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Masters Theses Student Theses and Dissertations

1969

An analysis of an audio amplifier utilizing an operational amplifier An analysis of an audio amplifier utilizing an operational amplifier

and negative feedback and negative feedback

Anthony Francis Lexa

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Recommended Citation Recommended Citation Lexa, Anthony Francis, "An analysis of an audio amplifier utilizing an operational amplifier and negative feedback" (1969). Masters Theses. 6954. https://scholarsmine.mst.edu/masters_theses/6954

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ANA .. LYSIS OF AN AUDIO At,JFLI?IER UTILI7.ING AN

OPEPATIONAL .ANP~.IFIEF AND NEGATIVE FEEDBACK

BY

ANTHOI\ry FR .. <\NCI S LEXA \ C\ ~s--\

A

THESIS

submitted to the faculty of

TBE UNIVERlSTY OF !1ISSQUl~I - ROLLA

in partic:l ft.:liullment of the reqnirements for the

Degree of

}'fASTER OF SCIENCE IN FLEC:TRICA.L ENGINEERING

Rolla, Hissouri

1969

ii

An audio a~plifier utilizing a monolithic operational

amplifier is analyzed. The analysis is a block diagram

approach where the amplifier is divided into three parts;

operational amplifier, power amplifier, and feedback

network. A Fourier series analysis is used to describe the

distortion and signal components of the amplifier out;put.

The results of this analysis demonstrate the advantage of

large negative feedback on frequency response and har­

monic distortion. Using relatively few passive components

a high quality audio amplifier is constructed with an out­

put power of 15 watts RMS power and negligible harmonic

distortion.

ACKNOWLEDGE}lliNTS

The author wishes to express his gratitude to his

advisor, Dr. Ralph s. Carson~ Professor of Electrical

Engineering~ for his guidance and assistance in this

Master's thesis.

Thanks are also due to Mr. Patrick Vennari for his

programming assistance.

iii

iv

TABI~E OF CONTENTS

ABSrfRA Cjf. . • • • • . • • • • • • • • . . • . • . • • • • • . • . • • • • . . • • • .. • . . . • • . • i i

4~CKNOWl..JEDGE?1ENTS. • • • • • • • • • • • • • • • • • • • • . • • • • • • • • • • • • • . • • • iii

LIST OF FIGURES. • • • . . . . . . . . • . . . . . • . . . . . . . • • . . . . . • • . . • • . vi

LIST OF TABI~ES. . . . . . . . . . . . . . • . . . . • . . . . • . . . . . . . . . . . . . . . . viii

LIST OF SYmOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

I • INTRODTJCTION. . . • • . . . . . . • . • . • • . • • . . . . • . • • • • . . • . • 1

II.

A. The Integrated Monolithic Operational Amplifier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

B. Audio Amplifier Using The Operational Amplifj_er.................................. 2

C. Method Of Solution......................... 5

DESCRIPTION OF POWER AMPLIFIER ................ .

A. Transistor Model Used For Power Amplifier Simulation ................................ .

B • ,.... • A t" . S :tmp ~ L.cy~ng s sump ~ons ................... .

C. Node Voltage Equations For PovJer Amplifier.

D. Solution Of Equations ..................... .

E. Block Diagram Of The Power Amplifier ...... .

7

7

14

15

18

21

III. ADAPTATION AND DESCRIPTION OF OPERATIONAL

IV.

A}1PJ_, ·rF I ER • • • • • • • • • • • • • • . • • • • • • • • • • • • • • • • • • • • • • • 3 0

A. Open-I ... oop Gain............................. 32

B. Frequency Compensation..................... 33

C B • • . ~as 1.ng ................................... .

D. Feedback Network .......................... .

COMPUTATION OF AMPLIFIER CHARACTERISTICS ...... .

34

36

41

A. Block Diagram Of The Audio Amplifier....... 41

B. Output Signal And Output Distortion........ 41

c. Adaptation Of Power Amplifier Analysis..... 46

v.

VI.

v

D. Audio .. tunp lifier Analysis Progr.am. . . . . . . . . . 52

EXPERI}lliNTAL VERIFIC~TION .................... .

A. Objectives ............................... .

59

59

B. Equipment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

C. Results................................... 61

DISCUSSION AND CONCLUSIONS •................... 67

BIBLIOGRA.PHY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

APPENDICES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

I. Least Square Curve Fit Of The Form y = y 0 ( exp (kx) - 1 ) .........•...•....•....

II. Newton-Raphson Method For A System Of Equations . .............................. .

III. Power Amplifier Analysis Program ....... .

IV. Derivation Of Fourier Series For Waveform Of Crossover Distortion ................. .

v. Audio Amplifier Analysis Program ......... .

78

81

85

Qq ..,_

101

VITA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ,. . . . 119

LIST OF FIGURES

Figure

1

2

3

4

5

6a

6b

7

8

9a

9b

Sirtlplified Block Diagram Of Audio Amplifier ..... ,

Schematic Diagram Of The Pcn·Jer Amplifier ........ .

Equivalent Model Of Tr2nsistor .................. .

Actual And Model Transistor Characteristics ..... .

Po·_,;er Amplifier Using Tra.._1sistor Nodels ......... .

Verification Of Po\·7er Am?lifier Frequency Independence (Theoretical) ................. .

Verification Of Power A~plifier Frequency Independence (Ac cua 1 ) .......•...• , •.....•...

Appi:cximate Haveform Of C;:-c:-.. ssover Distortion .....

Block Diagram Of Po·wer A:np lifier ................ .

'HLP.leform Of Distortion Signal J d 0 (Theoretical) ..

\.·Javeform Of Distortion Signal, d 0 (Actual) ...... .

10 Operationa.l Amplifier Schematic And Equivalent

vi

Page

3

8

10

11

16

19

20

22

25

27

28

Circuit . ........................ ~ . . . . . . . . . .. . . . 31

11

12

13

11+

l5b

16

1'7

18

Frequency Response Of Operational Amplifier ..... .

Exte.c:aal Circuitry Of Operational Amplifier. ..... .

Feedback Net-c;vork .....•..•..........•.............

Ba;;; ic Feedback .Amplifier ........................ .

Schematic And Block Diag1.·a:n Of Audio A!nplifier (Schematic) ................................... .

Schematic And Block Diagram Of Audio Amplifier (Block Diagram) . ........................... .

Theoretical Frequency Response Curve ............ .

Theoretical Plot Of Per Cent Total Ha rrnonic Distortion Versus Frequency ................ .

Theoretical Plot Of Per Cent Total Harmonic Distortion Versus Output Power ............. .

32

35

38

39

42

43

56

57

58

Figure

19

20

21

22

Pictorial Diagram Of Test Setup And Equipment ....

Experimental Frequency Response Curve ........... .

Experimental Plot Of Per Cent Total Harmonic Distortion Versus Frequency ................ .

Experimental Plot Of Per Cent Total Harmonic Distortion Versus Output Power ............. .

vii

Page

60

64

65

66

Table

1

2

3

4

5

LIST OF TABLES

Transistor Model Parameters ..................... .

Fundam~ntal Power Content Of Power Amplifier Output ....................................... .

Results Of Power Amplifier Analysis Program ..... .

Results Of Audio Amplifier Analysis Program ..... .

Results Of Experimental Analysis Of Audio A1nplifier . ................................... .

viii

Page

13

50

5!

55

63

LIST OF SYHBOLS

input signal to audio amplifier (sinusoid)

maximum value of e. ].

output signal from audio amplifier

E0 •.. maximlli~ value of e 0

. . .

input signal to power amplifier (sinusoid)

maximum value of e 1

output signal from power amplifier

maximum value of e2

feedback ratio

Q1 •·A transistor 2N 3904

Q2 transistor 2N 3906

Q3 transistor 2N 3791

Q1~ transistor 2N 3'715

Vee DC power supply voltage

Ra base limiting resistor

CB base lead capacitor

R1 load resistor ......

R. input resistor ].

Rf feedback resistor

cf feedback capacitor

ix

I 0 reverse bias saturation current for an ideal diode

transistor base current

empirically found constant describing transistor input characteristic

transistor turn on voltage

voltage across ideal diode in transistor model

X

vbe voltage from base to emitter in transistor model

Ic

vee

hFE

hoe

vl

v2

v3

v4

v5 T

A2

Bk

do

0 ok

de

0 ck

Eok

Zin

zout

. . .

. . .

transistor collector current

voltage from collector to emitter

h parameter DC current gain

h parameter output conductance

voltage at node 1 in power amplifier

voltage at node 2 in power amplifier

voltage at node 3 in power amplifier

voltage at node 4 in power amplifier

voltage at node 5 in power amplifier

peried of signal

amount of time that power amplifier output signal is near zero - delay time

radian frequency of input or output signal

radian frequency of positive and negative sinusoidal pulse for the approximate wave­form of crossover distortion

gain of power amplifier

coefficients of Fourier series representing crossover distortion

theoretical distortion signal

coefficients of Fourier series which represents d

0

theoretical distortion in output of audio amplifier

coefficients of Fourier series ~vhich resprsents de

coefficients of Fourier series which represents output of audio amplifier.

input inpedance of operational amplifier

output indedance of operational amplifier

PLOAD •••

DTOT ••.

Xi

mid-band gain of operational amplifier

frequency dependent gain of operational amplifier

upper cutoff radian frequency for operational amplifier

power delivered to load

per cent total hannonic distortion

I. Intro8uction

A. The Integrated Monolithic Operational Amplifier

The recent development of the monolithic integrated

operational amplifier has caused a great change in circuit

design theory. It is now possible to design circuits

around the operational amplifier as a basic building block

and use relatively few additional passive components. The

operational amplifier characteristically is a high gain

device. Because of thisJ negative feedback can be e~ployed

to provide good stability and increased bandwidth.

The techniques used for the manufacture of these

integrated_....mQ;HF~- circuits are repeated operations

of maski.ngJ photo etching, and dopant diffusion on a

single wafer of silicon. Because of this type of fabrica­

tion, it is actually easier to make a transistor or

diode than it is a resistor or capacitor. So it is

des:trable for the ratio of active to passive components to

be much higher in an integrated circuit as compared to a

discrete circuit of sirniliar function. This fact results

in the use of differential amplifiers in integrated

circuits. Coupling capacitors are not needed because the

differential amplifier is a DC amplifier. Hith this type-

l

of design a high ratio of active to p~ssive components can

be achieved. A more detailed descri.ption of the operational

amplifier circuitry will be discussed in Section III.

The use of differential amplification offers a number

of advantages~ such as, DC amplification, good stability,

ir.1munity to interference signals, and wide versatility to

~ention a few. However~ these advantages are dependent on

how well the two devices are matched. Since the transis-

tors are made from the same silicon chip and are physically

close to one another in the integrated circuit, they are

very closely matched in performance and temperature.

Therefore with these characteristics, the integrated

operational amplifier is attractive for circuit design.

Also, as manufacturing techinques are perfected , the cost

will be lowered making the integrated circuit economically

a tt·~.·ac t i ve.

B. At.idio Amplifier Using The Operational Amplifier

I

The circuit which this thesis v.,ill amalyze is an audio

a1nplifier using the type of operational amplifier discussed

above. The audio amplifier has three basic functional

divisions: operational amplifier, power amplifier, and

feedback network. The simplified block diagram is given

in figure 1.

The operational amplifier is a Motorola MC 1433

monolithic integrated operational amplifier. The power

amplifier is a complementary class B amplifier using two

complementary silicon driver transistors and two comple­

mentary silicon power transistors. The function of this

2

feed-oaci< ~------------~ -----· --~ network

ei() \ e ./0

Figure 1. Simplified Block Diagram of Audio Amplifier

w

circuit is to give the necessary power gain to drive the

load. The audio amplifier will produce 15 watts (&"'1S)

power into a four ohm load. This circuit will be discussed

more fully in Section II. The feedback network is a R-C

network which determines the amount of feedback from out­

put to input.

There are many advantages to this type of audio

amplifier design. One, there are fewer discrete components

than a similiar audio amplifier of the same po~7er rating.

The operational amplifier replaces many front end stages

of standard transistor design, where each of these stages

require many passive elements for biasing and signal

coupling. This causes simpler layout and fewer connections

resulting in higher reliability. Two, the load is directly

coupled to the output of the amplifier. Normally an out­

put transformer would be used, which is often a limiting

factor in an audio amplifier's fidelity. Three, the

frequency response is DC to beyond the audio range because

of the operational amplifier's differential amplification

and wide bandwidth. The power amplifier uses wide bandwidth

trans is tors wh:l.ch do not limit the frequency response.

Therefore, the upper cutoff frequency is adjustable by

selection of the passive elements used. Four, the amplifier

has high efficiency because the power amplifier is class

B. Also the operational amplifier's power requirement is

very small. Five, the amplifier has a very low level of

distortion. This is probably the most favorable advantage

of this type of audio amplifier design. Because of the

operational amplifier's high gain a large amount of negative

feedback is used. As ~vill be shown in Section IV, this

feedback results in a reduction in harmonic distortion by

approximately a factor equal to the gain of the operational

amplifier. In d high-quality audio amplifier the harmonic

distortion must be kept to very low level (less than 1 per

cent).

C. Method Of Solution

The audio amplifier will be studied by analyzing each

o:r the three functional blocks listed above sepa.rately and

then combining the results to describe the entire amplifier.

The power amplifier will be analyzed first. The

most prominent source of this distortion in the entire

amplifier is crossover distortion in this stage due to the

~lass B operation.

5

To analyze this circuit a transistor model is formulated

and used to describe the response of the power amplifier.

The digital computer is used to solve a set of five

simultaneous non-linear equations. \\Tith this information

a block diagram representation of the power amplifier can

be made consisting of an ideal amplifier in series with a

distortion signal generator. This ntodel discription of the

power amplifier makes a ·description of the entire amplifier

possible.

Next the operational amplifier will be studied. An

expression \·.7hich describes the gain as a function of

frequency will be found. Biasing and frequency stabilita­

tion by use of external circuitry will be discussed. \·Jith

this information it is kno~n how the operational amplifier

will function in. the audio amplifier.

6

By using the information. concerning the operational

amplifier, a description of the feedback network is deter­

mined. The usual symbol for the transfer function describing

how much of the output is returned to the input is~-

So a description of the feedback network is actually a

determination of$.

Using the information of the above three sections the

entire audio amplifier can be described. This description

is derived from a Fourier series of the distortion in the

power amplifier. Knowing the distortion introduced, the

gain of the operational and power amplifiers, and~, the

per cent harmonic distortion in the output signal can be

determined. By varying the input signal level (and thereby

the output power) and frequency, an evaluation of the

audio amplifier can be made.

Finally, experimental data of the same factors above

will be taken and correlation of theoretical and experimen­

tal results will be discussed.

II. Description Of Power Amplifier

A. Transistor Model Used For Power Amplifier Simulation

Figure 2 is the schematic diagram of the power

amplifier used in the audio amplifier. As was discussed

earlier, this is a class B complementary power amplifier.

All the transistors are silicon. Ql and Q2 are 210 mw

complementary driver transistors. Q3 and Q4 are 150 w

complementary transistors.

In order to evaluate the response of this circuit, a

transistor model was formulated. Since this is a class B

amplifier with no bias, the models must be accurate in

the cutoff region because the point of operation moves

fcom cutoff into the active region and back again as the

input signal varies around zero. The majority of the

distortion in .the power amplifier is due to crossover

distortion which is caused by operation in the nonlinear

region of cutoff.

The model used for this purpose is a "modified h

parameter" model. The output circuit for the transistor

model is a dependent current generator whose current is

equal to hFE Ib. In parallel with the generator is a

conductance h . This, so far, is identical to a standard oe h parameter equivalent circuit. The input circuit is

composed of an ideal diode in series "t-Jith a "battery 11

equal to the turn-on voltage, V~- The current through

7

I

+V cc 0

rff'J l--r-Q3

_LCs~ I . v I -

I I,L -----~ o---, i > R

I I '- L

nput

· RB Q4 9 L I ~~ ~ +Cs ~

~ -vee \,..

Figure 2. Schematic Diagram of Power Amplifier

Ql Q2

Q3 Q4

CB RB vee

2N3904 2N3906 2N3791 2N3715 0.1 mfd 20 ohms 15 volts

(X)

the diode is described by the diode equation:

( K v I Id = I 0 e be - 1)

where: Id = diode current

I 0 = saturation current at reverse bias

K - empirically determened constant

Vb~ = actual voltage across ideal diode

Figure 3 shows the transistor equivalent circuit

described above. Notice in the equivalent circuit the

use of a battery, V~. It should be stressed that Vy is not

an actual battery in the equivalent circuit but simply a

transposition factor. In order to fit the actual input

characteristics with the diode equation the starting point

on the horizontal base voltage axis must be shifted to the

right (increasing voltage) by an amount Vt . Thus Vt is

a transposition factor of Vbe. Therefore, in figure 4 for

the model input characteristics, the exponential equation

starts at v"6 0

In order to find the values of V'6 , K, hFE' and h 0 e

for a particular transistor a photograph of the input

9

(Ib vs Vbe) and output (Ic vs Vee) characteristics is taken.

From the output characteristic hFE and h 0 e are determined

and K and Vy is obtained fram the input characteristic.

I is found by measuring the current with the base emitter 0

junction reversed biased.

Measurement of one of the four transistors used in

the audio amplifier is demonstrated in figure 4. The

definitions of the parameters are:

b~--oc f"j.

'---!

tl -' e-~ ···-o ~

+

vt base +j L- ·-l o- il I

Ib ; ! I I vbe

o-

where:

Vbe \:7

T J

I (1\. "Jj hfe1b

I I .

I emitter

Ib = I 0 (exp(K Vb~) - 1)

vbe= vb~ + v'6 I0 ~I0 (exp(K(Vbe·V~)) - 1)

Figure 3. Equivalent Model Of Transistor

collector ·----{) l _.. + I

~hoe Vee ;

I

I o

t-' 0

Input · Charactertsttcs actual . (2N3906)

IIOdel

.,. ~1 •10

• . ' ·• ... Output

Cbaractertsttca Ic

(aa) I

J • ,

, '•

hFE -

hoe -

Ic !1:,--

Ic

vee

V - constant ce - (DC current gain)

Ib = constant (output admittance)

V - turn on voltage = voltage at which Ib

increases such that transistor goes into

the active region.

K -- empirically determined constant from input

characteristic data using a least sq~~are

fit. (Refer to Appendix I)

! 0 - current with base=emitter junction reversed

biased.

As can be seen from figure 4 a point must be choosen

on the output characteristic where the hFE and h 0 e can

12

be measured. If the point of measurement is changed the

values of hFE and h 0 e will also change. Therefore, an

approximation must be made. The points of measurement are

taken approximately in the center of the region of

operation. This is a compromise betv1een the extreme regions

of saturation and cutoff. Table I lists all of these

measured ct.'\nstants and points of measurements.

Once these constants are known a set of idealized

model characteristics can be formed. This is also done in

figure 4. In the left column the actual photo character­

istics are given, in the right column are the idealized

characteristi_cs obtained from the measured parameters.

Measurement in this fashion of all the transistors yields

IC I ----~--------T ____________ l ____ T__ ' --------1 j !Type 'I hFE 1 hoe l K : Vt 1 Io I points of measurement ;

t -----+ _4 !----+ I , --------Ql 2N3904- 167. o : 2 .xlo -u j 28.18078 1 o. 5 , o.l+ra 1 hFE Ib = 0.1 rna

lQ2 I 2N3906 165.0 I s.xlO-~-!30~940291--o.-sl O.lJ.ra I hoe at vee= s.o volts

~31-2N3"791 i 66. 7 j • 025 " ~-il: 91+ 721 I 0. 0 il.O,...a ~~hF;- Ib = 30. rna

}E 2N3'Tl5 I 56.7 I .020-v- j 12.2167U 0.0 11.~ hoe at Vee = 10 volts

Table 1. Transistor Model Parameters

j--l

w

Table I which lists all the parameters. With these

parameters and the transistor model the power amplifier

can be analyzed.

B. Simplifying Assumptions

Refering to figure 2, notice the capacitors from Q3

and Q4 bases to ground. The assumption will now be made

that t-7e c:-.=m neglect the GB capactors in this analysis.

14

The justification stems from the considerations that

CB is a 0.1 mfd capacitor designed only to attenuate high

frequency (mech higher than the audio spectrum). Therefore,

CB at audio frequencies does not present a low enough

impedance to attenuate the audio signal. Also notice that

the transistor models have no capacitance in the equivalent

circuits. The upper frequency cutoff from the transistors

themselves "'.vas neglected because the gain-bandwidth products

of the power transistors is 4 MHZ minimu.-rn and 250 HHZ for

for the driver transistors. With these gain-bandwidth

products it is easily shown that the transistors are not

frequency limited in the audio range.

This assumption is verified experimentally later in

this section. For the time being, by using these assump­

tions, the power amplifier analysis can be greatly simpli-

fied.

C. Node Voltage Equations For Po'tJer Amplifier

With the above assumptions, the power amplifier of

figure 2 can be redrawn using the equivalent transistor

models just discussed but neglecting the effect of CB.

Figure 5 is the power amplifier circuit using the equi­

valent models.

The transistor parameters Vt, K, hFE and h 0 e are

subscripted to match the number of the transistor Ql, Q2,

Q3, or Q4. Also the nodes of the circuit are numbered Vl,

V2, V3, V4, and V5. It should be noted here that the

transistor model is a representation for frequencies

15

from DC to the upper lim{t on the audio range (20,000 KHZ).

Therefore the Q3 and Q4 emitters are connected to the power

supplies and not AC ground since this is a DC model.

The circuit can be solved by writing the node voltage

equations. The base currents, which are exponential

functions of the base-emitter voltages, can be expressed

by substituting the voltage difference bet1;veen the proper

two nodes for Vbe" In this fashion the node voltage

equations can be written expressing the base currents as

exponential functions of the node voltages.

With these considerations we can now writ·e the

equations for all five nodes.

+Vee

e1 = input signal ( E1 sinwt) i,_eJ i r--· I :J. r ·-\ ,..,..

1_, (1 l '-. hoe3 .sz \.__'II _) ~ I Q3 ~E3Ib~ V ,+ _I I 2 R b3 ·~ ~ I

l B /v-o---tf\r, I _ c3 ~ v4 '>h3 ~FElib.~ h I' j < oel

1 v,. ' ( ~ Ql ? v l. \ +

+ \ ~---·--- R ' e

. ~~2Ib v i lb2 ±--.J I ~ hoe2 ~b4 ~ * 1>4+ -~ h 41b4

b2 - RB ~ Q4 !e4 1 oe

-~cc

e2 = output signal

Figure 5. Power Amplifier Usi.ng Transistor Models

1-' m

For node 1:

1o1 (hFE1 + 1) (exp(K1 Vbe1) - 1)

-!02 (~E2 + 1) ( exp (K2 Vbe2) - 1)

+!03 (hFE3 + 1) (exp(K3 Vbe3) - 1)

-Io4 (hFE4 + l) (exp(K3 Vbe4) - 1)

+(V2 - Vl) hoel -(vl.- V3) hoe2 +(Vee

-(Vl + v ) h ee oe4

For node 2:

For node 3:

-v3 + v5 = o

For node 4:

For node 5:

-(Vl I~) = 0

vbe2 = vl - el - vl2

vbe3 = vee - v4 - v~3

vbe4 = v5 - vee - v~4

- Vl) hoe3

(1)

(2)

(3)

These equations describe the response of the power

amplifier using the equivalent transistor model discussed

in Section II - A.

17

13

D. Solution of Equations

In order to solve the five simultaneous non-linear

equations, the well known NewtonRaphson interative techniq~e

was used. (Refer to Appendix II) A program was written to

solve the equations assuming a sinusoidal input voltage.

The reason for a sinusoidal input will become apparent

later in this discussion.

The period of the sinusoid is broken into 200 seg­

rnents and the equations solved for each of these segments.

Knowing the node voltages, the base currents, collector

currents, base-emitter voltages, and collector-emitter

voltages can be found by substituting back into the

proper equations. At each time increment the computer

calculates all of the above variables. Appendix III lists

the power amplifier analysis program, flow chart, the

actual program, and a sample of the output.

The only input variable into the program is the rnaxi.mum

value of the input sinusoid, E1 . From the previous assump­

tions the output waveform from the computer should describe

the actual circuit.' s wavefonn for any frequency in the

audio range.

To verify this, figure 6 compares the theoretical

waveform fran the equivalent model circuit to the actual

response of the power amplifier for three different

frequencies. Figure 6-a is the theoretical response

plotted by the computer. Figure 6-b is the actual response

19

Theoretical Response Using Transistor Models

10 sinw t (volts)

r :x, ' ·r •-:: J .,.. t ! I t : ..-.- t I _l 1 .._,~, l j " I

10;; ~~~ +

t

!/ :s 0 tl:+-~-+-~+-1-+-•-· > -+:· ~-+;_c-+++-+ ~-+ +' + ,_ + ++-· <-+--+ ' + -c + ,_ ~- ',"

~.; t ll~k \: //

-s.r- \ / t ' f ~ / -- -----------------

-10.1 Figure 6-a. Verification of Power Amplifier Frequency

Independence (Theoretical)

Act~al Po~r Amplifier Response

all verticle scales 5 volts/division

input (e1 )=lOsin w t for all cases

time(lO a/division)

20KHZ

20

Figure 8-b. verification of Power Amplifier Frequency

Independence (Actual)

21

at various frequencies. In all cases an input voltage

peak of 10 volts was used. From the observation of figure 6

it can be seen that the output signal appears to represent

a sinusoid except for a "flat" region near the zero volt­

age poj_nt. This is commonly called crossover distortion.

As can be seen from figure 6 the theoretical computer

description is a good approximation at low frequencies

(up to 10KHZ), but deviates at 20KHZ. This deviation,

for now, will be neglected and discussed in Section VI.

E. Block Diagram Of The· Power Amplifier

Thus far, the response of the power amplifier has

been obtained by use of the transistor models. Now, in

order to use this information in an overall analysis of

the audio amplifier it must be placed into a usable form,

such as a Fourier series.

Therefore, let us derive the Fourier series for a

general waveform of crossover distortion. The first step

is to make an approximate waveform of crossover distortion

so that the form of the Fourier series is reasonable.

Refering to figure 7, an approximate waveform is described.

The positive and negative sides of the waveform are assumed

to be sinusoids of a reduced frequency of the over all

periodic waveform with the zero regions at the crossover

point.

. -, ..:::__

r-=., . --{ f.-

rc c . ,..-, '-.J'

----, L........

cc ij_J

> c (.()

V"l 0 cc u

f(t) 1.0.,.

1

.f.. i

T

5 ! . -!

r

I

I I

I i

------...... / "'

// """ / \,

\ \

I

\ \

\

\ \

\

\ I . ( \ ~- I \ ~T.l · : \ I

0 • 0-~H+-t++-+-H-++++++H+t+h·' 1-+·H+++-r++++H+t-+t-i h->~•-H-1-;'- f+ H d+-H-+++·H-i H·i 1 t++ H-H+t·H-I++f I; H-H t t +I i-'-'

! I I I + ~\ I, T -- ----- -f-.... I r T :\/! c· I I r 1 1: 1 L 1 I I i \ I

I - 5' • T

T i I

t i

T

-1. QL

Figure 7.

\ ~ \ /

\ I \ I

\ . \ I \ : \ I

I ~'-'

Approximate Waveform of Crossover Distortion 1\) 1\)

Mathematically let

T = period of waveform

7' = "delay" time

Then the approximate signal is

0

sin w't

f(t) = 0

sin w't

0

' 27r ~vhe ... -e. r ,, = --.. · .......... T-2?"

J: <t < T-7' 2 2

T-r t T+r -< <-2 2 T+r .y - <t< T--2 2

,.,. T- ..1.. < t < T 2

For this signal the Fourier series is: 00

(6)

f(t) = [ Bk sin(27rkt/T) k=l k = 1,2,3 ...

(7)

where: Bk = (4/T) cos ( ~...,) [sin (b (T21'll sin(a(T2?')) -2b 2a

sin (b ( ; ) ) sin(a( J> J 2b + 2a

+(4/T) sin(w~r) [cos (b (T2 7')) cos (a (T~ ?" ) )

2b +-2a

-r' _ cos(a( J )) J cos (b (2))

2b 2a

(See Appendix IV for derivation)

a = 2?T('I' (k+l) - 2k r l (9) - T(T-2 )

b 27r(T (~-:ll__:_ 2k U (10) = - f('f-2--J

23

(8)

Figure 7 is actually a computer summation using the

above Fourier series with the series truncated at 50

harmonics. This explains the rounded edges near the zero

values. Notice that the peak value is normalized to a

peak value of one. Therefore multiplying all of the

Fourier coefficients by a gain factor will adjust the

output peak to the proper level.

24

To find the Fourier series of the output waveform of

the power amplifier all that need to be known are the

average ma.ximurn value of the waveform, 'land T . Since the

frequency is not a variable in the power amplifier, T andr

cannot be specified, therefore the ratio ofrto Tis

used in their place.

At this point the output waveshape and Fourier series

is known. This information can be used to form an

equivalent block diagram representation of the power

amplifier shown in figure 8.

Figure 8 represents the power amplifier by an ideal

amplifier with a gain ~· and a distortion signal, d 0 ,

being added to the output. If the input is assumed to be

a sinusoid and the Fourier series of the output is known,

the form of d can easily be found. 0

First assume

1 ) e 1 = E 1 sin w t

2) normalized Fourier series representing _s,

crossover distortion = L Bk sin k t k=l

;:--· (\v e+ \ 1

·--------------.

Power Amplifier

'-------·-----~

u e2 peak to peak

el peak to peak

_j--- ideal amplifier

o--- .

e~/ LJ.n = A2

1 --~

-~---·-----

----o -f\

___ )/\v ~

1-

Equivalent Block Diagram of the Power Amplifier

Figure 8. Block Diagram of Power Amplifier

25

iiJhere:

-- -·

peak~~ peak value of ~utput peak to peak va 1.ue or: input

As defined by figure 8:

A2el + do = e2 (12)

And 00

26

d 0 = e2 - e 1A2 = E1A2 L Bk sin kwt - A2E 1 sin t.ut (13) k=l Distortion d 0 can also be represented by a F · · - . our1er ser1es

d =~D k sin kwt (14) 0 k·l 0

Then for the first harmonic:

D01 = A2E1 (B 1-l) (15)

And for the remaining harmonics:

(16)

where:

k = 2,3,4,5 .....

Combining the two expressions 15 and 16, d can be 0

expressed as: 00

sin wt + E 1~ LBk sin k~ k=Z

(17)

Using tl1e info~1nation from the computer analysis of

the power amplifier and the Fourier series for crossover

distortion, d 0 and e2 can be computed. Figure 9 shows d 0

and e2 . Figure 9-a is the theoretical computer results

and figure 9-b is actual photograph response, when e 1 has

a peak value of 10 volts. d 0 was photographed by filtering

out the fundamental component of the output signal. Again,

since the theoretical result, figure 9-a uses only the

first 50 terms in the Fourier series and is calculated at

10. J..

I I

r i

-t-

5·T +

Figure 9-c:t.

27

Theoretical Computer Respollse

input(e 1 ) - 10 sino>t

f I f":E • ·H ++++ -l-..f-t--+1 ;.; t -t ·t +- t -1 1--, t-..t ;.- +! ~ + ; i t , r- r ; 1 ' r ~ ~ • ~ J ~ / ~

waveform of D1_stortion Signal, d 0

(Theoretical)

/(//

/ I

,/

*

Actual Power Amplifier Response

input (e1 ) == 10 sinw t

upper channel: outp~t(e2 ) vs time verticle scale: 5volts/division

lower channel: distortion signal(d0 ) vs time*

The verticle scale is not given because the distortion analyzer, used to filter out the fundamental, has a non-calibrated ~utput. This photograph is given primarily for a qualitative insight and not for­measurement purposes.

Figure 9-b. waveform of the Distortion Signal, d 0

(Actual)

28

discrete increments of time~ the distortion signal will

not be a perfectly smooth line.

In Stlffifi1ary~ this section has analyzed the power

amplifier by use of equivalent transistor models. From

these models~ node voltage equations could be written

and solved by numerical techniques. The non··dependence on .

29

frequency in the audio range was assumed and experimentally

verified. Then the Fourier series of amplifier output was

derived for the general case of any input signal level.

Hith this info·.cmation~ the power amplifier was represented

by an ideal amplifier with a distortion signal being added

to the output.

The next step is to investigate the performance of the

operA-tional amplifier.

30

III. Adaptation And Description Of O~erational Amplifier

The next functional block of the audio amplifier is

the operational amplifier. In this section the operational

amplifier will be investigated as to how it functions in

this particular configuration. The operational amplifier's

gain as a function of frequency, frequency stability, and

biasing will be discussed in that order.

One of the characteristic features of an operational

amplifier is its very high open-loop gain. It is because

of this. high gain that feedback can be used and all the

benefits of feedback obtained.

The dev~ce used in this circuit is the MC 1433

Motorola integrated operational amplifier. Figure 10 gives

the schematic and equivalent circuit of the operational

amplifier as given in the Motorola specification sheet.

Note that this circuit is not a circuit of discrete

elements but is fabricated on a silicon chip by planar

methods of photo etching and dopant diffusing. Figure

10 is sho~m to give the reader a better idea of the

operational amplifier. No attempt will be made to analyze

this circuit, but rather the amplifier will be viewed

simply as a high-gain amplifier as depicted in the

equivalent circuit. Also in figure 10 are some of the

pertinent p&rameters describing the operational amplifier.

The function and use of the various leads will be explained

as we proceed.

Motorola MC 1433 Monolithic Operational Amplifier

CIRCUIT SCHEMATICS

r~~f!_~J~T-1~ ____________ ~ _v~]~Y~A~~~J~T-., , o o c I G---

INPUT A~ '

INVERTING j I

'------------- ----1--1----------- __ _.J

V- 6 0 C 6 OUTPUT LAG

CIRCUIT SCHEMATIC

(taken from Motorola specification sheet) typ

F v+

G GAIN ADJ.

OUTPUT LAG

D l K C

EQUIVAlENT CiRCUIT

max unit

E

z. l.U

input impedance output impedance open loop gain

min 300 600 - kilo-ohm

zout A vol 20K

100

50K

150 ohm

Figure 10. Operational Amplifier Schematic and Equivalent Circuit

w 1-'

A. Open-Loop Gain

The first specification of an operational amplifier

is its open-loop gain, AVOL" Normally, the larger this

numbe:e is t:he better the operational amplifier performs.

32

If the open-loop gain is large, more feedback can be applied

and distortion is reduced and the frequency response extended.

For the operational amplifier in this discussion, the

minimum value of 20,000 will be used for gain. The minimum

value is used because our calculations should then reflect

the worst case.

The operational amplifier does have an upper cutoff

point. This point is essentially determined by the

external circuitry used for frequency stability. This

circuitry will be given later. For now, let us say that

the upper cutoff frequency, f 0 , for the case of interest

is approximately 500 HZ. Figure 11 is the Bode plot of

the operational amplifier as given in the specification

sheet for the case of interest. The gain is given in

decibels.

:O'j _ ____;2o t.. o~·..::o::;:o:!:o:J..)_.;__ _____ .....,.v.> __ o

~Co

~ -q: '0 \fl

Q.. 0 .do () ......,

~ZoL ~ 0

I

Figure 11 I

•• Frequency Response ..•

I

I I 1 I

i I soo IICIIC

of Operational

Ft2CQUENCY

~0 ICIII /MJKI/l

Amplifier

33

From figure 11 it can be seen that the gain expression

can simply be expressed as:

Then

(18)

Therefore we describe the open-loop gain magnitude of the

operational amplifier by equation 18.

B. Frequency Compensation

Now consideration will be given to the frequency

stability of the operational amplifier. By frequency

stability it is meant that the operational amplifier does

not become unstable over all of the frequencies of interest.

In other words, the phase shift is restricted so that at

high frequencies negative feedback does not turn into

positive feedback and cause oscillations. Again, by use

of the specifications sheet, values for these compensa­

tion networks are given.*

* For a detailed analysis of these networks see "A High

Voltage Monolithic Operational Amplifier"; Wisseman, L.L.

Motorola Application Note - AN - 248.

It is beyond the scope of this paper to do an analysis of

these frequency compensation networks because it requires

considerable investigation of the operational runplifier

itself. Figure 12 shows the operational amplifier with all

of its supporting circuitry.

Network R1 and c1 j_s an intermediate stage frequency

conpensation network which couples two stages inside the

operational amplifier. Network ~' c2 , and c3 is also

a frequency stabilization network which serves as a path

for the output signal to be returned to the internal

circuit of the operational amplifier. These networks

determine the cutoff frequency of the operational amplifier

and keep it stable over the audio range. Generally speaking

£0 can be extended greater than 500 HZ but at the cost of

less stability. These networks perform well and essentially

eliminate high frequency instability.

C. Biasing

The operational amplifier also has biasing require­

ments. From figure 12 it can be seen that the operational

amplifier has two inputs, one inverting and one non­

inverting. It is desirable to have the DC bias currents

flowing into these two inputs as equal to one another as

possible. This is necessary to maintain the balance of the

differential amplifier stage in the operational amplifier.

Without this balance the output voltage has a non-zero

R ,...._-------~ I I

I 1 l L' __jl I

I __ j: I di I

I

+I I

~ ein

-~

. < I l J i K~Fc,~~ I ~--l j ' I

---"'~ I peration ' E,

~f1mp1~- J1----l I L,-' I

l J C~_! C4::,:: ~ R3 _if y I C2 'r

&~ cc ~ ¢ ~c3

~ Figure 12. External Circuitry of Operational Amplifier

R. = 11 kJl 1.

Rf = 100 k.st

cf = 39pf

R1 = lO..R.

R2 = 910.5l

R3 = 10 k.2.

c1 = 0.1 mfd c -· 10 pf 2 -c - 200 pf 3 -c4 :.-:: 0.1 mfd

v 15 .. CC= VOltS

w \Jl

value for a zero voltage input.

To accomplish this~ resistor R3 is included. It can

be shown that if R3 = Ri Rf Ri+Rf

the input offset current

36

will be minimized. * Input offset current is defined as the

difference between the bias currents in each of the inputs.

The capacitor c4 acts as an AC short which places the

non-inverting lead (A) at ground potential. The input

voltage then is applied between the inverting lead (B)

and ground.

D. Feedback Network

The final functional block for analysis is the

feedback loop. Let us define ~as the feedback ratio~ which

is the ratio of ho\•7 much of the output is fed back intb

the input . ~ ~ as will be shown~ essentially determines

the gain of the entire amplifier as long as the gain of

the operational amplifier is large. Th~refore this

quantity is quite important and is found in all of the

gain and distortion equations in Section IV.

To derive~~ two assumptions must be made. First we

shall assun1e that the input impedance of the operational

amplifier is very high~ so that we can neglect the input

current of the cperational amplifier. Refering to figure 10,

* Blair~ K. "Getting More Value Out Of An Integrated

operation<.:tl A:nplifier Data Sheet" ; Motorola Application

Note - AN - 273.

it Shows a typical value of z. = 600 KS1.which J"ustifies ~n

our assumption. For the second assumption refer to figure

13. Since the open-loop gain (A1 A2 ) is very large~ the

value of e will be essentially zero in comparison to e. g ~

or e . 0 Note that since we are primarily interested in

the signal only~ the distortion from the power amplifier

was temporaily neglected.

Using these two assumptions ? can be found. Apply

Kirchhoff's current law at node eg.

Since

then

!in = If from asswnption 1

--·---

eo - e 8A1A2

e eg - 0

AlA2

~

e - e g 0

0 by assumption 2

Therefore by neglecting e : g 1

(19)

To relate equation 19 to standard feedback theory

refer to figure 14 which shows in block diagram form the

basic feedback amplifier. If A'?>>7 1 the gain expression is

1 simply - e . Now equate equations (19) to equation (20).

1 - T-

or for magnitude Ri

1ar = t" Rf {21)

37

38

,~~ f,_f ___,..., ..1-f_. I

.---------~--.~~~--------~

Rf

.!.i~ R. ~ o-----l\,'\lv-·-..... --~

J +l e. 1.

-I e -1

e

~~ ·-- -------·-----0

in

Assume:

1) z. - 00 1.0

so that rb--o

2) A1A2 --0

so that eo --·-- e ~ 0

AlA2 g

Figure 13. Feedback Network

39

+ + ;I 0-

t --o

t __ j f t e. e. +~eo Ae. +APe e

I ~ ~ 0 10 I I 0-·-- 0

A 1 -e~--- = -r...;....--A~ ~ _ ~ (20)

asst:une: A~>"> 1

Figure 14. Basic Feedback Amplifier

Equation 21 is then, the expression for I~ I. The

reason why this particular feedback network was used will

become clear in the next section.

40

IV. Computation. Of Amplifier Characteri sti.cs

Now that the three basic pa~ts of ~he audio amplifier

are mathematically described, a qualitative investigation

can begin.

!~1

First the audio amplifier will be assembled using the

equivalent block diagrams of the power amplifier,

operational amplifier and feedback network. Then the form

of the output signal and output distortion will be derived.

With these equations the theoretical response of the

amplifier will be fully described.

A. Block Diagram Of The Audio Amplifier

Figure 15-a is the total audio amplifier schematic

with the power amplifier fully drawn, all the operational

amplifier circuitry and the feedback network included.

Figure 15-b is the same audio amplifier using (1) the

power amplifier equivalent block diagram, (2) the oper­

ational amplifier with the gain expression of equation 18

and (3) the block diagram of the feedback network,~ . Let

us now derive the expressions for the output signal and

output diEtortion.

B. Output Signal And Output Distortion

Refering to figure 15-b we can write the output

·~~ l CB

I ~

J+Vcc

+ _j

... , R k ~·Q2 B 4 ·----.---1\/\l\t ~ Q

r., J.._ l6. VBT

~

Figure 15-a. Schem&tic and Block Diagram of Audio &~plifier (Schematic)

\ \

R~ ~ e

-!=' 1\)

feedback network r·-:;~

1 e . I

i _j

~· 1 I oper~~~ona i powetf. ~pl1t1er · ampl1 1er

~ : \A~ i ; I A2 I +

e.+He +d ) 1 0 c el

e . = E . sin wt 1 1

e = E sin wt 0 0

Figure 15-b. Schematic and Block Diagram of Audio Amplifier (Block Diagram)

4=' w

expressions. First the assumption will be made that since

there is distortion~ d0 ~ inside the loop, there will be

distortion~ de~ at the output. d 0 repiesents open-loop

distortion and de represents closed-loop distortion. Also

for the analysis e 1 and e 0 are assumed to be sinusoids and

the distortion signals are described by a Fourier series.

The two signals at the output can now be equated. The

signal component will be separated from the distortion

component.

A1A2 ei + (3A 1A2 e 0 + ~A1A2dc + d 0 = e 0 + de (22)

where:

signal distortion

A1A2 ei = e 0 (1 - (3A1A2 )

eo -e.

l.

Al =

AlA2 l - ~A1A2

(23) d c

AVOL / (1 + j (w/ w 0 )) (18)

A2 - gain of power amplifier determined from

computer analysis (A2 is not a function of

frequency)

Now substitute equation 18 into 23 and 24.

(25)

do - -,_-- ~ A:....2_,..A-V-OL_/...,...,...,( lr---::+---.j -r( \X)-::-:::'7-r-w'o)")- (26)

(24)

44

From 25 it can be seen that at low frequencies if

A2AvoL >> 1 the gain expression reduces to -1/(3, which "Jas

demonstrated before (equation 20). Also one of the bene­

fits of feedback can be seen in that the upper cutoff

frequency of the amplifier with feedback has been extend-

45

ed. The cutoff frequency of the close loop amplifier , uuCL,

now equals: (1-A2~AvoL)w0 . Therefore the frequency

response of the audio amplifier is greatly increased by

feedback.

The other advantage of feedback can be recognized in

equation 26 where the distortion in the output is equal to

the distortion in the loop, d , reduced by a factor of 0

~A2AVOL" Thus, since the factor is indeed large ( 2Xl03 ),

the distortion in the output is greatly reduced. This is

very desirable in audio amplification.

Again it should be stressed that ei and e 0 represent

sinusoid signals and d 0 and de represent non-sinusoidal

distortion signals described by a Fourier series.

From equation 14:

00

do =[ Dok sin kl.llt

k=l

d can be similiarly be expressed as: c

00

de =L Dck sin kwt (27)

k=l

where: k = 1,2,3 ...

Since the frequency of th~ fundamental component

(k=l) of the distortion signal equals the frequency of ei

or e 0 , the Fourier series representing the output can be

expressed as:

00

e + d =[Eok sin k~ (28) 0 c k=l

where: eo - Eo sin oJt

00

and: "d c - L D -k=l ck

sin kUo)t

so that: Eol = E + 0cl 0

Eok = 0 ck 0 ,

k = 2,3,4. 0 0

Expression 28 is useful in determining the harmonic

content of the final output signal.

C. Adaptation Of Po\ver Amplifier Analysis

In order to find the distortion signal, d 0 , informa­

tipn from the power amplifier analysis must be obtained

first. Specifically this information is in the form of

A2 (power amplifier gain) and ~T (ratio of delay time to

waveform period). When A2 and 1/T are computed a specific

input voltage e 1 must be given. (e1 = input voltage to

power amplifier). Therefore in order to use the analysis

of power amplifier and the analysis of the entire audio

amplifier together there must be a relation between e 1 in

figure 8 and e 1 in figure 15-b.

To find this relationship it will be ne~essary to refer

to both figure 8 and figure 15-b. For the first step

the two output voltages will be equated.

(29).

47

e2 is not~sinusoid, but is represented by a Fourier series.

From equation 28 the Fourier series of the right side of

the equation is known. Since the purpose of this deriva­

tion is to determine the relationship of e 1 , which is a

sinusoid, only the fundamental of the output will be

investigated. All of the harmonics (2nd, 3rd, 4th, 5th,

etc.) are grealty reduced due to the feedback, but the

fundamental is passed through the amplifier without reduction,

but actually gain. Therefore we shall rewrite equation 29

considering·only the fundamental harmonic.

e2 ,lst harmonic

where: E - peak to peak value of 1st harmoni~ of e2 • 21

Now the assumption must be made that Del can be

neglected in comparison to E 0 . We know that:

or

1 - (3A1A2

A2E1 (B 1 - 1)

1 - ?A1A2

from equation 24

from equation 15

where B1 is the first harmonic coefficient for normalized

crossover distortion.

Since (B 1-1) is very near to zero and this is further

reduced by the factor of the open-loop gain, the assump-

tion is valid.

Now_. rewriting equation 30 we have

E2 sin wt = E sinu.~t (31) 1 0

inserting equation 11 for E2 we have 1

Er~2B1 = Eo

or rearranging

E El =~

A2Bl (32)

48

Therefore it has been demonstrated that the fundamental

of the power amplifier output is essentially equal to e 0

and that the input and output of the power amplifier are

related by expression (32).

Normally the characteristic of the audio amplifier

(frequency response, per cent distortion in the output

signal, etc.) are eval~ated at some partic~lar power

output. The assumption will be made that the distortion

components in the output are of such low level that their

contribution to the total power delivery to the load is

negligible. The

p LOADRMS

power, E2

0

=2~

then, can be simply found by

(33)

where E is peak value of the output sinusoid (e = E sinwt). 0 0 0

Or we can use the fundamental of the output of the power

amplifier as described by equation (31).

p LOADRMS - (34)

To find E2 we can use the Fourier series of cross-1

over distortion. If the average peak value and ratio of

"l'T is kno~·m, B1 (magnitude of the fundamental) can be

found.

Mathematically:

where:

Eeak to peak value of e 2- ~

B1 = normali~ed magnitude of fundamental for e2

( r IT must be known)

Substituting 35 into 34:

(36)

Tl!e quantity (BJ. I 2 RL)_, as a function of "l'T has been

calculated and listed in table 2. (See equation 8 for

equation of Bk) Using this table and the power amplifier

analysis program_, the fundamental power content can be

49

found. Given an input voltage level to the power amplifier,

el, the output e2 will have an average peak value E2 and

a '1/T. Squaring E2 and using "r"IT in table 2, the power

the fundamental can be found by using equation (36). To

fing a particular power level this process is repeated

until the level is found to the desired accuracy.

in

The output power levels which will be used are 15, 12.5,

10, 7.5_, 5, 2.5, and 1 watt. By using the process described

above, table 3 was generated by using the power amplifier

gain of the power amplifier, A2 .

At this point the necessary information about the power

50

r B1 B2 1 -'f

2lr L

0.0 1.0000 0.1250· -

.01 0.9897 0.1224 ·--~- --------

.015 ~~9844 o. 1211 --

.020 0.9789 0.1198 f--- --1--·

.025 0.9733 0.1184 ----------·-

.030 0.9676 0.1170 ----------!----------- --

.035 Oo9618 0.1156 --- -----· --

.040 0.9558 0.1142 --·- ----- ---- ------- -- ·----------- -----

.045 0.9497 0.1127 ------ ------·--·- -

.050 0.9435 0.1113

Table 2. Fundamental Power Content of Power Amplifier Output

I I i I Fundamental ' Nominal I r

E E2Av E2AV2

1 ..:!_ I

1 Peak I B, I A2 = E2 Power Into Power I I T 8-"- l -y-· 4 Ohms I

I I 1 I

(Watts) I (Volts) (Volts) (Volts )I 1 (Watts) I

3.650 !2.964 8. 787. ·?_~_? ______ !14~ .8122--j-~-0035 1.0 -i --

5-321 14.620 21.35 .030 .1170 .8682 . 2.497 2.5 - ··-·· --- ·-f------- ---------- -----~ .. ----

7.170 6.458 141.704 l· o2o . 119s I . goo6 4. 996 5.0 ---

' -+ --8.588 7.869 61.92 '· 015 .1211 . 9163 7. 498

I 7·5 I ~ -- ---~ ·-·- ~·- .. ---- -------~--· ·--

g.81 9.086 82.55 .015 .1211 .9262 9.997 10.0 - ------- ------·

10.887 10.16 103.2 .015 .12~-9331 12.497 12.5 -

15=d 11.80 11.07 1122.5 .010 .1224 .9380 14.996 I -------- -~---~---L

Table 3. Results Of Power Amplifier Analysis Program

LT'I ......

amplifier has been obtained in tabular form in table 3.

And since the relation between e2 and e 0 is known, we can

proceed to find the characteristics of the entire audio

amplifier.

D. Audio Amplifier Analysis Program

From equations (25) and (26) the audio amplifier

gain and distortion are known. To find d , d must be c 0

kno"t>m. By using table 3 and equation 17, d can be 0

calculated. To perform these calculations a program

was written. Again, the audio amplifier will be examined

at the constant power levels of 15, 12.5, 10, 7.5, 5, 2.5,

and 1 watt.

So that these power levels are maintained, the output

voltage, e , is a fixed value in the program, adjusted so 0

that it yields the desired power output. From the gain

expression equation (25), the input voltage, ei, can be

found. Also by using equation (26) the output distortion

can be calculated. These results will take the form of a

52

Fourier series describing the output, e 0 + de. This series

is used to calculate the per cent harmonic distortion in

the output. To do this, all of the harmonic amplitudes

must be normalized with respect to the fundamental. Then

for per cent distortion for a particular harmonic multiply

by 100. This process is shown below.

Al = amplitude of fundamental of output signal

A2 - amplitude of 2nd harmonic

A3 - amplitude of 3rd harmonic . . . A so= amplitude of 50th harmonic

A2 Al ~ 100 = D2 = per cent 2nd harmonic distortion

.!3. Al x 100 = D3 = pe~ cent 3rd harmonic distortion

Aso • Al · x 100 = n50 = per cent 50th harmonic distortion

To find the per cent total harmonic distortion, the square

root of the sum of the squares of the individual per

centa.ges is found as follows:

per cent total harmonic distortion = ...( D2 + D3 ... n5~ Note that the Fourier series was truncated at 50 terms

because the magnitudes beyond the 50th harmonic are

negligible.

53

Since the gain and distortion expressions are functions

of frequency through A1 and~(see equations (18) and{21))

the characteristics of the audio amplifier must be calcul-

ated at various frequencies. Therefore, there are two

independent variables, power output and frequency; and two

dependent variables, per cent total harmonic distortion and

voltage gain. How these quantities will be plotted is

shown below.

voltage gain vs frequency at specified power

per cent total harmonic distortion vs frequency

at specified power

per cent total harmonic distortion vs power

at specified frequency

The audio amplifier analysis program is written so

that for a particular output voltage, (thereby fixing the

output power level) the voltage gain and per cent harmonic

distortion is calculated for a range of frequencies over

the entir~ audio spectrum (10 HK to 50 KHZ). Table 4

gives the results from this program. The program itself

and flow chart is described more fully in Appendix v.

With the tabulation of the theoretical results in

table 4, plots describing the performance of the audio

amplifier can be drawn. This is done in figures 16, 17,

and 18.

In summary, this section had described the entire audio

amplifier in a usable block diagram form. Then using

54

standard feedback theory, the expressions for output gain

and distortion were derived. To find the distortion signal,

d , the operation of the power amplifier had to be described 0

both outside and inside the audio amplifier. This meant,

specifically, relating e2 to (e 0 +de)· This essentially

integrated the two programs into one. Tables 3 and 4 give

the results at specific output power levels. The three

figures 16, 17, and 18, which sum up these results in

graphical form are the theoretical response of the audio

amplifier.

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Table 4. Results of Audio Amplifier Analysis Program

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Figure 18. Theoretical Plot of Per Cent Total Harmonic

Distortion versus Output Power

58

V. Experimental Verification

A. Objectives

In order to evaluate the validity of the theoretical

description above, experimental data must be taken and

compared to the theoretical results. The form of this

comparison will be essentially the same as the plots of

figures 16, 17, and 18.

59

Hopefully, there will be a correspondence between the

theoretical and experimental data. However, due to equip­

ment limitations and simplifing assumptions throughout the

analysis,. we can expect variation between the experimental

and calculated data.

B. Equipment

Figure 19 gives a block diagram of the test setup to

measure the response of the audio amplifier.

The measurements can be divided into frequency response

(gain vs frequency) and per cent total harmonic distortion

verses power or frequency. For the frequency response

measurements no equipment limitations exist. The voltage

gain is the ratio of the input and output voltages read on

the VTVM's. The desired output power can be obtained by

maintaining the output voltage at the desired level.

A problem does exist in the measurement of distortion.

audio oscillator

Instrument

power supply

band Qass filter

digital frequen meter

\}

audio oscillator band pass filter digital frequency meter VTVM oscilloscope distortion meter

I \J "----.-.r--

----" c;- Jt L-.---;-t- '1'-''-·----l i

~1 \7\

_I. ' '

1 '! distortion) oscilloscop, 1/TVlo! I , meter I I I 1~-~

~

Type

Electro model EFB Hewlett-Packard model 200 CD wide range oscillator Krohn-Hite model 3103 Hewlett-Packard model 5245 L electronic counter Hewlett-Packard model 400 D vacuum tube volt meter Tektronics model 516 oscilloscope General Radio model 1932-A distortion and noise meter range: 50HZ to 19KHZ accuracy: ± 5 per cent of full scale of each range, + a maximum residual distortion of 0.05 per cent below 7500HZ, and 0.10 per cent above 7500HZ.

Figure 19. Pictorial Diagram of Test Setup and Equipment

0')

0

61

Full-scale deflection on the General Radio distortion meter

at its lo\'Jest scale is 0. 3 per cent. When the audio oscil­

lator output is directly fed into the distortion analyzer

a reading of approximately 0.15 per cent is measured.

Therefore, the audio oscillator itself is introducing

distortion. From our theoretical results the distortion

in the output of the audio amplifier (0.01 per cent) is

far below the level introduced by the oscillator. To

measure this level of distortion is practically impossible.

Since the generally accepted maximum level of distortion

(2.0 per cent)* is well above the values discussed here,

measurement and calculations of distortion at such low

levels is purely academic. However, to reduce the distor-

tion from the audio oscillator, an adjustable band pass

filter is used as shown in figure 19. With this filter the

per cent distortion from the oscillator with the filter is

approximately 0.06 per cent. This is as close to a pure

sinusoid as can be expected. Therefore we can expect the

distortion level measurement to be never less than approxi-

mately 0.06 per cent because this is the level introduced

by the oscillator.

C. Results

Using the setup of figure 19 and following the data

outline in table 4 (theoretical results), the experimental

*Tremaine, Howard M., The Audio Encyclopedia, pg. 346.

results were obtained. The results are tabulated in

table 5.

And again in the same fashion of figures 16,17, and

18, the experimental results of table 5 are plotted in

figures 20, 21, and 22.

From these six plots (figures 16, 17, 18, 20, 21, and

22) a comparison between the theoretical and experimental

data can be made.

62

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* db = 20 Log10 (Av)

Table 5. Results of Experimental Analysis Of Audio Amplifier

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"I I I I J I I I I 'II , • I I ' I I I \ I I• I I ; I : I I I ' I I J,., •1. I 1· 1 '·· .1. 1 ·1···1· 1. j •• 1 ... 1. 1 , _,_LLJ, .... ILL .. j~~.l 1 ... 1 ... 1 ,.,.,. 1 .. 1 1 1·• I 1 1 .·0 •'I 'l I 1 ; I ' I I I! ,·' l I I ! I I i :I' I j I I I I i j I ' 1 1 ,I' '

50 100 1KHZ 10KHZ Frequency

Figure 21. Experimental Plot of Per Cent Total Harmonic

Distortion versus Frequency 0'> vi

.s s:: 0

"N • 7 .u ~ 0 .u Cll.6

"N e::l

c:J ·a. 5 0

~ Cll.4 p::

rl Cll .u.3 0

E-4

.o 1

Figure 22.

5 Output Power

10 15

Experimental Plot of Per Cent Total Harmonic Dlstortion versus Cuput Po"ii'1er

66

67

VI. Discussion And Conclusions

A comparison will now be made between the theoretical

and experimental data. This data has been plotted as

described in Sections IV and v. Ideally~ the two sets of

plots should exactly agree but in the actual case there are

discrepencies. An attempt will be made to explain the

sources of these errors.

First let us examine the frequency response plots.

The theoretical data (figure 16) yields a low-frequency

gain of 19.16 db and the upper cutoff frequency of

approximately 40 KHZ for all of the output power levels.

For the experimental data (figure 20) the low frequency

and 3 db point is 19.17 db and approximately 40KHZ for

1 watt output power; 19.50 db and approximately 35KHZ for

15 watts. For all cases~ experimental and theoretical~ the

gain is down approximately 1 db at 20 KHZ. This demonstra­

tes that the amplifier is well within the bandwidth require­

ments for high quality audio amplifier (gain deviation +

3 db from 20 HZ to 20 KHZ). The one watt level agrees

very closely. with the theoretical results. At 15 watts the

measured gain is approximately + 0.3 db from the theoretical

value and the 3 db frequency is 35 KHZ rather than 40 KHZ.

These small differences can be ignored and the theoretical

results accepted as a close description of the actual case.

Now let us examine the plots of per cent total harmonic

distortion versus frequency. From the theoretical data~

figure 17, the per cent distortion is reduced. This is

quite a unique characteristic. Usually the distortion is

greatest at maximum power output. This characteristic can

be accounted for by examining the response of the power

amplifier which is the source of the distortion present.

The power amplifier analysis is summarized in table 3.

Notice that the ratio ?/'T is greatest for small input

voltage swings. Physically this means that if the input

voltage is small, a considerable section of the output

signal will be in the crossover point, near zero volts,

until the driver transistor turns on. Since the turn-on

voltage of the driver transistors in the equivalent model

circuit is 0.5 volts, the output will not move from zero

until the input signal is greater than 0.5 volts. The

larger the ratio 1JT, the more distortion is contained in

the output. Therefore, as the input voltage increases,

17T decreases, thus reducing the distortion in the output.

Another feature of figure 17 is the general increase

of the per cent distortion as frequency increases. This

is very easily explained by considering equation 24 which

is the expression for the output distortion, de. Both

A1 and ~are functions of frequency described by equations

18 and 21 respectively. If we consider d 0 fixed for a

fixed output power, by substitution of equations 18 and 21

into equation 24, it can be shown that the output distor­

tion d increases as frequency increases. Or more simply, , c

the product ~Al decreases as frequency increases.

68

Figure 21 is the experimental measurement of per

cent distortion versus frequency and should be the same as

figure 17. However, upon investigation there is consider­

able deviation between the two (note the vertical scale

change). To compare the two plots let us first discuss the

low frequency portion of figure 21. From 50 HZ to 1 KHZ

the distortion levels are relatively constant. Note that

the curve starts at 50 HZ because that is the low frequency

limit on the distortion analyzer. Also, it can be seen

that just as in the theoretical plot, the 1 watt output

power has the highest distortion and the 15 watt has the

lowest distortion . This then agrees with theory; the

higher the power, the lower the distortion. However, the

1nagnitude of the per cent distortion is a full order of

magnitude less in the experimental data than was predicted.

This at first, seems to indicate that theory is in error,

but there is an explaination. Recalling from Section V,

that the audio oscillator with the bandpass filter has a

distortion level of approximately 0.06 per cent, we see

that the input test sinusoid has a distortion level of 0.06

per cent. Also from the manufacturer's specifications of

the distortion analyzer the full scale accuracy is ± 5

per cent + the residual distortion of the analyzer itself .

This residual distortion is 0.05 per cent below 7500HZ

and 0.10 per cent above 7500HZ. (See figure 19 for the

full distortion analyzer specifications). Therefore, even

without the amplifier considered, there are two sources

of distortion; the audio oscillator and the distortion

analyzer.

With these considerations the level of distortion

measured seems to be consistant with the predicted values.

There simply are no instruments available that can measure

distortion levels on the order of .005 per cent.

70

Next the section of the plot from 1 KHZ to 19 KHZ will

be examined. It is in this section that the most serious

deviation from the expected values takes place. Also

note that figure 21 stops at 19 KHZ, this is because the

upper frequency limit on the distortion analyzer is 19KHZ~

The theoretical data, figure 17, however evaluates the per

cent distortion up to 50 KHZ.

First let us compare the distortion curves for one

watt. At 19 KHZ, theory predicts 0.316 per cent and the

measured value is 0.42 per cent. Again, considering the

extrane01.:..s distortion introduced (oscillator and distor­

tion analyzer) as discussed above, this seems to be a

tolerable deviation.

From theory, if the power level increases we can

expect a drop in distortion. However, just the reverse

happens in the actual experimental data. For 5, 10, and

15 watts output power, the measured distortion level is

0. 1!3 per cent, o. 56 per cent, and 0. 73 per cent at 19 KHZ

respectively. These deviations cannot be explained by the

above considerations.

To justify these results we must examine more closely

71

the simplifying assumptions made previously. First, in the

audio amplifier analysis in Section IV, the method of

analysis assumed that if the input was a pure sinusoid, the

output would be composed of a fundamental signal and many

harmonics. These harmonics were considered to be the

distortion in the output signal. The magnitudes of these

harmonics are small compared to the magnitude of the funda­

mental. Therefore, the assumption was made that the harmonics

themselves would not generate further distortion, since the

output is fed back to the input through fJ. t"1hi.le this is

generally a good assumption, there actually is further

distortion generated by the original distortion signals at

the output. second, the operational amplifier was assumed

to be distortion free and actually is not. To analyze

this source of distortion is beyond the scope of this paper

and will just be mentioned. Third, the power amplifier was

analyzed on the basis of transister models which were a

good approximation of the transistors used. However as can

be s~en from figure 4, the actual transistor character­

istics can change considerably as the point of operation

changes. Thus, the value of hFE' for instance, might be 100

near cutoff and 50 near saturation. Although median values

were choosen, it can be seen that the theoretical power

amplifier response will deviate from the actual case. And,

to analyze the waveform of crossover distortion, the Fourier

series of an approximate waveform was found. Figures 6 and

7 show the actual waveform photographs and approximated

72

signal. As can be seen, the approximation is close except

at the higher frequencies. This then, would seem to be

another cause of the resulting deviation between the

measured and predicted values of distortion.

While all of the above sources of error contribute to

discrepancy found between figures 17 and 21, they do not

explain why there is such a large difference (e.g., at 15

'vatts, 19KHZ: theoretical - .069 per cent; experimental

0.73 per cent).

The author has concluded that high frequency oscilla­

tions are the primary cause of the deviation from the

predicted values. Since the operational amplifier is a very

high gain device it is very susceptiable to these high

frequency oscillations. These oscillations are caused from

excessive lead inductance, excessive power supply impedance,

and grounding loops. To reduce these effects, the circuit

layout used leads as short as possible and two 1.0 mfd

bypass capacitors were placed from the supply leads to

ground. For best results the circuit should be fabricated

on a printed circuit board to get lead lengths to a

minimum. The experimental tests used the amplifier con­

structed on standard vector board with the two power

transistors on a heat sink. Also during the distortion

measurements, diagramed in figure 19, the distortion

analyzer, VTVM, and oscilloscope were connected across the

load. This added approximately 100 pf across the load mak­

ing it still more susceptible to oscillation. During the

73

experimental measurements these high frequency oscillations

could be observed on the oscilloscope as a broadening of

the trace near the v;avefonn peaks. This effect was more

noticeable at higher power levels which was demonstrated

in the distortion measurements. To avoid pickup of noise

signals and 60HZ hum, shielded cable was used for the

input and output leads.

The frequency compensation networks, Cf and CB all

contribute to reduce these oscillations. Their values had

to be choosen on two criteria; frequency response and

per cent distortion. The distortion could be reduced, but

only at the cost of poorer frequency response. The values

cl1oosen v.1ere a compromise between reasonable distortion /"'') ; / r:·· v ; <.- (_,

levels and good frequency response. As given before, as ~

long as the distortion is less than 2.0 per cent, the ~

amplifier is considered to be a high quality audio amplifier.

Returning to figure 21, the effect of a rise in distor­

tion for a rise in power at high frequency, can be explained

on basis of these oscillations. At low power levels, the

voltages and currents are also at low levels and the

oscillation is at a minimum. However the reverse happens

as output power is increased. These effects therefore, are

not shown in t:he theoretical results simply because the

model system does not include thses operational difficulties

(power supply impedance, lead inductance, etc.).

Considering all of the factors above, the theoretical

model describes the distortion levels for audio amplifier

accurately only for low frequencies. It does however,

give a general approximation at higher frequencies when

the limitations discussed above are considered.

Figures 18 and 22 contain no new information but

simply hold frequency constant and vary the output power

level. The same considerations for figures 17 and 21

apply here.

Considering this last section, a general evaluation of

this analysis can be made. First it has been demonstrated

that a high quality audio amplifier can be constructed

using fewer components than previously possible before

thG advent of the monolithic operational amplifier.

74

Second the block diagram analysis is a straight forward

approach utilizing common Fourier series analysis and

transistor modeling. The equations describing the amplifier

are very similiar to those found in standard feedback theory.

Third, the use of the operational amplifier as a '~lock

gain'' is shown in this analysis. This type of design is

very favorable because the response can be determined

quickly by just analyzing the feedback network. Fourth,

the theoretical results of the analysis are reasonably

accurate for low frequency and a good approximation for

high frequency when the high frequency oscillations are

considered.

To make the analysis more accurate would entail consider-

ably more detail which would lose the present simplicity.

Therefore the audio amplifier and analysis described, can

75

be considered to be very useful.

BIBLIOGRAPHY

1. Ehrsam, B. (1967) Audio Power Generation Using IC Operational Amplifiers. 'Hotorola Applica­tion Note AN-275, pg. 1-8.

2. Blair, K. (1967) Getting More Value Out Of An Intergrated Operational Amplifier Data Sheet. Motorola Application Sheet AN-273, pg. 1-12.

3-

4.

5-

6.

Wisseman, L.L.and J.J. Robertson (1966) High Perfor­. mance Operational Amplifiers. Motorola

Application Note AN-204, pg. 1-16.

Wisseman, L.L. (1967) A High Voltage Monolithic Operational Amplifier. Hotorola Application Note AN-248, pg.l-12.

Stern, L. (1967) Analyzing Linear IC's, Part I. Motorola Monitor, Vol. 5, No. 2. pg. 12-19.

Hartin, T. L. (1963) Electronic Circuits. Prentice -Hall, Inc., pg. 247-283; 392-432.

76

7- Joyce, M.V. and K.K. Clarke (1961) Transistor Circuit Analysis. Addison- Wesley Publishing Co, Inc., pg. 28-65.

8.

g.

10.

11.

12.

13.

Chausi, M.S. (1965) Principles And Design Of Linear Active Circuits. McGraw-Hill Book Company, pg. 357-415.

Chirlian, P.M. (1965) Analysis And Design Of Electron­ic Circuits. McGraw-Hill Book Company, pg. 84-147: 401-443.

Lindmayer, J. and C.Y. Wrigley (1965) Fundamentals Of Semiconductor Devices. D. Van Nostrand Company, Inc., pg.29.

Javid, M. and E. Brenner (1963) Analysis, Trans­mission, And Filtering Of Signals. McGraw-Hill Book Company, pg. 40-75·

Johnson, c. L. (1956) Analog Computer Techniques. McGraw-Hill Book Company, pg. 7-12.

Conte, s. D. (1965) Elementary Numerical Analysis. MCGraw-Hill Book Company, pg. 19-48.

14.

15.

16.

17.

McCracken, D. D. and W. s. Darn (1964) Numerical Methods And Fortran Programming. John Wiley and Sons, Inc. pg. 263-283.

77

Scott, H. H. (1949) The Measurement Of Audio Distor­tion. N.A.B. Engineering Handbook. pg. (5-4-01) - (5-4-06).

Tremaine, H. M. (1959) The Audio Encyclopedia. Bobbs­Merril Company, Inc., pg. 346.

Standard Mathematical Tables (1964) The Chemical Rubber Company, pg. 328-329.

APPENDIX I

LEAST SQUARE CL~VE FIT OF THE FORM y = y 0 (exp(kx) - 1)

Gi··.;en a known constant, y 0 , and a set of data (y 1 , x 1 ;

y2,x2 ;y3 .x3 . · .x.. ·Y· · · .y ,x ) • let us derive the expression - 1· 1 n n ·

for k using the least square method. First,the assumption

will be made that for the values of interest,the above

equation can be approximated by y = y 0 (exp(kx)). Using

this equation greatly simplifies the analysis.

The next step is to take the natural logarithm of both

sides of the simplified equation thusly.

ln(y) = ln(y ) + kx 0

Now the square of the difference factor, S, is found.

S == t: ( 1 n ( y. ) - ln ( y ) - kx. ) 2

i=l 1 0 1

To find the minimum of S,the derivative with respect to k

is found. n

2 [ (ln(y.) - ln(y ) - kx.) x. = 0 0 1 1.

i=l 1

Solving for k, the result is: n Jl ... ~(ln(yj_)) (x. )= ln(y 0 ) .L 1.=1 . 1. 1=1

n

x. 1

11

+ k L (x. )2 . 1 1. 1.=

J_"; (ln(yi))(xi)- ln(y0 ) ~ i=l k = _1= 1 ----·-- -----

This, then, is the equation

f; i=l

for k.

(x. )2 1.

The program to calculate k is now listed.

conversion into Fortran is given in table form.

x. 1.

The variable

Notice that

the value of vb (x.) is converted to Vb~ by subtracting V--t· e 1.

79

Also, once the value of k is determined it is inserted into

the exact equation originally being fitted and y calculated

for a range of x so that the resulting curve can be compared

to the actual data.

Variable Conversion Table

least square derivation

Section II variables

1 ? .. 4 r;::

1 -,

l J 1 ? J ') , /"

l r; 16 1 7 t p. 1 9 ;:>(

vbe Ib Io K

v?f ::::::.1 ?· ·:~,-:-- ,Trr·c=~-? r>'\r~r:S=l50

'! I ·-• '- 1\ <:: J ,~: ' v f 2 ,- ) 9 X ( ? .') } f) ,, 1 r r == 1 , 1+ I(V=:TT

C' ;-:- ". :) c 1 , 1 r_ l qr-~.-, (l,'~·l :; ' /' r• ( l • t' , -, I 1)'1 -y ''-=, 1 ,,

~-· <; r ' V (~ "· ;.• r... { Y f I ) , Y C T I , r == 1 , "-i l

7 y ( .. l = y ( .• ) -\1 r, I' '/fl. - I :;> J T F c ) ' -:_ ' ' l 'J(~TC (',/1.;) . ! r: T T 1 ( , , ':) -~ } { V ( J l , X { T ) 9 T = 1 , "J ) ~lt;'t_YY:=f'· ')IJ ·-~X == ,-SttqYSt!=':· ,")("' ? J -:: 1 , ,, Y}=~l"r{v(.J))~Y(J)

7 SIJ'.'I_YY = Sl~'-'LYY+Xl [)"1 <. V-= 1' N '<?= Y(k'}

~ 'I I '··' y ~ s! I •• X + X ? nn ·4 , = 1. "'' X ' = X { I_ ) :::: Y ( L )

Fortran variables

X(I)

Y(I) sc B

VGAMA

"1 27 ?J 2 1f .,~

4 S I I ~ ' X ~ r"' == <:: t __ 1 ' ' Y <:: 0 f X '"l, ~-=- ( '1.'"1 vv.. - /.l nt,( sc }'T '3tJ'·1Y) I ( st_r~xs'.))

?{-_, ?7 ?P ?'! '3C -:1 1 ~? ?, "'l,

? '+ ~'1 '"l.f, 37 11'1 .,_q l~C·

1,!D 1 T!= ( -), ')r) !(!I

·> !') T T i:: ( ) • t. -~ , s (. ' J"l ' v G !l. \A~ :._, =' r r r c ')_, ? ') ,

l r_

'. .... :;~_ ... <.: . t."J ( _. c_

(~ c_ .. ::.,:: ("~·

(._·,~·c.' t:_.. t.:._~ l.,; c.:. C.·._::.,

If L.

L' lL

L.. U

. Ll .... L

lJ.. Ll

LL

1-· C

(. · ,~ .. ·c

C·C

c· C

Ut.":'

-t.

• ... · G

C c:.· ·.')C

.-(. C

C

c:-,:.:(~ ': c r_

,f. ·(-'•( .. ·C. c

>c. ·-

c cr .... ) ~-(.~>, ~~ c:. c·

r_

' :.~) c 0

' \.. ... (_~ (.:~ ~ .... ·

('~·a

li .. , __ .. '· (;

(.j'• "-' ~

!.{' ·-

::· ..:t· './-

C. r--

a:: ,_. 1"\... r<.

~' ,•

;r. lr

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" I"\. C

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" ' f"·, "­

C':'" •

• •

• •

• ••••

t .. · t_;: t..:~;: '.....) C.'· ~

1 L

. t _; L

~ C...'

r..r ·. -..t r< rr· ro-.~· rr. f".

~-. rt"'• .,__ c

:·oco

c>

c r.;r:::c

o

2 I

I I

I I

I I

I I

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: U

. I.L' U

. L!..' U. · U

.. L• . .' U_; u.,;

lY.. "--' l_

. ~--·1 <;.) t.: ~::::.. .... ~ ... , ·-

("_~

CY C

t..., C

•, • C

C

(.) ,...., ~:. C

=~ c· ·-:. . .:.. • ..

c c c

('•

t c

L

C

L ,_

, V

0 (

· 0 L..

L· C'

c· · ~--;;,_. c

· o ._ c.' , ::· c. o

L~ · o • .-....... c

,_. c c

: · o· V

; C

t .

...: .. t.r r:;-L.."

C, t-0 C

-~

0:::. c IJ

""..-<-.-C

'-.r'\..f'' ~-...;t 0:::

• •

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t: .• c·coc:.~c·c.·'· .c)c~

u.. c >

* 0" ""6

..... ~

cc ~

,....~

•

n X

u -.c I U

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81

APPENDIX II

NEv7TON-RAPHSON HETHOD FOR A SYSTEM OF EQUATIONS

In order to solve the five simultaneous non-linear

equations (Section II, equations 1,2,3,4,and 5) the Newton­

Raphson iterative technique was used.

The derivation for the approximation algorithm will now

be derived. Let the five equations have the following no­

tation.

A(Vl,v2,v3,v4,v5) = Iol (~El + l)(exp(K1 vbe1)- 1)

-Io2 (hFE2 + 1)(exp(K2 Vbe2)- 1)

+Io3 (~E3 + 1)(exp(K3 Vbe3)- 1)

-Io4 (~E4 + l)(exp(K4 Vbe4)- 1)

+(V2 - Vl) hoe1 - (V1 - V3) hoe2

+(Vee- Vl) hoe3 - (Vl +Vee) hoe4

-vl/~

B(Vl,V2,V3,VJ~,V5) = RB ~E1 Io1 (exp(Kl Vbe1)- 1)

+~ (V2 - V1) hoe1 -V4 + V2

c(vl,v2,v3,v4,v5) = RB hpE2 Io2(exp(K2 vbe2)- 1)

+ RB (V1 - V3) hoe2 - V3 + V5

D(Vl,v2,v3,v4,v5) = Ra Io3 (exp(K3 vbe3) - 1) -v4 + v2

E(Vl,v2,v3,v4,v5) = RB Io4 (exp(K4 vbe4) - 1) -v3 + v5

where: vbe1 = e1 - vl - v~1

vbe2 = v1 - el -v~2

vbe3 = vee - V4 - V}3

vbe4 = v5 - vee - v~4

82

Let V10 ,V20 ,V30 ,V40 ,V50 be an approximation to the solutio~

of the five equations. Now expand the equations about the

above approximation using the Taylor's series and neglecting

the higher order terms since vJe make the assumption that

the initial approximation is relatively close to the actual

sol1.1tion. To simplify notation let:

A ~ A(V1 ,v2 ,v3 ,v4,v5 ) etc

d;)A _ . JA _ ~ V 1 - A 1 , ~ V 2 - A2

Ao ~ A j vlo'v2o'v3o'v4o'v5o

etc

etc

etc ~-~I 1 vlo' v2o' v3o' v4o' v5o

The expansion is:

A~ Ao + Al(Vl-Vlo) + A2(V2-V2o)

+ As(vs-vso)

B = Bo + Bl(Vl-Vlo) + B2(V2-V2o)

+ B5(V5-v5o)

c - Co + cl(vl-vlo) + c2(v2-v2o)

+ c5(v5-v5o)

+ A3(V3-V3o) + A4(V4-V4o)

~ 0

+ B3(V3-v3o) + B4(V4-v4o)

~ 0

+ C3(V3-V3o) + C4(V4-V4o)

~ 0

D = Do + Dl(Vl-Vlo) + D2(V2-V2o) + D3(V3-V3o) + D4(V4-V4o)

+ Ds(v5-v5o) ~ o

By transposing the first term on the left side of the

h "d system of equations is formed. equations to the rig t sL e,a

To solve this system of equations Cramer's rule is used.

::::::

-B 0

-c 0

-Do D2 D3 04 D5

-Eo E2 E3 E4 E5

Al A2 A3 A4 A5

Bl B2 B3 B4 B5

cl c2 c3 c4 c5

Dl D2 D3 04 D5

El E2 E3 E4 E5

Det 11 ::::::

I Jacobian I

From this we can see that the next approximation is:

I Det 1 I vl = vlo + 1Jacobian1

This same process is repeated to obtain the next approx-

imation for v2,v3,v4, and v5.

v :::::: 2

Al -Ao A3 A4 A5

Bl -Bo B3 B4 B5

jJacobianj !Jacobian!

83

84

Al A2 -A 0 A4 A5

Bl B2 -B 0 B4 B5

cl c2 -c 0 c4 c5

Dl D2 -D 0 D4 D5

El E2 -E E4 E5 !net 3 v3 = v3o+

0 = v3o + Jacobian I !Jacobian I

Al A2 A3 -A 0 A5

Bl B2 B3 -B 0 B5

cl c2 c3 -c 0 c5

Dl D2 D3 -D 0 D5

El E2 E3 -E E5 + IDe!= 4 v4o+

0 v4o v =

4 = l I !Jacobian! Jacobian

Al A2 A3 A4 -A 0

Bl B2 B3 B4 -B 0

cl c2 c3 c4 -c 0

Dl D2 D3 D4 -D 0

El E2 E3 EJ+ -E +I

Det 5 I 0

v5 = v +- = v5o 5o I !Jacobian I

Jacobian

These then, are the recursion formulas which generate

successive approximations until the desired accuracy is

obtained.

The actual equations for all of the partial derivatives

and deterrninent expansions are listed in the program

described in appendix III.

APPENDIX III

PO,.JER AHPLIFIER ANALYSIS PROGRAM

The function of this program is to find the response

of the power amplifier by solving the five non-linear

equations given in Section II (equations 1,2,3,4,and 5).

The proceedure by which these equations will be solved

is the Newton-Raphson iterative method. The recursion

formulas are given in Appendix II. In order to meet the

requirements for Fortran coding "there must be a variable

name change. This is given below in tabular form.

SYl1BOL

Kl,~'K3,K4

1ol' 1o2' 1o3' 1o4

V¥1, Vy2, V'(3' V3'4

~El'hFE2'hFE3~hFE4

hoel'hoe2'hoe3'hoe4

RB

~ vee

El

vb~l,vbe2'vb~3 ,vb~4 vbe 1, vbe2 ,vbe3 , vbe4

;)A -:..v = A1, etc r:) 1

Det 1, etc Jacobian

1b1' 1b2' 1b3' 1b4

FORTRAN SYMBOL

Ql,Q2,Q3,Q4

SCl,SC2,SC3,SC4

VGAMAl, VGAl'1A2, VGANA3, VGA1.1A4

HFEl,HFE2,HFE3,HFE4

HOEl,HOE2,HOE3,HOE4

RB

RL

vee El

VBEl, VBE2, VBE3, VBE4 VBE5,VBE6,VBE7,VBE8

A1, etc

DET1, etc

AJAKE CB 1, CB2, CB3, CB4

SY1<:BOL (continued)

1cl' 1 c2' 1c3' 1c4

vcel'vce2'vce3'vce4

FORTRAN SYMBOL CCl.CC2,CC3,CG4

VCEl,VC~2,VCE3,VCE4

86

As was shown in Appendix II,the evaluation of the five equations, twenty five derivatives and and six determinents is necessary for each iteration. For clarity all of these equations are listed. Notice that the equations are in Fortran coding form.

A= (SCl)*(HFEl +l)*(EXP(Ql*(VBEl))-1) - (SC2 )* (HFE2 +1 )* (EXP (Q2.* (VBE2)) -1) +(SC3)*(HFE3 +l)*(EXP(Q3*(VBE3))-1) -(SC4)*(HFE4 +l)*(EXP(Q4*(VBE4))-l) +(V2'"'Vl )*HOE1- (Vl-·V3 )*HOE2+(VCC-Vl )*HOE3- (Vl+VCC) *HOE4- (Vl/RL)

Al = (SCl )* (HFEl+l )* ( -Ql )* (EXP (Ql* (VBE1))) -(SC2)*(HFE2+1)*(Q2)*(EXP(Q2*(VBE2))) -HOE1-HOE2-HOE3-HOEl~- (1/RL)

A2 = HOEl A3 = HOE2 A4- (SC3)*(HFE3)*(-Q3)*(EXP(Q3*(VBE3))) AS= (-SC4)*(HFE4)*(Q4)*(EXP(Q4*(VBE4))) B = (RB*HFEl*SCl)* (EXP(Ql* (VBE1) )-l.)+RB (V2-Vl).

-V4+V2 Bl - (RB*HFEl*SCl)*(-Ql)*(EXP(Ql*(VBEl)))~RB*HOEl

B2 = l+RB*HOEl B3 = 0

B4 = -1

BS = 0 C = (RB*HFE2·:+sC2 )* (EXP (Q2·* (VBE2)) -1 )+RJ3X"(Vl-V3)

*HOE2-V3+V5 Cl = (RB*HFE2'*SC2 )* (Q2 )* (EXP {Q2* {VBE2)) )+RB*HOE2

C2 = 0 C3 = -RB·*HOE2-1 C4 = 0

cs = 1

D = (RB*SC3)*(EXP(Q3*(VBE3))-l)-V4+V2 Dl = 0

D2 = 1

D3 = 0

D4 = (SC3*RB)*(EXP(Q3*(VBE3)))*(-Q3)-1 DS = 0

E = (RB*SC4)*(EXP(Q4*(VBE4))-l)-V3+V5 El = 0

E2 = 0

E3 = -1

E'+ = 0 E5 = (RB*SC4)*(Q4)*(EXP(Q4*(VBE4)))+1

•

Notice that many of the equations are equal to zero.

This fact reduces the size of the determinent expansions

considerably. These expansions are given below.

DETl = B2*D*"A4* ( C3*E5+1) -B2*D4*A * ( C3*E5+1) -B2*D4

*E*(A3-C3*A5)+D*A2*(C3*E5+1)-A*(C3*E5+1)+

DET2 =

DET3 -

DET4 =

C* (A3*E5+A5) -E* (A3-C3*A5 )+B2*D4*C* {A3*E5+A5)

+B*C3*E5* (A2*D4-A4 )+B* (A2*D1~-A4)

-D*Bl*A4*(C3*E5+1)-D*Al*(C3*E5+l)+D*Cl*{A3

*E5+A5)+D4*Bl*A*(C3*E5+1)-D4*Bl*C*(A3*E5+A5)

+D4*Bl*E*(A3-A5*C3)-D4*B*Al*(C3*E5+l)+D4*B

*Cl*(A3*E5+A5)

-E*Cl *AS* (B2*D4+1 )+E*Al * (B2.*D4+1) -E*Bl * (A2*D4-A4 )+ES*Cl*A2* ( -B-*D4-D )-E5*Cl*B2*

( ~A-*D4+D*A4 )+E5*Cl* (A+B*AJ+ )-E5*C*Al* (B2

*D4+l)+E5*C*Bl*{A2*D4-A4)

-A5*(-Bl*C+B*Cl)-(Al*B+A*Bl)-D*A5*(-Cl*B2)

-D* (Al*B2-A2*Bl) -E*Bl* (A3-C3*A5) -E5.*A3* ( -Bl

87

*C+Cl·x-B) -E5*C3* ( -Al*B+A*Bl)+E5*D*Cl*B2*A3

-E5*D*C3*(Al*B2-Bl*A2)

DET5 = -Cl*A2*(D+B*D4)+Cl*B2*(-A4*D+A*D4)+Cl*(A4

*B+A)-C*(Al*Bl*A4)-C*D4*(Al*B2-A2*Bl)-E*A4

*Bl*C3-E*(Al*C3-Cl*A3)+(E*D4*Cl*B2*A3)­

(E*D4*C3)*(Al*B2-Bl*A2)

AJAKE= -Cl*A5*(B2*D4+1)-(-Al-Bl*A4)+D4*(Al*B2-Bl*A~)

+(E5*Bl*A4*C3)+E5*(Al*C3-Cl*A3)-E5*D4*Bl

*A2*C3+E5*D4*B2*(Al*C3-Cl*A3)

The flow chart of the program is now given. Since

this program uses standard techniques no further explain-

ation is given.

POvJER AMPLIFIER ANALYSIS PROGRAM FLOW CHART

S I <:; T. ~-.·., Q ·. P 11. f.i i\ ''4 F T F R S ---·

P.JPf7P 0 2- 1 ( • 0 !.;.( 2 9 0~= llaP1',721 () t. .. = J. ? • 2 , h 7 5 SC l==C •'+!:'"-6 <; r ;:- = c • 4 F - 6 ') r ·:l = l • C F - A c; r 1, = 1 • ( r- A vr;,•. ·.• !' 1 =C. '5 VG"· '-'/\ ?=r • c; vr:".v.t~=c. o v r, r. '-~ " 4::: c • o Hi=t:l=lf7. t .. p::: i: 2 = i t:- c; • I·H· r 'l= f16 • 7 1-r= F i. = 5 6 • 7 ! ln F 1 == ? • r- 4 qn r ?: t:; F- 4 t-mr?=: o?5 ul)r4=.C·::>f

L _____ ._

r; JC CI!TT \'f.lll[S

f'Ul,= ?C. ?L = '• • VCC=l".

I"'f'UT \ll'l T/\G[ El

El=lf .0

DPOGP/\~ C~NSTANTS

I-=RQn):·=5.[-S PI==~.lttlS03 f '.! l T I ·~ L /-.. P P F f'J Y. I \A .1\ T I n~~ <; VJ=('• V2== 1'·. tt V~=-1'+.4 V4-=l4.4 vr::.=-!'•.4

~·J".J= ?C C• !\1 f\1 fJ = ~ J ~~ + 1 KTP1="0 KH-~::=l"C R. = r • c:

88

+

r -'·---...... i

v 1 ;-(' • 4 v?c=1'"· I. ·-'--1t •• t V -, ... . I

V4=1'·· ~4 V5-:::-llj. ··

I

----

K TD?} -Vfl, l II{ 1 • r > -- "c 2 '< r r. ? V. ( -',T)=. i':<Tt'?)

:_, T l == \• ( · ' ' ·;.; ? ) \I ( · ' · 1 ( I· 1<. l · • .. . Tl-\ •• Q...,) 1/ I /, • ' . -- ' ; r; v: T /' •I",T)-:::\, ., TD?) Vf • ,_\,, .~,v. · . ( 1--. T I -- • ' V. 'y·)-r . I rr.: ~- ·T -i> 2-1 KTf r:-K

-~---- ---~

-·_S2_ II\PFGY1 =V1 ~ P 1:' n X?= V 2 1\.-,rr-:v:?,.-=V""'· f.\CH: 0Y4=:\!4 ,~ p:: n y 5 = \1'3 Vrtf 1= VIr·!-Vl--\1'";1\M/\.1 VF';: /='Jl-V T"'-V~AM/\2 v~E~=vrr-v4-vraMa~ VQF~=VR+Vr~-VGA~A~ 'l =-(S~-1l>:·(!iFF1+1.l*(EV.P(0!~:'fV?.':l ) 1-1.1

- ( ' r 7 l ·~ ( } d=- f- ? + 1 • ) * ( c 'l( p ( o ? ,:, f v o c ? l ) _ ~ • ) • f s r: -;;, l .;, t 1- · J-:. r= '2 + 1 • l :!< ( ::: x o t n ? '" ( v r.. c ~ ) ) - 1 • 1

90

- ( ~ r t, l :': ( l .t c c /_. + 1 • ) >!' ( [ X p ( C) I+ >~< ( V !1.1.: 4 ) \ - l • ) + ( V ? - V 1 ) >!: H(} f l - ( \1 1 - V l ) ":< H f' F ':'? + { V (. C- V l } :< q '1 ~ ~~- ( V 1 + V r C ) *I l ,-, F '+ - ( V 1/ R I )

.'\ 1 = ( ~ C 1 } ~:: ( J-l r.-! 1 ·f- 1 • ) >:: ( - rJ ] ) ;':: { ::::: X D ( (' 1 '~ ( V PC l ) ) ) .. . - ( 5' C ? ) * { H f-= [ :;> + l . ) >!< { n. ? ) >!< { f: X P ( ~) :2 ,~, ( V R 1::: ? } ) )

, -l 4 n [ 1 - fl J' ~ ? - HG r= '3 -1--J C1 F 't- ( 1 • I r L l I J\.2=-l.l 0 ;:::1 .. 11--,-.;..wr'·:::z 1 •, /+ -::-c ( <: C 3 ) ~ { f ~ F ;:~ -~ ) * ( -f) 3 ) * { :: X 0 ( 0 3 >~ ( V ~ F '2 ) ) )

~~~=f-SC4)~fHF~4)*(0~)~(~YP(04*(VqF4 ))) / '::>, = f P n ,:, HF [ l ::: S r l ) >:< ( f X P ( 0 1 '~' ( V" F 1 l ) -1 • ) ; -t c r. ':< ( V ;:_ - V 1 l i<! 1 r• E l - V t.. + V :> i Ll l = ( F' :""1. ~:< f.J r r: 1 ::< Sr. 1 ) * ( - Q 1 ) * ( F X 0 { ') 1 ::::< ( V P ::: 1 l ) ) - P fl. t.< ~ '] E l : ·-~ ~~ = 1 • + P '"'- ''< 1-1 n E 1 ! r. = f ~- t) o~· ! l r c: ? ~-: <: r ? ) >:< ( E X ::> { r::; ? ,::: ( V ~ C 2 ) ) - 1 • ) [ + C' ~ '!: ( \f 1 - \' 7~ l o:< 1-l!i r ? - V -~ + V 5 I c 1 ·= { c• D ,:, ~ ~ f- r: / ;;: s r: ~ ) * ( (J 2 , ~:: { c \: p ( 0 ? * { v q r:= 2 ) ) ) + p R -!,: J ! 0 E 2 · (' -:> = - C'- ~J, o~: 1-W' I= ? - 1 ! < ~ ( ':' r- '~: <:: r ~- ) o:: f r; ~ ( 0 ~ >:' ( v ~~ F ~) l- 1. • , - \jiL + v? !~4=fsr~*c~l~(FVP(0;~(V0 ~~l}J*(-83)-l. ~=( 0 P*Sr~)*{FXD(04*(V~c4))-1.)-V~+V5 r-:; = ( c r. ~ c:: c t. ) ,:, ( r- 4} ~· ( E '< o ( n t~ ;}: r v q F £, l l 1 ._ 1 • ,~Ti~,7*r~~4*(C3*[~+1.) _,2*,4*A*CC3~E~•l.l - 0 ?*n4*~*(A1-C~*4~1+0*~?*(C3*F5•l.l-~*(C~*E5+1.J

I• f*(1~*f~ +~~) - F*('~ -C3*~5)+92*D4*C*{f3*E5+A~) + P*C~~r-5*(A?*D4-A4l +R*f42*~4 -~4) .D~T?= -n*f1*~4*(C3*F~+!.l-n*1l*fC3*r~t-1.J+n*Cl*{~J*ES+ft5)

+ f"' 1• ::< P. l ~- !\ 0:: ( c ?. >!< F 5 -f- 1 • l - fl 4 >!d\ l '!.< c * ( 1\ 3 '~: !:: 5 + ,.._ t:; l + 'J 4 1,: ·~ 1 * r: * (,"J':!-•\r:,:,r~l- ~~+"::~~~·t\1~· (r·3*F5 +l.l + r;/lt.•R-:'r:l,:<f,'\"::,*f'::S+ 11·~)

OET~= -~~fl*A~*CR?*D4+1.l~E*4l*(R?*n4+1.)-F*~l*(~~*D~-~4) · + !-= r::, ~ C ~ ,:~ ll 7 :::: f - P. >:' n 4- D l - r- l) ::: C l .:~ ~ ? *- { - 1\ -::' 0 ~ + r: -!:< !'.t+ l + F 5 ;"' C 1 * (t+P~~~J-F~*C*Al*(P2*~~+1.l+~5~C*Pl*C'\?*n4-A4) nF~~~-~~*(-~l*C+B*Cll-(-~l*R+~*ql)-D*~~*f-Cl*q7)-n*

{A]hP? -~?*nll-E*R]*( A~-C3*A51 -F5~~3*{-Rl*C+Cl*Rl -r~~r:~*(-~l*q+'\*~l'+~~*D*Cl* R? *6~ -f5*D*C1*(hl*R2-'<J~~"?) ') r: T r:; = - ( [ 1 ~.,_ !\ ? l ':< { f' + !3 ~:<~) 4 l +- ( C 1 >:: r\ ?. ) * ( - f. 4 '~ n + '\ >:' Q 4 ) + r: l ,q ~ 4 * B + ~ ) -(r}*f~J+ql*~41-CC*04J*(Al*~?-'\?*Rll-([*h4*~l*f~)-lEl* (A]*(~-Cl~~?)+Cr*D4*Cl*~~*A~)-(E*P4*C~)*(Al*~?-Rl*A?l

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7 R=rtl.

Dtn t\JT R ESIJI TS

!<: v :-: ,'"' ,)n ? 1 ~<'=1 'N'V'-1 'A r:' T T --: ( l , l J r) ) VTI\'-::-\I(l,Vl 'I 1 = v r ') , v ) \f':c-Vf' 7 K) 'J '=\fll. '\() \fl. -= v ( t; ' '< ' V'-:::V(f,,V) '.l = T T f) ( F )

vr-r')=VT~ .. -,,1 VPF·',=Ifl-V rrr \lf\ ~ 7="l.'- Vt. v r. F q = v ~ +\I r r v r-. r ,_ = '' "\ ~; (. - '' r~ 1'. ... ,, 1 II r"l. '7 :..· 'I r"l. ~- t,- 1/ r. ~ •.' 11 ?

II n (1-::- \I P r 7 -" r, ;, ; : /'. -~

If ') r ~~ = \I p r ,. - \j r. /\ . ~ r\ '· ', .f p T T r ( ..1, , T : I r ,I • ~: K ' ~! ~' r. ~·. 1 = ( ~; r l ) ,., ( L y () ( f) 1 '!r v R c 1 ) - 1 • ) r:~>.7= (~r.1 )':' (~ '(f1('l7~~yRr7)-i.) r \ l = ( v r.- '' ? l 1 ( ::- o. l ((?7 f\f::l-VC)}/('H.q r;-p,=rrl

~ '~'· = r r 7 \0 en

\4', 141· l41J 1 /+~ 147 l4° 149 1 c; i· l s, 1 ')~ , c: ~

l ') t .. la;r.;, l !)~ P~7 1C:.q 1SO 1 1-.!) 1 h 1 l ·" :-> lA~

1 (, '• u)s l ~f. lA7 ; r- r ln0 l 7~') 111 l 7? 171

-I. C ""'-:: r t-l F r ~ 1 "~ ( C F\ :7, l -t ( \1 C C - V L l t.~ ( ;. v; E i I CC4=f~FrG)*fCq~)+(Vl+VCC,~CH0~4) vrrJo::-\J">-\11 vr.r?=V1-\I? V\.F:7,=\ffC-V1 vr:F4=\il+vr:r l,.JP TTf: ( "),, 2C' uotr~(~,~r' Vl~,Vl,V2,V~ 4R ! T F ( 3 , P ;· } ~oJTr (~,lGJ V4,VS 1-J r? ) T r.: ( ~ , r ;~, )

't1 P T T r ( -, , "), r. ) V P. F ') , V P ~ 6 , V f\ t: 7 , '' ('\ F P l.l'? T T F ( ?- • l 7 C' } :F' TT~ (::-, ~C·l VfFl,VCf:7,Vrr:l,V(.f'+ ltJD T T c ( ?, • h r, ) • .• J\'..' 1 T !=' ( "), • •1• ( ) C 0 1. , ( !~ 2 , C a, 3, CRt+ !.Jf' T T r ( ~ , 7,- I ;.: r.; r T r r -\ , :-v· > c c 1 , r r ~ , r: r: ~; , r: r 4

71 '-\1<=1(1<+1 r.t.t I FXTT

1:• rn;.;:.~rr f///, 1 JTrPI\TT Vi~=•,Ttt,/, 1 TT~1r TNC.Pf;•1FNT=•,T4, 1 I' ' ~·q jlr'- ~ I) () r: T 1 '1 r I) T ~ n c, '::' ' J !!\

? c r r~ r I~ t : ( I I ' ~ 1 ( ' ' v T !\) ' ' T .> p ' • v 1 ' ' T /l 6 ' ' "? ' ' T L, It ' ' v "), ' ) '1 :~ ~ ; 1 r .·. ~ t, r ( t, c: 1 P • ,'l }

'~ :-, r-: r; ::- ~' f. T cr. r 1 ? • '• ) t:; (' c f 11 ~1 AT ( // , To , ' \ln. F- l ' , T "> 7 , t V n, r? ' , T 4 ":l. • 1 \f "-1-= \ ' , T 6 i. , ' \m r: It ' ) (. ,: c n t: •.· "' r f I I ' r 'l ' ' r "' 1 ' • T ? .,. • ' r : ~ ? ' ' r ~.. -~ • , r £1 ~ ' ' T ,) ? ' , 1 R 4 ' l 7 (o rn ~;.A f, T f // 'T ,, ' ' T r 1 I • T? 7' ' T r:? I 'T 1., -~' ' T r -~ • 'T f: ~' • I ('• ' ) h J r r r 11 r-. T ( 1 I , T 1 r , • v 4 • • T .., g , ' v s ' l

1 l (' r-:n f' '~ t\ T { ! I /I l 1 2 C r: n R M /1 T ( I I, T a , ' \1( i: l 1 , T 2 7 , ' V C r:? ' , TIt l , t V f F 1 ' , T b 3 , ' V C f 4 '

E\10

/()fiT !I.

~

OUTPUT SAMPLE

IT~?l\Tlr1~1S= 1( 1IMf 1NCPF~rM~= ?5 NlH~R[P. 'lf TJ!!E STFDS= 200

VIN n.707IC66C~ 01

V4 0.1412~6?0F 02

\!'1fJ :) • 7rl6l ""J.4CF or

vrFl . 0.707756ACF 01

!R1 o.?c~col~CF-a?

1 (. J G. 'V-tP3033Ci=-01

Vl O.A3'5045~0E C'l

v~ -n.t4?4~GlOF 02

\/~f"/' -r .??:.,..,,--~,_.or- cr;

vr~? r.?03Q?A58F C'

T0.2 -~.?coq~qQJ[-0~

Tr.? ~.1G13040JE-0l

V? 0.1347A010E 02

V~F~ r.R7S~7~R8c 0S

vr [~ 1 1 .RA495't6~F 01

rq3 ').14q3'113CJ:-01

Tr.~ n.25104?0CF 01

v~ -O.l4042410F 02

V~F4 o.7~4oanooF 00

VCE4 0.211~0440E C?

I fVt n.t~130' .. r·rE-Ol

TC4 0.trot4o~rF 01

\0 CD

APPENDIX IV

DERIVATION OF FOURIER SERIES FOR WAVEFORM OF

CROSSOVER DISTORTION

Using figure 7 and equation 6, the Fourier series

can be found by using the definition of the Fourier series

for an odd function.

For an odd function, f(t):

f(t) = Bk sin (2~kt/T)

T/2 where: Bk = (4/T) ! 0 f(t) sin(2ykt/T) dt

Substituting for f(t): T/2 [ ' r J Bk = (4/T)J0 sin UJ(t- ~ sin(2rrkt/T) dt

Nhere: I 2~

w = T-2 1-' .

Since the value of f (t) = 0 for; 0 < t < -2 and

T:_ r < t < T+ '1"" , new limits can be used. Also f ( t) 2 2

can be expanded thusly.

Bk = (4/T) cos(~1 )

- (4/T) sin ( ~·r)

Using integral forms:

f sin (rot) sin(nt) dt =

! sin(rot) cos(nt) dt

for both cases

Bk becomes:

sin (2~kt/T) dt

sin (2~kt/T) dt

sin(m-n)t 2(m-n)

cos(m-n)t 2{m-n)

sin(m+n)t 2(m+n)-

cos(m+n)t:__ 2(m+n)

100

(T-1)/2 Bk = (4/T)cos(w'r)( sin(b~ _ sin(at))

2 2b 2a 'Yj2

(T-?")/2 -(4/T)sin(~r)(- co~~bt) _ cos(at))

2a '"'72

where: a = m+n =

b = m-n =

Now substitute limits:

27T(Tfk+l) - 2k'i) T T-27)

27T(Tfk-l) - 2kr) T T-2'1)

sin(b C-f-)) - - 2b

T-r +(4/T)sin(w~"f)( cos(b( 2 ))

2b

T-....,..­cos (a ( 2 ) )

+ 2a

)

cos Cb c-f-)) 2b

_ cos (a(-f)) ) 2a

This then is the expression for the Fourier coefficients

for the waveform of crossover distortion .. v1hich is the same

as equation 8 in Section II.

APPENDIX V

AUDIO AMPLIFIER ANALYSIS PROGRAH

To describe this program the first requirement is

to give the conversion from the symbols used in this

paper to the Fortran symbols. The definitions of the

terms used in the program output are also given.

SYMBOL FORTRAN SYMBOL

Eo E0

A2 A2

r;T DELRAT

Rf RF

R. RI ].

cf CF

71" PI

A Al vol

f == uJ.,. FREQ0

0 . 271" I 0MGA uJ

r TAU

T TI

f FREQ

Al (kw) AV(I)

B (kw) BETA(I)

Bk C0D(I)

AlA2 GAIN(I) l-(3Al~

101

Ei

El

pout

Av

Av db

0 ok

0 ck

coefficients of Fourier series describing the audio amplifier output

f(t) using C0D(I) as the Fourier coefficients (normalized)

f(t) using DISLP(I) as the Fourier coefficients = d

0

f(t) using DIS0(I) as the Fourier coefficients = d

c

f(t) using E0Ur(I) as the Fourier coefficient = d +e

c 0

ei = E1 sin t

total harmonic distortion contained in the output of the power amplifier without feedback = e2

total harmonic distortion in audio amplifier output = e + d

0 c

EIN

El

PefWER

VGAIN

DBGAIN

DISLP(I)

DIS~(I)

E0UT (I)

SC0D (I)

SDISLP(I)

SDIS0 (I)

SE0UT (I)

SEIN(I)

DISTl

DIST2

102

Once the circuit parameters have been specified (eg: R. ~,

Rf,Cf,Avol'etc. ),the only input data is E0 ,A2 , and 7/T.

From this,the program will calculate the remaining vari-

ables.

So that the data points are evenly spaced on the log

103

frequency axis~ the increments in frequency are obtained

by r~peated multiplication of the last frequency increment

by a constant. To demonstrate this~ if it is desired to

have ten frequency increments from 10 HZ to 50 KHZ the

multiplier is 2.57. It works like this:

FREQUENCY INCREMENT

1st

2nd

3rd

9th

lOth

10 HZ 25.7 HZ 66.0 HZ

19031.1 HZ 48909.9 HZ

MuLTIPLY BY

2.57

2.57

2.57

2-57

This yields the desired results of equal spacing on

the log frequency axis, but has the disadvantage of having .

large gaps in the latter end of the scale; for example

from the increment from nine to ten. To avoid this, the

program is set up such that two frequencies can be choosen

at will. Therefore the program gives results at twelve

points in frequency; two of which are arbitrary and ten

of which follow the equal increment frequency sweep. The

two arbitrary frequency points are 30 KHZ and 40HKZ.

The program flow chart is given along with the actual

program and output sample.

FLOW CHART c (1 = n. 1 ? -? i\?=- Q?"?~ 'i F l n. ~ T,· - ;·. I" 1 r::

- - f·- - -' • ' :J OF=1 F~

PI=iir~;t.r. c~==~c.r-!2

Pic:-?.14J':-C1 I\ 1 =-?C( ('-r • FP~=.-:-:r=r::,.c.

c:p ~=0= ~c;:--:::r.

--------------~~------l-= __ 1_,,~_?_. ________ -J

~--------------·'~---------------TI=l./FI')f:0 r~u=r.f:t Pt:.T::e•rr OHG'"I~f?.~~rr J/fTI-2.*T\UJ T!_ I"=(TT-T!\ll)/?. r:q T''=T/I!J/?.

·--------------<:~· _______ r_=~J._s_,~_· _____ =r~ I

.'\ = f { ? • •:: :·) J ) ::~ { T I ~ { S + J • ) - ( ? • '~ S ::::: T f1 I J ) ) ) I ( ( T T ) ~' ( T T - ? • * f -"1 tJ J J 0 -= ( f ? • ::: ;.' T ) ':'- { T T i,< ( ~ - J .. l - ( ? • ::< s 0~ f l'J. u ) ) ) I f ( T T ) t,.· { T T - 2 • * T 'I u ) )

104

r: r> r~, f ! ' ::- f '~ • I i T ) t.< C ( 0 S ~ f t; . ~ r, !'. * T \ i ! ) I 2 • ) ) 'i< ( f t) I r,, ( Q, * T l ! ·.~ ) l I f ? • * 11 l -( ~ T ' · ( ') :!• T I. l f.' ) ) I { ? • ::: t ) - ( <; I r-! ( fl '/• tJ, ! _ I '' ) ) I { ? • ::: P, ) + ( S T '! C !'. "' ~~ L J '·' ) } I ( 'l • * " ) )

+ { 'i- • I T 1 ) ·~ f S I ~' ( ( r ~'' r~ ,~ ;~ T !\ 1_1 ) I 7 • ) ) ::< ( ( C n S t 0 * T L I '·~ J ) I { ? • :':: q ) + < r: r -~ s < r· ,., T '_ r , ~ , , 1 f 2 • -1:: A ' - c c , } s ( :~ ;~ ~~ L r : ·1 , , 1 c 2 • ":' ~ , - c c r 1 s < A~, ~ t I • ~ 1 l 1 ( ? • * !\ 1 J S= S + 1.

---··----------------.-----------------

[ -- ·< . yl<;O 1 t V ( J } = ( !I 1 ) I ( ~ Q R T ( 1 • + ( F R f7 ~1 ::!< c -R-.r::-Q-~--_X_*_Y_l_/_( _c_R_f:_(._1_C_*_F_Q-~--;m}ll

~-,:,Y+l. =~ -- f

J

105

.------- = < ~~~.50 --

11 L T :.r r 1 ~ ( "_ 1 1 c F ) ,:. s L~ o T t 1 • -~ 4 • ,x n r ;, o r ''< c F =:< c ~ ~ r. F :i: ~=: F * Y =>: v ,., F .':' r r. * r- ~~ F Q 3 Y=V+1. -~-----·-~--

---=---:_____ =T"O ? ~ r,11 I\' {I l = ( :~" ( T l*f\? l I ( 1 .-'\\I( I) '~fl.7·c:QE Tl-; r!) l Z=7+1.

J T l\ = r. n 1 r, -" I "! ( 1 ) 1 ;: r f- r· l 1 r {\ 7 :.!=_ con ( 1 > l " \·' t: r = ( :: r·, ::. F n l I q •

Lvr:~,r~·-==ffl/FIN ,... , 0"C,~P 1 =?·= .~<:''llf'<;lC{·\?t:.(V ,!'!..I ·'ll

~-'\frP=J ./ 0 f:T/I{K) J L I I --------~

-~ T F P , t. V ( '< l , S ft.. f ~-J ( K l , K

~

.I=2, ')("·

-~-~ 11 I ~ n ( T ) = n I <; l P f I ) I f 1 • - \ V C ! l ~'I\ 7 * q C: T .1\ ( I ) _l J

l

------""il ED tiT f I l =r_' T ~r C!) ~

----,~~---,----J

+

r--~------..., I r·:1r·.•r T~Jtlr l -· . . ·-

CAt L r: ,~ 1. L r ''-t t r i'.L L

- <:; u ._.' J n ( r: n'"' • T f , ~ (: ~-~ S 1/'•qJ o C r ! 5 t r- , T I , S r'l I <: L 1J l s! _i ·-1 u r c ~' r s IJ , T I • S ·'"'~ T S ,, l ~IJ''I!P ( ff 1 t1T, f T, S r'"'f IT l

q"=! •

106

+

[-;:R !'=Q=4~000.

L. _ _____,

Sl)1'1S 0 l)TTI''F 'StF4 UPfV.Y.1 .Ul

PT=~.l41_')r:J X= 1.' /l r,r. • T(JJ=X ·----

' ( r , =:: s 1 ~~ f c v * 2 • ~: r r ::': T c '- J ' 1 r "'-' ' , ~=SUi~+7_ ~-----------,-- ---------~

t If L l = SU'1 f(L+l)=T(t )+Y

r QET'JPN' --==r -J HJr I

107

S\.l'''J T <;=·'"'· C:

Y = l ( <:: T r: ~! l\ I f i + 1 } IS T G ~,. fJ. L ( 1 ) ) ~ l 0 (' • ) * ( ( C, I G t] A, L ( I + 1 l Is I c ~~ 1\l ( 1 ) l * 1 0 0 • )

.----__J_l --··-

1 ENn 1

108

1

? 'l,

4

') I-,

7

R

fl

1 ('

l 1 1? J ~

"· l" 11-. 17 1 R 10 2f'

AUDIO AMPLIFIER ANALYSIS PROGRAM 1\.J A iA r F 1 7 f"'l, .,, '-. , I I ._, F = 0 ? , P 1\ r; r- S ;,: 1 ':' C L F X 1\ T 0 NV

"' r M r-t ,' s T n "'' r, n n ( 5 ,.., ; , n T s L r> f s ,-. > , n r <:. n \ s c l , r. n 1 n < s o ) ., s r. nrH 1 o o l , lS!')l'-;LP( 1n01 ,SDTSP(l\C•l ,S[IlUT( lCCl ,~V(~C·l ,BETA{')(}) ,GAIN('50)

r r. svsn:,1 Vfl,r~t.'\RI !-S r c

r

r r r

r r r

r

r r (

r

r

'lUTI'lllT VflLTA\,( (1-\1\Xl r.n;:::n. 13r.7 f\7:.~"~.0?7r.,

nr-LI~'"T (f"\Ftl\V RATinl =TI\U/i>f:O f(lf) ~[i PJI.T='i,'1l'l

r.trr.1 1 lT Vfii_UFS

r.: F =' 1 • r r; ::o T -=- l 1 r. C •" • CF:-:-)f',r-1?

C""'S'""' r~ f) l -= ~ • lit 1 r, () 1

~[)CDIITTnf'J<\I t,'l.flllFTFR CO"J<:THJTS - nPt:~J Lnoo GAlN fiNO UPPER r • 11tH ~ r- !) r: r; 11 r "-! r v ~ 1 =- ?C ( ; ., • Fr';. '10: r. r.r 1

f'" " I= f) :: ~ ,- r r , .• r-n r, ' ':': 1 ' l ) T T :: l • /I· P r· "' T t-, I I = n !- L o 1\ T ::: T T , 1 •·• r "' = ' ., • ::: P r l 1 ( l r - 2 • * r .~ ,_ 1 , Tl T '·' :-: ( T T- T .~ II ) I? • ~~ l T •• :::-: T ~. ' I I ? • ·~ ::.~ ' T i= ( -.~,, 4 t l ·.~ o I T F ( • , 1 ~; t F P F o , T I , T fl ' 1 , q r , R I , r. F •,~o I r r ( 1, 4,.. l

J0!1 142

1-' 0 \0

?1 77 -., t.. ?4 2S

26

?7 ?~ /tl "30

·:q 3? 33 34

.,C) 1f, -:q 3P

,,., 4r 41 !....'? It"·

'•4 lt c; 4~ 't 7 4?.

c r;;:.;~:r::p.'\TF Ci:t-Ft=-tt.lH 1 T~. nF r-r1w1fr-~~ c-.r.nrr:s r:r1\ cRnssnvFF nrc;rn~nir:r-~ r

r r: r

' r (

r r r

r r c r

r

S= 1 • r \')q l ~ = 1 ' 5 ::. "'= { ( ? • ~, P ! l ~~ { T 1 :::, ( <::. + 1 • l - ( 7 • ~:~ s ~:~ T :-, t ~ l i ' : ( ( r ·~ 1 * ( T I - ., • ~:, T r, t 1 l l ",-= ( ( ? • t.~ o T ) >!': ( T I >:~ f S- i .. ) - ( .? ~ >:: C', :~: T !:. I I J ) ! I r ( T T } "' ( T t - 7 • >:: T fl t' ) ) r. n D ( T l '""' r !., • 1 T "! ) ::< ! r: r! \) f ' c •·· "': l '1.= T ~ ll i 1? • ) l ~:= ( { s I f·i ( n ;-: T 1 T r1 ) ) 1 ( ? • * !l ) -

1 I S 1 j\~ I f, '~ T I T 't ) ·, I ( ? .. ':: 1\ ) - I S t t : I \-\ •:' '-' I_ \ "- ) 1 I ( ? • ~< n. ) + i <; 1 N ( A :t.= n. L J ~" ) ) I I / • * /\ ) ) + ? ( t.,. • I T r I * ( ' i '\l ( ( r1 · ., r~ t ·:< T 1'1. I J l I ? • \ ' ':' ( ( C t 1 S ( f~ ~~ T L T "-" l l I ( 2 • * H l + ~ ( r n S ( ~ ~ ;-L T ~.~ J J I ( 2. *A ) - ( C fl ~ f ,q '!:: 11 f T M ) } I ( ? • "«A ) - ( C n S ( /1 * R l T M 1 J I ( ? • * 1\ ) J

1 S=~+-1.

~HU=PJ'.TF (;tdN nF ClPf-P.~T!r]!,!I\L l\',1DLIFTFP FOR l<:T ~0 HARMONT(.S

'<= 1 • '!n l 1 T = 1 , r; c ~ II f I I -= ( fl. 1 I I ( S n R T ( 1 • + ( r: q F n ·:, r I? r: 0 * X"" X } I ( F r r: () r * Fr~ E 0 n l ) J

)1 X=X+~.

12

13

14

r; U! F P AT E ~ ~ T 1\ F rJ R I S T r:; n 1·1 fl 0 :"! fl "J l C S

V: 1 • r-,n l 7 T ~ 1 , rj ('

~ET~fi)=(PJ/RFl*SORT(l.+4.*PT*PI*CF*CF*~F*RF~Y*Y*FREQ*FRfQ) Y=Y+1.

r,plf=PJ\TF r;AI~! F~CTOR Fno IST 5C H.~ 0 i'10NTf:S

7 = , • 'lfl l ~ T = 1 ' t; (I

r, ~. l 'l I f ) = ( !11J ( I l ):' .\ 2 ) I { J. • - ·' V ( T ) * 1\ 7 ~' f1 [ T 1\ ( T ) l 7:7_~1.

r. ll I r II I .!\T C P.! r II T V n L T A r; F ( "1 " X ) , E 1 C ·~ !\ X l , P n \" F R I N T n 4 0 H M L 0 A D , V11LT/'Gr r./\f".i :\f\!f) VOLT"GF: r./\PJ on.

r. T r,'= Ffl j:.,\ T ,,, ( l ) ~1=(rn)/lt?*(~'l(1 )) ~>~-w ,= P =! c:: IV' r- rn n~ .. vr.'l Tfi'=FI'/~ jr·: n ,, r: .\ T \' = ? i't • '' '\ I n r. l C ( 1\ R <; { V r. ft, Y "' l ) \,1 ;) 1 T r: ( l • 1 :> : J) r il , [ 1 , F P! , ;> n W '- R , V r. ~ P! , n f1 r; " I r-: ·.:oyrr (-:1,,7'.) nn J 4 K = 1 , r; 0 ATEl=1./f1Ff/\(~) ~~p ! T F ( 3, -:~, r. ) '\ T F f\ , A V ( l< I , r, 1\ Pl ( K ) , K ........

l-1

0

4Q c:;o '51

52 5?

54 5 e:,

56 ')7 5R C) ('I

6(• 61

o? 6~ A4 61.)

1-,1..,

1-,7 r,p. 1--.9 1n 71 T? 7"3

74 7r"i 76 77 78 70 q(')

,. \.

r.

c (. (

r. ( r

r.

\:.[l'JEf'I\TF. OJST\WTJfiN I"lSPH- t!lOP, fHSLP

OT SL Pf l l=l:l_,::f\?:>:.(\.00( l l-l. J DO 2 T =?, sn

7 f1I SLP( I l=F1 *1\?~~Cfll){ ll

GFNEPATF Cn~PONENTS OF OUTPUT SIGNAL

nn 3 I=l,c:;c. 3 fH S rl( T l = n T c; I. P C T l I ( 1 • -:\ V ( I 1 :t.: 1\? ~ p, E L1 ( I ) )

GtNFPt.Tt: niJTPUT l\NO 0\JTf>UT Ill ST\lR,Tl(lN

~nuT<ll=Fn+nrsnCll f)n t+ I = ? • '1 r

'+ ~ntJT( Tl=~"I~n( I l wo T T r f?, '+ 0 l 1rJ Q I T E ( ., ' r; C! ) ')f) 7 ~1= l' ')('

7 W ~~ T T F { 1 , () 0 ) I. 00 ( '1. ) , 0 1 c; l D ( i'A l , D l S fJ p .. q , F. OUT ( ~~ ) , M p:: (L-?l 15,l",lt1

C FPJll FfTl llSI~G 1ST 5n HfiRMONICS r.

r

15 C~LL f..'\ll. Cl\ I L C t'.l L

q P·~IIP <roo, r T, ~r ern (' l JVU D ( [H S L P, T T, Sn T <;t P) ~ !J •,~ \ Jf' ( n T S q , T I , S f)J S rn S II~~ II o ( F n U T , T T , S F: 0! JT )

1.·! r. r r F { "J, , '! r. l 1-j R 1 T F ( ? , 7 C } ~=!. !')f"' n r-.1 = 1 , ] cr. S i-= P! = F P 1 * S T f'.' ( C ? • .:~ P I ,>t f Q f- ')>:<! T '~ ;:? l 11< !\ • l t .. rpJTF (J,ROJ <\COf1nl),SI)ISLPCNl,Sni~non, SFOUT fNJ,SFIN,N

~ Q:=-r.'+l. 1 6 C n ~H I f\! U r-

r Fn1f1 TnT/\1 H~D')rJf'IIC DISTnPTTfJ"l WITH ANI) ~JTTHi1UT FFFOBAC:I( ,. (1'\IL H.~,.,nrqrn11,nJI\Tll ~J\ll Hf'll;lny<; n:ntiT,[)I<\TZ) ',~ o I T ~='" r ~ , t, ' l ':•I o T T ~ r 1 , 1 ·"' n ) n l s T 1 '"'o I ; r < ., , 1 1 n , n I s T? \~P l T ~ I .,, ' t.. ~· )

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119

VITA

Anthony Francis Lexa was born on July 19,1945 in

St. Louis, Missouri. He received his primary and secondary

education also in St. Louis. He received his Bachelor of

Science Degree in Electrical Engineering in Jm~e of 1967

from the University of Nissouri at Rolla. He has been

enrolled in the Graduate School of the University of

Missouri at Rolla as a graduate assistant since September

of 1967.