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The University of Alberta
Printing Department
Edmonton, Alberta
■ -
THE UNIVERSITY OF AL3ERTA
RELEASE FORM
Joachim F. W. Gurke
FM DISTORTION IN INJECTION-LOCKED
OSCILLATORS USED AS MICROWAVE AMPLIFIERS
M Sc
DEGREE FOR WHICH THESIS WAS PRESENTED .
YEAR THIS DEGREE GRANTED . . )97$ . . .
Permission is hereby granted to THE UNIVERSITY OF
ALBERTA LIBRARY to reproduce single copies of this
thesis and to lend or sell such copies for private,
scholarly or scientific research purposes only.
The author reserves other publication rights, and
neither the thesis nor extensive extracts from it may
be printed or otherwise reproduced without the author's
written permission.
NAME OF AUTHOR
TITLE OF THESIS
THE UNIVERSITY OF ALBERTA
FM DISTORTION IN INJECTION-LOCKED OSCILLATORS
USED AS MICROWAVE AMPLIFIERS
BY
JOACHIM F. W. GURKE
A THESIS
SUBMITTED TO THE FACULTY OF GRADUATE STUDIES AND RESEARCH
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE
OF MASTER OF SCIENCE IN ELECTRICAL ENGINEERING
DEPARTMENT OF ELECTRICAL ENGINEERING
EDMONTON, ALBERTA
SPRING, 1975
.
VJ*
^5 - A c\
THE UNIVERSITY OF ALBERTA
FACULTY OF GRADUATE STUDIES AND RESEARCH
The undersigned certify that they have read,
and recommend to the Faculty of Graduate Studies and
Research, for acceptance, a thesis entitled FM Distortion
in Injection-Locked Oscillators Used as Microwave Amplifiers
submitted by Joachim F. W. Gurke in partial fulfilment
of the requirements for the degree of Master of Science
in Electrical Engineering.
ABSTRACT
The dynamic behavior of injection-locked oscillators
(ILO's) operating as FM amplifiers has been investigated.
IMPATT diodes operating at X-band were used as the locked
oscillators. The ILO has been modelled mathematically by the
use of the generalized Adler's phase-locking equation, and
ideal FM modulators and demodulators at the input and output
respecti vely , thus simulating a baseband amplifier. This ILO
amplifier has been characterized in conventional amplifier
terminology by the use of FM input signals whose deviation
and modulation rate approach the locking bandwidth. Theoretical
results for the amplitude variation, the phase delay distortion
and the nonlinear distortion of the fundamental output signal
versus the modulating frequency have been obtained. The
nonlinear distortion curve has been verified experimentally for
a 34 dB gain case. The theoretical and experimental analysis
of the distortion in FM ILO amplifiers indicates that for the
output demodulated fundamental signal a) the distortion
characteristics are sensitive to changes in the input modulating
signal parameters but are not sensitive to changes in the locking
signal amplitude, b) the amplitude is frequency dependent, c)
the phase delay distortion is highly dependent upon all the
input modulation parameters, and d) the nonlinear distortion
iv
is dependent upon the modulating signal frequency. These
distortion characteristics will aid in the design of microwave
IMPATT diode oscillator-amplifiers for use as the output stages
of microwave FM transmitters.
v
ACKNOWLEDGEMENTS
The author wishes to express his appreciation for
the assistance of many people during the course of this research:
To Dr. P.A. Goud for his advice and encouragement
during the supervision of this work.
To members of the Microwave Electronics Laboratory at
the University of Alberta, who were closely involved in this
research, for their assistance, understanding and encouragement.
To the author's parents for their encouragement and
financial support during this work.
The author is also indebted to the following
organi zati ons :
To the National Research Council of Canada for their
continued support of this research and for the provision of
a research scholarship.
To the Department of Communications of Canada,
Communications Research Centre, (Contract D1 6R-36001 -1 -0513)
for the financial support of this research.
To the University of Alberta for financial assistance.
vi
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TABLE OF CONTENTS
I INTRODUCTION
Page
1
II THEORETICAL ANALYSIS OF AN INJECTION-LOCKED
OSCILLATOR (ILO) .
2.1 Introduction .
2.2 The Generalized Adler's Phase-Locking Equation
2.2- 1 Basic concepts and Approximations .
2.2- 2 Equivalent Circuit Represention .
2.2- 3 Phasor Analysis .
2.2- 4 Derivation of the Generalized Adler's
Phase-Locking Equation .
2.2- 5 Adler's Equation .
2.3 Steady-State ILO Characteristics .
2.4 Transient Response of an ILO .
4
4
4
6
7
11
12
16
17
19
III LOCKING TO A FREQUENCY MODULATED LOCKING SIGNAL
3.1
Introduction .
25
O • L,
3.3
Quasi -Stationary Equation Analysis .
26
3.4
Linearized Equation Analysis .
33
3.5
Analysis Based Upon Adler's Equation .
. 35
3.6
Analysis Based Upon the Generalized Adler's
Phase-Locking Equation .
. 39
3.7
. 41
.
.
•«v
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v^.
Page
IV CHARACTERIZATION OF AN FM ILO AMPLIFIER . 43
4.1 Introduction . 43
4.2 ILO Amplifier Model . 43
V FM DISTORTION IN A TUNED ILO AMPLIFIER . 51
5.1 Introduction . 51
5.2 Mathematical Modelling . 51
5.3 Results of the Tuned Calculations . 56
5.4 Conclusion . 57
VI FM DISTORTION IN A DETUNED ILO AMPLIFIER . 64
6.1 Introduction . 64
6.2 Mathematical Modelling . 65
6.3 Results of the Detuned Calculations . 68
6.4 Conclusion . 74
VII APPLICATION OF THE DISTORTION ANALYSIS IN THE DESIGN
OF A SINGLE-STAGE FM ILO AMPLIFIER . 75
7.1 Introduction . 75
7.2 Design Contours . 75
7.3 An Example of the Design Contour Applications . 76
7.4 Conclusion and Summary . 80
' •J8: Qa
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1
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Page
VIII DISTORTION MEASUREMENTS OF A MICROWAVE FM ILO AMPLIFIER . 82
8.1 Introduction . 82
8.2 Theory, Experimental Set-Up and Measurement Techniques. 82
8.3 Experimental Results and Their Interpretation . 96
IX SUMMARY AND CONCLUSIONS . 104
REFERENCES . 108
APPENDIX A SIGNAL DISTORTION IN TRANSMISSION SYSTEMS . 114
APPENDIX B DETAILED ANALYSIS OF THE TUNED AND DETUNED
DISTORTION IN ILO AMPLIFIERS . 119
ix
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LIST OF TABLES
Page
TABLE 2.3-1 A COMPARISON OF THE VARIOUS LIMITS PREDICTED BY
THE GENERALIZED ADLER'S PHASE-LOCKING EQUATION
(EQ. 2.2-20) WITH THOSE PREDICTED BY ADLER'S
EQUATION (EQ. 2.2-22) WHEN THE QUASI-STATIONARY
APPROXIMATION IS VALID . 20
TABLE 5.3-1 COMPARISON OF NONLINEAR DISTORTION FOR THE VARIOUS
PARAMETERS OF A TUNED ILO AMPLIFIER (DISTORT ION
EXPRESSED IN PERCENTAGES) . 60
TABLE 5.3-2 COMPARISON OF PHASE DELAY DISTORTION (EXPRESSED
IN DEGREES) FOR VARIOUS PARAMETERS OF A TUNED
ILO AMPLIFIER . 61
TABLE 5.3-3 RELATIVE AMPLITUDE VARIATION (EXPRESSED IN DB)
FOR VARIOUS TUNED ILO AMPLIFIER PARAMETERS . 62
TABLE 6.3-1 NONLINEAR DISTORTION(EXPRESSED IN PERCENTAGES)
IN AN ILO AMPLIFIER MODEL WITH A GAIN EQUAL TO
1 5DB , A MAXIMUM LOCKING BANDWIDTH OF 17.86 MHZ,
A PEAK FREQUENCY DEVIATION OF 0.75 A*, AND FOR
THE FOLLOWING NORMALIZED DETUNING FACTORS: 0.0,
0.1 , AND 0.24 . 69
TABLE 6.3-2 PHASE DELAY DISTORTION (EXPRESSED IN DEGREES)
IN AN ILO AMPLIFIER MODEL WITH A GAIN EQUAL TO
1 5DB, A MAXIMUM LOCKING BANDWIDTH OF 17.86 MHZ,
A PEAK FREQUENCY DEVIATION OF 0.75^ AND FOR
THE FOLLOWING NORMALIZED DETUNING FACTORS: 0.0,
0.1 , AND 0.24 . 70
TABLE 6.3-3 RELATIVE AMPLITUDE VARIATION (EXPRESSED IN DB)
IN AN ILO AMPLIFIER MODEL WITH A GAIN EQUAL TO
1 5DB, A MAXIMUM LOCKING BANDWIDTH OF 17.86 MHZ,
A PEAK FREQUENCY DEVIATION OF 0.75Z^, AND FOR
THE FOLLOWING NORMALIZED DETUNING FACTORS: 0.0,
0.1 , AND 0.24 . 71
TABLE 8.3-1 EXPERIMENTAL DISTORTION MEASUREMENTS FOR A 34DB
GAIN DETUNED SINGLE-STAGE ILO AMPLIFIER . 99
x
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*S£s
LIST OF FIGURES
Page
FIG. 2.2-1 EMBODIMENT OF THE INJECTION LOCKING
CIRCUIT AT MICROWAVE FREQUENCIES . 5
FIG. 2.2-2 EQUIVALENT CIRCUIT OF A "FREE-RUNNING"
MICROWAVE OSCILLATOR WHERE Z(oo) IS THE
CIRCUIT IMPEDANCE SEEN FROM THE DEVICE
(-Z (A)=THE DEVICE IMPEDANCE) . 5
FIG. 2.2-3 INJECTION-LOCKED OSCILLATOR (ILO) EQUIVALENT
CIRCUIT CONCEPT (WHERE Z =DIODE IMPEDANCE
AND Z =CAVITY IMPEDANCE) . 9
FIG. 2.2-4 EQUIVALENT PARALLEL IMPEDANCE REPRESENTION
OF AN ILO . 10
FIG. 2.2-5 EQUIVALENT PARALLEL CIRCUIT FOR THE ILO . 10
FIG. 2.2-6 THE PHASOR REPRESENTION OF THE SIGNALS
IN THE ILO MODEL . 13
FIG. 2.4-1 PLOT OF THE TRANSIENT PHASE RESPONSE EO. 2.4-3
VERSUS THE NORMALIZED TIME FOR SEVERAL
VALUES OF e0 . 24
FIG. 3.3-1 LOCATION OF RADIAN FREQUENCIES IN THE
FREQUENCY DOMAIN FOR A SWEPT ILO AMPLIFIER
WHEN Acoo<(ws-co0)<A0 . 28
FIG. 3.3-2 A PLOT OF THE PHASE SHIFT EQ. 3.3-4 VERSUS
THE NORMALIZED LOCKING BANDWIDTH (F0RAu>o=0.0) . 30
FIG. 3.3-3 A PLOT OF THE ENVELOPE DELAY DISTORTION
EQ. 3.3-7 VERSUS THE NORMALIZED LOCKING
BANDWIDTH ( FOR Ao)o=0.0, f =1.0 MHz) . 32
FIG. 3.4-1 PHASE VERSUS NORMALIZED FREQUENCY DEVIATION
FOR THE LINEARIZED APPROXIMATE SOLUTION,
EQ .3.4-4 . 36
FIG. 3.5-1 PHASE VERSUS NORMALIZED FREQUENCY DEVIATION
FOR ADLER'S EQUATION . 38
FIG. 3.6-1 PHASE VERSUS NORMALIZED FREQUENCY DEVIATION
FOR SEVERAL POWER RATIOS FOR THE GENERALIZED
ADLER'S PHASE-LOCKING EQUATION . 40
xi
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* fa Dll 4 - 3T Hi- 1
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Page
FIG. 4.2-1 SCHEMATIC DIAGRAM OF THE ILO AMPLIFIER
WITH AN IDEAL FM MODULATOR AND FM DEMODULATOR . 44
FIG. 4.2-2 LINEARIZED TRANSFER CHARACTERISTICS FOR
THE ILO AMPLIFIER MODEL OF FIG. 4.2-1 . 49
FIG. 5.3-1 FREQUENCY DEPENDENT DISTORTION CHARACTERISTICS
FOR A TUNED SINGLE-STAGE ILO AMPLIFIER MODEL
WITH A 34 DB GAIN . 58
FIG. 5.3-2 FREQUENCY DEPENDENT DISTORTION CHARACTERISTICS
FOR A TUNED SINGLE-STAGE ILO AMPLIFIER MODEL
WITH A 15 DB GAIN . 59
FIG. 6.3-1 FREQUENCY DEPENDENT DISTORTION CHARACTERISTICS
FOR A DETUNED SINGLE-STAGE ILO AMPLIFIER MODEL
WITH A 34 DB GAIN . 72
FIG. 6.3-2 FREQUENCY DEPENDENT DISTORTION CHARACTERISTICS
FOR A DETUNED SINGLE-STAGE ILO AMPLIFIER MODEL
WITH A 15 DB GAIN . 73
FIG. 7.2-1 CONSTANT NONLINEAR DISTORTION CONTOURS VERSUS
THE NORMALIZED PEAK FREQUENCY DEVIATION AND
MODULATING FREQUENCY FOR A POWER GAIN OF 15 DB
AND FOR NORMALIZED DETUNING FACTORS EQUAL
TO 0.0 AND 0.1 . 77
FIG. 7.2-2 CONSTANT PHASE DELAY DISTORTION CONTOURS VERSUS
THE NORMALIZED PEAK FREQUENCY DEVIATION AND
MODULATION FREQUENCY FOR A POWER GAIN OF 15 DB
AND FOR NORMALIZED DETUNING FACTORS EQUAL
TO 0.0 AND 0.1 . 78
FIG. 8.2-1 SCHEMATIC FORM OF THE EXPERIMENTAL SET-UP FOR
THE MEASUREMENT OF THE NONLINEAR DISTORTION
IN AN ILO AMPLIFIER . 85
FIG. 8.2-2 A PHOTOGRAPH OF THE FM ILO NONLINEAR DISTORTION
EXPERIMENTAL SET-UP . 86
FIG. 8.2-3 A PLOT OF THE VOLTAGE AMPLITUDE DEFLECTION OF THE
FREQUENCY DISCRIMINATOR VERSUS THE FREQUENCY
DEVIATION (SWEPT FROM 9.040 TO 9.060 GHZ) FOR
AN INPUT POWER LEVEL OF 0.2 MILLIWATTS . 92
FIG. 8.2-4 FREQUENCY SPECTRUM FOR AN FM SIGNAL WITH
CARRIER ZERO . 95
FIG. 8.3-1 AN EXAMPLE OF THE DEMODULATED INPUT AND OUTPUT
SIGNALS MEASURED FOR THE 34 DB CASE . 100
• ■ JB
.
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Page
FIG. 8.3-2 A PLOT OF THE EXPERIMENTAL VERSUS THE
THEORETICAL DISTORTION CHARACTERISTICS
FOR THE 34 DB GAIN CASE OF THE DETUNED
AMPLIFIER . 101
FIG. A-l ' ELEMENTARY BLOCK DIAGRAM OF A COMMUNICATION
SYSTEM . 115
FIG. A-2 FREQUENCY PLOT OF a )THE MAGNITUDE AND
b)THE PHASE VARIATION FOR THE DISTORTIONLESS
CASE OF EQ. A-3 . 117
• • •
xm
.
CHAPTER I
INTRODUCTION
Under certain conditions, it is possible to synchronize
the frequency and phase of a self-excited oscillator to a
1 -22
reference or locking signal . This may be achieved by
injecting the reference signal directly into the oscillator to
3 4
be controlled * . In the stable locked condition the oscillator
and reference signal frequencies will be identical and the
oscillator will be referred to as being injection-phase locked^’^
The locking signal amplitude is smaller than the self-excited
oscillator amplitude, therefore the injection-locked oscillator
(ILO) can be modelled as a limiter-amplifier"^’^’^. In the
stable locked condition, the reference and oscillator frequencies
will coincide exactly, however, this locked condition will
exhibit a definite phase relationship. The locking process and
mechanism when examined in terms of this phase relationship
depend upon
1. the initial frequency difference between the locked
and reference signal frequencies
2. the relative amplitudes of the locked oscillator
and reference signals (this ratio is called the
gain of the ILO)
3. the circuit parameters.
1
■
The injection-phase locked phenomenon can be further
characterized by analogies to an oscillator operating into
a mismatched load^’"*
The aims of this work include the examination of
the dynamic phase-locking properties of microwave IMPATT
diode oscillators when the reference signal is angle-modulated.
The ILO has been characterized within certain limits for the
1 4 -i
steady-state and transient conditions 5 . Several
simplifications are applied to the mathematical model such that
the analytic expressions can be used to characterize the ILO
under low frequency modulated conditions. However, when the
modulating frequency increases, the techniques discussed so
far will not be valid. Therefore, the aim of this work is to
characterize the ILO for modulating signals approaching the
locking limits. This will be accomplished by modelling the ILO
with the aid of the generalized Adler's phase-locking equation.
The amplifier model will treat the reference (locking) and locked
signals of the ILO as input and output signals of a baseband
amplifier, respectively. The resulting mathematical treatment
will be presented in the form of conventional distortion curves
for each set of ILO amplifier parameters. The distortion curves
are combined into design contours such that a circuit designer
will use them in selecting ILO limiting parameters. Finally,
•- .
I
3
the theoretical description of the ILO characteristics are
verified experimentally .
CHAPTER II
THEORETICAL ANALYSIS OF AN INJECTION-LOCKED OSCILLATOR (ILO)
2.1 Introduction
This chapter will derive the important relationships
and equations applicable to the modelling of the injection¬
locking phenomenon that follows.
2.2 The Generalized Adler's Phase-Locking Equation
Several authors have derived phase-locking equations
that may be used to model an injection-locked oscillator (ILO)
at microwave frequencies 1"4»17. jj- -j s propitious to discuss
the derivations of these phase-locking equations in some detail
so that the inherent assumptions will be understood in the
context of this work. The results of these derivations will
then be used as a basis for the description of the nonlinear
distortion added by an ILO amplifier to modulated locking
signals.
Let the simplified microwave circuit of Fig. 2.2-1
represent the typical schematic for the ILO considered in this
study ^ . The microwave "locking signal" and the "output signal"
are separated by a four-port microwave circulator. Therefore,
for practical purposes, the input and output signals may be
considered to be isolated. In the analysis to follow, it will
be shown that the locking signal has a tendency to "pull" the
4
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■
MATCHED LOAD
FIG. 2.2-1 EMBODIMENT OF THE INJECTION-LOCKING CIRCUIT AT
MICROWAVE FREQUENCIES.
FIG. 2. 2-2 EQUIVALENT CIRCUIT OF A "FREE-RUNNING" MICROWAVE
OSCILLATOR WHERE Z(w) IS THE CIRCUIT IMPEDANCE SEEN
FROM THE DEVICE(-Z(A) IS THE DEVICE IMPEDANCE).
D
frequency of the one-port microwave oscillator in a manner analo¬
gous to the frequency pulling which occurs when a microwave
oscillator operates into a mismatched load^”^*^.
2.2-1 Basic Concepts and Approximations
To facilitate an analytical mathematical treatment of
the reflection-type amplifier, several simplifying physical
assumptions need to be made at this juncture^. Firstly, it is
assumed that the bandwidth of the resonant circuit surrounding
the active device is much larger than the locking bandwidth or
than any other modulating frequency. For a single-tuned
resonant circuit the half-power bandwidth is given by
B. W. - u)0/Ql — co i -(a 2
(2.2
where B.W. = Bandwidth (radians/sec)
w0 = natural resonant radian frequency
= half-power point resonant radian frequency
Ql = loaded quality factor
Secondly, it is assumed that the amplitude control mechanism
of the oscillator acts much faster than the time period of the
frequencies of interest; namely, a) modulating frequencies, b)
beat frequencies, and c) "pulled" frequencies. Mathematically
expressed, this means that
T' « orl/“m or 1/A“o
(2.2
.
.
. V
7
where T1 = time constant of the amplitude limiting
mechanism
A<0 max = maximum frequency deviation
= initial frequency difference
«m = modulating radian frequency
As a first approximation, the results achieved by ignoring the
amplitude control mechanism are remarkably gooa\ As will be
discussed later, the above assumption may not always be
justified for the microwave oscillators used in this analysis.
Thirdly, it is assumed that the input locking signal power is
significantly smaller than the free-running oscillator power.
As will be seen, this basic assumption leads to a simplified
analysis that yields many of the basic ILO characteristics.
However, most practical microwave ILO amplifiers will find
application in an intermediate region, where the input and
output levels differ by 10 to 20 dB. Consequently, the
following analysis will contain an important improvement over
Adler's simpler theory l The next section will derive the
locking equations applicable to this intermediate power
range \
2.2-2 Equivalent Circuit Representation
The simplest equivalent circuit for a microwave
oscillator is given for the "free-running" condition.
Assuming for the moment that the device impedance is
frequency independent, let the equation for the free-running
*
VL
oscillator of Fig. 2.2-2 be given by
[ 2 M - Z0(A) H = 0 (2.
where Z0(A) is the device impedance, which is taken to be a
function primarily of the current amplitude A, and where Z(ui)
is the circuit impedance as seen looking out from the device
terminals 17.
To construct an equivalent circuit to provide a
description of the injection-locking circuit of Fig. 2.2-1,
let the input signal be represented by a current generator
(it could analogously have been a voltage generator). Next,
let the one-port microwave oscillator be represented by an
equivalent parallel circuit impedance , the tuned cavity by
an equivalent parallel circuit impedance. Thus it is seen
that the equivalent circuit of Fig. 2.2-3 easily becomes
Fig. 2.2-4 17» 25-30 # por purposes of this study, the
diode may be represented by its equivalent negative impedance
-G ^0,21 s with reference to Fig. 2.2-5, the following symbols
will be defined:
2 D ” "0 D
GD = Equivalent negative conductance of the free-
running oscillator
Zc = equivalent cavity impedance
Z' = equivalent cavity impedance plus the equivalent
impedance plus the load
j j
.
.
*
■
M
LOCKING SIGNAL
OUTPUT SIGNAL
I :n
FIG. 2.2-3 INJECTION LOCKED OSCILLATOR( ILO) EOUIVALENT CIRCUIT
concept(whereZ(a)= diode impedance and Zr = CAVITY
D ^
IMPEDANCE).
10
~l_ / = EQUIVALENT IMPEDANCE OF CAVITY
C AND CIRCULATOR
FIG. 2.2-4 EQUIVALENT PARALLEL IMPEDANCE REPRESENTATION OF
AN ILO .
FIG. 2.2-5 EQUIVALENT PARALLEL CIRCUIT FOR THE ILO.
vu
L - equivalent lumped shunt inductance of the tuned
cavity near resonance
C = equivalent lumped shunt capacitance of the
tuned cavity near resonance
Gl= equivalent lumped shunt conductance of the
load (initially assumed to be a match) plus
that of the tuned cavity near resonance
plus the circulator equivalent impedance
is = instantaneous injected current
i = instantaneous "locked" oscillator current
iR = resultant output current
Is = | U | = injected current amplitude
I = ji| = oscillator current amplitude
Ir= I Ul = output current amplitude
= unmodulated injected signal radian frequency
= "free-running" oscillator radian frequency
instantaneous phase shift of the locking
signal
/&)= instantaneous phase shift added by the tuned
circuit and other mis-matchs
Ql= loaded quality factor
Even though the circulator has been replaced in Fig. 2.2-4
and Fig. 2.2-5, it must be remembered that is and i are
isolated signals which add vectorially in the tuned output
circuit.
2.2-3 Phasor Analysis
Let the injecting signal of Fig. 2.2-5 be represented by
is = Is sin(u)5t +<x(t))
(2.2-4)
• ■
«
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■
m
12
where oc(t) is a time-dependent phase shift. When the locked
condition exists the locked oscillator current will be
i (t) = I s i n ( w5t +*(t) - e (t) ) . (2.2-5)
Here0(t) has been introduced to represent the instantaneous
phase shift of the locked oscillator signal with reference to
the injected signal. Thus the total instantaneous current
flowing through the resonant circuit will be the vector
sum of i(t) and is(t), which will be given by
v I* si n ( Wjt +«.( t) -e(t) + ^(t)) (2.2-6)
where
I
R
+ I
2.
+ 2I^cose(t)
(2.2-7)
and
-i Is • sin (t)
^(t)= tan I + Iscose(t) (2'2-8>
The various signals are represented in the phasor diagram Fig.
2.2-6.
2.2-4 Derivation of the Generalized Adler's Phase-Locking
Equation
Since the term ^(t) in Eq. 2.2-6 is the phase shift
introduced by the addition of the tuned circuit, it follows
.
.
-
-
Vk
13
FIG. 2.2-6 THE PHASOR REPRESENTATION OF THE SIGNALS IN THE ILO
MODEL.
'
.
'■
wl
that Eq. 2.2-8 may be simplified in terms of the other circuit
parameters. For a single-tuned parallel circuit with lumped
parameters, the phase shift across the circuit will vary as
14
J(o))=tan ‘[Ql( a)2- uf)/ o)2]
(2.2-9)
where
coo= 1//TC
In the locked condition, the frequency u wil 1 be very close to
w0 , so that Eq. 2.2-9 may be approximated quite accurately by
(2.2-10)
28
when w-o)q « . The phase shift introduced by the resonant
circuit will be opposite to the phase shift of the total current,
consequently the voltage across the resonant circuit will be
exactly n radians out of phase with the voltage introduced by the
injected signal. This principle will also apply to the
currents; this Eq. 2.2-10 may be equated with Eq. 2.2-8:
Issine(t) _ 2Ql(w -w )
- - - - - o
I + Lcoso(t) to
o
(2.2-11)
Solving Eq. 2.2-11 for ^yields,
w = w +
o
A0 s i n o( t )
(2.2-12)
l+Rcose(t)
where R/ 2Q u
R= ls/l
(2.2-13)
(2.2-14)
and
r i. »! ■$ 4
*
\ n i i 't '* H J'/£ ' ' t:,;:xe
Thus the output phase angle from Eq. 2.2-6 and Fig. 2.2-6 is
©R(t) = u)st +c<(t) - ©( t ) +&{t)
and the output frequency is given by the derivative of
Eq. 2.2-15:
U = -d eR(t). y + d<x(t) +d ^(t) _ d$(t)
dt
dt
dt
dt
(2.2-16)
Now, combining Eq. 2.2-16 and Eq. 2.2-12 yields
de(t) _ d^(t) + A0sina(t) = A^+ d*(t)
dt dt l+Rcose(t) dt
(2.2-17)
where Aio0 =
= the initial radian frequency difference,
Then differentiating Eq. 2.2-8 yields
(2.2-18)
d^(t) R( R + cos e(t)) de(t)
“ ' 9 ~
dt 1 + R2+ 2Rcoss(t) dt
(2.2-19)
which will be used to eliminate the ~~ terms of Eq. 2.2-17; i.e.
dt
de(t) n + R + 2Rcose(t)
dt
1 + Rcose(t)
(2.2-20)
[AU30 +
d<x(t ) A0si ne(t)
dt 1 + Rcose(t)'
' W
The solution to Eq. 2.2-20 yields the instantaneous change of
phase between the input locking signal and the output signal from
the locked negative conductance oscillator. Eq. 2.2-20 will
hereafter be referred to as "the Generalized Adler's Phase-
Locking Equation" because of its importance in the following
work. To solve for^(t), Eq. 2.2-20 may be substituted into
Eq. 2.2-8. Further, the output frequency will be found by
substituting Eq. 2.2-17 into Eq. 2.2-16 to yield
«.(t) ■ ^,+ -A-sin»(t) (2.2-21)
1 + Rcose(t)
2.2-5 Adler's Equation
In special cases, Eq. 2.2-20 must be simplified by
assuming a) a low level of injected locking power i.e.R«l
and b) that there is no significant modulation on the injected
signal (i.e. dc<t^— = o ). Then Eq. 2.2-20 reduces to Adler's
dt
phase-locking equation which (when expressed in terms of the
notation used in this thesis) becomes
(2.2-22)
= Ato0_ Acsine(t)
where
A„= EL-uy 2 - E0' Q L (2.2-23)
(where EL and E0 are the voltage amplitudes defined by Adler
which are equivalent to the expressions used here). The
■ mm mm gw j
.
-
.
. -
vs.
■
17
Generalized Adler's phase-locking equation that has been
derived in this section provides the basis for extensive
later work; in particular, it will be used to solve for the
distortion behavior of ILO's.
2.3 Steady-State ILO Characteristics
This section will examine the locking equations
derived in Section 2.2 by applying the quasi-stationary
approximation to all time-dependent terms. The elimination
of the transient solution will enable the locking equation
limits to be examined for the steady-state case. For Adler's
locking equation Eq. 2.2-22, the steady-state phase shift may
be expressed as a function of normalized initial frequency
difference; i.e.
© = sin (2.3-1)
when o. When the quasi-stationary approximation is
dt
applied to the generalized Adler's phase-locking equation
Eq. 2.2-20, the steady-state phase shift may be expressed as
9 =
-i
si n
/
u
J + uzRz
[1 ± R / l - u2( 1 - Ra )]
/
(2.3-2)
where u=^/^c and R= I5/I (from Eq* 2.2-14), the plus sign
applies for u>0 and, alternately, the minus sign applies for
u<0 . It is evident that Eq. 2.3-2 reduces to Eq. 2.3-1
for R « 1 . Now let these two "steady-state" results be
-
.
■*
*
-
Vk
examined for their limit variation when certain parameters are
changed.
18
Firstly, let the maximum initial frequency difference
occur when sin ©has its maximum value (i.e. sin 8=1) in
Eq. 2.2-21 ; i.e..
i AUt
= A,
max ' (2.3-3)
Utilizing a similar normalizing procedure, the generalized
Adler's phase-locking equation Eq. 2.2-20 yields its maximum
allowable initial frequency difference; namely,
Lax “ ^ < 1 -
Z, Vz
(2.3-4)
Secondly, the maximum allowable phase shift I© I that
max
an ILO will permit and. still maintain the locked condition
is^r for Eq. 2.2-21. For the generalized Adler's phase¬
locking equation I© | max is
0
max
JT
2-
+ cos 1 ( 1 - R2) 2'
(2.3-5)
It is interesting to note that the generalized Adler's
phase-locking equation predicts a) a maximum allowable phase
shift in excess of \ and b) a maximum locking bandwidth in
excess of A0 . In both cases the limits are directly proportional
to the injected signal amplitude ls and inversely proportional
to the "free-running" oscillator amplitude I. When the ratio
R is expressed in terms of power it becomes
-
'
19
R = (Ps/P)'4
where Ps refers to the injected power and P refers to the
"free-running" oscillator power. When R is expressed in a
different form, the ratio of injected to output power will
give the gain of the I LO as an amplifier, where gain is
defined as
G = -10 logto P$/P dB
or
G = -20 log,0R dB
Several authors have documented extensive in¬
vestigations of this "steady-state" analysis 1,8"'12’27>
For the purposes of this work the results predicted by
the "steady-state" analysis will be exploited in a simplified
theory to justify the experimental set-up. Therefore, in
Table 2.3-1 the maximum locking bandwidth and the maximum
phase shift limits predicted by the generalized Adler's
phase-locking equation are compared with the predictions from
Adler's equation for similar values of gain. An examination
of Table 2.3-1 shows that only for gains in excess of 20dB
will both equations predict essentially identical results.
2.4 Transient Response of an ILO
4
In this section Mackey's approach to describing
the transient behavior of ILO's is reviewed and an attempt
(2.3-6)
(2.3-7)
(2.3-8)
I '■
■
.
S3 I . 1
■
TABLE 2.3-1 A COMPARISON OF THE VARIOUS LIMITS PREDICTED BY THE GENERALIZED ADLER'S PHASE-LOCKING
EQUATION (EO 2.2-20) WITH THOSE PREDICTED BY ADLER'S EQUATION ( EQ. 2.2-22) WHEN THE
QUASI -STATIONARY APPROXIMATION IS VALID.
20
4s
4s
LU
C_>
ZT
LQ
OlI
UJ
u_
u_
I — I
o
co
zn
Cl
cc
O'.
o
o
CVJ
•
C\J
CVI
l
CVl
CVI
C T
UJ
4—
o
in
Z5
E
o
cu
U-
O-
cu
o
cr
CD
S-
QJ
4-
4-
1
II
CO
-o
CO
-O
CO
-U
CO
"D
CQ
"O
CO
■O
CO
*o
CQ
"O
LO
CO
XJ
CO
-D
CO
■a
*o
>5
O
o
o
'3-
CO
•
o
•
•vj-
o
LO
o
1 — 1
<c
CD
CO
co
co
CVI
CO
cvi
cvi
CO
1 — ■
c:
cu
Z5
c r
CU
s_ -
it
cvi
CVI
CVI
CVI
UJ
r>
CVJ
1 -
4->
1
1
1
f—
r—
r—
1 —
E
o
o
o
o
o
o
o
o
r—
1 —
1 —
1 —
1 —
1 —
00
C-
cu
4-
X
CO
X
X
X
X
X
s_
4->
CO
CO
LO
1 —
CO
LO
CVI
0J
r—
r-^
1 -
CO
o
CO
1 —
CO
JZ
O
•
•
+->
00
LO
LO
LO
1 -
CVJ
*3*
uo
CO
1 —
c -
CO
cu
<D
JZ
4->
CO
UJ
CVJ
Z3
C_>
1
c
zz
CVI
>1 —
UJ
•
E
DC
cvj
UJ
o
u_
•
cvj
u_
CT)
r-~
LO
CO
CO
CO
CVI
CVJ
1 —
1 —
CT
1
1 — i
1
i
1
1
1
1
1
1
1
l
1
UJ
CVJ
O
o
o
o
o
o
c
o
o
o
o
o
•
r~ —
i -
1 —
r—
r—
1 -
1 —
I —
r—
1 -
I —
So
CVI
-Q
CD
X
X
X
X
X
X
X
X
X
X
X
•
zz
-a
CT
UJ
o
o
o
o
LO
co
LO
1 —
CD
LU
=o
o
o
o
o
CVI
o
1 —
o
1 —
CVI
■+J
CT
03
So
UJ
LO
LO
LO
CVJ
r— ■
CVJ
LO
1 -
CVJ
I —
CO
o
JD
DC
•1 —
U_
-a
-o
a
\
*>
o
u
cu •
+-> aj
03
CJ
•5
<D
s-
Q-
<u
u
c:
<u
s-
CU
4-
4-
-o
cu
in
fO
•I —
-M
CO
CVJ
CVJ
CVJ
CVJ
1 —
r—
r—
r—
r—
•1 —
1
1
1
1
1
1
1
1
1
1
1
c
o
o
o
o
o
o
o
o
o
o
o
•r—
r—
r—
1 —
r—
1
1
r_
1
1
1
1
■O
X
X
X
X
X
X
X
X
X
X
X
CD
N
o
o
o
o
o
o
o
o
o
co
cvj
o
o
o
o
o
o
o
LO
o
1 —
CO
1 —
•
•
•
CVJ
LO
1 —
r_
cvj
CO
LO
E
C
>1
O)
E
O rtJ
TT C
4S
fO
+->
•r—
c
X
03
4s
4C
This result is expressed in degrees, i
■
A
.
■
»
21
is made to apply those concepts to the present study of
ILO's. The previous section discussed the ILO character¬
istics when the quasi-stationary approximation was valid.
However, when the ILO is subjected to a step change in the
locking frequency or to some other form of modulating signal
with a short rise time, the ILO is not expected to track
this change instantaneously. The transient phase response
of an ILO to such a frequency step will be discussed next
in order to find an upper limit on the modulation rate for
such an ILO.
When the locking signal changes at rates in excess
of Ao/zir Hz the quasi-stationary approximation will no longer
be valid. Consequently, the solution to the phase equations,
Eqs. 2.2-20 and 2.2-22 must take into account the time-
dependent terms. When Adler's equation is solved in the
time domain (as a first approximation of the expected
results), the integration must be performed as
(2.4-1)
where ©(0) = O0 » u = and u < 1. When © (t) is
integrated it becomes
u x ' 2
(2.4-2)
-
■
*
.
22
where ,,
C1 = tanh'1 l-utan(a»/2)/ (1-u2)
For the special case of u=0, Eq. 2.3-1 leads directly to
-i “ Aot
e(t) = 2 tan ( e tan %/2)
If 9(t) is limited to less than 60 degrees, then the
approximation tan 0= sis valid to better than 10 percent
of 6(t). Consequently, Eq. 2.4-3 may be simplified to
- A j*
e(t) =e„e
This approximate solution gives an elementary estimate of
the decay time of the phase shift 9(t). This time is
t1 = 1/ac sec.
for and ©(<*>) =u=0. The decay time t* suggests
that the upper limit on the modulating frequency appears
to be A0/2Tr Hz. The simplified solutions discussed thus
far do not apply if a) e6>1i/2b) u^O, or c) the generalized
Adler's phase-locking equation needs to be solved as the
phase equation (i.e. increased complexity for the lower gain
cases). Thus, these more general solutions will all have
decay times significantly in excess of those estimated in
Eq. 2.4-5. These more complex solutions are discussed in
great detail in the literature3,4 . For the purposes of
this study the result that may be used from the transient
(2.4-3)
(2.4-4)
(2.4-5)
'
'
■
-
.
response work is that normally the decay time is contained
in periods of time less than V* 5/AGsec.
A more practical approach to the quantizing of
a numerical value for the limit of the modulating frequency
in an ILO is the subject of the next several chapters.
To close this section a single graph of the time response
of Eq. 2.4-3 for several values of Qa is included to illustrate
the general concepts of the decay time for a step change in
frequency (see Fig. 2.4-1). Any degradation of parameters
from those assumed will result in a much longer settling
time and a lower limit to the modulating frequency. Lastly
this graph also indicates that this approach is not practical
in estimating accurately the numerical value of the distortion
that an ILO adds to a modulating signal.
*
PHASE ANGLE IN DEGREES
FIG. 2.4-1 PLOT OF THE TRANSIENT PHASE RESPONSE EO. 2.4-3 VERSUS
THE NORMALIZED TIME FOR SEVERAL VALUES OF eo<
CHAPTER III
LOCKING TO A FREQUENCY MODULATED LOCKING SIGNAL
3. 1 Introduction
In this chapter, the phase shift versus frequency
deviation characteristics of ILO's are used as an aid in
1-12
interpreting ILO behavior under modulated conditions . The
modulating signal ( t ) (defined in section 3.2) will be used
to sweep the ILO through its entire locking range (±A ),
starting with low sweep rates and progressing to higher rates.
Several procedures will be employed to describe the deviation
characteristics 8" 21 . Initially, elementary analysis will be
employed to describe the phase variation and more complex
solutions will be sought as a more accurate description of the
phenomena involved. Lastly, an examination will be made of
the assumptions upon which several of the phase-locking equations
are based, when the limits of their applicability are approached.
3.2 Modelling and Signal Analysis
For the work that follows, let the ILO model as shown
in Fig. 2.2-1 be used. The injected current (following Section
2.2) is given by
i = I sin (tat +*(t)) (3.2-1)
3 3 3
25
.
'
■
■
26
where is = the injected signal current at microwave frequencies
I$ = the magnitude of the injected signal
tos = unmodulated injected signal radian frequency
and the modulating signal is given by
*(t) = J(Amsinwmt) dt (3.2-2)
where Am = the peak frequency deviation (PFD)
= the modulating radian frequency
P = the modulation index
B = A /w
* nr m
Consequently, the output signal current will be, from Eq. 2.2-6
iR = IRsin( cj Qt + oc(t) - e(t) + £(t)) (3.2-3)
r 2 i Yz
where IR = I |_1 + R + 2Rcos ej (3.2-4)
and ^(t) = tan-"* (Rsin 0/1 + Rcose) (3.2-5)
where e(t) will be found by the solution of
de(t) .
l+R^+2Rcos<9(t)
4^0 ! d»c(t) ^0sin6(t)
dt
l+Rcose(t)
dt l+Rcose(t)
(3.2-6)
3.3 Quasi -Stationary Equation Analysis
Initially let the ILO be swept at low rates so that
'
r
27
only an approximate solution will be required to provide the
necessary accuracy for the description of the ILO characteristics
(recall the applicability of the quasi -stationary approximation).
To put these simplifications on a more mathematical basis, let
the gain of the ILO be greater than 20 dB (i.e. Reel, then
and IR = I). Then the output current may be approximated
by
iR = IR sin(o?st +*(t) - e(t))
where O-(t) is found by solving Eq. 3.2-6; namely
= + a? _Ao sin 6
(3.3-1)
(3.3-2)
Let the modulating signal Eq. 3.2-2 sweep the ILO through its
entire locking range (i.e. Am =A } at very low rates (i.e.
). Then de will approach zero and the quasi-stationary
approximation will be valid. It is possible to approximate the
cl
effect of at any instant of time as an instantaneous radian
frequency u> , i.e.
io =U) +
V1nu,mt
(3.3-3)
where us is defined as in Fig. 3.2-1. The phase shift ©will be
d ©
given by solving Eq. 3.3-2 for ^ = anc^ the result is a simple
analytic solution
& = sin
-1
w- u» +Ato
S 0
A
(3.3-4)
■ r
■
'
28
%
ws
Au)0
w
= "FREE-RUNNING" LOCKED OSCILLATOR RADIAN FREOUENCY
= "FREE-RUNNING" UNMODULATED LOCKING SIGNAL RADIAN FREOUENCY
= u)s-w0= INITIAL FREQUENCY DIFFERENCE
= INSTANTANEOUS RADIAN FREOUENCY OF THE ILO WITH A MODULATING
SIGNAL APPLIED
A = w0R = HALF-LOCKING BANDWIDTH
2Ql
G = -20 LOG R = GAIN EXPRESSED IN DB
FIG 3.3-1 LOCATION OF RADIAN FREQUENCIES IN THE FREOUENCY DOMAIN FOR
A SWEPT ILO AMPLIFIER WHEN Au)0<(u) -w0 )<AQ
- [
■
.
To aid in the proper visualization of the various radian
frequencies uj , and the frequency limitAo, at a
given instant of time, an example is shown in Fig. 3.3-1 for the
case ofAca < (Cj- u )< a .
o v o ' o
When the instantaneous injected signal Uj is swept
through the entire locking range of the ILO, the phase versus
frequency deviation characteristics is shown in Fig. 3.2-2
for the case of Adler's Equation. These phase shift
characteristics may be used to predict the useable symmetric
bandwidth
A_
m
max
and/or the permissible modulating frequency
that the modulating signal may have given a certainAcOQ. For
example, if the initial frequency difference is set at 0.15
then only 0.85 of the normalized deviation remains for a
symmetric sweep. Additionally, if the modulating frequency
is such that its effects may not be ignored then by standard
FM analysis the addition of the symmetric deviation and the
modulating frequency should total less than 0.85 (as a first
approximation) .
This basic analysis may also be used to predict some
of the distortion that may be involved in the transmission by an
ILO amplifier. Appendix A defines the transmission distortion
concepts that are relevant to this study. For the purposes of
this thesis, let the distortionless transmission for an ILO be
summarized as
.
.
- " /
.
•**
.
FIG. 3.3-2 A PLOT OF THE PHASE SHIFT EO. 3.3-4 VERSUS THE
NORMALIZED LOCKING BANDWIDTH( FOR Aco0=0.0).
31
a) a gain versus frequency characteristic that is
constant and,
b) a phase shift versus frequency deviation that
varies linearily with the frequency (or a constant
envelope delay characteristic) .
The envelope delay characteristic (EDD) of an ILO may
be approximated by expanding the phase shift of Eq. 3.3-4 about
cj in a power series; i.e.,
3 0 5
0-* + r + i- +
• • • •
where
(3.3-5)
A
o
The higher order terms of z produce the nonlinearity in the phase
shift. By approximating 6 with Eq. 3.3-4 and Eq. 3.3-5, it
is possible to obtain a simple expression for the EDD; namely,
d ©
= 1
d(w-u>0) A.
1
0 yi-?
sec
1 rad
(3.3-6)
When expressed in more conventional units of EDD (i.e. nanoseconds
per megahertz), the variable part of Eq. 3.3-6 becomes
EDD ^
159.2
max
1
- 1
W
(3.3-7)
where f
Ao
(expressed in megahertz). The EDD
max 2ir
characteristic is shown in Fig. 3.3-3 for the simplification
.
'
-
■
vw
32
NORMALIZED LOCKING
BANDWIDTH
FIG. 3.3-3 A PLOT OF THE ENVELOPE DELAY DISTORTION EO. 3.3-7
VERSUS THE NORMALIZED LOCKING BANDWIDTH (FOR Au)o=0.0
ARD fmax= l.OMHZ).
• -
w
33
as per Eq. 3.3-7.
In the work thus far, it has been assumed that
A(i^= 0 to simplify the results (here referred to as the tuned
case). However, it is unlikely that the injected carrier
signal frequency and the free-running oscillator frequency
will remain identical for a long period of time. Numerous
factors have been observed that cause the oscillators to drift
(including temperature dependences)^. Therefore, it would
be more realistic to work through the theoretical model for
the detuned case (i.e.Au^ 0). The only additional distortion
that the simple approaches to the distortion characterization
predict is a non-symmetri cal phase and EDD characteristic.
This suggests that less symmetric bandwidth is available to
satisfy a particular distortion criterion. When the modulating
frequency increases and approaches A0 , the approximation ^ = 0
is no longer valid and more sophisticated mathematical
techniques will be required to describe the ILO.
3.4 Linearized Equation Analysis
When the quasi -stati onary approximation is not valid
the dynamic characteristics of the ILO amplifier must be studied
by the use of time-dependent solutions to the phase equations.
-
i
.
Vi*
34
It will be recalled that Adler's equation is valid for the
case when R«l, 0«6and 1^ = I. When the modulating signal
is a sinusoid of the form of Eq. 3.2-2 then Eq. 2.2-22 becomes
d &
dt =AW0 + AmsinV ' Vine
(3.4-1)
Eq. 3.4-1 is nonlinear and may not be solved analytically for
the sinusiodal forcing function in a reasonable fashion.
However, in that region defined by the maximum frequency
deviati on
to + A
o m
and when this is kept less than 0.7AQ it
is possible to approximate sinG by 8 to better than 10 percent
Then the analytic solution to
4^+A 8=Aul + Asiniumt
dt o o m m
for 0(0) = 0Q is
G(t) =
^ + ^Um
0 A0 A 2+U> 2
o m
A1 4.AW0
e + -
A /A
m o
- s i n ( U) t -
— j x m
/tun
,A,
. -1 wm x
u" \ '
(3.4-2)
(3.4-3)
After the transient solution to 0(t) becomes negligible in
amplitude, the steady-state solution becomes
ss
m
tan"1 ^ )
Ao
(3.4-4)
1
■
In Fig. 3.4-1 the steady-state solution^Eq. 3 . 4 - 4 ^ i s displayed
as a function of instantaneous normalized frequency deviation
A* + V1nh,mt . , . .. .
- ) for the following set of parameters:
( i . e .
A
<^o/2lT = 10 GHz
Aui = 0
0
A_ = 0.5 A
m o
AQ/2Tt= 10 MHz
R = 0.1
Oj m/ 2 IT = 200 kHz, 2 MHz, 10 MHz
As the modulating frequency increases the phase angle versus
frequency deviation characteristics show an ellipse with an
increasing minor axis. This phenomenon for the linearized
approximation may be entirely explained by the arctangent term
of the phase in Eq. 3.4-4.
3.5 Analysis Based Upon Adler's Equation
When the frequency deviation is no longer limited
to the range discussed in Section 3.4, then, as Amo + Am
approaches Aq the approximation sin e - © will no longer be
valid. Therefore, Adler's nonlinear phase equation must now
be solved for the accurate representation of the frequency
'
(b)
(c)
FIG. 3.4-1 PHASE VERSUS NORMALIZED FREQUENCY DEVIATION FOR THE
LINEARIZED APPROZIMATE SOLUTION, EQ. 3.4-4.
36
■
deviation characteristics at the extremes of the range. Standard
numerical techniques were used to solve Adler's nonlinear
equation in the time domain. The results of these calculations
are displayed in Fig. 3.5-1 in a graphical form similar to that
of Fig. 3.4-1. Eq. 3.4-1 was solved for the following set of
parameters: lo Q/2 TT = 10 GHz, QL = 50, R = 0.1, £u)Q = 0,
Gain = 20 dB,A0/2iT= 10 MHz, = 0.99Aq, and for the
following modulating frequencies: UJm/2Tr = 200 kHz, 2 MHz,
10 MHz. Several observations may be made about these plots.
Since each modulating frequency cycle will trace out one circuit
around the graph it may be seen that at the higher modulating
frequencies :
a) it may take several cycles before a steady state
cycle is achieved
b) the major and minor axes of the ellipse (steady-
state) are displaced from those values they would
have with the linearized approximation.
As a consequence, it has proven laborious to attempt to use the
phase versus frequency deviation characteristics in this form to
describe the distortion characteristics in analytic or numerical
form. As will be shown in this study, the basis provided by the
examination of the phase plots may be applied in another manner
to arrive at an "engineering solution" to the distortion
characterization problem.
' -
.
■
*
■
o
o
o
o
o
FIG. 3.5-1 PHASE VERSUS. NORMALIZED FREQUENCY DEVIATION FOR
38
ADLER'S EQUATION.
3 . 6 Analysis Based Upon the Generalized Adler's Phase-Locking
Equation
The previous sections have examined the linear and
nonlinear phase dependent equations. Additionally, when the
approximation R<<1 is no longer valid, then the generalized
Adler's equation must be solved in place of Adler's equation.
For reference purposes it is repeated here with a sinusoidal
modulation term as
d e
dt
1+R +2Rcos6
1+Rcose
x
A si no ■
+ AmsinV -
(3.6-
The solution to Eq. 3.6-1 is nonlinear and similar numerical
techniques to Section 3.5 were required. The important added
dimension to the solutions is the variation of the hysteresis
in the phase display due to the gain. Eq. 3.6-1 is solved and
displayed (as in the previous sections) for the following para¬
meters: to / 2"^ = 10 GHz, Acoq =0, Ql = 50, Am = 0.99AQ,
A0/2tt= 10 MHz, tOm/2TT = 10 MHz, and the gains are 10, 20, and
30 dB. As can be seen from Fig. 3.6-1 decreases in the gain
of the IL0 produce a deterioration in the time required for the
phase characteristic to achieve its steady-state value. As
before^ the initial condition of & (0) = 0 has been used for
illustrative purposes. It is also observed that a greater
■
,
■
*
■ :
-
CD
CD
CD
CD
CD
FIG. 3.6-1 PHASE VERSUS NORMALIZED FREQUENCY DEVIATION FOR SEVERAL POWER
RATIOS FOR THE GENERALIZED ADLER'S PHASE-LOCKING EQUATION.
40
amount of steady-state phase shift occurs when the power ratio
is lowered. Unfortunately , this method of display does not
lend itself to a convenient and accurate solution to the
numerical distortion quantities required for the ILO
characteri zation. As will be shown in succeeding chapters, a
slightly different form of the same equations combined with a
different display will result in a reasonable distortion
characteri zation.
3.7 Concl usion
In this chapter the dynamic phase shift versus
frequency deviation characteristics of the ILO 1 s have been
displayed for a variety of parameters. These plots indicate
that the nonlinear distortion effect of the generalized Adler's
phase-locking equation is observed in the differences that
occur from the ellipses of Fig. 3.4-1. The amount of discrepancy
will be influenced by the modulating frequency, the peak frequency
deviation, and the circuit parameters of the ILO. These phase
shift deviation characteristics are a valuable aid in the
visualization of the manner in which the phase decays to its
steady-state value given certain changes in the variable parameters.
The possibility of using these plots on a basis for a detailed
description of the distortion in ILO's was investigated.
.
*<
'
•*»
I
After considerable effort it was found that the results
from the phase approach (as detailed in the previous sections)
did not yield the distortion quantities in the desired analytic
form. However, when a few changes are introduced into the
manner in which the ILO is modelled (as will be performed in the
following chapters) it will be feasible to arrive at numerical
quantities for the distortion effects. It is expected that
future work by other investigators will duplicate these numerical
results .
■ «' .
V4
CHAPTER IV
CHARACTERIZATION OF AN FM ILO AMPLIFIER
4.1 Introducti on
This chapter discusses in detail the frequency
modulated ILO amplifier model that is required for the distortion
analysis to follow. The dynamic behavior of the ILO has been
characterized in the previous chapters. The resulting
mathematical ILO model will then be treated with conventional
amplifier terminology and techniques.
4.2 ILO Amplifier Model
The modelling of the ILO amplifier in this section
will be based upon the concepts of injection-locking and
transmission distortion systems discussed in Chapters II and
3Z-40
III, Appendix A and in the literature . The ILO model will be
used to identify three main types of distortion; namely,
a) amplitude distortion
b) phase or envelope delay distortion
c) nonlinear distortion
Let the model of an injection-locked oscillator as shown in
Fig. 4.2-1 represent a microwave FM ILO amplifier operating
as a reflection type amplifier. When ideal FM modulators and
FM demodulators are added to the input and output respectively
43
. '
.
BASEBAND DEMODULATED
44
o
C — J
CD <C
UJ ei
tn cd
<C i— '
CQ C/D
CD
«=C —J
_l cC
Q CD
O •— '
51 C/0
+J
4->
ZJ
o
CD
to
CD
DC
O
h-
«=C
ID
CD
<c
O
IT
UJ
<
CD IE
UJ
i— 1 Ll.
CD
ni
1—
1 — 1
3:
oc
•
UJ
DC
1 — 1
0
u_
1—
1 — 1
C
_i
_i
CL
ID
SI
CD
C
O
2:
O
UJ
_i
0
1 — 1
UlI
DC
IE
>
UJ
UJ
U_
<
t— <
I c
u_
1—
Q
O
1 — 1
Z
oc
_l
u_
c
0 0
CL.
0
1— 1 _i
2:
DC
ie i-h
<c
2:
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<
1—
cc
<
CD
_J
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ID
1 — 1
Q
O
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IE
O
1— 1
IE
1—
U_
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IE
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<
HI
UJ
CD
Q
IS)
1 — 1
r-
CE
C\J
O
•
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_ j
_J
•
eC
ID
CD
UJ
Q
1 — 1
Q 5-7
O
U-
1 — 1 u_
s:
of the ILO amplifier of Fig. 4.2-1 it is seen that this
represents a baseband repeater. That this repeater has an
intermediate microwave stage does not alter the analysis
since it merely adds complexity and distortion at the
intermediate stage. Therefore, the FM modulator, ILO amplifier
model, and FM demodulator, will be an equivalent to a microwave
repeater link.
A comparison of the baseband output signal e ^(t)
of Fig. 4.2-1 with the baseband input signal e. (t) will
yield several conventional amplifier parameters. This justifies
the entire approach, since the simplicity of this model enables
the interrelationships of the various parameters to be understood
more clearly. Now, these conventional amplifier parameters;
namely,
a) the gain variation versus the bandwidth,
b) the phase and envelope delay variation versus the
bandwidth,
and, c) the nonlinear distortion versus the bandwidth
characteristics will be expanded upon in terms of the equations
of Chapters II and III. Let the baseband modulating signal e -n(t)
in Fig. 4.2-1 be given by a single-tone sinusoid (e.g. Eq. 3.2-2);
further, let the resulting FM injected signal be given by the
incident microwave signal Eq. 2.2-4, the output FM modulated
signal will be given by Eq. 2.2-6, and its frequency variation
' »
,
■
.
by Eq. 2.2-21. The demodulated output signal will be given by
eout(t> - Ko
-Aw + V1ne(t>
1+Rcos0(t)
(4.2-1)
where Kq is the gain of the frequency discriminator used in the
FM demodulator.
As is normal practice in analyses of this nature, a
simplified solution will be examined first, then the solution
will be expanded to more complex cases. Therefore, let the
generalized Adler's phase-locking equation Eq. 2.2-20 be
approximated for the high gain case (i.e. R^l , I^-I and © ).
Additionally, let the peak deviation frequency (PFD) be less
than seventy percent of the maximum locking bandwidth; i.e.
4.7/^. This will permit the nonlinearity from Adler's
equation, Eq. 2.2-22, to be approximated by i.e. sine^eto
better than ten percent. Thus the steady-state analytic solution
for 0(t) is
A + A.U)
m o
e(t)
= AiOc
+■
A*An
/
.sin
Ac
1nt -
tan~^
Aj
(4.2-2)
By applying similar approximations to Eq. 4.2-1 e t(t) becomes
(4.2-3)
Substituting Eq. 4.2-2 into Eq. 4.2-3 yields
*
■
w
47
A K
e (t) = m 0
si n
(
U) t - tan
V
-1 to,
(4.2-4)
It is noteworthy that the simple analytic solution for e ^(t)
does not depend onAu>Q. However, as will be shov/n shortly,
the Atoo term does play an important role in the even components
of the nonlinear distortion.
By using the above linearized locking equation solution
it is possible to represent the FM ILO amplifier and the ideal
modulator and demodulator by a simple transfer function; namely,
G(|V = eout(t)/e1n(t)
e-j tan'1 k'",/40
1
(4.2-5)
It is noted that the frequency dependence of the magnitude of
the transfer function is similar to the expression derived and
8-13
plotted in the literature . Additionally, Eq. 4.2-5 predicts
that the phase shift of the ILO amplifier model will not be a
linear function ofcom, especially whenOm approaches/^. It is
expected that there will be more phase distortion as approaches
(as shown in Chapter III).
To aid in the interpretation of Eq. 4.2-5, the
.
.
’
.
distortion-produci ng terms of the frequency-dependent magnitude
and phase will be isolated and displayed in simple visual form.
The purpose of this technique is simply to model the ILO in
conventional amplifier parameters. The magnitude of a
"distortionless transfer function" should ideally be frequency-
independent. Consequently, the normalized frequency-dependent
part of the magnitude of Eq. 4.2-5 will be plotted in dB as
G<“m>
-20 log10
1 +
(4.2
Next, the phase delay characteristic of the transfer function
must be compared to the distortionless linear variation; i.e.
this difference will be expressed in degrees as
^0= 180/ it 4tan-1 w m - Urn
^0 ^Oy
(4.2
Eq. 4.2-7 indicates that A&will produce a frequency-dependent
phase shift (or EDD) in the demodulated output signal of the
ILO amplifier when compared to the reference input signal.
Fig. 4.2-2 is included to display the two equations Eq. 4.2-6
and Eq. 4.2-7 as a function of modulating frequency.
It is interesting to note that the maximum modulating
frequency predicted from the transient response solutions
(from Section 2.4) suggests that the 3 dB bandwidth of the
linearized model is identical to the previous result; namely.
'• .
. ■
,
49
$33d03Q NI Q3SS3ddX3 NOUdOlSIQ AV13Q 3SVHd
aa NI Q3SS3ddX3 3aniI1dWV 3AUV13d
FIG 4.2-2 LINEARIZED TRANSFER CHARACTERISTICS FOR THE ILO AMPLIFIER MODEL OF FIG 4.2-
50
The model of the baseband ILO repeater developed
here will have application in the following chapters when more
accurate equations are used to describe the ILO characteristics.
The resulting description of the distortion of the ILO in
conventional amplifier terminology will justify this approach.
.
CHAPTER V
FM DISTORTION IN A TUNED ILO AMPLIFIER
5.1 Introduction
In this chapter the generalized Adler's phase-locking
equation is used to describe the frequency dependence of the
nonlinear and delay distortion of the FM modulation in an ILO
amplifier when the initial frequency difference is zero (i.e.
AxOq =0). This will be called the tuned case. A method of
approximate solution to the resulting nonlinear equations will
also be discussed in detail. Finally, the results of this study
will be presented in tabular and graphical form.
5.2 Mathematical Modelling
The analysis of the FM distortion produced by the ILO
modelled in Chapter IV will be presented in the following work.
Consequently, let baseband input and output signals, ideal FM
modulators and FM demodulators and the ILO amplifier of Fig.
4.2-1 be used for the description of the distortion effects.
The distortion will be found by comparing the output and input
baseband signals. Let the input baseband modulating signal be
modulated by a sing! e-frequency sinusoid, i.e.
O _ dy^t) = A^sinw^t
A L- . = Pr~ - m m
i n dt
(5.2-1)
51
.
'
52
where {J is the modulating frequency (simply referred to as the
frequency), Am = kAQ and k4l . Using Fig. 2.2-5 and Fig. 4.2-1
the injected modulated microwave signal will be
is = 1 s s i n ( t +oc(t)) (5.2-2)
while the output microwave signal will be given by
iR = IRsin(wst +ix(t) - 8(t) t)) (5.2-3)
where the symbols are identical to those defined in Chapter II.
Then the output signal frequency delivered to the load
of Fig. 2.2-5 (at microwave frequencies) will be given by
tUp=6o +dt>(-d0 + d^
K s dt dt dt
(5.2-4)
and the demodulated output frequency will be given by
= UJ0 - ws + Ao sin^t)
1+Rcos0 (t)
= -Au>0 + Aosin0(t)
l+Rcosd(t) (5.2-5)
Eq. 5.2-5 was solved by the use of Eq. 4.2-1, Eq. 2.2-16 and
Eq. 2.2-21. Finally, the solution for 0( t ) is given by Adler's
generalized (nonlinear) equation as
dS(t) =
l+R2+2Rcos6(t)
0
A„, + 04t) - V1n^'
dt
l+RcosS(t)
dt 1 +RcosB( t )
(5.2-6)
To evaluate Eq. 5.2-5, the solution to Eq. 5.2-6 must be solved
.
-
.
vi*
53
numerically and substituted into Eq. 5.2-5 to obtain the output
baseband demodulated signal. It must be recalled that ideal FM
modulators and demodulators have been assumed in this analysis
(and for simplicity their gains have been set at unity).
For most cases of practical interest anc* as
such the solution to Eq. 5.2-6 may be obtained by the technique
of successive approximations that is outlined in Appendix B.
The tuned case will be discussed in detail in this chapter while
the detuned case (i.e.Aa)Q^0) will be dealt with in the following
chapter. When the tuned case assumptions (i.e.Atd = 0) are
applied to the previous equations Eq. 5.2-6 reduces to
dO(t) =
l+R2+2Rcos6(t)
'tW(t) -AoSin®(t)
dt
l+Rcos&(t)
dt l+Rcos6(t)
(5.2-7)
while Eq. 5.2-5 reduces to
Co0ut(t) =Aosine(t)
l+Rcose(t) (5.2-8)
The demodulated input and output signals will be given by
(using the symbols defined with Fig. 4.2-1):
ein(t) =
V1nWnit
(5.2-9)
and
eout(t)
= K sins(t)
°( l+Rcosfi(t)
(5.2-10)
<
' c ** »
1 u
(here Kq is set equal to unity). To solve for e(t) in Eq. 5.2-6
a truncated series approximation technique, described in Appendix
B was used. As a result, it is possible to reduce Eq. 5.2-6 to
7
d Q(t) = y Pn ©r
dt n
Pi^O
(5.2-11)
where the coefficients PQ through P7 are defined as in Eq. B-ll .
The validity of this technique depends on the fact that for£<l
radian and for a given "weak" nonlinearity the first few terms
of the series approximation to Eq. 5.2-6 are quite accurate.
Additionally, when a recursive technique is applied to Eq. 5.2-11,
an approximate solution for @(t) is obtained (refer to Chapter
IV and to Appendix B). When the solution is limited to the first
ten harmonic terms it may be expressed as
io
©(t) = Ao + ^ [AksinK60mt + Bj<cosl<^mt
k-l L
(5.2-12)
where A , A^ and B^ are frequency-dependent coefficients whose
accurate calculation is performed as described in Appendix B.
Using a similar approach, the demodulated output signal
will be approximated by
eout(t)
KoA
r\~i
Rn 9
Zn-i
(5.2-13)
which results from substituting B-13 and Eq. 5.2-12 into Eq.
5.2-7 and the coefficients R-j through R4 are defined as in Eq.
B-13. The demodulated output will then be approximated by
'
'
»
55
io , -
eout(t) = c0 + 1 JhW sin(KV - tan_1
k=i L Sk '
(5.2-14)
where CQ, and are calculated by the techniques outlined
in Appendix B. The first few terms of the approximate output
solution contain most of the power spectrum. A comparison of
the output signal Eq. 5.2-14 and the input signal Eq. 5.2-1 for
the tuned case shows that CQ = 0 and that the power spectrum
is contained in the odd-numbered components (the third and fifth
harmonics are the predominant terms). In the more general
detuned case, Chapter VII will show that the output power will
also appear in the even harmonic components and in general,
CQ f 0. For certain circuit configurations and modulation
parameters, the second and fourth harmonics could contain as
much distorted power as the odd harmonics.
The complexity of the output waveform for the simple
sinusoidal modulation precludes studies with more complex
modulating waveforms using the techniques outlined in Appendix
B. As a consequence, several display techniques will be used
to aid in describing the distortion produced by the ILO
amplifier. Firstly, let the demodulated output fundamental
amplitude be compared to the input modulating amplitude and let
this ratio be expressed in dB, as
■
I
.
v>
= 20 1 og
(5.2-15)
(<o )
v nr
10
(S,2 +C12)1/2
Several tables and graphs will display this relation in detail.
Secondly, the phase delay distortion of the fundamental will be
expressed as the actual phase minus the linear (i.e. distortionless)
portion of the phase, i.e.
A0 (degree)= ( tan
(5.2-16)
Thirdly, the percentage of the output signal that is contained
in the harmonic components versus the magnitude of the fundamental
is a useful measure of ILO amplifier performance. This will be
expressed as a percentage nonlinear distortion (i.e. in a form
analogous to Eq. B-30)
ND(%)= 1 OOx
10
k=2
(Sk2 +Ck2)
2 2
S +f
bi h
1/2
(5.2-17)
5.3 Results of the Tuned Calculations
The formulas of Section 5.2 were programmed and solved
numerically . The results of these calculations; namely,
a) the nonlinear distortion
b) the phase delay distortion, and
c) the relative fundamental amplitude variation
■
■
versus the normalized modulating frequency are given in Figs.
5.3-1 and 5.3-2 and Tables 5.3-1 through 5.3-3 for a number
of variable parameters. Four cases have been chosen as
represent!' ve, and the amplifier parameters used are summarized
below (note that these are all tuned cases, AWQ=0 )
Case #1
Gain = 34dB
Ao/Zir = 2.0 MHz
Case #2
Gain = 34dB
A./^ 2.0 MHz
Case #3
Gain = 15dB
Ae/2TT= 17.86 MHz
Case #4 Gain = 15dB
4/2h= 17.86
O
is the normalized peak frequency deviation)
5.4 Conclusion
Examination of the figures and tables of the previous
section reveals the following ILO behavior in the tuned state:
»
*
AMPLITUDE
58
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LD
•
C\J
I
T
o
co
LD
•
co
i
n
S33dLJ3Q NI NOUdOlSIQ AV33Q 3S\/Hd
o o o o o o o o
OOGOOOOO
• •••••••
r^«sDLO'^'Coc\ji — o
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I
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Ln
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c
(S
co
a
co
3=
h-
i — i
LU
O
59
aa ni aanuidNV iviNawvaNru aAiiviaa
o
LO
•
•
o
o
i
i - r
o
i
"l
LO
i
T“
o
•
CM
I
~T~
S33U930 NI NOIiaOiSia
LO
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I
1
o
CO
I
nr
A\fl3G 3SVHd
LO
co
1
lN39U3d NI NOIlNOlSia NV3NI3N0N
FIG. 5.3-2 FREQUENCY DEPENDENT DISTORTION CHARACTERISTICS FOR A TUNED SINGLE-STAGE ILO AMPLIFIER
.
• ■
.
60
CO
oo
CXI
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1
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rx
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,
.
TABLE 5.3-2 COMPARISON OF PHASE DELAY DISTORTION (EXPRESSED IN DEGREES) FOR VARIOUS PARAMETERS
OF A TUNED ILO AMPLIFIER.
61
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o
CO
r—
r^.
r^.
CO
»-t ii
<C E
CD C
1 CO
LO
Cvl
CT)
CO
1 —
LO
1
LO
r—
LO
LO
i
1
CO
1
CO
1
CVJ
CVJ
1
1
, _
o
o
o
o
o
o
o
o
o
CO
1 —
1 —
r —
r—
1 —
1 —
1 —
1 —
1 —
"O
o
o
X
X
X
X
X
X
X
X
X
CO
•
o
o
CVJ
1 —
CVJ
r — ■
LO
*=3-
1 —
CVJ
*3-
II
o
cn
CO
o
CVJ
CO
CTl
LO
CP1
1 —
O')
CVI
z ...........
h-H ii i i — r-'.ooi — i — r^-c\ji — co
<C E
CD C
CD
i — i
I—
< —
— J o
=3 <3
Q \
O E
co
cvj
CVJ
CVJ
CVJ
CVJ
1 —
1 —
1
3
1
o
o
c
o
o
o
o
o
o
o
o
Q
LxJ >-
r_
•
1
1
1
1 —
1
1 —
1
Cvl O
X
X
X
X
X
X
X
X
X
X
X
»— i
_ 1 UJ
o
o
00
o
CO
CO
1 —
CO
CO
c£ ZO
o
o
LO
LO
o
CVJ
o
LO
LO
CT)
s: cy
CC LU
( _
, _
, —
cvl
«S}-
CO
r-
1 —
CVJ
CO
co
o cc
z: u-
■
■
,
TABLE 5.3-3 RELATIVE AMPLITUDE VARIATION (EXPRESSED IN DB) FOR VARIOUS TUNED ILO AMPLIFIER PARAMETERS.
62
LD
CO
CO
CO
CM
CM
>—
<—
r—
O
CD
o
o
o
o
o
o
o
o
r~~~
r~—
1 —
r—
r—
1 —
f— —
f—
1 —
1 — -
CQ O
•a <
X
X
X
X
X
X
X
X
X
X
LO LD
r—
r^.
On
CO
CO
on
LO
o
CO
LD
•
o
OJ
cu
r—
o
o
CM
o
CM
LO
1 —
II o
i
GAIN
A =
m
1 LD
i i
LO
i
1 —
1
CO
1
CO
1
CM
1
LO
1
1
CO
1
r^.
1
1 —
1
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1
LD
CO
CO
CO
CM
CM
1—
r—
1 —
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r —
1 -
r— —
f—
1 —
1 —
1—
r—
1 -
1 -
CQ
~o o
LO <1
X
X
X
X
X
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CO
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LD
1 —
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1 —
CM
1 —
r-.
an
r-^
CO
o
o
GAIN=
A = 0
m
1
1 CO
1 1
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CM
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1
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1 —
r—
1 -
1 -
r—
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1 -
1 -
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-O <1
X
X
X
X
X
X
X
X
X
X
^ LO
CO
CM
1 —
OD
LO
cn
CM
cn
o
C'-
•
CM
r—
CO
co
CO
o
CM
CM
e —
CM
LO
1 —
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1
•
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21
1 LD
LO
1 —
CO
00
CM
LO
r —
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1 —
co
t— 1 II
1 1
1
1
1
1
1
1
1
1
1
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CO
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1 -
r—
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X
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LD
LO
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LO
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r-
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1
1
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X
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CD
LO
LO
CD
LO
r—
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CM
CM
r—
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1
o
o
o
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■
.
1) the nonlinear distortion increases with increasing
normalized modulating frequency and increasing PFD
but only slightly with increasing gain,
2) phase delay distortion increases with increasing
normalized modulating frequency but changes only
slightly with increasing PFD and increasing gain,
3) the relative amplitude variation shows a decrease
with increasing normalized frequency but is affected
only slightly by increase in gain or PFD.
CHAPTER VI
FM DISTORTION IN A DETUNED ILO AMPLIFIER
6. 1 Introduction
In this chapter the solution to the generalized
Adler's equation for the injection-locking phenomenon is discussed
for the detuned case (i.e.AL30 f 0). The results of calculations
for both the tuned and detuned cases are compared in tabular
form and by the use of appropriate graphs. These results will
form the basis for the theoretical characterization of a single-
stage ILO amplifier with respect to its distortion effects on
the modulating signals involved.
The tuned condition for the signals involved in ILO 1 s
is not likely to be realized exactly in practice (it has been
suggested that, if negative feedback is used, it may be possible
to correct the amplifier free-running frequency ). For example,
the free-running frequency of the microwave oscillators used in
this study is dependent on the ambient temperature and its
variation, the temperature characteristics of the cavity, and
numerous other factors that are discussed in more detail in the
experimental chapter. In practice, these factors may cause the
free-running oscillator frequency to drift over a significant
portion of the locking bandwidth during an experimental run
64
.
*
l I
65
(i.e. a short period of time). These variations may be expressed
as part of the noise associated with these oscillators and the
drift will then be the predominant factor over a period of time.
It is vital that the dependence of the microwave oscillator
frequency on the numerous factors mentioned above be minimized
for the ILO to be practical for system applications. Consequently,
a study of the distortion characteristics of the ILO's in the
detuned case has a very practical application.
6.2 Mathematical Modelling
The previous chapter discussed the nonlinear and phase
delay distortion and the amplitude variations of the modulated
signals in a tuned ILO amplifier. The same technique that is
described in Appendix B will be used to solve the appropriate
nonlinear differential equations for the detuned case (i.e.
Au) f 0).
o '
As before, the input modulating signal to the ILO
amplifier model of Fig. 4.2-1 will be a single-frequency
sinusoid
e . (t) = 4^ = A simdt
inv ' dt m m
(6.2-1)
where the symbols of Chapter V apply. The output baseband
signal will then be (from'Eq. 4.3-1)
, . x , , a si n&
eout(t) = -^o — _
out 0 1+Rcose
(6.2-2)
where the demodulator gain has been set equal to unity and
V!*.
66
A *
When the output baseband signal's coefficient terms are
approximated by the first four terms of their appropriate power
series expansion (following Appendix B), Eq. 6.2-2 will become
Eq. B-13, i.e.
where the 1 R ' terms are defined as follows
R1 = 1/(1 +R )
R2 = (2R-1 )/6
R3 = (1-13R+16R2)/120
R4 = (-l+60R-279R2+272R3)/5040
The solution to © (t) will be given by the approximation to the
generalized Adler's locking equation Eq. B-ll, i.e.
7
de 5
dt
r\ = o
n
G
n
(6.2-4)
where the coefficients PQ through P -j are defined as in Appendix
B. The demodulated output will be found numerically by the
recursive solution of Eq. 6.2-4 and Eq. 6.2-3 as described by
the method of Appendix B.
When the output solution to Eq. 6.2-4 is limited to
the first ten harmonic terms it may be expressed as
:-W« tr-.O *“;*
.
.
67
(6.2-5)
where Aq, A^, and are frequency dependent coefficients
whose accurate calculation is performed as described in
Appendix B. The demodulated output baseband signal will be
approximated and rewritten as
(6.2-6)
where CQ, SR and Ck are calculated by the techniques outlined
in Appendix B. The evaluation of Eq. 6.2-6 resulted from the
recursive substitution of the linearized solution to Eq. 6.2-4,
using the procedure described in Appendix B and with the aid
of a computer program.
The results of these calculations will be tabulated
in the form of normal amplifier distortion parameters. Firstly,
let the demodulated output fundamental amplitude be compared to the
the unmodulated input amplitude and that this ratio be expressed in
in dB as r
G(u>m) = 201 og
A
(6.2-7)
m
Secondly, let the phase delay distortion of the fundamental be
expressed as the actual phase minus the linear (i.e. distortionless)
portion of the phase, i.e.
■
.
.
68
A0= tan"^ <4- - ~ (radians
Thirdly, let the percentage of the output signal that is
contained in the harmonic components versus the magnitude of
the fundamental be the measure of the percentage nonlinear
distortion, i.e.
r io
ND(%) = 100 x
ijs k2 + ck2>
2 2
S + f
U1
(6.2-8)
(6.2-9)
For the tuned case (ua =U3 ), only the odd harmonic amplitudes
o u
are nonzero in value while for the detuned case (agj f 0) all
the harmonic amplitudes are expected to have nonzero contributions.
In practice the second harmonic will often become comparable to
the third and fifth harmonic amplitudes.
6.3 Results of the Detuned Calculations
As examples of the results of the calculations that
are expected the Tables 6.3-1 through 6.3-3 and Figs. 6.3-1 and
6.3-2 are included as being representative of the ILO parameters.
Several cases were selected to display a) the nonlinear distortion,
b) the phase delay distortion, and c) the relative fundamental
amplitude variation versus the normalized modulating frequency.
The ILO parameters included a normalized peak frequency deviation
of 0.75, normalized detuning factors of 0.0, 0.1, and 0.24,
.
69
TABLE 6.3-1 NONLINEAR DISTORTION (EXPRESSED IN PERCENTAGES) IN
AN ILO AMPLIFIER MODEL WITH A GAIN EQUAL TO 15DB,
A MAXIMUM LOCKING BANDWIDTH OF 17.86 MHZ, A PEAK
FREQUENCY DEVIATION OF 0.75Aq, AND FOR THE FOLLOWING
NORMALIZED DETUNING FACTORS: O.O, 0.1, AND 0.24.
NORMALIZED
MODULATING
Ago =i
O.i
0
Ago =
0
.1A
Aw =
FREQUENCY
0
o
0
0
5.0 x 10'4
7.33
X
IQ'3
1.73
X
10“2
5.13
1.0 x 10"3
1.10
X
10'2
2.03
X
10“2
5.70
1.0 x 10"2
9.51
X
10"2
1 .24
X
10"1
2.90
1.58 x 10'2
1.50
X
10'1
1.95
X
10"1
4.55
2.5 x 10‘2
2 . 375x
10"1
3.07
X
10"1
7.13
4.0 x 10'2
3.73
X
10"1
4.82
X
10"1
1.11
6.3 x 10'2
5.8
X
10'1
7.47
X
10"1
1.66
1.0 x 10"1
8.76
X
10"1
1.13
2.40
1.58 x 10'1
1.25
1 .60
3.24
2.51 x 10'1
1.6
2.06
4.01
3.98 x 10'1
1.74
2.17
4.38
6.3 x 10'1
1.54
2.13
4.12
1.0
1.08
1.65
3.24
0.24a
o
x ICf2
x 10'2
x 10'1
x 10'1
x 10'1
.
■
,
.
70
TABLE 6.3-2 PHASE DELAY DISTORTION (EXPRESSED IN DEGREES)
IN AN ILO AMPLIFIER MODEL WITH A GAIN EQUAL TO
15 DB, A MAXIMUM LOCKING BANDWIDTH OF 17.86 MHZ
A PEAK FREQUENCY DEVIATION OF 0.75Aq, AND FOR
THE FOLLOWING NORMALIZED DETUNING FACTORS:
0.0, 0.1, AND 0.24.
NORMALIZED
MODULATING
FREQUENCY
A to =
0
0.
0
Acd =
0
O.lA
0
A co =
0
0.
24A
o
5.0 x
10"4
1.0 x
kt3
1.0 x
_2
10 L
3.33
X
10-5
3.7
X
10'5
6.3
X
10'5
1 . 58 x
10"2
1 .33
X
10"4
1.46
X
10"4
2.55
X
10-4
2.5 x
10'2
5.33
X
10'4
5.85
X
10'4
1.0
X
10'3
4.0 x
10"2
2.13
X
10'3
2.3
X
10"3
4.21
X
10* 3
6.3 x
10'2
8.44
X
uf 3
9.2
X
10"3
2.24
X
10"2
1.0 x
10"1
3.32
X
10' 2
3.6
X
10"2
8.19
X
10'2
1 . 58 x
10'1
1.29
X
10"1
1.43
X
10'1
2.7
X
10"1
2.51 x
10"1
4.84
X
10'1
5.2
X
10"1
8.64
X
10"1
3.98 x
10'1
1.71
1 .95
2.62
6.3 x
10'1
5.58
5.81
7.51
1.0
1.61
X
101
1.65
X
101
1.98
X
101
.
.
'
71
TABLE 6.3-3 RELATIVE AMPLITUDE VARIATION (EXPRESSED IN DB) IN
AN ILO AMPLIFIER MODEL WITH A GAIN EQUAL TO 15DB,
A MAXIMUM LOCKING BANDWIDTH OF 17.86 MHZ, A PEAK
FREQUENCY DEVIATION OF 0.75ao , AND FOR THE
FOLLOWING NORMALIZED DETUNING FACTORS: O.O, 0.1,
AND 0.24.
NORMALIZED
MODULATING
FREQUENCY
Aco = 0.0
o
= O.lA
O 0
A&) = 0.24A
o o
5.0
X
o
1
1.0
X
10~3
-6.7
X
icf6
-7.9
X
10"6
-1.09
X
icf5
1.0
X
10~2
-5.07
X
icf4
-5.3
X
10"4
-5.6
X
10'4
1.58
X
10"2
-1.29
X
10'3
-1.3
X
10'3
-1.42
X
10' 3
2.5
X
o
1
ro
-3.23
X
icf3
-3.3
X
10'3
-3.6
X
10'3
4.0
X
icf2
i
00
CO
X
10"3
t
CO
CO
X
10~3
-9.1
X
10'3
6.3
X
10"2
-2.04
X
C\1
1
o
r—
-2.1
X
10'2
-2.5
X
10'2
1.0
X
icf1
i
c_n
•
o
<x>
X
10“2
-5.2
X
10'2
-6.25
X
10"2
1.58
X
10'1
-1.26
X
10'1
-1.3
X
10'1
-1.5
X
10"1
2.51
X
10"1
o
CO
i
X
10"1
-3.1
X
10"1
-3.52
X
10_1
3.98
X
10"1
-7.2
X
10"1
-7.5
X
10"1
-8.0
X
10'1
6.3
X
10-1
-1.58
-1.6
-1.70
1.0
-3.16
-3.18
-3.31
•
72
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74
respectively , for each power gain of 34 dB and 15 dB.
6.4 Conclusi on
Fig. 6.3-1 and 6.3-2 show that both the relative
fundamental amplitude (i.e. frequency response) and the phase
delay distortion as a function of normalized modulating
frequency are not appreciably affected by frequency detuning,
PFD changes, or power gain changes (provided these changes are
comparatively small). However, the nonlinear distortion does
increase with increases in the normalized modulating frequency,
with increases in the normalized PFD and with increases in the
normalized detuning. The nonlinear distortion is not strongly
dependent on the power gain, so a few distortion curves suffice
to give a good estimate of the anticipated distortion.
.
.
CHAPTER VII
APPLICATION OF THE DISTORTION ANALYSIS IN THE
DESIGN OF A SINGLE-STAGE FM ILO AMPLIFIER
7.1 Introduction
The calculated results derived in the previous
chapters will now be used as the basis for a new display of the
distortion parameters. These "design contours" are useful
when utilized in the design of a single-stage ILO amplifier.
7.2 Design Contours
It is advantageous to use the distortion characteristics
of the nonlinear ILO amplifier model (i.e. Fig. 4.2-1) to
predict the allowable limits for the parameters of an FM injected
signal. That is, the ILO amplifier distortion added to a
modulated signal will be below some specified system objective
if the modulating signal amplitude and/or frequency remains
below certain limits. It is apparent that for each set of ILO
parameters, a new family of distortion curves (similar to those
shown in Figs. 5.3-1, 5.3-2, 6.3-1, 6.3-2) will be required to
accurately portray these features. For weakly dependent
distortion characteristics (e.g. relative amplitude variation,
etc.) only one of the family of curves will be required for a
good practical estimate of the expected performance. However,
75
.
■
r ■
.
76
for those parameters upon which the distortion characteristics
are strongly dependent, a more convenient form of display is
required (rather than using a multiplicity of graphs). It is
expedient to graphically display constant nonlinear distortion
contours and the constant phase delay distortion contours versus
the highly dependent variables. As a practical example the
constant nonlinear distortion of the ILO amplifier is displayed
in Fig. 7.2-1 for a constant gain of 15 dB (when the data points
are compared with other power gains, there is little change).
These contours are plotted as functions of the normalized PFD
and normalized modulating frequency, with the amount of
detuning as a parameter. A similar procedure was followed for
the phase delay distortion contours for the 15 dB gain case(pig. 7.2-2)
Finally, an examination of Table 5.3-1 indicates that only small
changes occur in these contours as a function of gain. It may
be predicted that these curves for the 15 dB case have wide
practical applicability in describing the ILO behavior.
7.3 An Example of Design Contour Applications
To demonstrate how the design contours of Fig. 7.2-1
and Fig. 7.2-2 may be used to define the limitations of the ILO
behavior a design example follows to illustrate the use of these
curves. Let it be assumed that the ILO amplifier may be char¬
acterized by a loaded Q of 50, a free running oscillator frequency
*
NORMALIZED MODULATING FREQUENCY («m/&
77
FIG. 7.2-1 CONSTANT NONLINEAR DISTORTION CONTOURS VERSUS THE NORMALIZED
PEAK FREQUENCY DEVIATION AND MODULATING FREQUENCY FOR A POWER
GAIN OF 15 DB AND FOR NORMALIZED DETUNING FACTORS EQUAL TO
0.0 AND 0.1.
.
■
GAIN = 15 DB
78
o
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o
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Vi.
of 10 GHz and that the power gain of the ILO is adjusted to
15 dB. Consequently, A0/2iTwill equal 17.8 MHz. Further, let
it be assumed that the "free-running" frequency drifts slowly
over the region bounded by f ± 1 .78 MHz, i .e. ^u3o/aq £ 0.1 .
Now if the maximum normalized modulating frequency is 0.336
(i.e. 6 MHz), there will be some maximum PFD that will be
permitted in order that the output modulation distortion will
not exceed the specified system values. For illustrative
purposes, let the following reasonable limits be chosen; namely,
a) 1.0% for the maximum nonlinear distortion
b) 1.0° for the maximum phase delay distortion, and
c) a maximum permissible relative signal loss of 3 dB.
Then, by examining the design contours in Figs. 7.2-1
and 7.2-2 the normalized maximum allowable PFD is estimated to
be 0.52 and 0.64 respectively. In this case, the nonlinear
distortion will limit the PFD more than the other parameters.
It is expected that, with proper baseband compensation networks,
the frequency response could be made quite linear over the band
of interest. Consequently, the relative signal loss limit will
not be the limiting factor in most practical cases.
*
■
■
[ >
'
Summary of the Assumed Design Example Conditions
Ql - 50
f = 10 GHz
Ao/2-n = 17.8 MHz
AuJo/Ao - J
f = 6 MHz
m
| G | = 15.0 dB
loss £ 3.0 dB
nonlinear distortion ^1.0 %
phase delay distortion 1.0°
Resultant Limiting Parameter
PFD/a^ < .52
To summarize then, the PFD should not exceed 9.25 MHz
in this example for the distortion to stay within the limits.
It is expected that Figs. 7.2-1 and 7.2-2 may form the basis of
the design procedure for a single-stage IL0 amplifier.
7.4 Conclusion and Summary
The previous three chapters form the basis for the
v •
■
theoretical description of the distortion added to the modulating
signal by a certain type of single-stage FM ILO amplifier model.
or or
The results reported in this work have been published ’ . In
summary, the most important results of the application of this
theory are the following conclusions about the ILO model; namely,
a) both the nonlinear and the phase-delay distortion
will increase with increasing modulating frequency
and increased peak frequency deviation of the
modulating signal.
b) both types of distortion increase significantly
when the sum of the PFD andA^ approach AQ.
c) the nonlinear distortion appears to have a maximum
when uj ^ ^ . The relative amplitude variation shows
m Z
that harmonics beyond A in frequency will be
attenuated severely.
d) power gain variations appear to have little effect
on the distortion characteristics.
e) if a small PFD or large phase delay are allowed, then
it is possible to modulate the ILO at frequencies
in excess of /SQ and still have some acceptable form
of transmission.
f) in the work it has been assumed that the sum of
the frequency detuning and the peak frequency
deviation total less than AQ •
'
.
CHAPTER VIII
DISTORTION MEASUREMENTS OF A MICROWAVE FM ILO AMPLIFIER
8.1 Introduction
In the previous chapters, the FM ILO amplifier has been
modelled theoretical ly by describing its effect on the modulating
signals (both at RF and as well at baseband frequencies). In
this chapter, the experiments that were used to verify a part
of these theoretically predicted distortion characteristics
will be presented, as well as the practical limitations of
these measurements.
8.2 Theory, Experimental Set-Up, and Measurement Techniques
Many basic ILO measurements have been reported in the
literature 3,8-14,27. jn this chapter, it will be assumed that
the reader will refer to the literature and in particular to
3
B.C. So's doctoral dissertation for a) the theoretical basis
of many of the basic measurements, b) the experimental set-up
for several basic experiments and c) the results of these
experiments. A short listing of the relevant experiments
will be included (particularly those which apply to the
experimental work of this chapter); namely,
1) measurements of the locking range
a) output spectra
82
■' .
’
*• ill | '»*»•*
b) locking range variation with power ratio
c) oscillator quality factor derived from the
locking bandwidth
2) measurement of the phase angle difference between
the locked oscillator output and the locking signal,
and the phase angle variation with initial frequency
di fference
3) intermodulation signals in injection phase-locked
osci 1 1 ators
Since most of the measurements were required for the calibration
and alignment procedures of the following experimental work, the
remainder of the chapter will build upon this base and detail
the additional work that was done.
The purpose of the following experimental work was to
verify the basic distortion theory derived in the previous
chapters and reported upon in the literature 3,8-14,18. The
types of distortion that are to be measured include:
a) loss of the demodulated signal amplitude with
increasing frequency,
b) the phase delay distortion of the demodulated
output signal versus the input signal,
and, c) the nonlinear distortion of the demodulated output
signal .
Several investigators have reported success in measuring the
. ;m ij jHok \m
$
.
■
84
relative power loss (most of them express it in slightly different
mathematical forms) . Thus it remains for the phase
delay distortion and the nonlinear distortion to be verified.
After extensive work using the available instrumentation, it
was only found practical to measure the nonlinear distortion.
Consequently, this chapter will now detail a method of measuring
the output signal's nonlinear distortion in an attempt to verify
the theoretically predicted distortion parameters. The nonlinear
distortion added to an FM modulating signal and amplified by the
ILO will be measured in a circuit configuration very similar to
the model shown in Fig. 4.2-1. Basically, the nonlinear
distortion added by the FM ILO amplifier to the modulating
signal is obtained by comparing the harmonic content present on
the demodulated output signal with that of the demodulated input
signal. By using demodulators with similar characteristics it
is possible to measure the main effect of the nonlinear
distortion — namely , the harmonic content of the demodulated
signal .
The experimental circuit used to verify these nonlinear
characteristics is shown in Fig. 8.2-1 and Fig. 8.2-2. A
microwave oscillator operating in the X-band frequency range
was used as the ILO (this oscillator is similar to the one So
used in his study). A baseband single-tone sinusoid is
used to modulate the voltage across a varactor diode in the
■»»
.
85
1.
LOW FREQUENCY MODULATING
9.
IMPATT DIODE OSCILLATOR
OSCILLATOR
10.
20 DB DIRECTIONAL COUPLER
2.
VARACTOR-TUNED GUNN DIODE
FM OSCILLATOR
11.
RF SPECTRUM ANALYZER
3.
ISOLATOR
12.
DIGITAL FREQUENCY COUNTER
4.
PRECISION ATTENUATOR
13.
WAVEGUIDE SWITCH
5.
POWER METER
14.
FREQUENCY DISCRIMINATOR
6.
10 DB DIRECTIONAL COUPLER
15.
LOW FREQUENCY SPECTRUM
ANALYZER
7.
30 DB DIRECTIONAL COUPLER
16.
OSCILLOSCOPE
8.
FOUR-PORT CIRCULATOR
FIG. 8.2-1 SCHEMATIC FORM OF THE EXPERIMENTAL SET-UP FOR THE MEASUREMENT
OF THE NONLINEAR DISTORTION IN AN ILO AMPLIFIER.
... .8
v^;
86
FIG. 8.2-2 A PHOTOGRAPH OF THE FM ILO NONLINEAR DISTORTION EXPERIMENTAL SET-UP
varactor tuned Gunn diode oscillator (2). The numbers in the
brackets here refer to specific equipment and to locations in
Fig. 8.2-1. The Gunn diode oscillator was used to produce
the incident microwave locking signal. This FM locking
signal was then injected into the cavity of an IMPATT diode
oscillator (9) through a 40 dB isolator (3), a precision
attenuator (4) set such that the desired locking power gain
was achieved, a 10 dB precision coupler (6) for power monitoring
purposes, a precision 3 dB coupler (7), and into port 1 of a
high gain four-port circulator (8). The locking signal power
level was monitored with a power meter (5) connected to the
auxiliary arm of the 10 dB directional coupler (6). The
locking signal power level was adjusted for different gain
ratios by adjusting the precision attenuator (4). This monitor¬
ing circuit was used to obtain an estimate of the correct
power level, the final adjustments were made by observing and
measuring the bandwidth relations to provide the most correct
answer. The power reaching the IMPATT diode was reduced by
about 3 dB, since the locking signal power is reduced by the
3 dB precision attenuator (7) and ports 1-2 of the high gain
four-port circulator (8). Paths 1-2 of the circulator have a
return loss of about 40 dB or better. The isolator (3) and the
adjustable attenuator (4) assure that the locking signal is
adequately isolated from the rest of the system. The locked
*
.*
88
oscillator output signal is dissipated in the matched 50 ohm
termination of the RF spectrum analyzer (11), after having
passed through the circulator path 2-3, and the two 20 dB
couplers (10). Any reflected power from the output circulator
circuit must pass through the circulator path 3-4 and is
dissipated in a matched load connected to port 4. This
circulator path 1-4 provides isolation in the order of 52 dB
while the insertion losses of paths 1-2 and 2-3 are about 0.2
dB. The IMPATT diode output power is monitored by a power
meter connected to the auxiliary arm of a 20 dB directional
coupler (10). The reason for the isolation is to provide
both the locking signal and the output signal individually
as well as for mixing purposes. Both signals must be
available without appreciably affecting the main injection¬
locking circuit. Consequently, the demodulated input FM signal
is obtained by monitoring through the auxiliary arm of the 3 dB
directional coupler (7) and by demodulating the RF signal by
the use of a dual-mode doubly-tuned cavity (14) as a
discriminator . The demodulated FM signal is displayed on a
low-frequency spectrum analyzer and an oscilloscope. To
demodulate the output FM signal, the waveguide switch is moved
to its alternate position, so that the output FM signal (coming
from the auxiliary arm of the 20 dB directional coupler) is then
demodulated by the same discriminator. By means of this circuit
arrangement it is possible to compare the input and output
•*fc
• U
■
.
89
demodulated waveforms with relative ease. In addition, since
essentially the same circuit components are used in both
measurements many of the errors in the measurements may be
assumed to be common to both input and output measurements and
may be eliminated by subtraction.
In Fig. 8.2-23arrow #1 points to the varactor-tuned
Gunn diode oscillator, which is used as the locking signal. It
has a nominal output power of about 10 milliwatts. Its frequency
may be tuned both mechanically (from 8.8 to 10.2 GHz) and
electrically (over a 50 to 75 MHz range). Arrow #2 points to
a Sylvania avalanche diode oscillator (model SY0-3200, S/N 140).
The IMPATT diode is mounted in an end-loaded coaxial cavity
and iris-coupled to a X-band waveguide (WR90). The output
frequency can be tuned both mechanically and electrically
throughout X-band. The oscillator used typically produces
about 21 milliwatts of RF power at 9.157 GHz for a reverse
bias of about 90 volts and a current of 20 ma. Arrow #3 in
Fig. 8.2-2 points to the doubly-tuned cavity which is used as
a direct down-conversion FM discriminator. To facilitate
using FM signals with a 2MHz deviation it was necessary to
increase the 3 dB bandwidth of the cavity and the detector
diodes (i.e. lower the Q). In particular, harmonics up to
10 MHz would have to be detected to make this method valid.
The cavity bandwidth was increased at the expense of discriminator
sensitivity by coating the tuning plunger face with a lossy
'
.
-
90
material (in this case ECC0S0RB powder was used). The bandwidth
of the detector diodes was also increased by a) reducing the
diode resistance by applying an appropriate forward bias on the
diodes, b) reducing the capacitive loading on the diodes, and
c) using high input-impedance emitter follower amplifiers.
The set-up described thus far will now be used to
verify the nonlinear distortion characteristics of a microwave
FM ILO amplifier. The system was characterized and aligned by
the procedure outlined and performed by B.C. So and as discussed
in the beginning of this chapter. In the specific experiment
performed in this study, the maintenance of an initial frequency
difference as close to zero as possible assumed major importance
since nonlinear distortion is highly dependent upon this factor.
Several other factors mentioned during the tuned and detuned
theoretical discussions (e.g. frequency drift etc.) necessitated
that the alignment procedures for the set-up be repeated
between each experimental point. The various power levels were
also monitored continuously to ensure that the circuit
parameters did not vary appreciably (this was necessary since
often this was the only "probe" available to a certain point).
Additionally, the combination of a low Q oscillator combined
with a higher Q discrimi nator necessitated the measurements
being performed at a fairly high gain to keep the bandwidth
smal 1 .
...
<
Vi*.
The discriminator used in this experiment (after the
cavity had its bandwidth extended) had a linear response
characteristic over ail. 75 MHz bandwidth about an adjustable
center frequency. The maximum detected frequency exceeded
5 MHz when allowance was made for some considerable nonlinear
behavior . A typical response curve for the frequency
discriminator is illustrated in Fig. 8.2-3. Consequently, a
modulating frequency in excess of 1.5 MHz could not have its
harmonics detected accurately. As mentioned previously, too
high a gain will preclude the accurate frequency alignment of
the system for any length of time. As a consequence, a practical
compromise was arrived at and the distortion measurements were
performed with a gain of 34 dB. The initial frequency difference
was maintained to less than fifteen percent of Aq in practice.
The frequency deviation was set such that the demodulated
output showed a frequency deviation of 1.125 MHz, i.e. 75 percent
of the maximum frequency difference; namely, 1.5 MHz (this was
maintained to better than ten percent by constant monitoring and
realignment where necessary). The RF spectrum analyzer and the
level of the demodulated output provided the means by which
these levels were monitored and maintained.
To obtain accurate experimental measurements the use
of the "zero crossing" method will now be explained. The
application of this technique will allow the selection of
'
■
*
.
■
^ . iiiii
AMPLITUDE IN VOLTS
92
FIG 8.2-3 A PLOT OF THE VOLTAGE AMPLITUDE DEFLECTION OF THE FREQUENCY
DISCRIMINATOR VERSUS THE FREQUENCY DEVIATION(SWEPT FROM
9.040 TO 9.060 GHZ) FOR AN INPUT POWER LEVEL OF 0.2 MILLIWATTS.
modulation parameters in the experimental set-up to give several
accurate experimental data points on the distortion graphs.
Let the FM modulating signal be expressed as
ein(t) = Einsin(toot +((TL)s1ntJmt)
' m' (8.2-1)
28
After some manipulation , Eq. 8.2-1 becomes
e. (t) = E. J J f
inx in \ o \
y
^m
sin u t
fA.
+J
m
1 O)
m
sin (W0 + ^ - sin<wo
2Am 2Am '
sin(u,0 + 1 3T}t + sin(w0 - Tr}t
+J
/Am
—
3 W
m
sin(oj0 + 3Am)t - sin(wQ - 3Am)t
^m %
+ * » »
(8.2-2)
where J , J-j , ... .represent the zeroth, first, second
Bessel function, respectively33. Now Eq. 8.2-2 has separated
the FM signal into its frequency components (or sidebands).
Each sideband will have a certain amplitude associated with it
and there will be an infinite number of sidebands which are
separated from the carrier frequency by integer multiples of
the modulating frequency. A very important characteristic
for this study is that the carrier amplitude will be reduced
.
.
V*
by ML • Additionally, the sideband amplitudes depend upon the
^ //\ \
value of J Ini ] where n = l,2, 3, 4... and these will diminish
^ A
rapidly when n > jn . Therefore, the sideband amplitudes diminish
m
rapidly outside the region of the maximum frequency deviation;
/\
namely, m/27t removed from the carrier. Fig. 8.2-4 illustrates
these principles for the following set of practical parameters:
= 10.0 GHz, Ao/2tt= 1.5 MHz, Am = .75 Aq, u)m = 2.4048,
Hr/^'r'= ^Hz. ^ 1S interesting to note that the power in
each sideband will be proportional to the square of the Bessel
coefficient while the total power on the spectra is proportional
to the sum of the squares of the carrier and sideband amplitudes.
Mathematically this will be expressed as
i 2
'A
m
0\lQ
i 2 oo
+
m ' J
t\=. \
fA
nlw
j i-HL
m
= 1
(8.2-3)
The important result of this study is that for certain
values of modulation index Am/u>m the zeroth bessel function will
go to zero. Consequently, the carrier amplitude will go to zero
when J (x) equals zero (i.e. for the following values of
x = 2.4048, 5.5201, 8.6537, 11.7915, 14.9309, 18.0711, etc.).
Therefore, it will be a relatively simple matter to measure these
points on the distortion curve accurately. When the maximum
frequency deviation is set equal to a constant, then certain
modulating frequencies will cause the carrier amplitude in the
frequency spectrum to go to zero.
' .
**
:
'
'
NORMALIZED AMPLITUDE
SET OF PARAMETERS
0.7 .
0.6 .
0.5 -
0.4 -
0.3 -
02 -
0.1 .
- , - | - 1 — L
- ao - 2.o
03 =
10
.0 GHZ
0
A /2n =
1.
5 MHZ
0
An =
0.
75ao
Am/o3 =
2.
4048
m m
ii
t=
OJ
E
3
469 KHZ
CENTERED ABOUT
10 GHZ
- , - . —
2.o ao
FREQUENCY IN
MEGAHERTZ
FIG. 8.2-4 FREQUENCY SPECTRUM FOR AN FM SIGNAL WITH CARRIER ZERO.
II. H I
.
96
As a result of the above theory, several experimental
runs were performed based upon this technique. The accuracy with
which the frequency deviation and the modulating frequency may
be maintained limit the accuracy of this experimental technique.
For the 34 dB gain case the experimental measurements were per¬
formed with relatively good accuracy (discussed in detail in the
following section) when the modulating frequency was less than
one half of the maximum locking bandwidth.
8.3 Experimental Results and Their Interpretation
In this section, the results of several experiments
are presented and discussed. The experiment used was that which
has been proposed to verify the model of Chapter IV. This
concept was used as the basis of the experimental set-up shown
in Fig. 8.2-1. In summary, the experiment was performed to
verify the nonlinear distortion curve of Fig. 6.3-1 for an ILO
amplifier with a 34 dB gain ratio. The plot shown in Fig. 6.3-1
was found by comparing the demodulated output signal with the
input modulating signal and arriving at a percentage distortion
by Eq. B-30. Therefore, a similar technique was used in the
experiment by the use of a common frequency discriminator on
the input and output frequency modulated signals.
The alignment procedures were described in detail in
3
the previous section and in the work of So . To summarize.
1
■
'
the free-running frequency of the oscillator (9) was adjusted
such that it coincided with the passband of the frequency
discriminator. The modulating signal was then adjusted such that
the initial frequency difference as measured by a RF frequency
counter was as close to zero as possible. The proper power
ratio was then adjusted for a 34 dB gain ratio (based mainly upon
the measured frequency deviation allowable for locking and the
frequency counter). Then the frequency modulation was applied
via an amplitude modulation signal to the voltage that tunes
the frequency of the varactor-tuned Gunn diode oscillator.
During the entire experimental runs periodic realignment of
the initial frequency difference was required to maintain it
close to zero. Then the demodulated input FM signal was
measured via the frequency discriminator for the parameters
necessary to coincide with the zero-crossing method. Then
comparable measurements were made of the demodulated output
signal and the two waveforms were then compared. For a 34 dB
gain ratio the following parameters characterized the system:
UJ / 2-tt= 9.57 GHz
£y27l= 1.5 MHz (±5%)
A = . 75 A
m o
Am/2TT = 1.125 MHz (to better than 10%)
M) /x £ .1 (i.e. to better than 10%)
o o
.
- ■
w
The results of a typical experimental run are presented
in Table 8.3-1 and Figs. 8.3-1 and 8.3-2. Fig. 8.3-1 displays
photos of the demodulated input and demodulated output signals
at the third carrier zero; i.e. 130 KHz. The large spike is
the "zero" starting pulse of the spectrum analyzer while the
second spike is the demodulated fundamental frequency. The
remainder of the sidebands are then the distortion products.
By use of Eq. B-30 it is a straightforward matter to compute
distortion quantities. Fig. 8.3-2 is a display of the
experimental points for a typical experimental run versus the
theoretically predicted results for the three frequency detuning
factors; namely, A<*;0= 0.0, 0.1 and 0.24. As can be seen the
experiment data follow the general shape of the predicted
curve until about 0.3 of the normalized modulating frequency.
As will be recalled., the zero crossing method may only be used
to accurately predict the parameters at those points shown in
Table 8.3-1. The highest fraction of the normalized modulating
frequency to which the zero crossing method is applicable is
0.313 (i.e. the first carrier zero). The experimental points
beyond this were measured by the alignment of the IL0 with the
frequency counter and the spectrum analyzer. However, the
amount of distortion measured at these experimental points does
not have the same degree of accuracy that the data coinciding
with the "zero crossing" points has.
.
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_J
CC
c
ZD
U_
CD
o
CD
1 — 1
i^i
CD
3
1—
■ — i
o
DC
DC
LU
_1
LU
CD
CD
— *
>
ZC
fO
< - ■N
>-
c
_I
CL
OO
1 — 1
CD
DC
LU
M
>-
o
<C
M
<
zn
CD
DC
<
- —
z
ZD
CO
CD
CD
DC
Q
CD
i — i
1 —
CM
oo
O
O
LU
r—
II
1—
Q_
ZD
OO
II
LU
CL
_J
>-
LU
c
1— H
CD
_l
CD
<
OO
CD
LU
CD
LU
ZD
OO
1—
cy
c
<c
LU
_l
1—
_l
DC
c
3
ZD
u_
CD
o
CD
i — i
I'M
O
3
1 —
i — i
o
oc
QC
LU
_i
LU
O
CD
*« — "
>
zn
FIG. 8.3-1 AN EXAMPLE OF THE DEMODULATED INPUT AND OUTPUT SIGNALS MEASURED FOR THE 34 DB GAIN CASE
.
1M33 d3d NI NOIiyOlSIQ yVBNUNON
.
■-
vJ
An examination of Fig. 8.3-2 indicated that the
distortion data points between 0.0413 and 0.313 of the normalized
modulating frequency follow the slope of the calculated
distortion characteristics closely. However, the numerical
values of the distortion data points do vary from those which
the theory predicts. In practice it was possible to maintain
the frequency dependent variables to better than 15 percent
( e.g. the initial frequency difference, the peak frequency
deviation, etc.). Several other factors that were not considered
in this analysis may have an effect (e.g. noise contribution
from both the locked oscillator and the locking signal).
Therefore, the distortion data points beyond 0.313 of the
normalized modulating frequency will not be accurate since
the harmonics of the distorted output signal are severely
attenuated by the ILO amplifier and these harmonics exceed
the linear region of the frequency di scriminator. In summary
then, the following factors must be considered in any error
analysis related to the experimental data of this work; namely,
1) the frequency drift of the "free-running" oscillator
2) the instabilities (expressed as noise) of the
locking oscillator
3) the limited linear range of the tuned cavity
frequency discriminator versus the larger bandwidth
requirement of the ILO amplifier
'
*
4) the noise and nonlinear behavior of the detector
diodes and amplifiers, and
5) the selection of a high gain case makes the relative
error calculations much more sensitive than for a
comparable lower gain case.
As a consequence, the experimental results presented in
Table 8.3-1 and Figs. 8.3-1 and 8.3-2 may be taken to be a partial
verification of the predicted nonlinear distortion characteristics
for an FM ILO amplifier. The experimental data appear to fit
the theory when the~i0= 0.24 theoretical curve is used as a
Ao
guide. It is difficult to arrive at a numerical value for the
error in these measured data points for the numerous reasons
cited above.
...
'
■
"
CHAPTER IX
SUMMARY AND CONCLUSIONS
A detailed systematic study of the injection-locking
properties of microwave IMPATT diode oscillators used as
amplifiers of FM signals has been performed. An injection-
locked oscillator (ILO) model based upon the generalized
Adler s phase-locking equation was used in the mathematical
treatment of the locking phenomenon involved. The analysis
has treated the ILO as a baseband amplifier whose locking
and locked signals are the input and output signals, respectively
(see Fig. 4.2-1). This analysis is an important addition to
the manner in which an ILO may be characterized--particularly
for the dynamic locking condition. It is now possible to
theoretically characterize the locked condition when the
locking signal is modulated at frequencies approaching the
maximum locking bandwidth. The differential equation involved
have been solved numerically. The results for the distortion
that a single-stage ILO amplifier produces have been presented
nr nr
in the form of design contours (see Chapters V, VI, VII; * .
The equivalent model for the ILO amplifier was based
upon the concepts and terminology common to baseband amplifiers
and repeaters. The theoretical analysis indicates that an ILO
104
' ; I
.
*v; , 3 , v:-*. Ho;$v I’tt* SuqM sis *r&#fra bns
■
105
amplifier behaves as a band-limited amplifier for baseband
modulating signals. The ILO amplifier was characterized by
the following three parameters; namely, a) the gain variation,
b) the phase delay distortion, and c) the nonlinear distortion
versus the baseband modulating frequency. This approach
indicated that the 3 dB bandwidth of the ILO amplifier model
is approximately A0/2tt Hz.
Based upon the previous theoretical description of
the ILO amplifier characteristics, the following conclusions
may be stated:
1. changes in the modulating signal frequency
parameters have a more pronounced effect upon
the distortion characteristics than variations
in the locking signal amplitude.
2. the variation in the demodulated fundamental
output amplitude is highly dependent upon the
modulating signal frequency; however, it is not
sensitive to changes in the peak frequency deviation,
the amount of detuning or the gain of the ILO.
3. the phase delay distortion of the demodulated
fundamental increases with increasing modulating
signal frequency, peak frequency deviation and the
amount of detuning.
<
.
'
.
106
4. the nonlinear distortion of the demodulated
fundamental signal increases with increases in
modulating signal frequency, peak frequency
deviation, and decreases in the gain of the ILO
amp! ifier .
These theoretically predicted results have been renorted in
the literature for dynamic locking conditions25,26. The
nonlinear distortion of the fundamental modulating signal has
been experimentally verified (see Chapter VIII) with a
microwave IMPATT diode oscillator as the ILO amplifier.
The following items (related to the characterization
of ILO's) represent areas for further research:
1. a more precise nhase-locking equation may be
required to effectively characterize the dynamic
phase locking characteristics when the amplitude
effect is included in the ILO modelling (i.e.
throughout the analysis it has been assumed that
the ILO acts as a perfect limiter. This is not
correct for certain ratios of locking powers,
modulating frequencies or peak frequency deviations
23
on the modulating signal)
2. the manner in which the dynamic phase variation
of the locking relationship decays to its steady-
,
.
**
-
107
state values throughout each modulation cycle
could be treated in more detail (see Chapter III).
Further mathematical analysis may uncover a
suitable analytic treatment of the distortion
added to the modulating signals of an ILO amplifier.
3. further experimental work could be performed with
improved experimental techniques to verify the
nonlinear distortion characteristics beyond 0.336
of the normalized modulating frequency and to
Verify the phase delay distortion curves (in
particular a much lower Q cavity must be
incorporated with a measuring system that will give
sufficient voltage deflection for correct
measurements) .
REFERENCES
1. Adler, R. , "A Study of Locking Phenomena in Oscillators,1
Proc. IEEE, vol . 34, pp. 351-357, June 1946
2. Paciorek, L.J., "Injection Locking of Oscillators,"
Proc. IEEE, vol. 53, pp. 1717-1723, Nov. 1965
So , B . C . , Injection Phase-Locking Properties of Microwave
IMPATT Diode and GUNN Diode Oscillators With System
Appl i cati ons , Ph. D. Thesis, Department of Electrical
Engineering, The University of Alberta, Edmonton,
Alberta, 1971
4. Mackey, R.C., "Injection Locking of Klystron Oscillators,"
IRE Trans, on Microwave Theory and Techniques,
pp. 228-235, July 1962
5. Tucker, D.G., "The Synchronization of Oscillators,"
Electronic Engineering, vol. 15, pp. 412-418, March
1943, pp. 457-461, April 1943, pp. 26-30, June 1943,
pp. 114-117, Aug. 1943
6. Stover, H.L., and R.C. Shaw, "Injection Locked Oscillators
as Amplifiers For Angle Modulated Signals," Digest of
Technical Papers, G-MTT International Symposium, 1966
7. Lee, T.P., and R.D. Standley, "Frequency Modulation of a
Millimeter-Wave IMPATT Diode Oscillator and Related
Harmonic Generation Effects," Bel 1 Syst . Tech. J . ,
vol. 48, pp. 143-161, Jan. 1969
108
-
Vi*
109
Isobe, T., and M. Tokida, "A New Microwave Amplifier for
Multichannel FM Signals Using A Synchronized
Oscillator," IEEE Journal on Solid-State Circuits,
vol. 4, pp. 400-408, Dec. 1969
9. Isobe, T., and M. Tokida, "Noise Loading Performance of
a Phase-Locked IMPATT Oscillator for Multichannel
FM Signals," Proc. IEEE, vol. 56, pd. 873-875,
May 1968
10. Isobe, T., and M. Tokida, "Effects of Phase Locking on
Modulation Characteristics," Proc. IEEE, vol. 56,
pp. 453-454, March 1967
11. Isobe, T. , and M. Tokida, "Noise Reduction of Oscillators
by Phase Locking," J. Inst. Electronics Communications
Engineers (Japan) , vol. 50, pp. 2093-2100, Nov. 1967
12. Isobe, T. , and M. Tokida, "Power Amplification for FM and
PM Signals with Synchronized IMPATT Oscillators,"
IEEE Trans, on Microwave Theory and Techniques, vol.
MTT-18, pp. 906-911 , Nov. 1970
13. Mastalli, P., et al , "A New Microwave Repeater for FM
Radio Links," Alta Frequenza(Italy) , vol. 37, pp.
85E-95E, May 1968
14. Hines, M.E., et al , "FM Noise Suppression of an Injection
Phase-Locked Oscillator," IEEE Trans, on Microwave
Theory and Techniques, vol. MTT-16, pp. 738-742,
Sept. 1968
15. Slater, J.C., Microwave Electronics, Princeton, D. Van
Nostrand Company, pp. 187-221, 1964
Wij
.
,
Kurokawa, K. , "Noise in Synchronized Oscillators," IEEE
Trans, on Microwave Theory and Techniques, vol .
MTT-16, pp. 234-240, April 1968
Kurokawa, K. , "Injection Locking of Microwave Solid-State
Oscillators," Proc. IEEE, vol. 61, pp. 1386-1410,
Oct. 1973
Udelson, B.J., and R.E. Hines, "Frequency Modulation of a
C.W. Avalanche Oscillator by an Injected R.F. Signal,"
Microwave Journal, vol. 14, pp. 25-34, Oct. 1971, and
pp. 42-46, Dec. 1971
Khohlov, R.V., "A Method of Analysis in the Theory of
Sinusoidal Self-Oscillations," IRE Trans . on Circui t
Theory, vol. 7, pp. 398-410, Dec. 1960
Hines, M.E., "Negative Resistance Diode Power Amplification"
IEEE Trans, on Electron Devices, vol. 17, pp. 1-9,
Jan. 1970
Cramer, N.B., "Character!' zation and Modelling of IMPATT
Oscillators," IEEE Trans, on Electron Devices, vol.
ED-15, pp. 838-846, Nov. 1968
Gray, W. , et al , " Applying IMPATT Power Sources to Modern
Microwave Systems," IEEE Journal of Solid-State Circuits,
vol. 4, pp. 403-413, Dec. 1969
Osborne, T.L., "Amplitude Behaviour of Injection Locked
Oscillators," IEEE Trans, on Microwave Theory and
Techniques , vol. MTT-18, pp. 897-906, Nov. 1970
•**
■
. ' 0 -V;H
1
Vi*
in
24. Van der Pol, B., "The Nonlinear Theory of Electric
Oscillations," Proc. IRE, vol . 22, pp. 1051-1085,
Sept. 1934
25. Nigrin, J., J.F.W. Gurke, and P.A. Goud, "Distortion of
FM Modulation in Injection Phase-Locked Oscillator-
Amplifiers," Proc. IEEE, vol. 60, pp. 458-459,
April 1972
26. Nigrin, J., P.A. Goud, and J.F.W. Gurke, "Distortion of
FM Modulation in Detuned Injection Phase-Locked
Osci 1 lator-Ampl ifiers ," Proc. IEEE, vol. 60, pp.
731-732, June 1972
27. Marazzi , E. , and A. Bellando, "Thin-Film Injection-Locked
Oscillators and Negative Resistance Amplifiers for a
2-GHz Radio Repeater," IEEE Journal of Solid-State
Circuits, vol. SC-17, pp. 23-32, Feb. 1972
28. Terman, F.E., Electronic and Radio Engineering, New York,
Me Graw-Hill Book Company, 1955, p. 503
29. Altman, J.L., Microwave Circuits, Princeton, D. Van
Nostrand Company, 1964, pp. 134-257
30. Van Valkenburg, M.E., Network Analysis, Englewood Cliffs,
Prentice-Hall Inc., 1964, pp. 228-374
31. Peirce, B.O., and R. Foster, A Short Table of Integrals,
New York, Blaisdell Publishing Co., 1956, pp. 44-46
Wozencraft, J., and I. Jacobs, Principles of Communication
Engineering, New York, John Wiley & Sons, 1965, pp.
645-665
32.
ni-i ■
112
33. Goldman, S., Frequency Analysis, Modulation and Noise,
New York, McGraw-Hill, 1948, pp. 141-204
34. Carson, J., "Notes on the Theory of Modulation," Proc.
IRE, vol . 16, pp. 966-975, July 1928
35. Fagot and Magne, Frequency Modulation Theory --
Applications to Microwave Links, New York, Pergamon
Press , 1961"
36. Members of the Technical Staff, Bell Telephone Laboratories,
Transmission Systems for Communications, Western
Electric Co., Fourth Edition, 1970, pp. 13-37, pp . 97-
122, pp. 237-278, pp. 423-432, pp. 450-468
37. Wang, S.C., " Distortion of FM Signals Casued by Channel
Phase Nonlinearity and Amplitude Fluctuation," IEEE
Trans, on Communication Technology, vol. COM-14,
pp. 440-448, Aug. 1966
38. Hammond, S.B., Electrical Engineering, New York McGraw-Hill
Book Company, 1961
39. Moreno, T. , Microwave Transmission Design Data, New York,
Dover Publications, 1948, pp. 210-213
40. Carlson, A.B., Communication Systems: An Introduction to
Signals and Noise in Electrical Communication, New
York, McGraw-Hill Book Company, 1968
41. Oliver, B.M., and J.M. Cage, Electronic Measurements and
Instrumentation, New York, McGraw-Hill Book Company,
1971
.
H
113
42. Di s hi ng ton , R.H., "Diode Phase-Discriminatior ," Proc. IRE,
vol . 37, pp. 1401-1404, Dec. 1949
43. Korn, G.A. , Basic Tables in Electrical Engineering, New
York, McGraw-Hill Book Company, 1965, pp. 18-44
44. Struble, R.A. , Nonlinear Differential Equations, New York,
McGraw-Hill Book Company, 1962
45. Holtzman, J.M., Nonlinear System Theory, Englewood Cliffs,
Prentice-Hall Inc., 1970
46. Rail, L.B., Computational Solution of Nonlinear Operator
Equati ons , New York, John Wiley & Sons Inc., 1969
*
■
■
Vi*'
APPENDIX A
SIGNAL DISTORTION IN TRANSMISSION SYSTEMS
The purpose of this appendix is to review the
definition of "distortionless transmission" through a
communications system and to define the various types of
distortion with reference to the ILO amplifier analysis^ ^ .
Let a communication system (often referred to as a
transmission system) be represented as in Fig. A-l . Then g(t)
will represent the linear system transfer characteristics.
When an input signal e. (t) is transmitted through a two-port
network, the output signal e ^(t) will often differ from the
input in several ways.
Let EjnM represent the Fourier transform of (t)
and Eout(l°) represent the Fourier transform of eou^-('t). Then
the linear system characteristics will be
G(w) =
E , (to)
outv '
E. (w)
in'1 '
GM
je(w)
(A-l)
where |g(ui)| is the frequency dependent magnitude of G(w) and
9 (to) is the frequency dependent phase shift of G(w). When
expressed in the time domain
G (t*i )
will be an amplitude
deM
variation and G (u>) will be a phase shift and ^ is a time
delay.
114
■
...
-SSc
. •
.
115
INPUT SIGNAL OUTPUT SIGNAL
g(t)= LINEAR TRANSFER
CHARACTERISTICS
TWO-PORT NETWORK
TRANSMISSION SYSTEM
FIG. A-l ELEMENTARY BLOCK DIAGRAM REPRESENTATION OF A
COMMUNICATION SYSTEM.
'
A transmission system is defined as being distortionless
if its output waveform is identical to the input signal with the
following two exceptions:
a) the magnitude is scaled by a constant factor, K
b) the output waveform is delayed by a constant time
factor which is expressed as t seconds.
Mathematically this will be expressed as
eoutW ■ K (A'2)
Then the generalized form of the transfer characteristics will
be
G(w)
j 0 M
Ke
(A-3)
where
6(u)) = -u>(t-to)
These distortionless transfer characteristics are displayed
in Fig. A-2.
(A-4)
In many practical communication systems the representation
G (to) will not be defined explicitly over the entire frequency
range of interest, (i.e.o^u^). The concepts presented thus far
will still be applicable if G(w) is defined over a frequency
range greater than the limits of both the input and output
signals (i .e. w, ‘'fe). Quite often this stipulation is also
.
■
■
00
SLOPE = -t
o
(b)
FIG. A-2 FREQUENCY PLOT OF a) THE MAGNITUDE AND b) THE PHASE
VARIATION FOR THE DISTORTIONLESS CASE OF EQ. A-3.
-
violated and in that case the linear representation for the
transfer function g(t) is no longer valid. This will be
noted by another form of distortion in the output; namely,
nonlinear distortion. Consequently, the main types of
distortion that occur in bandlimited transmission systems are
a)
b)
c)
amplitude distortion (i.e. GM J / constant)
phase distortion (i.e. 3(uf) ^-u>(t-to)) , and
nonlinear distortion in both the magnitude and
phase of the output signal.
' •
v- ,
, - y )Jfcn
fK , * 3
APPENDIX B
DETAILED ANALYSIS OF THE TUNED AND
DETUNED DISTORTION IN ILO AMPLIFIERS.
This appendix will detail the solution of the
generalized Adler's locking equation including nonlinear effects
in the time domain. The resulting solution will be used to
accurately describe the output demodulated signal from an ILO
amplifier in familiar distortion terms. After some rearranging
the locking equation of Eqs. 2.2-20 and 5.2-4 becomes
(1+Rcos0(t) + (l+R^+2Rcos o(t)) ^sin 0 (t) =
dt
(B-l )
= (l+Rcose(t))-(l+R2+2Rcos6(t)M + Aa*,)
dt
where the symbols defined in Chapter II apply. Eq. B-l may
further be rearranged to expand the coefficient terms, i.e.
[l+2Rcos e(t)+R^cos^e(t)] + [l+R^+2Rcoss(t)] ^sine(t)
dt
= [ AtJ0] [1+R2+(3+R2)Rcosg( t)+2R2cos2e(t)]
dt
(B-2)
119
-
.
The baseband modulation term
d«(t)
dt
will be given by
/.v _ d0<(t)
'in(t) ' dt = AmsinV
(B-3)
where Am is the peak frequency deviation, \i<1 , and w is the
modulating radian frequency. Now, let all the trigonometric
coefficient terms of Eq. B-2 be approximated by their four term
power series expansion, i.e.
cos 0=1- + JL-5 -
2 24 720
(B-4)
o
(here the truncation error is approximately 0/40,320 or
about 0.09% at 0= 1.57 radians), and
2
cos 0 =
1 + cos2e
= 1
3
6
e
45
o
(here the truncation error is about 0 /315 or 3.6% at
6 = 1.57 radi ans ) , and
(B-5)
• A a. - 6
sm 0 = 0 -
5
+
6 _
1 20
7
e
5040
(the truncation error will
be about 6/362,880 or about
(B-6)
.
' ■
.
i ?' 1
'
W
121
0.015% at 6= 1.57 radians) and.
(B— 7 )
3 15 315
(the truncation error is about 4e9/2835 or about 7.7 1 at
S= 1.57 radians) .
While the approximations of Eq. B-4 through Eq. B-7
may contain relatively large truncation errors near the extremes
of their range of applicability, these terms are often multiplied
by coefficients much smaller than unity. As a result these
approximations will be considered quite accurate when 0^<1.57
radians and less accurate when 0 approaches 7l/2-
Applying the approximations of Eq. B-4 through Eq.
B-7 to the coefficient terms of Eq. B-2, yields
1 = 2Rcos6 + R2cos2e = (1 + R)2 - (1 + R)R62
+ (R(l+4R)/2)e4
-(R(l+16R)e6/360
(B-8)
and
(l+R2+2Rcose)sinft= ( 1 +R ) 2 e -(1+8R+R2) e3/6
+ 0+32R+R2) e5/120
-(1+128R+R2) a7/5040
(B-9)
and
■
H
.
^ '
122
1+R3 + (3R2)R cos © + 2R2cos2@ =
(1 + R)3 - (3R+4R2+R3)02/2
+ R(3+16R+R2)G4/24
- R(3+64R+R2)o6/720
Then, substituting Eqs. B-8 through B-10 into Eq. B-2 and
rearranging the result yields
da (t ) n
- - = 2- p6
dt n=0 n
where the coefficients are defined as
PQ = (1+R) (a^q + Amsinc^t)
P-, = A0
p = _J_ PoR(3+R)
2 1+R dt 2(1+R)2
+ Ru'(R2+1 6R+3)
24 (1+R) 2
u’(R3+64R2+13R)
720 (1+R) 2
P3 = A0(1+8R+R2)
6 (1+R) 2
P. = -R(1+4R) de
12(1+R)2 dt
P5 = -A0(1+32R+R2)
120 (1+R) 2
P, = R ( 1 +1 6R) de
360(1+R)2 dt
P? = A (1+128R+R2)
5040 (1+R) 2
(B-10)
(B-l 1 )
V
.
.
The demodulated output frequency (for both the tuned and detuned
cases) will be giyen by
/ x AoSin (t)
eou t 1 ~ wout =[^+ - 7" ] (B-12)
1+ Rcose(t)
Applying the approximations Eq. B-4 to Eq. B-10 to Eq. B-12,
yields
eout(t) = Wo +\^i r/""1 <b-13>
where the 1 R ' terms are defined as follows:
Rt = 1/O+R)
R2 = (2R-1 )/6
R, = (1-13R + 16R2)/120
R4 = (-1 + 60R - 279R2 + 272R3)/5040
The series approximation Eq. B- 11 to the locking equation
Eq. B-2 will now be solved by the application of the "successive
approximation" technique ^ ^ # The nonlinear phase
equation Eq. B-2 is only weakly nonlinear for most of the input
conditions. Therefore, it is expected that only a few iterations
will be required to arrive at an accurate solution. The
mechanics of the technique are as follows. First the nonlinear
equation Eq. B-ll is linearized by solving
’
'
. -
124
P,e0= P
1 ° o
(B-14)
where the shortened notation is used i.e. & ~ e (t). The
o o
solution to Eq. B-14 after the transient has become negligible
will be
= (1+R)
+ sin [wmt - tan
/ Jm
/
l
(B-15)
Eq. B-15 may also be expressed in exponential form. To refine
this solution e , the recursive procedure will be to substitute
0Q into Eq. B-ll in the following manner.
de,
dt
+ P-,9, =
Po +
7 n
I Pne<,
n=2 n
where G-, = ©-, (t) and PQ through P7 have been evaluated at 0
The new solution 0,, will have the general form
(B-l 6)
o*
Go + Gk[ sinkumt +^]
(B-l 7 )
where G , G, and <2$, are the combined constants that result from
o k ~ k
the solution of Eq. B-l 6 . More accurate solutions forS(t) may
be found by repeating this procedure for @2* That is, substitute
G in place of in Eq. B-l 6, evaluate PQ through P 7 at©,, and
replace 0] by©2. This process continues until the "new"
.
.
xj
125
solution has the required degree of accuracy. This outlined
procedure will be rapidly converging for the "weakly" nonlinear
equations involved here. The resultant solution &(t) will then
be substituted into either Eq. B-12 or Eq. B-13 for the
solution of the demodulated output; namely , toouf
To mechanize the recursive technique to solve for6(t)
a few modifications were required to some of the equations.
The form of the sinusoidal solution for 0o(t) is rewritten in its
exponential form, i.e.
J’V -JV
e„(t)= Bo + Bie + C-j e 111 (B-18)
where the coefficients are defined as
B0=(l+R)
Br-jAm(l+R)/2(A0+j^)
To compute further solutions of &(t) in the exponential form
it is only necessary to automate the calculation of the product
of the following two series. Let
M . .
r- 3 rut -jra t
a=A +2. [A e m tie ^ ] (B-19)
r=l
.
...
.
■
126
and
M j rca t
b= Bn + H tBr e m + BN,
0 r=l r 1
]
(B-20)
where A refers to the coefficients of e and AN refers
I • ,
-jru t
to the coefficients of the conjugate e m . Let this
definition also apply for B^ and BN^ in a similar fashion. The
product of these two series may be written (for the case of M = N)
as
axb =
M
A B +>~ (A BN +AN B )
o o v r r r r
r=l
M
r=l
M
+ 1
r=l
r-1
M
i -
A B + YL
r's s s=r+l
+B A +A
o r
r-l
M
n
S = 1
AN BN +Y~
r's s s=r+l
s s-r s s-r
jnomt
s s-r r s-r
+B AN + A BN
or or
t
(B-21)
By the use of Eq. B-21, the product of 0 . (t) may be
computed automatically without the cross terms that would be
involved when Eq. B-17 is used. Therefore, when the technique
-
hi mm I
.
described in Eq. B-16 is generalized, the result is
^ * 'A*' ' * S "»<
(B-22)
where Pq through Pj are calculated at and©^ is the i+1
approximation to 6(t). Then the solution 0.j (t) may be expressed
in general notation as
. . M jkiL> t
■ D0 + H [ Dk e m
k=l K
where M is limited to 20.
-jk03 t
+ DNk e m ]
(B-23)
The following criteria were used to determine when the
successive approximation 0^+1 made for9(t) is sufficiently
accurate to halt further iterations:
a) comparing the relative change in the fifth harmonic
j5u) t
amplitude (i.e. the coefficient of e ) after
each iteration and stopping this procedure if the
new calculation produced a change of less than o.5%,
and ,
b) comparing the relative change in the fundamental
amplitude (i.e./D, + DN, ) and stopping the
calculations when this change is less than 0.02%,
and ,
c) limiting the highest order of M to 20.
V
•H
’
.
jrag
Then the calculated sol ution &.+1 is substituted
into Eq. B-12 for a quantitative solution of the output
frequency; namely,
AoSin 6.+1 (t)
enut (t ) +
out 0 1 + Rcos6.+1(t)
(B-24)
In terms of the series solution (which will now be used to
define the output frequency in terms of its harmonic components)
Eq. B-13 becomes
eout(t) wo Rn0i+1^
(B-25)
Both of these solutions for e t(t) may be considered as the
steady-state solutions for the output frequency. To compare
the output frequency with the input frequency it is convenient
to resolve the output into its magnitude and phase representation,
i .e.
M
eout(t) = G0 +£i [ Grsin (rtint + *P] (B-26)
where Gq represents the dc offset, Gr the amplitudes of the
harmonics and <f>^ the associated phase shifts, and M is limited
to 20.
The output demodulated frequency will now be compared
to the ideal input frequency by the subtraction of the input
Eq. B-3 from the output waveform Eq. B-26. Consequently, the
■
,
undesired part of the output will be given by
A t)= G + (G-, -A ) s i n [w +
outv ' o v 1 nr L m v 1 A
M<20
+ Z [G sin (rw t + jZ^)]
r=2
( B- 27 )
The three types of transmission distortion that Eq. B- 27
illustrates are:
a) the gain variation as a function of modulating
frequency, i.e.
|g|= 201og[G-,/Am] (B-28)
b) the phase delay distortion of the fundamental
( refer to Chapter IV, and Eq. 4.2-7), i.e.
AG (degree )= it ( ^
c) the nonlinear distortion of the fundamental
output (expressed in percentages) will be
M120
ND ( % ) = 100x[ XL
r=2
/ G
1
(B-29)
(B-30)
where the dc offset is not considered as part
of the harmonic content.
Chapters V, VI, VII display the results of the application
of the method of successive approximations as outlined here^
to the generalized Adler's nonlinear differential equation.
••• .
/
.