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Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Capacitors: Theory, Types and Applications : Theory, Types and Applications, edited by Alexander L. Schulz, Nova Science

Copyright © 2010. Nova Science Publishers, Incorporated. All rights reserved. Capacitors: Theory, Types and Applications : Theory, Types and Applications, edited by Alexander L. Schulz, Nova Science

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CAPACITORS: THEORY, TYPES AND APPLICATIONS

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Capacitors: Theory, Types and Applications : Theory, Types and Applications, edited by Alexander L. Schulz, Nova Science

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CAPACITORS: THEORY, TYPES AND APPLICATIONS

ALEXANDER L. SCHULZ EDITOR

Nova Science Publishers, Inc. New York Capacitors: Theory, Types and Applications : Theory, Types and Applications, edited by Alexander L. Schulz, Nova Science

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Additional color graphics may be available in the e-book version of this book.

LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Capacitors : theory, types, and applications / editor, Alexander L. Schulz. p. cm. Includes index.

ISBN: (eBook)

1. Capacitors. I. Schulz, Alexander L. TK7872.C65C38 2010 621.31'5--dc22 2010012152

Published by Nova Science Publishers, Inc. † New York

Capacitors: Theory, Types and Applications : Theory, Types and Applications, edited by Alexander L. Schulz, Nova Science

CONTENTS

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Preface

vii

Chapter 1

The Role of Capacitors and Capacitance within Plasma Processing Victor J. Law

1

Chapter 2

Voltage Stabilization Using a Storage Capacitor V.V. Tatur

27

Chapter 3

Ideal and Real Capacitors: How Their Behaviour Affect Energy Efficiencies Paulo Simeão Carvalho and Adriano Sampaio e Sousa

41

Chapter 4

AC Bridge Circuitry for the Capacitive Position Sensor inside the Superconducting Linear Motor System Shu-chen Liu

59

Chapter 5

Physical and Electrochemical Properties of Quaternary Ammonium Salts Based on Halogen-Free Chelatoborate Anions and Their Application to Electric Double-Layer Capacitors Noritoshi Nanbu and Yukio Sasaki

81

Index

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115

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PREFACE A capacitor or condenser is a passive electronic component consisting of a pair of conductors separated by a dielectric (insulator). When a potential difference (voltage) exists across the conductors, an electric field is present in the dielectric. This field stores energy and produces a mechanical force between the conductors. The effect is greatest when there is a narrow separation between large areas of conductors, hence capacitor conductors are often called plates. This new book reviews research on the role of capacitors and capacitance within plasma processing; voltage stabilization using a storage capacitor; disposal of PCB capacitors in Kazakhstan and how ideal and real capacitors affect the behavior of energy efficiencies. As explained in Chapter 1, low pressure plasma processing of semiconductor devices and engineering materials has become an established multi-billion dollar world-wide industry over the last 40 years, followed closely by atmospheric pressure plasma processing of engineering and biomaterials. Although, these two pressure regimes have developed differing plasma applicators (chambers) and processing formats there are many shared attributes. This is because electrical energy needs to be transferred efficiency to the plasma production zone where the material processing is performed. In Chapter 2 a problem is considered on pulsed voltage stabilization using a storage capacitor, as well as possible solutions of this problem. The stabilizer designed is intended for converting constant unstabilized voltage into pulsed voltage with stabilized amplitude. The stabilizer circuit is based on LC (inductance–capacitor) charging circuit. The principle of operation of the stabilizer is based on the possibility of preliminary charging of the storage capacitor to the voltage intended to compensate for an alternating component

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viii

Alexander L. Schulz

of the unstabilized input voltage. This is provided by supplementing the circuit with a subsidiary charging circuit. The paper presents grounds for the voltage stabilization method suggested, as well as a circuitry realizing this method. Detailed electrical circuit of the experiment is shown. The experimental results are presented as diagrams. Upto-date electrical elements are analyzed which can be used to realize the engineering solution suggested. In the authors’ case, the most important elements are stabilitrons, storage capacitors and charging inductances. Technical characteristics of the device are considered, in particular, voltage and time charging-discharging parameters. Some exploitation characteristics are presented. Possible applications of the device are considered as well. The engineering solution suggested allows one to make a pulsed voltage stabilizer on a storage capacitor with stabilization factor of up to 1000. Capacitors are generally considered “old fashioned” topics for scientific discussion. Classical treatment of these components tends to represent them as ideal components and forget their real behavior. Real capacitors are built by dielectrics with a finite resistance that result in a leakage electric current that affects the capacitor efficiency. In Chapter 3, the authors will focus their attention on how efficiency can be calculated and measured, giving alternative ways for the calculus of the time constant and the energies stored and supplied during charge and discharge processes in DC circuits, representing the electric power as a function of time in ideal and real capacitors. This will help to understand the difference between an ideal capacitor and a real one, and show why the efficiency of a real capacitor is always less than 50 %. Discussion about real capacitors will involve both DC circuits and AC circuits. As discussed in Chapter 4, there are many ways along with a bundle of equipments to measure the capacitance of a capacitor. To measure the capacitance change of a capacitor, the authors can use the conventional capacitor meter, capacitance bridge, or use a homemade 555 oscillator, a Qmeter, etc. For a fast changing but very small amount of capacitance (a few tenths of pF) of a capacitor within a few milliseconds at milliKelvin temperatures, it is a even much bigger challenge to have any measurable size of signal. The authors have designed and built a superconducting linear motor with an armature inside it which can be controlled the motion by the applied current pulse to the motor and monitored the motion by a capacitive position sensor. The authors have come out several solutions to monitor the motion of the armature via the position sensor with maximum movement distance 22

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Preface

ix

mm. Eventually, AC bridge circuit integrated with an analog lock-in amplifier provides the best solution. The use of nonhalogenated electrolytes for electrochemical devices is very important from the point of view of safety and cost. The authors have developed a series of quaternary ammonium salts based on halogen-free chelatoborate anions for electric double-layer capacitors (EDLCs): four quaternary ammonium bis(oxalato)borates (QABOBs) and tetraethylammonium bis[salicylato(2-)]borate (TEABSB). In Chapter 5 the authors investigated the physical and electrochemical properties of the quaternary ammonium chelatoborates and evaluated the performance of EDLCs using them as single electrolyte salts. These properties were compared with those obtained for tetraethylammonium tetrafluoroborate (TEABF4) as standard substance. Mass and van der Waals volume of chelatoborate ions are higher and larger than those of tetrafluoroborate ion (BF4–). Mass densities, viscosities, and surface tensions of propylene carbonate (PC) solutions containing the quaternary ammonium chelatoborate were higher than those obtained for TEABF4. As a result, conductivities of the PC solutions were lower. However, in the cases of QABOBs, electrochemical stability on an activated carbon electrode and gravimetric capacitances of three-electrode measurement and 2025-type coin cells were comparable to those obtained for TEABF4. The gravimetric capacitance of a negative electrode, at which the surface excess concentration of cations increases, dominated the magnitude of the gravimetric capacitance of a full cell.

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In: Capacitors: Theory, Types and Applications ISBN: 978-1-61668-972-8 Editor: A.L. Schulz, pp. 1-26 © 2011 Nova Science Publishers, Inc.

Chapter 1

THE ROLE OF CAPACITORS AND CAPACITANCE WITHIN PLASMA PROCESSING Victor J. Law

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National Center of Plasma Science and Technology, Dublin City University, Collins Avenue, Glasnevin, Dublin 9, Dublin, Ireland

Abstract Low pressure plasma processing of semiconductor devices and engineering materials has become an established multi-billon dollar world-wide industry over the last 40 years, followed closely by atmospheric pressure plasma processing of engineering and biomaterials. Although, these two pressure regimes have developed differing plasma applicators (chambers) and processing formats there are many shared attributes. This is because electrical energy needs to be transferred efficiency to the plasma production zone where the material processing is performed.

The premise here is that low pressure and atmospheric pressure plasma behaves like 2 series capacitors with an inter-linking resistive element. These elements constitute an equivalent electrical circuit of the electron and ion composition of the plasma; where the 2 capacitors are the plasma sheaths at the chamber wall and the material that is being processed. These sheaths are primarily 

E-mail address: [email protected]

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2

Victor J. Law

capacitive due to positive ions accelerating out of the glow region and the resistor defines the portion of the distributed dissipated power, due to inelastic collisions within the glow region. The equivalent electrical circuit topology and their values alter with: gas composition, gas flow rate, surface-interaction, plus chamber configuration and effective area. In these terms the plasma has dynamic load impedance which requires the transfer of electrical energy from the drive power source which invariably has characteristic impedance equal to 50 Ohm. Thus a means of impedance matching is required, ether by a lumped inductor capacitor network, or a drive circuit that operates at resonance. This condition is represented by equation (1) which gives the conjugate condition where maximum power is transferred from one complex-impedance to another.

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r  jx  r  jx

(1)

Equation (1) states when the real part (r) of the two complex impedances are of the same value and the magnitude of their imaginary (IjxI) is the same but of opposite sign, then maximum power transfer is achieved between the two impedances. To put it another way: maximum power transfer occurs at resonance. This impedance matching of the plasma load is subject to the physical boundary conditions of the plasma chamber, and/or, the electrical ground environment in the case of a plasma plume. Thus the reactance of the chamber is shunted across the plasma circuit: where the plasma is absent, this approximates to an open-circuit capacitance. When the plasma is generated the non-linear behavior of the plasma sheath capacitance generates integer harmonics of the drive frequency. These harmonics are of technological importance as they are used as an electrical probe of the plasma sheath interaction with the processed material that is being injected into the sheath. This knowledge of plasma and its impedance matching properties has lead to the construction of impedance bridges for the off-line (no plasma present) swept frequency measurement techniques that investigate the matching network and chamber impedance (otherwise called chamber state, or health), diplexers (that handle 1000 W) for real-time non-invasive passive measurement of plasma harmonics, and high-pass filters for real-time non-invasive swept frequency measurement of plasma reflection properties. In this chapter we shall deal with: the design rules for capacitor components within RF circuits used in plasma processing systems (section 1); high power diplexer ‘the Batlaw box’ (section 2); instrumentation Chebyshev filters (section

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The Role of Capacitors and Capacitance within Plasma Processing

3

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3); chamber and plasma impedance (section 4). Section 5 calculates the impedance transformation of L- Pi, and T-type matching networks for the chamber and plasma impedance as defined in section 4. . Section 6 is an example of how a diplexer positioned on the main power prevents spurious harmonic generation and so allow the true plasma harmonic to be measured. In section 7 we return to the plasma chamber and plasma to investigate frequency pulling of the drive frequency by the plasma capacitance and how the chamber capacitance can be estimated. Finally section 8 provides the conclusion of this chapter. Figure 1.1 gives the physical topology of system components within a typical plasma processing tool.

Figure 1.1. Schematic of RF external with respect to plasma chamber.

1. Filter Construction Design Rules Considerable care has to be applied to electrical filter construction. This is because some commercial bought inductors and capacitors suffer from non-ideal components losses which cause a reduction in the steepness of the filter frequency response and high insertion loss. These non-ideal component losses comprise series component resistance, distributed capacitance across inductor, distributed inductance across capacitors, and radiation lost due to poor ground plane. In the case of the capacitor, volume and contact resistance reduces the Q and increase the effective series resistance (ESR, typical expressed in milliohms). Here the component Q (Q = the ratio of power stored divided by the energy dissipated (per radian) is equal to the reciprocal of dissipation factor (tan-).

Capacitors: Theory, Types and Applications : Theory, Types and Applications, edited by Alexander L. Schulz, Nova Science

4

Victor J. Law

Q

1 tan  



Xc

(2)

ESR

The power dissipated (Pd) in the capacitor can be estimated by multiplying the ECR value by the square of the RF network current as shown in equation 3.

Pd  I 2  ESR

(3)

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For example, a current of 4.7 Amps circuiting in a capacitor with an ESR of 18 milliohms would dissipated 0.37 W in the capacitor. To overcome this deleterious effect ceramic based capacitors (figure 1.1) are the design choice for fixed values, and vacuum capacitors is the design choice for variable values. Variable vacuum and multiple plate air capacitors are mostly used in impedance matching networks. It should be noted however, multiple plate air capacitors are less expensive but at the cost of a lower working voltage. This generally means they are not used in high Q impedance networks.

Figure 1.1. A typical selection of ceramic capacitors.

To keep resistive losses to a minimum, a RF ground return and radiative shielding are essential. This are achieved by maintaining a low inductance ground of high conductivity (i.e. copper or phosphor bronze) and separation of circuit characteristic impedance of the input and output impedances of the filter circuit. All the filters described here are built using these design rules.

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The Role of Capacitors and Capacitance within Plasma Processing

5

2. The BatLaw Diplexer

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The BatLaw diplexer is designed to be operated from 10 MHz to 900 MHz and handle up to 1000 W into an impedance matched load of 50 Ohms. The diplexer is a combination of a 13.56 MHz low-pass and a 27.12 MHz high-pass filter with a band-pass common port. The name Batlaw comes from the people who designed the diplexer (Mr. Ivan Batty and Dr. Vic Law). A photograph of the BatLaw diplexer is shown in figure 2.1 and its circuit diagram is shown in figure 2.2. The diplexer design is derived from a normalized 1MHz, 5 elements, 0.1 dB ripple Chebyshev filter model that has a 50 Ohm input and output impedance, see tables 2.1 and 2.2. For more information on this design, see the references [1, 2, and 3] and reference [4] for a low-pass that can that can handle 10 W. The diplexer design has been modified in such way that the resulting zeros (minimum transmission) introduced (each separate circuit) appear in the stop-band of each respective filter and the insertion loss in each pass-band filter is restored. The diplexer is built into a compartmented aluminum box with 50 Ohm N-type connectors for the low-pass port and the common-port, and a 50 Ohm BNC connector for the high-pass port to minimize inductive coupling between the filter elements. Table 2.1. 1 MHz normalised Chebyshev low-pass filter model, and final design Cut-off Frequency 1 MHz 15 MHz

KL1 (H) 9.12 0.6

KL2 (H) 15.72 1.0

KL3 (H) 9.12 0.6

KC1 (nF) 4.367 0.3

KC2 (nF) 4.367 0.3

Table 2.2. 1 MHz normalised Chebyshev high-pass filter design, and final design Cut-off frequency 1 MHz 25 MHz

KL1 (H) 5.8 0.23

KL2 (H) 5.8 0.23

KC3 (nF) 2.776 0.11

KC4/5 (nF) 1.266 0.053

Capacitors: Theory, Types and Applications : Theory, Types and Applications, edited by Alexander L. Schulz, Nova Science

KC6 (nF) 2.776 0.11

6

Victor J. Law

Figure 2.1. Photograph of the BatLaw diplexer (the right hand section is the high-pass section, and the remaining three sections are the low-pass section).

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Figure 2.2. Circuit diagram of the BatLaw diplexer (version 1). The low-pass section of the diplexer is made from L1, L2, L3 and C1 plus C2. The high-pass section is made from L1, L2, C3, C4, C5 and C6.

Figure 2.3. Section view of shunt ceramic tile capacitor construction (left), plus 3-D image (right).

The shunt capacitors of the low-pass filter and the first capacitor of the high– pass filter are built from loss tangent ceramic tiles (ADS96R) which are fired with

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The Role of Capacitors and Capacitance within Plasma Processing

7

a 30 micron thick platinum and silver layer [5]. These capacitors are arranged with one contact heat-sunk to a copper phosphor bronze plate that directly seats on to the casing and positioned so that the inductors can be directly soldered to them without disturbing the screening of the inductors. Figure 2.3 shows schematics how these capacitors are put together. Using this design and mode of construction, the diplexer can handle up to 1000 W when inserted in to a power line which has a characteristic impedance of 50 Ohm. The first ceramic tile (C3) of the high-pass filter is located within the last section of the low-pass filter as full RF voltage is supplied at this common node. The remaining capacitors (C4 and C5) are of the ceramic disc type, rated at 5000 V and 1000 V, respectively. Two capacitors in series are used here to simply to meet the voltage requirement and get round the problem of non-preferred value availability. All the inductors are of the air-core coil design and use 1 mm diameter wire. To use the RF drive frequency (fo = 13.56 MHz) as a reference source, a mutual inductively-coupled magnetic loop is placed alongside the first inductive element of the low-pass filter section of the diplexer. The power coupling of this circuit is of the order of -30 dB with respect to the frequency response of the low pass section.

0 -5 -10 -15

Gain (dB)

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5

Low-pass section High-pass section Monitor port

-20 -25 -30 -35 -40 -45 -50 -55 0

25

50

75

100

125

150

175

200

Frequency (MHz) Figure 2.4. Measured frequency response of the BatLaw diplexer [1, 2 and 3].

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8

Victor J. Law

The coupling factor is chosen so that the reference signal is of similar amplitude to the expected amplitudes of those of the plasma harmonics at the high-pass port. The frequency response of the low-and high-pass sections and monitor port are shown in figure 2.4. Here it can be seen that each section has a flat frequency response providing frequency independent power transfer and minimum measurement distortion of the drive power and plasma harmonic amplitude and phase.

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3. Instrumentation Chebyshev Filters In the context of plasma processing, the purposes of the instrumentation Chebyshev filters is to define the frequency band to be measured and removal of unwanted signals at the input of measurement instruments. The term ‘instrumentation’ indicates that the filters are designed for power levels found in 50 Ohm input impedance of oscilloscopes and spectrum analyzers. The power levels range from +30 dBm to -100 dBm. The inductor and capacitor Chebyshev filter designs described here have been tested and work well for this purpose. The filters perform this task in the following two ways. Firstly, by reducing the filter bandwidth to a minimum for the task, the noise level entering measurement instrumentation is minimized. Secondly, using impedance reflection filters (inductors and capacitor), and not resistors, the theoretical noise floor is defined by the input impedance of the measurement instrumentation. Off cause the reader may wish to buy their own filter already made and calibrated filters [6]. However the purpose of this section is to describe the role of capacitors in filter design and construction. The following two sections describe filter-type and their response (section 3.1), and low-pass, high-pass and cascade band-pass Chebyshev filter design (section 3.2).

3.1. Chebyshev Filter Responses There are three main types of reflection filter designs and they are named after their inventers. These are the Bessel, Butterworth and Chebyshev. In this series their frequency roll-off (-dB.MHz-1) is inversely proportional to the ripple in their pass-band: Chebyshev filters have the highest circuit Q, frequency rolloff, and poor passband ripple. As there is no intentional resistance, these filters are

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The Role of Capacitors and Capacitance within Plasma Processing

9

reflective filters; outside the passband, it is mismatch that keeps power from reaching the noise sensitive part of the circuit. For plasma driven at 13.56 MHz, the Chebyshev response is the design of choose as it provides a rejection of 50 to 60 dB at the 2nd harmonic. The trade-off however is that a degree of ripple in the passband must be accepted. Operating at frequencies below a few MHz the filter response becomes critically dependent on the overall design.

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3.2. Chebyshev Filter Design For the Chebyshev filter, the design approach employed here is to use a normalized 1 MHz Chebyshev filter 0.001 ripple dB model with an input and output impedance of 50 Ohm [7]. The 50 Ohm impedance specification allows full integration into standard 50 Ohm instrumentation, for example oscilloscopes, spectrum analyzers. The designs are listed in tables 3.1 (low-pass), 3.2 (highpass), 3.3 (cascade band-pass). The Chebyshev low-pass filters has the inductors in series with the signal and the capacitors shunted to ground. For high-pass filters the inductors are shunted to ground and the capacitors are in series with the signal, see figure 3.1. The cascade band-pass filters are formed using low- and high- pass sections in series. The tables are used by dividing the 1 MHz component values (KL1, 2, 3, 4 and KC1, 2, 3, 4) by the desired cutoff frequency expressed in MHz. Worked examples have been given in each table for commonly used frequencies. The models scale from 1 MHz to 600 MHz for low-pass filter design, 1 and 1000 MHz for the high-pass design with stop-bands between 1 and few 100 MHz. The choice of a 7 element filter, with a ripple of 0.001 dB, rather than 3 or 5 element designs makes the filter more temperature stable as energy is dissipated over 7 elements rather than 3 or 5. The 0.001 dB ripple selection has the added advantage that practical RF inductor and capacitor can be purchased up in to the 250 MHz range. Table 3.1. 1 MHz normalised Chebyshev low-pass filter model, and design values for 2 MHz and 4 MHz -3 dB cut-off frequency

KL1 (H)

KL2 (H)

KL3 (H)

KL4 (H)

KC1 (nF)

KC2 (nF)

KC3 (nF)

1 MHz

4.69

1.22

1.22

4.69

3.95

4.92

3.95

2.5 MHz

1.8

4.9

4.9

1.8

1.6

1.9

1.6

4.5 MHz

1.0

2.6

2.6

1.0

0.86

1.1

0.86

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Victor J. Law

Table 3.2: 1 MHz normalised Chebyshev high-pass filter model, and design values for 13.56 MHz and 27.1 MHZ -3dB cut-off frequency

KL1 (H)

KL2 (H)

KL3 (H)

KC1 (nF)

KC2 (nF)

KC3 (nF)

KC4 (nF)

1 MHz

6.41

5.14

6.41

5.4

2.08

2.08

5.4

13 MHz

0.49

0.395

0.49

0.415

0.16

0.16

0.415

25 MHz

0.27

0.22

0.27

0.22

0.083

0.083

0.22

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Figure 3.1. Schematic of Chebyshev high-pass filter model.

Table 3.3 depict a low-pass and high-pass designs that can be cascade together to build a 13.56 MHz band-pass filter that has a -3 dB bandwidth of 1.5 MHz. Rows 2 and 4 provide the calculated values and rows 3 and 5 the preferred selected values. It is important to note, when cascading low- and high-pass filters the two filters become electrically coupled (through their zeros) as their respective stop-bands approach each other. This effect is a function of the preferred values used. To overcome this effect the circuit is tuned by additional shunt capacitors. Table 3.3. Chebyshev cascade band-pass filter design and preferred values for a 13.56 MHz band-pass filter -3dB cut-off frequency 14 MHz LP model

KL1 (H) 0.33

KL2 (H) 0.87

KL3 (H) 0.87

KL4 (H) 0.33

KC1 (nF) 0.28

KC2 (nF) 0.35

KC3 (nF) 0.28

Preferred values

0.33

0.82

0.82

0.33

0.27

0.33

0.27

13 MHz HP model

0.49

0.39

0.49

0.415

0.16

0.16

0.415

Preferred values

0.47

0.33

0.47

0.47

0.15

0.15

0.47

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KC4 (nF)

The Role of Capacitors and Capacitance within Plasma Processing

11

The following three figures provide a graphical and photographic example of the diplexer design as constructed from table 3.1 and table 3.2. Figure 3.2 shows the measured frequency response curves of the 2 MHz / 27 MHz diplexer constructed from table 3.1 and 3.2, and figure 3.3 shows a 4 MHz / 13.56 MHz diplexer also constructed from the tables. 0 -10

Gain (dB)

-20

2 MHz LP filter 27 MHz HP filter Reference level

-30 -40 -50 -60 -70 -80 0

10

20

30

40

50

60

70

80

90

100

Figure 3.2. 2 MHz / 27 MHz instrumentation Chebyshev diplexer frequency measured response. 0 -10

4 MHz LP filter 13.56 MHz HP filter Reference level

-20

Gain (dB)

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Frequency (MHz)

-30 -40 -50 -60 -70 -80 0

10

20

30

40

50

60

70

80

90

100

Frequency (MHz)

Figure 3.3. 4 MHz / 13.56 MHz instrumentation Chebyshev diplexer frequency measured response.

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Victor J. Law

The response curves were measured using a Hameg 5014 combined tracking generator and spectrum analyzer set at a RBW = 400 kHz. For completeness the through-line calibration are also shown. The insertion loss is the difference between the calibration curve and the passband level.

0 -10 -20 13.56 MHz BPF

Gain (dB)

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Figure 3.4. View of an instrumentation 4.6 MHz / 25 MHz diplexer construction depicting component layout and grounding.

-30

reference

-40 -50 -60 -70 -80 0

10

20

30

40

50

60

70

80

90

100

Frequency (MHz)

Figure 3.5. 13.56 MHz cascade band-pass filter response: made from low- and high-pass filter sections.

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The Role of Capacitors and Capacitance within Plasma Processing

13

It can be seen that for the two diplexers, the insertion loss of each arm is 1 to 2 dB and flat over the measured frequency range (100 MHz). In the case of the high-pass arms this response extends up to 500 MHz, (not shown here). In all cases the stop-bands have typical rejection levels of 65 to 75 dB and again extend up to 500 MHz (not shown). Of further note is the frequency role-off of each Diplexer arm, here they have a typical roll-off of -30 dB.MHz-1. A photograph of this filter is shown in figure 3.4. Figure 3.5 shows the frequency response curve of the 13.56 MHz cascade band-pass filter. The filter is constructed using the preferred values in table 3.3. Note the insertion loss is now some 6 dB which is due to the cascade nature of the design. Further note the stopband level is -50 dB, with respect to the passband.

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4. Capacitive Plasma-Tool Impedance This section describes the role of capacitors within a plasma tool impedance matching network. We do this by first determining the impedance of a generic 30 mm diameter parallel-plate plasma-tool driven at 13.56 MHz. First consider the open circuit impedance of the plasma chamber in the absence of plasma. This impedance corresponds to open circuit capacitive impedance at 13.56 MHz. The capacitive susceptance of the bottom electrode to ground is of the order of 240 pF, and equates to a capacitive reactance of 1

where

/ 240 x10-12 F = - 49 jOhm

(4.1)

 =  13.56 x106 Hz

(4.2)

The connection between the chamber and matching network can be up to 0.6 m in length of coaxial transmission-line. As the line is short with respect to the wavelength it can be represented as shunt1 capacitance of 100 pF m-1 x 0.6 = 60 pF, and equates to a capacitive susceptance of 1

/  60 x 10-12 = - 159.6 jOhm.

(4.3)

The combined capacitance reactance equates to 1

The term shunt is used to describe capacitor and inductor elements that connect the signal line with the return path.

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Victor J. Law (49 x 159) / (49 +159) = -39 jOhm

(4.4)

This value of capacitive susceptance forms the imaginary part of the chamber complex impedance. In practice resistive, radiation and dielectric losses from the bottom electrode and ground return strap inductance equates to a real part of = 0.1 to 6 Ohm. Aluminum chambers and mixed aluminum and stainless steel chambers such as the GEC reference cell have real values of the order of 0.5 to 1 Ohm, and pure stainless steel chambers have been reported to have real values of up to 6 Ohm. For our purpose, a resistance of 1.9 Ohm is selected as the real part of the chamber impedance. Thus, using the maximum power theorem as outlined in equation 1 the chamber impedance (1.9 - j39 Ohm,) requires a conjugate match of 1.9 + j39 Ohm, which the matching network must supply.

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5. Matching Networks We are now at the purpose of this section, the understanding of role capacitors in the process of impedance transformation in a simple three elementmatching network. Most traditional and modern 13.56 MHz impedance matching networks use variable air or vacuum capacitors and a fixed value air-coil inductor. This is because it is the simplest and cheapest method, although not the smartest from the power dissipation and frequency response point-of-view. [Contrary to the traditional approach, it would be better to dissipate power within the matching network over a larger number of elements and generate a single resonance (maximum transmission) at the fundamental frequency and a low, and flat, frequency transmission response at all other frequencies]. The task of the impedance matching network is primary three fold: (i) to provide sufficient potential (at the RF source drive frequency) to strike the plasma: (ii) to provide an conjugate match to the chamber and plasma and (iii) to protect the 50-Ohm resistive impedance of the RF generator from the variable reactive load of the chamber and plasma. A suitable primer for this section is can be in reference [7]. Here a simple lossless L-type matching networks with a target impedance transformation from 50-Ohm input impedance to target impedance of 1.9 + j39 Ohm is mapped. This target output impedance is the conjugate match impedance of the calculated plasma-tool chamber impedance of 1.9 - j39 Ohm. This simple analysis is useful as a first approximation for the design of a matching network at the source drive frequency; the analysis will progressively degrade at high

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15

frequencies when further (practical) elements are included. The settings for these matching networks are as follows.

5.1. L-Type Matching Network (Worked Example)

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For an L-type matching network the assumption is that the load impedance is less than the 50-Ohm output impedance of the RF source and greater than the resistive losses of the matching network. We introduce the quality factor (Q) of the match network; the Q is the ratio of the amount of stored energy in the system per cycle (voltage on the capacitors, current in the inductor), to the amount of dissipated energy per cycle (I2R). The design equations (assuming no resistance in the matching network) for a 3 element L-type matching network are:

Figure 5.1. L-type matching network.

Q-factor = square root (Z (source) / Re (load))-1

(5.1)

XCl = Re (load)*QL

(5.2)

And For our example:

QL 

50  1  5 and XC1 1.9*5  9.8 1.9

Transformation 1 (C1) The value of Cl follows is  found to be 1200pF (1/1200pF = - j9.8 Ohm). We can calculate the transformation of the impedance after the Cl (50 Ohm in parallel with –j9.8 Ohm): 1.9 - j9.8 Ohm.

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Victor J. Law

Transformation 2 (L1) We choose a series inductor with reactive impedance that exceeds the capacitance after transformation 1. A fixed 5-turn, and 60 mm diameter, air-core coil inductor (L1) has an approximate free-space inductance of 1 H and equates to positive reactance of 

1 x10-6 H = + j85.2 Ohm

(5.3)

The series inductance of L1 now transforms this imaginary part (without altering the real part) to 1.9 -j9.8 + j85.2 = 1.9 + j75.4 Ohm and appears to overshoot the target impedance.

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Transformation 3 (C2) The series capacitor (C2) is now adjusted to 320 pF (1/320pF = - j36.5 Ohm) so that the imaginary part is altered to 1.9 +j75.5 -j36.5 = 1.9 + j38.9 Ohm. Given the chamber has a real part of 1.9 Ohm, the conjugate value 1.9 +j38.9 Ohm has been reached in three transformations, each step relating to a individual matching network element, see table 1.2. Thus, each side looks through the matching network at is own complex conjugate impedance.

5.2. Pi-Type Matching Network (Descriptive) The Pi-type network can be considered as a back-to-back L-type network where each section transforms down to center impedance that is lower than either the generator (50 Ohm) or the chamber real impedance. For this example, the match down center impedance (real) around 0.9 Ohm, half of the real part of the load impedance. This means that the capacitance susceptance of (C1) is increased by a factor √2 with respect to the C1 capacitor of the L-type network. The value of C1 is adjusted to 1700 pF (1/1700pF = - j6.9 Ohm). Since the capacitive susceptance of C2 will increase the inductance of L1. The inductance of L1 is reduced with respect to the L-type network. In this case, a value of 0.4 H is used to produce a positive reactance of + j34 Ohm, which provides an imaginary transformation of + j34 - j6.9 = + j27.1 Ohm without altering the real part. The shunt capacitor (C2) is now adjusted to transform the real and imaginary components to meet the target impedance of 1.9 + j39 Ohm. In terms of C2 capacitance susceptance this equates to a value of 130 pF (1/130pF = - j90.2 Ohm).

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Figure 5.2. Pi-type matching network.

5.3. T-Type Matching Networks (Descriptive)

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This type of network can be considered as a front-to-front L-type network where each section transforms up to center impedance that is higher than either the generator or the conjugate match impedance. Hence the matching network has a free parameter (centre impedance).

Figure 5.3. T-type matching network.

For the T-type network, the first series capacitance (C1) increases the imaginary without altering the real part (50 Ohm). A series capacitance of 45 pF (1/ 45 pF = - j260 Ohm) produces a transformation of 50 - j260 Ohm. The inductor now must transform the impedance to undershoot the target impedance due to the next series capacitor. A 0.5 H (0.5H = + j42.6 Ohm) produces a transformation of 1.9 + j50 Ohm. The second series capacitor is adjusted to C2 = 1010 pF (1/1010pF = -j11.6 Ohm) to decrease the electrical length of L1 by +50 j11.6 Ohm to gain the target impedance of 1.9 + j38.6 Ohm.

5.4. Summary The worked L-type network example provides the transformation calculations for a given target impedance of 1.9 + j39 Ohm at 13.56 MHz. We have also

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Victor J. Law

described the changes in these values when the topology of the network is changed from L to Pi- and on to T-networks. Using this device a meaningful comparison can be made between the capacitor values within each network, see table 5.1. As a general the rule when the inductor is reduced by approximately by ½ from 1 H (initial value being common in low pressure plasma impedance matching networks) the initial capacitor value increases by a factor of 1.4 for Pitype network and decreases for T-type networks. For the T-type networks this is because C1 is now in series: the reverse effect being true for C2. Table 5.1. Nominal values for a three element L-, Pi-, and T-type matching network for a conjugate of 1.9 + j39 ±1 Ohm, for a source impedance of 50 Ohm and drive frequency of at 13.56 MHz Network

C1 (pF)

C2 (pF)

L1 (H)

L-type

1200 [9.8 Ohm]

320 [36.5 Ohm]

1 [85.2 Ohm]

Pi-type

1700 [6.9 Ohm]

130 [90.2 Ohm]

0.4 [34 Ohm]

T-type

45 [260 Ohm]

1010 [11.6 Ohm]

0.5 [42.6 Ohm]

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Values derived using an impedance and admittance Smith Chart.

6. Suppression of Noise within an external RF Power Circuits In this section we look at the performance of the Batlaw Diplexer and a 13.56 MHz Band-pass filter in suppressing RF amplifier noise and harmonic distortion on the main power line of the atmospheric pressure plasma micro-jet [8]. An atmospheric pressure plasma is used in this example has it should have relatively low harmonic content when compared to low pressure plasma. In this example (see figure 6.1) a function generator is used to provide the drive frequency signal and a broadband power amplifier is used to raise the power level from 0.5 to 20 W. The insertion of the diplexer after the amplifier and a lowpass filter is shown at the output of the function generator. Figure 6.2 provides the measured frequency response of the RF external circuit under varying filtering conditions. The drive frequency is 20W at 13.56 MHz.

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Figure 6.1. RF external circuit of a typical university atmospheric pressure plasma microjet [8]. The inserted filter and diplexer are shown in dashed outline.

Figure 6.2.

The top trace depicts the no filtering response. Here it can be seen that the measured signal has harmonics of the drive frequency from fn = 2 to 10 plus harmonics at fn = 17 and 19. The rolling average noise also appears to be between -50

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Victor J. Law

to -60 dBc for the most part with an enhanced noise floor in the150 MHz to 250 MHz region. Under these conditions if the operator wished to use harmonics as a means to monitor the plasma there would be many choices, but which one? The middle trace of figure 6.2 depicts the measured signal with the low pass filter inserted at the output of the function generator. Here it can be seen that the rolling average noise floor is round -60 dBc and the enhanced 150 MHz to 250 MHz noise floor is suppressed. The higher harmonics (fn = 12 to 19) are also suppressed. Finally, when the13.56 MHz Diplexer low-pass filter section is fitted to the broadband amplifier, see bottom trace of figure 6.2, a dramatic reduction in the harmonic content on the RF external circuit is observed. Only the second harmonic (fn = 2) along with the drive frequency is present. Our operator now has the simple task of choosing which signal (fo, or fn = 2 ) is to be monitored. This example is an extreme case of harmonic distortion arising from the use of a multi-functional function generator and RF broadband amplifier, both of which have spurious noise content due to the flexible design of their output stage. To remove spurious noise it is better to use an RF generator which has a spectral purity of ~ -70 dBc: a choice made by most semiconductor manufacturing faculties. However, in most universities this function generator scenario is all too common as many function generators and amplifiers are bought on a limited budget and employed in a variety of applications to justify their cost. The message is clear. If you are going to use multi-functional function generators and broadband amplifiers there is a cost to be paid in term signal misinterpretation and confusion when filtering is not employed.

7. Frequency Pulling This section is looks at the emerging field of frequency pulling of the drive supply oscillator [10] and it role in interpreting chamber capacitance and plasma capacitance. In this example a helium atmospheric corona discharge jet system called PlasmaStream™ which was developed by Dow Corning Plasma Solutions is examined. Frequency pulling of the drive is common in unlocked drive sources and has been reported to a lesser degree in RF 13.56 MHz generators [11]. The PlasmaStream™ system is driven by a Plasma Technics Inc [12] high voltage source operating in the 15 to 25 kHz range. This high voltage source uses a flyback transformer which produces a smoothed non sinusoidal voltage waveform peak-to-peak voltage of 20 kV on two parallel tungsten needles (electrodes) mounted at one end of a tube Quartz tube. The working gas helium is

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injected around the needles and flows out of the opposing end (nozzle). In this configuration a corona discharge is produced, and where the discharge visual morphology varies with gas flow. The plasma gas extends out of the nozzle to produce a plasma plume which is directed on to the work piece which may be a 2dimensional area or a 3-dimensional volume. A schematic of the device is illustrated in figure 7.1. Amongst the applications of this PlasmaStream™ device are: enhancing silicone adhesion to steel [13, 14], selectivity of cell adhesion [16], antimicrobial applications [17], deposition of soft plasma polymerised coatings [18].

Figure 7.1. The schematic of the PlasmaStream™ showing the positions of the current monitor and voltage probe.

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Victor J. Law

Irms Vrms

2.5

4.3 M 5.7

6.8 M

2.0

5.6 4.8 M

5.5

1.5 8.1 M

1.0

5.4 slope = 0.012 F/W Intercept Cc=0.19 F

Drive frequency (kHz)

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Applied voltage (kV)

Discharge current (mA)

Figures 7.2 a-b shows the RMS current and voltage, plus the resonant drive frequency of the device as a function of discharge power. In Figure 7.2a it can be seen that the discharge RMS current is negative and proportional to the discharge power, whereas the RMS voltage maps out three operating regimes. The system impedance ((pkpkV2/2)/I) is also seen to fall with discharge power. Of further note is that the resonant drive frequency (figure 7.2b) exhibits an inverse relationship to the discharge power. At 3.7 W discharge power the drive frequency is 18.3 kHz, and falls to 16.8 kHz at 7.6 W.

18.0

17.5

17.0

16.5 3

4

5

6

7

8

Discharge power (W)

Figure 7.2. The RMS current and voltage, and resonant drive frequency as a function of discharge power. Discharge impedance is shown on the voltage trace (3a). Electrical equivalent model Cc and Cp shown is in figure (3b).

The shift in resonant drive frequency (frez) can be explained by frequency pulling on the flyback power supply [9, 10], where pulling is controlled by tank circuit that is comprises the flyback transformer secondary coil inductance of (Lcoil), the chamber capacitance (Cc) and the plasma capacitance (Cp). The

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The Role of Capacitors and Capacitance within Plasma Processing

23

mathematical representation is shown in equation 7.1. This equation is an equivalent electrical model (EEM) of the plasma coupled through the chamber to the oscillator of the drive circuit power supply, and is used to understand the frequency dynamics of the discharge. In this equation; Lcoil, is the secondary coil inductance of the flyback transformer, Cc is the chamber capacitance and Cp is the plasma capacitance.

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f res 

1 2Lcoil Cc  C p 

(7.1)

Using this knowledge of the turn’s ratio of the secondary coil, an estimated value of between 2 to 10 mH was determined for Lcoil, The pulling of the resonant drive frequency is modeled using, Cc and Cp as fitting variables. The curve for Lcoil =2 mH from this EEM is shown as a red line in figure 7.2b. Here the extrapolated intercept is the chamber capacitance (Cc = 0.19 F) and slope is a measure of discharge capacitance increasing at a rate of 0.012 F / W. Thus to obtain the plasma capacitance at any given discharge power the value of 0.012 F is multiplied by the discharge power of interest. For example, at the maximum discharge power reported here (7.6 W) the total capacitance is 0.0912 F. The EEM simulation is consistent with the observation that the system impedance is falling with discharge power as the plasma volume fills the chamber. Under these filling conditions the plasma conductivity would be expected to reduce and capacitance increase due to increasing sheath area. Under increasing discharge power conditions or increasing helium flow rate the discharge expands to the work surface creating an additional capacitance to ground. For sake of simplicity this additional term is included in the term Cp.

Conclusions In this article we have looked at the role of capacitor construction and design for instrumentation filters, high power diplexers, and impedance matching networks that are employed in both, low pressure plasma, and atmospheric pressure plasma processing tools. It has been shown that the application in which the capacitor is to be used determines the type of materials to be used and the form of constructed to be employed. Proven models of low-pass, high-pass and band-pass filters have been presented. These models can be scaled from 1 MHz down to 100 kHz and up from

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Victor J. Law

1 MHz to 500 MHz. Given the appropriate mode of construction these filters can be utilized in instrumentation circuits or for high power diplexers. For impedance matching to a non 50 Ohm plasma capacitive dynamic load, three types of impedance matching have been examined (L- Pi-, and T) using a step-by-step systematic mathematical approached to the impedance transformation. This mathematical comparison provides the reader with a simple understanding of how capacitor and inductor values are altered between network topologies for a given load target. Section 7 of this article has shown how capacitor, inductor filters are used in the removal of harmonics that are related to the power source and broadband noise emanating power amplifier within the external drive circuit of plasma device. This differentiation between circuit noise and harmonics of the power source oscillator is a very important step in implementing Radio Frequency metrology; after all we want to explore the nonlinearity of the plasma impedance through its generative harmonics and not the external circuit. Finally section 8 has provided an example of drive frequency pulling which is a direct result of the plasma and chamber capacitance coupled back to the power supply oscillator. By implanting a simply equivalent electrical model, both the chamber capacitance and plasma capacitance can be obtained. This drive frequency pulling effect has now been recorded on a number of atmospheric and low pressure plasma tools academic [9, 10, 11] in industrial [19] environments. In the latter case the ease of use electro-acoustic probes has lead to low cost implementation of plasma process metrology. To conclude this chapter we have seen that capacitors play a major role in the plasma processing of materials. Their role ranges from the inclusion within instrumentation that are used to measure plasma process parameters to understanding of the plasma itself. The knowledge that the plasma capacitance is linked back through the external circuit to the drive oscillator within the power supply is now providing new and low-cost means of non-invasive plasma process metrology.

Acknowledgments The author wishes to thank Mr. I Batty, Dr T Gans and Dr. D P Dowling for allowing access to their plasma chambers. This work is in parted supported and the Enterprise Ireland grant CFTD/7/IT/304 and the Science foundation Ireland grant 08/SRC/11411.

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References [1] [2]

[3]

[4] [5] [6] [7]

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[8]

[9]

[10] [11]

[12] [13]

[14] [15]

Harmonic characterization of a plasma-tool using a diplexer. I Batty, M Cooke and V J Law. Vacuum. 52, 509-514, 1999. Remote-coupled sensing of plasma harmonics and process end-point detection. V J Law, A Kenyon, N F Thornhill and I Batty. Vacuum 57(4), 351-364, 2000. RF Probe technology for the next generation of technological plasmas. V J Law, A Kenyon, N F Thornhill. A Seeds, I Batty. J. Phys. D: Appl. Phys., 34(18), 2726-2733, 2001. Electrical isolation of radio-frequency plasma discharges, Miller P A, Anderson H, Splichal M P. J. Appl. Phys. 71(3), 1171-1176, 1992. The ceramic tiles where bought from: L E W Techniques Ltd, Cook Way, Taunton, Somerset, England, TA2 6BG UK. Minicircuit (Europe). Dale House 19-23 Wharf Road, Camberley, Surrey GU16 6LF. www.minicircuits.com. Reference Data for Engineers: radio electronic, computer, and Communications. 9th Ed. ed, M E Van Valkerburg and W M Middleton. pp9-14-18 (Newnes, Boston, USA) 1998. ISBN 0-7506-7291-9. Non-invasive VHF monitoring of atmospheric pressure plasma. V J Law, S Daniels J L Walsh, M G Kong, L Graham and T Gans. Plasma Source, Science and Technology 19(3), 034008, 2009.. Radio frequency metrology for mobile atmospheric pressure plasma devices. V J Law, N O’Connor, and S Daniels. PIERS Online 4(5), 561565, 2008: http://piers.mit.edu/piersonline/. V J Law. Process induced oscillator frequency pulling and phase noise within plasma systems.Vacuum 82(6) 630-638 2008. A. Pagliarani, A. J. Kenyon, N. F. Thornhill, E. Sirisena, K. Lee and V. J. Law. Process harmonic pulling in a RIE plasma-tool. Electronic Letters. 42(2), 120-121, 2006. www.plasmatechnics.com. The effect of plasma-polymerised silicon hydride-rich polyhydrogenmethylsiloxane on the adhesion of silicone elastomers. C Nwankire, M Ardhaoui and D P Dowling. Polym. Int. 58 996-1001, 2009. Polymeric coatings deposited from an aerosol-assisted non-thermal plasma Jet. L O'Neill, and C O'Sullivan. Chemical Vapor Deposition 15 21-6, 2009. Atmospheric pressure plasma polymerised primer to promote adhesion of silicones. L O'Neill, N Shephard, S R Leadley and L A O'Hare. J. Adhesion 84 562-77, 2008.

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Victor J. Law

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[16] Evaluation of cell behaviour on atmospheric plasma deposited siloxane and fluorosiloxane coatings. M Ardhaoui, M Nassiri, C Brugha, A K Keenan, M Al-Rubeai and D P Dowling. Submitted to J. Adhesion Science and Technology 24,889-903, 2009. [17] Anti-microbial coatings by agent entrapment in coatings deposited via atmospheric pressure plasma liquid deposition. L A O'Hare, L O'Neill and A J Goodwin. J. Surf. Interface Anal. 38, 2006 [18] Soft plasma polymerization of gas state precursors from an atmospheric pressure corona plasma discharge. P A F Herbert, L O’Neill, and J Jaroszynska-Wolinska. J. Chem. Mater. 21 4401-7, 2009. [19] Comparison of pilot and industrial scale atmospheric pressure glow discharge systems including a novel electro-acoustic technique for process monitoring. J Tynan, V J Law, P Ward, A M Hynes, J Cullen, G Byrne, S. Daniels, D P Dowling. Plasma Source, Science and Technology 19(1), 015015, 2009.

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In: Capacitors: Theory, Types and Applications ISBN: 978-1-61668-972-8 Editor: A.L. Schulz, pp. 27-39 © 2011 Nova Science Publishers, Inc.

Chapter 2

VOLTAGE STABILIZATION USING A STORAGE CAPACITOR V.V. Tatur∗

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Institute of Monitoring of Climatic and Ecological Systems, Siberian Branch, Russian Academy of Sciences 10/3, pr. Akademicheskii, Tomsk, 634055 Russia

Abstract In this paper a problem is considered on pulsed voltage stabilization using a storage capacitor, as well as possible solutions of this problem. The stabilizer designed is intended for converting constant unstabilized voltage into pulsed voltage with stabilized amplitude. The stabilizer circuit is based on LC (inductance–capacitor) charging circuit. The principle of operation of the stabilizer is based on the possibility of preliminary charging of the storage capacitor to the voltage intended to compensate for an alternating component of the unstabilized input voltage. This is provided by supplementing the circuit with a subsidiary charging circuit. The paper presents grounds for the voltage stabilization method suggested, as well as a circuitry realizing this method. Detailed electrical circuit of the experiment is shown. The experimental results are presented as diagrams. Up-todate electrical elements are analyzed which can be used to realize the engineering solution suggested. In our case, the most important elements are stabilitrons, storage capacitors and charging inductances. Technical characteristics of the device are considered, in particular, voltage and time charging-discharging ∗

E-mail addreess: [email protected]

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V.V. Tatur parameters. Some exploitation characteristics are presented. Possible applications of the device are considered as well. The engineering solution suggested allows one to make a pulsed voltage stabilizer on a storage capacitor with stabilization factor of up to 1000.

Introduction Pulsed power supply circuits are widely used in electronics and electrical engineering. Such circuits are based on discharge of power, stored on a storage capacitor, into some load. Many power supply sources for pulsed gas lasers follow the same pattern [1, 2]. In this case active power on the load, which is a gas discharge tube, is calculated by the following formula

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P = CU 2 f 2 ,

(1)

where: С is a storage capacitor capacity; U is a capacitor voltage; f is pulse repetition rate. As seen from the formula, active power on the load is proportional to a squared capacitor voltage. According to formula (1), +10% change of this voltage results in +21% change of the output power, while −10% change of the voltage results in −19% change of the output power. Such a change of power on the gas discharge tube of metal vapor lasers leads to significant change of output power of laser radiation. Therefore requirement on power supply stability is very important. To solve this problem, the simplest method is to stabilize input voltage applied to a storage capacitor charging unit. Application of a line stabilizer to stabilize a rectified line voltage in those circuits, where typical voltage at the storage capacitor varies from one to tens of kilovolts and consumption power varies from hundreds of watts to tens of kilowatts, does not find wide use, since such solutions are complicated and, as a result, not reliable. At present pulsed power supply sources are widely used, which include a line voltage rectifier, a higher-frequency generator, a transformer, a high-frequency rectifier and a filter. Such sources allow one to get practically any stabilized voltage. As for high voltages and powers, pulsed power supply sources are not quite suitable, since they have above-mentioned drawbacks and high cost. Solutions are known [3], where power switch is set in the storage capacitor circuit. In this case, power switch requirements are quite rigid, because this switch turns off at residual energy on the charging inductance and this process requires many guard circuits that results in additional power loss.

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Voltage Stabilization Using a Storage Capacitor

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In laboratory conditions supply voltage for charging units is stabilized with the use of an autotransformer or a line voltage stabilizer. There are different pulsed supply circuits with a storage capacitor, which have their own peculiarities, and their improvement goes on. The paper presents the engineering solution which allows one to solve the problem on voltage stabilization on the storage capacitor and can be applicable in many tasks.

1. Grounds

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A circuit with oscillatory storage capacitor charging is often used in pulsed power engineering (Figure 1).

Figure 1. Charging circuit for storage capacitor: С is a storage capacitor, L is a charging inductance, R is a resistance of charging circuit, D is a diode.

Storage capacitor charging in such a circuit is well known [4] and is described by the second-order differential equation. Differential equation for the voltage free component at the storage capacitor is given by

d 2U с RdU с U с + + = 0. Ldt LC dt 2

(2)

Solving this equation, one can derive change of storage capacitor voltage Uс with time t:

U с = U in −

U in e −α t sin (ω 0 t + ψ ) , ω 0 LC

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(3)

30

V.V. Tatur

where:

ω0 =

(R ) = 2π is angular frequency of enginmodes of 1 − T0 LC 4(L )2 2

the LC oscillatory circuit; Т0 is free period;

α = R 2 L is damping coefficient for free component; sinψ = ω 0 LC . Oscillatory charging is carried out provided that the charging time of the storage capacitor through the charging inductance far exceeds the discharge time into the load, while active resistance is far less than critical resistance of the charging circuit R