Circuit analysis and feedback amplifier theory

CIRCUIT ANALYSIS

and FEEDBACK

AMPLIFIER THEORY

© 2006 by Taylor & Francis Group, LLC

CIRCUIT ANALYSIS

and FEEDBACK

AMPLIFIER THEORY

Edited by

Wai-Kai Chen

A CRC title, part of the Taylor & Francis imprint, a member of the

Taylor & Francis Group, the academic division of T&F Informa plc.

Boca Raton London New York

University of Illinois

Chicago, U.S.A.

© 2006 by Taylor & Francis Group, LLC

The material was previously published in The Circuit and Filters Handbook, Second Edition. © CRC Press LLC 2002.

Published in 2006 by

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© 2006 by Taylor & Francis Group, LLC

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© 2006 by Taylor & Francis Group, LLC

v

Preface

The purpose of Circuit Analysis and Feedback Amplifier Theory is to provide in a single volume a

comprehensive reference work covering the broad spectrum of linear circuit analysis and feedback

amplifier design. It also includes the design of multiple-loop feedback amplifiers. The book is written

and developed for the practicing electrical engineers in industry, government, and academia. The goal

is to provide the most up-to-date information in the field.

Over the years, the fundamentals of the field have evolved to include a wide range of topics and a

broad range of practice. To encompass such a wide range of knowledge, the book focuses on the key

concepts, models, and equations that enable the design engineer to analyze, design and predict the

behavior of large-scale circuits and feedback amplifiers. While design formulas and tables are listed,

emphasis is placed on the key concepts and theories underlying the processes.

The book stresses fundamental theory behind professional applications. In order to do so, it is rein￾forced with frequent examples. Extensive development of theory and details of proofs have been omitted.

The reader is assumed to have a certain degree of sophistication and experience. However, brief reviews

of theories, principles and mathematics of some subject areas are given. These reviews have been done

concisely with perception.

The compilation of this book would not have been possible without the dedication and efforts of

Professor Larry P. Huelsman, and most of all the contributing authors. I wish to thank them all.

Wai-Kai Chen

Editor-in-Chief

© 2006 by Taylor & Francis Group, LLC

vii

Editor-in-Chief

Wai-Kai Chen, Professor and Head Emeritus of the Depart￾ment of Electrical Engineering and Computer Science at the

University of Illinois at Chicago, is now serving as Academic

Vice President at International Technological University. He

received his B.S. and M.S. degrees in electrical engineering at

Ohio University, where he was later recognized as a Distin￾guished Professor. He earned his Ph.D. in electrical engineering

at the University of Illinois at Urbana/Champaign.

Professor Chen has extensive experience in education and

industry and is very active professionally in the fields of circuits

and systems. He has served as visiting professor at Purdue Uni￾versity, University of Hawaii at Manoa, and Chuo University in

Tokyo, Japan. He was Editor of the IEEE Transactions on Circuits

and Systems, Series I and II, President of the IEEE Circuits and

Systems Society, and is the Founding Editor and Editor-in￾Chief of the Journal of Circuits, Systems and Computers. He

received the Lester R. Ford Award from the Mathematical Asso￾ciation of America, the Alexander von Humboldt Award from Germany, the JSPS Fellowship Award from

Japan Society for the Promotion of Science, the Ohio University Alumni Medal of Merit for Distinguished

Achievement in Engineering Education, the Senior University Scholar Award and the 2000 Faculty

Research Award from the University of Illinois at Chicago, and the Distinguished Alumnus Award from

the University of Illinois at Urbana/Champaign. He is the recipient of the Golden Jubilee Medal, the

Education Award, the Meritorious Service Award from IEEE Circuits and Systems Society, and the Third

Millennium Medal from the IEEE. He has also received more than a dozen honorary professorship awards

from major institutions in China.

A fellow of the Institute of Electrical and Electronics Engineers and the American Association for the

Advancement of Science, Professor Chen is widely known in the profession for his Applied Graph Theory

(North-Holland), Theory and Design of Broadband Matching Networks (Pergamon Press), Active Network

and Feedback Amplifier Theory (McGraw-Hill), Linear Networks and Systems (Brooks/Cole), Passive and

Active Filters: Theory and Implements(John Wiley), Theory of Nets: Flows in Networks(Wiley-Interscience),

and The VLSI Handbook (CRC Press).

© 2006 by Taylor & Francis Group, LLC

ix

Advisory Board

Leon O. Chua

University of California

Berkeley, California

John Choma, Jr.

University of Southern California

Los Angeles, California

Lawrence P. Huelsman

University of Arizona

Tucson, Arizona

© 2006 by Taylor & Francis Group, LLC

xi

Contributors

Peter Aronhime

University of Louisville

Louisville, Kentucky

K.S. Chao

Texas Tech University

Lubbock, Texas

Ray R. Chen

San Jose State University

San Jose, California

Wai-Kai Chen

University of Illinois

Chicago, Illinois

John Choma, Jr.

University of Southern California

Los Angeles, California

Artice M. Davis

San Jose State University

San Jose, California

Marwan M. Hassoun

Iowa State University

Ames, Iowa

Pen-Min Lin

Purdue University

West Lafayette, Indiana

Robert W. Newcomb

University of Maryland

College Park, Maryland

Benedykt S. Rodanski

University of Technology, Sydney

Broadway, New South Wales,

Australia

Marwan A. Simaan

University of Pittsburgh

Pittsburgh, Pennsylvania

James A. Svoboda

Clarkson University

Potsdam, New York

Jiri Vlach

University of Waterloo

Waterloo, Ontario, Canada

© 2006 by Taylor & Francis Group, LLC

xiii

Table of Contents

1 Fundamental Circuit Concepts John Choma, Jr................................................ 1-1

2 Network Laws and Theorems ................................................................................ 2-1

2.1 Kirchhoff's Voltage and Current Laws Ray R. Chen and Artice M. Davis..................... 2-1

2.2 Network Theorems Marwan A. Simaan................................................................ 2-39

3 Terminal and Port Representations James A. Svoboda..................................... 3-1

4 Signal Flow Graphs in Filter Analysis and Synthesis Pen-Min Lin.............. 4-1

5 Analysis in the Frequency Domain ...................................................................... 5-1

5.1 Network Functions Jiri Vlach ................................................................................ 5-1

5.2 Advanced Network Analysis Concepts John Chroma, Jr.......................................... 5-10

6 Tableau and Modified Nodal Formulations Jiri Vlach.................................... 6-1

7 Frequency Domain Methods Peter Aronhime.................................................... 7-1

8 Symbolic Analysis1 Benedykt S. Rodanski and Marwan M. Hassoun................ 8-1

9 Analysis in the Time Domain Robert W. Newcomb .......................................... 9-1

10 State-Variable Techniques K. S. Chao............................................................... 10-1

11 Feedback Amplifier Theory John Choma, Jr. ................................................... 11-1

12 Feedback Amplifier Configurations John Choma, Jr. ..................................... 12-1

13 General Feedback Theory Wai-Kai Chen ......................................................... 13-1

© 2006 by Taylor & Francis Group, LLC

xiv

14 The Network Functions and Feedback Wai-Kai Chen .................................. 14-1

15 Measurement of Return Difference Wai-Kai Chen ........................................ 15-1

16 Multiple-Loop Feedback Amplifiers Wai-Kai Chen....................................... 16-1

© 2006 by Taylor & Francis Group, LLC

1-1

1

Fundamental

Circuit Concepts

1.1 The Electrical Circuit......................................................... 1-1

Current and Current Polarity • Energy and Voltage • Power

1.2 Circuit Classifications ...................................................... 1-10

Linear vs. Nonlinear • Active vs. Passive • TimeVarying vs. Time

Invariant • Lumped vs. Distributed

1.1 The Electrical Circuit

An electrical circuit or electrical network is an array of interconnected elements wired so as to be capable

of conducting current. As discussed earlier, the fundamental two-terminal elements of an electrical

circuit are the resistor, the capacitor, the inductor, the voltage source, and the current source. The

circuit schematic symbols of these elements, together with the algebraic symbols used to denote their

respective general values, appear in Figure 1.1.

As suggested in Figure 1.1, the value of a resistor is known as its resistance, R, and its dimensional

units are ohms.The case of a wire used to interconnect the terminals of two electrical elements corresponds

to the special case of a resistor whose resistance is ideally zero ohms; that is, R = 0. For the capacitor in

Figure 1.1(b), the capacitance, C, has units of farads, and from Figure 1.1(c), the value of an inductor is

its inductance, L, the dimensions of which are henries. In the case of the voltage sources depicted in

Figure 1.1(d), a constant, time invariant source of voltage, or battery, is distinguished from a voltage

source that varies with time. The latter type of voltage source is often referred to as a time varying signal

or simply, a signal. In either case, the value of the battery voltage, E, and the time varying signal, v(t),

is in units of volts. Finally, the current source of Figure 1.1(e) has a value, I, in units of amperes, which

is typically abbreviated as amps.

Elements having three, four, or more than four terminals can also appear in practical electrical

networks. The discrete component bipolar junction transistor (BJT), which is schematically portrayed

in Figure 1.2(a), is an example of a three-terminal element, in which the three terminals are the collector,

the base, and the emitter. On the other hand, the monolithic metal-oxide-semiconductor field-effect

transistor (MOSFET) depicted in Figure 1.2(b) has four terminals: the drain, the gate, the source, and

the bulk substrate.

Multiterminal elements appearing in circuits identified for systematic mathematical analyses are rou￾tinely represented, or modeled, by equivalent subcircuits formed of only interconnected two-terminal

elements. Such a representation is always possible, provided that the list of two-terminal elements itemized

in Figure 1.1 is appended by an additional type of two-terminal element known as the controlled source,

or dependent generator. Two of the four types of controlled sources are voltage sources and two are

current sources. In Figure 1.3(a), the dependent generator is a voltage-controlled voltage source (VCVS)

in that the voltage, v0(t), developed from terminal 3 to terminal 4 is a function of, and is therefore

John Choma, Jr.

University of Southern California

© 2006 by Taylor & Francis Group, LLC

1-2 Circuit Analysis and Feedback Amplifier Theory

dependent on, the voltage, vi(t), established elsewhere in the considered network from terminal 1 to

terminal 2. The controlled voltage, v0(t), as well as the controlling voltage, vi(t), can be constant or time

varying. Regardless of the time-domain nature of these two voltage, the value of v0(t) is not an indepen￾dent number. Instead, its value is determined by vi(t) in accordance with a prescribed functional rela￾tionship, e.g.,

(1.1)

If the function, f(⋅), is linearly related to its argument, (1.1) collapses to the form

(1.2)

where fµ is a constant, independent of either v0(t) or vi

(t). When the function on the right-hand side of

(1.1) is linear, the subject VCVS becomes known as a linear voltage-controlled voltage source.

FIGURE 1.1 Circuit schematic symbol and corresponding value notation for (a) resistor, (b) capacitor, (c) inductor,

(d) voltage source, and (e) current source. Note that a constant voltage source, or battery, is distinguished from a

voltage source that varies with time.

FIGURE 1.2 Circuit schematic symbol for (a) discrete component bipolar junction transistor (BJT) and

(b) monolithic metal-oxide-semiconductor field-effect transistor (MOSFET).

(e)

(d)

(c)

(b)

(a)

R

C

L

E

I

− +

− +

v(t)

Resistor:

Resistance = R (In Ohms)

Capacitor:

Capacitance = C (In Farads)

Inductor:

Inductance = L (In Henries)

Constant Voltage (Battery):

Voltage = E (In Volts)

Time-Varying Voltage:

Voltage = V (In Volts)

Current Source:

Current = I (In Amperes)

Bipolar Junction

Transistor (BJT)

Metal-Oxide￾Semiconductor

Field-Effect

Transistor (MOSFET)

collector (C)

(a). base (b)

emitter (E)

drain (D)

(b). gate (G) substrate (B)

source (S)

v t fv t 0 i ( ) = [ ] ( )

v t fv t 0 i ( ) = ( ) µ

© 2006 by Taylor & Francis Group, LLC

Fundamental Circuit Concepts 1-3

The second type of controlled voltage source is the current-controlled voltage source (CCVS) depicted

in Figure 1.3(b). In this dependent generator, the controlled voltage, v0(t), developed from terminal 3 to

terminal 4 is a function of the controlling current, ii

(t), flowing elsewhere in the network between terminals

1 and 2, as indicated. In this case, the generalized functional dependence of v0(t) on ii(t) is expressible as

(1.3)

which reduces to

(1.4)

when r(⋅) is a linear function of its argument.

The two types of dependent current sources are diagrammed symbolically in Figures 1.3(c) and (d).

Figure 1.3(c) depicts a voltage-controlled current source (VCCS), for which the controlled current i0(t),

flowing in the electrical path from terminal 3 to terminal 4, is determined by the controlling voltage,

vi

(t), established across terminals 1 and 2. Therefore, the controlled current can be written as

(1.5)

In the current-controlled current source (CCCS) of Figure 1.3(d),

(1.6)

where the controlled current, i0(t), flowing from terminal 3 to terminal 4 is a function of the controlling

current, ii(t), flowing elsewhere in the circuit from terminal 1 to terminal 2. As is the case with the two

controlled voltage sources studied earlier, the preceding two equations collapse to the linear relationships

(1.7)

and

(1.8)

when g(⋅) and a(⋅), respectively, are linear functions of their arguments.

FIGURE 1.3 Circuit schematic symbol for (a) voltage-controlled voltage source (VCVS), (b) current-controlled voltage

source (CCVS), (c) voltage-controlled current source (VCCS), and (d) current-controlled current source (CCCS).

(a)

vi (t) vo(t)

vi

(t) ii

(t)

ii f[v (t) i

(t)] r[ii

(t)]

g[vi

(t)] a[ii

(t)]

vo(t)

1 1

2 2

3 3

4 4

io(t) io(t)

(b)

(c) (d)

+ + + + +

− − − − −

1

2

+

−

3

4

3

4

1

2

v t ri t 0 i ( ) = [ ] ( )

v t ri t 0 m i ( ) = ( )

i t gv t 0 i ( ) = [ ] ( )

i t ai t 0 i ( ) = [ ] ( )

i t gvt 0 m i ( ) = ( )

i t ai t 0 i ( ) = ( ) α

© 2006 by Taylor & Francis Group, LLC

1-4 Circuit Analysis and Feedback Amplifier Theory

The immediate implication of the controlled source concept is that the definition for an electrical

circuit given at the beginning of this subsection can be revised to read “an electrical circuit or electrical

network is an array of interconnected two-terminal elements wired in such a way as to be capable of

conducting current”. Implicit in this revised definition is the understanding that the two-terminal ele￾ments allowed in an electrical circuit are the resistor, the capacitor, the inductor, the voltage source, the

current source, and any of the four possible types of dependent generators.

In, an attempt to reinforce the engineering utility of the foregoing definition, consider the voltage

mode operational amplifier, or op-amp, whose circuit schematic symbol is submitted in Figure 1.4(a).

Observe that the op-amp is a five-terminal element. Two terminals, labeled 1 and 2, are provided to

receive input signals that derive either from external signal sources or from the output terminals of

subcircuits that feed back a designable fraction of the output signal established between terminal 3 and

the system ground. Battery voltages, identified as ECC and EBB in the figure, are applied to the remaining

two op-amp terminals (terminals 4 and 5) with respect to ground to bias or activate the op-amp for its

intended application. When ECC and EBB are selected to ensure that the subject op-amp behaves as a linear

circuit element, the voltages, ECC and EBB, along with the corresponding terminals at which they are

incident, are inconsequential. In this event the op-amp of Figure 1.4(a) can be modeled by the electrical

circuit appearing in Figure 1.4(b), which exploits a linear VCVS. Thus, the voltage amplifier of

Figure 1.4(c), which interconnects two batteries, a signal source voltage, three resistors, a capacitor, and

an op-amp, can be represented by the network given in Figure 1.4(d). Note that the latter configuration

uses only two terminal elements, one of which is a VCVS.

FIGURE 1.4 (a) Circuit schematic symbol for a voltage mode operational amplifier. (b) First-order linear model of

the op-amp. (c) A voltage amplifier realized with the op-amp functioning as the gain element. (d) Equivalent circuit

of the voltage amplifier in (c).

(a) (b)

(c) (d)

− −

+ +

vi

(t) aovi

(t)

vo(t)

1

2

3

−

−

+ + ECC EBB

vi

(t)

1

2

4 5

−

+

vo(t)

3

Op-Amp

−

+

vo(t)

3

Op-Amp

−

+

−

−

+ + ECC EBB

R2

R1

RS

vi

(t)

vs(t)

1

2

4 5

C

−

−

+ +

+ −

RS

R2

R1

vi

(t)

vs(t)

vo(t)

aovi

(t)

1 3

2

C

© 2006 by Taylor & Francis Group, LLC

Fundamental Circuit Concepts 1-5

Current and Current Polarity

The concept of an electrical current is implicit to the definition of an electrical circuit in that a circuit is

said to be an array of two-terminal elements that are connected in such a way as to permit the condition

of current. Current flow through an element that is capable of current conduction requires that the net

charge observed at any elemental cross-section change with time. Equivalently, a net nonzero charge,

q(t), must be transferred over finite time across any cross-sectional area of the element. The current, i(t),

that actually flows is the time rate of change of this transferred charge;

(1.9)

where the MKS unit of charge is the coulomb, time t is measured in seconds, and the resultant current

is measured in units of amperes. Note that zero current does not necessarily imply a lack of charge at a

given cross-section of a conductive element. Instead, zero current implies only that the subject charge is

not changing with time; that is, the charge is not moving through the elemental cross-section.

Electrical charge can be negative, as in the case of electrons transported through a cross-section of a

conductive element such as aluminum or copper. A single electron has a charge of –(1.6021 × 10–19

)

coulomb. Thus, (1.9) implies a need to transport an average of (6.242 × 1018

) electrons in 1 second

through a cross-section of aluminum if the aluminum element is to conduct a constant current of 1 amp.

Charge can also be positive, as in the case of holes transported through a cross-section of a semiconductor

such as germanium or silicon. Hole transport in a semiconductor is actually electron transport at an

energy level that is smaller than the energy required to effect electron transport in that semiconductor.

To first order, therefore, the electrical charge of a hole is the negative of the charge of an electron, which

implies that the charge of a hole is +(1.602 × 10–19

) coulomb.

A positive charge, q(t), transported from the left of the cross-section to the right of the cross-section

to right across the indicated cross-section. Assume that, prior to the transport of such charge, the volumes

to the left and to the right of the cross-section are electrically neutral; that is, these volumes have zero

initial net charge. Then, the transport of a positive charge, q0, from the left side to the right side of the

element charges the right side to +1q0 and the left side to –1q0.

Alternatively, suppose a negative charge in the amount of –q0 is transported from the right side of the

element to its left side, as suggested in Figure 1.5(b). Then, the left side charges to –q0, and the right side

charges to +q0, which is identical to the electrostatic condition incurred by the transport of a positive

charge in the amount of q0 from left- to right-hand sides. As a result, the transport of a net negative

charge from right to left produces a positive current, i(t), flowing from left to right, just as positive charge

transported from left- to right-hand sides induces a current flow from left to right.

Assume, as portrayed in Figure 1.5(c), that a positive or a negative charge, say, q1(t), is transported

from the left side of the indicated cross-section to the right side. Simultaneously, a positive or a negative

charge in the amount of q2(t) is directed through the cross-section from right to left. If i1(t) is the current

arising from the transport of the charge q1(t), and if i2(t) denotes the current corresponding to the

transport of the charge, q2(t), the net effective current ie(t), flowing from the left side of the cross-section

to the right side of the cross-section is

(1.10)

where the charge difference, [q1(t) – q2(t)], represents the net charge transported from left to right.

Observe that if q1(t) ≡ q2(t), the net effective current is zero, even though conceivably large numbers of

charges are transported back and forth across the junction.

i t dq t

dt ( ) = ( )

i t d

dt qt qt it it e( ) = [ ] ( ) − ( ) = ( ) − ( ) 1 2 12

© 2006 by Taylor & Francis Group, LLC

in the element abstracted in Figure 1.5(a) gives rise to a positive current, i(t), which also flows from left