Network Theory and Filter Design [2 ed.] 0470202254, 9780470202258

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Vasudev K. Aatre Naval Pbysical and Oceanographic Laborator, Ministry of Def~nce, India Formerly witb Nova Scotia Technical College Halifax. Canada

TKE07949 1i~le: Net-ùrk theo~y and fllte1 design

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PUBLISHING FOR ONE WORLD

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Bornbay Calcutta Guwahati

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Hyderabad

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To

Pror. K. Ramatrttbn.a a fr{end, phìlosopher and guide par e:iu:~Il~cc

Preface to the Second Edition

Sincethepublication of the first cdition of this book, teachers from scveral Universities have sent in their comments and criticisms of tbe contents of the book. On study of these comments, it was decided to modify the book incorporating their suggestions for its second edition. Though the philosophy of the book has not been changed, the book has been sufficiently modified to cater for specific comments. Severa! sections have been ddeted, ncw sections added and the presentation in some of thechapters have been changed. In chapter three, an additìonal section on networks with controlled sources has been added. Most electrical engineering curriculums cover the sinusoidal steady state in the first course on circuits Hence the chapter on sinusoidal steady state has been deleted and relevant parts of the chapters merged with Chapter 4. The application of Telegens theorem to evaluate the sensitivity of networks through Adjoint--Networks has been included in the chapter on network theorems. The chapters on Two-port and Indefinite admittance matrix bave been merged to a single chapter on multi-terminal net• works. The detailed formulation of state equions frnm graphtheoretic methods has bcen introduced. The chapter on Image para• meter filter design bas been relegated to an appendix with the section on frcquency transformation being appendcd to the chapter on approximation. The concept of energy functions has been introduced as a prelude to the PR character of driving point immittances. The realisation of ladder networks using zero shifting technique and of higber order active filters by simulated ladders bave also 'been included. With the preceding changes, the book has now 12 chapters and six appendices. These changes are incorporated with a view to improve the usefulness of the book as a text book for undergraduate

viii Preface · electrical engineering students. The second edition of the book is also aimed at gettiog rid of some or the notational inconsistancies and 'printers devii' mistakes. Prof. VGK Murthy of IlT, Madras and KVV Murlhy of IIT Bombay reviewed the book thoroughly and made recommendations "for the second edition of the book. The author is deeply indebted to these two. Dr. K. Souodararajan of ADE, Bangalore has spcnt hoursin rccasting thecbapter onActiveFilters. lgratefullyacknowledge bis help. The guiding force behiod tbe reviewed edition, as or the :fìrst editio11, was late Pwf. K. Ramakrishna of Indian Institute of Science. No amount of my acknowledgemeot can express my indebtedness to him. Indeed the book is dedicated to him, V. K. AATRB

Preface to tbe First Edition

The basis for most electrical engineering courses depends on two funda,m.ental theoretical concepts, viz.. the lumped-parameter circuit theory based on laws like those ofKirchhoft"s and Ohm's, and fl.eld tbeory based on Maxwell'a equations. Circuit and network theory courses may be considered a, prerequisi~s to most other courses in a normai electrica.l engineering curricula as many concepts developed in such courses form a basis for teaching electronics and signal processiog, control and linear systems. The two main branches of network theory are network analysis and network synthesis, the former being a prelude to the latter. Network synthesis naturally leads to filter design. The importance of the design and tbe application of filters cannot be over-emphasizcd as most signal acquisition and processing systems utilize filters-passive or active inductorless filters and more recently digitai filtcrs. Network analysis, network synthctis and ftlter design are well developed subjects, and are endowed witb many exccllent textbooka, tbough most of them are devoted to the discuasion of one of these topics. The stringcnt rcquirements placed by most electrical engineering curricula bave forced network analysis and syntbesis to a sequence of two courses. This book bas attempted a unification oC these two fundamental topics by discussing tbe required anaJytical methods and linking the essentials of approximation, synthesis and filter design. Following a brief review of basic circuit elements, fundamental concepts of graph theory and concept of tbe generali.ud network element are unified to yield both a formai ami tbe inspection metbod of writing various forms of network equations. Solution of network equatio~s based on Laplace transforms, sinusoidal steady-state analysis and resonance, network theor~s includìng Tcllegen', theorem,

x Preface concepta of natural frequcncy and poles-zeros of network functions. two-port parameters including analysia of tbe ubiquitous ladder net• work, indefinite admittancc matrix aoalysis of multi-terminal networks and equivalent source method of writing state equations are all discussed in sufficient detail to form tbe analysis part of tbe book. Essentials of approximation and synthesis leading to practical filter design are presented in the synthesis part of tbe book. Image parameter filter design is introduced for historical reasons and to briog out the concept of matching and cascading in filtcr design. Foster and Cauer form synthesis of two-element kind net• works and l'ealization of all-pole transfer function are discussed as prelude to Darlington's iosertion Joss synthesis. Comprehensive introduction to active filter design mcthods for single and multi amplifier filters, cascaded and GIC based structures, is also prosented. This book presupposes a knowledgc of differential equations and transform calculus, elementary circuit analysis covering Kirchhoff's laws, resistive networks, transients and three-phase AC circuits. lt is desigoed for use as a te.Il for undergraduate and postgraduate courses in network analysis and fundamentals of filter design. Contents of the book can be covered in a sequence of two courses, each with 30 bours of lectures aided by 20 hours of laboratory or problem sessions. The fìrst course can cover preliminary analysis (chapters 1-8) and the second one advanccd methods of analysis (chapters 9-10) aod syuthesis/desìga (chapters 11-15). In this form ìt has been used for severa! years at Nova Scotia Technical College, Halifax. A project of writing a book like this can only materialize witb tbe help of many. lt is a great pleasure to thank my undergraduate students who suffered through the earlier versions of this book and graduate students, especially Mr. W. D. Eckford and Mr. A. K. Mitra, who helped me in formulating many of my thoughts. I am indebted to my colleagues, Drs. V. Bapeswara Rao of Indian lnstitute of Technology, Madras, R. Hastings James and O.K. Gashus of Nova Scotia Techoical College, for many valuable criticisms. I am especially grateful to Dr. O.K. Gashus for providiog assistance in tbe prcparation of the earlier versions of tbc manuscript. For permission to use copyright materiai, I am grateful to: Prcntice Hall Inc., USA, for Figures3.9, 3.10, 3.11 aod 3.14 from O.E. Johnso.o: lntroduction to Filter Theory and Addison-Wesley

Preface xi Inc., USA, for Pro blems 3. 3 and 4. 3 to 4. 5 from S.P. Chan, S. Y. Chan and S.G. Chan: Analysis of Linear Networks and Systems. The final draft of the book was completed while the author spent bis sabbatica! ycar at the In)

The energy in the i_nductor, from eqns. 1.4 and I. 9, is I

W(t0 , t)

=

Jv(-.) it -r:)d-r: lo ♦ (I)

=

I

•

i(,p)d,f,

(I.9)

Jntroduction

ll

If tbe initial ftux, rf,(t0), is zero, then the energy stored in the inductor is ,j>(I)

W(t)

=

Ji(,p)d,j, o

F or a linear i nd uctor (4> = Li) this reduces to W(t)

= ~ r/,it) = ~ Li2(t)

(1.10)

1.2 Kirchholl's Laws

Figure I. 12 shows the model of au electric network (from bere on we use the word 'network' for its model). A basic problem in network theory is to analyse the network and find the voltages across and the currents through ali the elements of the network. Certain physical laws are fundamental to the algebraic formulation of such a problem. Two of the fundamental physical laws in network theory are the 'Kirchholf's current and voltage laws'. If we consider tbe junction points A and B of tbe network we can write at A : i = Ìc1 + h 1 at B : iL, = ic, that is, the current flowing towards a junction is equal to the total current ilowing away from the junction. This is Kirchhojf's current law (KCL). More formally, the KCL can be stated tbus: The a/gebraic sum of a/l instantaneous currents entering or leaving a junction is zero. If we consider the closed patbs consisting of the source and

+

.

•

mJ~

;e-I +

;,2

12

A

+ C1

v,2

+ Vc2

Vc 1

e Fig. 1,12

;°I B

C2

12 Network Theory and Fi/ter Design

capacitor C1 (say path l)and thecapacitors C1, C2 and theinductor L2 (say path 2) we can write path 1 : v1 = ve, path 2: ve, = v11 -1 ve, where v" ve,, ve, and v,1 are all voltage drops, i.e. the voltage drop between any two junctions is independent of the path between the junctions. This is Kirchhoff's voltage /aw (KVL). More formally, KVL states: The a/gebraic sum of a// voltage drops or rises in a closed path is zero. The KCL imposes linear constraints on the currents through the elements whereas the KVL imposes linear constraints on the voltage drops across the elements. Together, these two laws imply conservation of energy (Sec. 5. 4). 0

1.3 Classificutioo of Networks

The overall behaviour of an electrical network can be predicted by the constituent elements and their interconnection. The behaviour of tbe network, considered as a black box, leads to a number of classifications like lioear, nonlinear; time-invariant and timevariant; passive, active. (i) Linear and Nonlinear Networks

In a linear network, the relationship between the voltage and current is described by a linear equation. Consider two networks N 1 and N 2 as shown in Fig. 1. l3(a) and (b), respectively. Network (a) is made up of a linear resistor R, while (b) is made up of a semiconductor diode and linear resistor R. Let the cut-in voltage of the diode be 0.6 volt. In network (a) current 11 is given by V/R and exists for ali values of V. In network {b) if V is less than O. 6 volt the current / 2 is zero and for voltages higher than O. 6 volts 11 is given by (V - 0.6)/(R + RF), where RF is the forward rosistance of the diode. Obviously, in network (a) the current response is linear in contrast to that in (b). Let both the networks be excited by two serially connected voltage sources Vi and V,. Then it can be seen that l1CV1

~

+ V,)= I1CV1) + l1(V,)

~~+~~~~+~~

lntroduction

+· ,.

i1

N1 •• R

•

+

ii

1)

13

Nz

-

--

•• R

-

(al

(b)

Fig. 1.13

where / 1(V1) is the current into the tcrminals of network N 1 when excited by a voltage source V,. We say that network N 1 is linear and N 2 is nonlinear as the principle of superposition holds for N 1 but not for N 2 • Linearity of a network can be defined as follows: Let a network be characterized by F(x 1)= Y1 where x 1 is the input and y 1 the output, and F(.) denotes some function. Then the network is linear if, and only if, F(1X1 X1 + 0t 0 X2 ) = oc 1 F(x 1 ) + oc 2 F(x2 ) = (/.1 Yi + (l. 1Y2 where (/. 1 and tx 2 are arbitrary constants, and x1 and x 2 are any two allowable inputs. The principle expressed by this equation is called the principle of superposition and homogeneity. Hence, we conclude that a network is linear if it satisfics this principle; otherwise, it is nonlincar. (ii)

Time-invariant and Time-variant Networks

Lct a linear resistor be characterized by v(t) = R(t) i(t) where R(t) is a prescribed time function. This can be achieved, for examplc, by the sliding contact of a potentiometer being moved back and forth by a motor. Such a rcsistor is called a time-varying resistor. Similarly, it is possible to build time-varying capacitors and inductors. The clcments we considered in Scc. I. I were ali time-invariant in that they wcre characterized by parameters which were not dependcnt on time. A time-invariant network is characterized by :i constant coefficìcnt equation whereas the time-variant one by a time-variant

14 Network Theory and Filter Design coefficient equation. Mathematicall}', wc can describe a timeinvariant network by F[x(t - t0 )]

= y(t -

10)

when the network is characterized by F[x(t)]

= y(t)

i.e. the response (output) depends on the shape of the excitation '.input) but not on the time of application. A network composed of time-invariant elements is necessarily tiine-iovariant whereas 11etwork composed of time-variaot elements may exhibit time• invariant terminal behaviour.

'.iii) Passive and Active Networks Considera network made up of a single linear resistor. Tbe energy mpplied to (or dissipated by) the resistor, from eqn. 1.4, is I

W(t 0, t) =

Ji (t) R dt 1

'•

lf the resistor has to deliver power to the external world, R has to ~e negative. As long as R is positive the resistor will consume ,ower, and such a resistor is called a passive resistor. Let v(t) and i(t) be the voltage and current at the terminals of a 1etwork. Then the energy delivered to thc network is given by I

W(t 0 , t)

=

Jv(-r) i(-:) d-r lo

\ network is said to be passive if, and only if, W(f0, t)

+ E(t

0)

~

O

·or all t and /0 and for ali v(t} and i(t), where E(t0 ) is the energy in he network at t = t 0 • Otherwise, tbe network is said to be active. ·n other words, if the cnergy delivered to the network is non1egative for all time and input, the network is said to be passive. rhe conditions for activity and passivity of an element can also bé ,btained by a study of its characteristics. For cxample, we :an state that a nonlinear resistor is passive if, and only if, its ·haracteristic,for ali time, is in the .first and third quadrants of - i piane. Similarly, a capacitor (inductor) is passive if, and only f, its characteristic is in the first and third quadrants of the - q (t/> - i) piane.

-----

,_+

ONE-PORT

1S

.lntroductìon

;

i1

=

lj

,; Fig. 1. 14

The preceding classiflcation of networks was based on the basic behaviour of the response of the overall network. Often, networks are grouped without any reference to their response. Such a grouping includes n-terminal and n-port networks. If, for example, a voltage source is connected across the tcrminals k and k' of a network, then the current ik flowing into the network at terminal k must be necessarily the same as tbc current ì,:, flowing out of the terminal k'. Under these conditions the pair of the terminal k-k' constitute a port. A two-terminal or a one·port network has only one accessible terminal pair (Fig. 1 .14) and a two-port network has two accessible tenninal pairs (Fig. 1.1 S). This definition can be extended to n-port networks.

+ :, ['

--

~· i1

= i;

i2

TWO-PORT

i2

= ii

+

2

Vi

-

,,

i2

- 2'

Fig. 1 .15

Networks can be classified in a number of other ways-lumped and distributt>d networks; reciprocai networks; large scale networks and power distribution networks.

1.4 Nullators and Norators The lumped circuit elements introduced in Sec. 1. 1 are sufficient to model most of the networks. (Sometimes we may need coupled coils to model a network. Wc postpone the introduction of coupled coils till Sec. 3. 4.) But the modelling and analysis of

16 Network Theor)!\tlnd Filter Design active networks (i.e. networks with active components like transistors and operational amplifiers) is facilitated by two singular (or pathological) network elements-the •Nullator' and the 'Norator'. A nullator is a two-terminal device which is simultaneously open and short, i.e. the current through and the voltage across a nullator are zero. A nullator is represented by thc symbol shown in Fig. 1.16(a). A norator is a two-terminal device whose currcnt and volt-

·•----10,---.. (a)

-co-(h)

Fig, 1 .16

age are arbitrary, i.e. the current through and the voltage across a norator are determined by the external circuitry. The symbol for a norator is shown in Fig. I. 16(b). It is obvious that nullators and norators cannot exist in the real world. But the ideai behaviour of certain active components can be represented by a combination of nullators and norators. For example, an ideai common emitter transistor can be represented by the nullor (nullator-norator) model sbown in Fig. 1. I 7. In Chapter 12 we discuss the application of nullor models in analysing and io synthesizing active networks.

__

.,Rase

Colle-:tor

Emittcr

Fig. 1.17

As indicated earlier, modelling of the pbysical devices is basic to the analysis and the synthesis of signal processing networks and devices. In this chapter we bave learnt the basic models and the properties of certain circuit elements. The interconnection of these elements and the geometry of the network is the topic of the next chapter.

lntroduction

17

PROBLEMS J. I

The characteristic of a nonlìnear resistor is described by the i-v relationship

+

i - «o «1 lf the resistor excited by a signal I'

v + •zv, = A cos .,,,, find an expression for

the current i. If we are to generate a third harmonic signal what should be the characteristic of the resistor. 1. 2 Find the current through the resistors Ri, R 2 and R 3 in the network of Fig. P. 1.2.

.:..12 Volt~ Radio

R1

= Ri =

R1

=

IO Fig. P. 1.3

Fig. P. 1.2.

1. 3 A 9-volt transistor radio is to be run on a 12-volt battery (Fig. P. 1. 3). The radio draws a current of 6 mA. Pind R1 and Rs1. 4 The open-circuii voltage of a souroe was found to be 10 volts. A variable resistor R is connected across the terminals of the source (Pig. P. 1. 4) and the resistor R is adjusted such that voltage Vo is 4 volts. lf the value of the resistor is 400 n what is the internal impedance of the source. Voltage

Sourc.:

Pig. P. 1.4.

1 . 5 Find the currents I and /,, i - 1, 2, ... S in the network of Fig. P. 1 . 5.

R/2 /

,,

+ E

4R

R

2R

&R

SR Fig. P. 1.5.

18 Network Theory and Filter Design 1 . 6 Find the conditions for the passivity of time-varying capaci tor. 1. 7 Classiry the following elements as passive or active

(a) Resistor: i (b) Resistor: i (e)

= v2 = va

(d) lnductor: L(t) - a (e)

Inductor: i - - ♦ + ♦11 ( / )

Capacitor: q = Capachor: C(t)

+ b cos ,.,

V.+ v2 = acos ""' + b sin""'

1 8 The voltage across an inductor of tCO mH is shown in Fig. P. 1.8. Plot the current through the inductor assuming i1t0) = O. lf /1(0) ... 0.2S amps how would you modify the plot?

,: (Volts)

o

I (Scc)

-3

Fig. P. 1.8.

1.9 A nonlinear capacitor is ch1racteri7.Cd by lhe equation q =- 1 - e-•, Find tbc small sisnal capacitancc at • = , 1• 1.10 Fiaure P. 1.10 shows a •Darlington Pair'. Draw the nullator-norator equivalent diagrara and find tbc voltages across and cunents tbrough R1 and Ra-

+

F'i1, P. 1.10

lntroduction

19

NOTES AND REF'ERENCES Study of circuit elements and their models is fundamental to tbc, design of electronic devices and circuits. Here we have introduced a few passive circuits elemeots and the da,sification of Networks has bcen briefly reviewed. The coverage, by intent, is neither elaborate nor exhaustive. Severa! text books discuss the details of n,odelhng and of the ranae or application of these models. Desoer and Kuh's book 1 (Chap. 1 & 2) discusses in detail lumped circuit elements and their models. Chua's booka lChap. 1-3) on nonlinear networks introduces many nonlinear elements, their models and .signal processing abilities. Tbc concepts and models of R, L and C can also be developed from field point of view ami this is well discussed in Robrcr•s book• (Chap. 2-3). The concept of Nullators and Norators was introduced by Carlin&.

s. Kuh: Basic Circuit Theory, McGraw-Hill, 1969. (2) L. O. Chua: lntroduction to Nonlinear Network Theory, McGraw-Hill. 1969. (3) R. A. Rohrer: Circuii Theory: An introduction toState-variable Approach, McGraw-Hill, 1970. (4) H. J. Carlin: Singolar Network Elements, IEEE Trans. on Circuit Theory, CT-11, Marcb 1964, pp. 67-72. (1) C. A. Desoer and B.

2 Network Topology

Topology formalizes the formulation of the network equilibrium equations (loop equations, node equations and the like). Most of the computer aided analysis and design metbods utilize topolo1ical formulation. The derivation of the state equations of a network (Chapter I O) inherently depends on the topologica! matrices of the network. Consider the network shown in Fig. 2. l. The equations describin1 this network are: Vz

=

R. i1; Va

= Ra i.; ,, =

e, ~-

(b) 11

=

i 1 + i 1 , i 11

=

+ i,, v1 =

(a)

;3

i,, ; 1 = i2 R3

v1

+ v,

Fig. 2.1

The set of equations in (a) are the terminal relations of the various elements and these do not depeod on the interconnection of tbc clements. The 11et of equations in (b) are the constraints imposed by the two Kirchhoff's Jaws and as such do not depend on the elements but only on the geometry or the topo/ogy of tbc network. Also, there are seven equations in six unknowns. Of courso, in this particular case it is easy to recognize the redundant equation in (b). But in large networks this becomes a complex problem and a study of topology proves helpful in solving such a problem. Topology, or geometry of a network, is concerned with tbc interconnections of the elements in a network. The network

t,

Net~oril'"topology 21

is represented by a /inear graph, and in this chapter we discus1 the properties and applications of such a graph. 2.1

Basic Definitions and Properties

The linear graph of a network is obtaioed by replacing every element by a line-scgment and the junction points by nodes. For example, the lioear graph of the network of Fig. 2. I is shown in Fig. 2.2. By definition a linear grnph is a collection of line-segments and their end poinh. A line-segment with its end pointll is called an edge of the graph and the end points are called vertices (or nodes) of the graph. A linear graph or a graph of a network carries ali the information about the inter-connection of the elements of the network. The length, thickness and formation of tbe edges are of no consequence. Further, we impose the restriction that the end points of an edge are distinct. An edge is said to be incident at a vertex if that vertex is an end point of the edge. For example, edges l, 2 and 3 are incident at vertex v1 in Fig. 2. 2.

Fig 2.2

Sub-graph of a graph G is a graph whose edgcs aud vertices form a subset of the edges and the vertices of G. In particular, G is itself a sub-graph of G. Ifa sub-graph of G does not contain tbc entire graph G then it is called a proper sub-graph of G. Ifa subgraph contains all the vertices of G then it is called a spanning sub-graph of G. A complementary sub-graph of a sub-graph of the graph G is the graph formed by the edges not in the sub-graph of G. Consider a graph G shown in Fig. 2. 3. (The edges of the graph do not carry any arrows as in Fig. 2.2. The arrows on tbe edgcs indicate the orientation ·of the edge~; oriented edges and oriented graphs are discussed at a later stage.) This graph hu si~

22 Network Theory and Filrer Design vertices v1 ; i = 1 to 6 and nine edges e,. i = 1 to 9. The eh tbeory to electric networks besides providing proofs of many tbeorems cited in tbis Chapter. Additional information and problems can also be found in tbc booka by Seshu and Balbaoians {Cbap. 3) Desoer and Kuhll (Cbap . .S, 9-11). Graph theoretic concepts and methods provide a formai and algorithmic approach to the derivation of network equilibrium equations (Chap. 3) and are fundamental to the modclling of Iarse scale networks and systems. Sucb models find extensive use in communlcation and transportation networks'. Many non-circuit tbeory applications are discussed in Deo's book.1 Scrious students or graph theory may find books of Harary•, Swamy and Tbulaairaman' to be of interest. The former gives a mathematician 's point of view while the latter an engineer's. 1, S. Sesbu and M.B. Reed: Linear Graphs and Electrical Networks, AddisonWesley, 1961. 2. S. Seshu and N. Balbanian: Linear Network Analysis, John Wiley, 19.59. 3. C.A. Desoer and E. s. Kuh: Basic Circuit Theory, McGraw Hill, 1969. 4. H. Frank and I.T. Frisch: Communication, Transmission and Transportatlon Networks, Addison-Wesley, 1971. 5. N. Deo:Grapb Theory with Applìcation to Engineering and Computer Science, Prentice Hall, 1974. 6. F. Harary: Graph Theory, Addison-Wesley, 1969. 7. M,N.S. Swamy and K. Thulasiraman: Graphs, Networks and Algorithms, Jobn Wiley, 1981.

3 Network Equatioos

A linear network can be described by a set of linear equations. In this chapter we present methods of writing such equations. The basic questions to be answered before we bave a set of equations are: How many equations do we need? How do we write these equations? When are we sure that we bave enough equations? Some of these questions are answered by "Topology" of the network. In order to completely solve a network of n elements, we need to flnd n voltages and n currents. In ali, there are 2n variables. Hence, wc need 2n linearly independent equations in these 2n vari~ ables. The terminal relations (i.e. v-i characteristics) of the elements provide n independent equations. The second set of n equations is provided by the KCL and the KVL. We bave already shown that there are (v-1) independent KCL equations and (n - v + 1J independent KVL equatim~s and, thus, in total wc have n more equations. We bave thus divided the 2n equations needed to solve the network, i.e. to fìnd all the currents and voltages, into two groups: a group of n equations which depend on the elements but not on their interconnection, the second group of n equations which depend on the interconnection of tbe elements but not on the elements themselves. Different methods of combining these two groupP. of equations lead to different sets of equations, and tbis forms the subject of this chapter. 3.1 Node and Mesh Transformatioos

The Kirchhoff's current and voltagc laws for any generai network can be written as (3 .1) (3.2) I

SO Network Theory and Fì/ter Design

where Qr [(v - l) x e] and Br [(e - v + I) x e] are the f-cut-set and f-drcuit matrices, respectively, of the network graph of v-vertices and e edges. lf the edges of the network graph and the columns of tbese matrices Qr and Bi are re-ordered, then we can write

[Q11 U] [ :: ]

=

O

(3.3)

[U B12]

=

O

(3.4)

[

:: ]

wbere the subscript e indicates cbord variables and b the branch variables. Eq uation 3 .4 yields V.,= - B11Vb On using eqn. 2.18 we get Ve= Q~1Vb

(3.5)

By comhining this with the triviai equation we bave

V.=

[ Qh] U

Vb

or

V.

= Qj Vb

(J.6)

This equation iodicates that the edge voltages are expressed as a Jinear combination of (v - I) branch voltages, where vis the number of vertices in the network graph. Equation 3. 6 is called tbc branch iransformation. To verify the consistency of this equation we bave only to see that B1V. = B1(Q} Vb) = (B1Qj) Vb = O Instead of using the branch voltages as a set of independent voltages, we can also use the set of node voltages as tbe independent set. If io a graph we select a node as the reference node, tben we can write V, = A 1Vn (3. 7)

where V,. is the column vector of (v - l) node voltages with respect to tbc reference node and A is the reduced incidence matrix

Network Eqmtio,u SI

or thc graph.

(Tbc row corrcsponding to the re(eff:DCC node is deletcd from A••) T o prove this result we recognize two ldnds of edges; those which are incident witb the rcfen:nce node and those whicb are not. For the former, the cdge voltagc is the same as tbc nodc Yoltage or its negative depcnding upon the orientation of tbc tdge. For the Jatter tbe edge voltage forms a circuit with two node vollqes aad llencc the cdge voltage can be expresscd as a linear combinatioa of node voltages. In both cases tbe edge voltages are linear combination of node voltages. Let Y• be the edge voltage of the ktb edge and e1 the node voltage of tbe node i. If lhe kth edge connects the 1th node to the refermce node then. Y1 = e,. if edge k leaves node i. = - ir cdge k enten node I. On the other band, if tbc kth branch leaves node i and enten node j, then

e,.

Y,1:

=

e,- e,

Hence, we can write

r

r:

c•. -

-I l:

c., ,-1

L

where

C,1

= 1, if edge i leave~ node/. = - I, if edge I cnters nodej.

O, if edge i is not incident with tbc nodc j. From a definition of it can be easily noticed tbat tbe matrix

e = A• and bence

c.,,

V.= A'V,. 'Ibis k bown as notk trlJMjormation. We haTC cxpreued the edge voltages as linear combination of branch or node voltagcs. We can similarly express the edgie CVl'l'Cllts as Iiaear combinations of chord or mesh currcats. EqaatÌGtl 3 •3 yield1

I.= Qn~-

52

Network Theory and Filter Design

In view of eqn. 2, 18 we bave _Ib = B~2 le By combining this with the triviai equation I.= U le we bave

(3.8).

or

I., = B} le (3,9) This equation indicates that the edge carrents are expressed as linear combination of (e - v + 1) chord currents, wh~re the network graph has e edges and v vertices. Equation 3. 9 is called the chord transformation. Jf we are considcrìng a planar network graph, we can select a set of mesh currents instead of chord currents. lf Im is the vector of mesh currents, then we can easily show that

I~= B:_, Im (3.10) where Bm is the mesb matrix, indicating that the edge currents can also be expressed as linear combination of the set of mesh currents. This is known as mesh transformation, EXAMPLE

3.1

Consider the graph shown in Fig. 3. I. The branches are shown as solid lines and chords as broken lines. Vertex v4 is selected as the referen·ce node. There are three f-cut-sets, three f-circuits and three meshes, and these are marked on the graph. For this graph,

F1g. 3.1

Network Equations

" [

A=

r-

Q1 = 5

I

B1

o

-l

o

-1

I

2

3

4

l

O -I

O

3

O

O

3

;. l

2

3

4

o

o o

-1

-1

;, r fa

Thus,

v. = I. =

o o

3

4

5

o

o o

-I

-1

-1

-1

r

V1

v.

V,=

Va V4 V5

L v6 J

J

A•Vn, V,

= Q} vb

B'ml,,,,

=

I,

B1l,

o o

-~ j ~j 6

6

2

r6

Li«

o 5 -l

o

,,.

I,=

o

I

Bm= 2

5

o

-1

I

= 2

-I

o

o

-I

I [

o

6

5

o

o

Lo

I [

4

o

,_

4

6

3

I

:2 -~ J

2

1

o

~]

-1

6

]

53

54 Network T"'-7 ond Fllter Dalgn 3 . 2 ~.....

111c tenniDa1 equations for dle elementi of a network ca. be writtenu l'1 = Raia or i1 = G,. '• (3.11)

'•=t..!• '•=Ca I

or

I

i1 dt or

11,:=

l,.J '•"'

(3.12)

"·

(3.13)

i.,= Ca "iii

In a paiticular network thcso elementi may appear in seria or parallel u a 1ro11p. In ordcr to·formulate the network eqoatiom we introduce the concept of a goieralùed element. A generaliml elemcnt k haa an O/lff'atlonal impua,u;e z., or admittance 1• in lerics with a voltage sourc:e e1 and in parallel with a cunent IOllffll ia. as sbown in Fig. 3 .2. Let tbc opcrational impcdanc:e z1; be a 1erios combination of R, L and C and tbc admittance 11 a .-ra11e1 combioation or R, L aad C, iapcçtivelf. as shown in Fig. 3. 3(a) ancl (b). (In writiog tbc opcrational impcdance ali thc elemen.ta io series are groupcd togethcr to form a single impcda.oce. la writing tbe opcrational admittance ali elcmeota in panllcl ue groupcd together to form a single admittaoco. If wc want to write dlc operatiooal aclm.ittancc for the series combinatioo of Fig. 3 .3(a), cacb elcmeot is coosidered as a separato clomcot whilo for writiog tbeDQpedaDQlfor Fig. 3.3(b)each elemeotiscoosidcred separately.) From .Kirchholf's laws ,., the voltaao across tho cdge k with. an operational impedancc, is Z1c ~

R1c

Il e" (a)

+

.r:.,

f't

,,,,

j,.

.... ,.2

Yt

•

ES (b) l'fl,3.3

•

Network Equotton, 55 •l(t)

= e1(t) +Rt (ia(t) -/i(t)} +L1 :, {ii(t) .:,.. ~J{Mt) -

Jt(t)}

(3.14)

}t(I)} dt

The last three terms constitute tbc voltage drop across z1, whcre {IA(t) - /t{t)} is the current tbrougb z6 , and i1(t) is tbe edac earrent. Equation 3. 14 can be. rewritten as

•i(t) -= et(t)

+

{R. itlt) + Lt :, i,.C.t) + ~J dr}'

- {R,dt(t)

it{I)

+ Lt :,1k(t) + ~k

I

},.(t)

dt}

(3. 15)

This integro-differential equation can be tr~formed to an algebnic equation by Laplace transformation (Appendix B). Consider tbc cquation, Jtt)

=

R,,it(t)

+ Lt d':ìt) + ~k

I

ili(t) dt

Taking the Laptace transform wc bave F{I) =-R1lt(1) +Lt{llli{I}- i1k(O)}

+A{~+ fc~O)}

where 116 f iaitial co•di&klna aad, inptltl. ,tJleh network N baa a 1IDique ■olutloa •. ·

Of c:otano, a passive ~nvariantllLC network haa • maiqae aohition once the initial condition1 and input.I are apecifted. Again, consider tbe nodo equatioos. Tbe 1ol11tion to tbe node eqoations, i.e., a set or node voltages, is given by

V,,(a)= Y;1 {-AJ.{1)} . From equation 3.26

· J.{,) - J.(~)- Y.,(a)E.(,) -

c.v.,r, s.(1) ~ voltap tourcll vector, V-(0) tbe iDitial caiuitor-voltap vector,'11.(0)' the initial inductor eurreat •tor and -Y:,J,;) dJioi ect,e admittaaoe matrb,. aad c•. tbt • • ca,àclttMICUIUllrl,t.

.

We caa rearoup the torma ia eqa. 4_.3 u Va(,)•V.-(r)+ V,,J.1) wbtre

ud

v.,.-~ r.•" U1,.co)~·c.v-co)},

V~ oorrespoads only to tbe tollfCllli&IUl is tbe zero ttauueapome. Similarly V.. corretp011da to the · iaitial coDClitiont ònty, add il tbe zero input reapome. · Bea.ce, . · Laplace Trànsform _ Laplace Tra•o~Dl + Laplace Transform orthètotal reei,ou1t_ of Zlll' ' . . ofZSR. . In ordér to 8nd tlie tolal reaponae we have to' ftnd the inverse of Y,.; By deflnition .· . y;1 04/(Y,.) " == I r•• wl\ero ad.l(Y■) il tlle,adjoint ol. Y.(.r) ud IY,. / it the determinant or y J..•>·. We ean further write "' ·

I,

y;

m

'A(lf)

•(I) ,

wll.ere à(1) il a pol)'DOblial matrix and -.(1) a polynomlàl in 1. Sub1titutio• ot tll.it equation lo -eqn, 4 .4 yi~ VJ.11)- - ~/ .4{1.(1)-Y.(l)B.(1)}

-:=~ H .e

In

aeneral we bave

.

V '•) ""'

= 'h( P,(11) 1)

t,.(O) -

c. V.(O)}

.

and V: '•) =lii':

•t•)

f1 ,)

where P 1 aod P1 are vcctors and q1 aad q1 polynomials in , •. Jr wc contider tbe 1th nodo voltap, then

Yh) =p,.(,) +1'•(,) I\ fi(I)· q/.,1)

(4.5)

98 Network Theory and Fllter Design and

v,(t) = ,;c-1 {p,,(J)}+ _c-1 {p,,(s)}.

q1(s) . q1(s) Hence we can determine V,.(t). tbc node voltage vector in time domain. The node transformation

V.-A•V,, yields tbc edge voltage vector V.(t). The edge currents can be now determined by tbc v-i relationship1 of tbc corresponding elcments i.e. I.(s), = Y..(.t) V.( s) or i.(f) = .,C-1 [ Y.(.t) V.(s)J and tbe solution of the network is complete, as wc bave determined ali the cdge voltages and currenta. ,, From eqn. 4.$ wc see thatà .typical element for wbich wc hàve to find the inverse Laplace transform is of tbc form F(s)

= p(s)

q(s)

(4.6)

where p and q are potyhomials in s. In the next section we describe a method of :ftnding the inverse Lapiace transform of such rational funétions.

4.3 Partial-Fraction Expansioo Inverse Laplace tran~form of simple functions can be obtained from a table of Laplace transforms. But the inverse transform of a complicated function will not directly appear in the tables. If the given function is a rational function, then tbc inverse transform can be obtained by breaking tbe rational function into simple functions whose Laplace inverses are found in the tabica. This method is called parttal-fraction expansion (see Appendix A). Let tbe given rational function be F(s) = p(s) = bm .sm + bm- 1 sm-1 t ... + b1 s + b0 q(.t) a,. sn + Q,,_ 1 sn-l + ... al S + a 0 wherep aod q are polynomials in s and tbe coeftìcients a 0 , aJ ... a,., b0 , b1 , ••• hm are real num bers. Polynomials can also be expressed in factored form as m

K II (s-z,)

F(s)

= -·..,':,_1- IT (s-p,)

,-1

Solution of Network Equatlons 99 where z1 (l = 1, 2, ... m) are the roots of p(s) and p1 (i = 1, 2, ... n) are the roots of q(s). The z,'s are called the zeros of F(s) because F(z,) = O. The p 1' s are called tbe po/es of F(s) because F(p1) = oo (see Cbapter 7). Hère we assume that p(s) and q(s) do not bave aoy commoo factors. If p, a simple root of q(s), tben it is said to be a simple po/e of F(s). If p1 is a root of order r of q(s), then it is said to be a multiple po/e of order r of F(s). A rational function is called a proper ratiooal function, if the degree of the numerator· polynomial is less than the degree of the denomioator polynomial. If F(s) is not proper, we cao write

is

F(s) - F 1(s)

+ :~:~

(4. 7)

where p(s)/q(s) is a proper rational function and F1(s) is a polynomial in s. Inverse transform of F1(s) can be easily obta.ined. lt will contain delta functions and its derivatives. Hence in what follows we only coosider proper ratiooal functions. In obtaining the partial-fraction expansioo we oeed not factorize the numerator. Wc bave to factorize the denominator polyoomial and find the poles of F(s). We cons_ider three separate cases. Case 1: F(s) has simple poles F(s)

I

= pz(s) = p(s) 1-1 Il (s-p,) q\.SJ

(4.8)

(Here q(s) is assumed to be a monic polynomial.) Then the partial-fraction expansion is

=""' 2!._ Il

F(s)

(4.9)

" - s-p, 1•1

where k, fs the residue of F(s) at the pole p,. To obtaln k,, multiply both sides of 4.9 by (s-p,).

2 Il

i.e.

(s-p1) F(s)

= k1 +

J .. 1

k,

--(s-p,)

s-p1

J,.1

from which it follows that k, = (s-p1) F(s) ,_,, The inverse transform F(s) il

(4.10)

100 N1twork Theor1 and Fllttr De1lgn

I,-1• "ie'"

,et) -

i.e.

(4.11)

(This foUowa from tho linoarity pr_oporty of tho Laplaoo transfo~••>

Bx.ulPLB4.1 Lct

F{I)

.. Jl+6.r+8 ~ ($ -t- !}(I + 3-J{I + S)

Ff1\

== ~ + k, . + k.

From eqn. 4. 9 ' I

+}

I

I+ )

7D

Fromeqn.4.10

.

k,

k

= (1 + l)F(,) 1--i-. - (1.-+6.r+81 + 3X1 + 3) .....1 -13

• == (-?

I .-+w+s L. + +

6.r+ 8 '"""°1 1 + 3)~.r)I,...,.. = (z,-++l)(.r + 5) _.. -'::';f"=-4

ka = (1 + S)F(,)1,--, Fi(1)

Hoace

=

l)(s

(1

3)

3

·• ...,i

3/8 -+ . 1/4 + 3/8 .r+l 1+3 a+s

and from eqn. 4.11

/(t) ...,

3 1 3 sr' + le-4' + irl'

Cae Z; F(1) has multiple poles. . , Lct where

no

q(,) -

n c,-,,"'f,

(4.12)

,-s

ì,-1

n, = n.

partia1-fraction expanaion of 1'(1) is of tho form

F(s)

ks1 + k11 + k,"' + 1-p ..!iL... + (s-p kg -•-P, (s-p1)1 "'(.r-.-pJ• 1)' 1 .

I

Solllllon o/ N1twork E(JllQllonl 101

+

k.._

+ ... (1-pjiì•

+~ +

,

F(,) ==

i.e.

...

I - P,

.,

k,1 +

k,.,

~ ..(,- p,yI;

(1- P,)1 ,

2 _- 2 (-' ~•~,j1 1•1

(4.13)

/•I

In c:an bo 1bown that

-(n,~j)l~;~;{(,-p,)"rF(1) Tho invent•waaaform of F(.r) il k11

(4.14)

,,-1 ;,,

~ ~

/(t

}L,.,

(4.15)

'-' '-' k,,(J-1)1~ ·-·

/•1

BXAMPI.114.2

.

.~ + 1

F{I) - (I+ l)'(a + 2) From eqn. 4 .13 Jll(I\ k 11 k, +· ~ ~ .\,, - , + I+ (.r +1 1)1 (.r +,I,.+ .r + 2 From eqn. 4. 10 ·

k1 =- (• From eqn. 4.14 k11

I

-

,. +l)'11,,__. •.- S + 2)F(,) J.,.... =e,+ ·

d(I-I)

(1=])1 ~ ( , + 1)1 F(1)

,. + 'Ir__,, -(• + l)IF(a) t..-.,-= 7+T k 11

·

L 2

1d

-y "fii,•+ l)'F(•) 1,,.-i

I L ,. + L . · - -

u}

.

wbere n il the number of 1egments. Tbe smaller the segment width tbe better is the approximation. In tbo limit At ➔ O(n ➔ co) we bave an exact ropreientation of e{t). The preceding eqution ·can bo written as e(t)::::

I,,_,•

t(k At) 11(.t-k àl)-u~~t- (k+l) At}At

(4.23)

.

As At➔ O the summation becomes an integrai {infinite sum,. Also notiog that Lim u(t) - u(t - 6.t) _ S(t) 6,t

A...0

reduces the eqn. 4.23, in the limit, to .

e(t)

I

=

f

e{-r) 3(t- -r) d-r

{4.24)

D

Equation 4. 24 gives tbe representatlon of any function by the infinite oumber of weighted delta function. Tbis is also called the •siftlng property' of the delta function. We now proceed to derive the convolution integrai. Let a network N ,be described by its impulse responso h(t), i.e. 8(t) ➔ h(t) Here ➔ implica that h(t) is tbo rospon·so to 8(t). If the aystem is time-invariant, then 3(1- 1') ➔ h(t-T) If in addidon N is a linear network e(1') a(t- 1') ➔ e(-r) /(t - T) and conaequontly

' ' [ee-r> ac,- "'> tir➔ e h(t-"'> th

j

Tben from eqn. 4.24 we bave

e(t) ➔

J'

e(T) l,(._t- 1') fh

o

110 Network Theory and Filter Design

i.e. the responso r(t) to an arbitrary input e(t) ia given by t

r(t)

=

J

e(-:) h(_t - -r) dT

(4.25)

o

Tbis is called tbo con,olution integrai. The steps involved in. tbis argument are shown in Fig. 4.6. The right hand aide of eqn. 4.25 is generally written as e(t) • h(t)

where • indicates tbc convolution~ Henco r(t) = e(t) • h(t)

h(t)

8(1) 8(1 -

h(l - T)

r)

-T) J.r -

e(T)8(t - .-) I

I

Oe(T) 8(t -

e(t)

N

e(-r) h(I - T)

J:o e(T) /1(f -

--T) ~T -

J:e(T) h(t - .-) itrary constants•. To evaluate tbese constanrs some additional intorm!llion must either ~ givCrn with tbc cquatio1's or det~rmined from the physical µetwork itself. In the previods sections we bave tacitly assumed that this• foformation, the initial conditions, are •somehow' specified. In this seotiòn we evaluate these initial condltions. 1r network does not contalo any energy storage elements, then tbe question of"evahaating any iaitial conditions does not arise at ali. The initial condition tbat need to be evaluated, for an RLC network, aro tbc initial capacitor .v~ltages and initial inductor currents. The ini.tial conditions i11deed depend on tbe history or the network. Let us suppose that t~ network is connected ·to an energy source through a switcb (or switches) and let tbc switch (or switches) be closcd at. tbc instant or time t = O. We designate t,.,;. O- and t = O+ as tbe instant of time Just before and just after thc operation ·or tbc switch, respectively. · Tbc vaJues of the capacitor voltages and inducior currenta at t =O+, desi,gnated as Y0 (0 +) aod i1(0 + ), rcsp~ctivcly, :are called the inltial conditions of tbc network. ·(In the prcvious scctions and Chapter 3 wberever we bave v.,(O) and i1(0) we sbould interpret tbem as Vc(O +) and i1(0 rcspec.tively.) or course the values Qf;"~(O-) and i,(O-) at' t= O ~ are closoly rcl!ltcd to the correspoliding values at t =O+. Thc vatues at t =,O+ (i.e. t > 0) depend on botb the past history or the network and configuration for t > O. · Tbc terminal relationship for a capacitor can be written as

a

+),

lo(t) ·"".'" C di,(t)

or

,c(t)

=~

Hence wc bave

Yc(O-) -

~

J-..'

lc(x) dx

•-

f

--

f,(x)dx

Solution o/ Network Bqt111tlon, 115 and

...

1 e+I •.(O+) - C l,:(,x)dx o+

- ,,(O-)

+ hJ 14(.x)dx

·-

(4.,26)

For an inductor we can write the termh,al rdation u •,(t) == L d~~O)

,

or

=

iJ

v,(~)dx

1,C,O) ....

1....J

v,(x)dx

l,) -

j + i( o,C-dt)

(4.34)

The current and vo1tag«, will be. in pbase when 1

CJ>C

= ;l

Dcsignating tho resonantlr~quency as CJ>o we bave 1 · 1 (4.35) ea>o = -:-r=""- or /o= ,.;™ vLC · 2K LC Tbc admittance and susceptanee variations ate plotted in Figs. 4.14(a) and (b), roapectively. lt can be observed from the figure and cquation 4. 34 that for t,> < c.)0 tlÌe circuit is inducti'Ve and for CJ> > c.>0 it is capacitive. At low frequenmes most of the current ilows through the inductor and at higb frequencies tbrough the capacitor. At resonance ali the current goes through tho resistor and the loop formed by the inductor and capacitor bas a tocai clrcutating current (and heace the name tank circuit). Fig. 4.15 shows tbe locus of the impedance of the paralle1 circuit. Tbe ';-.

lmY

W

==

00

'

B

"'o,,--· ,.

, ,

I

(a) Fi• 4.14

I

,,

,

,,,,,. 1/wL

(b)

126 Network Theory and Filkr Design Jm Z

Fit, 4.U ocus is a circlewith radius R/2 and at thecentre at (R/2, O). lfthe ,aralie) circuit consists .of only capacitor and inductor, then at he resonance it will be ao open circuit and thus a parallel circuit 1as maximum impedance at resonance. The resooant frequency can àlso be found from the pole uro plot · ,f the admittance function. The admittanciì of the parallel circuit

s · . Y(s)

1

1

= R + sC + "iL =es•+ s/RC + 1/LC Il

rhe zeros of the function are at "1 =: -

Clt

+ j6)4

'•""' -«-jCfJ,1

,bere

« ....

l 2RC

and .,,

= VCIJI -

«1

(4.36)

~igure 4.16 shows the po/e zero p/ot ot'the admittance. Here « is

Jw

"•Cl---I ' I I

I

',"'o'

'

·ir... 4',16

Jw,

Solutlon o/ Network E(JUlltlons- 127 called the damping factor at1d t.>11 tbc damped natural fr,quency. Onco .r1 and .r1 are known «. Cùa and t.>1 can be easily dctermined. Observo that .._ is the radiai distance of the zero from the origin. Q of the circuit can be introduced througb tbe voltage response of the paral1cl circuit as sbown- in Fig. 4.17. Tbc maximum valuc of the responsc depends only on R, but tbe widtb and tbe steepness of tho response depcn,dl' on othcr elements alao. Tbe width of tbc V

IIIR. 111R1v2

-,--------=~~i....,..1.,__,..,., "'•

°'O

"'2

Fi.. 4.17

response is usually mcasured at the hai/ power point.r (or 3 dB points) and tbis width is called tbc Balulwidth of tbc circuit. Tbc bandwidth or the circuit depends on the quality factor or the circuit. Tbc quality factor Q is dcfined as Q

= =

2tr Max. energy stored at resonance Total encrgy lostÌcycle at resonancc 2 [WL 11

+ Wc]m.x Pa

wherc WL ad Wc aro.the energy stored in the inductor and tbc capacitors, and P~ the energy dissipateci per cycle by the resistor, respectively. . . Let l(t} - I., cos r.>t; ·then tbo voltate response at tésoD11Dèe is Y(t) ..;.

HCDCC

anc1

RI. COI "''·

Wc .... iCR1 P. ços' 6)1

wL -1cR• i:.••~ Pa -

P.R

i le

Tbuafinally, Q - 211

J! R' C/l P. R/1/o

==

6>0

RC

(4.37)

128 Nstwork Theoryand Flltsr De1ign From equationa 4.35 and 4.37 we havo Q ==

0t1

RC - R

Bquation 4.38 givea tho Q togetber witb 4. 36, h yiclda

or

JL .,.,. C

R fl»oL

(4.38)

the parallel reaonant circuit, and ·

·J

(4.39) ~ "'" - "°o 1 - 2Q « == 2Q' indicàting that tbc Q-factor is invenely proportional to the datnping «. From eglls. 4.34 and 4.38 it can be shown that the two ha/f-power frequencitfa 0> 1 and 1 (Fig. 4.17) are given by (Problem 4.17) · (i)J

=

e

(i)•

[J_1_+_r..,,~-,'-2~]

and

(4.40) For high Q circuita we can approximato the expressions further

1- 2~) ~ "°o ( 1 + m)

(1)1 · ~

(i).

and we see that c.., 11 bandwidth as

l'>d c.>,i,

BW

6to (

(4.4])

equation 4.40 (or 4.41) yields tbc 3 dB

= 1 -

~

=

Cllo

Q

(4.42)

showing the relatlonshii> ~tween BW, W, an4 Q, Now we are in a positÌon to draw some conclusio~a from the response curve of 4.17. If tbc circuit Q isamall, the BWi.largc, the response curve is flat and the /requency ae/ectivlty of the circuit ia poor. On the other band, a bip Q cirénit has a low bandwidth. the responsc ourve is b.igh pcaked and hencc the circuit is hilhty selective. The parallcl resonant circuit f11ndamcntàlly beha:vos Iike a barul-JHW filter (i.e. a network which transmits si,gnals witb frequencies between 1 and 1 with very little attenuation aod Rignals. at other frequencies with largo attenuation). As stated earlier, at resonance all the current ftows through the resistor. The capacitor and inductor currenta do not aft'cct the- .

Solutlon o/ Network Equation, 129

overall cam:nt input to tho circuit. Howevor, those curronts can be much larger tban tbc input current, depcading upon tbc Q of tbe circait. At resoaance the voltagc acro11 the parallel combination is given by ·v0 =- IR. The cmrents tbrougb tbe capacitor and inductor are

le

= it.>e CV0 -

jQl,

h

= ]o .... ..;re - 1 rad/sec

Q=RJ;_=S BW -

1

.

.

5 == o.s rad/sec

lH

2.5 O

2F

Pi1. 4.20

The impedanco and frequency normaliziog factors are

Y• -

Sx 10'

·

2.5 = 2xl(}I;

o..- s~ 10- Sx 10'

Hence the value of the elements of the deaormalized network uae R =- y,, R,,

a:;;

2>"< J(}lx2. S = S kJl

= ~ L,.== 0.2mH e C=.......!!--=200pF

L.

'Yn O,. Observe that the Q is tbc same as bcf'ore. denormalized circuit is

BW •

SX

The .bandwidtb of tbc

:()I, - 10' rad/sec.

PROBLEMS

•.1

The followina matricel ire 111pposed to bo file node .admittaace matrice1 ol ~ lloear time-invariant networb. Wbklb m. do you ICDlpC u oorroct, 111111 wbyl

134 Nt'twork thèory dnd Ftltrr ':D1slgn [

[

-1

+I s

[

+ 1/1 + 2, -(6 + 21.r)

1

4 -3

-~ ]· [ 4 6 -2

-3 ] 5

1·• .[ 6.r + 3/1 -4.r ]• +, :...4, Ss + 3 -(s + 2/s) ] [ 2 + s + 2/1 -2/s 1 + , + 3/1 . . 3/.r 21, + .4 -1

4

]

4.2 Pind cbo inverso t r ~ or '111 rouowina radonal functlons • 10, + (.r 5) (/) ,ÌI Ba T · ,L(il) (.r 2)(.r 4)(1 6l

.- + 'ics , · + +

,

,ur,

.rl'+31+2 1(1

+ 3)1(.r + 4) •+1

(P)

4. 3

(1

,

I·

(Y/)

Clirreots

Piad tbo elemental

Fia, P,4,3.

t> , e, +

:

+ 2)(.r1 + ,r + 1)

+ t)c, + + +

+

-

+ 1)

,a+2s+1

.

+ .2)1(11 + 4)(,' + Il + 1)

(I

and ,yottaaw,s IIÌ tbe nctfìorkl shown in

1.

J,

~:10,(,j

.r t)(.r

""-

2I

-,,(1)9 .

(a) ~

~1~,1~

2t ...

2f }2~11

(b)

~,.---

l

~~I)

(e) Pii, P, 4,3

,f

Solutlon o/ Network E(JIIIJtlon11 135 4.4

Sketch the step aod the impulse Fig. 4,4.

respoDfCI

or

the oirct,ùts showo in.

·O (a)

i

e

e

-

(b)

+

I

i

e

e

(e\

-

ti

t!

(d) Flg. P. 4.4

4,S

Find the hnpalse for the network ebòwn In Pig. P. 4.S.

~

•~ :

f

: c,f

R 1 -= 0.01 R2

=

100 L2

Fi1, P. 4.S

=

I C2

R,

=

I

136 Network Thwp and Ftlter Dutgn l.6

The impulle n,sponte

.ot a network ·il /1(I) = r 1, I ;;.,

= O,t
4k rad/aec ls 60 volts. Piod the parameten or the resonant clrcult. What 11 itl (l ?

4.15

=

4.17 Show tbat ror a para_llet rosonant cln:uit the uppcr and lower cut-off frequenc:1es are iiven by "'1

= 6>o

6>1 ,.;. 6>o

(J1 + :i. - 2~] 4

[j ~ + 4~

1

+ 2~ ]

resonant circult the Q,,factors 11· "="°•L_..!._ /L

4,18 Show tl'lat fora series

"'

R

R.Jc

U9 Deai,a a l)ll'allol resonant cltcu.it to resonate at 1 rad.fu.e with a Q ors IDlt a ffSistanco of 1 O. F'nqucncy and impedance scale it to bave an lmpedance or 10 kQ and a bandwidth of 500 Hz.

a' of tbc circuit shown In F'ia. P. 4,20 wbeD = O, (b) I== 3.Sxto-•, and (e) g = -4X10-1

,.20 l>ew-mine tbe (a) 6

+

v,

2k

4 mh

Fl1. P. 4.20

1,21 Dotormlne 6>o and Q of tbe èircuit 1hown In Fil- P, 4.21 wben (a) r - O, (b) r = 6 X 1()1 and (e) r = -99. What can you conclude? R

L

e

1000

SOmh

I :J

+

,1. PiJ. P. 4.21

+

...

Solution of Network Equationa

139

A parallel l'CIOnant circuit (R,, L and C) and a series resonant circuit (R., L ancl C) bave the same Q. Plnd the rolation bctwccn R, and R,. 4.23 Figure P. 4.23 1hows a parallel resonant clrcuit usina a pby■lcal lnductor. What i• its rOSODaDt rrequency ? Ir L .. mH; R1 =- 0.10, e·= 0.11 p.P and .R, 1 MO fuld the RIOD8Dt rrequency, impcdance at resonance, tbc maximum impedance BDd tbc frequency at whlch lt occurs and the Q of the circult. 4.22

=

Fla. P. 4.23

Solution or network.equatfons and subsequent detennination of network reaponse lnvolvcs thc ln\lCl'Slon of a matrix. Bxcept for vcry aimple· nat• works, llmited to 3 .or 4 indepcndent nodes or mcshes, the invenion or

.theac polynomial matrice■ by tona hand Js probibitive lC not lmpossibJe. Computor bas bccomo an indlspcnsabfé tool in o~inlns the respome of a network. Gaussian elimination, nwnericàl intearation, LU factorization and other nwnerical techniqucs bave become standard tools. Dlrectorl book1 (Chap. 3, 13, U and 16) lista a nwnber or ■implc c;_omputer proarams that can be implementcd oasily. Packaaod ptOll'IUDS llke ECAP, SPJCB are available for detcrminlns network responso. · Sinusoidal steady state respome i■ pnerally dlscussed in the 8nt couno on clrcuit theory. Bere it is introduccd for completeness and as a pRltide QIIW pheoomeoa. Tho concept and the measurement of tlle Q-factor ancl tllo bandwldth are fundamcntw to the desip of filten, A IOt of mitably desiped experiments are cssendal to crystalise theae concepts. A suitable set il : (1) · Parallol ~nce: varlablo capadtor (to contro! 6to) ancl varlab1' tesistance (to coatrol Q). Measorement of f'rcquency response, Q an4 bandwicltb. (2) ~ c n t (3)

or Q of an iron corod coll by ICl'fes re,onance.

Step rapome of a variable a re■onant circuit to brini out the e«. latlon betwecn the dampinl factor and Q-factor.

ne boob dtecl In Cbapter 3 may be ref'oreacod ror rurtlw inf'ormatloll ancl problema. 1. S.W. Dlnlctor: Circult Tbcc>rJ-A Q,mpatatloaal Approacb, ~ Wlley, !!>75.

5 Network Theorems

The loop and node equations completely characterize a network. On solving these equations, we can determine the current and voltage of any element of tbe network. Maµy times wc are interested in the current through and voltage across a particolar element of the network. In this case we can rcplace tbc rest of tbc network by an equivalent source and an impedance. This leads to the familiar Thevenin or Norton equivalent networks. In this chapter we study Thevenin-Norton's, Tellcgen's and other network tbeorems which are applicable to a large class of networks. The study of these theorems places in evidence many generai properties of the network response and belps us gain insight into many of tbc network problems. 5.1

Superposition Theorem

A linear system can be defined as a system wbicb satisftes the principle of superposition. The only requirement for the application of the superposition theorem is that the system or tbe network be linear. The network can be eitber time-invariant or time-varying.

Theorem S.1: At any point in a linear network the response to a number of excitations acting simultaneously is equal to tbc algeb• raie sum of tbc responses at tbat point due to eacb excitation acting alone in the network. We prove the theorem fora time-invariant network. A linear time-invariant network can be cbaracterized by a set of equations in the Laplace domain as

MX=N

where M is a matrix X and N are vectors (eh. 3). Without loss of generality, let us consider the node equations. For this case·

Network Theorem1 141 M - Y., X=- V,., N = - .4J, wbere Y,. il tbc node-admlttance matrix, v. a set of nodc voltages ((•-1) in number), À tbc lncldence matrlx of tbc network grapb .and J, tbe equivalent c:urrent soprce, i.e. Y.V,. .... - .4.J, from which (5.1) . V,. - - Y,;1 ÀJ, lf wc further considcr tbc ZSR only, tben (5.2) J, = [/.,t + J.,,.] = J.,-Y_E. wbere J., is tbc current source vcctor, E., tbc voltage source vcctor, Y, tbc edge admittance matrix and J,,. aod J.,,,. 1are the oquivalent current sourccs at thc ktb nodc due to indcpendent current sou.rces and indopcndent voltago sourccs, respectivcly, of tbc network. If 1 - [ll,1}, tbeo from eqns. 5.1 and 5.2 we bave we let

r.

.,., =

II -1

"

i-l

l•l

fJ,,. a,., U,, + J.,,,,]

wbere Y., is tbe itb node voltage and a,,1 is the (k, j )th entry of A (e is tbc number of elèmeots in tbc network). Tbc right band sido of this equation is obviously tbc sum of responso due to each

sourcc acting alone in tbe network. Hence tbc n()de voltage responses satisfy tbc superposftion theorem. tbat

Wc already know

v. = À'V. I.= Y.,V., wborc V., and I. are tbc edge voltage and current vectors, respectively. Hence tbc tbeorem. Notice tha& tbc theorem is true even if wo include the equivalent aources due to initial conditions. Wo bave omitted them bere due to notational simplicity. Theorem 5 .1 is true for any generai inl)ut. Let/tx) be a c0mplex periodic input to a linea, network. /(x) can be expanded into Fourier aeries as ·

/lx)

=~+

I-

{a. cos nx

+ b,. sin nx}

11•1

Let tho respome of tbc network to dc input

;o

be lflo and to

142 Network Theory and Filter Design sinusoidal input cos nx and sin nx be lj, (cos nx) and lj, (sin nx), respectively. Then tbe response lj,(x) to the input f(x) from superposition theorem is 00

lj,(x)

=

lji0

+~

{lj, (cos nx)

+

lji (sin nx)}

11•1

This demonstrates the usefulaess of .superpositioo tbeorem in tbc analysis of linear networks when tbe input is a complex function. 5.2 Substitution Theorem

The substltutlon cheorem allows us to replace any particular edge or element of a network by a suitably chosen independent source without changing any voltage or current in the network. This method is sometimes useful in solviog complicated networks. Consider an arbitrary edge of a network, say the kth edge. Let tbis edge be connected between the nodes a and b and the edge voltage be ek (Fig. 5. l(a)). Let us conoect a voltage source E, of

(al

(b)

(e)

f'iJ. S .1

value ek to node a (Fig. S. l(a)). Now the nodes b and e can be shorted as tbey are at the samc potcntial (Fig. S. lb)). Tbe kth element is in parallel witb a voltage source and hence ca.o be removed without aft"ecth;ig the rest of the network (Fig, S. l(c)). Tbus we bave replaced an e1ement by an indcpendent voltage source without affecting the rcmindcr of tbc network. Tbe e1ement cao also be replaced by a current source. Let the cdge current be i1: (Fig. S. 2(a)). Let us connect a current source '• between b and e (Fig. 5.2(b)), tbc valuc ofi1 being equa! to tbc edge current 11,.. Tbc current through the short-circuit is zero and hence tbc ahort~ircuit can be romovcd. Now tbc kth elcmeot is in serica witb a

Network Theorems 143 cunent source and hence tbi1 element can be removed soraras the rest of tbe network i• concemed(Fig. S..2(c)). Tbus wehave rcplaced the edge by a current source.

C;pO ), · wherc Y14 is the admittance of N°' at tbc terminala 11'. Obviously z,. = 1/Y,.. From Fig. 5. 12 il can be casily seen that

trfodon on tbe network is that it ·ahould

z.,,

Voc

= Z" I"

1111: =- Y-,. V0.,

+

z.,.

"··

V

+

t ,..

7.,.

. r ••

V

;,_ (11)

1' ~iS, S.12

Thevenin cquiv~eot is somctimes referred to as the voltage aource equi,alent and the Norton cquivalent as the · current aource equi1alent. Wc bave already used thia concept in Chaptcr 3 in formulating tbc notwork equations. Wo now demonstrate an application of Thevenin thcorem. Figure S.13(a) showsa linear time-invariant network, and wcwiah to determino the change of currcnt in tbc kth clcmcnl due to a amali change in tbc kth eloment impedance. Such problcms are desigtHlt"/ ed aa •eonsitivity studies' and play an important role In pract~ circait deeign. Lot tbe network N CO tho loft of tbc terminal kk'

ISO, Network Theory and Filter Design be replaced by its Thevenin equivalent. as ahown in Fig. ·S, 13(b). I.et us also assume that tho network is in sinusoidal ateady state. The phaaor of the currcnt ia the elemeut z,, is given by l(Z.,, Z1,;) = Voe (5. 3)

+

k

{.; i

...

N

V

z,, + Zt

v.,

"

k' (a)

+

(b)

Zt

.-

Pit, 5.13

Let the kth clement impedanc:c change by a small amount to Z1,; + t::.Z1,; and tbc current change to I' = I + AI. Then wc bave (I + Al) (Z., + Zr. + fl.Z.t) = Voc (5. 4) From eqns. 5.3 and 5,4 neglectlng the second order term Al•AZ", weget I AZ,. + Al(Z.. + Z)otO whlch reduce to /j.J

=_

lll.Z1

z.,, + z

This leads to a simple interpretation of àl. 111 is the current that would tlow in a circuit with impedance (Z., + Z,i) driven by a vohage source of lll.Z1c (Fig. 5.14).

+

z,.

fla. 5,14 ExAMPLB

5.2

· Determine tbc current througb ' the reaistor R1 of the network

Network Th,orana ISI shown in Fig. S. I S by rcplacing tbc rcat of tbc network by ili Tbevenin cquivalent. The circuit to detcrmine 11 is shown in a

R,

v,: e

i,(t)

R,

r,

av, R2

b PII, S.IS ·.

Fig. S.16 where J'"" ia tbe open circuit voltap at ob and the Thevenin cquivalent impcdancc.

z,. il

a

z,. + \

Y,.

R1 b Pia, S.16

the network. The network

(a) To flnd Y"' removò R1 and aolve witb Ra removed ia rewritten in Pia, S .17. l-l

y_.

=

Yi

=

•,e•

l.(,R1 + .H.) == Y1(1 Solving for / 1 in torma of 11 we bave a

and

v,

+

b

Pii, 5.17

+ a)

1S2 Network Theory and Filter Design 1• V

Hence

1

V.,,,.

l+a

= (1 + a}+

sC(R1

+

R1 R1 a) + sC(R1

= (1 +

+ R 1) 11

+ R2)

wbicb gives (b) To determine open R1 and the independent curtent source i1 and apply a voltage Y, as shown in Fig. S .18. As before

z.,,

V=V1

I-1 =-sC 1

The mesh equation for the / 1 mesh is V= (R 1 + R 1 )12 -aV Solving for V/I we bave z __ V_ R,+R1 •q -- T - (I + a) + sC(R1 + R 1)

+ y

+ Pi1, 5,18

(e) Thevenin equivalent voltage and impedance are given by Vob and z.,. Hence

.

i.e.

v.,,

'• = z.q + Ra t 1

= (R1

+

(R1

+ R )i 1

1

R1 Xl + sCRa) +~(1-+.,..--a~)Ra ....

5,4 TeUegen's Deorem This theorem, introduced by B.D.H. Tellegen, is extremely general and is applicable to any lumped network. The only restriction on :be application of the theorem ls that the voltage1 and currents of

Network Theortlll8

153

tbc elemeots of the network aatisfy the constraints imposed by tho Kircbhoff's laws. Al such, the nature of the elements is irrelevant. Tharem 5.4: Consider an arbitrary networkN whose graph Ghas l' nodes and e edges. Let 11 and vk represent the voltage and current of ktb edge. lf the edge voltages l'1:; k = 1, 2 ... e and edge currents i11, k = I, 2 ... e satisfy KYL and KCL respectively, then

(5.5)

Tbe theorem can be proved in two ways: (i) Without loss of generality we assume that G is connected and bas no parallel edges. Lct the nodes be .numbered I, 2, ... v

v,. 'X

e.

'·-

Il l':~

Pis, S.19

and node numbered 1 be selected as the reference node, Let e. and e, be the voltages of nodes II and ~ w.r.t. node .1 (el = 0). Let kth

edge connect node II and ~ (Fig. 5. 19). .Let the current ftow from · the node « to the nòde f!. Then vis 111 =- (e, - eis) ;,._ In façt, we can write tbis relation as Vk

11, = (el' - e~) 1,-

= - lii'%• Addlng the two equations yjeld1 . v"i1: = i[{e« - e,) I.o+(~-:- eJ i~] We can write one such equation for every edge. Su~mins over ali the edges of G we bave since i~

•

•-l.

,-1

~ 'ti1:=+ ~ · · ~ (ec- ~)ia,

In tbis equation

I '-Il •

and ~~are identically zero as theie /I

154 N,twork Th«Jry and Filter Design reprcsent KCL at nodes °' and ~. respectively. Hence the theorem. (li) The second proof is based on usiug node and chord transformations (Chap. 3). We can write the Jeft band side of eqn. 5. 5 as the product or two vectors,

•

211,,.,... v;i. ,_,

wherc v. is tbc column vcctor of tbi;; edge voltagcs and I. thc column vector of the edgc currents. From eqns. 3. 7 and 3. 9 wc bave

V.-= .4.'V., I.= B}lc Hence wc bave e

~

111ci1c

= (A•V,.)' (B}l

0)

1c-1

Hence tbe thcorcm. We now discuss some; of tbc implications of Tellegen's theorem. (i) In proving the Tellegen's Theorem only two laws wero used, namely, KCL and KVL whicb dcpend ooly on the topology ofthc network. Tbc cbaractcristics of individuai elements were not used and as such are not impoi:tant. Hence, Tellcgcn's Theorem is applicable to any network-Iioear or non•linear, active or passive, timevarying or invariant. as long as Kirchboff's laws are not violated. (ii) Since i,. is the power delivered to an edge of the .network at any time tbe theorem may be interpreted as: the sum of the power delivered to the network is zero wbicb is a re-statement of the law o/ conser,ation of energy. Hence wc conclude that Kirchhoff's laws imply law of conservation of energy. · (lii) Since the element cbaracteristics werc not used in deriving Tcllegen's Theorem, it is applicable even if tbe set of voltages and currcnts were evaluated at two difforent instanti of time, i.e .

11,.

r •

11,(1

1c-1

11.(tJ it(t1) =O

(5.6)

wherc 1). k = 1. 2, ... e and lt(t,.), k = 1. 2, ... are the edgc voltages and currents of tbc network at time t1 aod , .. respectivcly. This result is obvious as all that we nced in provini tho thoorcm is tbat l: l•fl be zero.

Network Theorema 15S (iv) We can further generalize the results of (ili). Let N1 and N 1 be two diffcrent nctworks having thc samc oricnted graph G .. Let v,. and ik, k = l, 2, •.. e be the edge voltages and currents of

N1 and

v,. and 'ì: be those of N

1•

Then it is easy to show that

I• vitJ 4(t.) = I• vk(,.) k-1

i,.(t1) = o

(S' 7)

k-1

Though Tellegen 's theorem holds good, tbc relationship expressed in eqn. 5. 7 does not directly imply conservation of energy as 111: ~ or 1,.;,. is not the energy of the edge k in either N 1 or N 1•

Ex.uiPLB 5.3 Consider a linear time-invariant RLC three-port net\\ork N shown in Fig. 5.20. Let the sinusoidal ateady state of the •·erminal

,'

/1

•

v.

Pia, S.20

variables. Y 1, j = I, 2, 3 and 11, i-' 1, 2, 3 be measured for- two sets of different voltages at tbc same frequency 6). Let the first set of measurements be representcd by V,/ and tbc secood set by

Y, f. respectively. Let the network N bave n elements. Then from Tellegen's theorem we bave a+a

~

•-1

n+•

Yk{6>)J.t(ò>) =-~ t't.(w) /11(.w)

,._1

Thia equation can be written in the expanded form as

156 Network Theory and Filter Design

I

n+a

+

Vt(6>) /k(Col)

(5. 8)

k .. 4

In the last cquation tbc first summation on botb sides of tbc equation is with respect to the terminal variables and second summation is with respcct to the elements of N. For tbc rth element of N we bave V,(oo) f,{Ct>)

= Z,(Cù)l,(w) /,(w}

and 'P,(w) 1,(61) = Z,(61) i,(oo) l,(Cù) wbere Z,() is tbc impedance of tbc rtb element at tbc frequency (,), Hcnce we bave V,(w) f.(w) = V,(Co1)l,(6>) Combining this witb eqn. 5. 8, wc bave

•

I•

V1(w)1J(6>) = ~

/•1

,-1

V1(w)I 1(1»)

wbicb cao be written as

v1 11 + v1 11 + v,t. = -P-1 11 + P-1 1, + -P,1, where tbc index (,) is dropped for convenience. Tbc last equation contains only tbc terminal variables. Tellegen's Theorem is also used in developing the sensltMty coefficienta for a network from tbc concept of adjoint networks (sce 5.6).

5.5 Reciproeit7 Theorem Reciprocity is a useful property found i~_many linear time-invariant passive networks. lntuitively reciprocity implica that tbc input and the output can be interchanged without altering the respome of a network to a prescribed input. The property of reciprocity is not only belpful in tbe analysis of networks hut also in measurement techniques. lt can be deftned as: •Tbe ratio of tbc response observed at a point to the excitation at anotber point is invariant to a cbange of position of observation • and exéitation as long as the topology of the network is unaltered'. Any network which satisfies this property is called a reciprocai network,

Network Theorems 157

Tbe property of reciprocity is satiafiod by a wide class of linear time-invariant passive networks. Ifa network consista of RLCM (resistors, inductors, capacitors and mutuai inductors) and excludes either dependent or independent sources. then the network satisfies the property of reciprocity. This can be stated in the form of a theorem. neorem 5,5: A linear tim~nvariant passive network containing only RLCM elements is reciprocai. Before proving the theorcm, we explore the implications of reciprocity a little further. Consider a linear time-invariant RLCM network with terminals 11' and 22' (Fig. S.21) whicb are acccssible for applying input and for mr..asuring the response.

I'

N

(RLOM) All ·initiaJ.

--

conditions

""O

-- 2

-

Fig, 5.21

For reciprocity three conditions of input and response are possible. (1) Let a voltage source , 1 be connected at 11' a.od the sbort circuit 11 be measured at 22'. Por the second set or measurements, let a voltage source ; 1 be applied at 22' and the short-circuit current '1 at 11' be measured. Then the reciprocity implies that

,.

-==-v,ii l'1

"

Here a combination of voltage source with zero internal impedance and sbort circuit current measurements assures that tbe same terminal conditions, i.e. short circuit, is maintaincd. (2) Let a current source 11 be applied at 11' and open-circuit voltage , 1 at 22' be measured. Then Jet a current source i. be applied at 22' and the open-circuit voltage 1 a& 11' be measured. The a.ssertion of reciprocity theorem implies that

v

.

.!!.=!.!.. il

A

i,

Here with current source of infinite impedance and open-

158 Network Theory and Filter Design

circuit the same terminal condition of open..circuit is maintained througbout tbc measuroment.

(3) Let a voltage source v1 be applied at Ir and opcn-circuit voltage v1 at 22' be measurcd. Then a current sourco of i. be applied at 22' and short circuit current at 11' be mcaaured. Tben reciprocity theorem uscrts

4

Ya ii -==,. Y1

"'

Here in ·both scts of mcasuremcnts terminal 11' is short circuit-, ed and terminal 22' is open circuited. Tbe reciprocity relationsbip as defined here ts applicable to any two pair of nodes of the network. To prove the reciprocity theorem consider a linear time-invariant passive RLCM network N shown in Fig. 5.22(a) Let the point of observation and excitation be J and k, respectively. Let a voltage

N

N

(a)

(b) Pig. S.22

sourcc be connected at the kth clement as shown in tbc figure. Lct tbc response be the current 11 through the element j. The mesh equation for the network can be written as Z,,.(s)lm(.t) = E, (5. 9) where O 1

E(s)

=

l

o

2

o

k - 1

Ek

k

o o

k

o

J

+l

Network Theorem1 159 Let I Z,,,(s) I = A(s) and the cofactor of the (k, j)th entry of Z.(s) be Ak.J(I). Then solving eqn. 5.9 by Cramer's rule wc bave I (s) 1

=

A1ci(s)E- 1 s) A(s) "'-

Hencc the ratio of the response to excitation is JJ(s) A,.,(s)

E,,(J) =

A(s)

(5.10)

Now Jet us piace a source E1 in the jth element and observe the responsc 1,., the current through the kth element (see Fig. 5.22(b)). Then in eqn. 5.10 we bave ·

I o

E(s)

=

1

o

2

o

j-1

E1

j

o

j

+

I

o The response J,. is, by Cramer's rule, given by I (s)

"

Hence

1,,(.s) E1(s)

= t:.,i(s)E (s) A(s)

=

A11c(s) A(s)

1

(5.11)

Tbe network N being Iinear time-invariant passive RLCM network, from Chap. 4, Z,,,(s) is a symmetric matrix and A11c(.r) = A1ci(_s) Tbc eqns. 5 . 10 and 5. 11 imply the theorem. Tbc theorcm can also be proved for other combination of cxcitation and of response. It ispossible tbat ccrtain active networksmay satisfy reciprocity relationship at tbeir tcrminals. But in generai active nctworks are not reciprocai. Tbe reciprocity theorem can also be proved using Tellegen'a theorem (Ptob. 5. 9).

160 Network Theory a,uJ Filter Design EXAMPLB

5. 4

Two sets of measurements are made on a linear passive resistive two-port network~ as shown in Fil, S.23(a) and (b), Ourobject is

N

N

(a)

(b)

P.ia, !I. 23

to find current through the 20 resistor. We solve this problem by finding the Norton equivalent at ob. To·fi.nd the Norton equivalent at ab o( Fig. 5.23(b) we piace a shortcircuit atab(Fig. 5.24). Now from reciprocity theorem (N is a linear passive resistive network and topology of the network is the same for Fig. S .23(a) aod Fig. 5.24)

E1' J,' i.e.

li'

=

= ~I E1' =

,;

E1 11

fo 30 -

N

3A

E' . 2

Pia, $,24

z.,,

To find for the Norton cquivalent of Fig. 5.23(b) wc short E 1 ' and measure the impedance at ab.

Hence ZMI

= !!. = 2,0 = 4 11

Finally the Norton cquivalent to determine tbc current througb 20 resistor is shown in Fig. 5.25.

Network Theorems

r--------,

I

a

I

I

161

I

I I

I

I I I

4n I

I

I

20

I

I I

I

L---------' Fig, S,25 Hence · 11'

=3

:+~•i==

b

2A

i.e. current tbrough the 20 resiistor is 2A.

5.6 Adjoint Networks Sensitivity of a network variable, current or voltage, with respcct to network parameters like R, L and C is an important consideration in the design of networks and filters. The concept of scnsitivity introduced in Section 5.3 will be c:laborated in' Chap. 12. Sensitivity can be computed by analysing the adjoint network N., of a given network N. Here adjoint network is derived as an application of Tellegen's theorem. Lct N and N., be two linear time-tune-invariant networks baving identica} linear grapbs. Let V and I be tbe voltage and the current vectors of N (Fig. 5. 26). In tb~ figure source variables are shown

V,

N

Fig. 3,26

separately and are considered as a part of tho network. Let É and J be tbe voltage and the current vectors of N .,. Applying Tellegen's tbeorem to N and N., yields

V'J=O and E'l=O or

(5.12)

162 Network Theory and Filter Design Due to the variation in the elem,ents of N let V and I change by !J.V and !J.I, rcspect_~vely. Now tbc Tellegen's theorem for N and N,1 yields (V+ !J.V)1 J-E' (I+ M) = O (S.13) From equations 5 .12 and 5. 13 tJ.V' J-E1 AI - O This is the diffcrential form of Tcllegen's theorem. This· can be written as l':

(J11 t,.Vi,-E,.M,.)

l•IOPr«I

+k••lomenq l': (J1,6.Vi,-E,.t,.1,.)..,. O

(5.14)

The elements of tbc adjoint network N, are so selectcd that the second summation in eqn. 5. 14 becomes independent of variatioos in element currents and voltages. This is illustrated by considering a resistive network N. Fora resistor V=Rl or !J. V = R t,.J + / AR Then from eqn. (5.14) we bave l': (J,.t,. Vi,-EtA],.)

.-,urcn

·

+ elemen11 l': {Jk (R,.M,. + J,.b.R,.)-E,.AJ,.} = o

or

I

{(R,.J,.-E,.)Al1i

•Jcmcnu

+ J1,l1,l:J.R,.} = aourcea I (E1,Al,.-J,.AV1c)

(5.15)

The elements of N,,, are now chòsen such that the LHS is independ• · ent or !J.l,. i.e. N, is dcfined by

R1cJ1c-E1c

=O

E,.= R1c J,., k = 1, 2, ... , n With this eqn. 5.15 rcduces to I J,, l1c b.R1c = :E (E,. b.l11 - Ì11 t.V1c) or

•l•manta

(5.16)

-

Tbc sensitivity compÒnent with respect to R,. is J 1c 11,,. By proper selcction of the sources in N and N,1 we can determine the sensitivity or any network functions. As N,. is the adjoint network, we can conclude that tbc adjoint network of a resistive network is another resistive network with identica! element values. Here we bave derived the adjoint network for a resistive network. Details of adjoint network and of sensitivity computation are discussed elsewhere (see notes).

Network Theorem

163

5.7 Maxinaam Pow• TrDlfer Theorem A problem that arises in the design of circuits is to match tbe

impedance of tbc load and tbe source so that maximum power traasfer takes piace. In tbc network shown in Fig. 5. 27 V, is tbc I

Fil- S.27

rmt value of tbc 1inusoidal voltage source at an angular frequency Z, is tbc given source impedanoe aod ZL is the passive load impedance. The problcm is to select ZL such that tbc power entering load is maximum. Tbc condition for such a power transfer is contained in the following theorem. (1)1

Theorem 5.6: The optimum load impedancc for maximum power transfer is equal to tbc complex conjugate of Z,. i.e. ZL = Z*s (5.17) Let Z, = R, jx, and ZL = RL + jxL be the source and load impedance at tho frequcncy (I). Let I be the rms current through the load. Then tht average powor delivered to tbc loacl is PL =- 111 •RL I V, since

+

- z,+zL

Iv.,.

RL (R.+Rd+(X.+XLf In this equation V,. R, and X, are known. RL and XL are cbosen to maximize P. Let RL be constant and XL varying. Then to maximize P PL

i.e.

=

164 Network Theory and Filter Design which yields XL= - X,. Under this condition the average power is

\ V, 11 Rr. Po= (R,+RJ• Now if RL is varying, to maximize P 0 we bave 8PL _ O

'òRr. .

i.e.

1

I v, I = R, =

R~-Rt (R,+RL)'

=o

which yields RL Hence the theorem. The power absorbed by tbc load is maximum when the load impedance is conjugately matched to the source impedance. Under matched conditio~ the power absorbed is

1• = Iv, 4R •

Pmo;J

(5. 18)

The average power delivered by the source is

= 1118 (R,+Rd

P,

Under matched conditions

IIl=l.!ti 2R, Heoce

P,

= Iv. 1·

(5 .19)

2R,

We may define the efficiency of the circuii as tbc ratio of thc average power delivered to the load and the average power delivered by the source, From eqns. 5. 18 and 5. 19 the eflìciency of the conjugately matched circuit is 50 per cent.

PROBLEMS

5.1

Let N be a linear passive network. sinusoidal steady state. Show that

•

~

k-1

½vk

*

lk

Suppose tbe network is in the

=o

where Vk is phasor of the edge voltage and lk is tho pbasor of the edga current and• lndicates compl~x conjugate.

Network Theorem, 165 5.2 Two sets or measurcments aro taken on a resistive network (soo Fig. P. S.2). Find V1 • (a) R 1 = 1 0, V1 = SV, 11 = 2A, V1 IV; (b) R, 10 0, V1 =6Y, 11 = 6A.

=

=

I, V,

N

Fig. P. S.2 S.3 Determine the current througb R8 and the volta,e across R 1 in Fig. P. S.3 by superposition theorem.

v, Pig. P. S.3

S.4 A set or measurements is made on a linear time•invarlant passivo net~ work, as shown in Fig. P. S.4(a). Tho network is then reconnected as shown in Pig. P. S;4(b), Find the current througb tbe S O resistor.

(a)

Hl 1;~P (b)-

FiJ. P. S,4

s.5

The terminal currcnts and voltaaos or a resiator network are shown in Fis. P. S.S(a). The network is reconnected. u shown in PIJ. P. S.5(b), If [9 o. find 11•

=

166 Network Theory and Filter De1ign

JOV

N

R

N

(a)

Plg. P. S.5

5.6 Pig. P. 5,6 shows tbe equivalent clrcuit of a common emltter transistor amplifier drlvins a resistive load Pind the Thevenin equlvalent net• work at tbe load termillàls and determino the load voltage.

'•

99

IGO k lk

+ V

10 k

Pig. P. 5.6 5,7

Two sets or measurements are made on a linear time-invariant resistive network (Pis. P. 5,7), Find tbe value of Ri•

IO

ia

= 0.3

i2

= 0.3 Js

Js

Fi1. P. S.7

5. 8 Prove Norton equivalent tbeorem. S.9

Show that a linear time-invariant passive RLC two-port network is rcciprocal by usin, Tollegen's theorem. (Show tlu1 fot a1l combiJla. tiODS or IOUtCCI at tbo porta,)

Network Theoren,s 1§7 S.10

A Jinear-lnvariant passive one-port network 11 driven by a current ,ource whosc phasor ls J 1• Show that

z

11 )

,•..,(,) -

2P.. +4/6>[E.v-Es)

l 1• 11

where J1 ls the peak amplitude or the sinusoidal Input currcnt. P 0 .,. E.v and Es are thc avcragc power disslpated, the averagc magnetic cncrgy stored and thc averagc electric encrgy stored ovcr onc period, respect~. .

NOTES AND REPERENCES.

Network Theorem exc:ept Tcll~en•a tbcorem 11 renerally cowred In the flnt courac on circuit tbeory. AppliClltion or Thevcnin-Nòrten's thoorem ~•11 bo dcmonstratcd t.y experi111 is the real frequency in radians per second. The network function is the same as the transfer function defined in sec. 4.4. Here we reserve the name transfer function for a particular network function (see sec. 4.5). We define two different sets of network fùnctions. (i) Driving point functioli

If the excitation and the response are measured at the same set of terminals, then the network function is called the driving point (DP) function. There are two DP functions; impedance and admittance.

zdp(s) = i.

cz2+f,2

cos

flt-j sin !I t ]u = 2~1 and / . H(j"')

= 1}

the frequency in Hz. Then

jt,)

= I H(t.>) I elf()

The magnitude I H(CJ>) I is

IH(jCJ>) I =

)1) w•

and the phase is ,f,(,CJ>) - - tan-1 o,

Figure 6.15 shows thc plot of the magnitude and phase as function of the radian frequency 61. These curves give the frequency response of the networ~. As can be observed from the figure the network passes lower frequency sigaals with little attenuatioa and

188 Network Theory and Filter Design

high frequency signals wìth a largo attenuation (low-pass filtor) and tbe phaso becomes progre■sively more Iagging. H(CJ>

rf,(w)

\ l/y'2

w

(a)

Fig, 6.lS

let

H(s)

=

p(s) q(s)

be any network function. Then H{j0>)

= p(j"!)

(6.9)

q(j"')

is the frequency response of the network. The frequency responso of the network is the value of the network function evaluated on the imaginary axis (real frequency axis) of the complex s-plane. The frequency response is in generai complex and hence can be ·written as t6,10) Most often the frequency responso curves are plotted with frequency (CJ> or/) represcnted on· logarithmic scale. To this end we define and

H(w)

=

20 Iog I H(j(J))

e/i( or log/ are called

Bode plots (Appendix Dj. There are other methods of represent• ing the frequency response (polar plots Nyquist plots), Figuro 6.16 shows typical Bode magnitudo plots for LP and HP networks. :M:ost of the filter speci:fications are generally given in tbc frequcncy domain, i.e. a fi.Jter is specificd by its frequency response. Indeed, tbc classification of filters is based on their frequency

Natural Frequencies and· Network Functions 189

f

'

H(w)

H(w)

in db

in db

FII- 6.16

response. From sec. 6.4 and 6.5 it is obvious that tbc timf, and frequcncy responses are correlated with the pote-zero pattern. The main concern of the approximation problem, which is a prelude to network 1ynthesis and :filter design, is to obtain the poles and r.eros of a network function which satisfy the specifled frequency or timo response. This problem will be discussed in Chap. 11. PROBLEMS 6.1

Plnd the natural rrequencies of tbe network shown in Flg, P. 6.1.

V

(a)

(b)

Fig. P. 6.1 6.2 6.3

Show that a network containins a circuit or ioductors can support a natural frequency at p 1 O. The natural frcquencies of a llncar timc-invariant network are Biven

=

by:

(a) a 1 (b) a1 (e) a1 (d) a1

= -2, '• = -3 = a1 = -2 = 2/, 11 = -2/ = -2 + J3, 11 =

-2 - /3

Glvc exprcsslons ror tbc :zero input rcspoosc in terms of real time functions,

190 6.4

Network Theory and Filter Design Calculate the driving point impedance of tbe networks shown in Fig 6.4 and plot their pote zero diagrams.

__

, _,' :0---111 ,f '] (a)

2 (e)

Fis. P. 6.4 6.S Find the voltage transfer function or tbe network shown in Flg. P., 6.5. R

~Ie

I

~

I e R

v.

l

+

o

+

KV3

Y3

V-.z

Fig. P. 6.5 6.6

The pole-zero plot of a voltage transfer function is shown in Fig. P. 6.6. The dc gaio is to be 10. Fmd tbc transfcr function.

Vm+ ••• aod tbe property immediately follows. In tbc network of Fig, 7 .19 if terminals 2 and 3 are coalesced (say, tbe new terminal is 2'), tben the 1AM of tbe resulting network is

1 Yr==

l [

5/7

2'. -5/7

2'

-5/7] 5/7

(,) Ifa terminal of the network is suppressed (say, kth terminal) theo the 1AM of the resulting network can be obtained by pivotal condensation of tbe originai 1AM with Yu as the pivot, (Tbe order of tbe new 1AM is one less ·than that of the originai

IAM.)

Let tbc new 1AM be designated as Y,', and its elements as y;1. Then we bave V YlkYkJ Y, -- .,o(7 .31) I} Ykk Supprcssion of a terminal implies that the current input at that terminal is zero. From eqn. 7. 28 we bave

I

Il

1,.

=

1-1 l~,r

y,., Vi + Ykk V,1:

220 Network Theory and Filter Design If we want to suppress the kth terminal tbeo /"

=

O. Thl1 yields

•

= - """Y1t1 Y,

. V1;

(1.32)

L,-1 Yklr. l;6k

The new 1AM is obtained by substituting for V" in tbe otber curtent equalions ofeqn. 7 .28. Consider current at tbe ith terminal

=

I,

[!,.,.,. v,] + Y11

V"

Y11r.

J-1

On substituting for Vk from eqn. 7 .32

i.e.

Il

I

~(

=~

I

YIJ -

YrtYkJ) V Ykk

I

i•l

j~k

and hence the property. In the example considered earlier the 1AM shown in eqn. 7 .29 should b~ the 1AM obtaioed after suppressiog terminal 4 or pivotal condensing the 1AM of eqn. 7 .30 about the (4, 4)th element. Hence from eqn. 7 .31 we bave (k = 4) y , IJ

=

y I/ -

y,.y41 7/2

y11 ' = l - (-l}(-I) = 1 - 2/7 = 5/7 7/2

2) - - 4/7 Ya '=0- (-l)(7/2 -

Yu Y11

,

= O-

(-1)(-f)

7/2

=-

1/7

'= 2- (-2)(-2) = 2- 8 = 6 7/2 7 7

Yaa' =0- t1)

236 Network Theory and Filter Design Thc solutioo of notworb by equicofactor matrb: was 6r&t considered by Sharp a.od Spainl. Weinberg'1 book' (Chap. 1) besides providing tbc proof for thi• property discu&ses tho 1AM in detail. Mitra's book• (Cbap. 4) consldcrs thc application of 1AM to active networks.

I.

G. Prabbavathi and V. Ramachandran: Topolo1ical Interpretatioo or Ladder Network :Formulas Proc. IEBB, Voi. 54, 1966, pp. 700-701.

2.

G.B. Shape and B. 'spain: On the Sollltion of Networks by Mcaos of Bquicofactor Matrix, IRE Trans. on Circuit Tbeory, cr-7, 1960, pp. 230-239.

3. 4.

L. Weinber1: Network Analysis and 8ynthesis, McGraw-Hill, 1962. S.K. Mitra: Analysls and Syntbll:la or Lii;u,ar Active Network■, John Wilcy, 1969.

8 State Equations

Tbe network equations as discussed in Chap. 4 are ali Laplace or frequency-domain equations. The analysis of the network through tbese equations involves the solutions of algebraic equations and then inverse Laplace transforming the solution to obtain the timedomain functions. As opposed to this, one can write tbe equations in the time-domain itself and solve for the network variable directly in the time-domain. To this end network equations are written in the form of a set of tìrst order differential equations. Such equations are called state equations. The concept of state and of state eguations form ao importaot part òf the study of optimal contro! systems. 8.1 Concept of State

As discussed io Chap. 4 response of a network consists of two parts: zero-input and the zero-state responses. Response to a given input depends on the zero-input response. Once the latter is known the overall response is uniquely determined. The zero-input response in an RLC network is completely determined once the initial inductor currents and capacitor voltages are known. Hence, we cali the initial capacitor voltages and inductor currents (inìtial conditions} as the initial states of the system. lndeed, the knowledge of capacitar voltages and inductor currcnis, at a given time of a giveo network is suffi.cient to calculateany of the network variables (currents and voltages) at that particular time. Heoce we call tbc capacitor voltages and the inductor currents at a specifled time, as the state variables of tbe network. We formally deftne the state of a network as a set of real or complex quantities that satisfy the foUowiog conditions: (a) the state at a~y time t 1, and tbc inputs from t1 to t (t > t1)

238 Network Theory and Filter De,1ign uniquely determine tbe state at time t. (b) tbc state at time t and tbe inputs at time t dctermine uniquely tbc value at time t of any network variable. For example. consider tbe network sbown in Fig. 8. J. lf tbe capacitor voltage is known at time t tben we can roplace it by a known voltage source vc(t) (recali Substitution Theorcm-sec. 5.2)

A V(f)

B

• Fig. 8.1

as shown in Fig. 8.2. Now tbc voltages across and the currents through tbe resiators can be found by solvìog a set of algebraic

A

+ v(t)

B

Fìg. 8.2

equations (say, mesh equations). Hence v.(t) is tbe state of the network. States can also be considered as variables wbich carry sufiicient information about the history of systems. Obviously, a purely resistive network bas no states at all. Only a reactive network or a network with energy storage elemcnts has states.

8.2 State Eqoations For the network shown in Fig. 8. 1 we can write a differential equation goveming tbc voltage across the capacitor. To tbis end

State Equations 239 0,6 ,4

0.8v(f)

B Pig. 8.3

we replace the network to the left of AB by its Thevenin's equi• valent (Fig. 8 . 3 ). The mesh equation of this network, in timedomain, yield · dv,,(t)

0.6dt dv., dt

or

+

v.,(t)

=

O.Sv(t)

4 = - -35 v,,(t) +· -v(t) 3

This equation is caUed the state equation of the network. Consider another example, the network of Fig. 8. 4. Here we bave two energy storage elcm.ents. We can write two first-order differential equations, one for the current through the inductor and

Pig. 8.4

the other for tbe voltage across the capacitor, as

dv. dt

=

l

(

(l/3) v-v.

)

.

+ IL

and or io matrix form (8.1)

240 Network Theoryand Fi/ter Design (Here we bave written, for Y(t). This simpliftes notation.) Let i, and i, be the outputs. Then we bave

[ ~] = [

=: :][;: ]+ [ : ]v

(8.2)

Equations 8. I and 8. 2 are called thoatate and the output equations of the network, respectively. The state equationa of a linear time-invariant network can be written as dX(t) dt

=

AX(t)

+ BU(t)

(8.3)

and the output equaUons

= CX(t) + DU(t)

(8.4) where X is the state vector, U(t) the input vector and Y(t) tbe output vector. A, B, C aod Dare mati"ices of appropriate dimensions. lf tho network is time-varying, then the entries of these matrices are functions of time. An advantage of the state equations is tbat sÌQl.ilar equations ~n be writton even for a nonlinear network where the· conventional network function technique is in generai not applicable. In writing eqn. 8.3 and 8.4 we bave made a tacit assumption in that tbc network does not bave any circuit (cut-set) of capacitors (inductors), or capacitor (inductor)-voltage (current) sources. lf it were not the case, the state and output equations take the generai form (bere the time t is not written to simplify notation) dX dIJ -=AX+ BU+ Edt dt (8.S) Y(t)

Y

EXAMPLB

=

CX

+ DV + F dU dt

8.1

Wc are rcquired to find tbc state equations ofthenetwork shown in Fig. 8.5. Evidently,

Vc2

Q

b

•2-

I,

+

+

v,,

JI

2

-•e,

Fis- S.5

""

8Dd bence there àre onty two 8te varialdos for tbe network (lince tbere is a circu.it of capaciton).. Sulllllling currents at nodea a and b, , ·r., at

a:

';;1

+ 2";;9 + (v,. -

2d, T, -

at b:

2d-,, T

+ (11._ -

11) - O ,)

=O

But

Heace we bave

The state equations are

[

~]· = dvc,

[-3/4 1/4] [

dt

-1/8 -1/8

1

Yc

]

Yea

+[

3/4] .,1/8

Let the output be i,; thco I,

i.e.

= (Y -

101)

+ (• -

i, - [ -2 1} [

fc 1 Yc,

]

v,J

+ 2• -

In aubtoquent sectiont w formalbe the dcrintioo

equatiou.

242 · Network 'I'Mory and Ftlter Dellp

1.3 Fonaalatloa of State &iatlea . éonsider the network shown in Fig. 8,6. Tbis network lias two

R

V

Fla,1.,

state variables, •ci (Xi). ,,.. (l".l, .

.

we only bave to see that(with

.

r-

To write tbc state eq11ationa dX.,

dt)

= current tbrough the capacitor -= X1 - i; L. i, .., volC.ge across the iaductor = " - X1 - R1 X1 C1 i,

The state equations are

[::J-[ =ic· -hl[J+L:J·

Bere we bave writtea the state equation by a mere inspection of the network. Obviously, this mctbod of writlag tbese equations (Jnspection Method) is limited to very simplc networks. more complicated networks we have to resort to more formai methoda. writiag the state equationsThere are two main methods Equivalent Source Mctbod and Oraph Tbeotetic Metbod.

For

or

8.3.1

EQuIVALBMT SoUI.CE Min'HOD

la this method capacitors and inductors are replaced by equivalent voltage or carrent sourccs and tbc c!tmmt tbrough the capacitors. and the voltage acrosa tbc inductors are found by application of Superposition theorem. if tbe circuit d* not contaio any degenera.te circuita (all capacito-: or capacitor-voltaai, source circuita) or degenerate cut-set (all inductor or iadoctor.current · sources èut-scts), thcà voJtages across aU the capacitors and currents through ali the induetors

S t à u ~ 243

constitute the ata.tes of the network. # The capacitora ilre replaced by voltage sourcea (P.,} and the inductors by c:urrent sources (li.). Thuì we obtain a resistive network witb voltage and current

soun:ea. _Tbe current tbrough thc capiù,,iton (Cd~) and tbc voltago across tbo inductors

(L) are

round by the application of

Suporposition tbeorcm to tbis resistive network. Tbc following uamples illustrate tbc metbod. BXAJIPLB8.2 Tbe network show:o in·Fig. 8. 7(a) does not. containany degenerate cut-sds or circuits. ·Wc are required te> find the state equations. Figure 8. 7(b) sbows tbe equi,a~i,t aource network obtajncd by replàcing tbc capacitors by voltàp sourccs ud tbc iaductor by

• (:i)

(b)

Fig. 8. 7

a current sourcc. This network is now to be analysed "by super. poaition. To tbis end wc rcmove ali sourecs (■hort voltagc spurces .and open current 1ourccs} cxcept one, and analyse the resultiµg network for the currcnt through the equivalent voltage sources tbc voltage across tbc equivalenc currcnt sources. Tbc final state equatioos are obtained bY combining tbe results of the analysis of

and

tbe individuai network&. Let us analyse tbe network with only the voltage source , in tbc network. The resultlog network is sbown in Fig. 8.8(a) lt is ca,ily 1eeo tbat tbe vollage &cross the iaductor due to tbc voltage

tource II is, itself, and tbe currents tbrough tbc capacitors are also "· Figure 8.8(b)1 (e) aod {d)· abow tbc aaalyses with respect to

244 N~twork

7:-or, and Filter Design o ......

..... V

..... y

+

x,f

x,

y

♦

(b)

-

2Kz

(d)

(e) Fis, 8.8

tbc othor sourca. From Fig. 8. 7 and Fig. 8.8 we bave the stato cquatioos aa

. 1

X [

~I

X3 EXAMPLB

j- [-2 -2

1

1 ] [ X1 ] -2 -2 0 X1 -1

O O

X1

+

r

1 ] 1

L

1

8.3

tn Fig. 8.9(a) the required output is the source currcnt and the states are X1 , X1 and X1• Figure 8. 9(b) is the equivalcnt source

(bi

(a)

Fis- 8.!>

Siate E(Jlllllu,n., :l4S

-x,

-o

~ .l.,Ì

-o

(a)

.,_ Xi

... o

1

~

X,•

2X1

tx.

4-

x,

fo

(b)

Xi

'

J

Xi

...

ix,

Xi

t

XJ

(d)

Fig. 8.10

network. Let us assume tbat all tbe elements are unity valued. Figure 8.10 shows the four netwotks neCCflsary for the applicatlon of superposition theorem. The state and output equations are

[!]=[ =: -: :][ ::J+ [ ;} 1, = 1- 1-1

1

X1]

L

x,

O]\ x, + ,

If the network contains degenerate circuits or cut-sets. the method is not directly applicablc. In such cases, we bave to fint select the state variables. The voltage of a capacitor is selected as a state, only if it does not form a circuit with a voltage source or capacitors whose voltages are already selected as state variabfes. The current of an inductor is selected as a state variablc, only if it does not form a cut•set with a current source or inductors whose currents are already selected as state variables. The capacitori and tbe inductors whoac voltages and currents, respectively, are not aelected a.s state variables are called excess elèments. Tbe

246 Network Theory and Filter Design voltage across an cmccss capacitor is tbe aJgebraic sum of the state variable voltages and the voltage sources. Similarly, tbe curreot in ao el[cess inductor is tbe·aJgebraic 1um of state variable currents and tho current sources. In obtaining tbe equivalent source network, tbè excess capacitors are replaced by curreat sources and excess inductors by voltage sources. The current through the Cl[CCSS capacitor is expressed in terms of state variable voltagcs and voltage sources. Similarly, the voltage across an exces1 inductor is expressed in tcrms of state variable currents and current soun:os. On eliminating these variables, we can obtain the flnal stato equations. Examples illustrate the state-equations or such networks. ExAMJ>LI!

8.4

The network shown in Fis. 8. ll(a) has a circuit or capacitors. Two of these capacitor voltages can be selected as state-variables. Here wc choose 111 and 111 as state variables. Figure 8.1 l{b) sbowa

{b)

Pig, 8.11

the equivalent source network for this selection. Fig. 8.12 shows1 tbe four networks we require 'for tbc application of Superposition tbeorem. From Fig. 8.12 we bave

'• - ,. + 2• 2ii. = "1 - '• + ,. -11 .,, = '• - '• ;1 = -

But

and

2,1 +

,. - i,, = 2 c'•1 -

i,.] Writing tbese equatioaa io matrix form we bave

State Equations 241

and

Combining tbc last two equations we get [ i,1 ]

v.

~-

=

r-2 1](fl, + [i.i" l] 1

]

1/2 -1/2

[2 -2]

1/~

[;v., ]+ [ 2]

l'

l

-1/2

\

[_: -:m: J-[; :1,M :: l L:12l· +

or

[;.'1•.} = [

-3/4

1/4 ].:·[

-1/8 -1/8

"1

]

"•

+[

3/4] 1/8

11

which is tbe required state eq~tion far the network.

11V-

-v, J2v,

(a)

(h)

o..-

o--.

o' +

/1

t

_,3_ /J~

(·!\

(e)

Fig. 8.12

EXAMPLB 8.5

In tbe network sbown io Fig. 8.13(a) tbere is an excess inductor. Let L. be tbìs excess inductor. Figure 8.13(b) shows tbe equivalcnt source network for tbis selection. Figure 8. 14 sbows the four networkl'required for the application of soperposition theorem.

•

•

241 Network Theory and Fllter Dulp 1

Fig. 8.13 . . . . 1,

2/

+

Fi1 8.14

Hence we bave

[ ; ]-[ -: _: ][ : f[ -: }+[ _: ], But

Combiniog the two cquations we bave

Stale Equations 249 i.e.

[ _: -: J[ ; or

H-:- _: J[ ~. H_: }

[; ]-[-: _::: J[ : ]+ [

: }

i$ t~ required state equatio'n. In all the preèeding examples the state ·equations are of the form

X= AX+ BU If the network has. circuits of capacitors and voltage source and/or cut-sets of inductors and current sources, tb.cn, and only then, tbc state equations take tbc form in eqn. 8. 5.

8.3'.2

OltAPHTHl!OllBTIC METHOD

Tbc method of equivalent sources is conceptually and algebraicatly a simple procedure for writin~ the state eqtià.tions. If the resistive network is a compliçatcd one, then the analysis of the individuai network becomes a niajor problem. Also, the method is not strictly · an algorithmic one. Tbc state equations can also be formulated by using graph tbeoretic concepts. But tbc method is algebraically quite col:Tiplicated. Many of tbc Jatest computer oriented analysis programs usè this method for writing the state equations. Wc outline this method bere. Tbc formulation of network equations by graph theoretic methods starts with tbc selection of a tree. To formulate the state equations a proper or modified proper tree is sc)ec;ted. A proper tree is a .tree which includes ali voltago sources and capacitors and excludes ali current sourcos and ioductors. A propor trce, if it exisb, implie1 that the network does not bave any excess elements. A modifted proper tree includes ali voltaie sourcea. as many as possible capacitors and excludes ali current sources and as many as possible inductors. Obviously every network has at lea■t one modified proper tree. The selection of tree as described exciudes

2SO Network Thepry and Filter Design the poasibility of /-circuita of capacitors having resiston and/or inductors aod /-cut-setl of inductors baving resistors and/or ca.pacitors. The notation to be followed in the development is that V0 represents voltage source vector, 11 the current source vcctor v,, and J,J are tbc element voltage and current vectors respcctively. The first subscript i refers to the tree or chord (b or e, respectively) and thc second subscript j to the element type (r, / and e). For cxample, Vb: is the branch-capacitor voltage vector and lc1 is tbc chord inductor current ve REFERENCES State-variable approach has only round limitcd applic:ation in network analyses and synthesis. An excellent exposition or state variable approach to circuit theoey is round in Rohrer's bookI. Application or state equations to time-varying and no&-Jinear networks are diecussed in Balbanian and Bickart's book• (Chap. 4). This book also discusses analytical and numerica! techniques or evaluating state tral'l,ition matrix {Chap. 4 and S). Qraph theorctic method simllar to the eme introduced here is appJlcable to the rormulation of state equation■ of non.electric network& also•. State cquations and state variablc approech finds·ntensive use in modero contrai systems and thc details can be round elsewhere•,1.

2.

R.A. Rohrer: Circuit Theory-An latrodaction to State--vaiiable Approach, McGraw-Hill, 1970. N. Balbanian and T. Bickart: Electrical Network Theory, Jobn Wiley,

3.

1969. H.B. Koenig, Y. Totàd and JLK. Kesavan: Analysis of Discrete Physical

l.

4.

S.

Systems, McGraw-HIII. 1967. B.C. Kuo: Automatic Control Systems, Prentice-Hall, 197S. LA. Za~ and C.A. Desoer: Linear S)'ltem Theory, McGraw-Hill, 1963,

9 Elements of Network· Synthesis

Problema in network tbeory fati into two categories-anatysis and synthesis. Analysis deals with the problem of determining the response wben a prescribed input is fed to a given ftetwòrlc. Tbc problem of syntbesis in cò1.1trast deals with thè design and fabrication of a network that satisfies tbc prescribed resp61lse speci• fication. The formcr i8 c:omparatively· simplc a11d ·ìl a solution exists it is generally onique. On the otber band tbe syntbesis pttiblem is not unique iii the tense thàt we be able to ftnd more than one network which provides tbe prescribtd :: 'respoosc. Indeed the synthesis prò'blem may not bave any solution at ali, A~alysis is a prelude to syntbesis and wè bave devoted Chap. 3-8 to various metbods of analysis of a linear passive network. In this chapter wc consider some aspects of pauive network: synthesis. Wc aro mainly concerncd with the syntbesis of driving point immittances and transfer functions.

may

9,1 P-.ftlYe Real Fuction A fun~mental theorem, in passive network synthesis states ihat "'a ratioJ,1&1 function of s is the driving point immit1-Jlçe of a passive network if, and only if, it is positive real." · A rational function F(1) is a po1lti've real function (pd) if it satisfics tbe c:onstraints F(,) real for real 1

Re[F(•)J ;.i: O for Re 1 ;.i: O If F(s) is a prf then the reciprocai of F(.r) (i.e, 1/F(s)) is also a prf. lf F(s) is expressed as F(s) = p(s) (9.1) q(s)

where P(s) and q(s) are polynomiaJ; in s, tben theac polyoomials

258 Network Theory and Filter De8ign

I

V

Fig. P. 8;7

8.8 Dçtermine lhe state equations of d1e network shown in Fig. P. 8,8 l)y graph theoretic rnethods.

V

Fig. P. 8~8

NOTES ANI> REFERENCF.8

State-variable approach has only round Iimlted application in network analyscs and synthesis. An exc:ellent exposition of state variable approach to circuit theory is round in Rohrer's bookl. Applicati on of state equa1ion1 to time-varying and non--linear networks are discussed in Balbanian and Bickart's bookl (Chap. 4), This book also discusses analytical and numerica! techniques of evaluatins state tra11,ition matrix (Chap. 4 and S). (iraph theoretic method simllar to the one introduced here is appJicablt to the formulation of state equation1 of non-electric nctworks alao•. State equations and state variable approach finds'ntensive utc io modcrn contro! syslems and the details can be fouod clsewhere•.•. l. R.A. Rohrer: Circ:uit Theory-An latroduction to Stat~iiable Approach, McGraw-Hill, 1970. 2. N. Balbanian and T. Bickart: Electrical Network Theory, Jobn Wiley, 1969. 3. H.E. Koenis, Y. Tokàd and H.K.. Kesavan: Analysis of Discrete Physical Systems, McGraw-Hill. 1%7. 4. B.C. Kuo: Automatic Control Systems, Prentice-Hall, 1975. S, L.A. Zad,b and C.A. Deaoer: Linar System Tbcory, McGraw-Hill, 1963.

9 Elements of Network Syothesis

Problems in network theory fall into two categories-analysis and synthesis. Analysis deals with the problem of determining the response when a prescribed input is fed to a giveo network. The problcm of synthesis in cò~trast deals with the design and fabrication of a network that satisfies the prescribed respon"se specification. The former is comparatively simple atid ifa solution exìsts it is gencrally. rinique. On tbe other band tbe synthesis pt6btem is not unique iii tbe sense that we may be able to ffnd more than one network which provides tbc prescribed · 'response. Jndeed the synthesis pro'blem may not bave any solution at ali, Analysis is a prelude to synthesis and wè bave devoted Chap. 3-8 to various methods of analysis ot' a Iinear passive network. In this chapter we consider some aspects of panive network synthesis. Wc are mainly concerncd with the synthesis of driving point immittances and transfer functions.

9.1 P()Sltlve Real Fvaetion A fund/lmental theorcm, in passive network synthcsis states that "a ratio~l function of s is the driving point immittance of a passive network if, and. only if, it is positive real." · A rational function F(s) is a posiUve real function (prf) if it satisffes the constraints F(s) real for real s Re[F(s)] ;;i: O for Re , ;;i: O If F(s) is a prf then the reciprocai of F(s) (i.e. 1/F(s)) is also a prf. lf F(s) is expressed as F(s) = p(s) (9.1) q(s) where P(s) and q(s) are polynomials in s, tben thesc polyoomials

260 Network Theory and Filter De.rlgn ·r:nuse have real coefflcieots. This ons11res that F(.r) is real wheo .r = a. p and q being real for .r ""' a, compie,: zeros must appear in conjugate paio. We fint considcr methods of testing a function F(•) ror pos_itive reality. . · We 6te se'ftral tbeorema, wtdlout &>roof (soe notes) and develop tbc test for pr nature of Et•)·

.,

'l1IINnllll 9.1: A prC F(.r) cannot bave aòy poles and zeros in tho riaht halt' (RH) .r-plane. Any zeros of p(.r) and q(_.r) on thej(l)-uis muat bo Jimplo•

.Jdti

ollary-.. 1_,.1 _: T_be resi_d.ues of F{.r)_ aad lÌF(.r) at tho.simple polos . .theJ.alis must be real and positive. . Tbe nece.ssary .coodition aiven in theoredl 9. 1 can be restated ~ "lhe n111De~~r ,-nel denominator polynoQlials of a prf must be Hurwll:t," A polynomialft.r) is a .rtrictly Hurwitz polynomlal if aJl ita a:eroa are in let't half(LH) .r-pla~. It is Hurwitz if its zoros ·are ila LH._plane or slmple on theJw-axi"

The ualyticity or F(.r) and 1/F(.r) or the Hurwitz nature òt' p(.r)

and q(a) can be test.cd in a numbei of waya. One of the methods is to com•rudt the Routh-Hurwitz array for the rcquired polynomiaL .

Let

= a,..r" + a_1,.-1 + a.......-a + ... + a1.r + a0 Fortf'(s) to be Hurwitz ali the coeftìcients a,, I = O, n, must be of the P(.r)

sa.mc sigo. No coeftìcient should be misaing fora strictly Hurwitz polynomiaL For a polynomial wbich has ali its zeros on the fa,>alternate_ coefticients must be zero, i.o. JJ(..r) ia either eTCD or odd · polyJiomial. The R011th•Hurwitz array i1 constructed as followi:

m

&"

a,. a,...s

a._, a,._•

a,...,

r•

b,. e,.

b-•

b,.....

,-1

.,.-a

• •

.r1 8'

clf-1

a...,

wbore

-· 1a,,

_,I ••

a.-.1

.

a-io...,. a__ ---..

•

b

-1=

a.,..

I

~ 0...1

1.._,.;..

--1 a..-·~

.,

•~ I

-

"-1. •--

b--:J• e-i-= b... • "-• b•

.,.,-..

b. ,

e■ = ~~~

.

Tbis array is a triangular oae aod tor a atrictly Hurwia polynomial there ahould not be any ebaàge of 111~ in the ~ a t s of tbe :Brst column, i.e. ~... "'-• 6-, A vanishing row in the array indicatcl that the polynomial canot be atrictly Hurwitz. In tbia case, the vaoishing row ia replaced by 1111 auxiliary polyaomial. Let the kth row; the row correspondiq be the row prior to the vanishlng row. Form the polynomial

c.,....

to ~,

p,l.s)

I

=r

ti-,A

+ cxa-.r• +~:,+

•1

where ar.,, are the coefftcionts of the kth row. Tbe polynomial p,J/J) wiU alwaya be an even polynomial. A vanishiog row shows the presence equal a:nd apposite roots ancl thé order of tbc polynomial p&{a) indicateflM mtmber of ~ual bllt oppoaite roots. An awdliary polynomial i• defined as

of

p',.= dp:1)

=

kc.t~-1

+ ( k - ~ +~.. . -

Tbe coe81cieats or (k + 1)th row are replaced by the c:oefli~cntl or tbe amiliary polyaomial and tlfe array il aow oontia• 1f thcre • a chaaac of sian in tlle :flrst column then tbc polynomial p(_s) la not Hurwitz. lf there are no chanaes of sip in the a,.. columa then the polynomial pa(.1) is testecl ror its zeroa aad if tbc :r.eros are simple on thc j.,..uis tben p(•) is Hanritz.

...

2'2 Network Tl,eory and Filler Deaign EXAMPLI! 9,1

p(s) = st

Let

+ 16.tl + 86.s1 + 176s + 105 l 16

a' s1

r

15

s1

153.6

s'I

105

~6 176 10S

10S

Hence p(s) is strictly Hurwita. Tbe actual factorised form of P(s) is p(s) = (s + l )(s + 2)(.r + S)(s + 7) (The factors cannot be extracted from Routh's array. Polynomial factorisation methods are neccssary for cxtracting tbe roots.) ExAMPLB

9,2

.P(s)

= 1• + 12.s' + 4Ssl + 60s1 + s• a' .,.

1 12

45 60

40···

40

48

48

44.t

+ 48

44 · 48

p,.(s)

O (vanishing row)

= 4881 + 48

p,.(s)=96s

.................................................. -. ............... s1

s8

48 96 48

48

p11(s) = 48(.s1+ l) The roots are .s= ±il

Hence the polynomiat is Hurwitz. The factored form of P(s) is

p(s)=(s+ 2) (s+ 4)(s+ 6)(.tl+ 1) A second method of testing for the Hurwitz polynomial is by conti,medfraction expansion. Let.P(.r) = m(s)+n(s) wherem and n are the even and odd parta of p(s). From a rational function p 1(s) given by P1(I)

=

m(.s) n(s)

(9.2)

The rational function of (9. 2) is cxpressed in continued fractions

as

Elements of Network Synthesis 263 P1(s)

I = ~1s + --------1- - - - ~.s+ ____ l__ _

~s+---~4s+ ........ . where ~1 may be zero. A necessary and sufficicnt condition for thc Hurwitz nature of JJ(,s) is that the cocfficicnts ~, be real and positive. EXAMPLE 9, 3

Le p(s) = s• + l 6s3 + 86s1 Then m{s) = s• + 86s1 + 105 and n(s) = 16s3 + 176s Pi(S) =

nm

=

+ 176s + 105 s5 -!- 86s1 + 105 16,a + 176s

75s/ 153.6

+

1/1.46-s

Hence p(s) is Hurwitz. Sometimes the continued fraction cxpansioo may end prematurely. For example coosider Bere

and Then

p(s) = s' + 12s' + 45s3 + 60s' + 44s m(s) = 12.s' + 60s3 + 48 n(s) = s 5 + 45.s' + 44s m(s) 12.t' + 60~ + 48 Pi(s) = n(s) = r> + 4Ss 3 + 44•

+ 48

1 s

IT +

l -

31 10

I

+ 5.r(s8+I)/6(s' + I) of tbc cancellation of factor (s1 + J) the expansion

Because eods prematurely. A premature termination indicates common factors between in and 11. These common factors form an even polynomial and has either roots on the jw-axis or roots in quadrantal symmetry (equal and opposite real roots are ruled out as this would result in a negative coefficient in tbc originai polynomial). In this particuJar case the common factor is due to a pair of jw-axis roots at

264 Network TIMory. ond Fllt,r De,Jgn

± Jl.

Hence tbc polyaomial il Hurwitz. In this eumplc

+

m(.r) - 12 (al + 1) (.r' 4) z(s) = .r (al 1) (al + 44) and P(.I) = (.rl + I) (,' + 12.r1 + 448 48) = (a1 1)(1 2) (.J + 4)(1 6) ~, Tbc prf P{.r), in addition to havins.P(.J) and q(1) as Hunritz prolynomial must satisfy tbc Cor. 9 .1. The conditions of the Thoorem and Corollary 9. 1 are necessary conditions for the positive realness of F(.r). conditions of tlle tbcorem imply that F(a) and its reciproca) are analytic io R.H s-ptaoe. With this we can reàtatc tho condition for a pd In tbc form of a theorem.

+

+

+

+

+

no

Tlleorem 9.2: A rational fuoctioo F(s) witb real coefflcients is a prf if, and only if, F(.r) is analytic in RH .r-plane with at ·molt simple polea on tbc jw-axis. The residues at any poles on the Ja,-axis are real and pasitive. Re [F(Jc.i)] ·~ O for ali (Il, The last conqition can be indircctly testecl •. .·

'

.

.

Let

F(1)

= mi

~

t "i ~

where m1• m1 , n1 and n1 are the even and odd parti of tbe numerator and denominator, respectively. lt can be easily showo that on tbc )lo>•axis the Re F(jw) ia tbe even part of F(jw}; Tbc even part of F(}6l) is

Bv. F(}fil) = On the /,.,.axis both ml and -

"'i"!• - 11in, L,,.. m -nl

nl

1

being positive, tbc R.o F{fa,,) ;;;ii O

for ali 61 if, and only if, A (c.i1) = (m1m1 - n1n_)I,..,. ~ Ofor ali (Il (9 .3) lf ali the coefficients of À(x) wherc x = fl>1 aro positive then A(x) is positive forali vaJues of x between O and o:,. .When all the coe11icicnts of A(x) are not positive wc, can re~rt to 'Siu,m•., teat. From A(x) a se& of functions (A. 0 , ..4 10 À 1 , ••• .4;;) are formed such tbat ,4 0 = À(x}, .4 1(.,) = dA.(x)/dx, The subsequent functions are formcd

as foUows.

Tbc ratio A,-.• + 30r.l1 + 24

Now

À 1 (.x)

Thcn

= ~ + &xl- 22x' + 30x + 24

A1(X)

="!i~~)=

8x' + 18x' - 44x + 30

A0(x) 2.x' + 6x' - 22xl + 30x + 24 A1(.x) = 8r + 18xl-44x + 30

= (1 x + .!.)- 11sx■ - 246x- t47 :ij

16

8A 1(X}

266 Network Theory a,ul Filter Design

Aa(x) = 115x1 - 246x - 147 At-axis is 7.ero. Tbc poles and zcros being on the imaginary axis tbe polynomials p(a) aod q(.r), wbere F(.r) = p(s))' must be either even or odd.

qls

Since tbc behaviour of the immittance function of an LC network must approach that of an inductor or a capacitor as s ➔ O or I ➔ IX), . WO. conclude that F(s) must bave a zero or a polc at the origin and at s = co. Ftirther it can be shown that the poJes and tbc zeros must alternate on the imaginary axis to satisfy tho condition that thè residues at tbe jw-axis poles be positive real. This forccs the slope of a reactance function to be strictly positive for all frequencies. Figure 9 .1 shows the plot of a reactance-function as a f'unction of c.'I, The preceding discussiom are summarized in tbe following

theorem.

·

1'eorea 9.4: Necessary and sufflcien1! conditions for a real rational functioo F(8) to be the driving point immittance of an LCnetwork are: (a) poles and zeros are simple and interlacing on tbe / ,raxis (b) at the origin and at iofinity. F(s) must bave a poJe or a zero. The propcrties of the LC driving point function are (many of tbese properties can be derived from tbc energy function concept):

270 Network Theory and Filter Design F(

/W)

il

I I

;:

I

I

:7··· -·· -:__. ,.__ ,_ _,_ ~ !•

I

I

I

II

I

I

Pig. CJ.1

(i) poles and uros are simple on the imaginary axis. (ii) poles and zeros are interlacing. (ili) at the origin tbere is a pole or a zero. (iv) at infinity therc is a pote or a zero. (v) Re F(j6>) = O forali Cil, (vi) the slope of F(jr.>) is strictly positive. (vii) F(s) is a ratio of even to odd polynomial or vice-versa. (viii) the residues al! the imaginary axis poles are positive and real. Tbere are a number of methods of reaJizing a reactance network. We only consider the four basic forms-Foster I, Foster II, Cauer I and Cauer II. The Foster forms are obtained by a partial fraction expansion of F(s) and theCauer forms by a continued fraction expansion of F(s). lf we combine the terms corresponding to a pair of conjugatc poles, tbc partial fraction expansion of F(s) can be writtcn as

F(s)

= p(s) = ~ q(s)

s

+"'

2k,s

~sa+

,-1

c,,f

+k

co

s

(9. 10)

where kfll k1 and krr, are the residues of F(s) at the poles at the origin, atjti>1 and at infinitv, respectively. These residucs are given by

= k1 =

k,

I .r-o

s F(s)

(s

1 wf) F(s) + Jt»1) F(s) I ,--1..1 = (s + 2s

.

I

··---:

(9.11)

Eltments of Network Synthesis 271 koo

=

F(s)I s

-

lf there is a zero at the origin (at infinity) then k 0 = O (k 00 = O). 1f F(s) in eqn. 9 .10 is an impedance function then it is the series connection of the elemental impedances (Fig. 9. 2) and the rcalized

-IMPEDANCE FlJNCTIOJ\i

NETWORK

..

c,,

ko

-,f

D

~1---o • I Co=,~

"•

Li

s2

~

2k 1 s +w;2

I 2k Ci =2k,i L, = W; 7 l,.,

~

K-.. s

L.,, = koo Fig. 9.2

network is shown in Fig. 9. 3. If F(s) has a zero at the origin then C0 will be missing and if it has a zero at infinity L 00 will be missing. Tbc structure shown in Fig. 9.3 is called Foster form I.

~v--or Li.

L,

e,

F(s)

=

.

L·

Z(s)

Fig. 9.3

272 Network Theory and Fllter Deaign

lf F(s) in eqn. 9. 1O is an admittance function. then it is the parallel combination of the elernental admittances (Fig. 9.4) and tbc realization is shown in Fig. 9. S. lf F(,) basa uro at tbe ADMITTANCE FUNCTION

I:~•·

~ s

1-~

2k1 s s2

NETWORK

2k1

+ ,,,12

t

e- 2k,

a

i - w;2

•

lc I -.. k - -

K.s

D

•

Pig. 9.4

origin, L 0 will be mtssmg. and if it has a zero at infinity,. C00 be missing. The network of Fig. 9. 5 is called Foster form l/. Tbc continued fraction expansion of F(r) is given by

F(.n

r

I..,

= >II) Pis. 9.5

F(s) = «1 s

1

+ ---·----1- -

«a s+ ----~.--«1

s+ ____,__ ... +--«,,,

(9. 12)

Bere F(s) is assumed to bave a pole at infinity. The removal of

Elements of Network Synthesia 273 tbis pole by the factor or.1 s leaves a function with zero at infinity and hence its reciprocai has a pote at. infinity which can be removed and the procedure repeated. If F(s) is a11 impedance function then the realization is shown in Fig. 9. 6 and is callcd Cauer form I. If the impcdance has no pole at infinity (then the admittance has a pole at infinity) thc inductance ac1 will be missing. lf the impedance has a zero at tbc origin tbc last shunt capacitance will be missing. o

o

I - I ------7

r r

I

Fig. 9.6

The Cauer Jorm II is obtained by removing the polc of F(s) at the origin. In tbis case the continued fraction expansion is given by

F(s)

= _!_ + ~1S

1 l

J...+-=--,\3

l

(9.13)

R+ ...

1-'as

If F(s) is an impedance, the realization is shown in Fig. 9. 7. Ir tbc impedance has no pole at infinity the last inductance will be missing.

Fig. 9. 7

We bave shown that a givcn reactance function F(s) can be realized in four different forms. Ali these forms bave tbe samc number of elcments and the number is equa) to tbc number of poles (:&eros) of F(s) including any at infinity. We cannot realize an LC network with lesser number of element and as sucb tbese forms are called canonic forms.

274 Network Theory and Fi/ter De.rig11 EXAMPLE

9,6 Ei(s) _ s6

Let

+ 5.r' + 6.75s1 _..i. 2.2.? s• + 3s + 2s 3

F(s) can be written as F(.r) -

f_J2_±_~:-~(s' s(s~

+

+__!_:_~!_(~~ :-!:_ ~~

I JlS~

+

2)

±jv'2, oo ±jv'0.5, ±jv'l.5, ±jV3

Poles: O, ±jl,

Zeros: F(s) satisfies the conditions of Theorem 9.4 and hence is a reactance function. Let F(s) be an impedance function. Foster I

Tbc partial fraction expansion of F(s) corresponding to eqn. 9. IOis

From cqn. 9. 11 ,r,r) I kO = lqS •-o

k l

= (0.5Xl.5)l3) = (1)(2)

1 . 125

= (-I +0.5)(-l t 1.5)(-l + 3) ==0.25 21 r••-1 2(-1)(-1 + 2) 2{(s)j - (-2+0.5)(:::Z+l .5)(-2+3) - O 187S

= (s•+ 1/\

k -

,

• - (s

+

2.s ••--• -

2(-2X-2

=

k =F(s)I ""

s

+

1)

1.

-

Tbe network is given in Fig. 9. 8. 0.5

O 1875

2

2.667

1.125

I

o-

r

F(s)

=

Z(.,J

Fig. 9.8

•

Elements o/ Network Synthesis 275

Foster li The admittance function corresponding to F(s) is F

_.

.

s(s:i

+ 1)(s

+ 2) l. 5Jls~

2

i(s) - (s~ 7 O. S)(s~ ,

:~3)

The partial fraction expansion is given by F ( 1 s)

k,

wherc k1=(s2+0.s/t(s)1 _

2

.

_

2

+

k _;

3

=

(-0.5+ 1)(-n.5+ 2). =0.15 2t-0.5+1.5)(-0.5+3) F/ s) \ _ ( -1 5 + I)( -1. 5 + 2) 1.5) 2s ••--1.s - 2(-1.5 + 0.5)(-1.5 +3) = 0,083 2s

k2 -(s

k~

= s2 + O. 5 + s 2 + I .3 + s2 ••--o.5

_ (-3 + 1)(-3 + 2) -2(--3+0.5){-J+ l.S)=0,267

F1(s)1

i\3-(S +3)2s ••--,

The network is given in Fig. 9.9. o

r-

1

JJJJ

!

6 024

~

o.6I o.. Il llJ

0

;

1.873

~

o.mf

Fig. 9.9

Cauer I

F(s) has a pole at infinity, which can be removed by long band division F(r)

= s8 + =3+ = s+

+ 6. 75s2 + '.'.25 s• -t- 3s 3 -t- 2s 2s4 + 4. 75s 2 + 2.25 s• + 3s3 -r 2s Ss-1

F1(s) F 1(s) has a zero at ìnfinity and hence l/F1(sl has a pole at infinity w hi.:h can be removed _l_ _ _ s5 + Js3 t}s ... = O Ss + 0.625s 3 + 0.875s Fi(s) - 2s4 + 4. 75s 2 -t- 2.25 · 2s4 + 4 .·, ~s• -r- 2.25

= 0.5s +

F 2(s)

276 Network Theory and Filter D,atgn 1/F1(s) has a pole at infinity

1

F 1(s) =

2.t' + 4. 75sl + 2 25 0.62S.! 1

+ O. 875.r =

.

3 .2.t

J.9Ss1

+ 2.25

+0.625s8 + 0.87S.S

= 3.2s + F1(s) Similarly, _1_ = 0.6258' + 0.875s FJ..s) l. 9Ss1 + 2.25 == 0.321s

= 0 _32 1s +

0.1533 l.95.fl + 2.25

+ F.,,(s)

ss•

- 1- - t . 9 + 2 · 25 = 12 ' 745s + 0.1533 ~~.!_ F,(s) 0.153s = 12. 74S.r + F.(s)

l

.

F,(s) = 0.068s

Hence the continued fraction expansion for F(s) (eqn. 9. 8) is

= s + - - - - - - -1---,-1- - - - - -

F(s)

0.Ss +

------.....::.--1- ________ ]__ _

3.28 +

0.321.r

+

I

12. 745.r + 0.068.r

The coefficients of the continued fraction expansion can · be obtained in a straightforward manner by Euclid's algorithm which employs synthetic division. The algoritbm is sbown in Table 9.2. Table 9.2

! 1

, !

Nurner-ator F(s)

Durominator- F(s)

8'+5.r'+6. 1s,a+2.2S

.sl+3a3+2.r

8'+3.s&+2.r1

s1+2.237Sal+l.12u

I

I

0.5s

2

0.32h

4

0.068.r

6

i

i

3

5

3.2.r

I

II

12. 75s '

2.s&+4. 7S.rl+2.25

0.625.,S+0.875.r

2.s&+2 .8.s2

0.62Ss•+o. 722s

1.9512+2.25

0.1$3.r

l.9!i.r1

0.153.r 2.25

o

Elementi of Network Symhesis 211 The first coeffi.cient is obtained by divìding the numcrator by the denominator. This step is indicated by 1. The second coefficient is obtained by dividing the denomioator by the remainder ofthe numerator obtained in step 1. This step is indicated by ~. This procedure of dividing the numera tor by the denominator and then the denominator by the numerator is continued tilt the remai ode:- becomc O, and steps 3, 4, 5 and 6 are obta.ined. The coefficient of the continued fractioo expansion are given by the corresponding quotients, 3.2

12.745

uo----l. . -lI~ r. . . 0

0 32 1 ·__ _ ......

_.s__

Fig. 9.10

Tbc Cauer I network is given in Fig. 9. 10.

Cauer Il In Cauer I the poles at infinity were removed. This is equivalent to dividing tbc numerator by the denominator to eliminate the higbest power in the numerator. In Caucr II we rcmove tbc poles Table 9.3

Numero or F(s)

1

3

2.25+6.7S

1

o.888s

Denominato, F(s)

,:a+ss'+sl

2.r+3.f'+8'

2.25+3.375,=+t.125.r

1 [294j

1 S J,6X}0-IJ

I I

1

2s+2.007.sl+O.S93st

3.375•'+3.87Ss'+.s41

0.993.sl+0.40711

3.37Ss2+J.383~

0.993s•+0.398s•

2.492s•+s1

0.009s•

2.49216

0.009,•

.

o

1.686.t

1

2-IDi

2

4

l

J.11 x IOl.s 6

278 Network Theory and Filter Design at tbe origin which is equivalent to dividing tbc numerator by the denominator to eliminate the lowest power in the numerator. Tbe division ia given in Table 9. 3. Tbe continued fraction expansion of F(s} is 1

F(s)=o.sss.,+

I

1

1

l.68fu+-.t,---~I---------0.294s+

l

2.513s+--:-1-----1--3.6 X 10"4.s.+ l.l l X l03s

The network is shown in Fig. 9. 11.

~----~\---..---e~, 0.888

0.294

1.686

3.6x 10--'

2513

I li

X !0-l

j

Fig.9.11

In ali the four realizations we bave six elements. The number ofpoles of F(s) is also six; thus demonstratingtheir canonic nature. 9.3 ·RC Functions A network with only one type of energy storage elemcnt (C or L) cannot produce any ringing and the network can be at best critically damped. As such an RC network cannot be designed with Q greater than l• Yet the study of RC network is of fundamental importance as they form tbe building blocks in active filters. High Q realization can be obtaioed by embeddiog an active element in an RC network in feedback conflguratioa (see Chap. 12). For an RC network, the energy function T,, is identically zero and hence from eqn. 9. 8 we bavles:

Zeros: - 1, -3, -S; 00 F(..r) is an RC-impedance (Zac). (i) Foster I

Expanding ZRc(s) in tho fonn of eqn. 9.17

ZRC(.r)

= ko+ ~ +.J:L a

+2J...

1+2 s+4 s+6 k0 = sZac 1- = 0.3125 2 + 5)=0.1875 2 +t)(- 2 + 3 k =( +2)Z I ..... 1 S RC _.I (-2)(-2+4)(-2+6)

>

,S\811 -t- 16) '.· z1 has a zero atjl as expected. Thc pole of y 1 atjl)s removed to produce the uro at Jl.

. i.e. Ya

= (s• +

s(s1 + 16)

15

4.2

--8

19.2 1) (s1 + :20~2) = a• + 1

--8

+

19.2 r..,,,_.+,.....,20.,,...""'2

15

The admittancc f)mction

!:; ; .produ~ the rcquired transmission

zero. Removal of t&is leaves an admittance function

4.2

19.2 8

Y■ = s• + 20.2 . s• + 20.2 b .• 1. . , .as a zer(!atJ(v 20. 2),. By ~ifµly removNow z1 = ·4 _2

19.2 8 iog the pole or z1 at the origin wc can shirt the zero to

j2, the

second transmission zero. Aa before we remove k from z1 where /f

. t.e.

k == [.r z11-n k 16.2 =- 4.2 {

19.2)

308 Network Th«1ry and Fllter Design wehave

k s' + 4 z,-z,- .... u--

19.2 3 and thus crcating a trammission zero at , - J 2. To produce tbis we coasidcr the admittanoc function.

4.2 19.2 8

Y1-.T+4 The network realizing the transfer fwiction c:an be obtained by referring to Figs. 9.2 and 9.4. The network is ahown in Fig. 9.30. Tbc constant multiplicr can be evaluated as indicated eatlier. 5/64

-----1-----~ Hl.2>1 ➔ pass-band edge 6>, ➔ stop-band edge "1 1 < t.> < ,-+ transition band Maximum pass-band attenuation rx,dB Minimum stop-band a(tenuation rx,dB i.e. where « is tbe attenuation of tbc ftlter in dB. Strictly speaking , rx being attenuation sbould be expressed as negative quanùty. Whcn wc specify a.,. and «, we should interpret it as rx,dB loss at 6> = a, and a.,dB loss at (Il = 611 • All the quantities in Fig. 10.29 are referred to O dB (unity gaio) axis and / T(jw)I in dB is negative for ali values of w > 6>1 • lf we plot tbc loss or attenuation in dB versus w tben and «, will always be positive (Fi,g. 10.2b). Tbc approximation methods solve tbe problem sclecting a realizable rational function whosefrequency response approximates that in Fig. 10.2. Let f(6>) be tbc required frequency response function and t(

E.;

6>b

be minimized. Here the criterion is to minimize the maximum error over the frequency band 6> 111 . . ; c.>..; w1,. Wc may also pick a mean-square error c:riterion, i.~. r(6>) is selec:ted sucb that it minimizes,

316 Network Tlteory an4 Filter Design f((I))

=

••

fI

/(f4)-t(.,)

I I fki

Anotber method of approximatina is to bave t(w) exactly equal to /(Gll) at a prescribed point c.,= W.• Let /(6>) and /(et) be cxpand• ed in Taylor•series about 6>9, i.e. j{w) =/(~)

+ / .(6>.)(111-(1)•) +rc~l ""'1! ((1)-fal.) + ...

and /(0) = t(w.)

+ t'(.;.)(w-,.,.) + ,·cc..> 21 (w-c.,o)1 + •·•

1

(wherc 'indicates dift"ercntiationwith respecttoc.,). lf/(fl>o) = t(t.>.) and as many as possible derivatives at c.t == Wo are set to zero we bave a Taylor series approximation. In ali the approximation the transfer functioo is selected as 1 (10. la) 1 K!(fl)) with

+ ••


) ► I, 6l > er>, With this selection, I T(jo,) 11 R:I 1 in the pass-band and approxi• mate]y zero in the stop-band. From eqn. 10. la the attenuation function i, «(r.1) = IO log [1 X:((l))J (10. lb) and tbe constant • determines tbc pass-baad and/or stop band attenuation. In what follows we consider two well•known metbods-Butte~ worth and Chebyshev approximations-in detail, and bridly discuss some other methods of approximations.

+ ••

10.2 Butterwerth ApprosloudiOII Butterworth approximation is a special form of Taylor series approximation in whicb tbc approximating function t(Ci>) and the speci6ed function f(w) are Jdentkal at er> = o. For tbis approximation K,,(m) is seleèted as K,, (w) = f3o + f316» + ll:i(,)1 + ··· + fl,.w' Fora Taylor series approximation the function K,. (w) must be maxirnolly flat at thc origin (i.e. fil = O). Hence as many deriva•

.4pproxtmation 317 tives of x. (..) as possible must vanisb at Butterwortb approximation

Ca>

= O.

Hcnc:e for

x. (c.>) = ... Tbe magnitude function I T(j,.) 11 and tbe attenuation function ll{(o)) are then gi~n by

I T(6>) 1· = 1 + ~ ... c((o)) = IO log [1 + cl ea>lil]

(10 .2) (Hcre we bave used e in piace of • in eqn. IO. I). In eqn. 10. I and 10.2., sboald be interpreted as thc frequency normalized witb respect to the pus-band edge 6>1 • i.e.

I 7lfa>) I' and

11((1))

ITC)a)I IO

l

+ cl( -..Il

r

(i)

= 10 log [ 1 + c1 (r,,:

(10.3a)

rJ

(10.3b)

.,__~..,..:!"lii:~-------,

L ________..:::::=;::::=::!~► w/°'~ I.O

Fia. 10,3

The frequency response of a Butterwortb filter for varioua values of n is shown in Fig. 10.3. AH tbe curves pass through the samc point at a = (I), and this point is determined by Iftbe spccifications are barcly satisfied, tben from eqn. 10.Jb wc bave

a.,.

318 Ne1work Theorv and Filrer Design

a,=

IOlog [ 1

eta= 10

The factor ed by k

6l 1

log[]

+ e• (~t']

+c1 t::)]

/w is called thc selectivily paramele, and is rcprese11t• 4

(10. 5)

i.e. Equation 10.4 can be rewritten as

c1 e' -kan

and

"""·

10•·1.otv - 1

=

10

o.1« a

oo:6)

-1

,.Dividing one part by the other wc have

k•• == 100,1s.,, _1 100· 1•. -1

lf we define [ 10"·111,.

-l]i = k

1oo.i•. -1

1

as the discrimination parameter, wc bave log k1 n=logk The order of the :fìlter n sbould be selected such that . ...,, log k1 n ,,.. log k • n a.n integer.

(IO. 7)

(10.8)

Jf n happens to be equa] to log k 1 /log k then the value of e obtained from botb the equations in 10.6 are tbe same. 1f n log ki/log k, then e -cao be selected to satisfy eìtber the pass• band edge or the stop-baud edge requirements exactly. (10.9a) In thefirst case: e =VI00-111JI -1 In the second case: e =v'k.,. (10°-111. - J) (10.9b)

+-

EXAMPLE

10, l

Tbe speci6cations for a LP filter are ci ~ 111

>

1 dB, for / ~ 3 MHz 60 dB, for /~ 12 MHz

Approxlmation 319 For this filter the selectivity and the discrimination parameters are 3

.

= 0.25,

k = 12 0

- ['°~-1 -,-]1 ]' -- 0.5089

k1 -

108 _

X

10-a

The order of the Butterworth polynomial frorn eqn. IO. 8 is

n, >- J,og k1 = -3.29H = 5 4702

,_. logk -0.6021 · i.e. n=6 . (i) If.c is to satisfy pass-band requirement (eqn. 10. 9a). tben =v'H)11 -~ -1 = 0.5089 (ii) Ife istò sitisfy stop-ban& r~quirement (eqn. 10.9b), theo 'C

e =v'(o.25)12 (I0.0

:-if

= 9.2441

{If n is chosen as 5.4702 then from eqn. 10.9 (a) and (b) we have e = O. 5089 as indicated earlier.)

Filter specification uften includes the c~t-off frequency which is (ù~ is tbe cut-off frequency then eqn. 10. 3 yields

aefined as the-3d8 point. lf

I

- v'2

e*(::)~= 1

ie.

and

CJ>p

(ù~

=,.ve

(10. IO)

For the previous example, ife is selected to satisfy the pass-band

requirement Cù

-

e-

filp

{1/0. 5089

= 1.12.c.>,

Ifthc pass-band edge is tbc cut off frequency, i.e.«, = 3dB, then e= I. ' From now on for convenience we consider a normalized Butterworth filter, i.e. = I and e = 1. In order to realize a filter we bave to determine its transfer function. From realiubility conditions, the poles of the traosfer function here ao ali pote transfer function, must be in the left half of a-piane. The poles of the transfer function are determined by

w,

320 Network Theory and FIJter De1lgn tactorizing the denominator of I T(J,,,.) 11 and properly allocatiq

roou.

ita

To tbis end let

·

T(}w) 1 = Bere I H(J,.,)

I H(~w)

= 1 + ..... and

11

r

(aec eqn. 11.17}

by analytical continuatioo we c:an

writo

I H(JCJJ)1 I =-

H(1) H(-.r) ~-I•= I+ (-s')• 1,.._. The 2n zeros of H(s) H( -,) are obtained by 10Mng 1 + (-1)" , .. = O (10.11)

As can be easily seen. the roots of H(s) H(-s) appcar in quadrantal symmetry. Hence tbe rootl in the LH piane can be separated aad assigned as the polca of T(.r). Two case■ ariae dependm, upon whetber n is eveo or odd. (i) Jf n i.r even: Equation 1.0.11 reduces to

r" = - 1 -

el(lk-1)11

The 2n-roots are

} P• = exp [ j 2k-l ~ff

k..,. l, 2,... 2n

. 1.e. P1e = cos (2k-l ~ n)

+ J. sin . (2k-l :rr) • k = l • 2,... ,.,. "-

(10.12)

~

Equation 10. ll reduces to

(ii) 1/n is odd:

Tbe 2n roots are

i.e.

P1e

=

exp[JtRJ. k = O, J, ...

p,.

=

cos [~~]

(~n - 1).

+ jsin{~n} k ==

(10.13)

O, 1, ••• (2n - 1).

Tbe transfer function is constructed from the LH piane rootl and is given by (see prob. 10. 3) I (10.14a) n even: T(s) = -,.,-1 ~ - - - - - II (s1+2 cos 8,. s+l)

.

•-1

Wlth

n odd: T(s)

84: -

2k-1 - 2-n- :rt

= ---=-,.--=1,=,1-1- - - - - - (.r

with 8,.= ~ n

+ 1)

Il + 2cos8.s + l) •-1 (s1

(10.14b)

Approximation 321 For ex.ampie for n = 2 T(s) =

I

I

- - - - - - - = """z------=--.s• + 2 cos ~ .s + l

s

4

+ v2s + 1

and for n =3

J

T(s)= (s

+ 1) ( ,• + 2 cos j s + 1) l

.. a8+2s1 +2s+ 1 ·11

cven

Jw

lì odd

jw

Fig.10.4

Thc poles of T(s) for even and odd n are shown in the Fig. 10.4. It can be seen that for odd n, there is always a pole on the negative real axis and for both tbe cases thcre c&nnot be a pole on thè: jCJJ = axis. Table 10.1 gives the denominator polyoomial of T(s) for various n. In the preceding discussion e was assumed to be 1. If it were not the case, i.e. when thc pass-band edge is not the 3 dB point, the radius of the Butterworth circle, is given by 11

cven! Pk

n odd:

=

veI

P• = Jc

!- and ve n

[.2k-l 1t]

exp J ~

exp

[1 zff].

,

k

the roots of H(s) are

= 1, 2, •.. 2n

k=.. O, I •... 2n -1

(10.15)

Tulel0.1 11

I

Factors

RooU

3

1-1,

4

i -cosk 8 ±Jsink 8

-cos ,r/3:t/ sin ,r/3 ff

W

':;±i sin~ (l:=l,2) I --cos k n,±1 sin k il (k-l,3,S)

7 I -1, -c:os k

s+l

.,a+v2.r+l

sz+v'2&+1

~

(s+ 1}, (&Z-ts+l)

s•+2s•+:zs+1

~

(.rl+0. 76S.r+1), (.rl+l.848.r+I)

st+2.613.Jl+J.414s+2.613s+l

(.r+I). (si+0.618.!+1), (.r•+t.618s+ 1) I .s5+3.236.r'+5.236s1 +S.236.sl+3.236.r+l

-1, -cos k i-"±J sin k

9

10

!

9

I

(s1 +0.44.r+I), (.t1 +1

.247.r+l),

(sl+l.802s+l)

(sl-t0,3.r+l), (.s'+l. I lts+J), (.r1 -tl.166.r+I), {s•+l .962&+1)

(k-1,2,3,4)! (i+l), (.r9 +0.347,-tl), (.sl+.r-t1). (s3 +J.S32.r+1), (s•+t.879&+1)

I -CO! k ii±l sin k !6 (k-I,3,S,7,9)

J i

~ ~

r±i sin k ~ (k=-1,2,3) I(&+I},

16±1 sin k f6 (k=l,3,S,7)

:i;

i

s•+J.86431 + 7,464.r4+9.142.Ja+ 7.464'+ 3 .864.r+t

(.sl+0.Sl8+1), (Jl+v'2.r+ I), (.rl-t 1. 932&-t ])

7

8 I -cos k

~

s+l

(k•l,3)

5 I -1, -cos 6

"''

(:j ~

l-1 2 ! -cos ,r/4±/ sin •/4 I

Pol;ynomiol

l

(& 7+0. 313.r-tl), (sl+0.908s+I), (s1 +v2.r+t),(s1+1.792.r+1), (.•1 +1.97Ss+I)

.r7 +4.494.r•+ 10.098.r'+ 14.592.r'+ 14.S92sll+ J0.098.r•+4. 4941+ 1 .ss+s.t26s7+13.137.r•+21.84fu1+ 2S.688s'+ 21 .8461•+13.137.r•+ S.126.r+ 1

1'+5.7S9.rl+J6.582.17+3J.J63s1+ 41,98f.r6+41.986&'-t31 .Hil.sl+ 16.S82a•+s. 7S9s+ 1 sto+6.393.st+20.432.r8+42.1!()2.t7+ 64.882.r'+ 74.233.r5+64.882.r&+ 42.802.rl +20.432&1+6.393.r+ 1

~

llil' ::s

Approximatlon 323 and the transfer function T(8) is obtaincd from tbe LH piane roots as iodicatcd earlicr. 10.3 Cbeltyshe, Approxl11111tioD From the standpoint of obtaining the best approximation to the ideal filter characteristic from a polynomial of a given degree, the Butterworth function does not do as well as may be done because it concentrates all of the approximating ability of tbe polynomial at 6) = O, instead of distributing it over tbe range O < c.> < 1. A better result in this regard may be obtained if we look fora rational function that approximates the constant value unity throughout this range in an oscillatory manner, rather than a monotonie manner. Ch~byshe, Approximation does exactly this. Chebyshe-, polyr,omials are defincd as linearly indepcndent soJutioos of the dift'erential equation 6>1)y

(1 One of tbe solutions is y

- co; - n'y

=O

= T.(w)=cos (n cos-1 w), I 6t I < = cosh (n cosh-1 fil), I I ~ I

l

6)

(10.16)

T.(co) is called the Chebyshev polynomial ofthe tirst kind. A power series expansion for T,.(6'J) can be obtained by rewriting eqn. 10.16 as

T,.{w) where

=

Re I .el"tf, I =Re I cos tf,

+J sin tf, l"

(10.17)

+= cos-16>

i.e.

cos if,

=

o,.

sin ~

=

yl -

6>1•

Binomia! expansion of eqn. 10.17 yields

T.,(6))

= .,_ n7 +432w1 - l 20w*+ 96>

+160w'-32(1)1 +1

512w10 -t280w 8 +11201~1.

(ii)

T.(O) = (-1)• 12 and T.( + I) = I for n even TJ..0) = o and I T.(±1) I = 1 for n odd. i.e. T,.(6>) ~ 1 for I w I ~ 1.

Approximation 325 (iii)

Beyond the interval I (l) increasing valucs of I ) -

10 log [ I

+ ,17! (.:J]

(10.21)

With the specification fora low-pass filter as given in Fig. 10.2, cqn. 10.21 yields

«, =

10 log [ 1 +

••T! (:;)) (10.22)

«.

=

10 log [ I

+ 111 7'! (:;)]

Thesc cquations can be solved to yield , and n. From the first eqn. in 10.22 ti = ]()O.llllp-1 (10.23) as T!(l) = I. From the same equation it can be shown that tbc order of the filter is given by (prob. 10.4).

t+

I

IT(iw)l 2

I

,2

"' Fia. 10 • .5

n

coslr1 (1/k1)

:> cosh-1 (1/k)

(IO.l4)

where k and kA are given as earlier _by eqn. 10.5 and 10. 70 reapectively.

326 Network Theory and Filter Design Figure 10.5 shows a plot of I T(Je,>) j• for two dift'erent values of n(n = 4 and n = 5). From eqn. 10.20 it can be sccn that 1 T(j(,)) 11 mai= J when T,.(w) ... O

I T(jru) 11 min = 1 ~ 1111

wben T ,J.w)

=±

1.

I

Hence the peak to peak ripple is 1 : "

and the approximating

function ripptes about the value 11+ •\2• Typically tbc allowable

+•

ripple ar., is specified in terms of decibels. The ripple in dB is given by (10.25) dB (ripple) = 10 log (1 + 111') alternate way of wri ting eqo. IO. 23). Figure 10. 6 (Tbis is only shows the relationship between dB (ripple) and E, The transfer function can obtained as io the case of Butterworth approximation by determining its poles. To tbis end we define I HLJw) 11 as I H(jf») 11 = 1 + r1T 1 () (10.26) The zeros of H(s) H(-s .• -,.. =I H((,)) 11 appear, as already indicated, in quadrantal symmetry and can be separated iato LH and RH piane roots. The LH roots are assigned as the poles of T(s) and T(s) is obtaioed from these poles.

an

J.O

.,

c. A.

ii: I O O2

0.4

0.6

0.8

I.O

Fil- 10.6

It can be shown that the polca Jt.r) are gi-ven by (prob. 10.6).

,,. =

a,.= sinh Y sin

011;

(2\: ,r). 1

+iw11: k

= O, l, 2, ... ,2n-l

(10.27}

Approximation 327 wi

=

cosh vcos

(2k+l) -i;,- 11:

where v = 1/n sình-1 1/t. In Butterworth approximation the poles were located ·on a circle of radius _1;-. The locus of poles of Cbcbyshev approxinv c mation can also be similarly determined. From eqn. IO. 27 we have

a,. = sin ·(2k+l) ~" - ~ = cos (. 2k + l 11:) cosh 1> 2n sinh v

and and finally

i

•

~ ~=l sinh1 v + cosh1 v

indicating that the poles of T(s) lie on an ellipse with tho major axis along the fa,>-axis and the minor axis along the a-axis. It is not necessary to evaluate eqn. 10.27 to flnd the poles of T(s). The poles can also be found grapbicaUy (Fig. 10. 7). The semi-major and minor axes of the ellipse are b (b = cosh v) and J log(0.5089 X

1~== 26 _36 log (0. 75)

i.e. n= 21. (ii) Cheb1shn jìlter:

From eqn. 10. 23

• = '\1100·1 -1 = 0.5089. From eqn. 10.24

n-"

cosb-1 {

1 } 0.508~ X 1~

= 10.4

cosh-1 ( O. 75}

n= H. From Fig. 10. 7 it is evidont that tho quaJity factor of the 8bebyshev· roots are higher tbaa tbose of tbe Butterworth roots.

A.pproxtmatlon 33 I Hence for a given ordcr of the filter transfcr tunctioo tho tramition band or tbc Cormer i• narrower tban the lattcr. Tbo two approximations can also be compared by tbeir bebavior at higb frequencies. Lea I T•(J6l 11 and I Te (Jo,) 11 be tbc Buttcrworth ami Cheby■bev mapitude fuactions. Aa 6l teads to ao.

I Ta Ut.>) 11 and

1

➔ e• 6) . .

1_

I Te (/t.>) 11 -+ E' (21 1 6l••)

i.e. 10 Ioa I T8 (Jto) 11

=-

(20 log e+ 20n log) dB

and 10 log I Te (}ti!) 11

::;= -

20 [log •

+ 20n log(I) + 3(2n-2)] dB

(10.29) Botb tbe responscs bave the same slope (-20n dB decode). Butterworth is oft'set by an amounl depending on e while tbc offset in Cbebysbev is dependent on both n and •· Because of tbe factor 3(2n-2) tho Cbebyshev reaponsc has a smaller value in the stop band for ali values of n and for e .... e-1 and consequently is a better approximation. Figure JO. 9 shows the phase responses of Butterworth and Chebyshev filten. As can be obse"ed from tbc graph a Butterworth tllter gives a fairly lioear phase response and is superior to Chebyshcv phase responso. ·

540

Fig.10.9

332 Network Theory and Filte~ Design As tbe order of the Chebyshev respoose becomes higher, the pbaac response becomes more non-1inear. lf linear phase response is a .requirement, neither Butterworth nor Cbebysbev is suitable (ace scc. 10. 7). From eqn. 1O. 29 it is evident that for a given order of the Chebyshev polynomia1 n, the increase of .r, i.e. the increasc of the pass-band ripple, will improve tbc stop-band performance. Tbc effect of variation of e (or cx,) is shown in Fig. 10.10. It is also evident from eqn. I0.29 that tbc increasc·ofthe ordcr ofthc polynomial will increase the loss in tbc stop band (Fig. 10. llJ. From T(jw)

0.35 dB} 0.9dB

. l,_75 dB . r1pple 2.5 dB

Pig. 10.10

Figs. IO.IO and 10.11 it can be concluded that a smallcr pass-band ripp1e and sbarper cut-ofl' requiremcnts cannot be simultaneously .achieved; if onc is decreased and other incrcases.

17liw>I I.O

'-------~_;:~IIÌÌÌ::::::::a-----w/w, I.O Fig. 10.11

A.pproximation 333

,, = 6, O. I dB ripple ,, = 6, 0.969 dB ripple n = 6, Butterworth

L----------11.--....::::.:..lliì.:=iiiàll---l► w/.wp

I.O

Pig. 10 .12

As a ftoal comparison Fig. 10.12 shows the response curves of a 6th order Butterworth ftltcr (oi:., e== 3d8) and two 6th ordcr Cheby-

sbev filters {at;, =O.I dB and «., = 0.969 dB). The conclusions we bave arrived at io the preceding discussion are confirmed by these curves. There is no better nth order all-pole filter than Chebysbev filter witb equal or better performance in both the pass and stop band.

10.S Frequency Trausformation In the earlier sections of this chapter we have only discussed tbc design of low pass (LP) :filters. There are many otbcr types of filters which find use in signal processing. Figure 10.13 sbows the frequency characteristics of ideai high-pass (HP) and band-pass (BP) and band elimination (BE) filters. Once the low-pass prototype filter has been designed, the other filters can be obtained by a simple frequency transformation. The frequency variable of tbc low-pass filter is transformed by a reactance transformation to obtain high-pass, band-pass and band elimination characteristics. In the reactance transformation the low-pass frequency variable n is replaced by a realizable, rational function X(w) of another frequency variable w. By a proper choice of this function we can transform the low-pass characteristfo to other characteristics. The transformation X(w) is selected as a reactance function (see Chap. 9). Let Q = X(w)

,~,

334 Network Th«,ry and Fllter Design

j

1~1 Bandpass

ffiJ~pass f,,

li

I

li

I

l~----Band elimiution

ì Pia. 10,13

be the required transformation where Xfc,)) - A c.,(c,)11 - c.>•) (Ca),• - I) \

-

(61 •• -

611) (Uai -

Ca)•) •••

and O< Ca)l 1 = Aa,, Cl.l1 = (I): 611

is tbc band of frcqucncies thqt i,· rcjccted. lf we are considcring a normalizcd· LP with O..~ l, then A spcciftes the rejection band. Thc elements of tbc traosformed network can be similarly obtained. Hcre L, transforms to a pàìallcl resonant cfrcuit and C, to a aeries resonant circuit. Figure 10.20 shows a low-pass prototype and tbc transformed BE filter. Band rejcction filter can also be obtained from a frcqucncy inversion of BP filter. (61 1 -fl>j)

L,

L,,

·•--'• ...r"·Blli--oo

Te,, o

-o Flg. 10.20

10.6 Bandpass Filter Approximation W e bave discussed the basic frequency transformation that

340 Network Theory and Fllt(!r Design transform a LP to HP or BP filters. lt is instructive to study the application of the aame to obtain the approximation for a BP filter.

ITUw)I

•t

O dB

Of.p

I

I

' J I

I

I

I

I. I I I I

---i-

I

I

I

I

"Wo

wi

oc,

J_ Wsl

Fig. 10. 21

Figure 10.21 shows the specificatfons (orthefrequency responsc) of a bandpass filter, w 1 and 0 is tbc centre frequency of the filter. From cqn. 10.31 we know that Cl>g

= V W1«.>1•

If the pass-band and the stop-band edge frequencies are such that Wl (1)'1

=

(10.36)

Ctllt (,)I

(i.e. «.>o = V 11 = 'V ,1g1) then the band-pass filter is called a geometrtcally symmetric bandpass filter. The frequency or the spectral transformation from low-pass to bandpass given by

n= -

A~•-(,)• with~ =A (,>

(10.37)

C111-i

always results in a geometrtcal/y symmetric filter. Here Op is the pass-band edge of the Iow-pass prototype. Jf we are to design a bandpass.filter our ftrst task is to convert the bandpass filter spccification to a corresponding low-pass fi.Iter speciflcation and then perform tbc approximation on tbis filter specification. This yields a low-pass filter transfer function whicb is then transformed to a bandpass filter transfer function by the transformation.

Àpproxlmation 341 8

➔ À 81 +s "'01

(IO • 38)

(the equation is obtained from 10.37 by substit11tiog s/j for O and ).· If the specifi.ed bandpass filter is not a geometrically symmetric filter, tben lower of two ratios in eqn. 10. 36 is used for tbe design. Tbis improves the other transition band. An all-pole low~pass transfer function cannot be transformed to a geometricatly asymmetric :filter. To generate the latter the corresponding low-pass fllter must necessarily bave traosmission zeros in tbe stop-band (see sec. 10. 7). 'Let the specifications f'or a geomettically symmetric band-pass filter be IX

~ Cip,

Il ~ C.,

6>1

E;i;

6) - -

c., ~ 1-C.,1

The stop-band edge of the low-pass prototype is now given by

0.- -A

Cr)..t.\

-6)·

r:1 . ,.

: (10.39)

(1). .

The specifications f'or tbe required lowpass filten is at=s:;;11,, 0«~. n;;,o. The low pass 8Jter approximatiou is petformed with tbese speclfications. ExAMPL8

10.4

The specUicatiom ror a bandpass ftlter are: 11, ~ 30 dB SO t rad < a, < 72 k rad, «. ;,:,: 40 dB < 30 k rad; 6) l> 120 k rad The specifications indicate that

= SO k rad/sec, tt.,, = 30 k nd/scc, toi

Hence

6>1 ~

m.. =

72 k rad/,cc 120 t rad/•.

342 Network Theor:, and Filter Design '°o == v'~ = v'~. = 160 k rad/sec. and the :filtcr i1 a geometrieally symmetric filter. Let the pass-band edge' of the low-pass prototype be 1 rad/sec. Then l 1 1

A=aw--=22k 6>w-Wl

- 0.0455 X 10--. The stop-han d edgc of the low-pass prototype from eqn. 10 .39 is

,

o = - o 045Sx 10-Sx(60x lOl)•-(120xlQI)• • . (120x101) = 4.095

Hence the low-pass filter apecifications are

< 3 dB,

ll < 1 rad/sec 40 dB, O ;;i,: 4.095 We can now approximate this by a Butterworth ti.Iter. 1 k = 4.093 - 0.2442 «,, ac,

k1 and

~

=

D~·~-;

1)

We have

= 0.0100

log k 1 , n ~ log k -3.21. Let n - 4.

Since tbe attenuation at the pass-band edge is 3dB, e = 1. From Table 10.1 the low-pass transfer fWlCtion (witb ., = 1 rad/sec) is 1 T,(3) = s' + 2.613.r' + 3.414sl + 2.613; + l Tbe transformation from LP ➔ BP from eqn. 10.38 is • ➔ 0.045Sx 1()-1 s• + (60 x IOI)••

.,

Yielding tbe bandpass transfer flmction ia

2,332x 1001'.r•

Ti,C,s) = (,e+ 2.197Bx IOis' + 1.4883 x l()lD.,. + 2.4795x tou;s

+ 8.1238 x 101'8' + 8.9272x lQIÌs' + l .92S9x ]O-si + t.025SxlO-s+ l.6798x1()18)

10.7 Otlaer Metheds

or Approxlmatla

In both Butt.erwortb and Chebysbev approximations, the filter responae in stop-band ia monotonie. Tbis, as we have seen, lead

..4.pproximalion 3oU

IT;,U'-")I

"i~.

-z--: wfw, "-------w...-,._;.__---4•~ Fig. 10,22

to an all•pole transfer function as the function K!(6l) is a poly11omial in o>, It is possible to approximate tl:e ideai filter charaoteristic by a function wbich has equiripple response ln stòp-band. For tbis case the function K!{ù>) is a l"lliioual function and the transfer function has finite zeros. 10;7. I

INVERSE CHBBYSHEV APPROXIMATION

8y manipulation of tbe Chebysbev approxiniation of eqn. I0.20it is possible to obtain an approximation which bas monotonie response in the pass-band and equiripple in the stop-band

Let

I T'(j6l) I' =

I -

I Tc(j(,)) 11

i.e. [Here we bave used the subscript C to indicate Cheby1bev approxi• mation of eqn. 10.20)' · -

.I.e.

I T'( .1;6> 1 .. ). 1· -

«'T..'(_=- 1. Now if we perform a HP to LP transformation on eqn. lQ.40, i.e. rep• 6) by 1/6>, wé bave

) 1• = I T'( "/ ) 1• = ctT.'(_1/w) I Te() 1 Ci> J l + E 1T,/(l/w) 6)

(10.41)

T,c(jru) 12 is a low•pass function with monotonie pau-band .ud equiripplc stop-band (stop-band edge being t,j · = I). This is called •Jnvern Chebyshev• · ai,proximation (bence tbe subscript le) and the rcsponse is sbown in Fig. IO. 22. In order to determino Che I

344 Network Theory and Filter Design transfer function, Iet (10.42) q(s) can be determined by first findiog qc(s) for the Chebyshev filter. Then, as ca.n be seen from eqn. 10.41,

q(.r) = snqc(l/s) To determine p(s) we observe that

p(s)P(-s) = "c,,1nr,,''(l/6l) An example illustrates the approximation.

I..•--••

(10.43)

ÈXAMPLJ! 10.5

,

Consider tbc Cbebyshev :filter of example 10,2 with n

=

S and

e= 0.5088.

qc{.r)

= (s + 0.2895)(s1 + 0.4684.r + 0.4294) (s'

Jlence q(s)

=

·,

+ 0.179a + 0.9902)

sa(!. + 0.289S (-;+o. 4684 + o.4294) 8

ti

S

(~ + 0.179s + 0.9902) = O. 1231(.r + 3.4S4)(.r' + 1.093.r + 2,329) (.r1 + o_. 18a + 1.009) From eqn. 10.43 and Table 10.2

p(s)P(-.r)

=

20 ,.,io ( -16 --+ (I,).

6)1

5) - 1 1

6)

----··

+ J(il 1,.,.;._.. = C"" + 20.rl + 16)1 p(s) - s.,t + 20.t• + 16. - (5C4t _

20) 11

(10, 52)

Hence 't'.(0) - l

(norma&ed delay).

Furthermore the :fìrst (n - J) derivativcs of T• with respect to .6> are zero at - O indicating the delay is maJ.imally flat at tb~ origin. Tbe deviation from this flat delay for otbcr values of Ci) is generally ex.pressed as .per cent of the ideai delay -r(O). Tbc only design parameter for a Bessel filter is its order n. The specific valuc of n is to be selected to satisfy botb the delay-error speciftca.tion and Jbe magnitudc deviation spcci:fìcation I T(j6>) IFigure 10.24 and Fig. 10.25 give graphs of delày ~or vs c.>T and magnitude error vs (r)-t •

or

.9Z

o

•

6'

2

Pig,. 10.24

.4ppro%imatilJn 349

Fig.. 10.2s EXAMPLB

10,6

Design a Bessel filter to satisfy tbe following specifications: T(O) = 8 ms, T - = 1 % for (I) ~ 200 rad/s. Magilitude deviation: I. 5dB for (I) ~ 200 rad/s

-r(O) Ci>= 1.6 From graph of Fig. 10.24 n = 4 From graph of Fig. 10. 25 n - S Hence

n=5

T(s) can be constructed from cqn. 10. SI. In this section we have briefly discussed tbree additional approximation techniques-two for magnitude and one for delay. The standard approximations discussed bere are fa.cilitated by filter design charts and grapbs. details of whicb are found elsewhere.

3SO Network Theory and Filter DeJlgn PROBLEMS

10.1

The specificatk>na fora Butterwonh LP filtor are Pass-band 0-2 M rad/sec Pasaaband loss e; 2 dB Stop..baocl loss e; 60 de at , M rad/sc:c, Find n and c. 10.2 Tbc apecifications for an LP filter are • 'lit l dB for / ~ 2 MHt ■;, 60dB for J;, 8MHz Find tbc tranlfer fun~ion of the Butterwortla filter whicb satisfies tbese specilìcationa. · Selec, e to satisfy tbe pau-band edg,e cbaracteristic exactly. 10. 3 Show that the t~sfer fllnction a11, nth order Butterwortb filter is

for

T(.s)

1

= 1118

2k-l , &1 -

~••

n evcn

D (.rl+2s cost.+l) .

k-1

1

=

and

K

, .e. - ;; , n odd. (.s1+2f COI ••+I)

Herc e is assumed to be I. Il e is not equaJ to l what modificatiom are to be made w tbe precedina equations. 10.4 Show that fora Chebyshev filter · n

> cosh-1 (1/kt)

cosh-J (1/k) wbere k and k 1 are given by eqn. 10.S and 10. 7. 10.S Find n and e fora Chebyghev filter witb the following specilcatiODS Pass-band rippl• 0.S dB Pass-band 0-2 M rad/sec Stop-band loss 60 dB at 4 M rad/sec 10.6 Show the.t thc poles ofthe trand'cr function fora CbebysbeV filter are given by

a,

,.- ... +i,,,.

wherc and ••• are givcn in eqn. 10.27. 10. 7 The poles of a Chcbyshev filtcr ttansfer function lie on an ellipse. Show that the semi-minor and semi-major alte& for this elJipsc are ai..r and where a - 1/2 (Alt•-A-1 1•), b -= 1/2 (Ali• + A-li•),

b..,,

A= 1/it + v'f+"i'/Ea and ..,, is the pass-band edge. Hencejustify the dctermination of the poles of tbc filter from the two Butterworth circles. 10.8 The specification fora LP fil~r are as given in prob. 10.2. Find the transrer function of the corresponding Chebysbev filtc:r. Compare tbis with tbe Butterworth filter of prob. l0.2.

Approximation 35 l J0.9 Tbe specificat!on fora bandpass filter are:

s, < 2,0 dB for 30 k ra4 ,s;; ,... < 30 k rad s, ;i, 50 dB ror ,... 10 k rad and,... 40 k rad.


) 11

=_!i_= 4RilV~•

P,

+ I K(jfJ}) 11

(11.13) Psma R1 WJ where t(.t) is called the tranrmif.tion coefficient. As P,. ,,. 0 is always greater tha"? P1 we can write

P,.,,.°" ""'·I

l

(11.14)

Here K(s) is called the lou or Characteristlc function (compare this witb eqn. 10.1). Uoder matcbed conditions (Fig. 11. 5) tbe load voltage is V'1

= ~ . (~•)

{11.15)

This is the Jargest possible load voltage. Tbc ratio V1'/V1 is sometimes called the transtfucer functlon and is given by

H'(s)

= l .~

..J°Ri

(V•) Vs

(11.16)

(observe that H'(a) = 1/t(J)) From eqn. 11.13, Il .14 and 11. 16 we bave 1H'(/6>) 1· = l + I K(jOJ) ,. (11. I 7) For a 1nmped, LTI network: t(s) and k(s) are rationaJ functions of s with real coetlicients and can be represented as

t(s)

=

q(.,)

p(s)'

k(s)

= /(s)

q(s)

Hence we can write from eqn. 11 .17 p(a) p(-.r) == q(.r) q(-.r) + /(s)/{-.t) (11 .18) Tbis is also callod the Felthkellers condition. From eqn. 11.16 it can be observed that the transmission coefficient differs from the overa11 voltage transfer ratio by on]y a multiplicative constaot. Hence tbc zeros of p(.r) and tbe zeros of q(.r) are tbe natural frcquencies and tbe transmission zeros of the network, respective]y. Tbc zcros of /(1) are the zcro--loss frcqucncies. Equation 11.18

358 Network Theory and Fllter Design

giftl tbe relation betweon tbese zoros. The speciflcatlon of eitber the transducer (or traasfor) fullCtion uniquely speci&es the zero-loss frequencin. The LC-notwork of Fig. 11. 3 can also be chatacterized by tho reflection coefficieot which il deflned as _ Z1(.r)-

PJ.(d - Z..(.s)

R:J

+ R,

(11.19)

where Z 1 is the input impedance at 1-1' of the LC-oetwork with the load R1• (fhe concept of reflection coefficient is quite familiar in the study of transmiaion linea. Here p1 Js the aource-end reflection coefficient. We can also deflne a load-eod reflection coefficient). Though we bave defined a number of o·ew network functions we can interrelate ali of them and obtain some bounds on these fuoctions. From eqns. 11. ti and 11. 12 wc bave

P 10 4R 1 R1 P 1 _ . = (R1 + RJ', aod from eqn. 11.8

(ll .20)

_-.!.!_ = I t(jot) jl ... 4R,R, r.. (11.21) P1 • 0 (R1 + R1 ~ Unde, equal termination, i.e. R1 .... R1 , P I is cqual to P 1 ,,,,.. and

I t(j0>) I' ... r• The bound on P1 also depends on the p" ~ PP• P I

t:ermination. As

from eqns. 11.8 and 11.20 we bave

•-

+

4R R,,

Pr ~ (Ri

R,)i

(11.22)

Under equa) termination, ehen P90 :;>P1 Tbe traosducer function, loss functìoo and the reflection coeflicieot can be similarly related. The power delivered to the LC-network cao be written as

P, ...

+

~~

z.r

R

where Z.. (/ot) - R JX. Sioce the network is Iosaleu this is also the power delivered to tbe load i.e. P 1 : : P 1• Hence from eqns. 11 6 and 11. 8 we bave

I,wnlon Lou Synthel/8 359

P, - .,. -

, R,

+ zl rR.J

(11.23)

(il1 + R.J1 R

Here the lnscrtion power ratio is expre,sed in term1 of tbe network paraineters only. From eqas. 11.19 and ll.22itcanbeeuily1bown

that

I - (R4R1 R. e-h = ~I 1 + RJ• 1 and 1111mg eqn. 11.21 we have

·-

- R,• + R

\..

== Pi.fa>m( - fa>) (11.24)

P1 I - p - -= P1 (/6>) Pi (-}11>)

t(J•> t•

.

I Pa (Jo>) 11 = 1-1 I P1 (lo>) 11 I t(./t,)) 11 - I (11.25) Equation 11. 25 iadicates tbat tbe sum. of the reftected powcr ...... power delivered to the load i• equal to tbe total power available. as it shoukt-be. lf Z 1 - ~ tbeo p1 -= O indicating tbat-P1 == P 1 ln this case eqo. 11. 22 bolds with an equality. In Chap. IO we defined tbe attenuation a filter as

i.e. or

+

or

«(c.,} == IO log (1 i.e.

+ I 1'(}6>) 1'}

«(c.,) - 10 log ~;;-

(11.26)

Prom rqns. li .8, Il.IO and 11.26 it can beshown (a.prob.10.11}

tbat «(6>) =IL+ lOlog

CR1 R+R. R,)• 4

1

(11.27)

With equal termioation. tb~ atteauation and iasertion loas are the same. For an LO low-pass fllter tbe low-frequency inscrtion loss approaches zerò because inductors become short and capa• citon open. With this condition

=

+ Rì)' 4R1 R1 implying tbat tbe attenuation at c., = O canoot be zero unless the load and aource resistance1 are equal. nough we bave considered tbe case of R1 =I: Ra. in practice filter networb are more often dcsigned with equal termination. «(O)

11.2.2

10 log (R•

JleALIZATION PaocBot,u

As pointed out in sec. 11.2.1 a doubly terminated filter network

360 Network Theory and Filter Design

can be specified the num ber of ways. In the Darlington . realization procedure these specifications bave to be converted to finally yield the squared transmission coefficieot I t(jCll) 1•. The spccified function can be

r,, [ r I~: 1· 1

'~: (j6>)

(jw)

or P1

(Jt.>) is tbc specified function then eqo. 11.13 yields I t{jCll) 11•

1f

lf

I~-

(jCJ>)

r

is tbc specified function then

I tUtJl> 1• = _i__ , v1 U(J)> ,• R R I, 1

(11.28)

1

(1'bis cquation is obtained from eqn. 11. 13 by substituting v. = R1 I.). If Pr (or e11) is specilied then cqn. 11. 21 yields I 1(16)) 11, We assume, without loss of generality. that tbe inscrtion loss charactcristic of the filter network is specified. In this case wc can writc from eqn. 1 I •24, that

P(s) p(-s)

=

1- (R~'R_;_

~:)*

F(s)

(11.29)

where F(s) is therational function obtained from the approximation methods ofChap. 10. If tbe specifted inscrtion loss characteristic

n approximated by Butterworth polynomia]s then K

F(.f) =I+ c•(-l}'s.,, and if approximated by Chebyshev it is K F(s) = 1 + ~•T!(s)

(11.30)

(11.31)

For Butterwortb and odd-order Chebyshcv the constant K is I and for even order Chebyshev it is (1 + «I). This can be shown by observing the behaviour of F(O) and recognizing tbe fact that the iasertion losses fora low-pass LC ladder is zero at s = O. Tbis coupled with equation 11 . 22 leads to tbe restriction -0n termina-

tion as

+

4R R

1 ~ (Ri

R:)1 Butterworth odd order Cbebysbev

1 4R1R, d Chb b e ys cv 1 + Et ~ (Ri + Ra)1 even or er

(11. 32)

Insertion Loss Synthesia 361

Heoce wc conlcude that we can realize Butterworth and odd order Chebyshev filters, filter fuoctioos which bave a reflectioo zero at c. =- O, with equal terminations, i.e. R 1 = R,. But we cannol realize even order Chebyshev, a filter function with no reftection zero at c. = O, with equal termination. The realization procedure is summarized as: (i) From tbe given pass-band and stop-band requirementa determine F(s) (see Cbap. IO). The normalized pass-band is assumed to be I rad/sec. (il) Froin F(s) obtain the reflectioncoeflicient p(s). Herewebave edge uscd pi.i piace of p1 • From cqn. 11.29 wc bave . p(a) 4R1 R1 p(s) p( -s) = q(s) = 1 - (Ri + R.)• F(s) (11. 33) R 1 and Ra must satisfy the restriction imposed by cqn. I 1.32. p(s) can be determined from this equation by observing that tbc .zeros of p(s) and q(s) appear in quadratral symmetry. Let p(s) = p 1(,)/q1(s). The zeros of q1{s) are tbc left half

piane zeros of q(s). Tbe zeros of JJ(,s) are equally distribu.tcd betwecn the zeros of p(a) and p(-.1), with tbc rcstriction that if z, is a zero of p(s) then. is a zero of p( -.r). The complcx zcros must be selected in conjugate pairs. (iii) From p(s) determinc Z(s). From equation I I. 19 wc bave

-z,

p(s)

= ± Zi(s)- R1 =± Z(.r)-1 Z 1(.s)

+ R1

Z(s)

+1

where Z(.r) is nonnali.;ced with rcspect to

li.e.!:

R1 = I = Z )• For the positive sign we bave Z(a)= I

+

p(.s) 1- p(s)

and for tbc negative sign I - pC's) Z(.s)= 1

+ p(s)

(il') Expand Z(.r) into continued fraction expansion about infinity

and obtain the laddcr. The two impedanccs dcftned in (iii) lcad to two (dual) laddcrs, ono with termination R. and tbo othcr with termination l/R1• (v) Denormalize the network to yield the required cut-offfrequcncy and terminations.

362 Ndwork Tlieory and Fllter Design We demonstrate the realization procedure by examp1es.

ExAIIPLB 11. 2 Let the insertion-loss characteristic be approximated by a thirdorder Butterworth fi)ter with a cut-off frequency of 1 rad/sec. Lct the terminating resistances R, and R1 be 10. We are required to desia,n an LP filter to satisfy this insertion loss characterlstic. (I) The cut-offfrequency is a1so the pass-band edge. Hence e = 1 and eqn. U. 30 yields I I F(s) - 1 + (-1)1, . - 1-,• The speclfted terminations satist'y eqn. 11. 32 (li} From eqn. 11. 33 P log (0.1005) = 3 _315 log (0.5)

n=4

Let I

Hence

F(s)

I

= 1 + t- 1,- si'= 1 + s8

The specified terminations satisfy eqn. 11.32 (ii) From eqn. 11.33 P(s) P(

-

S) -

~-

- q(s) -

I-

_l- = ~ l + s8 l + s8 eJ,.18, k = I, 2, ..• , 8. Hence

The zeros of q(s) are q,. = the poles of p(s) are el ..,,a, el 7•/8 , el imi• and ei lllf/8 i.e. the poles are p 1, 1 = -0.924 ± j0.383 and Pa., = -0.383±j 0.924. -

364 Network Theory and Filter Design p(.r) has a fourth order uro at s = O. Hence :r' P(.f) = (S- Pi)V-P1)(S -p,) (" - P,) s'= s'+ 2.613s1 + 3.414.s2 + 2.613s+ 1 (iii) From eqn. 11 . 34 ZI) 2.fl+ 2.613s■ + 3.4!4sl + 2.613a+ I \3 2.613.s*+ 3.414s*+ 2.613s+ 1 (i11) The continued fraction expansion is Z(s)

= O. 765.s+

I 1.848.r +

1 l

1.848s+

1 . 0.765s+ T

The normalized network is shown in Fig. 11. 7. I

O. 765

1.848

2

o. 765 I'

2'

Fig. 11. 7

(v) The frequency and impedance denormalizing factors are Oa ... 2n< 25 X IO-)= 157.08 X 103 I l X 101 y.=-1-

aod

The final values of the elements are obtained by

L=Lny,. and C=~

n. Yn!l• The final network is shown in Fig. 11. 8. I kn

4.87 mh

Il .i6 mh 2

V

11.76 nf

4.87 nf

,.

I,

Fig. 11.8

I ko I

lnsertion Loss S:,nthub 36S EXAMPLI! 11. 4 The iosertion Joss speciflcatiods for a LP filter are

Paas-band : O. 3 MHz Pan-band ripple: O. 5 dB High frequency discrimination: 40 dB at 6 MHz

R1 = R1 = 1000 (i) The specifications suggest a Chebyshev approximation. From eqn. 10.2S O.S = 10 log (l + Es) Hence t:1 = 0.122 and • =0.349. From equation 10.24 >-: cosh-1 (1/k1) n ,,_,, cosh (1/k)

Here,

1/k = 6/3 = 2

and

11/l = 286.263

I ()C0/10 - l 1/k1 = [ wu.a,io _ 1

n

and Let n = S.

cosh-1 286.263 cosb-1 2

>-,

""'

4 78

= •

Then

F(s)

=

l 1 + c1 T:(s)

From Tab]e 10. 2 r:(s) = -(16s1 (ii) From eqn. 11. 33

+ 20s8 + 5s)11 = 11{s) 1

p(s)

p(s) p( - s)

= q(s) = _

l - l - •'Ts•(s)

(16s' + 20s8 + 5s)1

+

+

- (I6sli 20s8 5s)11 - l/e1 The zeros of q(s) are the roots of the Chebyshev function filter with n = 5 and • = O. 349. From eqn. 10. 28

A=

o.! + J1 + ! 49

O. 2 ? = 5.597,

a = ½(5. 597 115 - 5. 597-116) =O. 351

:366 N 11-w,wk TJ,,ory and Fi/ter Duign and

b ....

½(5.597 1 + 5.597·· 11) 1 1

1.06

1

The polcs of p(I) are at -a, -a cos JC/5 ±jb sin K/S, cos 2n/S ±Jb sin 2n/5 i.e. the poles are at P0 =-0.351, P1, 1 = - 0.284±j0,623, P1, ,=0.109±} 1,008. The zeros of p(.r) are tbc roots of the equation

-a

(16.t' + 20s' + 5,')I = O These can be obtained by solviog for thc roob of T.(c.>) =aO and substituting 6) = s/J. T1()- O can be written •• COI

i.e. or

(5 cos-1 c.>) = o

S co■-16> == k2TC, k .. 1, 31 5, 1, 9

1m

= 1, 3, 5, 7, 9 c.>= O,± 0.588, ± 0.951 == cos 10, k

Hence and ftnally tbc zeros of p(s) are at Z1 = 0, Z1, t - ±j 0.588, ±} 0,951 Hence, p(8) = (8 - z,) (.r- z,,)'(.t - zJ (8-z.)(.t - z,) (8 - p 1) (a - pJ (a - pJ (.r - p,)(.1-p,\ . s'+ l.25J.rl+0.313a i.e. p(a)= a•+J.171.,.+l.935a'+l.307a•+0. 75s+0.167 (iii) From eqa. 11.34 zr > 2a1 +1.111at+3.186s•+t.307r+1.063a+0.167 \8 l.17Js4 +0.684s8+1.307.r1+0.437s+O. 167 (i•) The continued fraction expansion of Z(s) is

1- - - - - - - - - Z(s)= l.707s+ - - 1.298.t+ - - - - - - - - - 2.69.r + I 1.09.r

+

· 1. 944.r

The normalized ti.Iter is shown in Fig. 11.9. (•) The frequency and impedance scaling factors are Q 21t(3 X 10-) ]00 Il= 1 t "f■ = -. The denormalized network is shown io Fig. 11 . IO.

1

+ -.

Insertton Los:, Synthem 367 2.69

1.944 2

2'

I' Fig. 11.9

100 o

9.06 flh

14.27 µI,

I0.31 flh

•• I' Fig. 11.10

11.3 Discussioos

We have discussed the design of LP filters only. As already described we cao use frequeocy transformation technique (see scc. 10.S and 10.6). to design high pass, baodpass and band elimination filters. The insortion loss specification of HP, BP or BE filters is first converted to that of an LP filtcr and the LP ladder is realized as iodìcated. Then the LC elements of tbc LP filters aro trans• formed by the transformation given in sec. 10. 5. la the design of BP filters wc once again stress the point that the application of the spectral transformation technique leads to geometrically symmetric flltcr.

Example 10.2 to 10.4 bring out some interesting features of the Darlington design procedure. The Butterworth approximation of the insertion loss leads to a reflection coefficient with ali its zeros at the t:1rigin. If Chebysbev approximation is used instead, the zeros of the re:flection coefficient are all on the imaginary axis. Tbc transfer functions of tho filters generatcd. are ali all-pole functions. Ifwe need finite transmission zeros elsewhere-other than the origin-we bave to resort.to approximation techniques other tban the Butterworth and Chebyi;hev as ind1cated earlier. The continued fraction expansion of Z(s) by Euclids algorithm may nm into numerica) problems and to alleviate this the coefficients must be represented with the sufficient àccuracy. The

368 Network Theory and Filter Design

last term io tbc cxpa.nsion must be a constant equal to tbc load resistanec. Tbough we bave developed Z(s) into a ladder network, any other method of realfaing the input impedance is equally suitable. In the examples we bave consìdered ali the transmission zeros to be at infinlty and hence a ladder was the most convenient network to realize. We may resort to otber realization methods (other than continued fraction expansion) as loog as tbc network obtabed bas 1peciftcally tbe required transmissioo zeros aod the t~rmiuating resistances. If tbc specifications lead to an insertioo loss function with fln.ite transmission zeros elsewbere other tban tbe origin (Inverse Cbebyshev and elliptic approximations for example) but in the LH pian, then tbc ladder has to be developed by zero shift• ing or pole weakeoiog technique. If we coosider a non-minimum pbase network obviously we cannot resort to ladder structure. A largc part of Chap. 9 was devoted to the realization ofa driving poiot fÙnction. This is often the case in most network synthc1is books. As we bave seen from scc. 9. S and the Darlington procedure. the realization of the trànsfor or insertion loss function fundamentally depends on the realization of a driving point function. The coefficient matching techoique on the other band docs not depend on the driving point function realization. The realization of active filters mostly utilizes coeffi.cieut matching technique and this forms the topic for the next chapter.

PROBLEMS 11 .1

A doubly terminated two-port is shown in Fig. P. 11 .1.

Find the

lnsertion-voltagc ratio and power ratio. Also timi tbe lnsertion pbaso charactcristic.

Plot l'l(t.>) vs

61.

li . ..,. R1 =R2 Li= l C2 = I

=

I

Fig. P. 11.I 11.2

Find p(s) and Z(s) for the following ftlter spccifìcations: (a)

.-.

= K(l +

w 111)

lns~rtion Lo,, S~nthelis 369

=

(0 n n

(ii)

(b)

= K (1 + c1 ,:Cw)J (1) 11 = 3, e - 0.125, K = I, RJR.1 = 1 (li) 11 ... 3, f = l .Kl, R 1/R1 = l

~

(UO 11.3

U .4 11.5

11.f lJ,7 11. a

1, Jei/Ra = 3 RifR1 """ I

2, K -

= 3, K ..., I,

11

~ 2. e

=

0.125, R~/R, =- 1/2.

=

=

Deslln a low-pau Bu.ttei'Wm:tb filter ror n 3, 0>e1 3 t radi•· Ri R1 =- 1 kO b)' cae8loient matchins toclmiquo. If Rs - 2.Ri, ancl Ra 1 kO In prob. Il. 3; flnd tbe elanental valuel or 11ae :euu.worth &Iter. DtllJD a low-pas ftlter for tbe ..,.:fflcad11111 of prob, 10.J wltb Ril.Rt 600 0, DelilB a low-pus fiJcer ror tbo spec:i&ations of prob. 10.2 wttb Riflls 3, .1Ct - 100 O. Desiln a OIObylbeY fl.ltor· ror tbe speeillcationa or prob. 10.5 wlth i 1 1 W. Solect tho propor valuo or ~J>eailn a Cbebyahff atter tor the 1pccillcatlon1 or prob. 10 .1 with

=

=

=

=

= lii = dOO O.

.. Select tbc propcr value of R1 • Deslp a dllrd arder band-paas Butt.enrorth fìltcr with bandwidth == 20 ICBz, ccatre frequençy = 1 MHz and R1 = Re = 600. 11.10 Delisn a band-paa■ 11.lter for tho follow1D1 speciicatioas. Tbc llPPol' and Jower 3 clB poims aro at !MO ICRz and 1060 KHz. A loll ol at leatt 20 clB at 910 kC and 1090 kC R1 - R 1 = 600. 11.ll A doubly terminated LC Dcitwork with R1 and R1 •• terminatinl msiltanc:e1 il specified by tho inscrtion-loa IL and aiteouation ac(c.>).

11.9

Show that a.(r.u)

= IL + 10 Jos (Ri + R,J• 4RtR1

n .12

Delisn a third order Battenrorth film with 3 de paa-band loa an4 11. =- 3Jli.

11.13 Tfle IDM1ion loss specifìcatiom for a low-pus Butterworth iltcr aro the foDowms: Pau-band lou : I. 5 dB Stop-band Iosa : 2~ dB at 3,3M rad/tee

: O to IM

Pass-band R1

=

R

1500,::,

R,

=

rad/flJC

2

Desisn a LC laddcr for those spec:ifications. 11.14

A QiobyahcY.iltcr is rcqulrcd io satisfy the followina imertion IOls

a,qwRIIDCDtl: ~band

: O to 25 k radfsec

370 NdWOrk. Theory ond Fllter Dulp

Paa-band ripp)a: 1.~ dB Stop-band : 30 dB at SO k_ ra,lf,eç The filter il to be driVBD by • voi,-. ~ wlth &D in1erll&J impedanc:e of I SO Q. DmillD a fUter to aad.r,, tbcle ~

NOTES AND REPERl',NCES

'ne concept of llllertlon lou, iDtrodllOCII bere !or a lwnped Jlnear pushe ndwort:, pia)'■ an import role in tbe -■illD or VHP/UHF and mklmwalll also. Horo we haw, lbown tbe applic:adoll of ti. c:oncept in 11111' deslp. Cblln's book1 di,cusses 1hJa toplc In colllfdcrable delail, f'mther cle1ails am be bad from ~ of Chapter J. Insertlon loss synthesis wu introduced by Darllnpon In bis celebratocl paper on . synthesis or roactance 4-poles and bu to date atayod u tbo mmt powcrful mcthod of desl11Din1 doubly terminated filter netwod,1. I. W.H. Chen: Linea.r Notwork Dcsip and ayntheais, McGn.w-Bill, 197,. 2. s. Darllnaton: Syntbeals ol R.eactançe 4-pole which produce Pl'lllcribed Insctium, Loa Oaracleriatics. J. Math llbd Pllylicl,. Vol. 18, 1939. pp, 1SMS3.

mcwt

12 Active Networb and Fllters

A wide class of networks a.nd ftlters that bas rec:eived considerable attcntion in the Jast two decades is the aetiftRC aietworb. Some of the limitation1 of the passive ftlters, viz., the sbe of the inducton at low frequencies, the necessity of buffer or i10Jation ampli8ers to prevent loading while cascading sections of tllters, and the aeed for an extemal amplifler to adjust the required gain bave led to intensive research and development work in the field of .actiw ftlters, The present trend to microminiaturize the circuit in order to increue the package density .and the reliability of tlle cìrcuit bu also been an ~mportant factor. Unfortunately, it has not been possible to integrate inductors with practical value of in~, tances and reasonable Q-factors. The availability blgh quality inexpemive monolithic operational amplifier as an oft'-thc-shetf unit has made the active filter design an attractive one. A.a indicated earlier RC ftlters can at beat realize a Q of 0.5 and bencc cannot be used to design a bigh selcc:tivity filter. Most of the actiw ftlter structures are reali7.ed by cmbedding an operational amplifter in RC network in a feedback codguration, providiJla high g. networks. In this chapter we diséuss tbe design ot active RC fllters. Wc introduce the commonly employed activ.e elements. Tbe ~ncepi or 1e11sitivity wbich plays a major role 'ià thè acti,re Ster desip ia allo diacossed. Sevcral single amplifler and a few multiamplitler 81ten are also prcsentéd. lndu.ctance and ladder 1imula1ion for hqller ordor :O.lten is al10 int)'oduced.

or

13.1 ActlTe Ble...a.

Network cic,mcnts cao be clalsified as active aad p&llive. An element i1 comidcrocl activo if it caa sapply . . . . to elle extenm

370 NdWOrk. Theory ond Fllter Dulp

Paa-band ripp)a: 1.~ dB Stop-band : 30 dB at SO k_ ra,lf,eç The filter il to be driVBD by • voi,-. ~ wlth &D in1erll&J impedanc:e of I SO Q. DmillD a fUter to aad.r,, tbcle ~

NOTES AND REPERl',NCES

'ne concept of llllertlon lou, iDtrodllOCII bere !or a lwnped Jlnear pushe ndwort:, pia)'■ an import role in tbe -■illD or VHP/UHF and mklmwalll also. Horo we haw, lbown tbe applic:adoll of ti. c:oncept in 11111' deslp. Cblln's book1 di,cusses 1hJa toplc In colllfdcrable delail, f'mther cle1ails am be bad from ~ of Chapter J. Insertlon loss synthesis wu introduced by Darllnpon In bis celebratocl paper on . synthesis or roactance 4-poles and bu to date atayod u tbo mmt powcrful mcthod of desl11Din1 doubly terminated filter netwod,1. I. W.H. Chen: Linea.r Notwork Dcsip and ayntheais, McGn.w-Bill, 197,. 2. s. Darllnaton: Syntbeals ol R.eactançe 4-pole which produce Pl'lllcribed Insctium, Loa Oaracleriatics. J. Math llbd Pllylicl,. Vol. 18, 1939. pp, 1SMS3.

mcwt

12· Aetive Networb ud Fllten

A widc class of networks and ftlters that bas RCeivcd considera able attention in tbc Jaat two decadcs is the actiftllC networb. Some or the limitation1 of the passive ffltcn, viz.1 tbe size or the inducton at low frequencies, tbe nccessity of buffer or isolatioa ampli8crs to proveat loading whilo cucading scctions of ftltcrs. aad tbe aeed for an cxternal ampli&er to adjust the requlrcd gain have led to intensive research and developmcat work in tbc field of active 8Jten. Tbc present trend to microminiatorize tbc circuit in ·order to incrcue the pacbge density and tbe relfability of the cireuit hu allo been an \mportant raetor. Unfortunately. it has not been. pouible to integrate in.ductors with practical value of induc-' taneea and reuonable Q-factors. Tbo availability of blgb. quality inapemive monolithic operatioul ampliler as an oft'-the-shelf nit ba made tbc active ftlter design aa attractive one. 14 indicatcd earlier RC ftlters can at belt rcalu.e a Q of O. .Sud beace cannot be oaed to design a higb sclcctivity filter. Most of the active tilter stmctures are reamed by embedding an opcr11tional ampliler in R.O network in a feedback codgoratioo, providiJla biahQ.networks. In this c:hapt:or we diséals the desip of active RC :lllters. Wo introduco tho commonly employccl activ.e clemente. The ~ncept of 1C11sitivity which playa a major rolo'hi thè actiw-4ìl.- desip il· allo dilCtlSled. Soverat single amplifter and a rèw miattiampU&or 81ten are aJso presca.Séd. tnductaDce and ladder limulatioa for bialler ordor ftlten is allO int,oduced. IZ. I A.ctlYe BI•••

Network_ olcmèntl caa be clulilled •• activt ud paaive. Aa elemcmt i1 ~ ICliYe irit·oaa ~ .....-, to dle e1.....a

372 Network Tlllt,ry attd Fllttr Dt,Jgn world. lf •(t) and l(t) are the instantaneoua Toliage aad corrent.

respectively, at the terminals of tbc element tben tbe element la active if, and only if {W(t.. t)+E (t8)} ~ O for some t (see IICCl,3). From thia defìnition it isclearthat tbelumped R; L ml C elements defincd in cbapter one are all passive. In contrast a transistor 1s an actiTe element: Of the many active elements 118Cd in ftlters •. tbc most important one ia the operational amplifiet, Of late sevoral active elementi lite gyrators, generalised impedance convertors are becoming an intearal part of active ftlters. However a controlled source (ICC sec. 1.1. 2) may be c:oasidcred a.a the baaic activc elemcnt. Otber aetivc elements to be diacussed later in thia .section can be realized by 11Siag controlled aourcca. Controlled sourcca aro clasai8ed in fòur dift'erent ways. They are (i) Current-control/ed Cfll'l'fflt aource (CC'l'), e.a, a transistor. (li) Yoltap-controlled current aource (VC'I'), e.g. field e1roct

transistor. Yoliage-controlled ,olto,e aource (VVT), e.g. aa operatioaal amplifter. (ThQuah an operational ampliftor is reaU-S by usiog transistor or FBTs. we comider operational ampliler .as a buie element.) (lv) Cu"ent-controlled 110/tage aourCtJ (CVI). Piaure 12.1 shows the diqramatic representation of the ideal c:oatrollcd sources. The transmission parameter matric:es ud Che (Ili)

input ancl output impeduces (R, and R., respcctively) ol tàe four oontrollcd sowces are

'J

,J

.: . l f ,.' ~~

v,

,.,. :t./,

'+ J'3 -

,.v,.

,,.

+

l' -..,_ _ _ _ _ _ _.. VVT (ç)

VCT (b)

,.v,

; 11,

111',

,,-

CCT

(a)

2

I -,:

Vz

li- 1V1•

2·

:, ' • :.: ICVT~' ,1, Cd)

Pls,12.J

1'2 •

A.eme Network, l1IUI Fllter, 373 CCT:

[:

VCT:

[:

vvr: CVT;

[~ [!

R,-0

-D _}_]

R. = CIO R,

g •

= 00

o

Ro= CO

:1

R1 =0 Ro=O

R,-o

:}

Ro=O

lt can be observed from the transmiasion parameter matrices that we can generate a VVT by cascading a VCT and a CVT. Wecan 11.lso generate a CCT by cascading a CVT and a VCT.

12.1.1

()p11RAnoNAL AfilPLIFIBll

An..operational amplifier (op. amp) is a high gain differential input dc amplifier. ldeally it is considercd as a dual input infinite gain VVT. Tbe symbol and the equivalent cl.rcuit of an op. amp are shown in Fig. 12.2. Output voltage Ye is given -by V,= - .4.V = -.A(Y1 .:... V1) i.e. V0 = A(V1 -VJ (12.1) J .l

~~

t>

Vo

o

l

"•l

.e

2 Vi•

o0

+

J

+ AV

VQ

+

Fig. 12.2

lo an ideai op. amp À is considered to be infioity and the amplilìer ìs assumed to bave in:ftnite bandwidth. Also, the input impedance and tbe output admittance are assumed to be :···ilnit;y. Because of infinite gain A, the "differential voltage V is zcr. and

374 NdWOrk

r,-., t1Nl n,., »a1,,. or

if 011e or tbe tenninall an op. amp (i.e. either 1 or 2) in feedback conftguration i1 grounded·then tbc other ia a virtual p,Ulld. lt is abo aaaumed that an op. amp has .iero oft' set, I.e. the oatpat ft>bac Y• ia zero whenthe dift'ereotial voltap (Y1 - Y.) is made mo. Tbc current drawn by tbe two input · terminala of an op. amp ia zero as the input impedance i, auumed to be inftnify. J\2

Rt

v,

v. Rz Yo=--V1 R1

R, -

R1IIR2

(lnvertins amplifìer)

(a)

R:z R,-

~.!

VN

1

t> R2

I•

v.

V. _ R1 + Rz V. o--Ru 1-

(Non lnvertin& awplilier)

li'o

>---..... Vo

=-

•I

R1( 0

''!Il

V,

I

+ R-12 V,.) I •

(I nverting ,,ummer}

e

+

I .1CR1 V. (tnverring integrator) Vo= - -

:Pw, 12.3

Actt,e Networb llllll Fllter• 375 Op. 1111ps an rarely operated in open loop mode. Some of the feedback mode opcrations and tbe corresponding terminal relations ~ abown in Fig. 12.3. Op. amps·find extensive use in linear and nonlinear networb. A practical op. amp deviates considerably from its idcal behaviou.r and we discuss this in a later section. 12.1.2 NBGATIVE IMP&DANCB CONVEll'I'lllt A negati,e impedallce camerter (NIC) is a two-port devke whose iaput impedance when terminate = J
0 aod Q sensitivities for a second-order LP 1Uter. lt can be aimilarly shown that the relatioi;iship is truo for HP and BP filters. We bave deftned severa( sensitivity functions and interrelated them. However, in tbc :final analysis wc are interested in minimizing the deviation of the :filter response due to ìncremental variation of some network. parameter. In highly selective networks (i.e. bigh Q networks) tbe pole scnsitivity is an important factor to coosider because of the stability of the network. In the design of active :fìlters the sensitivity of thè filter transfer function to the variadon of the active parameters is a major coosideration. The study of 1eositivity is an integrai part of the design of active filters. In this section wc bave just introduced some the fundamental definitiona aad interrelation that in sensitivity ltUdics.

occur

or

394 Network Theory and Filter Design 12.A Single Amplifter FilteH

In Chapter Il, we considered the realisation of passive LC filters terminated in a load resistance. These filters still find extensive use in communication systems. However tbe present teodency is to design active RC filters to replace the passive LC :ftlters wherever possible. (Over the last decade and half, digitai filters ari finding wide use. Such filters are not considered io this book). This, as already indicated, is motivatcd by a desire to miniaturisc the filter circuits. Most active filters are generated by embedding an operational amplifier in a passive RC network. Active network design bas receivcd considerable attention from many authors (see notes). Though we can dcvelop filter realizations based on network syntbesis methods, here we resort io coefficient matching technique as an efficient metbod of filter design. The synthesis problem starts, as in the passive case, from the approximation technique. The tra.nsfer function so obtained is realised by an active RC network. The active element used is, in genera]. an operational amplifier. In this section we consider only single amplifier second order filter structures. Tbe generai configuration is shown in Fig. 12.29. Here N is a three port RC network I

+

N

2

3

1"2

v.

-

I'

Pig. 12.29

(porta are 11 ', 21' and 31 ') aod µ. it a finite gain ampJifier (sce Fig. 12. 3). Generai aaa)ysis of such networks are considered in books on active networks. Here wc consider tbe design LP, HP, BP and BE filtcrs only.

12.4.l

LoW-PASS FILTBR

A circuit reali.zing a second order low•pass .ftlter with a positive gaio amplifier is shown in Fig. 12. 30. The transfer function of the circuit is (see example 12.4).

Active Network& and Filters 395

e,

lt-1---

Ra ~

1!: I

[;>~l--..

r·

o

Fig. 12.30

T(a)= - - l s'+ [ Re

(L

I

(R1R:C1c) ~----I ] I

(12.21)

11 +Re11 +ReI l (l-µ) s + R114.clea

The generai transfer function of a second order LP filter is

=

T(s)

(12.22)

.vi!+ ùlo s/Q+ (I):

where ùlo, Q and K are the resonant frequency (peak response frequency), quality factor and dc gain, respectively, of the filter. On comparing the coefficients in eqns. 12.21 and 12.22. l 1 K= µ,(ilo=--;-=:==::;.:====:=-• Q= ___,==--~==-----~=2 2 +JR1 C3 + (l-µ) JR1 C1 'V R1RzC1 C2

JRR C C

R,Cs

R1 C1

1

1

The w~ and Q sensitivities of this filter with respect to the passive and the active elements are

s•o = s•o = S"c'1 = S"'' =Ra Ro C1

s;•-= O ((,)

0

and

1/2

is not a function of µ)

sj, = - sj. = Q

J;•~• - ~, 1

l

JCC (R•• R+RRa) - 21 S~ = Q [JRR CC+ JR,Ca + JR2C l R C R~C, o

a =

Se,=, - Se.

Q

1

1

8

2

2

I

V

I

l

2

1] -

1

1

Though the (ilo sensitivitics are low, the Q sensitivities are proportional to Q. As can be observed from the transfer function, the

396 Network Theory and Filter Design

structure will become unstable for values of µ greater than [ I

+ ~: ( 1 +

!:)J

In active filters built in thin :film form it is customary to keep the elemental spread (the ratio of maximum to minimum value) as small as possible. In the structure under consideration, if we select egual valued elements, i.e. R1 = Ra= R and cl = e, = e, we bave I 1 r.>o = RC' Q = 3 - tJ. The design equations for this second order filtel' circuit are:

RC

EXAMPLE

I

= rilo - ,

I

1.1.

= 3 - -Q

12.S

We are required to design a second-order LP Butterworth filter witb a cut-offfrequency of 2. 5 kHz. From Table 10. l the transfer function of Butterworth filter with a cut-off frequency of 1 rad/sec and a dc gain of µ. is T(s) -

µ.

- s2 +2s+1

Denormalizing the cut-off frequency to 21t(2. 5 x 103) we bave the desirea transfer function Jì( ) _ ~ - ~ - . µ (21t X 2. 5 X l{)ll)2 s - s 1 + s (21t X 2. 5 X 103) y2 + (2nx-t:S X 103) 1 From the design equations (Q = I/ y2, w 0 = 21t x 2. 5 x l 03) µ = 3 - y2 = I. 586 1 RC= 2.467x 108 If we select R = lO kn, then C = 0.405 pf. Instead of choosing egual valued capacitors as indicated we can select a unity gain amplifier instead (voltage. follower) and R 1 = R 2 = R. This selcction yiclds 1 Wo= --==== Q= ✓Ci RyC1C1 ' C2 Then design equations are 2Q 1 RC1 = - and RC1 = 2 Q Wo w0

Acti'le Network-a and Filter.r 397 and the Q sensitivities are

s~. = s~a = o, s~, = -s~. =

!. s;.

~Q•.

A positive gaio realization leads to a low-iain amplifier but the Q scnsitivities are large. This is a geueral !eature of ali positive gain structures. RJ

e,

Fig.. 12.31

A circuit realizing a second-order LP fìlter with a negative gain amplifier is shown in Fig. 12. 31. The transfer function is .. ( s•

l

)

R1R1C1C1

- IJ.

+ [..!..(...!.. + Ri _!_ + R.!_) + _!_(..!.. + ..!..)} C R C R1 R1 1 1

1

+R R ~ C

1B1S

1

[t + ;, (I + I

IJ.)

+!

1( I

1 + RR, I

+ RRt1 ) ]

The resistor R, can be considered as a part of tbc gaio producing network. Here again we consider equal elemental valued realiza• tion, i.e. R1 = R 1 = R 8 = R, = R and C1 = C1 = C. Tbc reso• nant frequency and Q-factors are then (,>o

= v'~. Q =

v5;+- p.

Hence tbe design equationa are 1.1.

=

25Q 2 -5, RC = 5Q c.>o

It can be sbown tbat the magnitude .of ro0 and Q sensitivities for Q > 1/2 are all less than or equal to 1/2. The closed loop gain ofthe amplifier for a reasonable Q is very large (for example Q = 10 yields 1.1. = 2495). This is a common feature of all negative gain realizations.

398 Network Theory and Filter Design

e, Fig. 12. 32

A second-order LP filter can also be realized by using multiple feedback infinite gain amplifter structure of Fig. 12. 32. The transfer function is 7i( ) _ S-

R 1R 9C1C,

1(1 l 1) 1 a +s- -+-+- +-~2

C1 R1

R2

R8

lfwe select R 1

=

R, = R 1 = R we bave

610

=

1 · I Rv'C1 C1 aod Q = 3

R1 R8 C1 C 2

JCe:

Themagoitude ofthe ro 0 and tbe Q sensitivities with respect to the passive elements are less than or equal to half. The element spread in this case is 9Q1 (i.e. C1/C 2 = 9Q2). We bave discussed three single amplifter LP filters. On the basis of sensitivity alone the infinite gain and negative gaio amplifter structures are superior to the positive gaio structure. The latter uses a low gaio amplifter. If we consider the gain-sensitivity product (i.e. µ, Sf) as a measure, the positive gain structure is superior to the other two. In many practical filter design situations the positive gaio realization is preferred to the other two. 12.4.2 HP, BP

AND

BE

FILTERS

The method of spectral traosformations developed in Chap. 11 is not applicable for designing active RC, HP, BP and BE filters as this would ncc•ssitate the inclusion of an ioductor in the circuit. The design of these filters is similar to that of the LP filters. The structures of various filters, their transfer functions, ro 0 and Q sensitivities and design equations are summarized in Table 12 .1.

Table 12,1

Trans/e, /lllll!tion

Structure

(I) HP: Positive gain

K=!J,, /(s)=s•

ao R,

e, o-;~~ (A)

Sen&itiVlties

1

st=Sa!=Sé!=sé: =-2

R1 =R 1 =A,

s;•=o.

µ.=3--,

1

•

C1 =C1 =C

R 1RaC1C1

1(1 1) 1-1,& '1i = ~ C + C. + R1C1

1

1 Q

l

Si--.ygR .,.,I =--2Q, 2

J

1

RC=(,)a

o = ~.....,=..,......,,,=-V(l i-i,)R1R 1C1C1

Q-

R1=R1 =R

SC:=-i

I

1

1

Duip equationJ

SensitMtita

1

-. ~

~

~

§ ~

-

Table 12.1 (Conthul,d)

;g

~

Struotur, (6) BP: Iunnito Pin

T,a,u/e, /111UJtlon

1

x .... - - . /(1)=1 R1C1

---

R1

(F)

1 11o• R1RaClC1

0i=2.. (.! + !.\, 11 C1 c"J 1

1

mo- vRiR.C1c1'

,/RJR'i

O=j.S+j~ c. e,

S,,utttvilie1

Ihilgn eq110lion.s __I

st=s;;=sc:=~=

1

=-2

a

C1=C1=C

~

J~='-0

:::,

!iEl,, ~

R,

1

sl=-si.=-2· 1

~

1

C,/RR,=I

s~ =sS.=o

"'"

~

619

I;:,

l

OQ

;,s

Active Networks Qltd Filler, 405 Tbc transfer function of these structures is given by Tf) K/(.t)

\, ... .r•

+ a,, + '1o

= __K::;...,,l{'-'.t)_ _ s• + !:!!, + 61o1 Q Hcnce

6)0

= V~,

Q

=

va, and /{.t) determines the zcrot. a.

Tbc transrcr. function for a notch (or band elimination fllter) is

T(,)

=

+ a1) + Q s + 611o -

K(al + b)

K(al

s•

6)•

,t

+ a1s + rfJ

There diff'erent situations may arise depending on a:, (I) High-pas.t notck: Dt < 6) 0 Cii) Low-pass notch: a. > ll>e (lii) Aclive notch: a.= Ci>9 • (Observe that low-pass and high-pass notch, transfer functions are tbc familiar low•pass and high-pass functions with a finite zero of transmission.) Table 12.2 shows tbestructuresand otherparticulars of these thrcc filters. In HP notch and LP notch tbc clemcmt spread is controlled by k and can be arbitrarily selected. ExAMPLB

12.6

Consider the LP notch transfer function

T,() 8

= K(s1 + 1.5) si+ 0.2r + l

From the design equations wc bave

and

a

=

11

= 2+

1 vl.5 1

e= (~ - 1)-1-1

k!

1

k

+

I ( \ 5-1-v'Oxo.2)

If wc select k as I, then a= 0.816, e= 0.250, ~ = 2.128 The finite gain realisations considered bere were the earliest to be introduced and are known as Sallen.Key structure,. The baod climioation structures using twin-T networks are known as Kerwin structures. There are many other structurcs realizing theso standard filters.

J.IIDIC 1.• .i.

(1) HP Notcb

1

K=l'•b=

+l

a

al

1

= -,

R

1] a•1 k+ 2-l'J di= - -l[-l + R-

i1o

k+ = [ l + --it. a

i~

k+ 1 = --.--

~

~

(l)•-1

ix

~

~

»~'6n eqaation

Transfer function

Structure

kfk

§

~

r,.1

"=

k

2

'-i

r(,)I

+ kTiL-;.-

1-

Wo] ~

2

l

~

f b

tl

rSQ·

(2) LP Notch

k/k

K - (k

+I

~_!_ + 1) C + 1' - al µ.

b

1 a=-,

«

I

48

•

•

•

= -all[-(k-+-1)_C_+_l] (k

•

+ l){C + (2 - fl,)/k] + l)C + l}

"i - - - a[(k

I'-

C=(:~ -1 )/ j (i.e., ali the entrics below the diagonal) tbeo such a matrix is an uppcr triangolar matrix. On the other band, ifa,, = O for j > i then il is a lower triangular matrix. If a matrix bas only one row it is called a row matrix or row vector while a matrix with only one column is called a column matrix or a vector. lf a matrix A is such that A = A', then A is called a symmetric matrix. If A= A 8 , then A isa Hermitianmatrix. rr A =-A'(-All),

a,

426 Network Tkeory antl Filter Design 3.

S.K. Mitra:

Analyail lllkl S)'Dthcsis of Linear Active Networks, Wl)ey,

1969. 4.

O.S. l,JOIIClaytz:

LiDcar IDtcpJted Networb, Voi. l AN 2. Van.

~

.s. 6.

.lleillhold, 1'74• c.s. Liadlqul1t: Active Network 0.ilà with Sisaal . Filtòrioa .. AtJPlications, Stew.tna and Som 1tn, L.T. Bruton: Activo Clrcuits Tbeoiy md Desiin, Pcentke Hall, Bu,iewoodCJUl's,19fO.

a.e

Sodra and Ji.o; Bro'utf:

Jl'iltcriDI Tb011 au4

De'iip..:.Adh'e

7,

A.S.

,.

K. llamakrlslma, K. 8olmduarajaa aD4 v .K. Aairo: Bfoct or AmPlller Imperl'octlons. on Ndve Networka, WBE Transactioas oa ·Qsuics a8' Systoms, Vot CAS 26, No. 11, 1979,.pp 1122-931,

ud Passive, Matrix, 1918'. · · 8. A.B. WiÌliams: Attive l'w. Doilp, Artsh Houie, l!175; · '

Appentlix A

A matrix is a rectaogular array of num"el;s representiog ,IÌ ti~asf«mation. Por example, tbc matrix '2u of 2.6 transforms set of cbord currents I,, to a se. of brancb currents l 0 of a given network with respect to a selected tree. · A matrix •.4• with m rows and n columns is represented as

eqn.

a:

A= [a11J.x. wbere a0 is the element in the itb row and Jth column of A. (Here A has no bearing on the incidence matrix of a graph). Tbe order of the matrix is (mx n). If m = n, then the matrix is called a •square matrix'. The matrix obtained by intcrchanging the rows and colwnns of a matrix is callcd the 'transpose' of matrix A and is reprcsented by À' À1

= [a,iJ,.,cm

witb A as de.fined earlier. If tbe elementi a, 1 of a matrix are complex, the conjugate of the transpose is called the •Hermitian traospose• and is represented by AH. Two matrices A and B of the same order defined by A = [ao],.x,. and B = [b11J,.x11 are said to be equal, ifa,,= b11 forali i= I, 2, ...m, andJ = 1, 2, ...n. Ifin a square matrix A, a;,= O forali I ,;éjthen tbe matrix is a •diagonal' matrix. If all the diagonal entrics of a diagonal matrix are I, then it is called a •unit or identity• matrix and is represented by U. lf in a square matrix À, 1 = O fori > j (i.e., ali the entries below the diagonal) then sucb a matrix is an upper triangular matrix. On tbc othcr band, if a11 = O for j > i then it is a Iower triangular matrix. Ifa matrix bas only one row it is called a row matrix or row vector while a matrix with only one column is caJled a column matrix or a vector. lf a matrix A is such that A = A', then A is callcd a symmetric matrix. I( A = AH, then A is a Hermitian matrix. If À = -A'(-A li),

a,

428 Network Theory 111111 Flller De&lgn theo it il called a skew (skew-Hermitian) matrix. lt can bo easily soea that tbc diagooal eotries or a skew-matrix are aD aro.

A .1 M'ATR.OC! OPBRATIONS

Addltion: Let

À and B be two matrice, defined by A. == (alJJ.x,. (b11:L.x. then tbe sum of tbc two matrices is another matrix of the salile order, (mxn), defined by C = [c,11,.,c,. wbere· c11 - a,1 + b,1• Ptopmlea of this operatio~ are (a) A +"B = B +, ..4 (oomD1utative) (b) A + (B + C) = (A B) e (associative) -(e) (A+ B)' = A. 1 B'

B

=

+

+

+

Scalar multlp/Jcation: The scalar multiplicadon of a matrix A. by

à scalar a is defined as «A

= I« au),,.x,.

Multiplic.ation: Let A and B be two .(mx n) and (n xp) matrice&, respectively. The product of the two matrices, AB, is another matrix of order mXp, defined by

C where

= AB = [c11J,,.>.U) is a singular matrix. The n scalar constants which render thc matri1 (A - ">.U) singular are called the eigenvalues of A (latent roots or characteristic roots) and the non-triviai solution vectors are called the 'eigenvectors• of A. The study of such a problem is classifled as 'Eigenvaluc problcm',

434

Network Theory and Filter Design

The n eigenvalues of A can be determined by the equation. det (A - ì.U) = O Tbc determinaot can be written as _\n

+

ot,._,An·•l

+ 0t1n-2Àn-1 + ... + IZ1À + «o =

O

and this is called the 'characteristic equation of the matrix. The n roots of this equation :>..,. i = 1, 2, ... , 11 are the eigenvalues of A. For each eigenvalue of the matrix we can form an cquation. AX1 ._. i\1X1 The solution vector X, is the eigenvector corresponding to tbe cigenvalue :>..1• ExAMPLE:

Let

A=[-:

-I 2

-1 Then the characteristic equation is

-1

\A-AU\-ddt: [ 2-A

i.e.,

2

1 -1

det -1

}~

À

o

j}-o

1

2-A 1

-1

o

I J=O

2-

i.e., ;.• - 6>.1 + 1b. - 6 = O The eigenvalues are 1, 2 and 3. i.e., (1 - l)(?. - 2}(?. - 3) = i\1 - 6?.1 + Ili. - 6 = O (Observe tbat tbc product ohbe eigcnvalues is tbc detcrminant of tbc matrix and tbc sum of tbc eigenvalues is the trace of tbc matrix. A trace of a matrix is the sum of its diagonal cntries.} To ftnd tbc ci,cnvcctors we bave to solvc the equation

[ _: -: -1

I

:] [::] = 2

x3

A, [::] X3

Appendix A 435 For

A1

;\I=

:>. 8

x. x1)' = [l 1 O) 2, [X1 x, Xa)' = [l 1 1] 3, [x1 x, Xa]' = [O l I]

= 1, =

[Xi

The following results can be easily derived. (i) The eigenvalues of a real symmetric matrix are all real. (ii) A skew symmetric matrix has imaginary eigenvalues.

(iii) lf a matrix is singular then it bas a zero eigenvalue. (iv) Tbc eigenvectors associated with distinct eigenvalucs of a matrix are linearly independent. (v} The eigenvectors associated with distinct eigenvalues ot a real symmetric matrix are orthogonal.

A matrix whose eigenvalue!I are ali positive (negative) is called a positive definite (negative definite) matrix. Otherwise, it is said to be indefinite. lf a matrix has à zero eigenvalue tben it can only be a semidefinite matrix.' Two matrices À and B of tbc same order are said to be similar if there exists a 1lon-singular matrix P such that

= P- 1AP

B

Simìlarly rclationship is an equivalence relationship. Similat matrices bave the same determinant, same characteristic equation and hence same eigenvalues, but not necessarily the same eigen• vectors. Ifa matrix has dis,inct eigenvalues, then it is similar to a diagonal matrix, the entries of which are the eigenvalues of the originai matrix, i.e., if A has distinct eigeovalues then,

S-lAS wbere b. = dia (À 1, À1 , i'-.3 , ....... >-n)• can be rewritten as, AS

If C1 are the columns of S, i

=

AC,

=

=

A

This

similarity relatioosbip

Sfl

l, 2, ...... , n, then we bave

=

1. 1C 1

indicating that C; is the eigenvector of A with respect to tbc eigenvalue ">..1• Hencc we conclude that "an ntb order matrix with distinct cigenvalues is similar to a diagonal matrix." The traos-formation matrix for this is the eigenvector matrix of the n iodependent eigenvectors of A.

4M Network Theory and Fllter Deaign For lhe exampJe consi(lered earlier

S

= [: O

:

:]

1

1

and ~•

=[ : -1

o 2

o

:J

If À does not havc distinct eigenvalues then in genera] it cannot be diagonalized by similarity transformation. In sucb a case the matrix can be reduced to its Jordan canonic form by similarity transformation. p-1AP = J i.e., O 0 ......... 0

where

J

=

o ...............

J,,.('>.1r.)

i= 1,2, ... , k are the eigenvalues of A and Àt repeats i.e. r 1 r1 + '" = n. The Jordan blocks J,1 (À/) are of tbc form

À1,

+ + ...

J,, (À,) =-

r;..,

1

o.~ .... o7

O

>.,

1. ..... O

r,

times,

:

1

LO ................. À,jflCr1 The proofs of many of the results considered bere and the otber aspects of matrix analysis bave been dealt with in great detail in R. BeUman's book: lntroduclion to Matrix Ànalysis, McGraw-HilJ.

Appendix B Laplace Transforms

The time•domain network equations in generai a.re integrodifferential equations. We can convert these integro-dift'erential eguations to higher order differential equations and solve these differential equations to obtain the time-domain network response. A method of solving the netNork cquations of a lioear time• invariant network is to resort to Laplace transform technique. Laplace transformation of an integro-differential equation witb constant coeffi.cients leads to an algebraic equation in the Laplace variable 's'. The algebraic equation is solved in the Laplace domain and the time-domain respoose is obtained by inverse Laplace traosforming the s-domain solution, (Compare tbis with the procedure of solving an equation x" = y by logarithms). The one-sided Laplace transform or a time function f{t); O< t < O'.), represented by F(s) = .l[f(t)], is deftned by the integrai

F(s)

=

f-

f(t)

e-" dt

(B.1)

o

lf the lower limit of integration is - oo then F(s) is a two-sided Laplace transform. We bave no occasion to use the latter. The Laplace transform F(s) of f(t) is a function of the cornplex variable s(s = a + jw). The Laplace transform of a function exists if, and only if, tbe integrai defined in (B. I) converges. The integrai converges. in generai, for some values of s in the right half plane. A sufficient condition for the existence of the Laplace transform is given by tbe Theorem B.l. Tbeorem B.1: Let/(t) be a function which is piecewise continuous on every finite ioterval in the range t ;;ii O and satisfying

; /(t) I ~ M & 1

(B.2)

438 Network Theory and Fllter Design f or some constants a and M. Then the Lapiace integrai converges absolutely for all Re s > a. A function satisfying the inequality (B.2) is said to be of exponential orde,. The order of the function is the smallest number a0 that satisfies (B.2). We say that the Laplace integra] converges absolutely for ali a > a11• a 0 is called the abscissa o/ com,ergence and the region e > a0 is the reglon o/ convergence. EXAMPLB

B.l

(i) /(t)

..

= u(t) F(a) ....

J,-•• ,

dt

= - ...!.. r' s

1• o

D

F(.r)

i.e.,

1 = -, a> s

(ii) /(t) - efAI u(t)

F(s) =

=

O,

-f

e,., r'dt

-•J

e411-•l' dt .... -•-. e - IX

F(s)

.. =f

F{s)

=

3(t)

t u(t) F(a)

a..

3(t) r ' dt

•l for ali s.

...

=

J e-•' t

dt

J tlO

, r' 1 + - -e-" =-dt S o • 8 00

'

l'° O

Àppendlx B 439

i,e.,

F(s)

(v) /(t)

=

1

=

a> O

31 ,

sin 6lt u(t) «I

F(s)

f sin

=

r»t

e-n di

o

j ["'6)1 2J

=

e

Jeot] r' dt

tJ

o

= ..!_ [' el-•+J•)l ,• 2) -s + }c., o I [

e-(ttJ(l;)I

-(s

+ }6>} o

1 1 ] s - jc., - s + Jw

= 2j

F(s)= s.--+ 1 •ai> O. (I) (,)

i.e.,

Table B.l gives some elementary time functions and the corresponding Laplace transforms. Alt time functions are detined for t ;?i:= o. TABLB B.l

/(t)

F(s)

3(t)

l

sin Cllt

"(')

1/s

cos 1

1+ a l

n! .,..+l

n!

(.r

+ a)"+l

Cùt

r"' sin cM

r

7 t"

T(1)

/(t)

01

COI 6)1

cosh c.>t

sinh c.>l

6)

s1

+ c.> 1

.,

s'1 +w1 (,)

(1

+ a)1 + c.>1

(s

+ a)11 + c.>1

s+a

s

7::-;• 6)

,. -

6)1

440 Network Theory and Filter Design Some of tbe important properties of Laplace transforms are: (1) Linearity: Letfi.(t) and/i(t) be two time f'unctions witb tbe corresponding Laplace transforms F 1(s) and F,(s), then ...l'[ac1_f.(t) + a 1/ 1(t)J = « 1F1(s) + ix.J't(,r) (B.2} where 0t1 and 0t 1 are two constants. (2) Shifting theorem: If .C[f(t)] = F(s), then the Laplace trans• forms of /(t- a) and e01 f(t) are e-as F(s) and F(s - a), respectively. (3) Transform of derivatives: If ..C[/(t)] = F(s), tben

.l[d1:>] =

s F(s) - ftO)

and .l[f(t)] = snF(s) - s"-1/(0) - s"- 1/ ' (O) •.• - /) vs log w. A typical transfer function can be written in the factorcd form as H(j)

=

Il (l+J-r.,1)1-1rÌ [ 1 + /21,.,1 -~ + (J .!!...)1] oo .., 6)•, (/)' ii (l +/r,:,-r,1) fi [1 + j2~,, ~ + (1.!!... ,-1 1-1 ,,..,,, p K

1-1

fJlp

rJ (D.1)

Thc polcs aod zeros of the transfer functioo are

zeros:

1

-r,., -ç_.l 6)zl

1 po/es: - ;- ,

,..,

i

=

J1 -

+ /r,:,zl i

1, 2, ... , m1

=

-~,, 6),, + /6),,

6):,,

i=l,2, ... ,m1

1, 2, ... , n1

J;_ ~!.

i= 1, 2, •.•, n1

I

rth order pole at tbc origin. For example, if H(s)

=

(s + a) (si + bs + d)

s'(s

+ ~)(s• + tT.S + 3)

446 Network Theory and Filter pesign then H(Jw) caa be written as H(jc,,)

= ad

( 1 +i

7) [ I + j ~ + (i ~ )]

J3B (jw )= ( 1 + j ; ) [ I

+j

1) = 20 + 20 log ù> 1 Heoce we conclude the term 20 log w has a slope of 20 dB/decade or 6 dB/octave. The plot of eqn. D.2 is a straight line with a slope of - 20 dB/decade. At w = 1 this line passes through tbc point 20 log K. This straigbt line crosses the -O dB axis at (o) = K1''· The angle contribution of thìs factor is - 90° r .focali frequencies. (ii) Factor 1 + j(f),e: For tbis factor we have

20 log I I

+

jw-e I

= 20 log {l +· w 1 -r1] 111 = 101og[l + 1-.8]

20 loglHUw)I

Fig. D.l

The plot is shown in Fig. D.1. If tbe factor is a pole factor then

448 Network Theory and Filter Design

tbc plot will bave a slope of -20 dB/decade and at tbe corner frequenc:y ita va1ue would be -3 dB. The phase corresponding to tbis factor is {,(CiJ) = tan-16)'f (D.4)

90°

45°

--------,--------

~·----------

l /-r Pig. D.2

and the plot is shown in Fig. D.2. For a pote factor the curve will be a mirror image of tbc one in Fig. D.2; tbc phase varying from O to _90•. (iii) Factor 1

+ j2t ~ (I),.

I (iJ 1 :

For this factor we bave

6),r

j + j2; 'c.>,,\ - Cù:

20 log 1

e.i,,

I=

20 log [(l-x•)11 + (~)9]1/1

10 log [(1-_xi)I + ~2.xl] where ;e = c,i/6),,. For x < 1 (i.e. 6> < Cù,.) the value is O dB and for .x ► 1 (i.e., (1) 11). the -value is 40 log x. Hence the low and high frequency asymptotes are the O dB axis and a line with slope 40 dB/decade, respectively. The corner frequency is at c., = 61,,

=

(i.e., x = 1). The error between the actual Bode magnitude plot and its asymptotic approximation depends on the value of t. lf the factor is a pote factor then the high frequency asymptote is a straight line with a slope of -40 dB/decade. Figure D 3 sbows the plots for a pole factor. The phase function for this factor is {,(©)

= tan-1 ( 1~:.}

.4ppendlx D 449 20. dB 10

l

o

-10

-20 -30 -40 O.I

2.0

2.0

I.O

IO.O

.... X

Flg. D.3

o~

+ -JOO

t

-60"

-90" -120"

- ISO" -180°

O. I

0.2

I.O

2.0 X

FJI. D.4

IO.O

450 Network Theory and Filter Design

Fot x < •• ~(r.,) = O and x < l, phase. 356 powcr ratio, 35~

voltap ratio, 356 lnvene bybrid parameters, 203,219 Lapw:e transforms, 98, 441

Gain-bandwlckb prod1,1Ct, 422

O.aeraliaecl

trammission Isbift, S9

p■rameters,

200, 219

lndex 415 Kerwin strueture, 405 KHN structura, 410 Kircbhofl''s eurrent law, 11, 32, 1S3 Kircbhoff's voltaae raw, l 1, 37, 153

Ladder similatecl, 416 Ladder network, 174, 207,291,415 Laarangian trce, 69 Laplace transforms, .SS, 437 LC laddors, 302, 3S5 Linoar graph, 21 Linear network, 12, 113, 140 Linear pba.,e filter, 346 Loop equations, 70, 73, 1S itnpcdance matrix, 70, 75 Low-pass filter, 183, 292, 314, 363,

394 Low-pass notcb, 405, 406 Lumped circuit elements, 2 Lumped Network, l S Matrix, 427

Ma.ximally ftat response, 316 Maximum power transfer, I 63, 35.7 m-derived filter, 465 Mesb, 39 equations, 65, 75 irnpedance matrix, 65, 7S matrix, 41, 65 transformatioo, 52 Minimum phase network, 309 Mutua.I inductance, 76 Natural frequency, 168, 442 Natural resPoDSe, 123, 169 Negative impedance converter current inversion, 376, 383 voltage inversion, 375, 383 Negativeimpedanceinverter, 379,386 Network activo, 14, 371 equation,49 limar, 12, 113, 140 lumped, 2, 1S DonlfDCIII', 12

one-port, 15, 195, 4.11 paaiVe, 14, 156, 2611

tboory, l timo invariant, 13, 113 titne variant, 13 topolol)', 20

two-J)Ol't, lS, 195,4SS Network functions, 173, 223 Nodo, 21 admittance matrix, 61, 1S equatlons, 60,72,75 tranaformatlon, 49, 154 voltaae, 61

Non-casual system, 314 Non-minimum pbaae functlon, 309, 423 Nora.tor, 15, 380 Nortoo's equivalent, 59, 146, 147 tbeorem, 146 Notch filtor, 405, 410 Nullator, 15, 380 Nullor, 380 One-port, 15, 195, 451 Open c:ircuit natural ftequency, 179 Open circuit stability, 182 Operation amplifìer, 208,225, 373, 394 non-ideai, 421 Operational impedance, 54 Order of complexlty, 172, 255, 2S6 Oriented graph, 28, l SS Partial functions, 98, 270 Passive network, 14, 156, 268 Path, 22 Phase response, 188} 331 Pivotal condensacion, 219 Planar graph, 28, 39, 40 Poles, 99, 176, 260, 319, 327 Pote removal, 272-273, 283 partial, 297 Pole-zero plot, J 78 Pole weakeniog, 302 Positive real function, 259, 264 Principle of conservatlon of cbar,e. t l 7 Principio or continuity of Dm link-

.,_, 11,

476 Network Theory and Fllter Design Proper uec, 249 modifiecl, 249

Prop111&don cOMtant, 462 Q•factor, 127, 330, 423 Quadrantal symmctry, 263, 361

RC function, 278

ladclen, 291 network, 394 Reac:tanc:e fwiction, 269, 333 Reciprocai network, 156, 198 R.eciprocity theorem, 156 R.Jftcction cOIIIBcient, 358, 452 Residue, 99, 270, 279, 280 Resistance, 3 incrementai, 4 Rcsistor current-c:ontrolled, 3 linear, 3

DODliru,a.r, 3, 14 'toltagc-controlled, 3 Resonance, 124 Resonaot,.frequeocy, 125, 467, 483 RL network, 289 Routh-Hmwitz array, 260 Sallen-key filter, 405 8':atter panimetors, 196, 451 Selectivity parameter, 318 Sensiiivity, 149; 161, 387,415 Sensitivity adjoint network, 161 gain, 387 pbase, 387 root, 388 Short circuit, natural frequencics, 179 stability, 182 Sinusoidal steady state response, 123 Source controlled, 5, 372 current, 5, 58 dependent, 6

independent. S voltaae, 5, .18 transformatìon, 58

State

equations, 23 7 tr1U1sition matrix, 254 vatlable filter, 410 variablea, 237, 408 State equationa, 238, 408 equivalcmt aource metbod, 242 graph theoretic method, 249 Step rasponse, l 05

Sturm's functiom, 266 test, 264

Subatltutlon theorem, 142, 147 Superpositlon tbeorem, 140, 148 Taylor series, 316 Tellegen"s theorcm, 152, 161 Thevmin equivalcnt, 58, 146 theorem, 146 Transduccr function, 3S7 Transfer functlon, 104-, 113,174, 2S4, 290, ]SS, -394, 395 Transform networ1ts, 84 Tranamissloo coefficicnt, 3S7 parameters, 191, 199, 207 zeros, 182, 291, 292, 302, 467 Tl'lle, 23,263 Tree, 23

, modified

proper, propcr, 249

249

Two-port fuoçtioos, 290 network, 15, 195, 355, 455 parameters, 196 Unit stcp, 105

Vertex, 21 Voltage transfer ratio, 175, 290 Vratsaoov's theOrcm, 231 Zero input respoose, 94, 113, 25S Zero state response, 94, 113, 25S Zcros, 99,176,260,357 Zcro sbiftiog, 298

Network Theory and Filter Design,. 2/e An altempt has been made lo include two aspects of network thcory, analysìs and synthesis under a single cover, Dasics of network theory - formulation and solution ol network equilibrium equalions., notwork theorems and natural lrequencies, mulititermma 1 networks, state rnodeils · ·- are ali discussed, Essential·s ol approx1mat1on and passive network synlhesis are presented as prelude !o fil1er design. A comprehensive i11troduction to active filter design with single and mulliamplilier is also provided. Higher arder filter design based 011 simulated ladder methods ìs also ,introduced, 1

Th1s book 1s intended lor use as text tor undorgraduato courses in network analysis and tundament□ ls ol passive and active lilter design tor a two semester course or a one semester coursc lor posi graduate

courses.

Dr. V.K.. Aatre was bom in Bangalore, India, He receiveci llis BE. frorn Universily of Mysore., M.C, !rom lhe lr1dia11 lnstiMe ot Science, Bangalore and Ph.D lrnm the Universitv o! Waterloo, Ontario, Canada, all in Flectlical Engineering in 1961, 1963 and 1,967 respec!ively. From 1967 to 1968, l'le was a post doctoral fellow at the Department ol Electrical Engineering, University of Waterloo. He ioined the Technical Uni,versity ot Nova Scotìa m 1968 ancl in 1980 he returned lo India. In 1977 he was a visiting professor al lhe lndia.n lnsti.\l.ltc of Science, In 1980 he joined Naval Physical and Oceanographic Laboratory, Cochin, where he is presently the Director o! the Laboratory.

WILEY EASTERN LIMITED New Delt1i, Bangalore Bomba\/ Hydernbad Pune luckno;

n' K rionK ijflll;,E rl)

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! 11

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