Electronic devices and amplifier circuits with MATLAB applications

Electronic devices and amplifier circuits with MATLAB applications

Electronic Devices and Amplifier Circuits

with MATLAB®Applications

Steven T Karris Editor

Orchard Publications

www.orchardpublications.com

Electronic Devices and Amplifier Circuits with MATLAB®Applications

Copyright ” 2005 Orchard Publications All rights reserved Printed in the United States of America No part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher.

Direct all inquiries to Orchard Publications, info@orchardpublications.com

Product and corporate names are trademarks or registered trademarks of The MathWorks, Inc They are used only for identification and explanation, without intent to infringe.

Library of Congress Cataloging-in-Publication Data

Library of Congress Control Number (LCCN) 2005901972

This book is an undergraduate level textbook presenting a thorough discussion of state-of-the artelectronic devices It is self-contained; it begins with an introduction to solid state semiconductordevices The prerequisites for this text are first year calculus and physics, and a two-semestercourse in circuit analysis including the fundamental theorems and the Laplace transformation Noprevious knowledge of MATLAB®is required; the material in Appendix A and the inexpensiveMATLAB Student Version is all the reader need to get going Our discussions are based on a PCwith Windows XP platforms but if you have another platform such as Macintosh, please refer tothe appropriate sections of the MATLAB’s User Guide which also contains instructions forinstallation Additional information including purchasing may be obtained from The MathWorks,Inc., 3 Apple Hill Drive, Natick, MA 01760-2098 Phone: 508 647-7000, Fax: 508 647-7001, e-mail: info@mathwork.com and web site http://www.mathworks.com.This text can also be usedwithout MATLAB.

This is our fourth electrical and computer engineering-based text with MATLAB applications

My associates, contributors, and I have a mission to produce substance and yet inexpensive textsfor the average reader Our first three texts* are very popular with students and workingprofessionals seeking to enhance their knowledge and prepare for the professional engineeringexamination We are working with limited resources and our small profits left after large discounts

to the bookstores and distributors, are reinvested in the production of more texts To maintainour retail prices as low as possible, we avoid expensive and fancy hardcovers

The author and contributors make no claim to originality of content or of treatment, but havetaken care to present definitions, statements of physical laws, theorems, and problems

Chapter 1 is an introduction to the nature of small signals used in electronic devices, amplifiers,definitions of decibels, bandwidth, poles and zeros, stability, transfer functions, and Bode plots.Chapter 2 is an introduction to solid state electronics beginning with simple explanations ofelectron and hole movement This chapter provides a thorough discussion on the junction diodeand its volt-ampere characteristics In most cases, the non-linear characteristics are plotted withsimple MATLAB scripts The discussion concludes with diode applications, the Zener, Schottky,tunnel, and varactor diodes, and optoelectronics devices Chapters 3 and 4 are devoted to bipolarjunction transistors and FETs respectively, and many examples with detailed solutions areprovided Chapter 5 is a long chapter on op amps Many op amp circuits are presented and theirapplications are well illustrated

* These are Circuit Analysis I, ISBN 0-9709511-2-4, Circuit Analysis II, ISBN 0-9709511-5-9, and Signals and Systems, ISBN 0-9709511-6-7.

The highlight of this text is Chapter 6 on integrated devices used in logic circuits The internalconstruction and operation of the TTL, NMOS, PMOS, CMOS, ECL, and the biCMOS families

of those devices are fully discussed Moreover, the interpretation of the most importantparameters listed in the manufacturers data sheets are explained in detail Chapter 7 is anintroduction to pulse circuits and waveform generators There, we discuss the 555 Timer, theastable, monostable, and bistable multivibrators, and the Schmitt trigger

Chapter 8 discusses to the frequency characteristic of single-stage and cascade amplifiers, andChapter 9 is devoted to tuned amplifiers Sinusoidal oscillators are introduced in Chapter 10.There are also three appendices in this text As mentioned earlier, the first, Appendix A, is anintroduction to MATLAB Appendix B is an introduction to uncompensated and compensatednetworks, and Appendix C discusses the substitution, reduction, and Miller’s theorems

A companion to this text, Logic Circuits, is nearly completion also This text is devoted strictly on

Boolean logic, combinational and sequential circuits as interconnected logic gates and flip-flops,

an introduction to static and dynamic memory devices and other related topics

Like any other new text, the readers will probably find some mistakes and typo errors for which weassume responsibility We will be grateful to readers who direct these to our attention atinfo@orchardpublications.com Thank you

Chapter 1

Basic Electronic Concepts and Signals

Signals and Signal Classifications 1-1Amplifiers 1-3Decibels 1-4Bandwidth and Frequency Response 1-5Bode Plots 1-7Transfer Function 1-9Poles and Zeros 1-11Stability 1-12The Voltage Amplifier Equivalent Circuit 1-16The Current Amplifier Equivalent Circuit 1-18Summary 1-20Exercises 1-23Solutions to End-of-Chapter Exercises 1-25

Chapter 2

Introduction to Semiconductor Electronics - Diodes

Electrons and Holes 2-1The Junction Diode 2-4Graphical Analysis of Circuits with Non-Linear Devices 2-9Piecewise Linear Approximations 2-13Low Frequency AC Circuits with Junction Diodes 2-15Junction Diode Applications in AC Circuits 2-19Peak Rectifier Circuits 2-28Clipper Circuits 2-30

DC Restorer Circuits 2-32Voltage Doubler Circuits 2-33Diode Applications in Amplitude Modulation (AM) Detection Circuits 2-34Diode Applications in Frequency Modulation (FM) Detection Circuits 2-35Zener Diodes 2-36The Schottky Diode 2-42The Tunnel Diode 2-43The Varactor 2-45Optoelectronic Devices 2-46

Summary 2-50Exercises 2-54Solutions to End-of-Chapter Exercises 2-59

Chapter 3

Bipolar Junction Transistors

Introduction 3-1NPN Transistor Operation 3-3The Bipolar Junction Transistor as an Amplifier 3-4Equivalent Circuit Models - NPN Transistors 3-6Equivalent Circuit Models - PNP Transistors 3-7Effect of Temperature on the - Characteristics 3-10Collector Output Resistance - Early Voltage 3-11Transistor Amplifier Circuit Biasing 3-18Fixed Bias 3-21Self-Bias 3-25Amplifier Classes and Operation 3-28Class A Amplifier Operation 3-31Class B Amplifier Operation 3-34Class AB Amplifier Operation 3-35Class C Amplifier Operation 3-37Graphical Analysis 3-38Power Relations in the Basic Transistor Amplifier 3-42Piecewise-Linear Analysis of the Transistor Amplifier 3-44Incremental linear models 3-49Transconductance 3-54High-Frequency Models for Transistors 3-55The Darlington Connection 3-59Transistor Networks 3-61The h-Equivalent Circuit for the Common-Base Transistor 3-61The T-Equivalent Circuit for the Common-Base Transistor 3-64The h-Equivalent Circuit for the Common-Emitter Transistor 3-65The T-Equivalent Circuit for the Common-Emitter Transistor 3-70The h-Equivalent Circuit for the Common-Collector (Emitter-Follower) Transistor 3-70The T-Equivalent Circuit for the Common-Collector Transistor Amplifier 3-76Transistor Cutoff and Saturation Regions 3-77Cutoff Region 3-78Active Region 3-78Saturation Region 3-78The Ebers-Moll Transistor Model 3-80

i C v BE

Summary 3-86Exercises 3-90Solutions to End-of-Chapter Exercises 3-96

Chapter 4

Field Effect Transistors and PNPN Devices

The Junction Field Effect Transistor (JFET) 4-1The Metal Oxide Semiconductor Field Effect Transistor (MOSFET) 4-6The N-Channel MOSFET in the Enhancement Mode 4-8The N-Channel MOSFET in the Depletion Mode 4-12The P-Channel MOSFET in the Enhancement Mode 4-14The P-Channel MOSFET in the Depletion Mode 4-17Voltage Gain 4-17Complementary MOS (CMOS) 4-19The CMOS Common-Source Amplifier 4-20The CMOS Common-Gate Amplifier 4-20The CMOS Common-Drain (Source Follower) Amplifier 4-20The Metal Semiconductor FET (MESFET) 4-21The Unijunction Transistor 4-22The Diac 4-23The Silicon Controlled Rectifier (SCR) 4-24The SCR as an Electronic Switch 4-27The SCR in the Generation of Sawtooth Waveforms 4-28The Triac 4-37The Shockley Diode 4-38Other PNPN Devices 4-40Summary 4-41Exercises 4-44Solutions to End-of-Chapter Exercises 4-46

Chapter 5

Operational Amplifiers

The Operational Amplifier 5-1

An Overview of the Op Amp 5-1The Op Amp in the Inverting Mode 5-2The Op Amp in the Non-Inverting Mode 5-5Active Filters 5-8

Analysis of Op Amp Circuits 5-11Input and Output Resistances 5-22

Op Amp Open Loop Gain 5-25

Op Amp Closed Loop Gain 5-26Transresistance Amplifier 5-29Closed Loop Transfer Function 5-30The Op Amp Integrator 5-31The Op Amp Differentiator 5-35Summing and Averaging Op Amp Circuits 5-37Differential Input Op Amp 5-39Instrumentation Amplifiers 5-42Offset Nulling 5-44External Frequency Compensation 5-45Slew Rate 5-45Circuits with Op Amps and Non-Linear Devices 5-46Comparators 5-50Wien Bridge Oscillator 5-50Digital-to-Analog Converters 5-52Analog-to-Digital Converters 5-56The Flash Analog-to-Digital Converter 5-57The Successive Approximation Analog-to-Digital Converter 5-58The Dual-Slope Analog-to-Digital Converter 5-59Quantization, Quantization Error, Accuracy, and Resolution 5-61

Op Amps in Analog Computers 5-63Summary 5-67Exercises 5-71Solutions to End-of-Chapter Exercises 5-78

Chapter 6

Integrated Circuits

The Basic Logic Gates 6-1Positive and Negative Logic 6-1The Inverter 6-2The AND Gate 6-6The OR Gate 6-8The NAND Gate 6-9The NOR Gate 6-13The Exclusive OR (XOR) and Exclusive NOR (XNOR) Gates 6-15Fan-In, Fan-Out, TTL Unit Load, Sourcing Current, and Sinking Current 6-17Data Sheets 6-20

The NMOS Inverter 6-31The NMOS NAND Gate 6-31The NMOS NOR Gate 6-32CMOS Logic Gates 6-32The CMOS Inverter 6-33The CMOS NAND Gate 6-34The CMOS NOR Gate 6-35Buffers, Tri-State Devices, and Data Buses 6-35Present and Future Technologies 6-39Summary 6-43Exercises 6-46Solutions to End-of-Chapter Exercises 6-49

Chapter 7

Pulse Circuits and Waveform Generators

Astable (Free-Running) Multivibrators 7-1The 555 Timer 7-2Astable Multivibrator with the 555 Timer 7-3Monostable Multivibrators 7-15Bistable Multivibrators (Flip-Flops) 7-18The Fixed-Bias Flip-Flop 7-19The Self-Bias Flip-Flop 7-22Triggering Signals for Flip-Flops 7-28Present Technology Bistable Multivibrators 7-30The Schmitt Trigger 7-30Summary 7-33Exercises 7-34Solutions to End-of-Chapter Exercises 7-37

Chapter 8

Frequency Characteristics of Single-Stage and Cascaded Amplifiers

Properties of Signal Waveforms 8-1The Transistor Amplifier at Low Frequencies 8-5The Transistor Amplifier at High Frequencies 8-9Combined Low- and High-Frequency Characteristics 8-14Frequency Characteristics of Cascaded Amplifiers 8-14Overall Characteristics of Multistage Amplifiers 8-26

Amplification and Power Gain in Three or More Cascaded Amplifiers 8-32Summary 8-34Exercises 8-36Solutions to End-of-Chapter Exercises 8-39

Chapter 9

Tuned Amplifiers

Introduction to Tuned Circuits 9-1Single-tuned Transistor Amplifier 9-8Cascaded Tuned Amplifiers 9-14Synchronously Tuned Amplifiers 9-15Stagger-Tuned Amplifiers 9-19Three or More Tuned Amplifiers Connected in Cascade 9-27Summary 9-29Exercises 9-31Solutions to End-of-Chapter Exercises 9-32

Chapter 10

Sinusoidal Oscillators

Introduction to Oscillators 10-1Sinusoidal Oscillators 10-1

RC Oscillator 10-4

LC Oscillators 10-5The Armstrong Oscillator 10-6The Hartley Oscillator 10-7The Colpitts Oscillator 10-7Crystal Oscillators 10-8The Pierce Oscillator 10-10Summary 10-12Exercises 10-14Solutions to End-of-Chapter Exercises 10-15

Appendix A

Introduction to MATLAB®

MATLAB® and Simulink® A-1Command Window A-1Roots of Polynomials A-3

Rational Polynomials A-8Using MATLAB to Make Plots A-11Subplots A-19Multiplication, Division and Exponentiation A-20Script and Function Files A-26Display Formats A-32

Appendix B

Compensated Attenuators

Uncompensated Attenuator B-1Compensated Attenuator B-2

Appendix C

The Substitution, Reduction, and Miller’s Theorems

The Substitution Theorem C-1The Reduction Theorem C-6Miller’s Theorem C-10

Chapter 1

Basic Electronic Concepts and Signals

lectronics may be defined as the science and technology of electronic devices and systems.Electronic devices are primarily non-linear devices such as diodes and transistors and in gen-eral integrated circuits (ICs) in which small signals (voltages and currents) are applied tothem Of course, electronic systems may include resistors, capacitors and inductors as well.Because resistors, capacitors and inductors existed long ago before the advent of semiconductordiodes and transistors, these devices are thought of as electrical devices and the systems that con-sist of these devices are generally said to be electrical rather than electronic systems As we know,with today’s technology, ICs are getting smaller and smaller and thus the modern IC technology is

referred to as microelectronics.

1.1 Signals and Signal Classifications

A signal is any waveform that serves as a means of communication It represents a fluctuating

elec-tric quantity, such as voltage, current, elecelec-tric or magnetic field strength, sound, image, or anymessage transmitted or received in telegraphy, telephony, radio, television, or radar Figure 1.1shows a typical signal that varies with time where can be any physical quantity such asvoltage, current, temperature, pressure, and so on

Figure 1.1 Typical waveform of a signal

We will now define the average value of a waveform

Consider the waveform shown in Figure 1.2 The average value of in the interval is

Period -

Figure 1.2 Defining the average value of a typical waveform

A periodic time function satisfies the expression

for all time and for all integers The constant is the period and it is the smallest value of

time which separates recurring values of the waveform

An alternating waveform is any periodic time function whose average value over a period is zero.

Of course, all sinusoids are alternating waveforms Others are shown in Figure 1.3

Figure 1.3 Examples of alternating waveforms

The effective (or RMS) value of a periodic current waveform denoted as is the currentthat produces heat in a given resistor at the same average rate as a direct (constant) current, that is,

(1.3)Also, in a periodic current waveform the instantaneous power is

(1.4)and

Equating (1.3) with (1.5) we get

or

(1.6)or

(1.7)

where RMS stands for Root Mean Squared, that is, the effective value or value of a

cur-rent is computed as the square root of the mean (average) of the square of the curcur-rent.

Warning 1: In general, implies that the current must first be squared

and the average of the squared value is to be computed On the other hand, implies thatthe average value of the current must first be found and then the average must be squared

In introductory electrical engineering books it is shown* that if the peak (maximum) value of acurrent of a sinusoidal waveform is , then

(1.8)and we must remember that (1.8) applies to sinusoidal values only

1.2 Amplifiers

An amplifier is an electronic circuit which increases the magnitude of the input signal The symbol

of a typical amplifier is a triangle as shown in Figure 1.4

Figure 1.4 Symbol for electronic amplifier

* See Circuit Analysis I with MATLAB Applications, ISBN 0-9709511-2-4, Orchard Publications.

Pave≠ Vave⋅ Iave v t ( ) = Vpcos ωt i t ( ) = Ipcos ( ωt θ + )

An electronic (or electric) circuit which produces an output that is smaller than the input is

called an attenuator A resistive voltage divider is a typical attenuator.

An amplifier can be classified as a voltage, current or power amplifier The gain of an amplifier is

the ratio of the output to the input Thus, for a voltage amplifier

or

The current gain and power gain are defined similarly

1.3 Decibels

The ratio of any two values of the same quantity (power, voltage or current) can be expressed in

decibels (dB) For instance, we say that an amplifier has power gain, or a transmissionline has a power loss of (or gain ) If the gain (or loss) is , the output is equal tothe input We should remember that a negative voltage or current gain or indicates thatthere is a phase difference between the input and the output waveforms For instance, if an

op amp has a gain of (dimensionless number), it means that the output is phase with the input For this reason we use absolute values of power, voltage and current whenthese are expressed in terms to avoid misinterpretation of gain or loss

out-of-By definition,

(1.9)Therefore,

represents a power ratio of

represents a power ratio of

It is useful to remember that

represents a power ratio of

represents a power ratio of

represents a power ratio of

Also,

Input Voltage -

Bandwidth and Frequency Response

represents a power ratio of approximately

represents a power ratio of approximately

represents a power ratio of approximately

From these, we can estimate other values For instance, which is equivalent

equivalent to a power ratio of approximately

current ratios become

(1.10)and

(1.11)

1.4 Bandwidth and Frequency Response

Like electric filters, amplifiers exhibit a band of frequencies over which the output remains nearlyconstant Consider, for example, the magnitude of the output voltage of an electric or elec-tronic circuit as a function of radian frequency as shown in Figure 1.5

As shown in figure 1.5, the bandwidth is where and are the cutoff cies At these frequencies, and these two points are known as the 3-dB down or half-power points They derive their name from the fact that since power

“halved”

Figure 1.5 Definition of bandwidth

Alternately, we can define the bandwidth as the frequency band between half-power points Werecall from the characteristics of electric filters, the low-pass and high-pass filters have only one

Vin -log

Iin -2

Iin -log

Voutω

Vout = 2 2 ⁄ = 0.707

1 0.707

ω

Bandwidth

Vout

cutoff frequency whereas band-pass and band-stop filters have two We may think that low-passand high-pass filters have also two cutoff frequencies where in the case of the low-pass filter thesecond cutoff frequency is at while in a high-pass filter it is at

We also recall also that the output of circuit is dependent upon the frequency when the input is asinusoidal voltage In general form, the output voltage is expressed as

(1.12)where is known as the magnitude response and is known as the phase response These two responses together constitute the frequency response of a circuit.

(1.14)

and thus the magnitude is

(1.15)and the phase angle (sometimes called argument and abbreviated as arg) is

R 1 j + ⁄ ωC -Vin

The magnitude and phase responses of the low-pass filter are shown in Figure 1.7.

Figure 1.7 Magnitude and phase responses for the low-pass filter of Figure 1.6

−90°

ω ω

V out

V in

1 0.707

Now, let us consider a circuit whose gain is given as

(1.18)

where is a constant and is a non-zero positive integer Taking the common log of (1.18) and

multiplying by we get

(1.19)

We observe that (1.19) represents an equation of a straight line with abscissa , slope of

illustrated in the plot of Figure 1.8

Figure 1.8 Plot of relation (1.19) for Then, any line parallel to this slope will represent a drop of We observe also that

if the exponent in (1.18) is changed to , the slope will be

We can now approximate the magnitude and phase responses of the low-pass filter of Example1.1 with asymptotic lines as shown in Figure 1.9

slope = −20 dB/decade

k = 1

20 dB decade ⁄

Transfer Function

Figure 1.9 Magnitude and phase responses for the low-pass filter of Figure 1.6.

1.6 Transfer Function

Let us consider the continuous-time,* linear,† and time-invariant‡ system of Figure 1.10

Figure 1.10 Input-output block diagram for linear, time-invariant continuous-time system

We will assume that initially no energy is stored in the system The input-output relationship can

be described by the differential equation of

(1.20)

For practically all electric networks, and the integer denotes the order of the system Taking the Laplace transform** of both sides of (1.20) we get

* A continuous-time signal is a function that is defined over a continuous range of time.

† A linear system consists of linear devices and may include independent and dependent voltage and current sources For details, please refer to Circuit Analysis I with MATLAB Applications, ISBN 0-9709511-2-4 by this author.

‡ A time-invariant system is a linear system in which the parameters do not vary with time.

** The Laplace transform and its applications to electric circuit is discussed in detail in Circuit Analysis II, ISBN 5-9, Orchard Publications.

bmd

m

dtm -vout( )t bm 1– d

m 1 –

dtm 1– -vout( )t bm 2– d

m 2 –

dtm 2– -vout( )t … b0vout( ) t =

and

n

dtn - vin( ) t an 1– d

n 1 –

dtn 1– - vin( ) t an 2– d

n 2 –

dtn 2– - vin( ) t … a0vin( ) t

Solving for we obtain

where and are the numerator and denominator polynomials respectively

The transfer function is defined as

(1.21)

Example 1.2

Derive the transfer function of the network of Figure 1.11

Figure 1.11 Network for Example 1.2

Solution:

The given circuit is in the *The transfer function exists only in the †and thus we redraw the circuit in the as shown in Figure 1.12

Figure 1.12 Circuit of Example 1.2 in the

* For brevity, we will denote the time domain as

† Henceforth, the complex frequency, i.e., , will be referred to as the

G s ( ) L 0.5 H

Poles and Zeros

For relatively simple circuits such as that of Figure 1.12, we can readily obtain the transfer tion with application of the voltage division expression Thus, parallel combination of the capaci-tor and resistor yields

func-and by application of the voltage division expression

or

1.7 Poles and Zeros

Let

(1.22)where and are polynomials and thus (1.22) can be expressed as

(1.23)

The coefficients and for are real numbers and, for the present sion, we have assumed that the highest power of is less than the highest power of , i.e., In this case, is a proper rational function If , is an improper rational function.

discus-It is very convenient to make the coefficient of in (12.2) unity; to do this, we rewrite it as

(1.24)

The roots of the numerator are called the zeros of , and are found by letting in(1.24) The roots of the denominator are called the poles* of and are found by letting

However, in most engineering applications we are interested in the nature of the poles

* The zeros and poles can be distinct (different from one another), complex conjugates, repeated, of a combination of these For details please refer to Circuit Analysis II with MATLAB Applications, ISBN 0-9709511-5-9, Orchard Publications.

1 s ⁄ × 1

1 s 1 ⁄ +

- 1

s 1 + -

=

Vout( ) s 1 s 1⁄( + )

0.5s 1 s 1 + ⁄ ( + ) -Vin( ) s

F s ( ) N s( )

D s ( ) -

=

N s ( ) D s ( )

F s ( ) N s( )

D s ( ) - bms

1

an - b ( msm+ bm 1– sm 1– + bm 2– sm 2– + … b + 1s b + 0)

1.8 Stability

In general, a system is said to be stable if a finite input produces a finite output We can predict

the stability of a system from its impulse response* In terms of the impulse response,

1 A system is stable if the impulse response goes to zero after some time as shown in Figure1.13

2 A system is marginally stable if the impulse response reaches a certain non-zero value butnever goes to zero as shown in Figure 1.14

Figure 1.13 Characteristics of a stable system

Figure 1.14 Characteristics of a marginally stable system

3 A system is unstable if the impulse response reaches infinity after a certain time as shown

Figure 1.15 Characteristics of an unstable system

We can plot the poles and zeros of a transfer function on the complex frequency plane of thecomplex variable A system is stable only when all poles lie on the left-hand half-plane It is marginally stable when one or more poles lie on the axis, and unstable when one ormore poles lie on the right-hand half-plane However, the location of the zeros in the isimmaterial, that is, the nature of the zeros do not determine the stability of the system

We can use the MATLAB* function bode(sys) to draw the Bode plot of a Linear Time Invariant(LTI) System where sys = tf(num,den) creates a continuous-time transfer function sys with numer-ator num and denominator den, and tf creates a transfer function With this function, the frequencyrange and number of points are chosen automatically The function bode(sys,{wmin,wmax}) drawsthe Bode plot for frequencies between wmin and wmax (in radians/second) and the function

response is to be evaluated To generate logarithmically spaced frequency vectors, we use the mand logspace(first_exponent,last_exponent, number_of_values) For example, to generateplots for 100 logarithmically evenly spaced points for the frequency interval , weuse the statement logspace(−1,2,100)

only, we can use the bodemag(sys,w) function

MATLAB requires that we express the numerator and denominator of as polynomials of indescending powers

Example 1.3

The transfer function of a system is

* An introduction to MATLAB is included as Appendix A.

=

a is this system stable?

b use the MATLAB bode(sys,w) function to plot the magnitude of this transfer function

The zeros and poles of are shown in Figure 1.16

Figure 1.16 Poles and zeros of the transfer function of Example 1.3

From Figure 1.16 we observe that all poles, denoted as , lie on the left-hand half-plane andthus the system is stable The location of the zeros, denoted as , is immaterial

b We use the MATLAB expand(s) symbolic function to express the numerator and tor of in polynomial form

denomina-syms s; n=expand((s−1)*(s^2+2*s+5)), d=expand((s+2)*(s^2+6*s+25))

1 – j 2

3 – 4 j

3 – + 4 j

G s ( )

For this example we are interested in the magnitude only so we will use the script

num=3*[1 1 3 −5]; den=[1 8 37 50]; sys=tf(num,den);

w=logspace(0,2,100); bodemag(sys,w); grid

The magnitude is shown in Figure 1.17

Figure 1.17 Bode plot for Example 1.3

Figure 1.18 Asymptotic magnitude response for Example 1.4

1.9 The Voltage Amplifier Equivalent Circuit

Amplifiers are often represented by equivalent circuits* also known as circuit models The

equiva-lent circuit of a voltage amplifier is shown in Figure 1.19

Figure 1.19 Circuit model for voltage amplifier where denotes the open circuit voltage gain

The ideal characteristics for the circuit of Figure 1.19 are and

The Voltage Amplifier Equivalent Circuit

Figure 1.20 Amplifier circuit for Example 1.5

Solution:

The equivalent circuit is shown in Figure 1.21

Figure 1.21 The circuit of Figure 1.20

The parallel combination of the resistor and capacitor yields

and by the voltage division expression

(1.25)Also,

(1.26)and by substitution of (1.25) into (1.26) we get

Vload( ) s 19.61 10× 14

107s 1.1 10 + × 14 -VS( ) s

=

(1.27)and with MATLAB

num=[0 19.61*10^14]; den=[10^7 1.1*10^14]; sys=tf(num,den);

w=logspace(3,8,1000); bodemag(sys,w); grid

The plot is shown in Figure 1.22 and we see that the cutoff frequency occurs at where

Figure 1.22 Bode plot for the voltage amplifier of Example 1.5

1.10 The Current Amplifier Equivalent Circuit

The equivalent circuit of a current amplifier is shown in Figure 1.23

Figure 1.23 Circuit model for current amplifier where denotes the short circuit current gain

The ideal characteristics for the circuit of Figure 1.23 are and

Gv( ) s Vload ( ) s

VS( ) s - 19.61 10

14

×

107s 1.1 10 + × 14 -

The Current Amplifier Equivalent Circuit

(1.29)Substitution of (1.28) into (1.29) yields

(1.30)or

In Sections 1.9 and 1.10 we presented the voltage and current amplifier equivalent circuits also

known as circuit models Two more circuit models are the transresistance and transconductance

equivalent circuits and there are introduced in Exercises 1.4 and 1.5 respectively

=

Rout+ Rload -Aisciin

1.11 Summary

• A signal is any waveform that serves as a means of communication It represents a fluctuatingelectric quantity, such as voltage, current, electric or magnetic field strength, sound, image, orany message transmitted or received in telegraphy, telephony, radio, television, or radar

• The average value of a waveform in the interval is defined as

• A periodic time function satisfies the expression

for all time and for all integers The constant is the period and it is the smallest value oftime which separates recurring values of the waveform

• An alternating waveform is any periodic time function whose average value over a period iszero

• The effective (or RMS) value of a periodic current waveform denoted as is the rent that produces heat in a given resistor at the same average rate as a direct (constant)current and it is found from the expression

cur-where RMS stands for Root Mean Squared, that is, the effective value or value of acurrent is computed as the square root of the mean (average) of the square of the current

• If the peak (maximum) value of a current of a sinusoidal waveform is , then

• An amplifier is an electronic circuit which increases the magnitude of the input signal

• An electronic (or electric) circuit which produces an output that is smaller than the input iscalled an attenuator A resistive voltage divider is a typical attenuator

• An amplifier can be classified as a voltage, current or power amplifier The gain of an amplifier

is the ratio of the output to the input Thus, for a voltage amplifier

f t ( )avea

Period -

=

or

The current gain and power gain are defined similarly

• The ratio of any two values of the same quantity (power, voltage or current) can be expressed indecibels (dB) By definition,

The values for voltage and current ratios are

• The bandwidth is where and are the cutoff frequencies At these

half-power points

• The low-pass and high-pass filters have only one cutoff frequency whereas pass and stop filters have two We may think that low-pass and high-pass filters have also two cutoff fre-quencies where in the case of the low-pass filter the second cutoff frequency is at while

band-in a high-pass filter it is at

• We also recall also that the output of circuit is dependent upon the frequency when the input is

a sinusoidal voltage In general form, the output voltage is expressed as

where is known as the magnitude response and is known as the phaseresponse These two responses together constitute the frequency response of a circuit

• The magnitude and phase responses of a circuit are often shown with asymptotic lines asapproximations and these are referred to as Bode plots

• Two frequencies and are said to be separated by an octave if and separated

• The transfer function of a system is defined as

where the numerator and denominator are as shown in the expression

• In the expression

where , the roots of the numerator are called the zeros of , and are found by letting

The roots of the denominator are called the poles of and are found by letting

• The zeros and poles can be real and distinct, or repeated, or complex conjugates, or tions of real and complex conjugates However, in most engineering applications we are inter-ested in the nature of the poles

combina-• A system is said to be stable if a finite input produces a finite output We can predict the bility of a system from its impulse response

sta-• Stability can easily be determined from the transfer function on the complex frequencyplane of the complex variable A system is stable only when all poles lie on theleft-hand half-plane It is marginally stable when one or more poles lie on the axis, andunstable when one or more poles lie on the right-hand half-plane However, the location ofthe zeros in the is immaterial

• We can use the MATLAB function bode(sys) to draw the Bode plot of a system where sys =

denomi-nator den, and tf creates a transfer function With this function, the frequency range and number

of points are chosen automatically The function bode(sys,{wmin,wmax}) draws the Bode plotfor frequencies between wmin and wmax (in radians/second) and the function bode(sys,w)

uses the user-supplied vector w of frequencies, in radians/second, at which the Bode response is

to be evaluated To generate logarithmically spaced frequency vectors, we use the command

magni-tude only, we can use the bodemag(sys,w) function

• Amplifiers are often represented by equivalent circuits also known as circuit models The mon types are the voltage amplifier, the current amplifier, the transresistance amplifier, andthe transconductance amplifier

F s ( ) N s( )

D s ( ) -

1

an - b ( msm+ bm 1– sm 1– + bm 2– sm 2– + … b + 1s b + 0)

1.12 Exercises

1 Following the procedure of Example 1.1, derive and sketch the magnitude and phase responses

for an high-pass filter

2 Derive the transfer function for the network shown below

3 A system has poles at , , , and zeros at , , and Derive thetransfer function of this system given that

4 The circuit model shown below is known as a transresistance amplifier and the ideal

characteris-tics for this amplifier are and

With a voltage source in series with resistance connected on the input side and a loadresistance connected to the output, the circuit is as shown below

RC

G s ( )

L 0.5 H

magnitude of for the range From the plot, estimate the cutoff quency.

fre-5 The circuit model shown below is known as a transconductance amplifier and the ideal

charac-teristics for this amplifier are and

With a voltage source in series with resistance connected on the input side and a loadresistance connected to the output, the circuit is as shown below

Derive an expression for the overall voltage gain

Solutions to End-of-Chapter Exercises

1.13 Solutions to End-of-Chapter Exercises

Dear Reader:

The remaining pages on this chapter contain solutions to all end-of-chapter exercises

You must, for your benefit, make an honest effort to solve these exercises without first looking at thesolutions that follow It is recommended that first you go through and solve those you feel that youknow For your solutions that you are uncertain, look over your procedures for inconsistencies andcomputational errors, review the chapter, and try again Refer to the solutions as a last resort andrework those problems at a later date

You should follow this practice with all end-of-chapter exercises in this book

We will use the MATLAB script below to plot versus radian frequency This is shown

on the plot below where, for convenience, we let

w=0:0.02:100; RC=1; magGs=1./sqrt(1+1./(w.*RC).^2); semilogx(w,magGs); grid

We can also obtain a quick sketch for the phase angle, i.e., versus , by

as ,

R 1 + ⁄jωC -Vin

Solutions to End-of-Chapter Exercises

For ,

For ,

As ,

We will use the MATLAB script below to plot the phase angle versus radian frequency This

is shown on the plot below where, for convenience, we let

w=−8:0.02:8; RC=1; argGs=atan(1./(w.*RC)).*180./pi; plot(w,argGs); grid