 Block diagram for a digital communication system.

A conventional NMOS ring mixer circuit that can be used in the system is depicted in Figure 10.71, illustrating the LO, LOC (complimentary), RF, and IF

Passive FET Applications

741

(Baseband) access. Note that the differential pairs 102, 104 and 106, 108 are driven by LO signals that are 180 out of phase, so only one of the FETs of each pair is on at a given time.

Figure 10.71 Schematic of NMOS ring mixer.

The mechanism of signal distortion in this mixer is observed in Figure 10.72(a), which represents the channel resistance modulation by the control gate voltage, VGS. Application of a LO sinusoidal voltage will result in a variable resistance within the time interval between the ON and OFF conditions around threshold voltage, Vt. This transition introduces distortion and insertion loss in the conversion process. The traditional solution is to apply a very large LO voltage, so that transition around the threshold is very fast. RON

RON

Vthreshold

VGS

VGS

(a) NMOS FET (b) N- P MOS FET combination Figure 10.72 RON as a function of VGS for N FET and combined N and P FET.

A couple of patents have been proposed to improve this feature, [37, 38] consisting of paralleling each NMOS device with a PMOS one. The combination of both devices results in the channel conduction represented in 10.72(b). The circuit schematic for this combination is in Figure 10.73, showing a ring composed of a PFET in parallel to a NFET at each branch. Note that the NMOS

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Microwave Mixer Technology and Applications

source is connected to the PMOS drain and vice-versa for the NMOS drain. The LO signal is applied in phase to the gates of M1 and M4 and in counter phase to the gates of M2 and M3. The application of LO to the PMOS devices is in counterphase, (i.e., the gates of M5 and M8 are in opposite phase compared to M1, M4). The on and off state are now controlled at each device by: Vgs,n = VLO – VRFin for the NMOS device Vgs,p = VRFin – VLO for the PMOS device

(10.46) (10.47)

During the positive sinusoidal LO half cycle, the voltage applied to FETs M1 and M4 turn them on, so that the IF mixer output 105 is coupled to the RF input via FET M1 and output 107 is coupled to FET M4.

Figure 10.73 NMOS + PMOS full ring mixer.

At the same time, the voltage applied to the other pair, M 2, M3 is in counter-phase so that the voltage is negative driving M2 and M3 off. Within the same cycle, M5 and M8 are driven by the LO complement making both devices to conduct simultaneously. At this moment, M1 and M5 couples the RF signal to the IF mixer output and so does M4 and M8 for the complement. The voltages applied to the other pair of devices M2, M6 and M3, M7 are turned off, appearing as an open circuit. The mechanism of signal distortion in this type of switch can be understood by observing the voltage waveforms of Figure 10.74 for the circuit in Figure 10.71. The waveforms are the same for each individual device. The voltage Vgs at any time is calculated from: V gs = VLO – VRFin. The figure represents three levels of conduction defined for the channel, namely: turn-on when the channel is shorted, turn-off when the channel is open, and a transition region where the channel presents a varying resistance. In general the LO voltage is much larger than the RF input signal. Thus, when the RF signal is low, the V gs amplitude is essentially determined by the amplitude of LO and Vgs will essentially track the LO voltage. However, when RF signal is large the amplitude of V gs becomes more

Passive FET Applications

743

dependent on the RF signal. During the first positive RF half cycle the LO signal is positive and the RF signal turns the device on only by the end of time t1.

Figure 10.74 Waveforms on NMOS ring.

Similarly, during the first negative half cycle of RF signal, the LO signal is positive and the FET is still turned on, since V gs = VLO –(-VRFin) > Vturn-on. The FET is turned on within the time interval, t2. During the second positive half cycle of RFin the LO is negative and the FET is partially in the transition region reaching the turn off state for most of t3. During the second negative cycle of RFin the LO is negative and the FET is only turned off for a small portion of t 4 and then enters the transition region. Due to the dependency of V gs on the time varying amplitude of the larger RF input signal, the stage prior to the passive mixer circuit will encounter variations in loading within each cycle. As a result the output baseband signal shows a larger time conducting compared to non-conducting, generating an asymmetrical waveform like the one represented in Figure 10.75.

Figure 10.75 Baseband waveform is asymmetrical generating distortion.

Let's now look into the same waveforms in Figure 10.76 for the circuit in Figure 10.73. The waveforms contained in Figure 10.76(a) are from a NMOS device with LO voltage and the one in Figure 10.76(b) for a PMOS device with LOC voltage. Note that during period t1 the NMOS FET is within the transition region for most of the period, while the PMOS FET is on. During period t 2, the NMOS FET is on for most of period t2 while the PMOS FET is within the transition region for most of the period. Since there is conduction in both periods,

744

Microwave Mixer Technology and Applications

the output waveform, represented in figure 10.76(c), will be more symmetrical, therefore, lower distortion is obtained with this configuration.

(b)

(a)

(c) Figure 10.76 Switching waveforms for a NMOS/PMOS mixer.

Following this approach, an 8 dB improvement was reported in patent [37], when compared with the classical NMOS topology. The results obtained for both approaches are depicted in Figure 10.77. Implementation of this technique has a technological drawback, the mobility of P devices is lower than for N devices.

Figure 10.77

IIP3 for conventional and proposed configuration.

So the P devices need to be of much larger size to provide the same equivalent resistance in order to match their characteristics. The larger PMOS

Passive FET Applications

745

device size increases the capacitance substantially, requiring a tuning inductance to provide a reasonable conversion efficiency. An alternative invention that also uses NMOS paralleled with PMOS is found in another patent [39]. The root cause for nonlinearity is the same, (i.e., when the RF signal becomes a significant fraction of LO voltage, according to (10.47) the gate source voltage decreases, which results in higher channel resistance at the instant in time when this occurs). To overcome this problem the inventor paralleled a NMOS with a PMOS device. The gates are driven by different paths in order to have them conducting simultaneously at each LO cycle. The schematic of this circuit is in Figure 10.78, illustrating the signal path for the first half cycle of LO. Note that FETs 327, 329 and 322, 324 are conducting. The other devices, 317, 319 and 332, 334 will conduct on the second LO half cycle. The objective of this invention is similar to the previous one, namely to decrease the on resistance, specifically within the transition region of open to low impedance. The on resistance is low for a longer part of the LO period improving distortion and minimizing losses.

VRF(+) VLO(+) VIF(+) VIF(-) VLO(-) VRF(-)

Figure 10.78

Alternative way to connect NMOS and PMOS FETs in parallel.

10.4.3 Subharmonic Topologies 10.4.3.1 CMOS Technology Circuit 3 An extension of the direct conversion invention from Figure 10.65 was proposed by the same authors, [40], applying a similar topology to subharmonic mixing, Figure 10.79. In this application a single LO operating at fLO is applied to drive the mixers.

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Microwave Mixer Technology and Applications

fRF

0 180 90

270

Figure 10.79 Subharmonic mixer application with Gilbert type of circuit.

The phase angles at each gate of the Gilbert cell are separated by /2 at the fundamental LO frequency resulting in  phase shift at 2fLO, as displayed in Figure 10.80. The time domain visualization shows the first level switch signal S1(t) and second level S2(t), resulting in an equivalent second harmonic LO at S(t)=S1(t)S2(t). Similar to the previous circuit, the S(t) represents the channel resistance variation. S1(t) S1(t) S(t) = S1(t).S2(t)

Figure 10.80 LO time domain waveforms.

The generation of complex I and Q signals is similar to the previous proposal, illustrated in Figure 10.81(a). The difference between both inventions is in the generation of LO signals to drive the gates. In this configuration eight LO signals are needed at the same frequency, with each phase shifted by /4. The poly-phase filters of Figure 10.81(b) power divide the differential input signal to obtain the four required LO signals with the proper phases. The block diagram of the complete RF front-end realization using a subharmonic drive frequency of 1200 MHz is shown in Figure 10.82 for operation at 2.4 GHz.

Passive FET Applications

I

747

V’45°

V0

V’315° 0 180 0

180

45 225 135 315

V’135° V180°

V’225°

Q

(a) Mixer block diagram (b) 45 Poly phase filter schematic Figure 10.81 Generation of I – Q signals with subharmonic drive.

Figure 10.82 Transceiver using subharmonic LO for 2.4 GHz.

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Microwave Mixer Technology and Applications

10.4.3.2 CMOS Technology Circuit 4 The invention proposed in [41], does not employ a balun to generate a single balanced resistive mixer. Instead, as depicted in Figure 10.83, the commonsource, common-gate feature of the FETs is used. The RF signal applied to input terminal 550 is delivered to two FETs in parallel, and the converted signal is extracted from terminal 570. The filters are diplexers that present the proper impedance to the RF and IF signals. The LO is applied at the first gate and the gate of the second FET is grounded for AC. An inductance is connected between the source and ground to isolate the gate from the DC bias supply. The circuit then acts as a balun transforming the single ended LO signal applied to the gate into a balanced signal at source and drain. Therefore, the LO signal pumping the gate does not interfere with the main line and the circuit functions as a subharmonic mixer. The capacitor connected between gate and source equalizes the LO voltages applied to each gate. An approximate version of this circuit was recreated and simulated in ADS using the device NE67300 with RF input centered at 10.1 GHz and LO at 5 GHz. Applying LO power, PLO = +13 dBm, it provided 17 dB conversion loss that compressed by 1 dB at input RF drive level of 0 dBm. An additional alternative to this invention proposed by the same inventor, [42], is to ground the gate of both devices and introduce the LO signal 180 outof-phase through the source terminals. The source is grounded for the IF signals through the inductances to ground. The RF signals are grounded by means of /4 open stubs at the RF frequency. This topology provides a good match for the LO generator and was found by the inventors to provide better conversion loss characteristics compared with the previous alternative.

Figure 10.83 Subharmonic mixer without a LO balun.

Passive FET Applications

749

A close facsimile of this circuit was also modeled using the ADS simulator, for the same conditions as the previous circuit, and provided a conversion loss of 11 dB when driven with P LO = + 23 dBm. The 1 dB compression point occurs at PRF = + 15 dBm. A high LO power is expected from a source LO injection circuit.

Figure 10.84

Subharmonic mixer with balanced LO injected to the two FET sources.

10.4.3.3 CMOS Technology Circuit 5 The application of FET switches connected as a successive mixer was proposed by the team from Berkeley, [43] to improve flicker noise in direct conversion receivers. The flicker noise is converted to baseband along with the desired signal degrading the noise figure by 20 dB or more. The target was a direct conversion architecture operating at 2.4 GHz, with a subharmonic approach. The work concluded the flicker noise is generated at low frequencies due to two inherent properties of direct conversion, namely: 1. Even order nonlinearities; 2. fLO and 2fLO leakage to the RF input. This is an interesting conclusion because a subharmonic mixer inherently has very good properties related to the second property. The study concluded the circuit becomes unbalanced during the crossover causing a 2fLO signal at the nodes. To further suppress 2fLO the authors proposed an additional signal path represented at the bottom, which is identical to the top, but connected in reverse

750

Microwave Mixer Technology and Applications

order. This causes the signal crossover at 4fLO, which is outside the bandwidth of the IF receiver, and provides further reduction of residual leakage at the summing point at the output compared with the conventional approach. The circuit schematic for this mixer implemented on 0.13 µm CMOS technology is represented in the figure. The differential input RF signal is applied to a LNA, degenerated by emitter inductance to improve matching without degrading noise figure.

Figure 10.85

Schematic of a successive subharmonic mixing. From [43].

Notice the drains are terminated into a reactive load that gives more voltage headroom for the devices and improves gain. The RF signal is coupled to the switches by means of capacitors to block DC, and the two outputs are DC connected to baseband amplifiers. The load is of the high impedance type resulting in large output voltage without the need for further signal amplification. The conversion loss for this type of mixer is obtained by multiplying the RF signal with an equivalent switching waveform given by (10.48). Vout  2VRF cos( RF t )

2



sin(2 LO t ) 

2



VRF cos( RF  2 LO )t  cos( RF  2 LO )t 

(10.48) The resulting fLO and 2fLO suppression are in Figure 10.86(a) showing a power level at -90 dBm or lower. The noise figure represented in Figure 10.84(b) confirms the hypotheses of the impact of 2fLO leakage. The noise figure of 11 dB extends down to 100 KHz with a gain of 5 dB. For the sake of comparison, the

Passive FET Applications

751

measured noise figure from other publication reported a corner flicker noise of 10 dB at 1 MHz, [44].

(a) fLO, 2fLO rejection

(b) Conversion gain and NF

Figure 10.86 Performance results illustrating: a. fundamental and second LO harmonic rejection; b. conversion gain and low frequency noise figure. From [43].

10.5 DISTRIBUTED GaAs APPLICATIONS As already covered in Chapter 9, the distributed approach has some desirable properties, such as the realization of a balanced mixer without the need of couplers. The resulting mixer has very broadband operation and is attractive to UWB - ultra wide band communication systems. It also performs well if realized in MMIC technology. The resistive approach brings an additional advantage, which is high linearity with a low LO power. 10.5.1 Single Gate GaAs Technology A single gate topology with three FETs is depicted in Figure 10.87, [45]. Two transmission lines are observed, the gate line where LO is applied to modulate the channel of each FET, and the drain line where RF is delivered to each one of the FET’s drain. Each line is terminated into its characteristic impedance, and at the output a diplexer (Lout, Cout in the figure) is employed to separate the IF from RF. Assuming IF frequency is much lower than RF, the phase of the gate line at LO, and the phase of the drain line at RF are same for best performance. The circuit was built with three mixer cells employing the Marconi F20 process. The conversion performance in Figure 10.88(a) shows 9.5 to 11 dB loss within the RF frequency range of 2 to 16 GHz. The IF is fixed at 100 MHz and LO set to + 10 dBm. The IF band is within DC to 500 MHz, with IIP3 on the order of +25 dBm.

752

Microwave Mixer Technology and Applications

RF

Lout Ld/2

Ld

Ld

Ld/2

Cout IF

Lg/2 Lg

LO

Lg

Lg/2

Figure 10.87 Distributed resistive mixer providing natural isolation between RF and LO. After [45].

-10 -11 -12

Gc_dB

-13 -14 -15 -16 -17 -18 -19 -20 0.0

3.0E9

6.0E9

9.0E9

1.2E10

1.5E10

1.8E10

2.1E10

LOfreq

Figure 10.88 Conversion loss, simulated with NE 67300.

A simulation in ADS of the same circuit using the NE67300 device gave the performance illustrated in Figure 10.88(b), achieving a relatively flat conversion loss of 12.8 dB from 1 to 18 GHz. In this simulation, IF frequency was set to 100 MHz and LO and RF frequencies were swept accordingly. The LO power is +13 dBm, and the input gate line return loss is better than 10 dB over the band. The gate and drain inductances are equal to 0.6 and 0.9 nH, respectively, and no external capacitance was added to the circuit. The small size allows integration of a distributed mixer into a compact image reject converter, depicted in Figure 10.89 [46]. In this particular example, two cells were considered for the distributed mixer, where an artificial transmission line is built with lumped elements.

Passive FET Applications

753

Figure 10.89 Compact image reject mixer topology designed with lumped elements. From [45].

The mixers are interconnected by means of a conventional Lange coupler at the LO side and a power divider at the RF side. The circuit was built with the 0.15 µm gate PHEMT process, and 2x33 µm2 devices were used in the mixer. The design was targeted for an RF band of 10 to 22 GHz and IF frequency to 1 GHz. 10.6 SUMMARY This chapter gave a brief summary of selected patents and articles on the application of using the modulated FET channel resistance to obtain the mixer function. The contents are more or less chronological and an attempt was made to cover different types of devices including Si-JFETs, GaAs MESFETs, PHEMTs and CMOS. Most of the applications use either MIC or MMIC technology. At the present time the single ended applications are increasingly being inserted into microwave and mm-wave using CMOS for performance, cost and simplicity. No other approach allows an efficient and simple balanced mixer using a single floating device. As a matter of fact, the linearity performance competes seriously with any of the other available topologies. The advantages of resistive FET mixers in direct conversion receivers either at fundamental or subharmonic were also described.

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Microwave Mixer Technology and Applications

REFERENCES [1] T. Kashiwa, T. Katoh, T. Ishida, Y. Kojima, Y. Yamamoto, M. Komaru, and Y. Mitsui, “A V-Band Monolithic InP HEMT Resistive Mixer with Low LO Power Requirement,” International Conference on Indium Phosphide and Related Materials, 1997, pp. 369-372. [2] R. Pena, J. A. Garcia, A. Brana, A. Jimenez, and E. Munoz, “Bias Selection for Conversion and Linearity Optimization in a GaN Resistive Mixer,” Integrated Nonlinear Microwave and Millimeter-Wave Circuits, 2006 International Workshop, January 30-31, 2006, pp 84-87. [3] W. K. Squires, “Mixer Circuit Employing Linear Resistive Elements,” US Patent 3,383,601, issued May 14, 1968. [4] Eugene A. Janning, Jr., “Orthogonal Passive Frequency Converter with Control Port and Signal Port,” US Patent 3,617,898, issued November 2, 1971. [5] Stephen Maas, “A GaAs MESFET Mixer with Very Low Intermodulation,” IEEE Transactions on Microwave Theory and Techniques, Volume MTT-35, No. 4, April 1987, pp. 425-429. [6] John J. Geddes and Paul E. Bauhan, “Resonant Loop Resistive Mixer,” US Patent 5,263,198, issued November 16, 1993. [7] Xiaohui Li and Michael Wendell Vice, “Unbalanced FET Mixer,” US Patent 5,678,226, issued October 14, 1997. [8] Ulrich Rohde, Antonio Almeida, Ajay Poddar, Vaseem Ahmed, Nilolay Ilkov, and Klaus Schoepf, “Passive Reflection Mixer,” US Patent 7,580,693, issued August 25, 2009. [9] J. F. Villemaze, M. Camiade, and J. Obregon, “Design of Low Conversion Losses Cold FET Mixers by Statistical Optimization of High Order Sidebands Loading,” IEEE 1993MTT International Microwave Symposium Digest, pp. 10291032. [10] Bahar M. Motlagh, Sten E. Gunnarson, Mattias Ferndahl, and Herbert Zirath, “Fully Integrated 60-GHz Single-Ended Resistive Mixer in 90-nm CMOS Technology,” IEEE Microwave and Wireless Components Letters, Volume 16, No. 1, January 2006, pp. 25-27. [11] Frank Ellinger, “26.5–30 GHz Resistive Mixer in 90–nm VLSI SOI CMOS Technology with High Linearity for WLAN,” IEEE Transactions on Microwave Theory and Techniques, Volume MTT-53, No. 8, August 2005, pp. 2559-2565. [12] Tsung-Yu Yang and Hwann-Kaeo Chiou, “A 28 GHz Sub-Harmonic Mixer Using LO Doubler in 0.18 µm CMOS Technology,” RFIC Symposium Digest 2006. [13] Sten E. Gunnarsson, “Analysis and Design of a Novel X4 Subharmonically Pumped Resistive HEMT Mixer,” IEEE Transactions on Microwave Theory and Techniques, Volume 56, No. 4, April 2008, pp. 809-816. [14] Carmine F. Vasile, “Multifunction Floating FET Circuit,” US Patent 4,705,967, issued November 10, 1987.

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[15] Jean Marc Mourant and Holly A. LaFerrara, “Floating FET Mixer,” US Patent 5,697,092, issued December 9, 1997. [16] Michael W. Vice, “Totem Pole Mixer Having Grounded Serially Connected Stacked FET Pair,” US Patent 6,064,872, issued May 16, 2000. [17] Aubrey Jaffer, “High Dynamic Range Mixer,” US Patent 4,727,596, issued February 23, 1988. [18] Philip Piro and Kevin Cornman, “Passive Balun FET Mixer,” US Patent 6,871,059B1 issued, March 22, 2005. [19] C. Daniel W. Stoffer, “Balanced Modulator with JFET’s Voltage Controlled Resistors,” US Patent 3,621,473, issued November 16, 1971. [20] Charles M. Puckette, “Linear Mixer with Reduced Spurious Response,” US Patent 4,352,210, issued September 28, 1982. [21] Stephen Maas, “GaAs MESFET Balanced Resistive Mixer,” US Patent 4,949,398, issued August 14, 1990. [22] Dennis A. Kruger, “Dual Quadrature Frequency Converter,” US Patent 5,006,811, issued April 9, 1991. [23] Doron Gamliel, “Double Balanced FET Mixer with High IP3 and IF Response down to DC Levels,” US Patent 6,957,055B2, issued October 18, 2005. [24] Pierre Dobrovolny, “High Level Wide Band RF Mixer,” US Patent 5,027,163, issued June 25, 1991. [25] Pierre B. Dautriche, “Mixer Arrangement,” US Patent 4,727,597, issued February 23, 1988. [26] A. Philippon, M. Campovecchio, J. C. Nallamtaby, P. Butterworth, and R. Quere, “Design Method and New Architecture of Sub-Harmonic Balanced Cold FET Mixer for MVDS Applications,” IEEE 2006 International Workshop on Integrated Nonlinear Microwave and Millimeter-Wave Circuits, pp. 90-93. [27] P-C Yeh, W-C Liu, and H K Chiou, “Compact 28-GHz Sub harmonically Pumped Resistive Mixer MMIC Using a Lumped-Element High-Pass/Band-Pass Balun,” IEEE Microwave and Wireless Components Letters, Volume 15, No. 2, February 2005, pp. 62-64. [28] K. S. Ang, A. H. Baree, S. Nam, and I. D. Robertson, “A Millimeter-Wave Monolithic Sub-Harmonically Pumped Resistive Mixer,” IEEE 1999 MTT International Microwave Symposiurm Digest, pp. 222-225. [29] K. S. Ang, M. Chongcheawchamman, D. Kpogla, P. R. Young, I. D. Roberton, D-S Kim, M-C Ju, and H-C Seo, “Monolithic Ka-band Even Harmonic Quadrature Resistive Mixer for Direct Conversion Receivers,” IEEE–2001 Radio Frequency Integrated Circuits Symposium, pp. 169-172. [30] Eric W. Lin and Walter H. Ku, “Device Considerations and Modeling for the Design of an InP-Based MODFET Millimeter Wave Resistive Mixer with Superior Conversion Efficiency,” IEEE Transactions on Microwave Theory and Techniques, Volume MTT 43, No. 8, August 1995, pp. 1951-1959. [31] Scott M. Weiner, Donald A. Neuf and Steven J. Spohrer, “Double Balanced Mixing,” US Patent 4,947,062, issued August 7, 1990.

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Microwave Mixer Technology and Applications

[32] Michael W. Vice, “Balanced Reflection Transformer,” US Patent 5,551,074, issued August 27, 1996. [33] Raymond M. Waugh and Mahesh Kumar, “Monolithic Double Balanced Mixer with High Third Order Intercept Point Employing an Active Distributed Balun,” US Patent 5,060,298, issued October 22, 1991. [34] K. S. Ang, I. D. Robertson, D-S Kim, M-C Ju, and H-C Seo, “K-band Monolithic Double-Balanced Resistive Mixer with Integrated Balanced Oscillator,” IEEE 2001 MTT International Microwave Symposium Digest, pp. 1329-1332. [35] Wolfram Kluge and Dietmar Eggert, “Mixer,” US Patent 6,970,687B1, issued November 29, 2005. [36] Lawrence Connell, Yan Cui, Poojan A. Wagh, Patrick L. Rakers, “High Linearity and Low Noise CMOS Mixer and Signal Mixing Method,” US Patent 7,177,616 B2, issued February 13, 2007. [37] Sining Zhou, “Ultra-High Linearity RF Passive Mixer,” US Patent 6,847,808B2, issued January 25, 2005. [38] Arya Reza Behzad, “High Linearity Passive Mixer and Associate LO Buffer,” US Patent 6,972,610B2, issued December 6, 2005. [39] Said E. Abdelli, “Passive Mixer with Improved Linearity,” US Patent 7,113,755B2, issued September 26, 2006. [40] Wolfram Kluge and Dietmar Eggert, “Harmonic Mixer,” US Patent 7,085,548 B1, issued August 1, 2006. [41] Antonio Romano, “Sub-Harmonic Mixer,” US Patent 20040104758 A1, issued June 3, 2004. [42] Antonio Romano, “Sub-Harmonic Mixer,” US Patent 7,084,693 B2, issued August 1, 2006. [43] H. C. Jen, S. C. Rose, and R. G. Meyer, “A 2.2 GHz Sub-Harmonic Mixer for Direct Conversion Receivers in 0.13 µm CMOS,” 2006 IEEE International Solid-State Circuits Conference, February 2006, pp. 1840-1849. [44] Hsiao-Chin Chen, Tao Wang, and Shey-Shi Lu, “A 5-6 GHz 1-V CMOS Direct-Conversion Receiver with an Integrated Quadrature Coupler.” IEEE Journal of Solid State Circuits, Volume 42, No. 9, September 2007, pp. 19631974. [45] K. S. Ang, S. Nam, and I. D. Robertson, “A 2 to 28 GHz Monolithic Resistive Distributed Mixer,” 29th European Microwave Conference, 1999, pp. 222-225. [46] K. Pha, K. Fujii, H. Morkner, and A. Rixon, “An Integrated Low Noise Amplifier and Image Rejection Mixer in a Surface Mount Package for Low Cost 10-22 GHz Applications,” First European Microwave Integrated Circuits Conference, September 2006, pp. 122-125. [47] Pat Hawker, “G3SBI’s High Performance Mixer,” Radio Communication, October 1993, pp. 55-56.

Chapter 11 Active FET Applications When JFETs became available, mixers were typically built using triode tubes or BJT devices, both having complex relations between output current and input voltage. In contrast, JFETs demonstrate an almost true square law relation when gate voltage is between pinch-off and zero. In spite of this ideal property, the technology to process JFETs was crude, and obtaining good devices in high volume was difficult, thus JFET technology declined in prominence. The wide spread use of FETs had to wait for the development of Silicon-IC CMOS technology driven by digital applications, and emerging Gallium Arsenide technology driven by RF and microwave applications. CMOS technology continues to advance in the RF arena, and is generally considered the preferred technology for high volume low cost applications. This chapter summarizes selected active FET mixer applications found in scientific journals and patent reports. Biasing in the active region allows merging the two functions of frequency conversion and gain in a single device, potentially reducing the number of required components, and in some cases providing lower noise figure. Other functions including self oscillating mixers and distributed mixers are also described as having some surprising properties.

11.1 SINGLE ENDED Single ended mixer topologies, as mentioned in the previous chapter, can provide isolation between mixing signals only by filtering. There is no circuit balance to provide cancellation of even or odd harmonics, nor of unwanted mixing products falling within the desired output band. This limits applications to systems that require no overlap between RF, LO, IF, and image bands, and very clean frequency plans with little or no spurious in the output band. However, the simplicity of the single ended mixer is compelling and desirable for situations where these deficiencies are not relevant. Single ended mixers topologies are usually implemented with discrete devices and are rarely seen in IC technology.

757

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Microwave Mixer Technology and Applications

11.1.1 Single Function Topologies 11.1.1.1 JFET Technology One of the first applications of a FET to down-convert RF signals is disclosed in a 1968 patent, [1]. The schematic diagram shown in Figure 11.1 reveals a JFET mixer circuit with LO voltage applied to the gate, RF applied to source, and IF extracted from the drain. The purpose of this invention was to generate low spurious levels with a high conversion gain and low noise figure. +B

IF Output LO Input

Signal Input R1

Figure 11.1

Schematic of a JFET mixer with gate LO injection using tank circuits to couple signals to device terminals.

A tank circuit is attached to each terminal to provide filtering and impedance matching. The tank circuit can be considered as a PI network with a parallel capacitor, series inductor and parallel inductor. By varying the inductor tap location, the impedance transformation ratio and the signal voltage level on the mixer terminals are adjusted. The resonance frequency of the source and gate tank circuits are adjusted simultaneously to maintain the same IF frequency. The author’s purpose is to maintain the mixing device operating within the square law region to avoid gate to source conduction, and operating below pinch off voltage to minimize distortion. Therefore, the maximum LO voltage swing corresponds to VP/2 and its magnitude can cause the conversion gain to vary. The high gate impedance of the JFET allows a minimum of LO power to obtain a high conversion gain. The source impedance is low and the RF tank circuit matching trades off best noise figure with conversion gain. Resistor R1 is responsible for setting the bias current and for stabilizing the operating point over temperature and supply voltage variations. It is bypassed by a parallel capacitor to minimize it’s effects at RF frequencies. Since the FET exhibits manufacturing variations in pinch-off voltage, the bias resistor should be designed to minimize its effect on mixer performance. A good practice is to adjust R 1 to obtain the lowest pinch-off voltage condition.

Active FET Applications

759

The tank circuit connected at the drain is tuned to IF frequency, matching the high IF impedance to the output load. At all other frequencies it shows low impedance, nearly a short to ground. For operation with weak signals, maximum conversion gain is obtained by conjugate matching the drain impedance at the IF frequency. The L – R isolation in this mixer is provided by the tank circuit impedance at the source side at the LO frequency. The L – I isolation is provided by the drain tank circuit impedance which is low at LO and RF, assuming a low IF frequency compared to LO and RF. In case higher conversion power gain is required, an alternate connection of the FET was proposed as shown in Figure 11.2. Here the RF and LO connections are interchanged, the RF signal is connected to a higher impedance, and the LO to a lower impedance. +B

IF Output

LO Input

Signal Input R1

Figure 11.2

Schematic of a JFET with source LO injection.

The LO level can be adjusted by changing the tap point on the LO tank circuit. The RF tap also needs to be adjusted to accommodate different impedance levels. 11.1.1.2 JFET Dual Gate MOS1 Metal oxide semiconductor (MOS) technology found its way into RF applications before the GaAs FET revolution. An example of such, [2], is a dual gate MOSFET used in a mixer for TV receivers, displayed in Figure 11.3. In such a circuit the RF and LO signals are added by two capacitors to gate 1. The IF signal, leaving the drain is filtered and applied to an emitter follower stage. The drain impedance at LO and RF are shorted to ground by a capacitor in the IF filter. The second gate is RF shorted to ground, and the bipolar device 30 acts as a high gain low noise common collector IF amplifier. Therefore, two functions are included in the device: mixer and IF amplifier, resulting in high conversion gain compared to a

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Microwave Mixer Technology and Applications

single gate device. One problem found with TV receivers is the wide percentage bandwidth the circuit must cover from low to high frequency channels. As a result, it is difficult to maintain a constant LO voltage level over the band, which impacts conversion gain.

Figure 11.3

Schematic of a DGFET mixer for TV receive with constant conversion gain.

One reason for the LO amplitude variation is the Q factor of reactive elements used in the matching. In this specific work the LO voltage ranges from 0.1 V to 0.6 V proportional to frequency. A circuit to correct this problem consists of a bipolar 18 current source connected in parallel to a biasing source resistor 28 and controlled by the tuning voltage. When the tuning voltage increases, more current is drawn from the source so that the effective source resistor becomes smaller and more current is driven from the FET increasing conversion gain thus maintaining the IF level stable. 11.1.1.3 JFET Dual Gate MOS2 In this patent example, [3], the dual gate is a cascode converter, where RF is applied to gate 1 and LO is applied to gate 2. The schematic is found in Figure 11.4 illustrating the bias circuitry. Gate 2 is often used in amplifier applications to control power gain, and the same function is available here. This is a convenient way to build an AGC function, where conversion gain is controlled by the bias applied to gate 2, which is supplied by detecting the average or peak signal level later in the receiver chain. Note this is a dual conversion circuit for FM receiver applications, where the band-pass filter inserted between first and second mixer rejects image noise and also presents a low drain impedance for optimum mixer performance.

Active FET Applications

Figure 11.4

761

Dual gate FET active mixer/combiner.

11.1.1.4 MESFET MIC Technology A single MESFET active mixer is found in this patent [4], where transmission lines are used to inject the LO to the source, the RF to the gate, and to extract the IF signal from the drain. The fact that only one mixing element is used means savings on couplers or baluns which usually add losses and complexity. The transmission lines act both as filters and impedance matching elements. The circuit proposed in Figure 11.5 has three elements that receive special attention: The first is the open stub dimensioned to be a quarter wavelength at the RF frequency, shorting the source to ground for the incoming RF signals; the second is a circuit short circuit stub close to a quarter wavelength at the LO frequency slightly detuned to resonate the first circuit; The third consists of a RF trap to apply a low impedance to LO and RF signals at the drain followed by an IF impedance matching network. RF

Matching Circuit

LO

Matching Circuit

RF trap

2nd circuit Figure 11.5

1st circuit

FET mixer using source diplexer for LO insertion.

IF

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Microwave Mixer Technology and Applications

The schematics in the next figure illustrate the equivalent circuit of the FET at each frequency, to simplify impedance analysis. The performance results for a low noise 0.5 m gate length FET inserted into a ceramic package are in Figure 11.7, which depicts gain and noise figure. The SSB noise figure obtained at center band is equal to 6 dB, the associated conversion gain is 10 dB, and the 2 dB bandwidth ranges from 11.5 to 11. 7 GHz. D

RF

D G

G

S

S

D G

IF

S

LO Figure 11.6

Equivalent FET circuit at different frequencies.

The final implementation of the proposed circuit is in Figure 11.8, displaying two versions built in MIC technology. In the first version, part (a), the RF matching circuit employs a capacitive impedance at the gate followed by a PIsection of transmission line.

Figure 11.7

FET mixer performance in terms of conversion gain and noise figure.

The DC is de-coupled through a pair of open circuit shunt stubs to the gate. The RF couples to the gate through a low pass filter, followed by an edge coupled capacitor for DC blocking. Packaged FETs typically have two source connections; in this circuit one was used to DC ground the device, and the other to inject the LO signal. A simple open circuit shunt stub is used at the drain to trap the RF and pass the IF. In the second version, part (b), a semi-lumped element circuit is proposed where the meander line inductors and inter-digital capacitors are designed in a single structure to guarantee the required resonance frequency.

Active FET Applications

Figure 11.8

(a) Distributed version Mixer schematics showing microstrip layouts.

763

(b) Lumped version

11.1.1.5 HJFET MMIC Technology A MMIC version of the previous invention employing Hetero Junction FET, or HJFET, technology is found in two publications [5, 6]. The circuits were built using coplanar waveguide, instead of the conventional microstrip line. In the first report, the mixer was designed for operation within the 20 to 23 GHz band with an IF between 1.5 to 2.5 GHz. The up-converter exhibited maximum conversion gain of –6 dB for an IF at 1.9 GHz and LO drive of 10 dBm at 21.5 GHz. The isolation between L–I and L–R are better than 22 and 20 dB respectively. The schematic of this circuit is in Figure 11.9.

Figure 11.9

MMIC upconverter operating between 20 to 23 GHz. From [5].

The second mixer, in Figure 11.10, was also designed for HJFET technology but for operation within the 60 GHz band. The conversion gain with +7 dBm of LO power ranges from –7 to –12 dB, depending on filter losses.

764

Figure 11.10

Microwave Mixer Technology and Applications

MMIC upconverter operating at 60 GHz. From [6].

The stability analysis carried out by the author consisted of the reduction of three to two ports by terminating the LO terminal with 50 ohms. The lines connected to the source were then iteratively optimized by linear and nonlinear simulations to tradeoff electrical stability and conversion gain respectively. 11.1.1.6 MESFET Plus BJT MIC Technology An analysis of injecting LO into a FET drain mixer is covered in this proposal, [7]. The schematic for the LO injection employing a transformer is shown in Figure 11.11 and the I/V curve describing the bias and load line for the FET is in Figure 11.12. The transformer couples LO voltage to the drain. Note the drain circuit contains two filters used to separate LO from the IF and RF signals. The drain electrode is biased by a power supply E 0, whose load line is vertical in the I/V plane. The static biasing Point P, in the absence of signal, is situated at the intersection of the I/V curve for the gate voltage -Vg1 and the vertical line through Vds = E0. The LO generator impedance RL appears in the secondary multiplied by the transformer impedance ratio, n2RL. This is the load line for the LO signal on the drain side. If RF or IF signals are applied to the gate, then the straight dynamic load line D moves parallel to itself around quiescent point P, lines D 1 and D2, as the amplitude of the impressed LO voltage changes. When the amplitude becomes large, point P sweeps along line –Vg1 from point O to PM.

Active FET Applications

765

Figure 11.11 Schematic for transformer LO coupling.

Point PM is situated on a straight line DM of slope -1/n2RL. When the gate voltage of transistor 1 is modulated by generator ω, the dynamic trajectory on the I/V plane may, as the result of both signals, cover the entire zone Z 1, limited by the three lines namely, DM, Vg =0 and Vg = VP.

Figure 11.12

Dynamic trajectory on I-V plane for the circuit of Figure 11.11.

The inventors proposed to replace the transformer with a silicon BJT device in the manner represented in Figure 11.13. The LO is applied to the BJT base and modulates the collector current, and therefore the FET drain current. The bipolar device acts as an emitter follower to the LO generator, therefore, a low impedance DC supply voltage is applied to the drain.

766

Microwave Mixer Technology and Applications

Figure 11.13 LO injection with emitter follower bipolar device.

The drain voltage is determined by the base voltage, VBE considering the junction active and the resistive divider 19, 20. The quiescent point is determined by the intersection of the drain voltage (VB0 - VBE) and the gate voltage –Vg1, given static current I0. In the absence of gate voltage the application of LO voltage makes the bias point move from P1 to P2. When both LO and RF are simultaneously applied to the FET, the trajectory can be found to be within zone Z2 represented in Figure 11.14. The biasing mode allows complete sweeping of the non-linear zone of the FET, limited along the line –Vg1 from O to PM.

Figure 11.14 Dynamic trajectory for emitter follower.

A third option is to use a bipolar pnp device as an amplifier to inject signal at the drain as depicted in Figure 11.15. The drain voltage is no longer determined by the BJT, but instead by the superposition of the DC characteristics of both devices. There are now two current controlling devices, the BJT current source controlled by the emitter base voltage determined by the resistive divider 31, 32, and the current determined by the gate voltage applied to FET gate. In this case one of the devices will control the current and force the other to be a variable

Active FET Applications

767

resistance. The LO load line is now dependent on the magnitude of the signal applied to the base that appears as a variable resistance to the FET, modulating the bias point from P1 to P2 in the I/V plane. When both signals are applied to the gate and drain, the resulting dynamic area is zone Z3, which is limited between the pinch-off voltage – VP, I/V and the line crossing point PM.

Figure 11.15

(a) Schematic Drain LO injection with PNP bipolar device.

(b) Load line

11.1.1.7 MESFET Gate MMIC Technology A low cost MMIC mixer designed for high volume requirements is described next. Consumer applications are usually portable where battery life is important, requiring MMICs to be designed for best power efficiency. Since MMIC cost is driven by circuit area, the selected topology must use a minimum number of components having small size. Circuit miniaturization calls for IC integration where lumped elements or active devices can replace traditional functions like couplers. The gate mixer disclosed in patent [8], meets this requirement and was developed for operation at a RF frequency of 1.9 GHz and IF of 100 MHz.

Figure 11.16

Simple lumped element power combiner, for a MMIC gate mixer.

768

Microwave Mixer Technology and Applications

The circuit, represented in Figure 11.16, contains a lumped element Wilkinson combiner that also provides impedance transformation when combined with shunt inductor L60. The FET is self biased near the pinch off region by the source resistor R70, so that transconductance is modulated. The IF signal is obtained at the drain of the FET by appropriate filtering through a shunt capacitor (4 pF), to isolate the IF from all high frequency components. In addition, by varying the values of the components in the power combiner, the ratio of the RF to LO power may be changed. For instance, by selecting inductor L48= 10.7 nH, resistor R50 = 150 Ohms, capacitors C52 = C54 = 0.6 pF, C58 = 0.3 pF and L60 = 5.2 nH, the ratio of RF to LO power is 2:1 at the gate of FET, with the RF input terminal 46 equal to 50 Ohms and the LO input terminal 44 lower than 15 ohms, for a 0.25 µm gate FET. 11.1.2 Multifunction MMIC Topologies The application of IC technology opened the way to add RF and IF amplifier functions on the same die, and in some cases an LO amplifier as well. 11.1.2.1 MESFET Drain MMIC Technology The circuit illustrated in Figure 11.17, represents the schematic of a mixer [9], which successfully addresses the target specifications for a MMIC converter. The LO signal is applied to a common gate amplifier that delivers power to the mixer device, FET40. The RF signal is applied to another common gate amplifier before being delivered to the mixer device. Both amplifiers provide a low VSWR to the RF and LO signals, and high L – R isolation from the reverse gain of common gate amplifier. The mixer device is unique in the way the signals are injected. The RF and LO signals are both applied at the gate and to the source through R 38. Without the resistor the mixer is merely a diode mixer. If the resistor is large compared to the channel resistance, then it behaves as an active gate mixer. If its value is between those extremes, it can be classified as an active gate mixer, with drain conductance and transconductance modulated by the LO signal. The mixer device also acts as a constant current device biasing both the LO and RF common base amplifiers. The average nonlinear resistance value of FET 40 in series with R38 varies as a ratio of VDG/I, where I is the constant current through FET 40. The generated IF signal is applied to a linear class A source follower, FET 50, which provides impedance transformation from the mixer to a lower value more suitable for IF filtering and subsequent amplification. R52 and R54 form an appropriate voltage divider for negative DC and RF feedback applied to the gate of FET 30 to establish a low distortion class A operating point that is stable over temperature, and to compensate for normal foundry variations in capacitor and resistor values. The source of FET50 is attached to the junction of R52, C56 and C57. The other end of C56 contains the RF, LO, and the converted sum and difference frequency

Active FET Applications

769

signals and all mixing products including the desired IF. Port 15 is used to control the amount of negative RF feedback applied to FET 20 from the source output of FET50 via R55 and coupling capacitor C57. Normally, this port is grounded, shorting out the gate resistor, so that no RF feedback is applied. When large signal operation is desired with minimum mixer output distortion, a fixed or active variable resistance is connected at port 15. Increased feedback significantly reduces inter-modulation and cross-modulation products at the cost of conversion gain reduction and noise figure degradation. In the preferred embodiment of this design, transistor 50 has twice the area of transistor 40. FETs 20 and 30 are identical and one half the area of FET 40. Resistors R18 and R28 are equal and twice the value of R38. Resistor R52 is 24 times the value of R54. The basic ratio of values of R18, R28 and R52 are set for best conversion gain and optimum port VSWR performance consistent with minimum power drain.

RFin

IFout LOin

Figure 11.17 MMIC consumer FET mixer with transconductance and drain conductance modulation.

The experimental results for a circuit representing a nominal configuration with no RF feedback and minimally sized FETs are shown in Figure 11.18. In part a, PIF as a function of PRF is plotted for three different LO drive levels. RF input drive varies from -30 dBm to -10 dBm with LO power, PLO, at 10 dBm, 6 dBm and 0 dBm, respectively, for lines 70, 72 and 74. The lowest LO drive level provides linear conversion loss up to -15 dBm input drive, which is a good tradeoff relative to the results of higher LO drive given the need to conserve battery life. Above the -15 dBm level, conversion gain decreases due to normal circuit saturation characteristics. In part b of the figure, conversion gain and return losses over the RF band are represented, respectively, by line 76, 78, 80. Performance is optimized over the 0.8 to 0.9 GHz band. Return loss is dependent upon the design values selected for the port coupling capacitors C 16, C26 and C56. For the aforementioned frequency range, these capacitor values in the preferred

770

Microwave Mixer Technology and Applications

embodiment are each 0.5 pF. Maximum conversion gain occurs where the port return loss is minimized. Higher frequency performance is possible with smaller FETs. Low frequency and higher power level operation require larger area FETs. This MMIC FET mixer described in 1996 overcomes the problems of low DC efficiency, low conversion efficiency, low port-to-port isolation, high input and output VSWR, high inter-modulation distortion, high cross-modulation distortion, and large circuit area.

(a) Gain compression (b) Gain and return loss versus RF frequency Figure 11.18 Gain performance in terms of PRF versus PIF with PLO as a parameter and in terms of frequency.

11.1.2.2 MESFET Drain Mixer Plus IF Amplifier An alternative important application of MMIC technology was in the development of components for the direct broadcast satellite (DBS) market, which imposes severe constraints on performance and cost. Typically, MMIC mixers for DBS applications contain a low noise amplifier to receive signals within the range of 11 to 12 GHz, a down-converter mixer, and an IF amplifier operating from 950 to 2050 MHz.

IFout

RFin LNA

Ga1, F1, IIP31

Block Filter

Ga2, F2, IIP32

MMIC

Ga3, F3, IIP33

Ga4, F4, IIP34

Figure 11.19 Typical block diagram of a satellite receiver with a performance budget in terms of gain, G, noise figure, F, and third order distortion, IIP3.

Active FET Applications

771

Since the entire IF band is converted, this type of mixer is called a “block down-converter”. To optimize the trade-off between performance and cost, the number of on-die components needs to be minimized and the specifications of each one needs to be properly optimized. The block diagram of a typical receiver is in Figure 11.19, consisting of an LNA designed with discrete low noise HEMT devices, an image reject band pass filter, and a down-converter MMIC. The specifications for each of the components in the receiver are optimized by trading off gain, G, input intercept point for third order distortion, IIP3, and noise figure, F. The equations for the cascade are given by (11.1) and (11.2), and are also discussed in Chapter 3. Ga1 Ga Ga Ga Ga Ga 1 1    1 2 1 2 3 IIP3t IIP31 IIP32 IIP33 IIP34

Ft  F1 

F2  1 F3  1 F4  1   Ga1 Ga2Ga1 Ga3Ga2Ga1

(11.1)

(11.2)

The down-converter MMIC, depicted in Figure 11.20 and disclosed by invention [10], proposes to optimize the receiver parameters at low cost. It contains an active drain mixer, a LO generator and an IF amplifier. The mixer is a single gate FET (32) receiving RF signals within the range 10.7 GHz to 11.8 GHz at the gate. The LO at 9.75 GHz is applied to the drain through another FET device (34). The applied LO power modulates the transconductance and output conductance of the mixer FET 32, and the output conductance of the LO driver FET 34. An ensemble of frequencies, including the sum and difference of the RF and LO signals plus harmonics, are present at the output of the mixer. The low pass filter following the mixer is implemented using a series inductor and a shunt capacitor. All signals coming from the mixer are attenuated by this filter except the difference frequencies located within 950 to 2050 MHz. The IF amplifier, besides magnifying the IF signal, also functions as an active load for the mixer. The FET 62 is a common gate amplifier with an active drain load by means of FET 60. The capacitor grounding the gate of FET 62 also compensates the gain versus frequency response of the active load circuit. The drain voltage of the IF amplifier FET is set by means of resistive dividers. The total mixer current or equivalent DC load is determined by the LO drive power, but the drain voltage is dependent on the bias supply. Since the load is active, it will provide a wide range of current while keeping bias relatively insensitive to the LO power.

772

Microwave Mixer Technology and Applications

RFin

IFout

Figure 11.20 Down-converter MMIC block diagram depicting LO driving FET 34 and high impedance filter 46, 48.

The mixer drain voltage, terminal 64, is biased at about 1 volt, experimentally determined to be an optimum point for mixer operation. The gate voltage of FET 62, VG is around 0.7 V, but the FETs are depletion mode, so the source voltage is higher than the gate, in this case by 0.3 volts. 11.1.2.3 Modified Dual Gate A conventional dual gate, or cascode mixer, is depicted in Figure 11.21 where a LO buffer amplifier has been added. The large signal LO causes output conductance and transconductance modulation of M2 and M1. The resulting IF signal is amplified by M2, producing higher gain compared to using a single device. The drawback of this approach is the nonlinearity of the common gate amplifier that limits third order IM performance measured by output intercept point, OIP3. High gain and low OIP3 equates to a reduced maximum input signal that the receiver can sustain. This property frequently discourages usage this approach.

OIP3  IIP3  Gain

(11.3)

Active FET Applications

773

IFout LOin RFin

Figure 11.21

Conventional dual gate mixer.

A patent that addresses these issues is found in [11]. The contribution here is to change the connection of the output signal from the drain of M 2 to the connection of M1 and M2, point X in the Figure 11.22. The device is no longer a true dual gate, instead it becomes a drain mixer with LO injection by means of a source follower. The IF amplification is delivered by a separate common source amplifier, operating linearly over a wide range of signal power. The drain terminal is now directly connected to a power supply, so one can interpret this configuration as M1 operating as a drain mixer and M2 as an LO pump source follower amplifier.

LOin

RFin

Figure 11.22 Modified dual gate mixer.

IFout

774

Microwave Mixer Technology and Applications

To highlight the usefulness of the preferred embodiment of the present invention, simulation results comparing conventional and modified circuits are tabulated in Table 11.1. The frequency of operation is 0.8 to 5.8 GHz. Table 11.1 Simulation Results for New and Traditional Down Converter Mixer

_____________________________________________ Modified Traditional down-mixer down-mixer Conversion gain (dB) 9 to 11 7.5 Image band att. (dB) 8 to 9 7.8 Drain current (mA) 11 to 13 11.5 OIP3, Pin = - 40 dBm 19 to 23 8.6 IIP3, Pin = - 40 dBm 10 to 12 1.1 ________________________________________ The simulation results show that higher gain and more robust intermodulation distortion suppression are obtained with the new proposed topology. The results shown are for devices built with MESFET technology, but they are also equally applicable to other FET technologies. 11.1.2.4 HEMT X4Subharmonic HEMT technology with 0.1 µm gate length was applied, [12], to the design of a V-band (50 – 70 GHz) subharmonic converter. The approach is to use a X4 multiplier with a quasi cascode configuration comprising a RF buffer amplifier and a single ended gate mixer as depicted in Figure 11.23(a). The multiplier operation is described as a fundamental LO frequency applied to the gates of both devices M1 and M2 generating second and fourth harmonics at the drain of M 2. In addition to this generation from the fundamental, one can also think of the second harmonic generated by M1 at high level mixing at the drain with the fundamental delivered by M2, resulting in third harmonic energy enriching the generation of the fourth LO harmonic tone. The signal from the multiplier and the RF amplifier are combined and applied to the mixer M4. The performance is in part (b) of the figure, expressed as conversion gain and IF output power, as a function of IF input power. The LO power applied to achieve the performance was 10 dBm at 14.5 GHz, and the converted IF carrier is at 2.4 GHz. The conversion gain is nearly 4 dB and 1 dB gain compression takes place at an input power of - 9 dBm.

Active FET Applications Vg1

/4(2f0)

775

Vd 2

IF & Vd3

M3

RF

M4 /4(f0)

M2

LO

/4(f0)

Vg3

M1

Vg2

14

-5

7

-10

0

-15

-7

-20 -25

-14

-25

-20

-15

-10

-5

IF power – dBm at 2.4 GHz

Conversion gain - dB

(a) Schematic.

0

RFin (dBm) at 60.4 GHz (b) Gain and power performance. Figure 11.23 Quasi-cascode multiplier generates LO power to be delivered to the gate mixer M4. After [12].

11.2 SINGLY BALANCED 11.2.1 Hybrid Topologies 11.2.1.1 Substrate LO Injection Using JFET Technology An integrated version using JFETs as a balanced mixer where the substrate is used as a signal port appeared in 1973 [13]. The invention relates to a novel balanced mixer circuit using two JFETs built on a single die with off-chip elements to match impedance. The source terminals of the two transistors are tied together. One input signal is applied to the gate electrodes differentially, while the other input is applied to the substrate so it is in phase to each device.

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Microwave Mixer Technology and Applications

Figure 11.24 Dual monolithic JFET mixer applying LO at the substrate.

The load is a transformer tuned to either the difference or sum frequency, representing a low impedance for both f1 and f2 signals. For practical purposes there is no leakage into the gates. For the topology displayed in Figure 11.24, the channel conductance is controlled by the transverse electric field created across the channel between the gate and the substrate. The field is dependent on the reverse voltage applied to the gate, which is a reverse biased p-n junction. The channel is doped with high resistivity layer so the depletion region extends into the channel in direct proportion to the applied reverse bias. The same effect can be obtained by reverse biasing the substrate-to-channel interface. Increasing the voltage applied across either p-n junction will increasingly constrict the channel until there is no more current in the channel. Therefore variations in the potentials applied across the gate or substrate will modulate the channel current as long as the total applied voltage is less than the pinch-off voltage. To optimize performance as an up- or down-converter, the frequency range of the drain transformer is set to operate over the RF or IF bands. Let’s initially assume up-converter operation, where the base-band signal is applied to the substrate gate. In this case the drain transformer is designed for RF frequency. The IF signal is applied to the substrate in phase to the gates and the LO signal is applied to the control gates of the two FETs in a push-pull manner. The corresponding drain currents are filtered by the output parallel resonant circuit. The RF signals are in counter phase and are combined by the tuned transformer. For down-converter operation, the drain transformer is designed for IF frequency. The drain currents on the two devices still follow a square law function regardless of whether the input signal is applied to the substrate gate or the diffused control gate or both. Therefore, the channel current will generate mixing products at the sum, fLO + fIF, and difference fLO – fIF. The mixing signals are out of phase at each drain but are reversed and added by the transformer action at the output port. In addition either the sum or the difference frequency can be selected by filtering out the unwanted signals. The main advantage of this approach is the

Active FET Applications

777

isolation between LO and RF built into the device structure. Therefore, there is no need for an external coupler to apply the signals to the channel. 11.2.1.2 Complimentary MOSFET The N-channel, depletion mode MOSFET device has been widely applied to amplifiers and mixers. In this application, [14], complimentary enhancement mode symmetric FETs are employed taking advantage of CMOS technology. Operation in common source mode permits optimum gain to be obtained from both stages of the mixer, and the circuits are relatively simple and inexpensive. The work developed by Merle V. Hoover describes the design details of this complimentary MOSFET mixer/amplifier circuit. In the schematic displayed in Figure 11.25, the transistors PA and NA are of complimentary conductivity and are connected in series between a bias supply +V and ground. In operation, resistors, 38, 42, and 44 bias the CMOS amplifier at or close to the center of its linear operating range. In other words, the biasing is such that transistors N A and PA both quiescently conduct. The signal EIN1, may be considered the LO at frequency f1 and is applied to the gate of NA through a tuned transformer. The second signal EIN2 at frequency f2 maybe considered the RF signal and is applied to the gate of PA with a similar transformer.

Figure 11.25 Schematic of complimentary MOSFET mixer.

The bias for the gates is derived from the voltage at the common drain connection of both FETs. The large signal E IN1 modulates the transconductance of NA and the output conductance of both NA and PA. The small signal EIN2 is amplified and mixed by the modulated output conductance. The output circuit of the mixer is a series tuned circuit at either the upper (f1 + f2) or lower (f1 – f2) sideband frequency. The desired IF signals are linearly amplified by the CMOS pair PB, NB after proper filtering. In this circuit P B, NB are biased automatically by

778

Microwave Mixer Technology and Applications

the common drain voltage of the previous stage. An additional advantage of CMOS is the complimentary nature of the devices that can provide the proper 180 phase shift to the signals without a balun transformer. 11.2.1.3 MESFET_Technology The contribution of the following patent, [15], is to obtain a high conversion gain mixer using FETs biased in the active region. The FETs are common gate for the AC signals, and common source for DC, as depicted in Figure 11.26. Self bias is employed by means of resistors 46 and 48. The combination of RF and LO signals are by means of wire wound transformers. The RF signal applied to the transformer center tap reaches the FET source in common mode. The filter 60 is transparent for RF due to the equal phase of both arms.

Figure 11.26 Active singly balanced FET mixer.

The unbalanced LO signal is transformed into differential mode and applied to the FETs in counter phase. The resonant circuit, 60, shorts the source for the IF signals avoiding dissipating IF power into the high frequency elements. The resonant circuit is transparent to the RF signal and introduces some loading on the LO signal, but it is minor if the LO and IF frequencies are sufficiently separated. On the drain side, the low impedance for the LO signal is obtained by means of capacitor 66. The RF signals in this configuration are terminated with a high impedance at the drain. On the other hand the capacitor 66 and inductances 62 and 64 comprise a low pass transformer from high drain impedance to a lower impedance compatible with the IF impedance of the next stage. The differential IF signals are converted to unbalanced by means of the output transformer. 11.2.1.4 FET Technology A floating active mixer that follows the same basic topology described in Figure 10.18 is addressed in this patent [16]. The circuit schematic illustrated in Figure

Active FET Applications

779

11.27 reveals that RF and IF baluns are added to the basic topology, creating a quasi-doubly balanced mixer. The term quasi-doubly balanced is used to describe a mixer that resembles a doubly balanced where both RF and IF are balanced and simultaneously resembles a singly balanced since RF and IF frequencies cannot overlap. The LO signal is applied between gate and source terminals through a parallel LC circuit combination and a series capacitor. The inductor resonates the total capacitance presenting an open circuit impedance to the LO source. The series DC blocking cap is used to self bias the FETs, providing a small DC rectification at the gate. The voltage developed is dependent on the magnitude of the applied LO. Distortion in resistive mixers is generated by leakage of LO signal into the FET channel, perturbing its conductance and creating signal distortion. According to (10.6) of Section 10.2.2, this effect can be minimized using the basic topology of Figure 10.18. In addition, by applying a low bias to the channels of the FETs it is possible to optimize inter-modulation. In the case of this work, a 5V supply is applied to the center tap of transformer T 1, which connects to the drain through a combination of parallel LR circuits. The resistor gives an appropriate termination to the FETs; the inductance L bypasses the DC current and can also be used to resonate the drain capacitance. The RF signal is applied to the drains via a transmission line balun and a diplexer consisting of a simple LC circuit provides isolation between RF and IF signals. Similarly, the IF signal is extracted from the mixer by means of diplexer and a discrete wire wound balun. At the drain terminals, both RF and IF signals are balanced, and their respective baluns reject the common mode LO signals. It is clear that the diplexer precludes the overlapping of RF and IF bands. In the time domain, signals propagating in the transmission line transformers are mismatched at the drain terminal. The transmission line impedance of 50 ohms is either much higher than the FETs on-impedance of ZON = 8 , or much lower than FETs off-impedance. 10 100 D1 LO

50

2.5

10

10

G

D1

10 100

T1A

T2A

T1B

T2A

T1B

10

10 85

Figure 11.27

IF

T1A

Quasi doubly balanced Vice type of mixer.

RF

780

Microwave Mixer Technology and Applications

The off-resistance is high for a small size MESFET, say 1000 , but it is shunted by the device output capacitance. Assuming Cds = 1 pF and fRF = 1600 MHz, then ZOFF = 1/(Cds) = 100. Therefore, a standing wave pattern appears in the transformers due to the impedance mismatch and incident and reflected signals. In the frequency domain, an equivalent average impedance appears at the drain terminals, which requires computer analysis for its determination. A rule of thumb that provides reasonable results is to average both impedances geometrically, so that Zeq = 28 . The IF signals generated in the channels, say at 200 MHz, are also mismatched and equivalent impedance is higher, around 80 to 100 ohms. A simplified version of this mixer was proposed by the same inventor [17], where the RF and IF baluns were replaced by a single one as depicted in Figure 11.28. The frequency range is dependent on the frequency response of baluns T1A, T1B. The balun employed by the inventor allows operation for IF frequencies from 10 to 100 MHz and RF/LO in the 2000 MHz range. Another simplification is in the diplexer that is now localized in the unbalanced area, so only one circuit is required. The behavior of this mixer is essentially similar to the previous circuit with differences pertinent to the lower frequency of operation. For instance, at lower frequencies the off resistance is higher while the on resistance is approximately the same. If the RF/IF source impedance at the drain is made higher than 50 ohms, the circuit will operate more like an ideal switch with improved performance. 10 100

LO 5

50

T1A

100

RF

T1A

T1B

100

100

IF 10 6

100

T1B 10

100

10

Bias 10 Figure 11.28 Schematic of the simplified quasi-doubly balanced mixer.

The reason is the higher source impedance will make the on-resistance relatively closer to a short. Therefore, connecting a balun as a 4:1 transformer causes the RF/IF impedance to increase to 200 ohms. Simulations carried with the values provided by the inventor and using the NE673 device are represented in Figure 11.29. The LO frequency was set at 2 GHz for an IF at 100 MHz. The obtained conversion loss is 5 dB for a lossless circuit. The Guanella transformers were modeled as ideal transformers. If losses are added, then conversion loss will

Active FET Applications

781

increase by at least 1 dB. The third order suppression is in the plot on the right for two conditions, Vbias = 0 V and Vbias = 1V, showing an improvement of 4 dB on the intermodulation level and an increase in power of approximately 0.5 dB. 0

dBm(fsweep_NE673_Vic..Vload) dBm(Vload)

-1 -2 -3

gain

-4 -5 -6 -7 -8 -9 -10 1.80E9

-14 -16 -20 -24 -28 -32 -36 -40 -44 -48 -52 -56 -60

1.85E9

1.90E9

1.95E9

2.00E9

2.05E9

0

25

50

RFfreq

75

100

125

150

175

200

freq, MHz

(a) Conversion gain (b) Third order suppression at PLO = 7 dBm Figure 11.29 Performance of Vice mixer in terms of conversion loss and third order intercept point.

11.2.1.5 MESFET Up Converter A GaAs FET gate up-converter for 20 GHz operation is found in the technical literature, [18], with the following main items: (1) method to analyze mixers; (2) use of slot line baluns; (3) performance. This up-converter proposal reverses the classical converter by making the IF signal a large signal modulating the gate voltage and LO a small signal voltage that is up-converted to upper and lower sidebands. To avoid confusion this section refers to the LO as the carrier, and the IF input as being large signal. That is a convenient way to up convert, since the digital voltage when applied to the FET's gate can efficiently modulate the drain current. The transconductance modulation is assumed to be of the square wave type, with maximum value given by gmm. (1) Method to analyze mixers. The method is a simplification of the conversion matrix, and adequate for circuits where the parallel feedback is important and series feedback can be disregarded. jB

ZS

jX

Vg YO

ELO

Ri

YIF YRF,YIM

IRF

IU

IL

Figure 11.30 Simplified model for a gate mixer with parallel feedback.

782

Microwave Mixer Technology and Applications

The model is therefore transformed into the one represented in Figure 11.30. The load comprises an open stub with low impedance at RF and an admittance Y0 at other frequencies. The FET used is 0.5 µm gate length, 750 µm gate width and the authors assumed large signal device output impedance to be equal to 1/Y0. The current generator comprises three components: RF, Upper, and Lower sidebands whose relations to gate voltage is described by the conversion matrix (11.4). The output conductance, Y0, is assumed constant and equal to the average value over the LO cycle. The matrix in (11.5) relates gate voltages to the collector currents, plus the excitation signal at the gate.  1  2  I LO   I   g  j mm  U     I L   j   VLO   A2 V    0  U   VL   0

0 A1 0

j



1 2 0

0 0 A2

j    VLO   0  VU   1  VL  2   I LO  A3   I U  0   I  0   L   E LO 

(11.4)

(11.5)

The coefficients A1, A2, A3, A4 have the following functions: A1 = admittance coefficient relating gate voltage due to output upper and lower sideband currents; A2 = LO admittance coefficient relating gate voltage due to LO output current; A3 = LO dimensionless coefficient relating gate voltage to input LO voltage. The coefficients for (11.5) are defined as follows: A1 

A2 

A3  

(11.6a)

BX  1  Ri  jX 2Y0  jB  2Y0  jB  j 2 BY0 Ri  jX    Z  S  BX

(11.6b)

1  Y0  YL  jB  jB Y0  YL Ri  jX    Ri  jX Y0  YL  jB  Z S  

1 jX Y0  YL  jB  Z S Z S  jX Y0  YL  jB   Ri Y0  YL  jB   jB Y0  YL Z S Ri  jX 

YL  jY0 cot 

(11.6c) (11.7)

Active FET Applications

783

Once the coefficients are determined they can be entered into the matrix equation and solved for the output currents as a function of drive voltage. The output sideband currents can then be expressed as a function of circuit parameter and input signal drive.  1 1  A2  g  mm 2 j  A   2  j   A2  



j



A1

1 1  A1 g mm 2 0

 1   2    I LO      I   E A  j  0 LO 3  U      j  1 1 I L     A1  g mm 2     j



A1

(11.8)

(2) Slot line baluns The block diagram indicated in Figure 11.31 denotes the need for two baluns at the gate, one for the LO carrier and the other for the large signal IF. The LO voltages at the drains are therefore in counter phase so that any point joining the drain circuits at equal distance from the drains is a virtual ground. The IF input is also in counter phase at the drains, so the upconverted signals are in phase and can be combined at the output as shown. G

180

D S

Output

LO G

D S

180 Hybrid

IF Figure 11.31 Block diagram for the up-converter system. After [18].

The circuit realization of this block diagram in MIC technology is depicted in Figure 11.32. The figure details the input LO balun created by the slot line power divider. The output shows the drain lines in parallel before combining. Since the LO is in counter phase, the virtual ground creates a short circuit stub termination in the drain for the LO frequency, whose inductance is determined by (11.8). The virtual ground phase, or inductance to ground, is optimized by varying the position of bonding ribbon on the lines. The IF input signal at the drain is

784

Microwave Mixer Technology and Applications

shorted by the drain bias line. Decoupling of DC to the load is obtained through open ended parallel coupled lines. IF - (0) microstrip

LO

slotline

MN

MN

MN

MN

gold ribbon

DC Block

RF

Combiner 180 Divider

IF - (180) Figure 11.32 Implementation of the up-converter in MIC technology. After [18].

(3) Performance Conversion loss of the mixer is about 7 dB. With a 140 MHz IF input at +13 dBm power, mixed with a 21 GHz LO carrier at +10 dBm, output power at the upper sideband is +3 dBm. 11.2.2 Singly Balanced in IC Technology 11.2.2.1 MESFET Types E and D Another converter design for satellite TV receivers is disclosed in patent [19]. The proposal for this work is to employ a singly balanced gate mixer built using low cost MESFET technology, where packaging parasitics are absorbed into the RF, LO matching circuitry. The RF and LO are applied to an external folded ratrace hybrid that adds those signals in phase for LO and counter phase for RF before being applied to the packaged device. The block diagram for the packaged receiver device in Figure 11.33 depicts an external hybrid used to combine RF and LO signals that in turn connect to the converter differential input. The RF and LO signals are delivered to both mixers. The mixer outputs each couple to a filter, and then combine in a three stage differential IF amplifier that is followed by a two stage single-ended amplifier. A representation of the single ended mixer is shown in Figure 11.34. The input shows a “T” type network consisting of wire bond inductances, 82 and 84, and package lead inductance 80. The low impedance for the RF and LO frequencies on the drain side is depicted by capacitor 104. The association of shunt capacitor and series inductance comprises a low pass matching network transforming the 50 ohm load impedance to 100  at the drain terminal.

Active FET Applications

785

RF+LO Input Zin

RF

Input

LO

Hybrid

IF Input Zin RF+LO

Figure 11.33 Block diagram of receiver concept. The mixers are built with enhancement devices and the amplifiers use depletion devices on the same die. 100 80

84 10 4

82

Figure 11.34 Equivalent circuit for the single ended gate mixer.

The mixer MESFETs are of enhancement type, measuring 0.5 X 320 µm2 built with ion implant technology, which resulted in high transconductance compared to similarly built depletion devices. The IF amplifier MESFETs are of depletion type. The die was inserted into a miniature plastic package, having six leads, three on each side. The LO can be set at two discrete frequencies, say 9.5 and 10.5 GHz to convert two RF frequency bands covering from 11 to 12 GHz. 15.0

FdB, (1.5 dB/div)

Gain, (2dB/div)

20.0

PLO = 3 dBm VDS = 3.0 V IDS = 5.67 mA

0.0 950

Frequency, MHz

Figure 11.35 Conversion gain and noise figure.

2050

0.0

786

Microwave Mixer Technology and Applications

The IF frequency band is capable of covering from 900 to 2500 MHz, a bandwidth useful to block convert, for example, TV information having 500 MHz of bandwidth. A plot of conversion gain and noise figure for the balanced mixer is in Figure 11.35. The effects of package parasitic on the high frequency RF and LO are reduced by minimizing package lead lengths. And after down-conversion, the low frequency IF is much less sensitive to package parasitics. 11.2.2.2 CMOS mmWave Application of CMOS 0.13 µm technology to build mmWave components is a challenge. A gate mixer having respectable performance was reported by the Berkeley group [20] at 60 GHz. The bias supply voltage is limited to V ds = 1.2V, and power is limited to 0 dBm, which means RF performance will be constrained to this level of power. Since this component was designed in standard digital process where losses are high, a coplanar structure was employed in the design to minimize substrate losses.

Figure 11.36 Quadrature mixer designed with microstrip lines. From [20].

The I/V characteristics of this technology and measured transconductance as a function of gate voltage are found in Appendix 9B. The circuit schematic shows the gate of each FET connected to a branch line coupler, allowing application of LO and RF simultaneously, with reasonable isolation between the ports. This whole circuit was built on a 1.6X1.7 mm2 die. Its performance is illustrated in the following two figures, the first shows conversion gain as function of LO drive, about – 2dB at PLO = 0 dBm; and the second shows the conversion gain versus RF frequency. If 3 dB is the reference gain, then the bandwidth is 54 to 62 GHz, all obtained from a power supply of 1.2 V.

Active FET Applications

787

(a) Conversion gain versus LO drive (b) Conversion gain versus frequency Figure 11.37 Quadrature mixer conversion gain. From [20].

11.2.2.3 CMOS Linearized and Compact A linearized RF amplifier was proposed in [21], where two CG (common gate) devices with different properties are paralleled. The differences are in bias or size, so that distortion generated by one device can be compensated by the distortion of the other. The plot in the figure shows the second order transconductance of each device biased at different voltages, and the composite, showing a reduction within a certain range of gate voltages, which reflects in lower third order distortion. One problem pointed out by the author is the feedback of second harmonic distortion components, demonstrated through Volterra series analysis.

(a) Circuit schematic. Source [21] ( b) Transconductance compensation Figure 11.38 Paralleling two devices biased at different gate voltages to compensate third order distortion.

This effect is minimized by adding a RF current source or tank circuit at the feedback frequencies. The resulting mixer schematic in the next figure shows

788

Microwave Mixer Technology and Applications

how the linearized RF current feeds the PMOS switched mixer. The IF is extracted from the drain in differential form. The results applied to a 900 MHz CDMA mixer biased at 1.2 V and 5.4 mA, gave IIP3 of +20 dBm and conversion gain of 7 dB. Noise figure, however, is high, on the order of 15 dB.

Figure 11.39

Linearized compact mixer. From [21].

11.2.2.4 CMOS X2 Subharmonic A compact subharmonic mixer built in 0.13 µm CMOS technology was reported, [22], for operation within the 58 to 66 GHz band in the up-converter mode. The switching devices M3, M4 are composed of 2X36 µm2 gates. The schematic in the figure shows only 4 devices are used along with a 90º hybrid built with lumped elements. The IF input at 4 GHz is applied in single ended mode to the gate of M 1.

Figure 11.40 Schematic of X2 subharmonic converter. From [22].

The lumped element 90 hybrid is built by broadside coupling two spiral inductors as illustrated in the figure. The coupler measures only 85x75 µm2.

Active FET Applications

789

Figure 11.41 View of lumped element hybrid employed in the design. From [22].

The M1 output at 180º is applied to switch M3 and also to the M2 gate where an additional 180º is introduced in the IF signal before being applied to switch M4. Hence M3, M4 receive IF signals in differential mode. The measured conversion gain represented in the figure is for LO power of 5 dBm and shows better than 5 dB gain over the 58 to 66 GHz band. In addition, the RF input power saturates at –5 dBm and the L – R and 2LO – R mixing products, respectively, are better than 40 dB and 25 dB down.

Figure 11.42 Conversion gain versus LO frequency. From [22].

11.2.2.5 PHEMT X3 Subharmonic A subharmonic X3 mixer was reported in [23] to provide a conversion loss comparable to X2 mixer. A 0.15 µm GaAs PHEMT process was employed in this design to translate a 29 GHz carrier to 2 GHz. The circuit schematic is in Figure 11.43(a), containing a 90º hybrid power splitter at the LO port followed by an additional phase shifter to guarantee a 120º phase difference between the gates.

790

Microwave Mixer Technology and Applications

The third harmonic is obtained by driving the FETs with high amplitude so that the output conductance is nearly a square wave. VGS

VDS

90 Hybrid /12@LO LO L=0.88 nH C=0.35 pF

HPF

LO0

RF

VGS

VDS

LPF

IF HPF

LO120

LO Band Reject

(a) Circuit schematic. From [23].

Conversion Loss, dB

18

Measured ( Simulated (

16 14

) )

)

12 10 8

LO = 9 GHz @ +10.5 dBm

6

1.0 (b)

2.0

3.0

IF, GHz

4.0

5.0

Conversion loss versus IF frequency. After [23]

Figure 11.43 Subharmonic X3 mixer schematic and conversion loss.

In such a case the switching function contains a high third harmonic to perform the mixing, and the fundamental LO component is far enough from the third that it can easily be shorted to ground. The minimum conversion loss is around 13 dB and maximum at 16 dB at the high end of the band. Additional relevant performance parameters are the 3L – I isolation is better than 45 dB, and 1 dB gain compression occurs at 6.7 dBm at the IF side.

Active FET Applications

791

11.3 DOUBLY BALANCED In this section, dual singly balanced mixers applications are described. Since this topology requires very similar devices, the majority of applications are in IC form, either for fundamental or subharmonic types. 11.3.1 Discrete Floating MESFET A floating active double balanced mixer that follows the same basic topology described in Figure 10.18 is addressed in the patents, [24, 25]. In that invention, two FETs were assembled in back-to-back form (i.e., gates tied together, sources tied together, and LO is applied between gate and source, floating above ground). Here, two singly balanced mixers similar to the previous invention are connected in parallel, resulting in a doubly balanced mixer as shown in Figure 11.44. Therefore, four FETs, Q1-Q4 are used as the mixing elements and special baluns are required to provide balanced signals to the FET connections. For instance, the LO balun is required to deliver signal at the gates of Q 1-Q2 at 180 out of phase in relation to the gates Q3-Q4. Bifilar wire wound on a ferrite core can provide this function at L – band frequencies.

Figure 11.44

Floating active DBM with core mixer after Vice.

On the RF/IF side the balun has to provide the same function for both channels. This is obtained by employing a special trifilar transformer, which the author called a reflection transformer, shown in Figure 11.45. There are six transmission lines T1-T6 in this transformer, T1 is composed of wires 32 and 34,

792

Microwave Mixer Technology and Applications

and T2 is composed of 34 and 36. The additional transmission line T 3 shares wires 32 and 36 with T1 and T2. The similar notation holds for the additional three linesT4-T6. The RF signal is previously balanced by balun 18 and applied to the RF ports of this structure that subdivides the signals additionally generating the four ports to connect to the FETs. The resulting impedances for each transmission line are listed in Table 9.2 along with the phases at each terminal.

Figure 11.45 Schematic showing transmission lines of the reflection transformer.

This structure behaves similarly to a center tapped transformer. The RF signal is applied at the “primary” lines 34, 44 inducing RF signals at the “secondary” lines 32, 36 and lines 42, 46. The RF signal appears at the drains on the top in counter phase relative to the drains at the bottom. Table 11.2 Impedance for Each Section of Transmission Line

T

Z

0

1

2

3

T1 T2 T3 T4 T5 T6

Z0 Z0 2Z0 Z0 Z0 2Z0

0 0

180 180

0 180

0 180

180 180

0 0

180 0

180 0

Due to the phase reversal connection, the IF current returning to the transformer, adds up at the center tap and cancels at the primary, due to the opposing IF voltage phases. According to the transmission line impedances, one can verify the impedances presented at each terminal. The RF impedance consists of the parallel impedance of T 1 and T2 in series with the parallel impedance of T4 and T5, so that ZRF = Z0. The IF impedance consists of the parallel combination of T3 and T6, which is equal to ZIF = Z0. However, the balanced impedance seen by the FETs is equal to 2Z0. The bias to the devices is applied through the balun 16

Active FET Applications

793

and the DC return is through the current sources 30, 32 of Figure 11.44. The FETs used as current sources have an IDSS = 23 to 30 mA and exhibit a high impedance over the LO frequency band when saturated. The gate bias is developed upon application of LO power, letting a small amount of rectified gate current flow into Q1, Q4, charging capacitors C1, C2. Thus the FET terminals are still floating with the application of bias. The supply voltage employed is 5V and the bias elements are R5 = 25  and C5 = 0.01 F. The resistors R3 and R4 plus capacitors C1, C2 generate a small positive bias to the gates. The resistors R 2 and R3 are equal to 100  and their objective is to match the gate impedance. 11.3.2 Doubly Balanced MMIC Technology 11.3.2.1 JFET IC Technology A SSB modulator in monolithic form was introduced in 1988 [26]. The circuit uses two differential amplifiers as shown in Figure 11.46. The devices M5 and M11 are the current sources for the differential devices. The LO voltage is applied to one of the gates of a differential amplifier and the RF or modulating signal at the other. Therefore, the LO injected at the gate of one device, is also applied to the source of the other device, acting as a common gate to the LO. The LO voltage at the drains of the differential pair are in counter phase and are zeroed. The RF is similarly applied, so that the drain current at each device contains the product of both LO and RF signals.

M5

M11

Figure 11.46 Schematic of the JFET MMIC mixer.

The inventors proposed various ways of injecting the LO and RF. The case in the figure below uses a 90 coupler added to generate LO (–sine and

794

Microwave Mixer Technology and Applications

cosine) signals and a simple 90 coupler on the RF to generate (sine and cosine) signals. The mixing products of importance are in the figure at each drain. Since the drains are connected together the only component that is not cancelled is the – cos(R-A) corresponding to a single side band converted signal. By swapping the inputs of the RF drive the sum cos(R+A) is obtained instead, following the same approach as depicted in Figure 11.47(a). The LO in this case is a low frequency signal, 12 KHz and the RF a higher frequency, 1.6 MHz. The capacitor at the gate of one arm in the chip limits the frequency of operation of this chip. The bias is applied externally, between the drains of the differential pair to ground of current source. In addition to the different ways of applying RF and LO signals, the inventors also proposed a dual-gate version shown in part b of the figure.

(a) Signal components Figure 11.47 Schematics of proposed mixer.

(b) Dual gate version

The rejection of carrier, sidebands, and other spurious signals is shown in Figure 11.48 to be more than 40 dB. The results are for a JFET IC, but the concept is applicable to other technologies and frequencies.

Figure 11.48 Spectrum of generated SSB signal depicting sideband and carrier suppression

Active FET Applications

795

11.3.2.2 FET Technology A patent applying FET devices to differential topology was disclosed in 1987, [27]. This hybrid circuit proposal contains a FET quad die connected to transformer baluns for the RF, LO and IF signals. The quads are biased by a current source also acting as an IF amplifier. In this work, represented in Figure 11.49, the IF signals leaving the quad are transformed to single ended and applied to an IF amplifier input through a filter.

VDD

LO

RF

IF Filter

IF

Figure 11.49 Differential type of FET mixer.

The author proposes the use of a SAW filter. Signals in the mixer components are isolated from the IF amplifier by an RF choke and a resistive “PI” type of filter connected on one side to the IF amplifier drain and DC connected on the other to the sources of the ring-connected devices. The circuit showed 6.5 dB conversion gain for an IF band covering 100 to 800 MHz.

796

Microwave Mixer Technology and Applications

11.3.2.3 MOS Technology In conventional FET mixers, the gate is modulated with a large signal LO to create the required squared characteristic at the drain. This invention, [28], proposes to use a nonsymmetrical differential amplifier (i.e., the devices on the left and on the right of a differential amplifier have different transconductance). With the approach the inventor demonstrated it is possible to generate a multiplier with good mixing properties, requiring low LO signal amplitude.

Figure 11.50

Schematic of an anti-symmetrical differential mixer.

In order to demonstrate this possibility, consider the differential amplifier of Figure 11.50, where the drain currents of each device are given by (11.9), (11.10).

I D1  1 (VGS1  VTH 1 ) 2

I D 2   2 (VGS 2  VTH 2 )

(11.9) 2

(11.10)

In these equations,  is the transconductance parameter defined by:

  n

Cox W , 2 L

n = electron mobility Cox = gate oxide capacitance per unit area W/L = ratio of gate width and gate length Making the assumptions as follows

VGS1  VGS 2  VRF  VLO

(11.11)

Active FET Applications

I 0  I D1  I D 2 V  VRF  VLO  VTH

797

(11.12) (11.13)

It is possible to develop the current of each device as a function of the transconductance, bias voltages, and currents. The currents are divided into two terms to facilitate the development.

I D1  I D1A  I D1B I D2  I D2 A  I D2B I D1 A 

1

1   2

I D1B  2

I D2 A 

I0 

1  2  2  1 V 2 1   2 2

1 2 V    I (1  2 )  V 2   1  2  1 2  0 1 2  2

1   2

I D 2 B  2

I0 

1 2  2  1 V 2 1   2 2

12 V    I (1  2 )  V 2   1  2  1 2  0 12 

(11.14)

(11.15)

(11.16)

(11.17)

Note the first term in (11.16) is essentially DC related. If the devices are equal then the first term of ID1A is equal to I0/2. The second term contains a V2 term which is dependent on the difference between the device transconductance values. If they are equal then there is no square term. Therefore, if the RF and LO voltages are defined by (11.18) and (11.19), then they are multiplied in the V2 term and mixing products are generated.

VRF  VRF cos(2f RF t )

(11.18)

VLO  VLO cos(2f LOt )

(11.19)

The second part of the equation ID1B essentially is proportional to the input voltage and does not contribute to mixing. The frequency components contained in the drain current can be extracted by means of voltage developed on

798

Microwave Mixer Technology and Applications

RL. The same principle was applied to the circuit below that contains two sets of dual gate devices. In this case the transconductance parameter of the device is proportional to the bias applied to the second gate.

11  11(VB  VG )

(11.20)

12  12 (VG )

(11.21)

Where β11, β12 are the transconductance parameters of M11 and M12, respectively, and are the same for the devices M 14 and M13. By selecting VG and VB one can set the desired transconductance for the devices.

Figure 11.51 Topology of anti-symmetrical differential mixer with dual gate devices.

11.3.2.4 Four Quadrant MOS Multiplier This type of multiplier was previously invented in bipolar IC technology in the form of a current multiplier by Gilbert. Its application in MOS technology was disclosed in [29] with several circuit modifications compared to the bipolar topology, required because of the basic differences in the technologies. The present invention proposes creation of a multiplier based on the equation:

Vo 





1 2 V1  V2 ) 2  V1  V2   V1V2 4

(11.22)

Such a function is expressed in terms of a block diagram represented in Figure 11.52, consisting of a voltage adder, followed by a squarer that provides the desired product. Two sets of circuits are used to add the input voltages, one in phase and the other in counter phase.

Active FET Applications

799

Figure 11.52 Functional block diagram of a CMOS multiplier.

An adder can be obtained by implementing a cascode topology with a current to voltage converter. VDD R 1

Q3

Vout V1

Q1

V2

Q2

Figure 11.53 Simplified CMOS adder circuit.

The output voltage of the circuit is a function of the following voltage relations:

VDQ1  VDD  R1 I D1

(11.23)

Vout  VDQ1  VGS 3

(11.24)

The currents in Q1 and Q2 are given, respectively, by:

I D1  1 (V1  VTH ) 2 I D 2   2 (V2  VTH ) 2

(11.25) (11.26)

800

Microwave Mixer Technology and Applications

The current in device Q3 is the same as the current in device Q2, therefore, assuming devices Q1 and Q2 have same area, and solving for ID1 = ID2:

VGS 3  VTH 

1 V2  VTH 2  VTH  1 V2  VTH  2 2

(11.27)

The output voltage is given by:

Vout  VDD  R11 (V1  VTH )2  VGS 3 Vout  VDD  R11 (V1  VTH ) 2  VTH 

1 V  V   2 2 TH

Vout  VDD  VTH  R11 (V1  VTH  2V1VTH )  2

Vout  VK 

2

1 V  V  2 1 2

(11.28) (11.29)

1 V  V  (11.30)  2 2 TH (11.31)

With

VK  VDD  VTH  R11VTH  2

1 V  2 TH

Therefore the circuit of Figure 11.54 acts as an adder plus a constant term. In order to eliminate the constants, the circuit was inserted into a differential topology, and the output voltage now is given by:

Vo 

Vo 

1 1 (VGS 3 V GS1)  (VGS 4  VGS 2 )  (V V )  (VGS 2  VGS1 ) 2  2 GS 3 GS 4

1 (V V 2) 2 1

(11.32)

The next function is to obtain a circuit capable of generating the square of two input voltages. To implement this function, two MOS transistors with source and drain tied together are used as in Figure 11.55(a). In normal operation the currents at the drain cancels out due to the differential of two equal voltages. However, if the device is biased in the saturation region the currents can provide the multiplication function as will be demonstrated.

Active FET Applications

801

Figure 11.54 Full differential CMOS adder.

For this circuit the following voltage equations are valid:

Va  VGS1  VS  0

(11.33)

Vb  VGS 2  VS  0 VOUT  VDD  R0 ( I1  I 2 )

(11.34)



( I1  I 2 )   3 VGS1  VT   VGS 2  VT  2

2

(11.35)



(11.36)

VS, is the voltage at the source connection of both devices. Developing (11.36) the sum of current is shown to be defined by (11.37), with Vi = VGS2 VGS1.





( I1  I 2 )  3 Vi   2(VGS 2  VT )VGS1  VT  2



(11.37)



VOUT  VDD  R0 3 Vi   2(VGS 2  VT )VGS1  VT 

(11.38)

VOUT  VDC  R0 3Vi 2

(11.39)

2

With

VDC  VDD  2R0 3 (VGS  VT )2 and VGS1  VGS 2  VGS is a DC term

802

Microwave Mixer Technology and Applications

(a) (b) Figure 11.55 Simplified squarer circuits. (a) Single ended squarer. b) Differential squarer.

The DC terms can be eliminated by the use of a differential configuration. The output voltage is then expressed as:



VOD  R0 3 VX  VY 2

2



(11.40)

With those basic functions the full four quadrant multiplier can be assembled as shown in Figure 11.56. The desired multiplied output voltage, V0 is finally given by (11.41).

VO  3

1 R VV 2 0 1 2

Figure 11.56 Full IC CMOS multiplier.

(11.41)

Active FET Applications

803

The drawback of the NMOS multiplier is a reduced dynamic range compared with the bipolar option. This is due to a series of limitations in each of the voltages, in particular:

Vi max  2(VS  VT )

(11.42)

11.3.2.5 MOS Micromixer A FET version of the micromixer was addressed in [30], with additional improvements. The schematic of the FET IC is in Figure 11.57, showing two main parts, the active RF balun and active doubly-balanced mixer. The active balun is similar to the bipolar version, where a common RF input applies signal to a common source and a common gate amplifier simultaneously. The main difference here is the replacement of the current mirror by a high inductance at the source of the common gate device. Elimination of the current mirror requires an additional biasing circuit represented by “bias circuit 6” in the figure. The common gate voltage gain is equal to gmRL and the common source gain is adjusted to be the same through the feedback resistor.

Figure 11.57 Schematic of a FET version of the micromixer.

804

Microwave Mixer Technology and Applications

The load impedance is the composite of the impedance of the amplifier output matching circuitry and the mixer input. In this invention, enhancement mode FET (EFET), technology was used at the RF stage for its higher gm and no need for a negative gate voltage supply. LC elements are used in the drain to impedance match between the mixer and amplifier, thereby increasing conversion gain. The resistor 28 in the circuit balances the gain and phase response of both amplifiers. The active mixer uses depletion mode devices (DFET). The RF amplifier acts as a voltage to current converter and a biasing element. Current mirrors are used to bias the devices so that they can track bias over temperature and process variations. The FET devices present a low impedance (1/g m) to the RF signals, resulting in good linearity performance. The LO buffer provides a constant voltage drive for a wide range of LO input power. The tank circuits are tuned at the LO frequency (1.65 GHz in this case) and help reject out of band noise from adding through the conversion process. The tank circuits in parallel with the differential output reject the second harmonic. 11.3.2.6 MOS Image Reject Mixer The application of 0.2 µm gate length in MOS technology to the realization of an image reject mixer is found in this invention, [31]. The conventional block diagram for an image reject mixer is given in Figure 11.58. The proposal considers designing both mixers and LO phase shifter on a single die, and the additional IF components are on a PCB.

Figure 11.58 Conventional topology of an image reject mixer.

The level of image rejection is a function of accuracy in the amplitude and phase balancing of the vectors constituting the output IF signal from each mixer. A metric for this degradation, given by (11.43) obtained from reference [31], specifies the parameter image rejection ratio (IRR), based on the amplitude

Active FET Applications

805

and phase imbalance of the IF signals. If the two mixers have an error of 2% in the amplitude or about 2 error in the phase, the image rejection ratio is approximately 40 dB; these are unbalance levels that can reasonably occur in Gilbert type of mixers.

1  A IRR   IF 4  AIF 

2      IF 2   

(11.43)

A possible circuit implementation of an IRM to improve IRR is shown in Figure 11.59, where two Gilbert MOS-cells have been paralleled on the same die, with differential application of RF signal and differential quadrature application of LO. The LO-I on the left represents the in-phase differential LO input, and the LO-Q represents the same for quadrature LO. There are two IF signals available at the output, the IF-I in phase and the IF-Q in quadrature, which are shifted by 90 and added at the output.

Figure 11.59 Gilbert type image reject mixer.

In addition to the parallel connection, a single differential RF amplifier was used to drive the parallel set of quad switches, so that gain differences between amplifiers are minimized. On the other hand the IF becomes more sensitive to LO phase errors, originated by the dependency of phase on bias voltages. The IF phase error related to LO phase error in the circuit, obtained from HSPICE simulation is illustrated in Figure 11.60. The plot was obtained for a device technology with a gate length of Lg = 0.2 µm. For a threshold voltage of 0.3 volts, a minimum error in IF phase and LO phase is obtained. The authors proposed to isolate the DC between both circuits using a pair of decoupling

806

Microwave Mixer Technology and Applications

capacitors, C, so the bias for the RF amplifier and mixer cells can be adjusted independently to optimize IRR. The purpose of the LC circuit connecting the differential amplifier sources to ground is to present high impedance to the RF and LO signals and a short circuit at DC. The tank circuit could be replaced by high inductance, but the tank circuit requires lower inductance implying a smaller circuit area. The impedances connected at the drains of RF amplifier, ZL1 and mixers cells, ZL2 are both used to pass DC and isolate the signals from the rail.

Figure 11.60 IF phase error normalized by LO phase error as a function of LO DC bias.

The invention requires differential quadrature LO signals for driving the mixers, which are not shown in Figure 11.59. They can be obtained from polyphase filters described in Chapter 8 or be generated actively by simple differential amplifier circuits as shown in Figure 11.61(a).

(a)

(b)

Figure 11.61 Proposed active balun. (a) Schematic. (b) Dependency on ratio K.

The authors showed the size of device M3 relative to M1 or M2 contributes to the improvement of phase balance, illustrated in the plot of Figure

Active FET Applications

807

11.61(b). The best performance obtained for a k = 2.2 expressed in terms of the ratio (Vout1+Vout2)/Vout1 is 2%. The performance is limited by losses in the tank circuit that affect the signal balancing. 11.3.2.7 MOS Low Flicker Noise The objective of this invention, [32], is to modify the Gilbert cell mixer to reduce flicker noise in direct conversion receivers. The proposed principle is to minimize the circulation of DC current through the ring mixer itself, which is detailed using the schematic of Figure 11.62. In this figure, transistors T 2 and T3 and the cell containing the quad T6 to T9 can be identified as the Gilbert cell. The devices T 2, T3 are voltage to current converters acting as amplifiers for the RF voltage. The LO and the counter-phase LOX are applied to the control gate of the mixing FETs. During operation there is a common mode DC level at the input, between the terminals 10, 11 to ground and at the output 13, 14 there is also another common mode DC level. The common mode DC levels are sensed at the input by resistors R3, R4 and at the output by resistors R7, R8. Those voltage levels are compared and fed into the current sources T 4, T5. By proper adjustment of the circuit parameters, the difference between the input DC level and output DC level is approximately zero, so that there is no common mode current flowing into the mixer cell. VDD VDD T5

T4

R5

23 CM - ctrl 21

51

T8

10

R2

C2

C4

T7

C1

T9

LOX

R7 61

11 T3

RF

C3

T6

T2 T1

T11

13

R 3 R4 R1

IF

Vb2 T10

Ib

R6 C5

LO

14

R8

Vb2 T12

T13

Figure 11.62 Schematic of flicker noise cancellation circuit.

This reduction of the common mode currents, leads to a large reduction of the flicker noise without degrading the converter linearity. In spite of the bias applied to the quad it operates in near resistive mode. Additionally, the circuit

808

Microwave Mixer Technology and Applications

includes capacitors C3, C4 that suppress the generated upper sidebands. The output signals are delivered to an output amplifier T 10, T12 and T11, T13. The output current is converted to voltage by means of resistors R5, R6. 11.3.2.8 MOS Linearized gm An alternative to improve the distortion of RF amplifiers in Gilbert cells was proposed in [33]. The schematic of the invention is in Figure 11.63, where it is observed to be a modified RF amplifier.

+ Iref

IF

VA

LO

-

R3 VB

170 +VA

R2

176 +VB

VG R1

-

+

172 174

210

160

162

166

+VG

168

+VG

RF Figure 11.63 Linearized RF amplifier/mixer.

The invention introduces a compensated input trans-conductor stage (cascode stage), consisting of two stages in parallel, one biased in the saturation region and the other in the triode region. The cascode 160, 170 are in parallel with 166, 172 for the left part and similarly 162, 176 in parallel with 174, 168. The opposite gain characteristics of the first and second transconductor cancel each other providing a flat response and improving overall linearity of the transconductor stage. The sources of transistors 160, 162, 166, 168 are connected to ground. The linearization is best explained by Figure 11.64. The authors verified the gm for the transistors operating in saturation behaves in the manner shown in Figure 11.64(a). The device gm increases with gate voltage peaking at a specific gate voltage and shows a slight decrease when VGS = 0 V. The gm for the device operating in the triode region, depicted in Figure 11.64(b), increases with gm and peaks at VGS = 0V. Therefore, the saturated device shows gain expansion

Active FET Applications

809

and the device in the triode region shows gain compression for a signal differentially applied. Since those devices are paralleled, the combined gain of both transistors forms a substantially flat gain response shown in Figure 11.64(c).

Figure 11.64

(a)

(b)

(c)

(d)

Transconductance for (a) linear, (b) triode and (c) combined FET circuit

As the differential input signal increases beyond the limit of linear amplification, VLin in Figure 11.64(a), the input transistor on one side of the triode pair ceases to be in the triode region and enters the saturation region. When this happens, the transconductance of the triode pair, which is in saturation, decreases as the absolute value of VLin increases. In other words, the transconductance characteristic of the triode pair becomes compressive. The combined compensation takes place within VGS = VLin and VGS = 0 V. The single cascode configuration of Figure 11.64(d), can be used to define the limits of operation of the compensated transconductor. As long as the bottom device is in saturation (VD > VG-VT), the transconductance increases with VG. This is demonstrated by means of the equations below. The current at saturation is defined by the equation:

K1 (VG  VT ) 2 2 W  K1  Cox   L

ID 

(11.44) (11.45)

810

Microwave Mixer Technology and Applications

Therefore, gm = K1(VG – VT) and increases proportionally to VG. On the other hand when M1 enters the triode region (VD < VG-VT), the trans-conductance decreases as a function of VG. The current equation is (11.46) and the transconductance is in (11.47). 2

I D  K 2 [(VG  VT )VD  gm 

VD ] 2

(11.46)

dI D dVD V  K 2 [VG  VT  VD ] D dVD VG VG

Since (VG  VT  VD )  0 and

(11.47)

VD  0 , then gm is inversely proportional VG

to VG. The cross over point occurs when VD = VG-VT. At that point one can equate both currents. Replacing the gate voltage by V B from figure:

K2 K (VB  VD  VT ) 2  1 (VG  VT ) 2 2 2

(11.48)

The external applied voltage at the initial bias is VG0 = VG and the linear input range is given by: V Lin = VG0 -VG. Working the equations one can find the input linear range voltage, VLin, which is given by the equation:

V Lin 

VG 0  VT 1

K2 K1

IR

K2 K1

1

K2 K1



 aV Dsat  const

(11.49)

Therefore, the linear input range is proportional to V Dsat for the M1 device plus a constant. The proportionality constant "a" is a function of the relative size of M1, M2 and the second constant is a function of device size ratio and the voltage developed on the bias circuit resistor. The combined transconductance is kept constant if the VDsat is constant. To guarantee transistor 160 operates in the saturation region the following condition should be met:

VDS _ 160  VGS _ 160  VT _ 160 VA  VDS _ 170  VG  VT _ 160

Active FET Applications

811

VA  VG  VDS _ 170  VT _ 160 I ref ( R2  R3 )  VDS _ 170  VT _ 160

(11.50)

and,

VB  VG  VDS _ 172  VT _ 166

I ref R2  VDS _ 172  VT _ 166

(11.51)

11.3.2.9 MOS Multiswitch A modified Gilbert cell mixer, targeting a high IP2, for direct conversion applications, was disclosed in 1995, [34]. Among the specifications for this type of receiver are a low I-Q mismatch and high IIP2. The first specification relates to how much the vectors deviate from quadrature and the second is the sensitivity to second harmonic. The spurious products from second harmonic signals fall adjacent to the base band during conversion. Additional problems caused by the second harmonic include the self mixing, created by leakage of LO to the RF port or vice versa, generating an interfering time-variable DC component. Therefore the second order inter-modulation level in direct conversion is more important than the third order.

Figure 11.65

Example of a single balanced version of the mixer.

In the configuration depicted in Figure 11.65, the two pairs of switches on the left are driven by two LO signals in quadrature. Similarly, the two pairs on

812

Microwave Mixer Technology and Applications

the right are driven by an additional LO pair in quadrature with respect to each other and shifted 180 relative to the first LO pair as indicated in the figure. The converted I+ and Q+ signals are added at the drain on the left and I- and Q- are added at drain on the right. Since I and Q are orthogonal they do not interfere with each other. The first improvement is the fact that fundamental I, Q signals are in counter phase and second harmonics are generated in phase. When passing through a wide band balun, the fundamental is enhanced and the second harmonic is rejected, improving mixer IIP 2. Paralleling two singly balanced mixers and employing eight LO signal sources, a doubly balanced I-Q mixer is obtained per Figure 11.66. The odd signals generated by the switches S1, S3, S5 and S7 are grouped together, as are the even ones generated by the switches S2, S4, S6 and S8. The resistors R1 and R2 are the I+, Q+, and I-, Q- loads, respectively. The RF amplifier is differential, composed of transistors QN5_1 and QN5_2, which also are current sources for the mixing switches.

Figure 11.66 Doubly balanced I-Q mixer.

11.3.2.10 MOS CMOS Modulator A simple mixer capable of operating at low power, low bias is the object of this invention, [35]. The basic operation is explained by means of Figure 11.67, displaying a complimentary MOS arrangement with N- and P-channel devices (CMOS) and a N-channel active load. A first look at this schematic reveals it is a cascode type where RF signal is applied to the CMOS devices and the LO is applied to the top NMOS source follower. Due to the complimentary connection of P, N devices they operate as a class AB trans-conductor. The total

Active FET Applications

813

transconductance is obtained from the derivative of current with respect to voltage and is defined in (11.53). It shows the inventor’s choice was to bias Vin at Vdda/2.

I dst  k Vgs  Vth 

2

Gm  gmp  gmn  2 K (

Figure 11.67

(11.52) Vdda  Vth ) 2

(11.53)

Basic proposed mixer topology.

The voltage applied to the gate of source follower also sums up to the Vdda voltage, so the total transconductance is modified as follows:

Gm  2 K (

Vdda  Vd  VT ) 2

(11.54)

The output current is given by (11.55), as a multiplication of Gm as a function of LO voltage expressed in terms of Vd, and the input voltage Vin. The desired multiplication property of Vin and Vd voltages is in (11.56).

I out  GmVin  2 KVin (

I out  GmVin  KVinVd

Vdda  Vd  VT ) 2

(11.55) (11.56)

A differential version of the mixer is shown in Figure 11.68, where gate bias is applied by means of resistor R1 and R2, from a charge pump bias control, and the signals are coupled to the circuit by means of capacitors C1 and C2.

814

Microwave Mixer Technology and Applications

Figure 11.68 Differential version of proposed mixer.

The output current expressed in terms of the differential gain modulated by applied voltage Vd, is in Figure 11.69.

Figure 11.69 Gain modulation property.

11.4 SUBHARMONIC APPROACH Subharmonic mixers using the Gilbert topology and built with FETs are classified similarly to those built with bipolars: QLT where the quad LO is applied to the core cell that is stacked on top of a differential RF amplifier, and DRT where the differential RF amplifier is stacked on top of the core cell. QLT.-The first type, [36], is represented in the figure where the core cell is similar to a quad ring where each device has been replaced by a parallel

Active FET Applications

815

combination of two FETs. The RF amplifier at the bottom is a modified cascode type amplifier where bias is applied to the joining of drain source terminals by means of a LC filter circuit. Besides bias insertion, this filter is a trap for LO improving the switching and at the same time the L - R isolation. With higher isolation the self mixing common in this type of mixer is minimized. This is called an LC folded amplifier, with the goal of a low voltage bias supply, in this case 1.5V.

Figure 11.70 Subharmonic with quad LO on top. From [36].

The results of this work are in Figures 11.71 and 11.72. In the first figure, the conversion loss and noise figure is represented as a function of LO power. The plots were obtained at an RF frequency of 5.2 GHz and an IF of 20 MHz. One of the issues in direct conversion receivers is the conversion of various type of noise into base band, in particular the one resulting from self-mixing is represented in the second figure.

Figure 11.71 QLT Conversion gain and noise figure versus LO power. From [36].

816

Microwave Mixer Technology and Applications

In this work the authors claim a noise power of -110.7 dBm that is below kT (thermal), which allows the baseband lower cut off frequency to extend below 1 MHz with a reasonable noise figure.

Figure 11.72 QLT base band noise figure for two conditions: STD - standard CMOS process; ICP CMOS compatible process where silicon is selectively removed underneath spirals and transformers. From [36].

DRT - In the second type of differential subharmonic mixer, DRT, the RF and LO ports of a conventional mixer are interchanged, so that RF is applied to the top and LO at the bottom. The LO switches at the bottom are frequency multipliers that generate second harmonic and reject the LO fundamental. +VDD

+ IF RF

+

M1 M2

-

M3 M4

rd M5 LO 0

rd

rd M6

rd

M7

M8

LO 90 LO 270

LO 180 RC Figure 11.73 Subharmonic with RF applied to top quad. After [37].

Active FET Applications

817

The circuit from [37], was built on a 0.18 µm CMOS process and achieved 8 dB conversion gain and -8.5 dBm IIP3 when converting a RF input at 1 GHz to an IF at 100 MHz. To improve linearity in the quad, degeneration resistors were added to the source, with a penalty to the noise figure. This basic X2 mixer was extended by the authors to form a X4 mixer, where the multiplier is represented in Figure 11.74, [38]. L

 0

 90

iT1

 180

L

 270  0

 90

iT1

 180

 270

Figure 11.74 A times four multiplier circuit. After [38].

Approximating the device model for a linear I D/VG relationship for a short channel device, the authors made the analysis in two steps: first analyze a frequency doubler, then analyze the output of the first doubler applied to the second doubler. This process is detailed in Figure 11.75 where the bold trace represents the X2 generation. By superimposing this signal with another shifted in phase by 90, it is seen that the final waveform has a high X4 content.

Figure 11.75 Waveform for X4 construction from fundamental. From [38].

In analytical terms the two initial sets of doubled currents are given by (11.57), (11.58) valid for a conduction angle between 0 and 180. Within the addition 180 to 360 the output current is zero. Therefore, each current contains a second harmonic term. The coefficient A is the magnitude of the applied

818

Microwave Mixer Technology and Applications

fundamental signal. Adding both currents in quadrature, the resulting magnitude is given by (11.59), which can be approximated by a Taylor series. Disregarding terms higher than the first, the output current is approximately given by (11.60). Notice the argument t actually contains second harmonic due to active current rectification. The fourth harmonic is therefore contained in the output current iT1. Inductor L connects the drain to the positive supply is to increase the magnitude of the multiplied signal, improving conversion efficiency. 1  n CoxWE sat A cos t 2 1     n CoxWE sat A cos t   2 2 

i0,180 

(11.57)

i90, 270

(11.58)

1 iT  nCoxWEsat Acos t  sin t  2 cos t  sin t  sin 2t  1 

x 1   0

(11.59)

(1) n (2n)! n x (1  2n)(n!) 2 4n

1  15 1  iT 1  nCoxWEsat A  sin(2t )  2  16 2 

(11.60)

An alternative compact subharmonic mixer, [39], uses the triple level structure represented in Figure 11.76 in two versions, singly balanced and doubly balanced. The key of this contribution is the application of 4 FET switches in a ring topology with LO properly phased. It can be demonstrated that the differential output voltage is expressed by the following frequency components: VBB  k1 cos RF t  k2 cos RF t cos 2LO t  k3 cos 3LO t  k4 cos 3RF t cos 2LO t  ... (11.61)

The term associated with k2 will perform the down converting function and there is no odd mixing conversion. The output contains high order frequency components that are filtered by conventional low pass filters. The published results give a conversion gain of 11.6 dB and noise figure of 12 dB for an RF signal at 2 GHz. For direct conversion applications this design achieved +40 dBm IIP2 when the gm mismatch between the devices is lower than 1%. The DC offset cancellation performance due to self mixing is quantified by (11.62), where GRF is the conversion gain when input is at RF port, and GLO is the conversion gain when the input is at the LO port.

Active FET Applications DCoffsetcancellation 

819

G RF G LO

(11.62)

The results contained in the reference are: GRF = 11.6 dB, LO self mixing = -77.07 dB therefore, DC offset cancellation is equal to 86.67 dB, obtained when the FET gm mismatch is about 1%.

-BB LO270

+BB

LO90

LO90

LO0

LO0

RF+

LO180

RF-

(a) Singly balanced SHM.

-BB LO270 LO90

LO90

LO0

LO0 LO180

LO270

LO90

+BB LO270

LO0

LO0 LO180

Vb RF+

RF-

(b) Doubly balanced SHM. Figure 11.76 Triple level subharmonic mixers in singly and doubly balanced configurations. After [39].

820

Microwave Mixer Technology and Applications

11.5 SELF-OSCILLATING FET MIXER The design of a circuit to perform this function requires trading off the objectives for a mixer circuit with that of an oscillator circuit. The final circuit is expected to give inferior performance compared to the classical solution of independent functions. For example, self-oscillating mixers suffer from higher cross modulation interference and degraded inter-modulation characteristics. In addition conversion loss/gain is inferior since energy is required to maintain the circuit under oscillation. However, from the point of view of cost, self-oscillating mixers, compared to separate mixer and LO, are simpler and can be implemented by a single active device, and thus are lower in cost. 11.5.1 Single Ended-Gate MESFET A patent where Autodyne was used to denote a self-oscillating mixer was filed in 1970 [40], employing a dual gate JFET device. This device type allowed the design of FET oscillating mixers with lower harmonics compared to bipolars. The design of self oscillating mixers became practical with the use of a high “Q” dielectric resonator coupled to a FET device to generate stable and low phase noise oscillations at low cost. The performance obtained was sufficient to meet the requirements of many microwave systems, and in particular this approach became popular in TVRO applications. The circuit of Figure 11.77, [41], illustrates a FET with a common source (or common gate) connection and a feedback path between output and input ports.

Figure 11.77 Dielectric resonator self oscillating mixer.

The drain termination and the feedback from drain to gate induces a negative resistance at the gate port to initiate oscillations. The coupling of a dielectric resonator to a microstrip line is equivalent to a parallel RLC circuit in series with the line, imposing an open circuit at the resonating frequency. At all other frequencies it is ideally terminated into 50 ohms, resulting in a very stable oscillator. The coupling point is located at the closest point between the resonator and the line. By appropriately selecting the distance of the dielectric resonator from the FET port, the phase of the open circuit is adjusted to meet the oscillation

Active FET Applications

821

conditions at the desired frequency. The high Q of the dielectric resonator causes the LO to be isolated from the RF input, but allows the RF input to pass to the gate. The higher the Q, the lower the leakage of LO into the RF input port. Employing the FET in common source configuration is very convenient since the DC power (heat) generated in the device is easily dissipated through the ground connection. 11.5.2 Balanced FET The integration of the LO and a singly balanced FET mixer while increasing bandwidth is the subject of the circuit [42] shown in the figure. Figure 11.78(a), represents a conventional singly balanced FET mixer, with a LO balun transformer to feed the gates differentially and an RF balun transformer to feed the drains differentially. The IF signal is extracted from the center tap of the drain balun transformer. The FET switches mix the RF and LO signals and deliver the difference frequency to the output. A self-oscillating mixer is obtained by inserting a LO circuit in place of the LO balun, and merging the terminal impedances. Since the inputs to the mixer are in counter phase it is more efficient to insert a differential oscillator.

Figure 11.78 Conventional singly balanced FET mixer.

The design starts with a single ended LO, similar to the one represented in Figure 11.79. It is essentially a Colpitts topology with the capacitive divider formed by the source capacitor and internal C gs capacitance to create a negative resistance at the gate. The condition of oscillation must be met at the joining point of the varactor tank circuit and negative resistance circuit; that is: Real(ZT) + Real(ZR) < 0 Imag(ZT) + Imag(ZR) = 0

(11.63) (11.64)

822

Microwave Mixer Technology and Applications

Besides the oscillation conditions it is desirable to have a negative resistance expressed in terms of reflection coefficient in dB on the order of dB(T) = 3 to 6 dB. A large signal oscillator design procedure provides estimates for both frequency and power. For more information on GaAs oscillator design see [43, 44]. Lbias

ZT = -RT + jXT

LS

CS

Lp

CV

Cb

RS ZR = RR - jXR Figure 11.79

Single ended oscillator.

The single ended circuit input is converted into a differential oscillator by adding two devices connected by a tank circuit containing tuning varactors as represented in Figure 11.80. The circuit now contains two modes of operation, the odd mode where the gate voltages due to the left circuit are opposed in phase to the gate voltages from the circuit on the right. At the center point those voltages add to zero, making that fictitious point a virtual ground. In the even mode of operation, these voltages are in phase and add constructively at the center point. The conditions for oscillation however are designed for only one mode, say the odd mode, where the oscillations are excited. The even mode does not satisfy the oscillation condition as its eventual oscillations are quenched after the initial transient. Vbias

V+

VLS Cb

CS

VV

RS

Figure 11.80 Differential oscillator, varactor tuned.

LS

CS RS

Active FET Applications

823

11.5.3 Complimentary FET Complimentary MOSFET devices were also applied to a self-oscillating mixer. The example found in [45], targeted operation at frequencies where the FET capacitances are low so that the gate shows a high impedance. In such conditions tank circuits can be attached to the gate without degrading their Q-factors. The Nchannel FET is operated as an oscillator, at frequency f LO, with drain-gate feedback given by the resonant circuit, L2, C2. The RF signal is applied through a tuned transformer L1 and capacitor C1 tuned at frequency fRF to the gate of an Nchannel FET that acts as a low noise amplifier.

Figure 11.81 Self-oscillating MOSFET mixer.

The oscillations generated by the N-channel device, modulate its drain current and the P-channel device current as well since they are in series. The LO voltage at point A is heavily distorted since the channel impedance of a saturated device is high. The amplified RF voltage at point A is therefore switched by the developed LO voltage. In this peculiar circuit, part of the oscillation voltage is applied to the gate of P-channel device through the connecting impedance L4, making this device a gate mixer. Therefore there is more than one mixing mode in this circuit. A tuned transformer is used to extract the fIF signal and deliver it to a load. 11.5.4 SOM DGFET MOSFET One can think of a dual gate configuration as appropriate for frequencies where gain is not enough to sustain oscillations. Also, one of the ports can be used to apply feedback to generate the oscillator function, or an additional port where the down converted signal can be fed back to improve linearity. A general topology for such mixers is in Figure 11.82 [46] developed for TV receivers, which

824

Microwave Mixer Technology and Applications

confirms the higher linearity of FETs compared with BJTs used for this function. It shows an RF input terminal, TRF, which is isolated from the oscillator function though inter-gate capacitances. The oscillating signal at the drain is reactively divided by C10 and CE10, and fed back to the gate by means of inductance LE2. The oscillation is determined by the PI-type of circuit, CE10, CE20, LE2, and C5. A low impedance, C5, is applied to the drain to improve conversion efficiency without shutting down the oscillator. It is also part of another PI-filter designed to extract the IF signal and reject all high frequency signals. Determination of the elements that guarantee stable oscillation can be done using standard oscillator design procedures, for instance found in [43, 44]. The former uses simple empirical equations to estimate saturated power and simple models to estimate the currents and voltages at the FET terminals. Then the elements of a PI- or T-network are determined in terms of input and output voltages on the active device.

Figure 11.82 Schematic of a DGFET self-oscillating mixer.

The latter reference uses large-signal S-parameters for the FET, and solve the network elements that satisfy the oscillation conditions. Additional examples of the application of this topology found in the patent, are detailed in Figures 11.83(a, b). The varactor has been introduced as part of C E10 in the shunt arm of the feedback network to control the oscillation frequency. The control voltage is applied at T1 and the IF output is taken from T IF. All other elements in the schematic are essentially for biasing the FET and the varactor. In the compact version, capacitor C5 was absorbed by C E1 and the IF signal is extracted by a similar low-pass filter.

Active FET Applications

825

(a) Varactor tuned converter

(b) Compact version Figure 11.83 Detailed schematic of DGFET self-oscillating mixer. 11.5.5 Dual Gate DRO MESFETs The phrase tetrode FET was initially used for JFET devices having two gates. Later with the advent of planar technology, the two gates were photo engraved near to each other, and dual gate became the descriptor of preference for this type of device. Some useful circuit topologies and performance advantages are provided by the dual gate FET, (DGFET), which cannot be realized by other devices. As seen in Chapter 3, for DC analysis, dual gates can be treated as a cascode of two single gate devices. However, at high frequencies it is difficult to explain what happens within the channel when a large signal LO is impressed in

826

Microwave Mixer Technology and Applications

any one of the gates and the small signal RF and IF, respectively, are injected and extracted. The application of a Dual Gate GaAs MESFET as a singly balanced frequency converter using a dielectric resonator was patented in 1987, [47]. Two single ended dual gate mixers were combined so that the gates are in counter phase and the drains are connected in phase.

Figure 11.84

Dual gate MESFET converter.

The counter phase connection at the gate is obtained by the inductive coupling of dielectric resonators, whose magnetic fields are 180 out of phase at opposite points. Therefore, there is no need for an LO balun. In a balanced mixer two out of the three signals (RF, LO, IF) must be 180 out of phase, so here either the RF or the IF needs to be balanced. In the case of this patent the RF signals are applied to both FETs in phase so that the drain circuit must be in counter phase to add the resulting IF currents. The drain circuit must also provide an in phase connection between the drains, such as a coupling capacitor, canceling the LO at the output port. This is a desired feature in down-converter receivers. The RF and LO signals are applied at different ports so that RF can be matched for noise while LO port can be matched for power, and the L – R isolation is solely determined by the device isolation. The inventors proposed manufacturing this topology in two substrates, the gate circuit containing the dielectric resonator in alumina substrate, and the FET and remaining matching circuits on a GaAs MMIC substrate. 11.6 DISTRIBUTED APPLICATIONS 11.6.1 MESFET Gate Mixer The first application of a distributed topology for the purpose of converting RF signals made use of distributed amplifier topology with application of LO and RF at the input port by means of a coupler, [48]. A basic theory was laid down in that

Active FET Applications

827

work, and the practical application used an alumina substrate and a packaged NE710 MESFET device by NEC. IF 50 

V

Cp

Cp

Cp

Cp

Cp

DS

50  RF Exponentially tapered microstrip line

100 

LO Figure 11.85 Gate mixer MESFET schematic. After [48].

The schematic shows the circuit contains 5 single MESFETs and the output circuit contains a diplexer to absorb high frequency signals into a 50 ohm termination, and extract the converted IF by means of a low pass filter. This circuit provided 3.5 to 5.0 dB conversion loss over the 2 to 10 GHz RF band with a constant IF of 1.5 GHz. The LO drive was only 6 dBm. It works well for low frequency IF, in which case the LO and RF are close and the phase shift of each cell in the distribute input line will be approximately the same for LO and RF. 11.6.2 MESFET Drain Mixer A simple way to obtain LO to RF isolation is to use a drain mixer with RF applied to the gate and LO to the drain. The output line is shared by the RF, LO, and IF, with the major issue being simultaneously matching the RF and LO, since the IF is lower in frequency. An example application of this approach is illustrated in the next figure, coming from [49]. In this design, four 2X0.15X75 um2 PHEMT devices were built using coplanar transmission lines to serve as the gate and drain lines. Their dimensions are designed to follow the phase conditions in (11.65). The signal is tapped into and out of the lines by means of series connecting lines. The gate and drain lines are terminated by a resistor DC decoupled by a capacitor.

 drain( f LO )   gate ( f RF )   drain( f IF )

(11.65)

828

Figure 11.86

Microwave Mixer Technology and Applications

Drain mixer MESFET schematic. From [49].

The load capacitor must present high impedance to IF frequencies that are collected by an external circuit. The conversion loss of this circuit, depicted in in Figure 11.87, for the RF frequency range of 5 to 35 GHz. The applied LO power is +15 dBm, and the fixed IF output is at 1 GHz.

Figure 11.87 Conversion loss for mmWave mixer. From [49].

11.6.3 Dual Gate Mixer A better solution to isolate the signals LO, RF, and IF is presented in this patent [50], where a dual gate solution is employed. Thus, three independent lines are available making it easier to fulfill the (11.65) requirement. The patent employs

Active FET Applications

829

four dual gate devices operating over the RF band of 14 to 20 GHz, IF band of 2 to 8 GHz, and a fixed LO frequency at 12 GHz. The input RF signal is applied to a gate line and the IF signal is extracted from the drain line. The LO in this particular invention is applied in phase to gate 2 of all dual gate devices, as in Figure 11.88. During operation, the input signal Vrf is coupled to gates G1a to G4a through capacitors C2 to C5. This capacitor along with the intrinsic gate to source capacitor forms a reactive voltage divider delivering a selected amplitude and phase to Vrf1 to Vrf4, at the gates of FET 1 to FET4. The phase shift is related to the number of taps (or FETs) on the input traveling wave structure. The LO voltages are applied in similar amplitude and phase to the gates G 1b to G4b, generating the intermediate frequency signals Vif1 to Vif4 at the drain. The output signals contain frequency components corresponding to the sum (RF + LO), the frequency difference (RF - LO) and inter-modulation products of the components (nRF +/- mLO). The output coupling network parameters allows propagation of (RF - LO) and blocks the RF, LO, and all other mixing products and harmonics. The operation of a distributed mixer is similar to an ideal distributed amplifier, where the signals add constructively in one direction, for RF > LO, at the output terminal 19b and cancel at the reverse direction, terminal 19a. The difference in mixers is the presence of negative frequencies that change the output compared with positive frequencies. For RF < LO, the signal adds at terminal 19a and cancels at terminal 19b. The explanation for the discrimination comes from the traveling structure property. The transmission lines important to describing this effect are T 3 - T5 and T8 - T10, providing phase shifts equal to rf and if, respectively. In this analysis it is assumed both rf and if have the same phase shift equal to 90 at the respective RF and IF frequencies. Case where RF >  LO Let’s assume Vrf1 is referenced at 0 and applied to FET1 where it is mixed with LO at the same phase, so Vif1 also has the same phase shift. The Vif1a signal appears at 19a with 0 and Vif1b appears at 19b with 3IF (reminder: only T8, to T10 are considered). The RF signal Vrf2 is shifted by RF and mixed with LO at phase 0. The intermediate signal Vif2a has a phase shift of rf relative to Vif1. The signal Vif2a reaches 19a with a phase shift of rf + if = 2 and the signal component Vif2b reaches terminal 19b with a phase equal to rf + 2if = 3. Thus at terminal 19a the signals Vif1a and Vif2a are in opposite phase and are cancelled. The signals Vif1b and Vif2b are in phase at terminal 19b so that they are combined. The same effect happens for the signals Vif3 and Vif4.

830

Microwave Mixer Technology and Applications

Figure 11.88 Distributed mixer with in-phase LO.

Case where RF <  LO In this case the phase of IF signals is negative, so that for the first signal we have 0 for Vif1a at 19a and - 3IF for the Vif1b at terminal 19b. For the second FET the Vif2a at terminal 19a is rf - if = 0, which is in phase with Vif1a so they add. The phase of Vif2b reaching terminal 19b is rf - 2if = - which is opposed in phase to the phase of Vif1b, so they cancel at terminal 19b. The inventors provided a table that summarizes the several phases at the IF and translated frequencies. Table 11.3 Phases of IF Relative to LO and RF

FET 1

rf LO + IF

2

LO + IF

3

LO + IF LO + IF

4

RF > LO shift IF freq IF 0 IF  2 3

IF IF

shift 0

rf LO - IF



LO - IF

2

LO - IF LO - IF

3

RF < LO shift IF freq IF 0 IF  2 3

IF IF

shift 0  2 3

The details of the construction on a 4 mil thick GaAs substrate are shown in Figure 11.89. The gate periphery of each FET is equal to 200 µm. The capacitors C2 – C5, and the transmission lines T3 – T5 length and impedances are designed to deliver equal amplitude and desired phase at each gate. The lengths

Active FET Applications

831

and impedances of T8 – T10 are designed to provide the phase shift estimated above for a wide band of frequencies. Note that the output artificial transmission line is composed of the folded transmission line T S8 – TS10, which behaves as a spiral inductor and shunted by capacitors C8 – C10. They are all calculated to provide the IF phase shift provided by Table 11.3. RF input is located at point 13 on the top right side. The conductor 45 is used to apply bias to gate 1 of each dual gate FET through a 2k ohm epi resistor. The termination resistor for the artificial transmission line is a thin film resistor. The LO circuit is located at the bottom, consisting of transmission lines 18, which are connected to the second gate. Bias for gate 2 is applied through the LO line, not shown in the figure. The IF circuit is also located at the bottom, with two outputs, one at each side of the die.

Figure 11.89 Layout of mixer on GaAs substrate.

The distribution of voltage at the gate of each FET at different frequencies is in Figure 11.90(a), and the relative differential phase shift between succeeding adjacent FETs as a function of frequency is in Figure 11.90(b). The phase difference is expressed as a lag between FETs 2 and 1, by curve 62, FETs 3 and 2 by curve 64, and FETs 4 and 3 by curve 66.

832

Microwave Mixer Technology and Applications

(a) Simulated input voltage for each FET

(b) Simulated relative phase between the input of adjacent FETs Figure 11.90 Voltage and phase distributions versus frequency.

The IF responses are shown in Figure 11.91, where line 68 indicates signal conversion at port 19b, and line 70 indicating signal rejection at port 19a.

Figure 11.91 IF signals at output ports.

This performance was obtained by optimizing the values of input line capacitance for RF signal frequency higher than the LO frequency. However, the input circuit could be optimized for the case of RF frequency lower than the LO frequency, and designed for an output signal at terminal 19a and a null at terminal 19b. The FET is biased in nonlinear operation to generate the desired mixing products. However it also generates the second harmonic of the LO signal, which can mix with the incoming RF signal and create the familiar distortion signals at (2LO - RF), (2RF - LO), the first one being the image frequency, IM. Generation of these signals is undesirable as it increases the conversion loss of the mixer. Assuming the image and the desired signal generated by the mixer have

Active FET Applications

833

the same phase and are of negative frequency relative to the main input signal, they propagate in reverse direction compared to the main signal. They are therefore attenuated at the output port minimizing conversion loss of the mixer Additionally, the mixer topology provides substantial cancellation of noise introduced into the IF band from the LO generator. In order to explain the cancellation, let’s take the spectrum of LO noise in Figure 11.92. The LO excitation of each FET is equal in phase and amplitude, so that the downconverted noise to the IF band from both upper and lower side bands are also equal in phase and amplitude. The IF noise voltage component at the IF port is denoted by en(fm) where fm is the offset frequency of the noise component from the carrier as shown in the figure. All en signals are from same LO, so that the components are fully correlated and add algebraically. Taking into consideration the phase shift introduced by the IF circuitry, the noise voltage at a corresponding offset frequency fm is given at each output port by:

e19a ( f m )   en ( f m )e  j ( n1) n 1

e19b ( f m )   en ( f m )e  j ( N  n ) n 1

Where N is the number of devices and n=1,2,3,4 is the device order. The inventors propose that the noise signal at either 19a or 19b ports are e19a = e19b, which provides (11.66) for the noise at either port.

e19b ( f m )  e19a ( f m )  e( f m )e(  j ( N 1) ) / 2

sin( N / 2) sin( / 2)

(11.66)

This valid for all frequencies around the LO carrier. Complete noise cancellation of the frequency noise occurs when  is chosen to satisfy the condition  = 2/N for (N>1). In the case of four FETs, N = 4, the required phase shift is 90. Therefore, the distributed approach behaves as a balanced mixer where down-converted LO noise is suppressed.

Figure 11.92 Noise spectrum of LO voltage.

An alternative embodiment of the distributed mixer proposed by the inventors is in Figure 11.93. In this case, instead of feeding the LO in phase at

834

Microwave Mixer Technology and Applications

gate 2 of each dual gate device, it is fed with a phase controlled by the transmission line elements T12 - T16. Several combinations are possible, and the 90 phase shift in particular, allows cancellation of LO at the output ports. T15

T16 R2

VLO1

T12

T13

T14 VLO2

VLO3

VLO

VLO4

VGG T1

VRF

T2 C1

RG

T3

RG

19a

T8

VIF1a VIF2a VIF3a VIF1b VIF2b VIF3b Figure 11.93 Distributed mixer with distributed LO line.

T6

VRF4

VIF3 T9

R1

C5

VRF3

VIF2

VIF1 T7

RG

T5

C4

C3 VRF2

C2 VRF1

RG

T4

T1

VIF4 T1 1

0

VIF4a VIF4b

19a

An additional patent on a distributed dual gate mixer was proposed by Anthony Pavio, [51], where a resistor was attached to the FET sources for self bias operation. 11.6.4 Differential CMOS One of the first applications [52] of a differential core mixer cell embedded into a distributed topology, made use of 0.18 µm CMOS technology. The mixer employed 5 differential cells to achieve a conversion gain of 3.8 dB over the 3 to 22 GHz RF band, and dissipated 130 mW using a 1.2 V voltage supply. The schematic of the proposed mixer illustrated in Figure 11.94 represents the first and last of the 5 cells. The topology consists of 5 lines, the input gate line for the RF signal, two gate lines for the balanced LO signal, and two IF lines for the balanced IF output signal. The lines are terminated with a load matched to the characteristic impedance. The principle of operation is similar to the dual gate and is easily described if one considers lossless lines. The input RF voltage propagates on the RF line and is absorbed by the internal termination. At each tap the signal is rotated by the RF phase, and is converted to an output IF current, by the differential cell switched by the LO voltage.

Active FET Applications

835

IF+

VDD

VDD

IF-

LO+ LORF

Unit cell

Unit cell

Figure 11.94 Differential mixer with coplanar lines. After [53].

The output IF current travels in two directions, with positive phase shift IF in the direction of load and to the internal termination with reverse phase. If the phases obey the relation, IF= LO - RF, then the IF signals add at the external load and cancel at the internal termination. Another application of the same mixer technology for ultra wide band (UWB) receivers appeared in a paper employing two cells and artificial lines built with spiral inductors and parasitic capacitors of MOS devices, [53]. The conversion results for an IF of 528 MHz are in Figure 11.95, for a bias setting of 1.8V and 6 mA. The RF band covered is from 3.0 to 8.72 GHz and the IF selected at 528 MHz corresponds to a sub-band of an OFDM UWB transceiver. Average gain was 3 dB, obtained for an LO power set at 9 dBm. Additional results are a noise figure of 6.8 to 7.3 dB over a 7.5 GHz wide band and an IIP3 of 5 dBm at 5 GHz. If the number of cells increases, then the losses in the artificial lines on CMOS technology cannot be neglected. That causes increased losses due to un-equal signals at the output ports. Also the losses in the LO line results in different power applied to each cell as the signal propagates down the line.

836

Microwave Mixer Technology and Applications

Figure 11.95 Conversion gain as a function of RF frequency. From [54].

The analytical treatment for a circuit with n cells becomes complex and was addressed in a recent publication, [54]. As reference, Figure 11.96 represents the case for n = 4, illustrating the number of spiral inductors required. One of the important equations in this paper is the power conversion gain reproduced in (11.67), relating the transmission line impedances at IF, RF, and LO. The corresponding gate line and RF line cutoff frequencies are taken into account in (11.68). The parameters of LO line in terms of cut-off frequency and attenuation constant are similar to the ones from RF line, simply substituting the RF constants by LO constants.

Figure 11.96 Differential distributed on 0.18 µm CMOS. From [54].

Z IF  LIF CIF  LRF CRF  LLO CLO

(11.67)

Active FET Applications

G

1 4

p11 2 g mRF

LIF CIF

2

 C  1   RF P   g mLO 

2

  1   IF   C 

     RF 1      g RF

   

   2

2

837

 L  RF e n ( IF  RF )  CRF 

      RF  1      C

  

(11.68)

2 3/ 2

  

Where, C = cutoff frequency of the delay lines IF, LO, RF if the LC delay lines are identical. Otherwise a different cutoff frequency is defined at each frequency. gRF = 1/(rgRFCgsRF), RF transistor input cutoff frequency dIF = 1/(rdsCds), drain cutoff frequency RF = input RF frequency IF = output IF frequency

2 sin( f LO ) , is a Fourier coefficient from the time domain  f LO  current gain between tail and output of differential pair  = interval of time between switching on and off of FETs. gmLO = instantaneous transconductance of the switching pair gmRF = small signal transconductance of RF transistor p11 

 RF 

 IF 



C

/  gRF X RF

   1  1   C    gRF 

   

2

2

, RF attenuation constant of input line

  X RF 2  

dIF / C  , IF attenuation constant of output line 1  X IF

2

XRF = RF/C XIF = IF/C, with IF = 1/(rdsIFCdsIF) CP = parasitic capacitance from the floating node of the differential configuration to ground n = number of cells

838

Microwave Mixer Technology and Applications

11.7 SUMMARY This chapter has provided a series of examples of active FET mixers, starting from early designs using JFET technology up to advanced CMOS RF technology. One will notice most of the chapter deals with GaAs technology including: MESFET, HEMT and PHEMT since those were the technologies originaly employed in the early microwave and mmWave mixer development. RF CMOS was also covered reflecting the change in technology trends for active mixers. However, the topologies and ideas developed for a specific FET technology, are equally applicable to others since basic FET theory is similar. Most of the popular topologies were covered with a brief summary, which will help the reader in developing new ideas and improving the results from original work. In certain cases additional analysis relative to the original publications were added to improve understanding of concepts. Besides the single ended, singly balanced, and doubly balanced topologies, the less well known approaches of self-oscillating mixers, subharmonic mixers, and the recent advances on distributed mixers were included. REFERENCES [1] Lawrence W. Fish, Jr., Hudson and Daniel R. von Recklinghausen, “Signal Mixing and Conversion Apparatus Employing Field Effect Transistor with Square Law Operation,” US Patent 3,348,154, issued October 17, 1967. [2] Jack K. Kreng and Sanjar Ghaem-Maghami, “Bias Optimized FET Mixer for Varactor Tuner,” US Patent 3,976,944, issued August 24, 1976. [3] Katsuji Kimura, “Receiver Including FET Frequency Mixer,” US Patent 4,541,122, issued September 10, 1985. [4] Hiroshi Ohnishi and Sahahiko Yamashita, “Microwave FET Mixer Arranged to Receive RF Input at Gate Electrode,” US Patent 4,592,095, issued May 27, 1986. [5] M. Madihian, L. Desclos, K. Maruhashi, K. Onda, and M. Kuzuhara, “A Monolithic AlGaAs/InGaAs Upconverter IC for K-Band Wireless Networwks,” IEEE Transactions on Microwave Theory and Techniques, Volume 12, No. 12, December 1995, pp. 2773-2777. [6] M. Madihian, L. Desclos, K. Maruhashi, K. Onda, and M. Kuzuhara, “60 GHz Monolithic Down–and Up-Converters Utilizing a Source Injection Concept,” IEEE Transactions on Microwave Theory and Techniques, Volume 46, No. 7, July 1998, pp. 1003-1006. [7] A. Benussan, P. Birot, C. Ban K., and J. C. Curtinot, “Transistor Mixer for Ultra High Frequency Transmitters,” US Patent 4,490,854, issued December 25, 1984.

Active FET Applications

839

[8] Joel D. Kirkeland and Vijay K. Nair, “Frequency Mixing Circuit with Impedance Transforming Power Combiner,” US Patent 5,325,000, issued June 28, 1984. [9] Lyle A. Fajen and Michael Dydyk, “MMIC FET Mixer and Method,” US Patent 5,517,668A, issued May 14, 1996. [10] Wayne Kennan, “MMIC Downconverter for a Direct Broadcast Satellite Low Noise Block Downconverter,” US Patent 5,649,312, issued July 15, 1997. [11] Kai Tuan Yan and Junichi Shibata, “Mixer with Higher Order Intermodulation Suppression and Robust Conversion Gain,” US Patent 6,351,632, issued February 26, 2002. [12] D. An, S. C. Kim, W. S. Sul, H. J. Han, H. S. Lee, W. Y. Uhm, H. M. Park, S. D. Kim, D. H. Shin, and J. K. Rhee, “High Conversion Gain V-Band Quadruple Sub-harmonic Mixer Using Cascode Structure,” IEEE 2003 MTT International Microwave Symposium Digest, pp. 911- 914. [13] Donald L. Wollensen, “Integrated Circuit Balanced Mixer Apparatus,” US Patent 3,727,078, issued April 1, 1973. [14] Merle Vincent Hoover, “Complimentary Symmetry FET Mixer Circuits,” US Patent 4,032,851, issued June 28, 1977. [15] Duane C. Rabe, “High Gain Balanced Mixer,” US Patent 4,449,245, issued May 15, 1984. [16] Michael W. Vice, “Quasi-Double Balanced Dual Transformer Dual FET Mixer, Which Achieves Better Isolation by Using a First and Second Diplexer, and a Transmission line RF Balun,” US Patent 5,732,345, issued March 24, 1998. [17] Michael W. Vice, “Quasi-Double Balanced Dual Transformer Dual FET Mixer,” US Patent 5,799,248, issued August 25, 1998. [18] Tetsuo Hirota and Hiroyo Ogawa, “A Novel K-Band Balanced FET UpConverter,” IEEE Transactions on Microwave Theory and Techniques, Volume MTT-32, No. 7, July 1984, pp. 679-683. [19] Wayne Kennan and Edmar Camargo, “Single Balanced Frequency Downconverter for Direct Broadcast Satellite Transmission and Hybrid Ring Signal Combiner,” US Patent 5,903,827, issued May 11, 1999. [20] A. Emami, C. H. Doan, A. M. Niknejad and R. W. Brodersen, “A 60-GHz Down-Converting CMOS Single-Gate Mixer,” 2005 IEEE RFIC Symposium, pp. 163-166. [21] Tae Wook Kim, “A Common Gate Mixer with Trans-conductance Nonlinearity Cancellation,” 8th IEEE International Conference on ASIC, 2009 ASICON, pp. 459-460. [22] P-S Wu, C-H Wang, C-S Lin, K-Y Lin, and Huei Wang, “A Compact 60 GHz Integrated Up-Converter Using Miniature Transformer Couplers With 5 dB Conversion Gain,” IEEE Microwave and Wireless Component Letters, Volume 18, No. 9, September 2008, pp. 641-643. [23] Jin-An Hou and Yeong Her Wang, “A Ka Band Balanced Third LOHarmonic Mixer Using a Lumped–Elements Quadrature Hybrid,” IEEE

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Microwave Mixer Technology and Applications

Microwave and Wireless Components Letters, Volume 18, No. 6, June 2008, pp. 404-406. [24] Michael W. Vice, “Biased FET Mixer,” US Patent 5,513,390, issued April 30, 1996. [25] Michael W. Vice, “Quasi-Double Balanced Passive Reflection FET Mixer,” US Patent 5,799,248, issued August 25, 1998. [26] Anthony M. Pavio, Jr. and Scott D. Thompson, “Monolithic Double Balanced Single Sideband Modulator,” US Patent 4,768,000, issued August 30, 1988. [27] Yawata Seiji Sakashita, “Frequency Conversion Apparatus,” US Patent 4,677,692, issued June, 30, 1987. [28] Katsuji Kimura, “Frequency Mixer Circuit Using FETs,” US Patent 5,306,969, issued April 26, 1994. [29] Jesus S. Pena-Finol, “Quarter-Square Analog Four–Quadrant Multiplier Using MOS Integrated Circuit Technology,” US Patent 4,546,275, issued October 8, 1985. [30] Paul R. Andrys and Philip H. Thompson, “Double Balanced Differential Active Mixer with Current Shared Active Input Balun,” US Patent 6,057,714, issued May 2, 2000. [31] Mamoru Ugajin and Tsuneo Tsukahara, “Mixer Circuit,” US Patent 6,871,057B2, issued March 22, 2005. [32] Werner Schelmbauer and Josef Zipper, “Mixer Arrangement, Use of the Mixer Arrangement and Method for Frequency Conversion,” US Patent 20070,010,228, issued January 11, 2007. [33] King Chun Tsai and Lawrence Tse, “Mixer Constant Linear Range Biasing Apparatus and Method,” US Patent 7,177,620 B1, issued February 13, 2007. [34] Young-Jin Kim, “Mixer Circuit for Direct Conversion Transceiver with Improved IP2,” US Patent 20050,170,806, issued August 4, 2005. [35] John B. Hughes, “Analogue Mixer,” US Patent 20060,211,397A1, issued September 21, 2006. [36] H-C Chen, T. Wang and S-S Lu, “A 5.6 GHz 1-V CMOS Direct Conversion Receiver with an Integrated Quadruple Coupler,” IEEE Journal of Solid-State Circuits, Volume 42, No. 9, September 2007, pp. 1963-1975. [37] Brad R. Jackson and Carlos E. Saavedra, “A CMOS Subharmonic Mixer with Input and Output Active Baluns,” Microwave and Optical Technology Letters, Volume 48, No. 12, December 2006, pp. 2472-2478. [38] Brad R. Jackson and Carlos E. Saavedra, “A CMOS Ku-band 4X Sub harmonic Mixer,” IEEE Journal of Solid State Circuits, Volume 43, No. 6, June 2008, pp. 1351-1359. [39] S. J. Fan, S. T. Lee, D. J. Allstot, and A. Bellaouar, “A 2 GHz CMOS Even Harmonic Mixer for Direct Conversion Receivers,” ISCAS 2002, IEEE International Symposium on Circuits and Systems, Volume 4, pp. 807-810. [40] D. L. Wollensen, “Crystal Controlled Autodyne Converter using Field Effect Transistors,” US Patent 3,510,781, issued May 15, 1970.

Active FET Applications

841

[41] Keiko Shinkawa, Hiroji Shoyama, Chuichi Sodeyama, and Mitsuhisa Shinagawa, “Self-Oscillating Mixer Circuit,” US Patent 4,219,779, issued August 26, 1980. [42] Rakesk Sharma and Boleslaw M. Sosin, “Mixer,” US Patent 4,977,617, issued December 11, 1990. [43] Christen Rauscher, “Large-Signal Technique for Designing SingleFrequency and Voltage-Controlled GaAs FET Oscillators,” IEEE Transactions on Microwave Theory and Techniques, Volume MTT-29, No. 4, April 1981, pp. 293304. [44] Rowan J. Gilmore and Fred J. Rosenbaum, “An Analytic Approach to Optimum Oscillator Design Using S-parameters,” IEEE Transactions on Microwave Theory and Techniques, Volume MTT-31, No. 8, August 1983, pp. 633-639. [45] Merle V. Hoover, “Complimentary Symmetry FET Frequency Converter Circuits,” US Patent 4,334,324, issued June 8, 1982. [46] Hiroshi Miyamoto and Mitsuhisa Shinagawa, “Self Excited Mixer Circuit Using Field Effect Transistor,” US Patent 4,112,373, issued September 5, 1978. [47] A. M. Pavio and K. J. Anderson, “Single Balanced Self Oscillating Mixer,” US Patent 4,658,440, issued April 14, 1987. [48] Tang O. S. A and Aitchison, C. S., “A Microwave Distributed MESFET Mixer,” 14th European Microwave Conference, 1984, pp. 483-487. [49] C-H Chiu, K-H Liang, H-Y Chang, and Y-J Chan, “A 3-34 GHz GaAs PHEMT Distributed Mixer with Low DC Power Consumption,” CSIC-Compound Semiconductor Integrated Circuit Symposium, 2006, pp. 73-76. [50] Yusuke Tajima, Robert A. Pucel, and Ward S. Titus, “Frequency Conversion Circuits,” US Patent 4,662,000, issued April 28, 1987. [51] Anthony M. Pavi, “Monolithic Distributed Mixer,” US Patent 4,751,744, issued June 14, 1988. [52] Xisohua Fan and Edgar Sanchez-Sinencio, “3-22 GHz CMOS Distributed Single Balanced Mixer,” Proceedings of the IEEE International SOC Conference, 2004, pp. 93-96. [53] P. Heydari, D. Lin, A. Shameli and A. Yazd, “Design of CMOS Distributed Circuits for Multiband UWB Wireless Receivers,” 2005 IEEE Radio Frequency Integrated Circuits Symposium, pp. 695-698. [54] A. Q. Safarian, A. Yazdi, and P. Heydari, “Design and Analysis of an Ultrawide Band Distributed CMOS Mixer,” IEEE Transactions on Very Large Scale Integration (VLSI) Systems, Volume 13, No. 5, May 2005, pp. 618-629.

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Microwave Mixer Technology and Applications

Appendix Sampling Mixers In contrast to fundamental and subharmonic mixers, sampling mixers use high order LO harmonics to convert high frequency RF signals down to the IF. They originated as a separate category of mixers for use in microwave instrumentation where the emphasis is on down-converting signals over a very wide input frequency range, where higher conversion loss and lower dynamic range can be tolerated. They are also used extensively in phase locked oscillators as sampling phase detectors [1], and recently they have been applied to ultra-wideband (UWB) receivers [2]. In addition to higher conversion losses, their pulse shaping and frequency conversion circuitry adds complexity and can introduce ringing and spurious. So, while sampling mixers use the same heterodyne process as conventional mixers for frequency conversion, they have not been widely used in communication systems. Basic sampling theory says that when a signal, f(t), of bandwidth  (Hz) is sampled at a rate, fLO, that is greater than or equal to twice the signal bandwidth, fLO ≥ 2, the information contained in the original signal is maintained. This is embodied in the uniform sampling theorem, or equally in the Nyquist criterion, [3, 4]. The output sampled signal, fs(t), is obtained by multiplying the time domain input signal, f(t), by the periodic sampling function s(t).

f s (t )  f (t )s(t )

(A.1)

The sampling function is a series of pulses of duration,, amplitude, K-1, and period TLO, as depicted in Figure A.1(a). The Fourier series of the sampling function is in (A.2), which gives the spectrum, S(f), plotted in Figure A.1(b).

s(t ) 



c e

n  

n

j 2nfLOt



 c0   2cn cos(2nf LOt ) n 1

843

(A.2)

844

Microwave Mixer Technology and Applications

Where

cn  Kf LO sin c(nf LO ) ,

sin c( x) 

sin(x) x

s(t) fLO = 1/TLO

c2 Kτ

-1

τ •••

Figure A.1

-TLO 0 TLO

c2 c3

c3

••• -3TLO

|S( f )| c0 c1

c1

t

3TLO

c4

c4

-4fLO

τ-1 4fLO

-2fLO

0

2fLO

f

(a) Time domain, s(t) (b) Frequency domain, S(f) Representation of the switching function. After [3].

Per (A.1) the output sampled signal equals the input signal multiplied by the sampling function in the time domain. This equates to convolution in the frequency domain.

Fs ( f )  F ( f ) * S ( f )

(A.3)

Equation (A.2) shows S(f) comprises a series of discrete frequency signals whose amplitude equals the Fourier coefficients. Convolution of a signal with an impulse function produces a replica of the input signal that is shifted in frequency. Thus the output spectrum of the sampled signal is given by (A.4).

Fs ( f ) 



 c F ( f  nf

n  

n

LO

(A.4)

)

|F( f )|

-β fc β Figure A.2

|Fs( f )|

fn = fc + nfLO

f

f f-4 f-3

f-2

f-1

fc

f1

f2

f3

f4

(a) (b) (a) Input signal spectrum. (b) Output sampled signal spectrum

The output spectrum comprises the original input spectrum, F(f), plus copies of it offset by nfLO , n = 1,2,… with amplitude weighted by the Fourier coefficients. For the case of the sampling mixer, a conversion matrix is involved

Sampling Mixers

845

to include effects of varying terminations to the different mixing products, so their levels will deviate from the theoretical Fourier coefficients. Usually the desired output signal is the IF centered closest to zero frequency. And the sampling frequency, fLO, is normally much lower than the RF input center frequency, fc, so the baseband IF output is obtained by mixing the RF input signal with a high harmonic of the LO. As depicted in Figure A.2(b), the amplitudes of the signal replicates generally follow the sinc function profile, resulting in increasing conversion loss for higher LO harmonics. A low pass filter is applied at the sampling mixer output to select the baseband IF signal and reject the higher frequency products including the RF, LO harmonics, and spurious. The generation of periodic narrow pulses is normally achieved using either a step recovery diode, SRD, combined with a tank circuit, or a nonlinear transmission line using a series of varactor diodes. These mechanisms also introduce ringing and distortion that transfer to the down converted signal. Once generated, the pulse width can be further reduced using short circuit transmission line stubs to create an inverted wave that causes destructive interference. The sampling process can be modeled using the circuit of Figure A.3. The pulse drives the switch that gates the RF source. The circuit comprises a RF voltage source, VS, its internal impedance, RS, a switch, and a hold capacitor, Ch. When the switch is closed, the capacitor charges to a voltage proportional to V S. The switch then opens and the voltage on the capacitor becomes independent of the transient effects of RS. While the switch is open (off), the charge on the capacitor is applied to the load RL. The charge in the capacitor equates to the integrated current, so the capacitor voltage is proportional to the average value of the input signal over the sampling aperture time, TS, [5]. The switch is usually realized using Schottky diodes in a balanced configuration. Gate RS

RL Ch

Buffer

Vsint Pulse Figure A.3

Ideal subharmonic sampling. After [5].

This process limits the ability of the capacitor to charge to the full value of VS depending on the amount of time the switch is closed (on). If the on time is too short, then there is not enough time for the capacitor voltage to reach the maximum value. This loss is called sampling efficiency [5], which is defined by:

846

Microwave Mixer Technology and Applications

Ns  1 e



Ts Rs Ch

(A.5)

An intuitive understanding of the sampling mixing principle is depicted in Figure A.4, where the top waveform indicates the RF signal to be downconverted. The LO pulses in the middle waveform are multiplied by the incoming signal and the resulting waveform is shown at the bottom constructed from sampled points 30, 32, 46, 48, 50, 34, 52, and so on.

Figure A.4

Sampling principle of operation.

The bottom waveform is an IF with the same phase as the RF signal, and proportional to the RF in amplitude. Its frequency is fIFn = nfLO ± fRF, where nfLO represents the LO pulse repetition rate multiplied by the harmonic number n. The sampler architecture represented in Figure A.5, combines various functions to operate as a mixer, [6].

RF

RF Matching

Sampling Gate Balun

Pulse Generator LO Figure A.5

Sampling mixer architecture. After [6].

IF Matching

IF

Sampling Mixers

847

Pulse Generator The step recovery diode (SRD), is the device most commonly used to generate high order harmonics. During forward conduction the SRD stores charge in the depletion region, causing the SRD to have a low impedance. When reverse biased, the charge is quickly removed, and once it depletes conduction abruptly halts resulting in the SRD having a high impedance. Thus the nonlinear capacitance of the SRD causes abrupt variations in impedance when driven by a sinusoidal waveform. The typical C(v) for the SRD is similar to that of the varactor diode; the difference lies in the magnitude of low and high capacitance values. A typical SRD harmonic generator circuit is shown in Figure A.6, [7], containing an inductor, load resistor, and a bias supply. The battery E0 represents the contact potential of the SRD.

L Vsint

SRD

Vout

RL

E0 Figure A.6

Step Recovery multiplier circuit. After [7].

L

iL i0

iL Vsint E0

t Rd  0

Vout

0

Vout 0 0

Figure A.7

t

Equivalent circuit and I/V waveforms before SRD charge is depleted.

During the conduction interval, the equivalent circuit comprises the drive inductance, L, and the forward conducting SRD impedance that is nearly a short circuit. As depicted in Figure A.7, as long as the current i L is positive, the SRD

848

Microwave Mixer Technology and Applications

stores electric charge, illustrated by the i L(t) plot. When the applied voltage reverses phase, a transition state exists during which the current becomes negative, but the SRD impedance stays low while the charge is being removed. During this interval of conduction, the output voltage is constant and equal to the diode contact potential since the diode resistance is assumed to be zero. At the instant in time when the diode charge is fully depleted, the source and battery voltages are equal and opposite, so as depicted in Figure A.8 the equivalent circuit becomes the parallel combination of the inductor, L, SRD reverse capacitance, Crev, and the output load resistance, RL. This forms a tank circuit where the energy stored in the inductor generates an impulse current i 1, flowing through the tank circuit that oscillates until the energy is dissipated, or until the driving voltage reverses phase and the process repeats. The result is that a voltage pulse is created every other half cycle of the driving signal, and depending on the values of L and Crev oscillations can accompany the pulse. iL i0

Crev

t

0 L

RL i1

Vout Vout 0

t

0 Figure A.8

Equivalent circuit and I/V waveforms after SRD charge is depleted.

Optimum efficiency may be achieved by using either a self biasing resistor or an external bias source. Self bias is preferred as it can provide temperature compensation, increased bandwidth operation, and feedback to reduce effects from variation in input drive power. The diode impedance is much lower than the standard 50 ohms, so the circuit of Figure A.6 is usually modified to include the low pass elements required for impedance matching.

Figure A.9

Lowpass input matching circuit for SRD. After [6].

Sampling Mixers

849

Pout - dBm

An example [8] of a circuit matched to a 50 ohm generator to provide an output spectrum with nearly constant amplitude up to the 10th harmonic is depicted in Figure A.9. The SRD has a Crev capacitance of 2 pF and provides a pulse width of 180 ps. The resulting “comb” spectrum is in Figure A.10.

10 0 -10 -20 -30 -40 -50 -60 1.2

2.4

3.6

4.8

6.0

Frequency - MHz Figure A.10

Comb spectrum for input drive at 200 MHz. From [8].

Nonlinear Transmission Line (NLTL) An alternative way to generate a sharp pulse for sampling purposes is to employ a nonlinear transmission line, [9]. It comprises a transmission line along which varactor diodes are inserted at fixed or variable intervals. Its operation relies on the fact that the output signal will be a distorted version of the input signal due to variation in the phase velocity of the line. The phase velocity varies with capacitance, which in turn varies with the voltage level as the wave progresses along the line. The effect is to sharpen the pulse. The speed, or phase velocity, Vp, of the signal propagating on the transmission line is given by the equation:

Vp 

1

(A.6)

LC (v)

The phase velocity is therefore inversely proportional to the square root of the varactor capacitance, C(v), which in turn depends on the magnitude of voltage at that specific location on the line. The expression for the varactor capacitance depends on how the junction is built, determined by the value of m, which is equal to two if the junction is abrupt, and greater than 1 for hyperabrupt junctions.

C (v ) 

C jo  V 1   

  

m

(A.7)

850

Microwave Mixer Technology and Applications

Due to its inverse relationship with voltage, the capacitance will be high for low voltages and decreasing as voltage increases. Therefore, as an impressed signal propagates down the line, the voltage increases at each capacitor cell causing the capacitance to decrease, which in turn increases speed. The result after a number of cells is a very sharp pulse with large amplitude. Note that as the pulse’s velocity increases its amplitude also increases and its width decreases. Figure A.11 depicts the pulse width decreasing and amplitude increasing as the pulse propagates along the NLTL [10].

Figure A.11 Nonlinear transmission line. From [10].

Figure A.12

Output waveform of the NLT with 1 volt peak and 65 pS wide input pulse applied. From [10].

A NLTL was built using BiCMOS 0.18µm technology with 100 capacitors and 100 inductors. It was simulated with an input pulse having 1 volt peak amplitude and pulse width of 65 ps resulting in an output pulse as narrow as 2.5 ps wide with 0.8 volt amplitude. Measured results in Figure A.12 show a pulse rise time of 9 pS.

Balanced Sampler A sampler that became popular for network analyzer applications is depicted in Figure A.13, [11]. It consists of two matched Schottky diodes, coupled in balanced form to the pulse generator by means of a balun. The pulses are applied

Sampling Mixers

851

in opposing phase to the diodes with sufficient amplitude to drive them to a low impedance state (on). The pulse width is narrowed by launching the pulses at the open circuit end of two short circuited slot lines. The pulse travels along each slot line and reflects off the short circuit. The reflected wave is inverted in phase and propagates back to the input of the slot line, where it cancels with the incident wave causing zero amplitude for the remainder of the pulse. When the amplitude goes to zero, the diode switches immediately go to a high impedance state (off). The RF signal to be converted is applied to a microstrip line in parallel to the slot line so that while the diode switches are on, the RF signal charges capacitors C 1, C2. When the diodes switch off the stored charge flows into the IF terminals, -V and +V. The thin film circuit is depicted in Figure A.14, with the ground plane side up toward the reader, and the RFin microstrip (dotted) line on the backside. The slot line etched into the ground plane runs parallel with, and above, the microstrip line. The slot line is short circuited at the left and right ends, comprising two slot lines connected in parallel, the left side and the right side. The LO pulse input goes to a piggy back substrate mounted on top of the main substrate. A ribbon connects to the piggy back substrate and jumps over the slotline, and its other end connects to the load resistor.

Figure A.13

Sampler equivalent circuit. From [11].

The two Schottky diodes are connected in series, with the common point connected to the microstrip line with a plated hole through the substrate (via). The other two ends of the diodes connect to capacitors that in turn connect to the opposite sides of the slot line. The positive and negative IF output lines also connect to the capacitors, whose voltage is proportional to the average signal voltage during the sampling interval. Various versions of this circuit have been used over the years, a recent one using coplanar waveguide (CPW) is given by [2].

852

Microwave Mixer Technology and Applications

-VIF Ground Plane

RFin

RL

Left Slot Line

Right Slot Line

LOin Figure A.14

RFout

via

+VIF

Approximation of sampler thin film layout. After [11].

The 3 dB bandwidth obtained by a sampling device can be estimated from the pulse width using (A.8), [12]. If the capacitance shunting the diode junction and the series inductance are known, then the cut off frequency is determined by the familiar (A.9). The diode capacitance given in [11] is 0.1 pF and the lead inductance is 250 pH, which results in a maximum frequency of operation around 30 GHz.

BW (GHz) 

fc 

1 2 LC

350 t g ( ps)

(A.8)

(A.9)

SMD Sampler Monolithic samplers are now commercially available from several companies. In general they follow the schematic of Figure A.15, obtained from [13]. The surface mounted device (SMD) component comprises two matched Schottky diodes capacitively coupled to a SRD. The capacitors, 33A, 33B, besides AC coupling, differentiate the pulse applied to the Schottky diodes. The pulse forward biases diodes 34A, 34B, allowing them to conduct and creating a pulse at node 37 whose magnitude is proportional to the RF signal. If the LO is harmonically related to the RF signal at the same point in each RF cycle, then the IF signal becomes a DC level. As the RF frequency shifts away from equaling an LO harmonic, the IF frequency becomes equal to the difference between fRF and

Sampling Mixers

853

the closest nfLO harmonic. The IF sampled voltage is available from terminals 40, 41, which are terminated by large resistors to avoid loading the pulse.

Figure A.15

Commercial sampling phase detector (SPD) module.

The complete sampler is shown in Figure A.16, illustrating the SMD component, and a balun transformer connected to the LO input line 50. The RF signal in this particular patent is coupled to transmission line 51, and applied to node 37. The resistors 42, 43 provide ground returns for the diodes and in general are in the order of Ks. Terminals 40, 41 should be grounded at RF to allow the RF signal to be completely absorbed by the Schottky diodes. Resistors at terminals 35, 36 are 50  to match the LO signal and minimize ringing and other transient signals. Other patent examples are found in the literature; for instance, in reference [14] a similar approach to the one from Figure A.15 is disclosed.

51

SMD

Figure A.16

Thin film version of circuit from Figure A.15.

854

Microwave Mixer Technology and Applications

NLTL Sampler Also a monolithic sampler using a nonlinear transmission line producing a narrow pulse, called a shock wave is disclosed. The circuit is depicted in Figure A.17, [15]. The pulse from the shock wave turns on diodes 58, 60, allowing RF applied at terminal l to flow into the IF circuit. The pulses travel lines 46, 48, and encounter a low value resistor 50, preferably a short circuit that reverses polarities of the pulses and reflect them back to the non-linear transmission line, shutting diodes 58, 60 off.

Figure A.17

Phase detector with commercial nonlinear transmission line pulse generator.

Sampling Mixers

855

REFERENCES [1] R. A. Liman, J. L. Muraro, P. Lautier, O. Llopis, J. Graffeuil, “Noise in Sampling Phase Detectors for RF PLL,” Proceedings of the 39th European Microwave Conference, 2009, pp. 480-483. [2] J. Han, C. Nguyen, “Coupled-Slotline-Hybrid Sampling Mixer Integrated With Step-Recovery-Diode Pulse Generator for UWB Applications,” IEEE Transactions on Microwave Theory and Techniques, Volume 53, No. 6, June 2005, pp. 1875-1882. [3] Bernard Sklar, Digital Communications Fundamentals and Applications, Upper Saddle River, New Jersey, Prentice-Hall, Inc., 2001. [4] Scott R. Gibson, “Gallium Arsenide Lowers Cost and Improves Performance of Microwave Counters,” Hewlett Packard Journal, February 1986, pp. 4-10. [5] Kenneth Rush and Danny J. Oldfield, “A Data Acquisition System for a 1GHz Digitizing Oscilloscope,” Hewlett Packard Journal, April 1986, pp. 4-11. [6] K. Madani and C. S. Aitchison, “A 1 to 20 GHz Microwave Sampler,” 20th European Microwave Conference, 1990, pp. 617- 622. [7] Harald T. Friis, “Analysis of Harmonic Generator Circuits for Step Recovery Diodes,” Proceedings of the IEEE, Volume 55, No. 7, July 1967, pp. 1192-1194. [8] “Nonlinear Modeling of Step Recovery Diodes Using Verilog-A,” Agilent Application Note, May 2004. [9] W. C. Whiteley, W. E. Kunz, and W. J. Anklam, “50 GHz Sampler Hybrid Utilizing A Small Shockline and an Internal SRD,” IEEE 1991MTT International Microwave Symposium, pp. 895-898. [10] Ehsan Afshari and Ali Hajimiri, “Non-Linear Transmission Lines for Pulse Shaping in Silicon,” IEEE Journal of Solid-State Circuits, Volume 40, No. 3, March 2005, pp. 91-94. [11] J. Merkelo, “A DC to 20 GHz Thin-Film Signal Sampler for Microwave Instrumentation,” Hewlett Packard Journal, April 1973, No. 8, pp. 10-13. [12] W. M. Grove, “Sampling for Oscilloscopes and Other RF Systems DC Through X-Band,” IEEE Transactions on Microwave Theory and Techniques, Volume MTT-14, No. 12, December 1966, pp. 629- 635. [13] Christopher David Grondhal, “Sampling Phase Detector and Multiple Frequency Band Termination Circuit and Method,” US Patent 5,953,645, issued September 14, 1902. [14] Herbert Brauns, “Sampling Phase Detector,” US Patent 5,900,747, issued May 4, 1999. [15] C-Y Su, M. R. Ty Tan and W. J. Anklam, “Monolithic Sampler,” US Patent 4,956,568, issued September 11, 1990.

856

Microwave Mixer Technology and Applications

About the Authors Edmar Camargo received his master and Ph.D. degrees from University of São Paulo, Brazil, in 1976 and 1985, respectively. Within this time he took a leave of absence to work at the Centre National d'Études des Telecommunications (CNET) in Lannion France in 1977 and 1982. He was a teacher assistant and research engineer in the same university in Brazil, until 1993 when he emigrated to the U.S. He worked for Hewlett-Packard on mmWave Transceiver design and for Fujitsu Compound Semiconductor Inc., where he took the lead on frequency converter projects, in particular the patented FMM5107 for satellite receivers and high linearity mmwave mixer FMM5116/17. From 2000 – 2004 was director of engineering at Fujitsu coordinating the mm-Wave and handset developments. From 2005 to 2009 worked at iTerra Communication, Watkins Johnson Communications on power amplifiers for infrastructure applications and at RF Micro Devices on power amplifiers for handsets. From 2009 to 2012 was a MMIC design consultant in the San Francisco Bay Area, and in 2013 became a Principal engineer at Quinstar Technology in Torrance, California. Edmar Camargo is a senior member of the IEEE-MTT Society, served on the technical symposium committee's from 1996 to 2008 and served as a member of the MTT-22 Signal Generation and Frequency Conversion from its foundation until 2007. He and his wife, Marcia, reside in Lomita, California, and have two sons, Marcel and Regis. Bert Henderson is a Technology Fellow with Cobham Defense Electronics in San Jose, California, where he has worked since 2002 on microwave and millimeter-wave multifunction integrated subsystems. From 1994–2002 he worked on commercial millimeter-wave radios, first with Endwave Corporation and then with Bridgewave Communications. From 1979-1994 he was with Watkins Johnson Co., where he held various design and leadership positions. He received the BSEE from the University of California Davis in 1978, and MSEE from the University of California, Berkeley, in 1979. While with Watkins Johnson Co. he designed numerous mixers including the M50, the first mixer with 2-26 GHz RF/LO and 2-18 GHz IF coverage. He led development of the SMC1844, the first uniplanar mixer with 18-40 GHz RF/LO and DC-18 GHz IF coverage. His technical articles include “Reliably Predict Mixer IM Suppression”, which provides a closed form approximation for spurious suppression used in various commercial simulators. He has six patents for mixers and filters, and in 2007 was a recipient of the Tyco Electronics Key Innovator Award. He has written various spurious and system analysis programs in C and C++. He serves on the IEEE MTT-22 committee for Signal Generation and Frequency Conversion, and was a presenter for IEEE MTT-S workshops on AM Noise in 2005, Subharmonic Mixer Circuit Designs in 2009, and Analytic Concepts for Low-Noise and Low-Distortion Mixers in 2012. He and his wife, Ann, reside in Sunnyvale, California, and have three daughters, Katie, Kim, and Emily.

857

858

Index 180 Hybrid couplers  Waveguide, 205  Bifilar hybrid, 206  Ratrace hybrid, 207  Lumped ratrace, 211  Coupled line , 213  Broadband four port, 217  Suspended microstrip, 222 90 Hybrid Couplers  Branch line, 224  Lange hybrid, 225  Broadside coupler, 791 Active rectifier, 4 Active mixer, 6  Tuned transformer, 534  Differential with tank circuit, 536 Active 180 combiner, 199, 201. Active current based combiner, 202 Active ratrace BJT modulator, 521 Active power splitter, 730 Alexanderson, 3 AM detector, 6 Angelov model, 129 Anode, 4 APDP, 301 AM modulator linearized by feedback, 533 Armstrong, 5, 9, 10, 11, 13, 14 Anti symmetrical MOS, 796 ASK, 56 Audion, 4

Autodyne, see self-oscillating mixer Avalanche breakdown, 107 Available conversion gain, 443 AWGN, 77 Balun structure, 156  Guanella transformer, 157158  Ruthroff bifilar line, 160  Three coupled lines, 176  Tapered coupled line, 183  Lumped elements, 187  Microstrip, 169  Asymetrical coplanar, 191  Slotline, 191  Active single FET, 194, 196, 198.  Active CG, CS, 196  Active differential, 198  Distributed, 730 Balanced-unbalanced mixer, 560 Unbalanced-balanced mixer, 562 Base transit time, 116 Baseband physical layer, 50 Baseband signal, 55 Beam lead diode, 106 Beat receiver, 5 Bessel function, 260, 303, 333, 339 BER, 50, 64 Bifilar line, 28, 157 Bilinear model, 617 Bipolar junction transistor (see BJT) BJT, 105, 109 859

860

Microwave Mixer Technology and Applications

BJT cold, 120 BJT active load to FET mixer, 764 Boltzmann constant, 78 BPSK, 59 Bridge combiner, 18 Bridge quad mixer, 406 BSIM, 146 Cascaded noise figure, 81 Cascaded phase noise, 90 Cascaded current amplifier, 548 Cascaded voltage amplifier, 549 Cascode  Mixer amplifier, 21  Transistor, 144  Mixer, 458, 515, 615, 637  Cascode BJT, 457, 8  Linearity, 638  Distributed mixer, 672 Carson modulator, 30 Cathode, 6 Cellular mobile radio, 53 Chalmers FET model, 129 Classification of diode mixers  termination (Saleh), 251  configuration, 300 Classification BJT mixers, 430 Coherer, 2 CMRR, 198 Child-Langmuir equation, 4 Chroma modulator, 556 CMOS, 146 Coaxial tank circuit, 20 Coaxial mixer, 346, 347, 349, 350 Cold FET, see resistive mixer Comparing MOSFET with GaAs FET, 589 Complimentary BJT, 521  Collector transformer coupled, 524, 526, 527.  Emitter coupled, 526  Emitter coupled output signal, 527, 528



Complimentary subharmonic, 581 Complimentary MOS,  Pmos, Nmos, 776  With buffer, 777 Continuous receiver, 13 Conversion matrix diode, 248  BJT, 441  FET, 596 Conversion loss, 254 Conversion gain,  BJT, 438, 443, 444, 469, 502  FET mixer, 594, 599, 605, 620  resistive, 629, 634  cascode, 640  differential amplifier, 654 Converter  Single BJT, 534, 537  FET, 733 Coplanar mixer, 383 Cross bar mixer, 359, 363, 366 Cross coupled even order, 566 Cross coupled mmWave MMIC, 582 Cross modulation, 289, 516, 769 Curtice model, 122 Curtice-Ettenberg model, 126 Darlington SOM DRO resonator, 537 Design study,  CDMA downconverter, 453  Cascode BJT, 458  Wi-FI 2.45 GHz Gilbert mixer, 490  X-band FET mixer, 612  Drain mixer at 10 GHz, 622  Quasi linear mixer synthesis, 630

861



Single balanced gate mixer, 646 Dielectric resonator, 536, 820, 826, Differential  Tube, 26  BJT with transformer, 537, 538 Digital modulation, 54 Distributed mixer, 46 Diode  Vacuum tube, 4  Schottky, 105 Direct conversion  Receiver, 13, 68  Subharmonic, 668  Weaver receiver, 74  Gilbert cell, 715, 721, 734, 738, 745, 749, 807, 811, 815, 818 Distributed mixer  UWB, 669  Conversion gain, 671  Cascode topology, 673  Image reject, 753, 830  Differential cell, 674, 834 Differential mixer  Linear mode, 650  Non-Linear mode, 653  Linearity, 655  transformer coupled, 795 Doubly balanced  Star mixer, 29, 298, 320, 387, 388, 394, 397  Ring mixer, 247, 320, 386, 391, 396, 408  Quadrature, 413  Differential BJT 475, 544, 553, 556, 558, 559, 562, 564, 565, 568, 570, 573  Passive BJT ring mixer, 529  transformer coupled BJT, 529, 535, 536, 539



FET differential, 661, 726, 728, Drain mixer, 616, 770 DRO, 537, 820 Dual conversion system, 13 Dual IF mixer, 41 Dual gate MESFET  device, 142  modified, 773 Dual mode tube, 24 Duty cycle, 416, 508, 635, 693, 711 Dynamic range, 102 Early effect, 112 Ebers-Moll model, 112 EEHEMT HEMT model, 137 EVM, 67 Exponential diode model, 107, 266  Saleh, 275 FET mixer LO injection modes, 590 FET quadratic gate mixer, 592 Finline, 363, 365, 366. Flicker noise, 72 Floating  MESFET, 648, 698, 716, 728  MOS, 701 Friss equation, 81 FSK, 57 Frequency to voltage converter, 16 Frequency translator circuit, 22 FM demodulator, 11 Fmax, 118 Fourier transform, 56 Frequency modulated transceiver, 14 Front end receiver, 515, 557 fT, Transition frequency, 118 GaAs MESFET, 121 Gain compression  exponential model, 290  Switching model, 290 Gilbert cell, 475, 483 Guanella balun, 159 Gummel-Poon model, 112

862

Microwave Mixer Technology and Applications

H-Mixer, 271, 275 H-mode FET mixer, 730 Harry Houck, 42 Harmonic balance simulation, 276 Hartley modulator, 18, 33 HEMT, 121 Heterodyne, 7 High permeability material, 164 Ideality factor, 106 IIP3, 75, 101, 470, 656, 663 Image reject  Circuit, 30  Voltage tuned filter, 572  Quasi-zero IF, 575  MOS, 804 Image noise, 68, 83 Inter carrier sound system, 39 Intermodulation, IM,  Definition, 74, 76  Cascaded, 101  Exponential model, 293  Switching model, 293  Order, 299 JFET, 121 Kirk effect, 114, 472 LAN, 54 Lange coupler, 223 Lecher line, 24 Lee de Forest, 4 Light wave link, 52 Linearity,  BJT, 451, 516  BJT single ended, 449  BJT singly balanced, 469, 470  BJT doubly balanced, 478, 483, 538  Gate mixer, 607, 609  Drain mixer, 621  Resistive, 635  Dual gate, 640  Differential FET, 642, 655, 662

Linearization, 516  Emitter degeneration, 486, 564  Feedback, 532, 569  Feedforward (GasComp), 551  Micromixer, 488  Multi tanh, 489  Transformer coupled, 539  Amplifier bypass, 559  Transconductance amplifier, 564  RF amp with feedback, 568  Transconductance compensation, 569  Vice approach, 700  Paralleling FET, 786, 806  P, N Gates, 740 Local Oscillator, 8 LO AM Noise, 87 LO chain, 97 Low frequency mixer, 433 LO injection  On BJT, 432,  On FET, 590  Gate, 591  Source, 614, 758  Drain , 616 Low noise Gilbert Mixer, 553 Low voltage operation, BJT, 566 Marconi, 3 Marchand balun  Definition, 164  Order, 168  Microstrip edge coupled, 176  Suspended microstrip, 179  Broadside coupled, 183  Spiral transformer, 186, 394 Micromixer,  BJT, 488, 550

863

 FET, 661, 803 Morse Code, 2 MOSFET, 121, 146 Multiple circuits, 411 Multiplier (BJT) Gilbert type, 479 Multiplier CMOS, 796 Nyquist theorem, 55 Noise, 76 Noise figure,  Definition, 80  System cascade noise figure, 81  BJT single ended mixer, 450  BJT singly balanced, 461  BJT doubly balanced, 483  FET gate mixer, 607  FET differential, 662 Noise reduction, 20 Nonlinear conductance mixer, 248 Nonlinear currents method, 609 Nonlinear BJT capacitance, 448 OIP3, 76 Orthogonal, 688 Paramixer, 408, 409 Parasitic losses, 264 P1dB, 76 Phase noise,  Definition, 91  System cascaded phase noise, 92 PHEMT, 121 Point-to-point links, 51 Point-to-multipoint, 53 Poly-phase filters, 576 Power combiner active  Single, 730  Distributed, 732 Power splitter, 730, 732 Pumped nonlinear conductance, 259 Push-push  Operation, 27



SOM, 583, 709, 733, 821, 826 Push-pull modulator, 519 QAM, 63 QPSK, 60 QPSK constellation, 61 Quad transformer  Emitter coupled, 570  Source coupled, 659, 729  Packaged FET, 726 Quadratic diode model, 262 Quasi doubly balanced Vice mixer, 779 Quasi static assumption, 105 Regenerative receiver, 9, 514 Reginald Fessenden, 3 Resistive mixer  Channel conductance, 625  Cold FET, 135, 140  90 couplers, 642  Linearity, 640  Conversion gain, 630  MOS floating mixer, 648  Tuned Cgd, 690  Self biased, 691 Richardson constant, 107 RF physical layer, 50 Root model, 139 RS232, 50 Saleh theory  Switching model, 274  Exponential, 275 Sampling mixer with BJT, 524 Sandbox, MESFET, 697 Self-oscillating  Single ended mixer, 514, 531  Balanced mixers, 44  Converter, 13 Singly Balanced 180,  Diode, 315  Diode waveguide, 359, 388  BJT differential type, 461

864

Microwave Mixer Technology and Applications

 FET, 646 Singly Balanced 90  diode, 314, 381  FET, 645  resistive, 707 Slotline mixer, 374, 376, 378 SNR, 20 SOM DRO resonator  Bipolar, 538  FET, 820  Dual gate, 826 SOM varactor tuned,  FET, 821, 822  FET CMOS complimentary, 823  Dual gate, 823 Spark gap, 2 Spectral regrowth, 62 Square wave LO, 278, 710 SSB concept, 30 Stripline, microstrip, 368 Subharmonic mixers,  Tubes, 42, 416  APDP, 300, 417, 418  Diode coplanar, 421  Diode quadrature, 422  BJT singly balanced, 473  FET singly balanced, 657  QLT type, 666  DRT type, 667  Triple Level type, 668  2X Subharmonic FET, 656  2X Quadrature balanced, 657  2X CMOS, 694  4X subharmonic, FET, 658  4X mHEMT, 695  Successive mixing, 750 Stability,  BJT, 452  FET, 611 Substrate LO injection, 776

Super Heterodyne  Concept, 11  Receiver, 68 Super regenerative receiver, 10 Synthesis  Diode, 323  FET Sparameters, 630  FET Yparameters, 718 Switch dual transistor transformer, 528 Switching model  Simple mixer, 243  Saleh, 274 TDR, Time domain reflectometry, 162 Tesla, 3 T-Model, 109 Temperature stabilization in bipolars, 439 TIM, 409 Transcapacitance, 138 Transconductance modulation, 444 Transconductance parameter, 148 Transducer gain, 254 Tree mixer, 554 Trifilar transformer, 715 Triode, 5 Triple level, BJT, 495 Triply balanced,  Diode, 322, 401  Subharmonic, 421 TV Receivers, 30  Dual Gate, 760 TVRO, 53 Up-converter, 642 UWB, 669 Variable reactance tube, 15 VSAT, 53 Waveform analysis, 55