P‘G‘WER QUTPUT AND EFFECLENCY QF PUL§E
DURATESN MSSULAEQ AMFLEFEERS

Thesis for fire Degree of DH. D.
MICHEGAN STATE UNEVERSETY
William C. Hoim

1961

This is to certify that the
thesis entitled
POWER OUTPUT AND EFFICIENCY OF PULSE
DURATION MODULATED AMPLIFIERS

presented by

William C . Holm

has been accepted towards fulfillment
of the requirements for I

JILL—degree in_Ele.c1:r_i.ca1 Engineering

W K

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/1/’L//\)L/L‘¢

./ Major professor

Date August 18, 1961

 

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ABSTRACT

POWER OUTPUT AND EFFICIENCY OF PULSE
DURATION MODULA TED AMPLIFIERS

by William C. Holm

The output power capabilities and efficiency of conventional
linear untuned amplifiers of the vacuum tube or transistor type are a
function of, among other things, the waveform and magnitude of the
signal being amplified. A square wave input signal results in highest
efficiency and greatest possible output power for a given dissipation
rating of the driven device. By modulating the pulse duration of a
constant amplitude square wave and using the waveform to drive output
tubes or transistors, it is possible to attain high efficiency and high
output power regardless of the waveshape of the modulating signal.
To recover the modulation signal, a low-pass filter is usually necessary
at the output of the driven device. The filter efficiency must be con-
sidered in the output efficiency. With certain types of filters, pulse
duration modulation (PDM) amplifiers offer no improvement in efficiency
over conventional amplifiers. Constant-resistance filters fall in this
category; however, distortion is very low, theoretically being independent
, of the characteristics of the driven device. With the use of a constant-
resistance filter, power output capabilities of the PDM amplifier would
be higher than for the conventional amplifier because losses in the
filter do not contribute to dissipation within the driven tubes or transis-
tors. - It is often desirable to obtain high output power for a given

dissipation, with efficiency being considered of minor importance.

Abstract William C. Holm

If high efficiency is desirable, PDM amplifiers with L-C low—pass
filters may be used. High efficiency as well as high output power may
be obtained by using L-C filiers, though usually at some increase in
distortion.

This thesis analyzes the performance of PDM amplifiers with
constant-resistance low-apass filters. Power output, efficiency, and
dissipation equations are derived for various PDM amplifier configur-
ations. These equations are experimentally verified for each type of
amplifier. Performance equations are derived for PDM amplifiers with
one type of L-C filter, and tests are made which substantiate the con-
tention that high efficiency as well as high output power may be obtained

with-PDM amplifiers which use L-C filters.

Copyright by
WILLIAM C. HOLM
1961

POWER OUTPUT AND EFFICIENCY OF PULSE
DURATION MODULATED AMPLIFIERS

BY <

X‘"
William CY‘Holm

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Electrical Engineering

1961

AC KNOWLED GMEN TS

The writer wishes to express his appreciation to
Dr. ~Myri1 B. Reed for his encouragement and guidance,
to Dean John D. Ryder and Dr. Lawrence J. Giacoletto
for their suggestions and interest in the study, and to
Ian O.‘ Ebert for his careful reading of the manuscript
and his many helpful suggestions.

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CHAP'.

III.

IV.

REFER

 

 

 

TA BLE OF CONTENTS

CHAPTER
I. INTRODUCTION .....................
II. GENERATION OF MODULA TED PULSES ........
III. POWER OUTPUT AND EFFICIENCY ..........
‘IV. EXPERIMENTAL RESULTS AND CONCLUSIONS. . . .

REFERENCES ......................

iii

Page

17
29

107

LIST OF TA BLES

TABLE Page
I. Performance of Conventional Amplifiers and PDM
Amplifiers with Constant-resistance Filters ...... 23
II. Power Output and Improvement Ratio with Utilization
of Full Plate Dissipation Rating ............. 24
III. Performance of PDM Amplifiers with L-C Filter 25
IV. Performance of Series-fed Class A PDM Amplifier
with Constant-resistance Filter ......... 37
V. Performance of Shunt-fed Class A PDM Amplifier with
Constant-resistance Filter . . . . . . . . . . . . . . 38
VI. Performance of Series-fed Class A PDM Amplifier
with L-C Filter .................... 39
VII. Performance of Class A Single-ended Push-pull PDM
Amplifier with Constant-resistance Filter ...... 40
VIII. Performance of Class A Single-ended Push-pull PDM
Amplifier with L-C Filter ....... . . . . . . . . . 41
IX. Performance of Class B PDM Amplifier with Constant-
resistance Filter ............ . . . . . . . . . 42
X. Performance of Class A Complementary Pair Transistor

XI.

XII.-

XIII-

PDM Amplifier with Constant-resistance Filter . . .

-Performance of Class A Complementary Pair

Transistor PDM Amplifier with L-C Filter . .. .. .........

Effect of Shunt Coil Reactance on Shunt-fed Amplifier
Performance . ..................... .

Deviation of Measured Values of Power Output and

Efficiency from Theoretical Expectation for Amplifiers
with Constant-resistance Filters ...........

iv

43

44

45

46

FIGURE

2-4

3-1

LIST OF FIGURES

Page
Block diagram of pulse generator. . . . . . ..... 11
Pulse generator waveforms for d-c modulating
voltage ............ . . . ......... 12
Pulse generator output waveforms for sinusoidal
modulation .......... . . . .......... 13
Monostable multivibrator ............... 15
Periodic voltage function applied to filter 17
Constant-resistance low-pass filter . . ..... 18
L-C low-pass filter . . . . . . . . . . . . . 47
Attenuating network between pulse generator and
transistor amplifier ...... . . ...... 47
Complementary pair transistor configuration . . . . 48
Grid voltage waveform ............ 49
Series-fed Class A conventional amplifier. . . 52
Load line. . . . . . . . . ..... . ........ . 52
Series-fed Class A PDM amplifier. . ....... . 53
Plate voltage waveform . . . . . . . . . . . . . . . . 53
Shunt-fed Class A conventional amplifier configur-
ations. .............. . ...... . . . . 61
Load line . ...... . ..... . ........ 61
(a) Switching circuit configuration, (b) Duty cycle for
switch ......... . . ............... 63

 

13-3

17-1

F-Z

F-3

LIST OF FIGURES - Continued

FIGURE

Shunt-fed Class A PDM amplifier configurations. .
Transformer coupled push-pull PDM amplifiers . .

Plate to cathode voltage waveforms . . . . . . . . .

(a) Common push-pull configuration, (b) Single-
ended push-pull ..................

Single-ended push-pull a-c amplifier . .. .. . .. . . .

Load line .....................

(a) Transformer coupled Class B PDM amplifier.
(b) Single-ended push-pull Class B PDM amplifier

Series—fed PDM amplifier with L-C filter . . . .
Single-ended push-pull . . . ...........

Complementary pair transistor configuration . .

Waveform at input to filter .............
Filter and load configuration ............

Waveform at input to filter .............

vi

Page
67
67

7O

79
79

79

94
97
98
101
101

103

LIS T OF A PPENDIC ES

APPENDIX
.. A.. Pulse Modulation Linearity ..............
B. Comparison of Series-fed Class A Amplifiers . . . .
C. Comparison of Shunt-fed Class A Amplifiers .....
D. Comparison of Class B Amplifiers . . . .......
E. PDM Amplifiers with L-C Low-pass Filter ......
F. Determination of No-modulation Duty Ratio for

Maximum Output Power and Maximum Efficiency . .

vii

Page
49
52
60
79

92

101

 

 

 

 

Output «

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LR.

LR.

DR.

DR.

DEFINITIONS AND SYMBOLS

Output efficiency: The ratio of a-c signal power delivered to the load

over power supply power furnished to the plate or collector
circuit. When Specified, output efficiency is the ratio of a-c
signal power delivered to the load over total input power to the
output stages, including base driving power for transistors.

Filter efficiency: The ratio of a-c signal power delivered to the load

over total input power to the filter.

Maximum P: Greatest average power.

PDM

I. R.

'IoR-B

D.R.

Puls e duration modulation.

Improvement Ratio. The ratio of maximum power output
of a specific class PDM amplifier to the maximum power
output of the corresponding conventional amplifier, with

neither amplifier exceeding rated tube dissipation at this
maximum signal level or any lower signal level.

Improvement Ratio relative to Class B PDM amplifier.

Dissipation Ratio. The ratio of tube dissipation of a PDM
amplifier to tube dissipation of a conventional amplifier
when both amplifiers are operating with the same circuit
parameters‘(same Ebb" Emin' R, etc.) and delivering
the same power to the load.

Dissipation Ratio relative to Class B PDM amplifier.

Duty Ratio. The ratio of on-time to the time of one period,
the period being that of the repetition frequency. This
may be a time function variable due to modulation.

Duty ratio with no modulation.

A real positive number (0< M < 1) giving the fraction of
maximum permissible amplitude of an a-c signal for a
conventional amplifier. With a PDM amplifier M is the
modulation factor and represents the fraction of maximum
permissible pulse duration swing.

viii

 

 

 

min

Output efficiency of a Class A series fed amplifier.
Output efficiency of a Class A shunt-fed amplifier.
Output efficiency of a Class B push-pull amplifier.
Output efficiency of driven device (tube or transistor)
exclusive of efficiency of filters or coupling networks
between device and load. This applies only to PDM

amplifier 8 .

Output efficiency of Class A series-fed amplifier with
L-C filter.

Output efficiency of Class A shunt-fed amplifier with
L-C filter.

Filter efficiency for constant-resistance filter with ..
Class A series-fed PDM amplifier.

Filter efficiency for constant-resistance filter with
Class A shunt-fed PDM amplifier.

Filter efficiency for constant—resistance filter with
Class B PDM amplifier.

Filter efficiency for L-C filter with Class A series-fed
PDM amplifier.

Supply voltage for capacitor charging in R-C pulse
timing circuit.

Cutoff bias for tube.

Maximum negative going grid voltage excursion of
monostable tube.

Time duration of generated pulse.
2th harmonic distortion component.
Plate or collector supply voltage

Minimum plate to cathode or collector to emitter voltage.

- For conventional amplifiers this is at maximum signal level.

ix

ot
P
op

pt

A time function variable, minimum plate to cathode or
collector to emitter voltage for PDM amplifiers with
L—C filter.

Modulating frequency

2 17 times the modulating frequency.
The pulse repetition frequency.

The filter cutoff frequency.

Current intercept point of load line.

~Maximum plate current.

Maximum rated plate current.
The ratio of I to E .
m min

See Equation (C-lZ)

Average signal power per tube delivered to the load for
a conventional amplifier.

Total signal average power delivered to the load for a
conventional amplifier .

Average signal power per tube delivered to the load for
a PDM amplifier.

Total signal average power delivered to the load for a
PDM amplifier.

Plate dissipation per tube (or transistor) for a conventional
amplifier.

Plate dissipation per tube for a PDM amplifier.

"Rated plate dissipation per tube.

Plate power supply power per tube.

Total plate power supply power.

Farr:

Sg

58

Screen dissipation per tube for a conventional amplifier.
Screen dissipation per tube for a PDM amplifier.

Average power delivered by a PDM amplifier to the load,
including d-c power.

Input power to the filter of a PDM amplifier.

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I. INTRODUCTION

In the design of logic circuits for digital computer applications,
the active elements need only exhibit two stable states; on and off.
The power dissipation of the active elements isilow for both of these
states, being high only during the time of switching, and the switching
time is purposely made short. Linear untuned voltage and power
amplifiers of the vacuum tube or transistor type (audio, control, iso-
lation, and servo amplifiers) must be operated with Class A, Class AB
or Class B bias, and the active elements (tubes or transistors) are
operated in a region of high dissipation a large percentage of the time
for most driving signal wave shapes. Efficiencies for output stages
can be derived in terms of class of bias (A, AB, or B), plate or
collector supply voltage, minimum instantaneous output voltage, wave-
form of the signal being amplified, and type of load. These efficiencies
are usually expressed for sinusoidal signals. General efficiency
expressions are derived in Appendices B, C and D for conventional
amplifiers. By setting M=l and Emin = 0 it can be seen that the
maximum theoretical plate circuit or collector circuit efficiencies for

these amplifiers are

Maximum ”AR = 25% (1 " 1)
Maximum ”A = 50% (1 - 2)
1r

71" x 100% = 78.5% (1 .. 3)

Maximum 27
B
These efficiencies can be approached but never achieved in practice.
Actual efficiencies in most cases are considerably below these upper
efficiency bounds because other factors, such as minimum distortion

and maximum power output for a given plate loss, are usually con-

sidered more important.

If the driving signal waveform is of a shape that puts the active
element into heavy conduction (with low plate or collector drop) during
on-atime and cuts the device off (little or no plate or collector current)
the rest of the time, higher device efficiencies may be achieved than
are possible with sinusoidally driven Class A or Class B amplifiers.
This type of drive has many of the attributes of Class C Operation and
the maximum theoretical efficiency is 100 percent. Average conduction
current is proportional to the length of time the device is turned on
provided the on-current repetitively assumes the same respective value
and the off-current is zero. A nonzero off-current will. merely add a
constant to the output current. Linear amplification may be achieved
by using a constant amplitude square wave to turn the output elements
on and off, the onwtime being determined by the instantaneous value of
the signal that is to be linearly amplified. This is pulse duration modu-
lation and for satisfactory results requires a square wave pulse
repetition frequency that is high compared to the highest frequency
component of the modulating signal. For nonperiodic modulating signals
the necessary requirement is that the signal change but a small amount
in amplitude in the time of one period of the repetition frequency of the
square wave. For most applications this pulse duration modulation
method will require an averaging or lowupass filter at the output. The
cutoff frequency of the filter would lie between the highest modulating
frequency that it is desired to amplify and the repetition frequency of
the square wave that turns the output elements on and off.

Many applications for PDM amplifiers suggest themselves. With
a pulse repetition frequency of 100 K. . C. and a low—pass filter cutoff
frequency of 10 K. C. , an audio amplifier would result which has a
frequency response from dmc to 10 kilocycles. -Use of a lower pulse
repetition rate, say 1 K. C. , would provide a dmc amplifier meeting the

requirements of isolation, control, and servo amplifiers in many systems

applications. Here d-c amplifiers are often required, and the a-c
response need be good only over very low frequencies. In many of these
applications linearity of modulation of the pulse generator need not be
good, thus allowing simple pulse generator designs and lower cost
equipment than would be necessary for linear amplification.

Some applications require no low-pass filter at the output. One
such application would be the driving of a resistive heating element in
which the heating effect is to be a function of some low power Control or
error Signal that must be amplified.

Tubes and transistors are obvious choices of output elements.
Other devices such as silicon controlled rectifiers would find application
if conditions unique to their operation are met. For example, if a due
source is used for silicon controlled rectif‘iers provision must be made
for periodically removing the source from the device so that the trigger
element may regain control. Use of an a-c source would result in a
nonlinear amplifier, but with some applications this may be satisfactory.

All references to tubes and plate circuits apply also to transistorsi‘
and collector circuits even though not Specifically mentioned.

In order that PDM amplifiers and conventional amplifiers may be
compared on the basis of power output, efficiency and dissipation, the
analysis of PDM amplifiers is carried out for sinusoidal modulating
signals. In many cases the analysis is valid for d~~c signals as well
since most of the circuit configurations are dmc amplifiers. 1-1 this
analysis it is assumed that the frequency of the modulating signal is low
compared to the pulse repetition frequency. This restriction is not

‘

necessary from a mathematical point of view but it allows many very

3

important relations to be easily established.

From a practical point of View the above restri-zztion is an absolute
necessity if the output waveform is to be an accurate reproduction of
the input. The. waveform of a PDM signal may he represented lay a

Fourier series.1 The spectrum components include: a possil‘lle du-c

 

 

yr, ”i in..- .. . _ ..
With PNP tran51stors voltage and current polarities are reversed
from those expressed in equations.

 

 

 

 

 

term (this is not present with some circuit configurations), a term at
the frequency of the modulating signal, terms at the pulse repetition
frequencyand its harmonics, and terms at the sums and differences of
these frequencies. Some of the latter terms will lie within the pass
band of the low-pass filter and may show up as unwanted components at
the load unless their amplitude is small. The coefficients of these
terms are small for modulating frequencies that are low relative to the
pulse repetition frequency. A finite number of these intermodulation
terms lie within the pass band of the filter, and the peak voltage of the
waveform represented by their sum is small if the amplitudes of the
individual terms in the sum are sufficiently small.

- Experimentally it is found that a pulse repetition frequency to
modulation frequency ratio of 10:1 or greater is satisfactory and that
no zero beat phenomena is apparent for modulation frequencies below
this 10 percent figure. Zero beat interference is observed when the
frequency of modulation is 12. 5 percent of the pulse repetition frequency.

Many advantages result from using a constant-resistance filter
for the low—pass output filter? 'It exhibits a driving point impedance which
is resistive and constant at all frequencies. This helps greatly in
maintaining waveshape at the input to the filter. This type of filter is
essential for output circuits which at times have all driving elements
cut off and where there is no provision for a conducting element across
the filter input terminals. A disadvantage of using a constant-resistance
filter is the fact that it is an R-L-C device and has loss.

‘ As derived in Appendices B, C and D for various PDM amplifier
configurations with constant—resistance filters, the efficiency of the

output tubes or transistors is

Emin

ntzla-Eb‘g... (1‘4)

 

This expression is valid no matter what the waveshape of the

modulating signal; the modulating signal does not have to be sinusoidal

or even periodic. Equation (1 - 4) does not include the efficiency of the
output filter.
. The following advantages result from this switching method of
operation in conjunction with a constant-resistance filter at the output.
1. Most of the circuit configurations are d-c amplifiers.
Modulating frequencies may range from d-c to 10 percent of

the pulse repetition frequency with good linearity of operation.

2. Greater output power is possible with given components than can

be obtained in conventional Class A or Class B amplifiers.

3. Amplifier linearity is independent of the characteristics of the

 

device being driven. The linearity is a function only of the
pulse generator and its modulating system, and performance can
be made very good. This is extremely important. Output ele-
ments with very poor linearity (those unsuited for operation in
Class A or Class B linear amplifiers) may be used, or those
elements which are normally chosen for Class A and Class B
amplifiers may be used over a wider range of values on their
volt-ampere characteristic curves than is possible for linear
operation in conventional amplifiers. ' And it is usually possible

to use a lower value Em

in°
4. There is good gain stability as a d—-c amplifier and instantaneous
recovery from overdrive or clipping. These features are a

function of the pulse generator design and its modulating system.

5. There is no degeneration due to screen to cathode voltage vari-
ation for cathode follower type output stages and for single-

ended pushwpull amplifiers. 3

All of the advantages listed for PDM amplifiers with constant-
resistance filters also apply when LuC filters are used except for

point 3. The linearity of amplification is not independent of the

characteristic curves. The expression

_1_ eb,min

derived in Appendix E takes the place of Eq. (1—4). The term eb min

is not a constant as is E but varies as a function of the instantaneous

min'
value of the modulating signal. This introduces an amplitude distortion.
The amplitude distortion may be kept small by keeping the maximum
value of eb’ min small or by attempting to hold eb’ min at a constant
value with clamping diodes or by using feedback to the amplifiers
furnishing the driving pulses for the output tubes. Transistors are
capable of supporting a very low ratio of maximum eb, min to Ebb'

Although'L-C filters show promise of providing very high plate
circuit efficiency, extremely difficult circuit problems arise in their
practical application. The driving point impedance must be controlled
in the block band as well as the pass band, perhaps to 100 or 200 times
the cutoff frequency of the filter. A low driving point impedance at
frequencies above cutoff would result in high plate loss and would lower
the power handling capability of the amplifier. An L-C filter with a
series inductor at its input would provide a desirable high driving point
impedance at and above the pulse repetition frequency, but provision
must be made for current continuity in the series inductor. This last
requisite results in circuit design requirements that seem impossible
without recourse to either clamping diodes or to a feedback system
which alters the grid or base driving signals so that proper waveshape
is maintained at the input to the filter.

The subject of this paper is confined to an investigation of the
performance of PDM amplifiers with constant-resistance filters.
A research area that shows additional promise is an investigation of the

use of L-C filters with-PDM amplifiers. For comparison to conventional

amplifiers and to PDM amplifiers with constant-resistance filters, the
power output and efficiency expressions are derived for a PDM amplifier
with one type of L-C filter. A PDM circuit with this type of L-C filter

is tested to illustrate the difficulties involved in practical application.

~ II. GENERATION OF MODULATED PULSES

Two convenient types of duration modulated pulses are those having
a no-modulation duty ratio of 50 percent and a no-modulation duty ratio
of zero. With the first type, positive going modulating signal would
increase the duty ratio, the upper limit on duty ratio being 1. 0. .Negative
going modulating signal would reduce the duty ratio, the lower limit
being zero. With the second type two outputs would be needed. . The no
modulation condition would provide a zero duty ratio (or no output) at
both output terminals. » Positive going modulating signal would cause the
duty ratio to increase at one of the outputs, while the duty ratio of the
second output remains zero. Again, the upper limit on duty ratio would
be I. O. . Negative going modulating signal would increase the duty ratio
of the second output, while that of the first remains zero. The second
type of pulse modulation requires that two output elements be driven,
whereas the first type may be used to drive either a single output element
or, with the addition of an inverting network, two in push-pull.

If the output tubes of a PDM amplifier drive a resistive load such
as the input of a constant-resistance low—pass filter, there is an
interesting parallel, insofar as power supply requirements are con—
cerned, . between the use of these two types of modulated pulses and the
operation of Class A and Class B amplifiers. With the first type of
pulse duration modulation the d-c power supply current is constant,
whether a single stage is being driven or two elements in push-pull.

. This is identical to the power supply demands of a converitionalC‘lass A
amplifier, whether it be single stage or push-pull. The second type

of pulse duration modulation results in no current with no :modulation
and an increasing d-c current as modulating signal amplitude increases.

This then corresponds to the power supply demands of a conventional

push-pull Class B amplifier. Because of this parallel, these two types
of pulse duration modulation will be referred to respectively as Class A
modulation and Class B modulation.

A There are no commercially available pulse generators capable of
providing Class B modulation or pulse duration modulation over the range
of duty ratios from zero to one. This necessitated, at no small expendi-
ture of time, the design and construction of a pulse generator capable of
meeting these requirements. Since this pulse generator is aninstrument
to provide a particular form of driving signal to various circuits which
are to be tested, it is designed for precision of adjustment and versatility,
with no consideration for efficiency. The instrument is much more
complex (there are 20 tubes) than would be called for in any one application
since it must be capable of driving many types of output circuits having
either tubes or transistors. Distortion in modulation is held much lower
than would usually be required, and pulse amplitude and waveshape are
very closely maintained. In order to achieve short rise andfall times and
to allow large output voltage excursions, tubes rather than transistors
are used. The pulse generator meets the following specifications:

1. The repetition frequency may be anything up to 100 kilocycles.

2.. Either Class A push-pull or Class B pulse modulation may be
obtained.

3. The pulse duty ratio may be anything between zero and l. 0.

The modulation factor may be 100 percent.

4. The voltage excursion of the output pulses is 160 volts (out-
put voltage is either 90 volts or 250 volts).

5. The rise time and fall time at the outputs of the AND and OR
stages are in the range of 0. 2 microseconds to 0. 4 microseconds,
depending upon the class of modulation and the load on the output.
The 11. 5 micromicrofarad input capacitance of the scope probe

must be considered part of the load.

10

6. Themodulator is d-c coupled and has an input impedance of
approximately 3, 000 ohms. ‘ A balanced 600 ohm source may
be used to drive the modulator. An a-c input voltage of 1. 95
volts results in 15 volts a-c signal at the grids of the clamp
tubes and provides 100 percent modulation.

7. At 100 percent modulation, the combined distortion of the

modulator and Hewlett Packard ZOO-CD driving oscillator is
O. 3 percent second harmonic, 0. 1 percent third, and 0. 005
percent fourth. These figures are the average for three
different amplifier tubes, with and without triggering of the
monostable circuits. The distortion of the oscillator at the
input of the modulator is 0. 25 percent second harmonic, 0. 1
percent third, and O. 005 percent fourth.

Figure 2-1 is a block diagram of the pulse generator. Each of
the four monostable circuits is adjusted for a 25 percent duty ratiOiwith
no modulation. The outputs of OR-l and NOT-2 constitute a push-pull
Class A modulated signal. Likewise so do the outputs of OR-Z and NOT—l.
For single ended use, the output of OR-l or OR-Z alone may be used.

A push-pull Class B modulated signal is obtained at the outputs of AND—l
and AND-2.

Figure 2-2 gives the waveforms for Figure 2—1 with various fixed
value input signal conditions on the input to the modulator. Figure 2-3
gives the waveforms for Figure 2-1 with sinusoidal modulating signal on
the input to the modulator. The binary states are triggered by negative
trigger spikes.

The Class B modulated pulses have both leading edge and trailing
edge modulation. The centers of the pulses are evenly spaced on the
time axis. The Class A modulated pulses at the outputs of either of the
OR stages have trailing edge modulation.

Four monostable circuits are used rather than two so that the duty

ratio demand on any monostable circuit will be 50 percent or less under

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14

all modulating conditions. This allows the duty ratio of the AND circuit
outputs to be 100 percent at the peaks of the modulating signal, and yet
allows 50 percent minimum recovery time for each monostable circuit.
Monostable circuits of the type employed in this pulse generator use
R-C elements to determine pulse duration, and at the completion of the
pulse enough time must be allowed to re-establish the initial conditions
on the capacitor before another pulse may be started. The monostable
circuits in this pulse generator are capable of a 75 percent duty ratio
(recovery time 25 percent of the period of the repetition frequency).

A commercially available pulse generator which enjoys extensive use is
capable of duty ratios of approximately 70 percent (no provision is made
for pulse duration modulation on this unit), and the recommendation of
the manufacturer is to not exceed duty ratios of 60 percent.

Figure 2-4 illustrates the method of modulating the pulse duration
of the monostable circuits. Clamp tube V3 determines the magnitude of
the negative excursion of the V1 plate voltage and hence determines the
duration of the generated pulse. The plate voltage of V1 can never drop
below the voltage of value given by the instantaneous control grid voltage
of V3 plus a voltage of an amount that represents the working bias of V3
when it is conducting.

The linearity 0f pulse modulation may be made very good by return-
ing R to a high positive voltage and by restricting the voltage change on
the capacitor during on-time to a small percentage of this voltage.

In Appendix A an equation is derived for the instantaneous time function
demodulated signal at the output of the filter as a function of a sinusoidal
modulating signal. For the derivation of this equation it is assumed

that the period of the modulating signal is long compared to the period of
the pulse repetition frequency. This equation may be used in conjunction
with the five point method of determining the percent second, third, and
fourth harmonic distortion.2 Choosing ES = 400 volts and Emm: 60 volts

with 100 percent modulation, the harmonic distortion is l. 746 percent

15

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16

second, 0. 041 percent third, 0. 001 percent fourth, and 1. 747 percent
total harmonic distortion. Modulation factors below 100 percent provide
less distortion.

An experimental method of determining the percent harmonic
distortion for the first four harmonics makes use of precision d-c volt-
meters. The instantaneous value of a modulating cosine voltage is
determined at each of the required angles (0, 60, 90, 120, 180 degrees)
of the five point method for determining harmonic distortion. 2 The ampli-
tude of this cosine voltage is such as to provide the desired percent
modulation. For each calculated instantaneous value a d-c voltage is
applied at the input of the modulator and the d-c voltage at the output
of the low-pass filter is read. These output voltages are used in the
equations of the five point method of determining distortion to determine
the percent second, third and fourth harmonic distortion.

Another method of determining the magnitude of the harmonic
distortion components is to drive the modulator with a sinusoidal voltage
of such amplitude as to provide the desired percent modulation and to
determine the magnitude of the harmonics at the output with a wave

analyzer .

III. POWER OUTPUT AND EFFICIENCY

The driving point impedance of a constant-resistance low-pass
filter is resistive and constant at all frequencies.4 Apply the periodic
voltage waveform of Figure 3-1 to the input terminals of this type of

filter. e(t) 4

E

 

 

 

 

> t
T, T
Figure 3-1. Periodic voltage
function applied to filter.
Let
T
= —-L- . -1
a T (3 )

The time function current at the input terminals of the filter is
. em
1( ) R < )

where‘R is the driving point resistance of the filter. The instantaneous

power delivered to the input of the filter is
PM = e(t)i(t) (3-3)

The average power delivered to the filter is

l 1
Pin = Tr— IOTpMdt = -— foT[e(t)]zdt

TR
T, E’- E2

: — -— :: ——- 3-4
T x R a R ( )

The filter will average the voltage which is applied to the input and
provide an output voltage to a load resistor R of

Ti.

T E: (LB (3-5)

1 T
Eave = -,f, f0 e(t)dt=

l7

 

 

 

 

18

The power delivered to load resistor R is then

E2 E’-
p0,,t = fixe- = e?- x Ti' (3-6)

The filter efficiency factor is

nf = gain- ‘ a (3-7)
From Equation (3-7) it is seen that the filter efficiency factor is equal
to the duty ratio 0.. Here a is a constant in the range 0 i o. _<_ l.

A consequence of Equation (3-?) is that constant-resistance filters
cannot be of the'L-C type. With Class A and Class B pulse modulation
0. is a time function variable, and efficiency must be evaluated over
one period of the modulation signal. The efficiencies for these classes
of modulation with various circuit configurations are evaluated in
Appendices B, C, and D. In Appendix F it is proved that the highest
efficiency for Class A modulation results when the no-modulation duty
1
ratio is '2’.

The constant-resistance filter is designed for whatever load
resistor R is to be used. If the load resistor is to be changed to a new
value for some reason, such as to increase output power, the filter
must be redesigned for the new load value. With ratios of pulse
repetition frequency to modulation frequency of 10:1 or greater a filter
with an ultimate attenuation slope of 12db per octave is Satisfactory.

. Figure 3-2 shows a constant-resistance low-pass filter of this type.

 

 

 

 

 

 

 

C o
D C

Input Output

c} 43

A Figure 3-2. - Constant-resistance low-pass
filter.

19

 

L hie (3-8)
1
c -
2 1”~12. fCR (3 9)
L _ z
c _ 2R (3-10)

The cutoff frequency fc is the frequency at which the output power

is 3db less than the input power. The output is to be terminated in a
resistive load R. The driving point impedance of the filter is equal
to R at all frequencies.

' Any actual filter device that is constructed will of course not
have a resistive input impedance of R at all frequencies. However,
the pulses of an actual PDM signal will have finite rise and fall times
and the relative amplitude of harmonics will be small above a certain
frequency. This frequency is several multiples higher than that
which would have a period equal to the rise or fall time of the pulse.
, There is no 'merit in designing the filter to have a constant input
resistance above the frequency of the highest significant harmonic.
The choice of cutoff frequency for most of the investigations of this
paper is 10 percent of the pulSe repetition frequency. If the input
resistance of the filter is to be constant for frequencies up through
the 10th or 20th harmonics of the pulse repetition frequency, it is
required that the input impedance of the filter be controlled to 100 or
200 times the cutoff frequency of the filter.

. For a given fc’it can be seen from Equation (3-8) that large R
values require large L values in the filter. It would seem that for
satisfactory performance, the natural resonant frequency of the
inductors should be higher than the frequency up to which the filter

input resistance is to be held constant. The coils do not act as

20

inductors of design value L at their natural resonant frequency, and
above resonance they present capacitive reactance. The R values
generally used for the filters in the investigations of this paper result
in'L values which are so large that the natural resonance of coils
tested for use falls between 5 and 50 times the cutoff frequency of the
filter. . There is a definite upper limit to the natural resonant frequency
that may be achieved for a coil wound to give a specific inductance,
regardless of the type of winding used or the means employed to keep
interwinding shunt capacitance to a minimum. However, an analysis
of the filter of Figure 3-2 shows that it performs very well even‘with
coils that do not maintain the design value inductance over the whole
frequency range for which the filter is to be used. The filter consists
of two legs connected in parallel at the input terminals. ‘ At high
frequencies the inductor between input and output presents a high
reactance and the filter input impedance is determined essentially by
the resistor R within the filter. If the reactance of the inductors is
abnormally high, their contribution in parallel with R is little different
than it would be if they had the correct reactance. ‘ If the impedance of
the coils is so high at resonance as to completely remove them from
the circuit (the worst possible case), the input impedance of the filter

at this frequency fr will be

21,, = R(1 - jfié‘i (3-11)
When f1. = 5 fC

zin = 1.039 RZ— 15°45' (3-12)
When f1. = 10 fC

21,, = 1.01 Rd— 803' (3-13)
When f1. = 50 fC

zin = 1. 0004 RZ-1037' (3-14)

21

Thevoltage transfer function of the filter is affected by the improper'L
values at high frequencies. However, the result is greater attenuation
at these high frequencies than that eXpressed by the design equations,
and this results in an improvement in performance since it is intended
that these high frequency components do not pass through the filter.

At extremely high frequencies (well above the natural resonant frequency
of the coils) the capacitive reactance of the coils becomes low and will
affect the filter input impedance and the voltage transfer function. The
filter 'must be designed so that the capacitive reactance of the coils is
not low relative to R at frequencies for which voltage components are
present at the input of the filter.

‘ In Appendices B, C, and D equations are established for the
average plate dissipation of the output tubes. These dissipation
expressions are based on the assumption of sinusoidal modulating signal.
The peak dissipation is greater than the, average, and for those circuit
configurations which are used as d-c amplifiers, operating conditions
must be chosen so that the peak plate dissipation does not exceed the
rated dissipation of the tube.

' A problem is encountered when using push-pull Class A modulation
with single-ended push-pull circuits}: If the switching times of the grid
driving pulses are not extremely short compared to half the period of
the repetition frequency, there is a common on-time for the tubes.

The ZEbb supply delivers a large current during this common on-time.
This represents a power loss from the supply that does not contribute
power to the load, resulting in lower plate circuit efficiency. . The tube
dissipation is increased and the amplifier has a lower power handling
capability.

~ At high pulse repetition frequencies the switching time of tran-
sistors may be very muchlonger than the switching time of the driving
pulses. -If the switching time of the transistors is an appreciable

fraction of the half period of the repetition frequency, the transistor

22

operating point may be in a region of high dissipation for a greater
percent of operating time than is desired for maintaining low dissipation.
. Unless very fast transistors can be obtained, it is necessary to Operate
transistor PDM amplifiers at very much lower pulse repetition rates
than may be used with tubes.

The complementary pair transistor circuit configuration of
Figure 4-3 offers some advantages over push-pull and single-ended
push-pull configurations (the diodes are not needed when a constant-
resistance filter is used). The base leads may be tied together and
driven from a single-ended source; no phase inversion or push-pull
drive is needed. No degeneration occurs with either transistor; they
both operate as grounded emitter amplifiers. However, when the base
leads are tied together, the peak to peak signal driving the transistors
must be greater than the peak to peak output signal at the junction of
the collectors. This situation can be remedied by driving the bases
separately through some d-c coupling element such as a battery or
zener diode. An NPN and a PNP transistor may be used together in
the complementary pair circuit configuration even if they do not have
similar characteristics. They should be able to carry the desired
on- current and they should have about the same collector to emitter
voltage when on. The base currents may be greatly different for given
on-time collector currents.

Tables I, II, and III compare the performance of the various type
amplifiers. . The equations are derived in Appendices B, C, - D, and E.
The performance equations for single-ended push-pull are the same as
those for push-pull of the same class.

. The plate circuit efficiency for each class of PDM amplifier with
constant-resistance filter is identical to the same class conventional
amplifier. This correspondence between plate circuit efficiencies is

further justification for the definitions of Class A and Class B pulse

23

 

 

 

 

 

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m I . M
Class A Series-fed Class A Push-pull
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"This table is based on the assumption that the maximum value of
eb, min is negligible compared to Ebb- The L-C filter investigated
is the low frequency leg of a constant-resistance filter. See

Appendix E.

26

duration‘modulation. The fact that plate circuit efficiency expressions
for’PDM amplifiers with constant-resistance filters are no better than
forthe corresponding classes of conventional amplifiers does not
preclude the possibility of obtaining more output power from the PDM
amplifiers. , The dissipation, except for the contribution due to 111:, is
in the filter rather than the output tubes, and more power supply power
may be delivered to the circuit without exceeding the dissipation rating
of the output tubes than can be handled by a similar class conventional
amplifier. » Maximum output power for a given plate dissipation is
usually considered more important than efficiency. - It may be noted
from Table III that the maximum theoretical efficiency of a PDM
amplifier with‘L-C filter is 100 percent. A In addition it should be noted
that the percent efficiency is independent of the percent modulation.
Both of these factors are very important when efficiency is a con»
sideration. . For many audio amplifier applications the average power
output ability of the amplifier should be 50 times the power level
necessary to handle the normal program 1eve1.5 Then according to the
plate circuit efficiency equations, a Class A conventional amplifier
would normally be operating at less than 1 percent efficiency and a
Class B conventional amplifier at less than 11 percent. The plate
circuit efficiencies of the corresponding classes of PDM amplifiers
~with constant-resistance filters would be reapectively the same, but
the plate circuit efficiency of a PDM amplifier with‘L-C filter is
independent of signal level and the efficiency may be close to 100 percent.
' In view of the fact that plate circuit efficiency is independent of
signal level when an L—C filter is used, an economy in the design of
the modulated pulse generator may be effected. - In order to obtainhigh
efficiency it is not necessary to swing the duty ratio of the modulated
pulses from 0 to 1 at maximum signal level. 'A simple pulse generator

can then be built with allowance for recovery time.

27

In an actual amplifier circuit it would be difficult to hold
eb, min constant at the value Emin and there would be a small itth loss
within the tubes due to filter input current variation about the established
signal current value. _ This loss does not appear in the equations for
‘PDM amplifier performance with-L-C filter because of the forced
condition that eb’ min = Emin- — Equation (E~23) indicates that there is
no power furnished by the power supply when M = 0. However, when
eb, min is not held constant at Emin during on-time, there will be a
power‘loss in the tubes due to the build up of current in the input
inductor of the filter each half cycle of the switching signal. This loss
may be'made smaller by increasing the size of the input inductor of.
the filter, but this creates new problems. The cutoff frequency of the
filter would be lowered, perhaps putting too low a limit on the highest
modulating frequency. ~ In addition it would be necessary to maintain a
high natural resonant frequency for the inductor so that a high‘impedance
would be presented to the upper frequency components of the waveform
which is applied to the input of the filter, and this requirement would be
very difficult to meet.

The expressions for Inaprovement Ratio and for output power in
terms of Pdr have no maximum theoretical limit as Emin is made
smaller. .However, the maximum current and voltage ratings of the
tube impose a limit on the maximum power obtainable. Expressions
relating to these limitations are derived in Appendices B,~ C, and D.
The power output and dissipation equations should be used to determine
the operating conditions for PDM amplifiers. . There is no reason to
refer to practical operating conditions for conventional amplifiers.
Parameters chosen for Optimum results with a PDM amplifier would
very likely result in over dissipation and distortion if employed in a
conventional amplifier. ' In the expressions for Dissipation Ratio and

Improvement Ratio the circuit parameters for the two kinds of amplifiers

28

are the same for comparison purposes only. .Much greater increases
in power handling ability for PDM amplifiers over conventional
amplifiers may be achieved than are indicated by the Improvement
Ratio expressions.

Since aClass B PDM amplifier has the same linearity as a
Class A PDM amplifier, and since this linearity is better than that of
- a conventional Class A amplifier, a Class B PDM amplifier may replace
any class of conventional amplifier with'no increase in distortion. To
evaluate the improvement in power handling ability when this is done,
Improvement Ratios (LR. B) are given in terms of a Class B PDM
amplifier. 'Expressions are also given for Dissipation Ratios (D.R. B)
in terms of a Class B PDM amplifier. It may be noted from, Tables
I and 11 that the D. R. and the 1. R. are reciprocals for amplifiers of
the same class.

‘ Screen grid dissipation is given by the product of the average
screen current and the screen grid voltage. . For'PDM amplifiers the
on-value screen current is Isgt Then for Class A PDM amplifiers the
average screen current per tube is $- ‘Isg and for Class B PDM
amplifiers it is EELS-L. For conventional amplifiers the average screen
current is represented by 153' and the peak screen current by leg
The peak screen current of a conventional amplifier may be the same
as the on-value screen current of a PDM amplifier, but the average
screen currents are not at all likely to be the same. ~ Screen current
tends to be very large for low plate voltages (Bruin): resulting in
greater screen dissipation for PDM amplifiers than for the same class

conventional amplifiers operating with the same Ebbr Emin: and load

resistance.

IV. EXPERIMENTAL RESULTS AND CONCLUSIONS

All data presented is for a 10 kilocycle repetition frequency, a
1 kilocycle filter cutoff frequency, and a 100 c.p. s. modulation
frequency. Multiplication Of all of the above frequencies by a factor
of 10 results in the same power output, efficiency, and distortion
characteristics for the PDM amplifiers; however,the rise and fall time
of the 100 kilocycle square-waves limits the modulation to 92 or 93
percent. Circuit Operation with M = l is instrumental in demonstrating
the validity Of the derived equations for power output and efficiency.

Table XIII indicates how well experimental results compare with
theory for amplifiers using constant-resistance filters. For applications
using an L-C filter, no comparison Of measured results is made with
the theoretical equations of Table III because these equations do not
take into consideration the losses associated with the circuit which was
tested.

Tables IV through XI give the performance Of. each type Of circuit
for which performance equations were derived. For those cases using
a constant-resistance filter, these tables also give the theoretical
expectation of performance based on the equations appearing in Tables
I and II. These equations are derived in the appendices. The upper
number Of multiple value entries given in the tables for each value Of M
is the theoretical value. The lower number is the actual value.

All theoretical entries for output voltage, power output, and efficiency
take into consideration the transfer loss of the filter employed. All
filters used, including the L-C, have a voltage transfer function of

O. 98 at the frequency of modulation for all signal levels used. The

omission of an entry in a harmonic distortion column means that the

29

3O

particular distortion component is so small and erratic as to be unread-
able. The magnitudes Of frequency components clustered about the
repetition frequency and its harmonics are read in decibels below the
full power output of the recovered signal. These decibel readings

show how well the filters perform in rejecting the unwanted high frequency
components. As is seen, the decibel levels of these particular filters
are quite satisfactory for most applications, especially when the load is
insensitive to signals at these high frequencies. - It might be pointed
out, though, that these filters all have an ultimate attenuation slope of
12 decibels per octave, and if an application required more filtering
than that of the filters used, constant-resistance or L-C filters may be
used which have an ultimate attenuation slope of 18 decibels per octave,
24 decibels per octave, or higher.

In Tables X and XI (for transistors) the dissipation column is
omitted because the derived equations for dissipation do not take into
consideration driving power. A total transistor dissipation may be
Obtained by subtracting the input power to the filter from the total input
power to the transistor circuit.

The 75 microfarad (not electrolytic) capacitor used as a return
for the filter in the push-pull circuits does not affect the magnitude or
phase Of the output voltage. ~ At 100 c.p. s. the magnitude of the signal
across this capacitor is 1 percent Of the magnitude across the 2, 000
Ohm load resistor, and the phase angle is such that the capacitor voltage
contributes essentially nothing to output voltage Observed with respect
to ground.

- Since the oscilloscope was used tO measure the value of Emin'
several percent error might be expected in the evaluation of theoretical
output power and efficiency. . For example, in Table V the actual value
Of Emin could easily have been either 45 volts or 50 volts. An Emin of

45 volts results in a theoretical efficiency which is Z. 5 percent higher

31

than Would be the case if Emin were 50 volts, and in a theoretical output
power which is 5. 06 percent higher.

The oscillOSCOpe was used for setting the value Of M. The time
base was uncalibrated and set so that one period Of the repetition
frequency covered 10 centimeters. The degree Of pulse duration
modulation may be observed, the value of M being the fractional part
that this is of the 10 centimeters. One realizes then that several percent
error may exist in calculated values which are a function of M, especially
with expressions that are prOportional to M2.

Actual efficiencies are a function of the average pulse duty ratio
or of the pulse duty ratio with no modulation. Variations Of these duty
ratios from O. 5 would cause actual efficiencies to be somewhat different
than the theoretical efficiencies which are based on a 0. 5 nO-modulation
duty ratio. This explains in part the magnitudes of percent deviation
fro'm theoretical efficiency in Table XIII. The power output is not
affected by a slight variation of nO-modulation duty ratio from 50 percent
as long as over-modulation does not occur. Then less variation should
be expected in percent deviation of actual output power from theoretical
than would be expected for percent deviation of efficiency. This is
verified by Table XIII.

Precautions were taken in the design of the single-ended push-pull
circuits of the vacuum tube type to prevent grid current Of the upper
tube from contributing power to the load. As with conventional amplifiers,
the equations for theoretical efficiency do not take into consideration
the contribution Of grid driving power. It is desirable to drive both
tubes to grid clamp, but excessive grid current is prevented by the use
Of grid current limiting resistors. These resistors are bypassed with
capacitors in order to maintain short switching times. The result is a
grid-leak bias which automatically adjusts to the level of grid voltage

drive and allows the grid to be driven to clamp voltage with a minimum of

32

grid current. With the transistor circuits the base driving power is
considered part Of the input power and no effort is made to have it be
negligible. ~As a matter Of fact, the on-value base current must be
enough to put the transistor into saturation if high output power, high
efficiency, and low distortion are to be attained. The single-ended
push-pull stages of the vacuum tube type which are tested then have a
relatively high Emin° This is why the push-pull circuits Of TablesxVII
and IX have a lower output power per tube than do some of the other
circuits. These push-pull circuits are capable Of more power output
merely by increasing the grid drive.

Output power Of the upper tube of single-ended push—pull stages
is a function of cathode current, and it is immaterial whether this be
plate current or grid current. - Efficiency would be higher than that
given by the theoretical expression because the input power to the grid
is not accounted for. In the test of the Class B push-epull circuit
(see Tables IX and XIII) it is estimated that the power contribution of
the upper grid amounts to about 3. 5 percent Of the power delivered tO
the filter input at full modulation. At lower modulation levels the
percent contribution would be slightly more. This helps explain the
high positive percent deviations Of efficiency for the Class B amplifier
in Table XIII.

The same effect would tend to make the efficiency deviations Of
Table XIII for the push-pull Class A amplifier (see Table VII) be
positive and high. However, the effect would be much less than in the
case Of the Class B amplifier because Of the larger Class A plate
current. ~An estimate Of the contribution of grid current power to the
filter for this circuit is 2. 25 percent. This percent contribution to the
filter input power should be the same at all modulation levels.

The major distortion components for a 200 c.p. s. modulation

frequency in Table XII are the same as those for M = 1 in Table IV.

This indicates that the shunt coil reactance at 200 c.p. s. is satisfactory.
When a 2., 000 Ohm load is used, a shunt coil should have at least this
much reactance at the lowest modulation frequency for which it is desired
to Operate the amplifier. Lower values Of reactance result in higher
distortion, as is indicated in Table XII.

When a 100 percent modulation signal is recovered with a filter
connected directly to the output terminals Of the pulse generator, there
is 1.81 percent total harmonic distortion. The oscillator which provides
the modulation signal has 0. 2 percent second harmonic distortion and
0. 1 percent third, with no other significant harmonics. The calculated
total harmonic distortion for pulse generator modulation is 1. 747 percent
with M = 1 (see p. 14 and p. 16). It is seen then that the distortion
produced in modulating the pulse generator is very close to that which
is theoretically predicted. A look at the distortion figures in Tables
IV through XI shows that the output stages Of the tested PDM amplifiers
contribute a negligible amount Of distortion when they are resistively
loaded.

The power output can never exceed the tube dissipation for a
conventional Class A amplifier, whether series-fed or shunt-fed. The
series-fed Class A PDM amplifier which was tested (Table IV) delivers
7. 5 watts Of sinusoidal signal to the load with a tube dissipation Of 4. 2
watts. The rated tube dissipation is 5. 5 watts. This strikingly illustrates
that more output power may be Obtained from a PDM amplifier for a
given tube dissipation than may be obtained from a conventional
amplifier.

. In Table VIII it is seen that the use of an L-C filter with a vacuum
tube PDM amplifier results in an efficiency that is independent of signal
level, as was theoretically predicted. With the transistor amplifier
and L-C filter Of Table XI, the efficiency falls Off slightly with signal

level. This is because Of fixed losses associated with driving the

34

transistors to saturation. These losses are not accounted for in the
derived efficiency equations for PDM amplifiers with L-C filters.
The efficiencies in both of these tables (Table VIII and Table XI) are
very high compared to efficiencies of conventional amplifiers,
eSpecially at the low distortion levels involved.

. The O. 37 milliamperes plate current for no modulation in Table
VIII is the plate current due to common on-time Of the tubes during
switching since it remains the same when the filter and load are removed.

The tube dissipation Of a Class A PDM amplifier with constant-
reSistance filter is constant and does not vary with the signal level.
This is not true of a conventional Class A amplifier. Use Of an L-C
filter with a PDM amplifier gives the lowest tube dissipation.

The transistors in the complementary pair circuit configuration
Of Figure 4-3 need not have matched or even similar characteristics.
It is only necessary that the square-wave at the input furnish the proper
base currents to turn each transistor on to saturation. This is a
definite advantage over a conventional complementary pair transistor
amplifier where a matched pair of transistors must be used in order .
tO Obtain low distortion amplification.

‘An interesting feature of the amplifier of Figure 4-3 is the fact
that it puts power into the collector supply voltage source at low signal
levels. It may be Observed in Table XI that there is a negative
collector supply current at zero signal level. This indicates that the
base driving power (which is constant for all signal levels) exceeds
the circuit losses by 6 milliwatts. It might also be noted that no
diodes are needed in conjunction with the L-C filter Of the amplifier
Of Figure 4-3. The transistor junctions themselves provide the
necessary diode action.

Though no data is given, it has been Observed with a Class A

vacuum tube PDM amplifier that there is good linearity between output

35

signal amplitude and plate supply voltage for a given modulation signal
level. This affords a means of controlling output signal level other
than by adjustment of the modulation level, and efficiency is not
sacrificed. A product circuit may be formed for multiplying voltages
by using one Of the voltages for the plate supply. The other voltage in
the product is that used for pulse duration modulation.

. In view Of the reasons given for the discrepancies in the tables,
it may be concluded for’PDM amplifiers that:

l. The circuits which were investigated and tested do perform
according to the derived theoretical performance equations.

2. Greater output power may be Obtained for a given tube or
transistor dissipation when amplifying sinusoidal signals than
may be Obtained with conventional amplifiers. This is true
whether constant-resistance or L-C filters are used.

3. The harmonic distortion contribution of the output stage is
negligible or extremely low when constant-resistance filters
are used or when Emin is small.

4. High efficiency, which is independent Of signal level, may be
Obtained with the use of L-C filters. This latter application
merits further investigation.

The "results of the investigations in this thesis suggest that further work
be carried out in the following areas:

1. The use Of’L-C filters with PDM amplifiers.

2.. The design and construction of a simple transistorized pulse
generator which is capable Of pulse duration modulation.

An aid to achieving a simple pulse generator design is the

fact that the efficiency Of PDM amplifiers with L-C filters

is independent of modulation level, therefore the pulse
generator need not be capable of 100 percent modulation.
» Recovery time may then be allowed for the monostable circuits

of the pulse generator. An effort should be made to achieve

36

relatively high efficiency levels for the whole system,
including modulator, pulse generator, PDM Output stages,

and low-pas 8 filter .

37

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46

Table XIII. Deviation of Measzired Values of Power Output and Efficiency
from Theoretical Expectation for Amplifiers with Constant-
resistance Filters.

 

Per cent deviation
from theoretical

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PDM. Amplifier Type Table M Pop 77
Series-fed Class A IV 1.0 +0.40 +7.6
0.8 -l.67 +5.2
0.5 ~1.l7 +5.67
Shunt-fed Class A V l. 0 -3.1 -5. 98
0. 8 -Z.95 -7.6
0.5 ~3.ZZ -6.3
Push-pull Class A VII 1.0 v =3. 06 +3.6
0.8 -3 5 -2.08b
0.5 -3.01 +5.55
Push—pull Class B IX 1.0 0 +5.03
0.8 +0.45 +5.63
0.5 +0.115 +7.92
Complementary Pair X 1. 0 -0. 32 +2. 31
Transistor Configuration 0. 8 +0. 25 +2.. 96
0. 5 0 +2.52

 

 

47

 

 

 

 

 

 

 

 

 

 

L
Input C ;: Output
G {)
Figure 4-1. L—C low-pass filter.
0
i::‘:::“:1: “ A
p O— pulses
pulse
generator
_<> e
47 K 10 K
1, 334 ohms
2 'K * To input of
complementary
10 K 10 K pair transistors
J-
- «.150 v. :- ‘

Figure 4-2. Attenuating network between pulse generator
and transistor amplifier.

 

 

 

48

 

IBM ’ b
075 U

 

 

 

 

 

 

 

 

 

 

 

 

 

V 2 Ebb = +20 volts
NPN
Low-pass
Filter R/z = 2,000 ohms
IBM
025
z: 75 mfd.

‘5 PNP

 

 

Figure 4-3. Complementary pair transistor configuration.

 

APPENDIX A
PULSE MODULATION LINEARITY

A- 1. Linearity of Modulation of the Monostable
Multivibrators (see Figiire 2-4) .

 

 

Circuit parameters are chosen to provide a zero duration pulse
when the grid voltage of V3 is at the maximum positive value of an
input signal which is of such amplitude as to provide 100 percent
modulation. This requires the negative plate voltage excursion of
V1 to be equal in magnitude to the cutoff voltage of V2 at this particular
instant of the modulation cycle. As the instantaneous value of the
modulation voltage at the grid of V3 changes, the clamp voltage for the
V1 plate changes. The instantaneous value of the demodulated signal

at the output of the low-pass filter is proportional to the pulse duration

tm'

 

 

V2 Grid Voltage

 

Figure A-l. Grid voltage waveform.

Em z (Em ' Eco) (A'l)

Em = (Es + Eco ‘1' Em) [1" 6- tm/RC]

49

 

‘E-

 

 

50

 

 

Let

xm = tm/RC (A-Z)
Then

Em a = l _ 6 .){1n
Es + Eco + Em
Em
= 1 a ,
xrn 1111 +Es+Eco] (A3)

Let Ernm be the maximum value of Em and let Em vary sinusoidally
in the range 0 _<_ Em g Emm at a frequency which has a long period

compared to the longest pulse duration. Then

 

Em = E__r_§_;_n (1 + cos wmt) (A-4)
- Emm(1 + C08 ‘9) _
xm .. 1n [1 + 2(Es + Eco) —~ (A'S)
Emm(1 + cos 0)

tm=RCln[1+

 

2(2. + Eco) ”'6’

The five point method of finding distortion may be used to find the
percent harmonic distortion for the. second, third, and fourth

harmonics . 2 Let

x. = x... (0°)
x2 = xm (600)
x3 = xm (90°)
x. = xm (1200)

x5 = xrn (1800)

Then since x5 = O,

x 1
Ao=-61- ‘1'“;(3‘2‘1'3‘41'3‘3

x

A1331" 30‘; 3‘4)
5.1.-331.

A2=4 z
x 1

A.=-g— --_.;<x.-x.)
maul. 25;.

A4-12 3(x2+}c4)+2

 

 

 

51

D2 = 12?; 100%

D3 = %x100%

D4 = 32-3.: 100%

D: ’D22+D32+D4z+'°

 

In most cases EC0 may be neglected since it is usually very small

compared to Es'

 

 

 

APPENDIX B

COMPARISON OF SERIES-FED CLASS A AMPLIFIERS

B-l. ‘Series-fed Class A Conventional Amplifier.

 

It is assumed that the bias is chosen at the value which allows the

maximum possible output voltage amplitude for a given R.

I'V
m

 

 

p——————-

—--

 

R 1m ., - - -
Ebb
.. Emin

Figure B-l. Series-fed Figure B-2.

Class A conventional amplifier.

 

I _ E:bb " Emin
m R
Eb = Ebb +2- Ernin

Instantaneous signal power delivered to load R is

 

 

 

Ebb " Emin - 2
pm = [M ( 2 ) smwmt]
R
Let
9* r: wmt
Then
13(9) = Mz(Ehb - Emin)z Sinz '9

4R

Average signal power delivered to R is
1 17
P. = g I. panda
o
Mam}... - Eminfi
BR

52

Eb eb 'Ebb

Load line .

(B-l)

(B-Z)

(13-3)

 

 

53

Average tube current is

... I1’1'1 : Ebb " Emin

 

 

 

Idc - 2 2R (5'4)
Power supply power is
- E
Plate circuit efficiency is
.130
"AR ’ 138
z
_ 15.4. [ 1 .. 5331111.] - (13-6)

Ebb

Maximum Pd occurs when M = 0, and for the best utilization of the

 

tube this maximum Pd should be equal to the rated plate dissipation.

 

. E - - M3
Pd = P. - Pdc - P0 =, bb4REmm[Ebb + Emin - 'é—(Ebb ' EminH

Mammmfi Pd _(Ebb- E11113) (Ebb + Emin) _ Pdr (13-7)

P = M2(Ebh - Errlin)
° 2(Ebb + Emin)

”(Ebb 5min)

Pd.- (B-8)

 

 

 

 

 

 

 

 

Maximum P P (3'9)
°= 2“"3191; + Emin) dr
B-Z. Series- fed Class A PDM Amplifier with Constant-
resistance Low- -pass Filter.
eb A
Ebb‘i
Low—pass
Filter R
g £— Emin . .
4 . it + t
Ebb 0 T1 T
Figure B-3. Series-fed Class A Figure B-4. Platevoltage
PDM Amplifier. waveform.
Let
T
a z T}- (B'lo)

54

The voltage waveform at the input of the low-pass filter is identical to
the plate voltage waveform of Figure B-4 minus Ebb. The time

function demodulated voltage at the output of the filter is then

80(t) = " O. (Ebb - Emln) (B-ll)

For Class A sinusoidal pulse duration modulation

1

(i=2- (1+Msin-9) , (B-IZ)

 

1 .
eo(t) = - -2- (1 + M 8111 O) (Ebb - Emin) (B-13)
. e t
1m) = Jig—’- -~ (3.14)
The average tube current is then
-1 2n,
Idc — 7 IO 10(t)dt
E - - 217
: 1 '
41rR f0 [ +M51n-9]d-G
Ebb " Emin
= -1
2R (B 5)
Power supply power is
E — E ' E

2R

Instantaneous power delivered to load resistor R at the output of the

filter is

_ [3005112
P“) - ‘71—"

- . z
(Ebb4R Emm) (1 + 2M sin-9 + M2 sinz 9)

 

Average power delivered to R is

I 1 211'
Pop = “—211 L, 13(0)“?

55

(Ebb ' Em' )2
1 8R 1n (2 + M2)

Pdc + Pop .-.. IécR .+ Pop

, 2
=(Ebb'Emm) + P

 

 

 

4R 0P
2 _ . Z
, 2
Maximum Pop = g- ): (Ebb REmm) (B-IB)

Plate circuit efficiency is

 

P M2 Em-

If Emin is set equal to zero, Equation (B-19) expresses the efficiency

of the filter and load combination. Therefore

M2
anR = T (5-20)

and the tube efficiency is

m; = [1 "gain—1 (13-21)

”AR = anR'It (B-ZZ)

Tube dissipation is independent of M.

The input power to the filter is

p =_13_op_ = W
"fAR ZR

The tube dissipation is

 

E _ .
Pdp = PS - Pt bb ZREmln [Ebb " (Ebb " Emin’]
= (W (3-23)

 

56

1:- best utilization of the tube, circuit conditions should be chosen so
1: Pdp = Pdr' Then from Equation (B-l7)

p = M2(Ebb - Emin) x (Em2 - Eminmmjn
op 4 Emin 2R
M2 Ebb
= ~— --- -1 P 13-24
4 [ Eruin ] dr ( )

ssurning that the maximum current rating of the tube is not exceeded,
he maximum possible power output is

1
4 1n
By using the full rated plate dissipation and the same Ebb and Emin
for the two types of amplifiers, the Improvement Ratio for the PDM
amplifier over the conventional amplifier for any particular tube is

I. R. _ Maximum Pop __

_ a 1. _ .1. [Eb—b.— + 1]
Max1mumPo 2 Emin

(B-26)
The increase in power handling ability of the PDM amplifier over the

conventional amplifier is achieved by using a lower value load
resistance. If Emin' Ebb: R and M are the same in both types of
amplifiers, the output power and plate circuit efficiency are the same

for both; however, the plate dissipation of the PDM amplifier is less
than that of the conventional amplifier by the factor

Pdp = Pdp g : Ernin
Pd (1 ' "Anlps

_ '11 z

(5—27)
Ebb + E'min - 7- (Ebb - Emin)

The tube dissipation of the PDM amplifier is the same for all values of

M. The conventional amplifier has its greatest dissipation when M = O

 

 

 

57

and it must be designed to handle this. The Dissipation Ratio is

 

D.R. = Maximum Pde = Esm- (13—28)
Mammum Pd Ebb +7 Emin

From Equation (B- 18) it can be seen that the lower the value of

R, the greater will be the maximum output power for given values of
Ebb and Emin- The lowest permissible value of R Swill then be limited
either by maximum allowable plate current Imr or by the rated plate
dissipation Pdr' whichever results in the higher R value. The plate
voltage waveform of Figure B-4 and the input resistance R of the low-

pass filter result in an on-time tube current of

= Mm. (3-29)

Im R

Since Ln must not be greater than the rated maximum plate current

Irnr'

R > W. (13-30)

And since Pdp must not be greater than Pdr: then from Equation
(B-23)
‘- E _ E . .

- (Ebb - Emin) Emin
..R _>_ ZPdr , (B-31)

In most cases Emin willbe partly determined by the value of R.
. For example it might be convenient to have Emin lie on a chosen grid
voltage line of the plate characteristic curves such as the zero grid

voltage line. Then Ernin will be determined by the point of intersection

 

 

 

58

of the load line and the chosen grid voltage line. Another line upon
which it might be convenient to have Emin lie is the diode line. 7‘
The intersection of this line and the load line would then determine
Emin- . The diode line and grid voltage lines pass through the origin,
and each line of practical use is usually straight enough to be expressed

algebraically as a line of slope K, where

Equations (B-17), (B-29) and (B-3Z) give

P .- MzEbbz x KZR (3-33)
01’ 8 (1 + KR):

Taking the derivative of FOP with respect to R and setting it equal to

zero shows that the maximum output power occurs when

1
R — '1?- (3‘34)

Power output becomes less as R is made smaller than the value given
in-Equationr(B-34). However, other considerations such as Equation
<(B-30) and Equation (B-3l) might require R to be greater than the
value expressed by Equation (B-34). .. For‘maximum power output
without exceeding the ratings of the tube, R should be chosen as low
as possible commensurate with the following inequalities, and Ebb

should be less than or equal to the maximum allowable plate voltage.

 

A Ebb ' Em'n
R > ———————-L-— (B-30)
‘— ‘ I'mr
R > (Ebb " EminlEmin (3'31)
— ZPdr
R > i (B 35)
— K

59

An alternative to the inequality given in Equation (B-35) is the following
inequality which results from a combination of Equations (B-29), (B-32)

and (B-35).

Emin: Egb- (B-36)

It is to be noted that the a-c plate resistance of the tube is no consider-

ation in the choice of R.

If the tube has a screen grid it is necessary to assure that the
screen dissipation rating is not exceeded. . The on-value screen current
may be obtained from the plate characteristic curves once Emin and Im

have been determined. Then instantaneous screen grid power is

p(t)= 0. leg E8g
Screen grid dissipation is then

I 'E 211 1
Psg =-§'gZ—;T'§'g' f0 0. d0: '2' Isg Esg (B-37)

 

 

 

APPENDIX c ,,

COMPARISON OF SHUNT-FED CLASS A AMPLIFIERS

C-l. ' Shunt-fed Class A Conventional Amplifie_r.

It is assumed that the bias is chosen at the value Which allows
the maximum possible output voltage amplitude for a given R. The
circuit configuration of Figure C- 1d is known as single-ended Push-
pull. 3 When the circuit balance and grid drive are proper for this
amplifier, the supply voltage Ebb to which the load resistor-is
returned furnishes no net power but must be able to support currents
of both polarities. In cases where d-c amplification is not required,
the Ebb supply to which the load resistor-is returned may be replaced
by a large bypass capacitor to ground. . This is commonly done. . The
time constant of the load resistor and capacitor combination would
have to be long compared to the period of the lowest frequency to be
amplified.

Each tube of the amplifier configurations of Figure C—l has an
effective plate load resistance of R (shunt-fed) and an effective supply
voltage of Ebb. Therefore Figure C-Z may beiused for evaluating
the performance of each amplifier‘in terms of individual tubes. . The
total output power of any amplifier is then equal to the product of
the number of tubes and the power output per tube. At full signal
level the maximum current and the minimum plate voltage are the

same for all tubes.

60

 

 

 

61

 

 

 

 

 

 

 

 

 

Figure C-Z. Load line.

.. Ebb
(a). (b)
s e
ZR
R2
‘: Ebb
(C) (d)
A.;.. Figure C-l. Shunt-fed Class A conventional amplifier
configurations .
I
1m o
In. « ----- 'F
ib I
l
ire. w- --J _______________ Q
2 l I
I I
I I
l A l G
_Emin ‘ . Ebb eb ZEbb-Emin

 

Nun

 

 

 

 

62

 

R = 2(Ebb " Ernin) (0,1)
Im
z(Ebb " Ernin) (C-Z)

 

Irn = R
Signal current into R is

1m = M 5—29 sin comt

Let
(CS-3)

Then
1(0) = M—If-L sin-0

Instantaneous output power per tube is
pm = R 12m

Average output power per tube is

 

11' Z Z 11'
P0: ‘11-'[0 p(-Q)d-9= RM4TrIm f0 sing-Gd‘G

z z z I . 2
M 1m R __ M (Ebb '- Emin) (C-4)

8 2R

—

The d-c power supply current per tube is

I E - '
Idc = _r2r_1___ ___ bbR Ernin (C-S)

The power supply power per tube is

E E - E '
PS : IdCEbb : W (C-6)

A separate consideration of the circuit of Figure C-ld (or this circuit
with the load resistor returned to a capacitor) yields a total power
supply power PsT given by the product of ZEbb and the d-c current of
the upper tube. Then the power supply power per tube is half of
1381‘ and is still given by Equation (C-6).

 

 

 

63

Plate circuit efficiency is

Z .
P° “—4-— [1 23;?) ((3-7)

"A: P: = 2
Maximum Pd per tube occurs when M = 0, and for the best utilization

of the tubes this maximum Pd should be equal to the rated plate

 

dissipation.
Maximum Pd = PS 2 Pdr (C-8)
P. - nip. - ig— [ 1 - Egg? 1 .1. (6-9)
1 Emin ] Pdr (c-10)

 

Maximum Po=-2- [1 - Ebb

C-Z. , Derivation of Average Current Furnished by a Constant- Volta e
Source Which is Periodically Switched to a Parallel L- R

 

 

 

 

Combination.
it)": e - . '2
£11,“) on
E T-
R * L l
:1:

 

off «Tl-v-
<———--—-— T——->|
(b)

 

 

(a)
(a) Switching circuit configuration. (b) Duty

Figure C-3.
cycle for switch.

The voltage across the L-R combination is not zero during the
off time of the switch but is determined by the L-R circuit and the

value of coil current at the instant of Opening the switch.

64

i(t) = iL(O) + f?- +

blt‘l

t for0<t<iT1

uiy-)=11}01+ +

tilt!)

E
135R T1
. . " 1:03 " T1)
IL“) = 1L(T1)€ for T1 < t < T

R
——'rr - T I
.Lm = 111m. L ’ = iL(0)

 

__ i(0) + i('I'1) 'I'
Iave " W 2 T x T1
. E . E E
11(0)+F— +1I(0)+ — + - Tl

- T #0 2

 

x T"
T

E

1L(T,)=1L(0)+ 1'3 Tl

R

" --- T n T
iL(T) ‘-' iL(0) = [11,(0) + 2 T1] 6 L ( I)
R
E -f' (T - TI)

 

11(0) = I“ , 1

=EL1+EL[1-€ , ]

Let B be defined by

Let the duty ratio of the switch be

T1

0"?

Then

 

(C-ll)

(C—lZ)

(c-13)

(C-14)

65

If L is chosen large enough so that the 1%: time constant is very large

compared to T, then (3 becomes very small.

Assuming

0 _<_ o. < 1 (C-15)
And

L .

— >> T ‘ ..

R ‘_ (C 16)

Then

 

 

 

 

 

I _1im eE[1+9_Bxl-I- “5(1’°)] [-

ave (3 -—> 0 R 2 1 -§('1-a) ;
_ (1E ' o 2 "j
_ R [ 1 + 2 x 1_ o. ] .

E o. .
_ 131 x 1-o. LCLI?)
The change in coil current during on-time T1 is given by
. . . E ET T

A1L(t) = 1L(Tl) - iL(0) = I: T1 = L'II'

= 0.13 11221 . (C-18)

It can be seen from Equation (C-18) that the change in coil current is
negligible during on-time when Equations (C-15) and (C—16) are
satisfied. The current during on-time is then constant and is given by

' E 1
Irn = -R- x 1- O. . ’ (0-19)

Since the on-current is a constant, the source voltage looks into a

 

 

resistive load during on-time. Let

1
O. = '2'" (C'ZO)
Then
' 2E
I = T (C-Zl)
E

 

66

~ Since the on-current I'm is twice the value that is taken by the resistor
alone during on-time, the magnitude of coil current during on-time is
equal to the resistor current during on-time. During the time the
switch is Open the coil current does not change, resulting in a resistor
current which is equal in magnitude but opposite in direction to the
resistor current during the on-time of the switch. This means that the
voltage existing across the L-R combination during off time is -E.

If a, is varied sinusoidally about an average-value of 1/2 at a rate which
is fast enough so that the coil current does not have time to change,

the average. current will still be given by Equation (6-22) and the
magnitude of the voltage excursions across the coil will be the same as
with no modulation. When a tube is used for the switch-(same circuit
as Figure C-la, but with a square wave grid drive to switch the tube on
and off) and when Equations (C-16) and (C-20) are satisfied, the on-time
voltage driving the L-Rci’rcuit is '(Ebb - Brain) in place of E, the on-

time tube current Ln is given by -I'm, and the d-c power supply current

is given by 'Iave' Then
-. 3 E - ' )

Im- R (C-23)

IdC = (M (6-24)

R
These equations are identical with Equations (C-2) and (C-5) respectively.
If the tube is switched on and off with a Class A PDM signal, Equations
(C-23) and (C—Z4) are still valid. A requirement to be met is that the
11 time constant be long compared to the period of the lowest modulation

R
frequency.

C-3. Shunt-fed Class A PDM Amplifier with Constant-Resistance
Low-apass Filter. 1

 

 

The amplifier of Figure C-Sb is not a particularly good configur-

ation. A very low repetition rate for the modulated switching voltage

67

 

  

 

 

 

 

 

i p:
R '"F Low-pas? . R R", how-pass R
Filter "'" “Filter
—. _‘1 ‘ , .f d|
Ebb _ ‘ E

(a) A (b)
s
I; *Iiow-pass-
$2 "" filter
I}...

' (C)

 

 

 

 

 

le
ha
0 .
‘ s
I
"U
m
m
I "I
who

 

 

 

 

WFigure C-4. Shunt-fed Class A‘ PDM amplifier

 

 

 

 

 

configurations... . ' '
R._.’J LOW-p388. ,R _....‘...
g Filter R2
' Iy-Ebb
'1' p (a)

 

 

Filter "

 

 

if" ..

 

 

% Rag—L.» Low-pa d: g

Eb V

 

 

 

 

(b)

4 Figure C-5. Transformer coupled push-pull PDM
amplifiers.

68

would be necessary because of output transformer limitations, even
with expensive high quality transformers. 'In addition this is not a d-c
. amplifier. The range of modulation frequencies is then very limited
and the circuit has little utility. '

The circuit configuration of Figure C-4b is preferable to that of
Figure C-4a. ~ In Figure C-4a the inductor would have to havea large
inductance (L/R large relative to the period of the lowest modulating
frequency) and an extremely high natural resonant frequency. Shunt
capacitance between coil turns presents a low capacitive reactance at
frequencies much above'resonance, and the inductor‘must present a
high reactance at all frequencies within'the spectrum covered by the
significant frequency components of the Fourier representation of the
PDM signal. The significant frequency components may range to 100
or‘ 200 times the cutoff frequency of the low-pass filter. When placed
at the output of the filter, the inductor need present a high reactance
only over the range of modulation frequencies,. the highest of which
is the cutoff frequency of the filter.

The Ebb supply to which the filter is returned in Figure C-4c
furnishes no net power when the PDM amplifier is modulated with an
a-c signal; however, it must be able to handle currents of both polarity.
When d-c amplification is not required the circuit of Figure C-4d may
be used in place of that of Figure C-4c. . The RC product must be long
compared to the period of the lowest modulating frequency.

The two single-ended amplifiers of Figure C-4 and the amplifier
of Figure C-Sa have plate loads which meet the conditions of the
derivation of Section C-Z. The constant-resistance low-pass filters
are the unbalanced type with a common lead between one input terminal
and one output terminal. Between the other input terminal and output
terminal there is d-c continuity through one or more inductors. The
on- current and d-c power supply current are given by Equations

(C-23) and (C-24). The on-time plate to cathode voltage for the tube

69

is Emin- The Off-time plate to cathode voltage is given by the sum of
the supply voltage and the voltage Ebb'Emin which existsacross the
L-R combination during Off-time. The off-time plate voltage is then
ZEbb'Emin- It is seen then that during on-time, the Operating point
of the single-ended PDM amplifiers is given by point F on the load line
of Figure C-Z. The on-current 1m is the same as the maximum current
Of a conventional shunt-fed Class A amplifier that has the same plate
load resistor R, the same Emin» and the same Ebb- During Off-time
the plate to cathode voltage is given by point G andis the same as the
plate voltage Of the conventional amplifier‘at the instant the grid is
driven tO cutoff. The point Q has no meaning for the PDM amplifier
since the Operating points are either G or F. It can be seen also that
the linearity Of the demodulated signal at the output Of the filter is
independent Of the characteristic curves of the tube.

- Each individual tube Of the amplifier configurations Of Figure
C-4 and Figure C-5 has an effective plate load resistance Of R (shunt-
fed) and has an effective supply voltage Of Ebb- For this reason
Figure C-Z may be used for evaluating the performance of these
amplifiers. Power evaluated on a per tube basis is then multiplied by
the number Of tubes to Obtain total amplifier output power. .The plate
voltage waveform for Figure C-Sa, for the single- ended amplifiers of
Figure C-4, and for the lowertubes Of the single-ended push-pull
amplifiers is given in Figure C-Qa . The plate to cathode voltage wave-
form for the upper tubes of the single-ended push-pull amplifiers is

given in Figure C-6b.

70

   

 

 

 

 

eb A ebA
ZEbb‘Emin‘ ZEbb-Emin
Ebb . Ebb 1»
Emin . , Emin ' I t
. . > t 1 1 > t
o T, T o T, T

(a) (b)

Figure C-6. Plate to cathode voltage waveforms.

 

 

_ Ebb " Emin
R — Im/Z (c-25)
1m z 2(EbbR- Emin) ((3-26)
0. =1 Tl/T (C-Z7)

The voltage waveform at the input Of the low-pass filter is identical
to the plate voltage waveform Of Figure C—6a minus Ebb- The time
function demodulated voltage at the output of the filter is then

eo(t) = ‘ (20 - 1) (Ebb - Emin) (C-28)
For Class A sinusoidal pulse duration modulation

1 .
a = -2- (1 + M81n0) (C-29)
ea“) = ' M(Ebb ' Emin)sin-9 (C'30)

For the single-ended PDM amplifiers the average tube current

is given by Equation (C-Z4).

Idc = EbbiEmin (c-31)

The total power supply power is given by the product of Ebb and

71

Equation»(C-3l). The power supply power per tube is the same since

there is but one tube.

Ebbuabb '" Emin) (C-32)

PS : PST R

The instantaneous power delivered to the load resistor R is
leg" my M2(Ebb - Emin)231nz'9

W) = R ~ R

Average power delivered to load R is

 

 

1 217
Pop — 5;— IO pwm
z _ z
. 2
Maximum Pop = (Ebb ZREmin) (CL-34)
Plate circuit efficiency is given by the ratio Of POp to PsT'
_ M’- [ 1 _ Emin] (c-35)
nA - 2 Ebb

The filter efficiency is found from Equation (C-35) by setting Emin = 0.

M2
an : "2—— (C‘36)
The tube efficiency is
Emin
n = 1.. .————.— (C-37)
t Ebb
"A '-' "m "t (C-38)

The tube dissipation is independent Of M. . The input power to the

filter is

72

 

P = POP = (Ebb ‘ Emin)z
t an R

The tube dissipation is

E -E - -
Pdp: PS .. Pt -_- ( bb Rmin)Em1n (C-39)

For best utilization Of the tube, circuit conditions should be chosen

SO that Pdp = Pdr' Then from Equations (C-33) and (C-39)

L43 Ebb

 

 

: - 1 ..
pOp 2 Emin ]pdr (C 4O)
1
Maximum Pop = -2-[ :23: - llpdr (C-4l)
in

By using the full rated plate dissipation and the same Ebb and Emin
for this amplifier and the conventional shunt-fed single-ended amplifier,
the‘Improvement Ratio for the PDM amplifier over the conventional

amplifier for any particular tube is

 

__ Maximum Pop __ Ebb
I'R° - Maximum Po _ Ernin (C 42)

As was the case with the series-fed Class A amplifiers, the increase
in power handling ability of the PDM amplifier over’the conventional
amplifier is achieved by using atlower value load resistance, so that
the tube dissipation in both amplifiers is the same. - If Emin, Ebb

and M are the same for the two amplifiers and if the load resistor
used in the PDM amplifier is the same as that used in the conventional
amplifier, then the plate circuit efficiency and power'output are the
same for both; however, the plate dissipation Of the PDM amplifier is
less than that Of the conventional amplifier by the ratio Of Pdp Of

Equation» (C-39) to the Pd Of the conventional single-ended amplifier.

 

 

Pdp ___ Pdp : Ezmin (CZ-'43)
Pd (1 - 77A)Ps E [1_ _1\_4_ ‘1 _ Emin )]

73

The tube dissipation Of the PDM amplifieris independent of M, but
the tube dissipation Of the conventional amplifier is highest when
M = O and'the amplifier must be designed to accommodate this great-

est dissipation. The Dissipation Ratio is then

Maximum Pdp __ Emin

D°R° = Maximum PdT' _ Ebb

 

(CI-44)

For the push-pull amplifier of Figure C-Sb one tube is conduct-

ing while the other is Off, and either tube during. conducting time has
I“Ebb “ Emin)

an on- current of R

. .The d-c power supply current is

then

2 E - E '
Idc : ( bbR min) (C‘45)

The total power supply power is given by the product Of Ebb and the

d-c power supply current.

PsT ___ 2Ebb(ERbb -' Emin) (046)

The instantaneous power delivered to the load is

Z
p(t) = 2[e0ét)]

Reference to Equation (C-30) yields

2M2(Ebb — Eminfsinze

p(t) = R

Average power delivered to the load is

 

 

 

1 217
PpT — 2 1T Io P(‘9)d9
z . , z
-.- M (EbbR Emm) ((3-47)
, 2
Maximum PpT = (Ebb REmin) (C-48)

74

Average output power per tube is

_ _ M2(Ebb - Emin)z
Pop“ 2 PpT ' 2R »(C-49)

Plate circuit efficiency is given by the ratio Of PpT to PST

M2 *E -
nA=—Z-[1--§§§-1 <c-so1

This is the same as that given in Equation (C-35). The filter
efficiency and the efficiency of the tubes are as given by Equations
(C-36) and (C-37) respectively. The dissipation per tube is as given
by Equationv(C-39) and the output per tube within its dissipation
rating is given by Equations (C-40) and (C-4l). A The Improvement
Ratio with maximum utilization Of the tubes is given by Equation
(C-42). ‘If the PDM amplifier and conventional amplifier are Operated
with the same R, Emin, and Ebb, the tube dissipation Of the PDM
amplifier is less than that of the conventional amplifier by the ratio
expressed in Equation (C-43), and the Dissipation Ratio is given by
Equation (C-4'4).

. For the single-ended push-pull amplifiers the on-time current
for the individual tubes is given by Equation (Cv26). The‘Ebb supply
furnishes nO net power and nO d-c current. . The average power supply
current for the ZEbb supply is the same as the average current of

the upper tube and is given by

1 2:"
IdC = 'Z—T'r— f0 Im(1-a)d-9

 

 

Ebb -' Emin 2" L _ _IL .
TTR Io ( 2 2. SinO)d-0
= Ebb - Emin ‘ (C‘SI)

R

The total power supply power is given by the product of ZEbb and the

75

current expressed in Equation (C-Sl).

ZEbb(Ebb -‘ Emin) (C-SZ)

PST: R

The instantaneous power delivered to the load is

p(t) = Z [ egt) 12
Average power delivered to the load is then the same as is given in
‘Equations (C-47) and (C-48). 'Average output power per tube is then
as given inEquation (C-49). ~ Plate circuit efficiency is as given by
Equation (C-SO). Filter efficiency is given by Equation (C-36) and
the efficiency Of the tubes by (C-37). The dissipation per tube is given
by Equation (C-39) and the output per tube within its dissipation rating
is given by Equations (C-40) and (C-41). Maximum utilization Of
tubes results in an Improvement Ratio which is the same as that
expressed by Equation (C-4Z). By Operating the PDM amplifier with
the same R,- Emin. and supply voltage as is used in a conventional
single-ended push-pull amplifier, the tube dissipation Of the PDM
amplifier is less than that Of the conventional amplifier by the ratio
expressed in Equation (C-43), and the DissipationRatio is the same as
that Of Equation (C-44).

, The following equations then apply tO all the PDM amplifier

circuits illustrated in Figure C-4 and Figure C-S.

 

 

 

 

1m = 2(Ebb ;%in) (053)

Rs. z Ebb(E‘tI>;o - Emin) (054)
M2 E - - 3 M2 E

POp :: ( beR Emin) = 7 [ Elfin .. ”de (c_55)

. Z
M aximum P W :

1
Op _ ZR '2' [ - 1]Pdp (C-56)

76

 

”A =._I\1IZ_2_[1_‘§_E}1:1§1] (c-57)
”LA: 3;; (c-ss)
nt = 1 _gfiLIL ((3-59)
Pdp = (Ebb - Eénin)Emin . (c-eo)
1.3. = 2% «2-61)
D.R. = %“;§‘— (0'62)

When R is chosen tO provide maximum utilization Of the plate

dissipation rating, the following equations are valid.

M2 Ebb
Op 2 [ Emin ] dr ( )
Maximum P 1 E———-bb (C-64)

0p: :7:[ Emin.1]1:3dr

Equation (C-Sés indicates that the lower the value Of R, the
greater will be the maximum output power for given values Of Ebb
and Emin' The lowest permissible value Of R is limited by either
*maximum allowable plate current Imr or by the rated plate dissipation

'Pdr» whichever'results in the higher R value. - Since

1m <_ Imr
Then from Equation (C-53)

R > MERE" Emir). (C-65)

" Imr

 

Since

Pd 5. Pdr

P

77

Then from' Equation (C-60)

R: (Ebb " :?in)Emin (C-66)
r

The discussion in Section B-Z of Appendix B relative to Equation (B-32)

is equally applicable here. Let
Im = K Emin (C‘67)

Equations (C-53), (C-55) and (C-67) give

_ M_2Ebb7‘ KZ R
Pop- “—2—" X W (068’

Taking the derivative of Pop with respect to R and setting it equal to

zero shows that the maximum output power occurs when

2
R — }? (CI-69)

Then in order to Obtain maximum output power-without exceeding the
ratings Of the tube,. R should be chosen as low as possible with due
consideration for the following inequalities, and Ebb should be

chosen in accordance with the recommended plate voltage for the tube.
The peak plate voltage ZEbb - Emin should not exceed the rated peak

plate voltage Of the tube.

 

 

V_ R > 2(Ebb ' Emin) (C-65)
- Imr

R > (Ebb " Emin)Ernin (Ci-66)
"' 13dr

R '> E (C 70

_ K " )

An alternative to the inequality given in Equationi(B-70) is the follow-
ing inequality which results from a combination Of Equations (C-53),

(C-67) and (0-70).

78

E
Emin _<. -—]§b (071)

As was pointed out for the series-fed Class A PDM amplifier, the a-c
plate resistance of the tube is no consideration in the choice Of R.

The considerations for tubes with screen grids are the same as
those discussed in SectionB-Z of Appendix B for series-fed amplifiers,
and Equation (B-37) applies. For single-ended push-pull configurations
Esg must be considered to be the screen to cathode voltage Of a tube
when it is on or conducting.

In conventional single-ended push-pull amplifiers provision
must be-made for keeping the screen grid Of the upper tube at a fixed
voltage with respect to the cathode or degeneration will result. This
problem does not exist with single-ended push-pull PDM amplifiers
since the tube currents and voltages dO not change during on-time.

This is a definite advantage for PDM amplifiers Of this circuit configur-
ation. - A disadvantage however is the fact that the screen to cathode
voltage Of the upper tube is very high during the Off-time of the tube.
Screen dissipation is no problem during the time the tube is cut. off as
there is no screen current, but the large interelectrode voltages
require the use Of tubes with very high screen‘voltage ratings.

In addition, the cutoff voltage for a tube with a given plate to cathode
voltage will be higher than would be the case witha lower screen tO
cathode voltage, and a greater amplitude gridedriving signal will be

required in order tO assure cutting off the tube.

APPENDIX D
COMPARISON OF CLASS B AMPLIFIERS

D-l. Class B Conventional Amplifier.

 

Figure D-l shows the two important circuit configurations for

Class B push-pull amplifiers.

 

1
N
P!
U
U

 

8..
fl

 

 

 

 

 

 

:- . 'Ebb
(a) (b)
Figure. D-l. (a) Common‘push-pull configuration.
(b) Single-ended push—pull. '
ZEbb ' r .
1m \
' 1m ..-- F
e §R - =
1 l
.13 .
1‘3 ' l “ G H
T __ Emin eb bb ZEbb-Emin

Fis'aers-Z: 7' Sinéle"-1€nded Figure D-3. - Load line.
push-pull 3;"? amplifier. -

When the circuit Of Figure 13-15 has an appropriate a-c driving signal,
the Ebb supply t-O‘which the load resistor is returned furnishes nO d-c

current and no net power; however, it must be able to handle currents

79

80

of both polarities. When d-c amplification is not necessary, the
circuit of Figure qu is commonly used in place of that of Figure D-lb.
For all the amplifiers shown, each tube individually has an effective

plate load resistance of R and an effective supply voltage of Ebb-
— Min
I - R (D- 1)

The instantaneous power delivered to the load is

t) z [M(Ebb - Emin)Sin‘°mt]z

 

 

 

 

p( R
Let _

‘9 = (amt (D-Z)
Then

M2 E - - zsinz-G

P69) = ( bbR Emin) -

Average power delivered to the load is
- 1 17 __ Mz(Ebb " Emin)z -

PoT - g I, p(0)<19 - 2R (D-3)

Average tube current is
_ MIm 7" . : MI!!! __ M(Ebb' Emin)
Idc - 2 1T f0 sure-d9 1r - - - YwR _ (D-4)

The average power supply current for Figure D-la is twice the value
given by Equation (D-4). Total power supply power for Figure D- 1a

is given by the product of Ebb and the average power supply current. ~

2M E - - E
PST : ( hwaEn'un) bl_b (D_5)

For the single-ended push-pull configurations the total power supply
power is given by the product of the average current for the upper
tubeand ZEbb. This power thenwis also given by Equation (D-S).

Plate circuit efficiency is

=3..le MIL -E_Ln_i_l_1_ ..
"B Par 4 [1 Ebb (D 6)

Total power dissipated in the tubes is

81

_ 2M(Ebb'Emin)Ebb M2(Ebb'Ernin)z

 

 

 

PM = PsT ' PoT ~ «R ' 2R

(D-7)
To find the maximum Pdt, set £131:- = O and obtain
2 .
M = D-8
Emin ( )
1T[ 1 - ---]
Ebb

Two cases arise, since we must have

 

0 E M _f 1 (D-9)
Case 1
When[ 1 - Egg-S- ] _>_ -— , maximum PdT occurs at
2
M = 13-10
W [ 1 _ Emin ] ( )
Ebb
Then
. 1
Maximum PdT = (1 - nB) PST = —2— PST (13-11)
2
It R
The dissipation Pd per tube is half the total dissipation PdT.
Then
. _ E 7'
Max1mum pd - fig— (p-13)

The dissipation per tube should not exceed the rated dissipation

Pdr per tube.

Maximum Pd : pdr (D— 14)
E ’- ‘
Pdr 3 7.31%" (13-15)

For best utilization of the tubes the maximum Pd should be

equal to Pdr .

2
Maximum Pd: %-§—— = Pdr (13-16)
_ 1 _ M2(Ebb- 1);
P0 - Z PoT - 4R (D-l?)

82

M2 1T2 Emin z
— T- [1- *‘E—gg] pdr (13-18)
2
. _ _1_T_ _ Em' 2 v _
Maximum 20- 4 [1 Ebb ] Pdr (D 19)
Case 2:
Emin E -

When[ 1 - Ebb ] _<_ 17 , max1mum PdT occurs at

M = l (D-ZO)
Then

Maximum PdT (1 - ”BlpsT

 

2 Ebb'E ' n
= ( min) [Ebb ' "' (Ebb-Emin)] (13-31)
11R 4
The dissipation per tube should not exceed the rated dissipation

per tube.

E -E ‘
r z ( “,Rm‘” [Ebb - {f (Ebb-3min” (p-22)

 

Pd

For best utilization of the tubes the maximum Pd should be equal

 

Maximum Pd = Pdr (D-Z3)
— 1 ._ M2(Ebb' Emin)z
Pox. " ’2' P01“ ’ 4R " (D’PA)

M2W(Ebb" Emin.) g
4[Ebb - 7:: (Ebb'Emin)]
fiEbb-Egiin) w x pdr (D-26)
4[Ebb - Z (Ebb' Emin”

x Pdr (D-ZS)

Maximum P =

Case 1 and Case 2 both give the same result when

- 2
[I'EE’BSE]:? (D-Z?)

Then

83

 

 

E z.
Pdra fi‘i— (D-28)
2 Z
pOT = 3%,gbha ZMZPdr (D-29)
. 2 2
Po - Mfifil = szdr (13-30)
2
P3T — 9-32?— = 4Mpdr (11—31)
2 Z-M ME 3 '
PdT- ( "I; bb = ZM(2-M)Pdr (13-32)
Z-M ME 2
Pd - ( 132R bb = M(Z-M)Pdr (D-33)
. _ 2E ’-
Max1mum PdT VIE-h— ZPdr (D-34)
. E ’-
Max1mum Pd = $z%- — Pdr (D-35)
Maximum PO - Pdr (D-36)
Maximum PoT - ZPdr (D-37)
M
, 1
Max1mum ”B: -2- (13-39)

D-Z. Class B PDM Amplifier with Constant-resistance Low-pass
Filter. P W I v

 

For reasons discussed in Section C-3 in conjunction with
Figure C-Sb, the PDM amplifier of Figure D-4a is not a particularly
good one.

When d-c amplification is not required, the Ebb supply to which
the filter is returned in Figure D-4b may be replaced by a large
capacitor to ground. The discussion of Section C-3 relative to the
Ebb power supply requirements and the use of a capacitor applies

equally well to the circuit of Figure D-4b with Class B modulation.

84

 

 

4b nga LOW-page- R
Filter L 2

$7_ Eh

(a)

 

 

 

 

 

 

 

 

 

  

 

 

’ R"——>. 'f‘low- pas s-
g ' Filter R

(b)

Figure D-4. (a) Transformer coupled Class B PDM
amplifier. (b) Single-ended push-pull Class B PDM
amplifier. '

Each individual tube of the amplifier configurations of
Figure D-4 has an effective plate load resistance of R and an effective
supply voltageeof Ebb- Thus Figure D-3 may be used for evaluating
the performance of these amplifiers in terms of individual tubes.
Output power evaluated on a per tube basis is then doubled to obtain

total output power. During on-time the operating point of the PDM

85

amplifier tubes is given by point F on the load line of Figure D-3.
The on- current 1m is the same as the maximum current of a con-
ventional Class B amplifier which uses the same load resistor and
has the same Emin and Ebb. During off-time the plate to cathode
voltage of the PDM amplifier tubes is given by point G or H,
depending upon whether the -other“tu’be.7istonducting or not.

As was. the'eas'e with Class'A- FDIC/lilamplifiers,Iit‘ can be seen '4'

that the linearity of the demodulated signal at the output of the filter
is independent of the characteristic curves of the tube. During the
half cycle of modulation in which a tube is switched on and off, the
on-time plate to cathode voltage is Emin and the off-time plate to
cathode voltage is Ebb' - During the other half of the modulation cycle
there are plate to cathode voltage excursions between Ebb and

ZEbb' Emin but the tube does not conduct.
9 = (amt (D-40)
0.: -;-[1+Msin-G] (D-4l)
The magnitude of voltage excursion at the input to the filter is
Ebb - Emin during both half cycles of modulation, the sign being

different during each half cycle. The time function demodulated voltage

at the output of the filter is then

60(9) -(2a - 11<Ebb - Ernin) (13-421

- M(Ebb - Emin)sin-G (13-43)

Average tube current is given by

 

Idc = - 7,1}?— ffeotome (13-441
- M(Ebb'Em° )
- "R 1“; (D-45)

The total power supply current for Figure D-4a is twice the value

given by Equation~(D-45). The product of Ebb and the total power

86

supply current yields the total power supply power for Figure D—4a.

ZMEbb(Ebb- E‘min) (D-46)
TTR

 

PST:

The total power supply power for Figure D-4b is given by the product
of 2Ebb and the d-c current of the upper tube since the Ebb supply to
which the filter is returned furnishes no net power. This total power
supply power is then also given by Equation (D-46). The instantaneous

power delivered to the load is

[eo(t)]z : MZ(Ebb- Emin)zsin20
R R

 

p(t) =

Average power delivered to the load is

1 217

 

 

 

ZTr
_ M2(Eb§)I;Emin)z (D-47)
Maximum PpT = (Ehbgfiminfi (D-48_)_'
Plate circuit efficiency is given by the ratio of PpT to PST

The filter efficiency is found from Equation (D-49) by setting Emin = 0.

Mn

 

 

The efficiency of the tubes is
‘ Emin
n : 1 - (D-Sl)
t Ebb
nB = nant (D-SZ)

The tube efficiency is independent of M. The input power to the filter
is
P T : 2M(Ebb-Emin)z
t 17R
"£13

 

(D-53)

87

The tube dissipation is

2M E - - '
PdT = PST _ Pt .3 ( bbflflfmin)Em1n (D_54)

Maximum PdT = “E ’Emin) 'n (D-55)

11R

This maximum total tube dissipation should not exceed twice the rated

dis sipation per tube

Maximum PdT _<_ ZPdr (D-56)

Maximum Pdp < Pdr ’(D-57)

For best utilization of the tubes, circuit conditions should bechosen

so that maximum Pdp = Pdr' Then from Equations (D-47) and (D-55)

 

 

Mzw Ebb
PpT - 2 [ in- 1] Pdr (D-58)
_ M v Ebb
Pop _ 4 [ Emin- 1] Pdr (D-59)
. 11 E
Max1mum PPT = T [ £111 - l] Pdr (D-60)
Maximum Pop = -;1T—[ PEib - l] Pdr (D-6l)
1n

There are two cases to consider in evaluating the Improvement Ratio
for the Class B PDM amplifier over the conventional amplifier. In
each case the same Ebb and Emin are used for both amplifiers and
the maximum tube dissipation for each amplifier is the same, though
the maximum tube dissipation may not occur at the same fraction of
maximum permissible signal level in the two amplifiers. As with _
the Class A amplifiers, the greater output power ability of the Class B
PDM amplifier over the conventional Class B amplifier results from

using a lower value load resistor.

88

 

 

 

 

Case 1:
Emin Z
- --—-—— > -" D’62
1 Ebb _ fl ( )
Maximum P T EbbZ
LR. = . D = D-63
1 Mammuxn PoT TrEmin(Ebb'Emin) ( )
Case 2:
Emin Z
- —— < — - 4
1 Ebb - 11' (D 6 )
Maximum PDT Ebb 7' 7‘
. . = . a — = —— 1 - —— —- D-65
I R 2 Max1mum PCT Emin( 4 H 4 ( )
When
1 _ Emin : _2_ (p-66)
Ebb 1T
Then
LR. = Ebb—- = ———’1—— = 1.376 (D-67)

23min 2(11-2.)

The Improvement Ratio is greater than one under all circumstance
since Emin can never be chosen greater than Ebb- A choice of

Emin = Ebb results in an Improvement Ratio of one, but the output
power of either type amplifier is then zero. In a practical amplifier
of either type Emin will probably never exceed half of Ebb. A choice
of Emin = 0.1 Ebb is practical and results in an‘I. R. = 3. 537. When
Emin: Ebb, and M are the same for both types of amplifiers and when
the load resistors are the same, the output power and plate circuit
efficiency are the same for both. , Two cases must be considered in

an evaluation of the dissipation ratio.

Case 1:

1 - Emin. > .12? (D-62)

D.R.1= Mammum PdD _—. ___—W E (E ‘E ' ) (D-68)

Maximum Pd bb

 

89

 

 

 

Case 2:
1 .. 15310 < .2. (D-64)
Ebb — 1T
EEEE Em = Emm w»
d Ebb - '4: (Ebb'Emin)
When
1 - Emin = :7: (D-66)
Ebb 1T
Then
D.R. = 2 ___1n = 2 -— = 0.7268 (D-70)
Ebb 11

When 1 - 1%?)363- _<_ f: the maximum tube dissipation occurs a; 2
maximum signal level for both types of amplifiers. When 1 - 3:111:33- > -;
the maximum tube dissipation occurs at a signal level which is
less than the maximum signal level in the conventional Class B
amplifier.

Equation (D-48) indicates that the lower the value of R, the
greater will be the maximum output power for given values of Ebb
and Emin- The lowest permissible value of R is limited either by
the maximum allowable plate current Imr or by the rated plate

dissipation Pdr, whichever results in the higher R value. ‘In accord-

ance with Figure D—3 it is seen that the tube current during on-time is

: M11 (13-71)

1m R
Since
Im 5. Imr
Then from Equation (D-7l)

R > Ebb ‘ Emfin

r (D-7Z)

In addition, the maximum dissipation per tube must not exceed the

rated dissipation per tube.

90

Maximum Pdp _<_ Pdr

Then from Equation (D-SS)

R: (W (D-73)

1T Pdr
The discussion in Section B-Z of Appendix B relative to Equation (B-32) .

is applicable here. Let

in = KEmin (p-74)

Equations (D-47), (D-7l) and (D-74) give
KZR
_ z 2 _

POP - M Ebb x (m (D 75)
Taking the derivative of Pop with respect to R and setting it equal to
zero shows that the maximum output power occurs when

- -1- (D 76)
' K

The load resistor R should be chosen as small as possible in accordance
with the following inequalities and Ebb should be chosen so that the peak
plate to cathode voltage ZEbb- Emin does not exceed the rated peak plate

voltage of the tube.

R: Min. (13-72)

Imr
R > (Ebb — EmjnlEmin (11-73)
- 1T Pdr
1
R: 1(- (13-78)

An alternative to the inequality given in Equation(D-78) is the following
inequality which results from a combination of Equations (D-7l),
, (D-74) and (D-78)

1
Emin _<_ 3 Ebb (D79)

As pointed out for Class A PDM amplifiers, the a-c plate resistance

of the tube is no consideration in the choice of R.

91

The considerations for tubes with screen grids are the same as
those discussed in Section B-Z of Appendix B and in the last two
paragraphs of Section C-3 of Appendix C. Equation (B-37) does not

apply however. Instantaneous screen grid power for one tube is

P“) = (20- 1)1‘58Esg for O i G _<_ 11

= 0 for n<e< Zn

The screen grid dissipation per tube is then

Peg PM 2}: [Ice - nee: M—K—I—ISfiES (D-80)

APPENDIX E

PDM AMPLIFIERS WITH L-C LOW—PASS FILTER

E-l. General Considerations in the Use of L-C Filters.

 

The driving point impedance of any L-C low-pass filter which
is used with a PDM amplifier should have a pole at infinite frequency
so that the quadrature currents delivered by the high frequency voltage
components of the Fourier representation of the waveform will be small.
This implies a series inductor. at the input of the filter. A disadvantage
of using any type filter with a series inductor at the input is that provision
must be made for continuity of current in this inductor, either through
the device driving the filter or through some element (not an R, L or C)
connected across the input terminals of the filter. The off-time
current at the filter input terminals is not zero. The current at the
input terminals of the filter changes during on-time and varies also
as a function of the instantaneous modulation signal. None of these
disadvantages are encountered when a constant-resistance filter is
used. However, a constant-resistance filter is not lossless. Except
for small losses associated with the actual elements used in the
construction of a filter, an L-C filter has an efficiency of 100 percent.
All of the plate circuit loss is then either in tube dissipation or as
unused power in the load. The series—fed circuit configuration comes
under the category of having unused power in the load since it has a
constant d-c load current in addition to the signal for any particular
modulation factor M.

Since the on-time current at the filter input is not constant, the
minimum plate voltage for the tube or tubes driving the filter is not
constant. The characteristic curves of the tube are an important

factor in determining the relation between on-time current and on-time

92

93

voltage. For this reason no general equations can be written for plate
circuit efficiency or maximum output power for a given tube dissipation.
Expressions for instantaneous efficiency and dissipation can be given

in terms of the minimum instantaneous plate voltage eb, min- The time
function volt age eb’ min is unknown until a specific tube and operating
conditions are chosen. The instantaneous power delivered by the tube

to the filter is

put) = (Ebb - eb,min)is(t)

The instantaneous power furnished by the power supply is

ps(t) = Ebbism
Then at any instant during the time of tube conduction the tube efficiency
is given by

=1-f_11..m.i_a_ (E4)

"t1 Ebb

During the time the tube is not conducting, the power supply furnishes

no power and efficiency is of no concern.

E-Z. Use of the Low Frequency Leg of a Constant-resistance
Filter.

 

 

The low frequency leg of a constant-resistance low-pass filter
constitutes an L—C low-pass filter. It has d-c continuity between
input and output and meets the requirement of having an inductor at the
input. In addition it has a voltage transfer function which is identical
to that of the constant-resistance filter from which it is taken. This is
the type of L-C filter which will be considered, even though the driving
point resistance over the pass band may not be as well controlled as
with some other types of L-C low-pass filters.

If the maximum value of eb, min is small compared to Ebb: the

waveform distortion at the output of the low-pass filter will be negligible.

94

Then a sinusoidally modulated PDM signal will result in an approxi-
mately sinusoidal signal current in the resistive load at the output

of the filter. If a single-ended circuit configuration is used, it will

be necessary to use Class A modulation. Some means must also be
provided for current continuity at the input of the filter. A diode
connected across the input terminals of the filter will provide for
continuity of current in the input inductor and will maintain the wave-
shape of voltage at the input of the filter during the off—time of the tube.

This circuit configuration is shown in Figure E-l.

 

 

 

 

 

_ Low-pas?
Filter Q R
e

Figure 15-1. Series-fed PDM amplifier with L-C
filter.

For sinusoidal Class A modulation

0. 541+ Msin-G) (E-Z)
9 = ”Int 1' .(E'3)

II

To evaluate the performance of the filter and load combination exclusive
of the tube, set eb’ min = 0. The resulting equations will quite
accurately specify the performance bf the amplifier when the maximum
value of eb, min is small compared to Ebb- Any exact analysis depends
upon the characteristics of the tube and can only be carried out once a

specific tube is chosen. Let

95

The instantaneous current in resistor R is

io(t)= LEE- : 132% (1+ MsinO)

The instantaneous power supply current is

Z
ism = a[ 'io(t)] = - ‘1 Ebb

 

The d-c power supply current is then

1 2 .
Idc= 5—,,— I “lemme
_ Ebb 2
- 8R (2 + M) (E-S)

The power supply power is

E 2
P5 = Ebbldc = :1]: (2 + M2) (IE-6)

 

Total power delivered to the load is

 

' _ R ZTI , 2
Pop — 2‘" f0 10 ('9)d'9
E 3 WE 2
= f]: + ———8Rbb Pdc + pop (E'7)
Signal output power is then
MZE 2
Pop = "é‘fihl— (E'S)

Efficiency of the filter and load combination is

P M2
: —-Q-P— : ———- ..
nflR PS 2 + M2 (E 9)
1
Maximum nflR = 3 (E-lO)

If eb, min can be held constant at the value Emin during the on-time

of the tube, the following equations specify performance.

96
E - - .
10(t) = ——-§m—bb2R 1“ (1 + Ms1n0)

az(Ebb -* Emin)

 

 

 

ism = - R
E _ .
Idc = bgREmm (2 + M2) (12-11)
E E - E - 2+ Mz

__ (Ebb ' EminV' Mz(Ebb - Emin)z

 

 

 

!
Pop - 4R + SR = PdC + Pop (E-13)
_ M2(Ebb - Emian w
- ~ P0 — M2 Emir.
"1R ‘ ”flRnt ‘ 12>s ‘ W) [1+ Ebb ] (13-15)
E .
= 1 _m1n_ _
”t + Ebb (E 16)
M2
anR = ___-22 + M (E-17)

If the single-ended amplifier configuration is used with a diode
across the input terminals of the filter, it is necessary that the tube
and load be series—fed. . However, as can be seen by Equation (E-lO),
the series-fed configuration does not offer as much as might be
desired in terms of efficiency. Push-pull and single-ended push-pull
configurations eliminate the d-c current in the load and provide higher
plate circuit efficiency. There is no merit in using Class B modulation
since Class A modulation provides an efficiency which is just as high.
When there is no modulation the power supply furnishes no power.

The push-pull configuration of Figure C-5b has the same drawbacks
when an L-C filter is used as are discussed in Section C-3 of
Appendix C. The single-ended push-pull configurations of Figure C—4
swould seem to be good choices for use with L-C filters; however,
even with one tube presumably on at all times, provision still must be

made for current continuity at the input to the filter. It would not be

97

efficient to meet this requirement by allowing or forcing the normally
off tube to conduct. During the normal off-time for a tube the plate

to cathode voltage is large, and plate current during this time would
result in high plate dissipation. This would lower the efficiency.

In addition the power handling capability of the amplifier would be
greatly reduced by the need to keep the plate dissipation within the tube
ratings. The use of diodes as shown in Figure E-Z and Figure E-3 is
preferable, but unless the ratio of the maximum value of eb, min to Ebb

is very small, waveform distortion will occur at the input to the filter.

 

 

 

 

 

 

Low-pass " R
L-C Filter 7

Ebb or bypas S

 

 

' capacitor to ground

Figure E-Z. Single-ended push-pull.

This happens because the diodes permit eb, min to be zero during part
of the normally on time of the tubes. With transistors.'§the ratio. of the
maximum value at eb’ min to Ebb is usually very smallfiand waveform
distortion is much less severe than with tubes. For the circuits of
Figure E-Z and Figure 15-3 the instantaneous efficiency of the tubes
is given by Equation (E-l). As was the case with the single-ended

amplifier, no general equations can be written for plate circuit

efficiency or maximum output power for a given tube dissipation.

Itis

eb,r

hold

tube

'The

 

"The

Sin.

the

Sin

81v

98

 

PNP i +Ebb

Low-pass ‘ ‘R
l L-C Filter '2-
_ NP

 

 

 

 

 

 

 

 

Lt
_ 1' 'Ebb-

Figure E-3. Complementary pair transistor configuration.

 

.u-

It is necessary to select a tube and determine the functional form of
eb,min- { For sinusoidal Class A modulation Equations (E-Z) and (E-3)
hold. If eb’ min can be held constant at Ernin during the on-time of the

tubes, then the instantaneous voltage across the load resistor is

30“) = (20 " l)(Ebb ' Emin) (13-18)
The signal power delivered to the load is

_ M2(Ebb " Emin)2
PpT ‘ R (Ia-19)

 

The output power per tube is

2 _ , 2

0P ZR

Since an L—C filter is employed and since there is no d-c current in

the load, the efficiency of the filter and load configuration is 100 percent.

Since eb, min is being held constant at the value Emina Equation (E-l)

gives the tube efficiency as

”t

The ove

The tot

The po*

For the
returne
supply

d-c cu:

The po
equalh
The d-
(E-Zé)

 

By re.

 

 

99
— Esau;

The overall plate circuit efficiency is then

B .

The total power supply power is

 

P 3 - -
PST : _21. = M Ebb(Ebb Emin) (E-24)
771 R

The power supply power per tube is

MzEbb(Ebb - E

mi 11)
2R (E-ZS)

PS=

 

For the circuit of Figure E-Z, the Ebb supply to which the filter is
returned furnishes no net power for sinusoidal modulation. The 2Ebb
supply then furnishes the power expressed by Equation (E— 24). The

d-c current furnished by this supply is

Idc ZEbb 2R (E 26)

The power supplies for the circuit of Figure E:-3 share the load
equally. Each furnishes half the power given by Equation (E-24).
The d—c current furnished by each is then also given by Equation

(E-26L

 

 

Nfizs - E - .E -
Pdp : PS _. Pop : ( beR mln) mln (E'Z?)
Maximum Pdp = (Ebb - §?in)Emin (Fl-28)

By referring to Equation (C-8)

_ MaXimum PdD ___ Emin (E_29)

D" R' — Maximum Pd- ZEbb

 

If, inst!
on-til’m
is mad.
Figure

througl

 

100

If, instead of holding eb, min constant at the value Emin during the
on-time of the tube, the ratio of the maximum value of eb’ min to Ebb
is made small, the performance of the amplifiers of Figure E-Z. and
Figure E-3 is expressed by setting Emin = 0 in Equations (Co 19)
through (C-26). Then
Idc = M_—:§bb

MZE Z
PST - —-,—,‘9-‘—°-— (Is-31)

(rs-30)

MZE Z
PPT = ——Rt-’-‘9— ~ (E-32)
MzEbbz
ZR

MZE 2
Pop 2 jib-_- (E-34)

PS : (E-33)

711 = 1 (E-35)

DI

F-l.

e(t)

Figt
to fi

Let

 

Le

Le

Tl

 

APPENDIX F

DETERMINATION OF NO-MODULATION DUTY RATIO FOR
MAXIMUM OUTPUT POWER AND MAXIMUM EFFICIENCY

F-l. Series-fedC1ass A PDM with Constant-resistance Filter.

rVfi’

 

 

 

 

 

 

 

 

e(t) A
.E ;
6“) Constant-resistance .
' Low-pass Filter
I- I d
‘ ’ ~ 4 t
0 T; T
Figure F-l. Waveform at input Figure F-Z. . Filter and load
to filter. configuration. '
T
= _.1.. -
a T (F l)
0 i u i 1 (F‘Z)
Let the duty ratio with no modulation be A
T .
k = __1_ l , -
T . (F 3)
0 _<_ k _<_ 1 (F—4)
Let
Let
0. = k (1 +_'Msin-9) (F-6)
Then
k(1+M-)_<_l (F-7)V
l
M_<_ .1: - 1 ‘ (F'B)

101

Instanta'

p<
Average

F”

The a-

The ef

Case

 

 

102

Power input to the filter is

E2
PM‘°E
l 211 E2
Pt - 3.7!, PM“; - ki—

Instantaneous output power to load R is

5 z
p(9)= “LE-L

Average output power is

I i 1 211'
Pop 3 2,", I0 P('G)d'9
kZEZ MzszZ
=T + "Tr: Pdc+Pop

The a-c signal output power is then

(MkEY‘

P0P -_- 2R

The efficiency is

P kMz

-_- JR _
anR Pt 2

Case 1:

(F-9)

(F- 10)

(F-ll)

(F-IZ)

(F-l3)

- (F-l4)

For a fixed value of M the output power and efficiency become

greater as k is made larger. The greatest possible output

power and efficiency occur when M = l.

1
k - '2'
E2
Mammum Pop -"'- TE—
1

Maximum anR = Z

(F-15)
(F-16)

(F-l7)

Ca:

.' 0

(IO

Th'

103

 

Case 2:
1
—Z- :k_<_1 (F-18)
M: 11: .. 1 . (F-8)
2 2. 2
MAR: 1%: : gig—)2- (1120)

The lower the value of k in the range given by Equation (F-18),
the greater will be the upper bounds on output power and
efficiency. Then the greatest possible output power and

l
efficiency occur when k = E .

 

1
k = 2— (F-Zl)
. E3
Max1mum Pop 2 8R (F-ZZ)
Maximum_ anR = % (F-23)

Therefore the best value of k in the range given by Equation (F-4) is

l
E. This value of k allows the greatest possible output power and the

highest efficiency. Let

 

 

1
= _ -2
k 2 (F 4)
Then
_. 1 1 ° F 25
(If? ( +M51nwmt) ( - )
M2
anR = '4— (F-26)
F-Z. Shunt-fed Class A PDM with Constant-resistance Filter.
E1
e(t)
0 T1 T "t
-E-

 

 

 

 

Figure F-3. Waveform at input
to filter.

Th

In

T]

104

The filter and load configuration are the same as Figure F-Z.

1-3

[a = 711— ~ (F-27)

0<a<1 (F—ZB)

 

 

 

T
k 2 7f;- (IF-29)
0 5 k <_1 (JP-30)
Let
.9 = “mt (F-31)
Let
a = k (1 + MsinG) ' (F-32)
Then
k(1+ M)_<_l (F-33)
l
M _<_ E - 1 (F-34)
Power input to the filter is
E’- E" E2
PM— “ E1— ”14’s" '13?
l 211 E2
Pt- 2'; f0 P(9)d‘9‘-_' ff- (F-35)
Instantaneous output power to load R is
__ [qE - (1 - (1)312 (20. - 1)2E’-
13(0) " ff 3 R 1 1
Average output power is
1 1 2'"
pop " ‘2'"; I0 P('9)d9
z 2
= sz fo"[2k + ZkMsinG - 1]’-de
z
= g—uzk - 1)2 + ZkzMz] = PdC + Pop (F-36)

The d-c output power is then

105

E2
Pdc = T (2k - 1)2

The a-c signal output power is

_ 2(MkE23
Pop - R
The efficiency is

- Po _ 2

Case 1:

5—4

k<—-
--2
2.0 _<_M :1

.n
o 0-.

(F-37)

. (F-38)

(F-39)

(F-40)

(F-41)

With M fixed, the greatest output power and efficiency occur

1 .
whenkt = 2-. The greatest possible output power and efficiency

occur when M = 1.

1

1“:
E2
MaximumPop- TR
Maximuman =-:—
Case2:'
l
—-< <
2 _ k __1
l

M <_1: -1

2(Mk1-3)z < ~2(1- 103132

 

Pop: R .—

an = ZMzkz 5. 2(1 - 1oz

(F-42)
(F643)

(F-44)

(F-45)

(F-46)
(F-47)

(F-48)

The greatest possible output power and efficiency occur when

l
k--2-°

106

l
k - '2'

. E:z
Max1mum Pop = 7R-

Maximum an = ~21-

(F-49)

(F-SO)

(F-Sl)

Therefore, as in Section F-l, the best value of k in the range given

1

by Equation (F-30) is -- This value of k allows the greatest possible

output power and the highest efficiency.

2 0

loss in the load is zero. Let

Then

1
k- '2‘
1 .
o. = 2- (1+ M51nmmt)
M2
an= '2—

Pdc= 0

In addition, the d-c power

(F-SZ)

(F-53)
(F-54j

(F-55)

REFERENCES

1. Black, H. 5., Modulation Theory, D. Van Nostrand Company, Inc.,
New; York, 1953.

 

2. Ryder, J. D., Engineering Electronics, McGraw-Hill Book Company,
Inc., New York, 1957.

 

3. Chai Yeh, “Analysis of a Single-Ended Push- Pull Audio Amplifier, "
Proc. of I.R.E., June 1953, pp. 743-747.

 

4. Crowhurst, N. H. , "The Basic Design of Constant Resistance
Crossovers, " Audio Engineering, October, 1953.

 

5. Reference Data for Radio Engineers, Fourth Edition, International
Telephone and Telegraph Corporation, 1956-1957.

6. Landon, Vernon D. , "Theoretical Analysis of Various Systems of
Multiplex Transmission, " RCA Review, Vol. IX, Number 2 and
Number 3, June and September, 1948.

 

7. Feldman, C. B., and Bennett, W. R., "Band Width and Transmission
Performance, " Bell System Technical Journal, Vol. 28, pp. 490-595,
July, 1949.

 

107

 

 

 

mate's!

 

   

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