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Research Collection
Doctoral Thesis
Synchronisation of reflex-oscillators
Author(s): AbdelDayem, Aly Hassan
Publication Date: 1953
Permanent Link: https://doi.org/10.3929/ethz-a-000099179
Rights / License: In Copyright - Non-Commercial Use Permitted
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ETH Library
Prom. No. 2165
Synchronisation of Reflex-Oscillators
THESIS
PRESENTED TO
THE SWISS FEDERAL INSTITUTE OF TECHNOLOGY
ZURICH
FOR THE DEGREE OF
DOCTOR OF TECHNICAL SCIENCE
BY
Aly Hassan Abdel Dayem
of Egypt
Accepted on the recommendation of
Prof. Dr. F. Tank and Prof. Dr. M. Strutt
Zurich 1953
Dissertationsdruckerei Leemann AG.
Leer - Vide - Empty
Table of Contents
Preface 5
Chap. 1. Synchronisation of Oscillators 7
1.1. Introduction 7
1.2. Summaries of Some Simplified Theories on Synchronisation ... 8
1.3. Calculation of the Steady State Amplitude and Phase by Usingthe Energy Equation 14
Chap. 2. Synchronisation of Reflex-Klystron Oscillators 26
2.1. Introduction 26
2.2. Choice of the Equivalent Circuit 31
2.3. The Energy Equation and Steady State Solution 34
2.4. Amplitude and Phase Behaviour of the Synchronised Reflex-
Oscillator 37
2.5. Calculation of an Example 43
Chap. 3. Mutual Synchronisation of two Klystrons 44
3.1. The Equivalent Circuit 44
3.2. Steady State Equation Under Mutual Synchronisation 49
3.3. Synchronisation of Two Identical Klystrons 52
a) Effect of the Coupling Phase Angle 53
b) Case of Small Coupling . 59
3.4. The Long Line Effect 63
Chap. 4. Synchronous Parallel Operation of Reflex Klystrons 64
4.1. General Requirements 64
4.2. Magic-T; Scattering Matrix 67
4.3. Alternative Combining Networks for Synchronous Parallel Ope¬ration 72
1. Parallel Operation with an External Synchronising Signal . . 73
2. Symmetrical Combining Networks; Coupling through Reflec¬
tion 76
3. Combining Network Composed of a Single Magic-T with Compli¬
mentary Bethe-Hole Coupler 78
3
Chap. 5. Experimental Results 80
5.1. Introduction 80
5.2. Cold Test to Estimate the Q-Factors of the Klystron Cavity . . 87
5.3. Mutual Synchronisation 91
5.4. Synchronous Parallel Operation of Two Reflex-Oscillators. . . 101
5.5. Synchronisation by a Signal from a Harmonic Generator.... 106
Literature 110
4
Preface
The early experiments and theoretical treatments of the problemof syncohinisation have led to a considerable interest in the possiblepractical applications of the synchronised oscillator. There is a
continually growing literature on the possible applications, especi¬
ally on the subject of using an oscillator as a synchronous-amplifierlimiter for f. m. reception. Also in the microwave region, synchroni¬sation has already found application in the "linear accelerator".
Here a chain of synchronous high power magnetron-oscillators are
used to drive the linear accelerator for the production of high-
energy atomic particle.In the present work a theory is presented which predicts the
behaviour of any self-limiting oscillator, when synchronised by an
external signal of any magnitude and any waveform. The theory is
based on the principle of conservation of energy and enables the
calculation of the steady state amplitude and phase, when the
nonlinear characteristic of the oscillator is representable by a
simple mathematical function. The theory is then extended to
include the mutual synchronisation of two reflex oscillators of arbi¬
trary properties and with any degree of coupling between them.
This part of the work is included in the first three chapters.In the fourth chapter some bridge-circuits are suggested, which
enable the synchronous parallel operation of 2 or 2n reflex oscilla¬
tors. The experimental work is then described in chapter 5. Com¬
plete verification of the predictions of the theory presented has
been established by the experimental results.
5
Leer - Vide - Empty
Chapter 1. Synchronisation of Oscillators
1.1. Introduction
The nonlinear theory of electrical and mechanical oscillations
has been extensively studied by a great number of authors. It is a
well known fact that in any self-exciting, self-limiting source of
steady harmonic oscillations there must exist some form of a non¬
linear relation between the acting forces and the resulting harmonic
motion. This nonlinear relation is responsible for the self-excitation
as well as for limiting the steady state amplitude to a finite value.
Although the formulation of the differential equation for the
general case of an oscillatory system disturbed by some external
signal is a rather simple matter, it is very difficult and often impos¬sible to find its exact solution. It is often advantageous to trans¬
form the obtained equation, if possible, into the form of the Hill's
differential equation, for which many useful approximate solutions
have been developed and in some very special cases its exact
solution is known. In this way it is sometimes possible to gain a
clear insight into the behaviour of the system or at least to be able
to discuss its general features.
In treating the problem of synchronisation it was, therefore,
found more appropriate to discuss the behaviour of the oscillator
from a physical point of view without primarily laying stress on
mathematical rigor. Such discussions have led to simplifying
assumptions which enabled to get a solution for the special case
where the amplitude of the external signal is small compared with
that of the undisturbed oscillation. In what follows it is intended
to give a short summary of some of the theories developed to studythe synchronisation problem. A simple method based on the prin¬
ciple of conservation of energy is then described. This method
enables to calculate the steady state amplitude and phase, if the
nonlinear characteristic can be represented by a simple function.
7
1.2. Summaries of some Simplified Theories on Synchronisation
Van der Pol [1], in his well known paper on "The Nonlinear
Theory of Electrical Oscillations" considered the case of a triode
oscillator disturbed by an external signal. He assumed a solution
of the form
v = b1 sin co11 + b2 cos w11,
substituted it in the differential equation, and was thus able to
calculate the steady state amplitude as a function of the detuning
w1 }'An interesting work on the synchronisation of oscillators by
modulated signals was published by F. Diemer [3]. He treated this
problem by transforming the differential equation into the Hill's
form and derived a solution in an integral form. This enabled him
to discuss many general features of the problem and to show the
range of validity of other theories based on the assumption of a
linear characteristic. He could further derive the condition of
stability of the synchronised oscillation in terms of the charac¬
teristic exponent and the auxiliary phase — these are parameters
generally used in the solution of the Hill's equation. The exact
solution, however, was impossible, except for the case of a very
small signal having a small degree of modulation. Therefore he had
to resort at last to a more physical consideration. Like Van der Pol
he plotted curves for the steady state amplitude and phase as
functions of the detuning for the case of an unmodulated external
signal. He then based the rest of his discussion on these curves by
considering the synchronised oscillator as a filter having the above
curves as gain and phase characteristics. Representing the filter
curves in the synchronisation region by a simple approximate
equation he could also calculate the amount of distortion producedin an FM signal while transmitted through the filter.
In the above two papers the transient state has not been con¬
sidered. The study of the transient state is especially important if
the applied signal is modulated in the frequency. If the oscillator is
capable of following the frequency deviations of the impressed
signal, we have a state of a continually disturbed oscillation. Let us
8
assume that the amplitude of the impressed signal is big enough so
that synchronisation can be obtained with an unmodulated signalwithin a band of frequencies greater than twice the maximum
frequency deviation. The knowledge of the time constant of the
pull-in process is still required to determine the highest modulation
frequency allowable in order to attain steady locking. It is evident,
however, that if the changes in frequency of the external signaloccur in a time long compared with the pull-in time constant, the
oscillator can be assumed in equilibrium at each instant of time
and a succession of such steady states are a good enough approxi¬mation to the actual situation.
Turning now to the theories on synchronisation by a small exter¬
nal signal, we begin with that published by H. Samulon [2]. He
studied the synchronisation of a triode oscillator by a small signal
of frequency o)x=— (a>0 + A a>0) where - is any rational ratio.
According to his treatment the process of synchronisation in this
case can be described briefly as follows: "Cross-modulation, occur-
ing between the impressed signal and the oscillator voltage, pro¬
duces components of frequency given by cofc= i^tuj ±q0cj0. Those
components which have a frequency approximately equal to oj0,
can be considered as a synchronising signal for locking with a one-
to-one frequency ratio". This is valid if the frequency of the
oscillator is controlled by some tuned circuit which presents a very
low impedance to all harmonics other than those having a>k~
coQ.
Superposing the anode current component 70 of frequency cd0 with
those components Ik of frequency u)k^.io0 a beating current is
obtained which, under the assumption Ik-^I0, gives a current of
constant amplitude and varying phase. The oscillator tuned circuit,
when driven by this current, oscillates at a variable frequency.From the phase characteristic of the tuned circuit together with
the phase relationship between anode current, anode voltage, and
grid voltage the instantaneous frequency deviation from co0 can be
derived. The solution of the differential equation thus obtained is
either periodic or aperiodic depending on the circuit parameters, the
initial frequency deviation and the relative magnitude of the cross
modulation product f-pl .The aperiodic solution leads to a steady
9
state where the oscillator frequency is equal to —co1; i. e. synchroni¬
sation, and is attainable within certain limits of the initial frequencydeviation A a>0 given by
A ^ A « \ ** fr ^ft
d«^A" =
\f0\-2%'
Q being the figure of merit of the tuned circuit.
The periodic solution, occuring if A o)^>Aojm, shows that the
oscillator should vary its frequency periodically between the limits
tu0— A o)m^ojt^oj0 +A com, a), being the instantaneous value. The
period Tf of this cyclic variation of the frequency is a function of
-j-4^- and tends to infinity as -r-^5- approaches unity. SamulonA 0)m A U)m
plotted curves showing the relative instantaneous frequency devia¬
tion ~~^-—
*".'~ a)°
for different values of.
°i".These curves have
A com A (om A utm
a nearly sinusoidal form for large values of -—-, so that the average
value of (at taken over a complete period Tf is very nearly equal to
cu0. If -r-^-^l, the curves show that the oscillator frequency
remains nearly constant over the greatest part of T, at a value equal
to <x)0 + A <om or co0— A (um depending on whether cu1 =
— (a>0 + A a>0)n
or (x>x =— (o>0— A oj0) respectively. In this case the average value ofro
io( over Tf is approximately equal to (oj0 ± A com).
Returning to the synchronising component Ik of cok ^ w0 it is
obvious that its amplitude is dependent on the form of the nonlinear
characteristic present. Representing this characteristic by a power
series it can be easily proved that for the case m = 1, the lowest
order term which enables the required Ik to be produced is the ra-th
order term. All terms of lower order will contribute nothing to the
amplitude of Ik. The coefficient of the w-th order term is therefore
the decisive factor in the synchronisation process. Rememberingthat the coefficients of such a power series diminish rapidly with
increasing order in most practical cases, it becomes directly evident
that synchronisation becomes more difficult for larger values of n.
However, it has been experimentally observed that such oscillators
like multivibrators and relaxation oscillators synchronise easily
10
even for larger values of n. Samulon explained this behaviour as
follows: Such oscillators contain higher harmonics of amplitudecomparable with that of the fundamental and may thus have a
considerable contribution to the amplitude of the synchronising
component Ik. Remembering that the contribution to Ik originatingfrom cross-modulation between the external signal and one of the
higher harmonics requires a term of the power series of an order
much lower than n, we notice that this contribution may even be
bigger than that due to the fundamental component itself, dependingon the rapidity with which the coefficients of the power series
decrease with increasing order. As an example consider the case
n = 25. If the oscillation is purely sinusoidal the lowest order term
necessary is the 25-th, while if it contains e. g. the 4-th harmonic
then the 7-th order term already contributes to the required com¬
ponent. This may lead to an increase in the amplitude of Ik and
consequently to an improved ability to synchronise.
A treatment, quite similar to the previous one, has been pub¬lished by Adler [4]. Here only the special case of o^^coq has been
treated. The differential equation derived as well as the results
obtained are essentially the same. However, Adler studied the
requirements which an oscillator must meet so that the above analysis
may be applicable. These requirements are fulfilled if the different
elements of the oscillator circuit are dimensioned in such a manner
that there are no aftereffects from different conditions which may
have existed in the past. To explain this we quote a part of his
discussion. "If an oscillator is disturbed but not locked by an exter¬
nal signal, we observe a beat note — periodic variations of fre¬
quency and amplitude. If these variations are rapid, a sharply tuned
circuit in the oscillator may not be able to respond instantaneously,or a capacitor may delay the automatic readjustment of a bias
voltage. In either case the above assumption would be invalid. To
validate it, we shall have to specify a minimum bandwidth for the
tuned circuit and a maximum time constant for the biasing system.''
Thus, if a tuned circuit is to reproduce variations of phase (i. e.
frequency) and amplitude without noticeable delay, its decay time
constant must be short compared to a beat cycle l-j—1, or stated
11
in other words, its pass band should be wide compared to the
"undisturbed" beat frequency (Acu0), i.e. the frequency of the
external signal should be near the center of the pass band. Also,
any amplitude control mechanism present in the oscillator circuit
should have a time constant short compared to one beat cycle, in
order that we may be able to assume that amplitude variations are
reproduced instantaneously everywhere in the oscillator circuit.
But when the amplitude control mechanism acts too slow to acco¬
modate the beat frequency, phenomena of entirely different charac¬
ter appear. Such an oscillator would fall outside the scope of our
mathematical analysis.We consider again the phenomena occuring outside the limits of
synchronisation (A w0 > A cum) where the oscillator is disturbed but
not locked by a small external signal of frequency t^. The above
theories have shown that the oscillator frequency u>t varies periodi¬cally with a period Tf between the limits co0
— A cvm S cot S coQ + A u>m,
and that the average value wt taken over one period Tf has the
values:
oi,~ a>n for —. ;=- 1
A conand o)( -> a>„ ± A wm as -.—- -> 1
The instantaneous beat frequency A<x> = o>1—
cot = Acu0 — A«ot is
therefore always lower than A oj0 and varies periodically with the
same period Tf between the limits A io„ — A cum g A u> ^ w0 + A wm.
The average beat frequency Aco, taken over T^, has the values
A oj( ^ 0,
A~oj ^ A cu0 for -r^- ^ 1A wn
and A cot -> A wm ,Jcu^O as -—- -> 1
Thus, as the value of o1 approaches one of the synchronisationlimits (w0 + A com), the average frequency of the oscillator approachesthe same limit and it can be said, that the oscillator frequency is
being "pulled" towards that of the external signal. Such a "pulling"phenomenon would not exist, if the oscillator were linear. In a
12
linear system of natural frequency cd0 the application of an external
signal of frequency iox does not disturb the frequency oj0 of the
"free" oscillation; large amplitude changes may occur if ojx is near
enough to w0 but the output will contain no other frequencies than
o>x and a>0. In the above theories the nonlinearity of the oscillator
characteristic has been accounted for by assuming that the appli¬cation of the external signal results in negligible change .of the
amplitude of the output voltage. Such an assumption implies the
existence of an oscillator which has developped its "undisturbed"
amplitude up to the saturation value determined by its nonlinear
characteristic. This means that the internal source of energy which
maintains the oscillation has a limited power-carrying capacity and
that the amplitude of the self-excited oscillation has been developed
up to this limit. Hence, the application of the external signal cannot
affect any further increase of the amplitude, yet it can change the
instantaneous rate at which energy is supplied to the oscillatorycircuit. This results in disturbing the phase balance of the oscillator
associated with a subsequent change in the frequency. As long as
the frequency of the oscillation is not equal to that of the impressed
signal, the instantaneous rate of energy supplied is continuallydisturbed i. e. this rate does not correspond to that necessary to
maintain an oscillation of a frequency either equal to cj1 or to o>,.
Thus a>( should vary. But as the frequency of the external signal is
fixed, its contribution to the rate of energy supplied may be expec¬
ted to favour any oscillation of a frequency nearer to its own and
thence the observed "pulling" of the average frequency of the
oscillator away from w0 towards cuj. This pulling increases (i.e.
-^- becomes smaller, tending to zero byw°
= 1) as a>1 appro-
aches io0 ± A a>m, till at the limit locking occurs and the oscillator
attains a fixed frequency a>1. Thus the synchronisation phenomenon
may not be described as that state where the "free" oscillation is
being suppressed by the external signal and only the "forced"
oscillation exists. It is rather that state in which the phase balance
of the oscillation can only be attained at the frequency o>1. Such a
state, in which the oscillation does not take place at the natural
frequency of the oscillatory circuit occurs, if the feed back circuit
13
introduces a phase shift other than 180° between the anode voltageand the voltage fed back to the grid. Here the self-excited oscilla¬
tion adjusts its frequency to a value w#oj0 at which the phasebalance is fulfilled. Thus the small synchronising signal may be
replaced for example by a ficticious impedance in series with the
feed back circuit which causes the oscillation to take place at a>x.
A similar concept is used by Huntoon [5] in his general treatment
of synchronisation.All the above theoretical treatments of oscillator synchronisation
have been concerned with the internal mechanism within a triode
oscillator which accounts for synchronisation. The theory presented
by Huntoon discusses certain features of synchronisation without
reference to the internal mechanism which accounts for it. His
theory is therefore generally applicable to all types of oscillators. He
defines a set of compliance coefficient which show how the amplitudeand the frequency of the oscillation depend upon the load impedance.The values of these coefficients may be derived theoretically or
measured for the particular oscillator. He considered the external
signal voltage as equivalent to the IZ drop on a ficticious increment
in the load impedance. The oscillator's frequency and amplitudeshift in accordance with its compliance coefficients and the magni¬tude and phase of the incremental load impedance. He obtained a
differential equation similar to that developped by Adler but more
general. In addition he was able to discuss the amplitude behaviour
of the- oscillator. An important value of this theory lies in the fact
that it can be easily extended to include the mutual synchronisationof two oscillators of arbitrary properties, if the coupling between
the oscillators is weak.
1.3. Calculation of the Steady State Amplitude and Phase by Usingthe Energy Equation
The method of calculation that we have developed, is explained
by applying it to the pentode oscillator shown in Fig. 1.1. The non¬
linear characteristic is assumed to be representable by the third
degree parabola
i = -avg + bv(l3, (1)
14
where i is the variable part of the anode current, and vg the variable
part of the grid potential. vg is the sum of the voltage — kv fed
back from the output plus the voltage vx of the impressed signal,so that
vg = -kv + v1 (2)and
i = oc(v + u) — y (v + u)3, (3)with
a k, y = bk and u =
k(4)
Fig. 1.1. Simplified circuit of a pentode oscillator with an external signal
applied in series with the grid coil.
The differential equation for the oscillatory circuit driven by the
current i is
v 1 fjA
„dv(5)
The principle of conservation of energy states that the amount
of energy supplied to a system during any time interval (t —10)should be equal to the sum of the energy consumed in the system
during the same time interval plus the increase of the energy stored
in the system. Calling Pt the instantaneous power supplied to the
system, Pc the power consumed and WSo, Ws the initial and final
energy stored, the energy equation will be given by:
\ptdt = {Ws-W,o) + \pcdt.
Differentiation gives
rt~dt
+r°
(6)
(7)
15
Eq. (7) could have been used to derive the differential equation for
the oscillatory circuit. For the simple case under consideration (5)
can be easily transformed to (7) by multiplying both sides by the
voltage v, thus yielding
L)vdt + Cvdt\ + K'(8)
with Pt = iv, P<- =R'
(9),
dW, v C, ^
dvand —r-1 = -=- \vdt + (v~.
dt L J dt
If in the steady state the system oscillates with a fixed period T,
the functions Pt, Pc, Ws and -~ are all periodic functions of the
time having generally a period — \T\ whereas the functions $ Ptdtt u
and J Pcdt are monotonically increasing functions of time indicatingu
the continuous energy supply and energy consumption. Thus, inte¬
grating (7) over a complete period T yields
(10)J" PtdtU
U+ T
= J PedtU
since, because of the periodicity of^s
wsito+T) - WS%) = Q.
Equation (10) says that the energy supplied to the system over a
complete period is equal to the energy consumed; the energy
storage assumes again its initial value. Thus if v in Eq. (8) has a
period T, the integration of the different terms over a completeperiod, using the notation
gives
Sf(t)dt = f(t).T,T
J ivdt = iv • T,T
16
JV
($vdt)dt = ^T[($vdt)*]=0..T
and -=dt = -^--T.T
This yields -%
iv=^. (11)
It is obvious that Eqs. (10) and (11) are equivalent, and that theywill give us the steady state amplitude of v. It is still required to
derive, in a similar manner, another equation which gives the
frequency. It may be argued, that differention of (7) gives an
equation containing the frequency as a multiplying factor and then
integrating over a complete period gives the required expression.If this is applied on (7) the result obtained will be 0 = 0 because of
the periodicity of all the functions contained. But if we observe
that the energy equation is always associated with a "force" equa¬
tion, the differentiation of the energy equation and the cancel¬
lations of those terms which satisfy the force equation will lead to
an expression which, if integrated over a complete period will not
lead to a result 0 = 0. Thus differentiating (8) gives
di v dv v2„
dv dv [ v If, ^,dv .).,_,
VTt=RTt+L+Cvdi + -dt\B+L)vdt+ Cdt-l\- (12)
Because of (5) the expression between the brackets {} vanishes
and we are left with
di v dv v2^
dv,,„.
v-r =-^ -T- + T- + Cv~. (13)
dt B dt L dtK '
Since
\V%dt = \vd{ty[Vt]T-lttdV
-'-I®'*--®'-*-the integration of (13) over a complete period yields
^i^V-Cv'2, (14)Li
17
18
n<a>2v2-=3 {G=yve'3
—ave'
(20)=ve3y
—xve
»2
(18)
v2
yields(14)and(11)in(19)Substituting
(19).ye3—ote=%
bygiveniscurrentthe(3)From
.C7sin<p=sini/iE
<pcosU+V=if)cosE
with
(17)</>),(t^t+sin£=u+v=e
thatso
(16)9)+(<o11sinU=u
bygiven
besignalexternaltheletFurther,negligible.areharmonicshigher
henceandsinusoidalnearlyveryisitindevelopingoscillation
thethatsodamped,slightlyiscircuitoscillatorythethatassumed
haveweMoreover,place.takecansignalexternalthebyoscillation
theofsynchronisationwhereo>1oflimitsthosewithinvalidonly
issolutionaSuchsignal.externaltheoffrequencytheisu>1where
(15)w11,sinV=v
formtheofsolutionavvoltagethefor
assumeweconsiderationundercasetheto(14)and(11)applyTo
(14).incontainedtermstheto
meaningfamiliaranyassigntodifficultisitpower,averagegive
(11)incontainedtermsthethatrecognizeeasilycanweAlthough
i.functiondriving
theofnaturethewithaccordanceinandsystemtheofquency
fre¬naturaltheoftermsinoscillationoffrequencythegivesThis
(14a)C(u>Q2v2-v'2)=Ti'
formtheinputbecan(14)y^,=w02Callingfactor.
multiplyingaasfrequencythehavingthusandtimethetorespect
withdifferentiatedtermscontainsitthatfacttheby(13)Eq.from
differentis(14)Eq.respectively.-=-and-3-meanv'andi'where
As the frequency of the oscillation is assumed to be given Eqs. (20)
are expected to give the steady state phase cp and amplitude V as a
function of the detuning, as well as the limits within which the
solution (15) is valid.
Using (15), (17) and (20) and performing the integration over
2-7Tthe period T = - we get
-= = t COS i
K .*(«-3^)
(^-l)=^sin^(a-^).Separating V and i/r yields
L
(21)
v i/- +-J— l^~wA2 = e(*-^bA
Putting
tan^^/^-^V(22)
w0L
7i
Q =
—r= Q factor of the oscillatory circuit
co0L
§ = (j^-^L detuning\o)0 wx/
in Eq. (22) we get
^ / 3
(23)
tan i/j = §
(22a)
which give the steady state phase and amplitude as functions of
E, a, y, 8. It is, however, required to find V and <p in terms of U
and 8. Thus using (18) and (23) in (21) yields:
V = {V+Ucos<p)
8 =
tB-3-**
U sin<p
{(V+U cos <p)2+U* sin2?}](24)
V + U cos <p'
19
From Eqs. (24) we can discuss the phase and amplitude behaviour
of the synchronised oscillator.
The undisturbed amplitude V may be obtained by puttingU = 0 giving
F°2 =37 (*-]!>)' <25>
which yields also the familiar condition for self-excitation (with a
parallel resonant circuit), namely:
1
a>iT
The effect of the nonlinearity of the oscillator characteristic in
limiting the steady state amplitude to a finite value is also indicated
in (25); putting y = 0 (i.e. linear characteristic) yields F0 = oo.
Normalising (24) by using the relative amplitudes v =
-^-and
u =
jj-and putting
v„
and
we get
O.K -A=X
V 2' 0
4X
3yR
(26)
v = (v+ ucosy) [1 -x{(v + ucosip)2 + u2sin29> -1}] (27)
8 =USmy
. (28)V + UCOS<p
Eliminating v between (27) and (28) and denoting § by tan tfi, the
phase <p as a function of u and tfi is given by
sin3© sin qs 1 cos<p ..„,
u-—7--r—y =~. (29)
sm^i/i smi/i x cosy
Putting <p = 0 in (28) gives 8 = 0. Thus the application of an external
signal of frequency w1 = w0 and having any initial phase angleagainst the oscillator voltage results also in a transient state where
the oscillator phase rotates towards that of the external signaluntil both voltages are in phase. It is also obvious from (28) that
values of <p different from zero correspond to finite values of 8,
20
denoting that in the steady state v and u are not in phase if w1 4= oj0 .
This is the natural result to be expected for, as the oscillation
occurs at a frequency w1 =#o)0, the current and voltage in the reso¬
nant circuit must have a phase difference & which, for a singletuned circuit is given by
tan 0 = 8, (30)
with the current leading the voltage for cox > a>0 or 8 > 0 and laggingfor o)x < w0 or 8 < 0. Remembering that the fundamental componentof the anode current i is in phase with the total grid voltage e, the
vector representation in Fig. 2 shows that @ must be equal to t/t,
Fig. 1.2. Vector diagram showing the phase relations between disturbing
signal u, anode voltage v, and current i and total grid voltage e for io^Wq.
as is also obvious from (22a) and (30). This vector representationshows the new phase balance to be established under the influence
of the external signal, if synchronisation takes place. But synchroni¬sation can only take place for those values of 8 which satisfy (28).The limits are determined by the maximum value of S given by (28)and occurs at <p= +90°. Thus
\K\ = ~ (31)m
where vm is the value of v corresponding to <p = 90°. For A com<co03
8 ~2QA
so that
Result (32) is the same as that obtained by the theories discussed
above, if Vm is replaced by V0. Thus if Vm is less than V0, the
21
result (32) shows that synchronisation will take place over a band
of frequencies broader than that given by the mentioned theories;
the deviation between the two results being larger for larger values
of U, where the external signal can produce a considerable change
in the amplitude of the output voltage.To calculate the steady state phase we use (28). Putting
x = ^? and y =?"Z (33)
sm ijj cosi/r
(28) gives
u2 x3 — x = — y (34)
which is the equation of the third degree parabola illustrated in
Fig. 1.3. The above discussion has shown that <p and ft have always
Fig. 1.3.
the same sign with <p>ft, and that for 8 = 0, 99 = 0 and S = Sm,
<p= ± 90. Thus only the portion AB of the parabola is required,where at B
y=\, 9 = 0, ft = 0 and'8 = 0,
and at A
y = Q, (p = 90°, sini/fm = u and 8 = 8m.
As to the amplitude behaviour of the oscillator, the assumptionof a third order parabola for the nonlinear characteristic yields
amplitude variations which cannot hold for an actual oscillator.
The assumed parabola is a good approximation of the actual oscil-
22
iator characteristic between the points P and P' (see Fig. 1.4),
beyond which the parabola gives a decrease in the current for
increasing v while in the actual characteristic the current further
increases up to some saturation value. Thus the results obtained
below will describe the amplitude behaviour of an actual oscillator
only for values of u small compared with unity.
flcl-ual
characherishc
Fig. 1.4. Deviation of the assumed third degree parabola from the nonlinear
characteristic of an actual oscillator.
Calling v0 and vm the values of v for any u at 95 = 0 and <p = 90
respectively. (27) and (28) give
(v0 + u)3-(v0 + u) = —,
v^ + u2 = 1, (35)
and 8„, =u
Vl-u2'
These relations are plotted in curves in Fig. 1.5 and Fig. 1.6. Here
we notice the effect of the assumed characteristic on the amplitudebehaviour by the fact that the amplitude of the output voltagedecreases if the amplitude of the external signal increases beyonda certain limit. For x < 0,5, v0 increases with increasing u up to a
maximum value given by
= 3(1+*> 3*(36)
23
at a value of u given by
l-2Xl/l+v3X
(37)
and then decreases for larger values of u. We also find that v0max is
attained at u > 0 for x < 0,5 and at u = 0 for x = 0>5 and would be
attainable at u<0 for x>0,5. Thus it may be concluded that the
form of the assumed characteristic does not allow the development
Fig. 1.5. Relative amplitude of the
oscillation v as function of rel. amp.
of the ext. signal u for 8 = 0 and
different values of x = «R — 1.
-U
F„Fig. 1.6. Relative amplitude .
and limiting value of the detuning
amp. of ext.
signal u.
8m as functions of rel
of an output voltage of amplitude greater than ~v0max; this maximum
value occurs when the driving voltage applied to the grid has a
certain amplitude, say e0. The value of e0 can be easily determined
by using Eq. (19). The current maximum occurs at a voltage
E* =^
'
3y(38)
Relating now all amplitudes to Es we get from (26), (36), (37) and
(38) the values:2_
* 0
'2
Es2 1 + X'
vL(a= 4(l+x)v,2 "27v
(40)
(41)
24
andV,2 27 y
(42;
Remembering that the driving voltage applied to the grid is pro¬
portional to (v + u), the value of e0 may be taken to be equal to
(vom«* + uc')- This gives4
2_ c. 2
_ (Vo, • + u« (43)
From (40) and (43) it is obvious that vr2 = e for x = 0,5. Thus for
X > 0,5 the undisturbed amplitude develops to a value which drives
the grid voltage in excess to e0, so that the application of an exter¬
nal signal can only result in a decrease of the output voltage. This
decrease is due to the falling part of the assumed characteristic
which is shown dotted in Fig. 1.4. We may therefore conclude that
the amplitude behaviour of an actual oscillator will be similar to
that of the oscillator under consideration for x < 0,5 with u ^ u0; for
u > uc the amplitude does not fall but rises slowly to its saturation
value.
In Fig. 1.6 the dropping of vm to zero by u = l is due to the
infinite value of the detuning accompanying it.
0 0J 0,2 0,3 0A 0.S S
Fig. 1.7.
0,1 0,2 0,3 0A 0,5 6
Kg, 1.8.
Calculated Curves showing phase (Fig. 1.7) and (Fig. 1.8) behaviours of a
synchronised oscillator, whose nonlinear characteristic is assumed ta be a
third order parabala.
v0 = undisturbed aplitude. u = relative amplitude of extenal signal.
S = detuning =Q(---^.\m0 w!
25
Figs. 1.7 and 1.8 show the behaviour of our hypothetical oscil¬
lator when synchronised by an external signal. The data used in
calculating the curves shown are
a = 1,5 X 10-3 A/V b = 0,6 X 10"3 A/V
k = 2,5 X 10-2 a = 3,75 X 10~5 A/V
y = 9,4 X 10-9 A/V3 Q = 120
R = 34 kQ x = °>275 < °>5
Bearing in mind the range of validity of the various assumptionsused above in representing the properties of an actual self-exciting,self-limiting oscillator, the above discussion together with the cal¬
culated curves enable us to have a clear idea of the behaviour of
such an oscillator when synchronised by an external signal.
Chapter 2. Synchronisation of Reflex-Klystron Oscillators
2.1. Introduction
The theory of velocity-modulated tubes, operating as amplifiersor oscillators, has been given by various authors [6—9]. Althoughthe theory differs somewhat among the various presentations, all
of them give essentially the same results. One useful form of the
theory has been developed by using a number of simplifyingassumptions, some of which are justified by the special design and
simple geometry of the tubes used in practice. In such a tube the
electron beam passes down the axis through a succession of regionsseparated by plane grids. Some of these are regions of acceleration,drift and reflection, which are relatively free from r-f fields. Others
are gaps forming the capacitive portions of the resonator circuits,where interaction between the r-f gap fields and beam current takes
place. These gaps have depthes that are usually small comparedwith the diameters of the gap areas. Moreover, the excitation of the
resonators is generally such that the electric fields in the gaps are
directed parallel to the axis and are nearly uniform over the gap
areas. If, in addition, the beams are nearly uniform and fill the gaps,
phenomena in the gaps are approximately one-dimensional. This
26
idealisation of the gap phenomena to uniform fields and a uniform
beam composed of electrons moving parallel to the axis of the tube
is a tremendous simplification that makes possible the analysis and
discussion of tube behaviour. There are, of course, many limitations
to a treatment of gap phenomena based on the assumption of
uniformity. Since all gaps have finite areas and all beams have
limited cross sections, there are edge effects. Uneven cathodes,fluctuation in emission, nonparallel grids, grid structure, and
uneven reflector fields make the beams nonuniform. In addition,
the conduction current is carried by the electrons, which are finite
charges with local fields and hence contribute to the unevenness in
the gap currents and fields. Electrons have transverse velocities,
and the electron velocities must be well below the velocity of lightif magnetic forces are to be neglected. Yet, the simplifying assump¬
tions have considerable validity in most practical tubes and have
CaHiode Resonator Grids Reflector
Region oFd
acceleraKo
gap
Fig. 2.1. Schematic drawing of reflex oscillator with the d.c. voltages appliedto the different electrodes.
27
made possible much of the theoretical treatment of these tubes.
Moreover, they enable the further development of the theory to
include the effect of one or more of the different factors, which have
been neglected in the simplified form of the treatment.
In the following chapters the simplified theory of velocity modu¬
lated tubes is going to be applied. As mentioned above this theoryis valid under the assumption of uniformity together with the
assumptions of linear reflecting field, negligible space charge effects
and negligible thermal velocity spread. A short summary of the
main relations is given in the following paragraphs.
Fig. 2.1 shows a schematical representation of a reflex tube
together with the potential distribution in the different regions,
neglecting space charge effects. Thus, if all electrons are emitted
from the cathode with zero initial velocity, then under the action
of the d. c. accelerating voltage V0 they arrive at the r-f gap all
having the same velocity u given by \m u02 = e V0, with e and m
the mass and charge of an electron respectively. If the cathode
emission is uniform, the input current I0 from the accelerating
region into the r-f gap is constant in time. Between the grids of the
r-f gap the injected current will be velocity modulated under the
action of the gap fields. Thus if the r-f voltage in the gap is givenat any instant by
v = V sin cot, (1)
electrons arriving at the gap at any instant t' gain (or lose) a' kinetic
energy eVsinait' during the transit time T1 through the gap, if
this time were negligibly small compared to the period of the r-f
voltage i.e. w T1<2tt. With a small but finite transit time T±, the
energy gained will be eM V sin cot', where the factor
M = -§f-, (0! = ^) (2)
T
is termed the "beam coupling coefficient" and is included to account
for the reduction in the energy gained by the electron stream due to
the change in the r-f field during the time of passage through the
gap. Thus the velocity at exit is given by
28
, ,
MV.
u = un { 1 + -
=— sin to t
(3),MV
.
1 +—.y-
sin w t
for-„y-
*< 1> i-e- if the "depth of modulation ly-"
is small.
In the reflector region the electrons will be decelerated by the
reflecting field, stopped and then returned back to the r-f gap to
make a second transit. As a result of the initial velocity modulation,the stream is density modulated on returning to the gap. The pro¬
cess, that an initial velocity modulation of an electron stream
results by drift action in a density modulated stream, is known as
"bunching".It can be easily shown that those electrons, which have made
their first transit at the instant t' return back to the gap at a time t,
given in terms of t' by the relation
/ MY \cot = wt' + @0l 1- -—=-sintat'\ (4)
which again holds for a small depth of modulation i. e. WyF <H 1.
The quantity <90 is the d. c. transit time through the reflector region,measured in radians of the input frequency co:
0o = ^, (5)
where d is the depth in the reflector region (measured from the
center of the r-f gap) attained by the center-of-the-bunch electron,i.e. by that electron which has made its first transit through the
gap at an instant where the r-f voltage is zero and changing from
one which decelerates to one which accelerates the electrons. It is
also convenient to define
x_mv&0
r =cot; (7)
X is a dimensionless quantity known as the "bunching parameter".The transit time relation is thus given by
29
t = t' + @ — X sin t'. (4a)
Now let the instantaneous density modulated beam current return¬
ing to the gap at the instant t be denoted by i (t). The charge car¬
ried by the electrons arriving at the instant t during an interval of
time A t will thus be (i (t) -At). But the same electrons have departedfrom the gap at an instant t' during an interval A t' and the current
was constant and given by the d. c. current I0. Thus
i(t)-At = -I0-At',
or dividing both sides by A t and using t instead of t we get
,-(t) = -/0|J, (8)
Eqs. (4a) and (8) give the instantaneous beam current i (t) and
show its nonlinear dependence on the voltage.
During the second transit this modulated current interacts with
the gap fields. If the relative phase of this current and the r-f
voltage lies in the proper range, power can be delivered from the
stream to the resonator. If this power is sufficient for the losses
and the load, steady oscillations can be sustained. It is clear, that
for the phase to be optimum the center of the bunch should arrive
at the gap when the field exerts the maximum retarding effect on
the electrons at the center. Thus at optimum phase the d. c. transit
angle © should have the value
@w = 277(7* + !)'
(9)
n = 1,2, 3, . . .
As has been mentioned above, a finite gap transit angle @1causes the gap voltage to be not fully effective in producing
bunching and similarly the bunched current not fully effective in
driving the resonator. The latter effect arises due to the partialcancellation in phase of the current in the gap. This introduces
again the factor M defined by (2). Thus if the beam current is givenat any instant by i (r), the driving current i. e. the current in the
external circuit induced by i(r), will be Mi(t).
Now, if an external signal is injected into the klystron resonator,
30
the new voltage appearing at the gap will disturb the phase and
amplitude balances between the bunched current and gap voltage.If the disturbing signal frequency is near enough to the free-runningfrequency of the klystron oscillator, a new state of balance may be
achieved at the disturbing' frequency associated with a change in
the output power. In the following sections we are going to studythe behaviour of the reflex klystron oscillator when synchronisedby an external signal. Again the energy equation is going to be
used to derive the steady state phase and amplitude.
2.2. Choice of the Equivalent Circuit
The equivalent circuit to be chosen should consist of elements
which can be easily measured for a given tube under test. Moreover,these elements should be so arranged that the applied voltages and
currents may be easily related to the power of the external signalinjected into the klystron cavity as well as to the power outputfrom the klystron.
Term.
Fig. 2.2. Block diagram of a possible circuit arrangement to study the
behaviour of a reflex-oscillator synchronised by an external signal.
Let us, therefore, first consider a circuit arrangement which
enables an experimental investigation of the klystron behaviour
when synchronised by an external signal. In Fig. 2.2 let 8 be the
signal source and assume that its power carrying capacity is some
100 times greater than that of the reflex klystron K under test, so
that a large line attenuation is allowable between K and 8, to
inject a signal power to K of the order of magnitude of the output
power from K. The coupling between 8 and K takes place over a
calibrated attenuator and a directional coupler of known couplingcoefficient. Power output from K is led over a second calibrated
attenuator to a crystal detector which serves to measure the relative
31
Receiver
or CRT1
Direct.3
- Coupler2 4
Calib. Aften. — Det.
Calib. Atten. _(~)s
output power level from K. A known part of the output power
from K couples through the directional coupler and the first
attenuator into the signal source 8. This power, being very small
compared with the output power from S, is assumed to produce no
effect on the signal source. The output from the crystal detector
may either be a beat note or a d. c. voltage depending on whether
beating or synchronisation takes place. The existing state can be
detected by supplying the crystal output to a spectrum analyser,a cathode ray tube or hetrodyne receiver. When synchronisation
occurs, a d. c. meter together with a calibrated attenuator serve to
determine the relative output power level from K. Further, it is
assumed that 8 is provided with some device which indicates
accurately its frequency of oscillation.
Assuming that 8 delivers a constant power output over the
frequency range necessary for our investigation, the known couplingcoefficient of the directional coupler together with the reading of
the calibrated attenuator will enable to determine the power
incident on the klystron K. Call this incident power Pi. Now, if at
signal frequency the klystron is matched to the line, the whole of
P( will be absorbed into the klystron cavity and thus contribute to
synchronisation. Such a match can only exist if the klystron cavityis in tune with the impressed frequency. Thus if P=yeja denotes
the reflection coefficient at the signal frequency as seen in the line
looking towards K, that part of P{ which couples into the cavityis given by
Pt=(l~Y*)Pt, (10)
the rest is reflected and absorbed in the different attenuators. Thus,
Pi is a function of the frequency for a given Pt. Although only the
fraction P, will contribute to the synchronisation, it is obvious that
we should take Pt as a measure of the magnitude of the disturbingsignal, especially because Pt is independent of the frequencydeviation and can be easily measured.
It has been tacitly assumed that all circuit elements are matched
to the line over a wide frequency band. From Fig. 2.2 it can be
easily seen that the klystron supplies its power to a very nearlymatched line.
32
Thus our equivalent circuit should contain some current or
voltage source (independent of the frequency) to represent P{, and
its elements should be so arranged that the power absorbed into the
elements representing the cavity is given in terms of Pt by (10).Moreover the circuit must allow the output from the klystron to be
supplied to a matched line. This requirement can be easily fulfilled
by shunting the output terminals of the equivalent circuit by an
admittance equal to the characteristic admittance of the line.
It is usual to consider the reflex klystron as a parallel resonant
circuit (GB, C and L) driven by a negative electronic transadmit-
tance. For our case a similar equivalent circuit can be used.
M~) V I lL f" fatfi}
Pig. 2.3. Equivalent circuit of a reflex oscillator disturbed by an external
signal represented by the current source 2 i2 so that incident power is given
The simple circuit in Fig. 2.3 is found to fulfil all the above require¬ments. Here Mix is the driving current due to the electron beam
and is given as a function of the gap voltage v by the nonlinear
relation contained implicitly in Eqs. (1), (4a) and (8). C, L and GRare the equivalent capacitance, inductance and conductance respec¬
tively as seen at some reference plane in the waveguide. Y0 is the
characteristic admittance (real) of the waveguide; thus the klystron
supplies a matched line. A simple calculation will show that the
incident power Pt is given by ~ and that Eq. (10) is fulfilled for-*
0
the above equivalent circuit. It is also obvious that the power
output from the klystron is equal to the power absorbed in Y0 due
to Mix and v. The elements GE, C and L can be easily measured
by the cold test procedure.With the help of this equivalent circuit our problem reduces
to a simple parallel resonant circuit driven by 2 current sources.
33
One current source is independent of the terminal voltage v and
represents the disturbing signal. The other current source is depen¬dent on v as given by velocity modulation and bunching. It is quiteobvious that the voltage v is the result of the simultaneous action
of both dependent and independent currents; it must be considered
as a whole and not as the sum of two voltages, each being the con¬
tribution of one current source. (Due to the nonlinear relation
Mi1 = Mi1(v) the law of superposition does not hold.)
2.3. The Energy Equation and Steady State Solution
Reference to Fig. 2.3 shows that the differential equation and
energy equation for the system are given by
Mi1 + 2i2 = CC^ + v(GR + Y0) + ~ [vdt (11)
and
ft fj f) j
(Mi1 + 2i2)v = v*(GR+Y0) + Cv~ +j-
\vdt (12)
respectively. Putting
@ = GB+Y0
and using (7) we get
Mi1 + 2i2 = toCv' + vG-\ f\vdr (Ha)a>L J
and
(Mi1 + 2i2)v = v2G + coCvv' + ~\ \vdr. (12a)a>LJ
with v' =j-. Differentiating (11a) with respect to t and using (12a)
yields
v' C(Mi,+2i2) v' = Gvv' + cx)Cv'2-\ = vd->
uLJ(13)
If in the steady state v is a periodic function of t and has the
period T, then integrating (11a) and (13) over a complete periodyields
Mi^v + 2t2~v= GV2 (14)
34
and Mi^v'-Vli^v' = a>Cv'2 + —T v'ivdr, (15)
where the horizontal bar denotes again the average of the quantityin question taken over a complete period.
In Fig. 2.3 the klystron cavity has been represented by the
parallel resonant circuit (GR, G and L), which has one natural fre¬
quency of oscillation given by &>02 LC= 1. But, in a lossless resonant
cavity free oscillations can take place at any of an infinite number
of resonant frequencies which correspond to the infinite number of
normal modes ofthe cavity. Thus in Fig. 2.3 it has been assumed that
one of the resonant frequencies oj0 refers to the mode of particularinterest and that all other modes are widely separated from it. This
assumption is safe for ordinary reflex oscillator cavities, providedthat the transmission line out to the chosen reference plane is not
too long. For a long line, coupling between cavity and line may
result in modes that are close together. For our case we suppose
that the modes are widely separated. Further, due to the high Qof the cavity the effect of all higher harmonics may be neglected.Under these assumptions the steady state terminal voltage v is very
nearly sinusoidal. Thus, if synchronisation takes place by the
impressed current
»8 = /asin(T + j8), (16)
the steady state terminal voltage is v = V$mr, which gives rise to
the steady state bunched current given by (4a) and (8).Let us consider first the energy supplied by the bunched beam
current to the oscillatory circuit over one cycle. This is given by the
first term of (14)
Mi1v2rr = M \ixvdr.o
Using (1), (4a) and (8) this expression yields
2w
27r-Mi1v= - MI0V J sin(r' - X sinr' + 6) dr'o
= - MI0V { cos © /"sin (T'- X sin t') dr' + sin s[cos (t'~ Zsin t') dA
= - JfJ0Ffo + sin©-27r JX(X)),
36
where Jx is the first order Bessel function. Thus the average power
is given by
P1=-i-7-2Jf/0J1(X)-sin© (17)
Here V is the amplitude of the gap voltage, 2 MI0 J1(X) is the
fundamental current component induced by the bunched beam
current and (? + ©) is the phase angle between them. Similarly, the
first term in Eq. (15) gives
Miy = -^-V-2MI0J1(X)-cos@ (18)
Here again, only the fundamental component of the bunched beam
current appears.
Using (1) and (16) to perform the integration over a complete
cycle, (14) and (15) yield respectively
- 2MI0 JX(X) -V sin© + 2/2Fcos£ = V2G (19)
- 2MI0J1(X)-Vcos© + 212V sinp = V2(coC ^ (20)
We notice that (19) and (20) could have been directly ob¬
tained by applying the steady state circuit theories to the circuit
of Fig. 2.2, with Mi1 replaced by its fundamental component.
Eqs. (19) and (20) are then nothing else than stating that the vector
sum of the currents flowing into any node of the network is zero.
However, when applying the methods of linear-circuit analysis to
our nonlinear problem, we must bear in mind that the law of super¬
position does not hold. Thus all current sources should be applied
simultaneously to the circuit and the resulting voltages are due to
this simultaneous action and cannot be considered as the sum of
the voltages resulting due to successive application of the current
sources — each alone — to the circuit.
That only the fundamental component of the bunched beam
current appeared in our equations, arises from our assumption that
the gap voltage is purely sinusoidal. Under this assumption the
higher harmonics contained in the bunched beam current cannot
contribute to the average power supplied by the beam to the
cavity, although they affect the instantaneous power supply.
36
2.4. Amplitude and Phase Behaviour of the Synchronised Reflex
Klystron
In discussing the information contained in (19) and (20) we use
the following abbreviations:
Ge ...
M*6I0
g = -~ with Ge = ——° (21)
where Ge is known as the small signal electronic transconductance,
Q =
-79—= loaded — Q of the cavity,
8=e(^-^U2Q("^), (22)\<X)0 w / \ w0 /
© = ®n + <p = 2tt(»i + |)+(p,
sin ® = — cos 9 ,cos 0 = sin 9 .
Further, as a measure of the external signal we define a parameter—
similar to the bunching parameter — by
M&0 i2
^-^Tv'W (23)
Substituting these quantities in (19) and (20) we get
^7^gcos«p + ^-2cosi3 = l (24)
and ^—-g sin 9 + -^sinjS = 8. (25)
These equation give the amplitude of the oscillation voltage (X)and its phase (j8) relative to the impressed current in terms of the
impressed signal amplitude (X2) and the frequency deviation of the
impressed frequency from the resonance frequency of the cavity (8).The angle cp is the deviation of the reflection transit angle from the
value @n = 2n(n + %). The power output from the klystron is
Yobtained by multiplying (17) by -^ ; this gives
P = ^-^-°-XJ1(X)cos9, (26)
37
which can be considered proportional to X J1(X), if we neglect the
variation of © (and <p) with the frequency.To get a clear idea of the behaviour of the synchronised klystron
as described by the above equations, and to understand the effect
of the different parameters contained therein, we should first con¬
sider some special cases with reduced number of parameters and
then calculate the amplitude, power output and phase for some
hypothetical tube.
The Undisturbed Behaviour
In the absence of an external signal (X2 = 0) the above relations
reduce to:
^^gcos =1, (14a)X
— tan93 = 8, (15a)
2)-XJ1{X)cos<p = P, (16a)
Y 2 T V
VG ®
' ( '
where
Inspection of these expressions show the effect of the reflector
voltage and the conductance parameter g on the output power and
on the frequency of oscillation. If the reflector voltage is adjustedat the center of some mode i. e. such that 93 = 0, the oscillation takes
place at the resonance frequency of the cavity and maximum power
is delivered to the load Y0. Changing the reflector voltage to either
side of the center results in a change in the frequency and in a
reduction of the power output. The effect of the conductance para¬
meter is quite important. At the center of any mode (95 = 0) the
power output is proportional to the product X Jx (X), X being now
solely determined by g as given by (14a). The product XJX (X) has
a maximum = 1,252 at Xop = 2,40, which is the value of the
bunching parameter for optimum power conversion. The necessary
value of sr as given by (16a) is g =2,31. As g =7r~^~, it is seen
that the power conversion to the load can be varied by changingthe load conductance. Again, if the conductance parameter is
38
changed by increasing Y0 so that gcos<p = l, equation (64a) givesX = 0 and the output power is zero. Here the r-f voltage drops to
zero due to overload.
Beflector Voltage Initially Adjusted to y = 0
The reflection transit angle as a function of the frequency is
given by (5). Calling @0 = —— and using (5) and (22) we get
<P = 0o(^~1) + (6>°_0J=^S + <?'0 (28)
where <p0 = @0 — @n = relative reflection transit angle, if the oscil¬
lation frequency is a>0. If the reflector voltage is adjusted to the
center of the mode in the absence of an external signal, then (p0 = 0.
@At any frequency other than co0 we have 95=5-^8. This gives
(p p& 5° for the following typical values: ®0 = 50 and Q — 200 when
the frequency deviation is 15 Mc/s from a central frequency of
8830 Mc/s. Thus as a first-order approximation we may put cos <p = 1
and sin 99 = 99 = 6 where ^ = oT>- Ge wul a^80 vary with the fre¬
quency, but it is readily seen that its variation is very small
\Ge = Geo 11 + yo) an^ mav ^e neglected.
Substitution in Eqs. (24)—(26) gives
2^£>g+2^cos^l (24b)
S{l + l^l^j = ^sin^ • (25b)
p-XJ1(X) = P (26b)
These expressions are quite simple and enable to deduce easilysome important results.
Consider first the case where 8 = 0, i.e. case of synchronisation
by an external signal in tune with the undisturbed frequency.
Eq. (25a) gives j8 = 0 or the gap voltage is in phase with the injectedcurrent. The external signal, being in tune with the oscillator fre¬
quency, results simply in a transient state where the oscillator
39
phase rotates towards that of the external signal until they coincide.
The transient state then disappears and the oscillator phase remains
"tied" to that of the external signal. This simple fact may have
some important applications. For instance, let it be required that
a number of klystrons run in synchronism so that their outputsshould have prescribed phases at some definite reference planes.This can be easily accomplished in the following manner. Let the
synchronising power be supplied by a klystron through a length of
waveguide with a reflectionless termination at its other end. In the
guide we have thus a single travelling wave. If at the appropriate
planes some sort of coupling device (a hole in the common wall
with directive properties or a probe etc.) is provided to couplepower from this travelling wave into the different klystrons — all
klystrons being pre-tuned to the same frequency with <p = 0 — theyrun in synchronism having the fixed phase relationship determined
by the fixed phases of the injected signals.Now consider the amplitude of the gap voltage and the power
output for 8 = 0 as a function of the external signal. Eq. (24b) gives
x{l-2^-g) = 2X2. (29)
2 J (X)Remembering that —^—- = 1 at X = 0 and decreases with in¬
creasing X, Eq. (29) shows that the amplitude of the gap voltageincreases continuously with increasing X2. At -3l=3,83, Jt(X) = 0
and (29) gives 2X2 = 3,83. At this value of X2 the power outputis zero. Thus if a strong external signal of 2X2 = 3,83 is injectedinto the klystron cavity no oscillation can take place. For optimumpower output (X = Xop =2,4) the amplitude of the external signalshould be
Thus, if for some tube g<gop, the application of the external signalresults first in increasing the output power with increasing X2 up
to X2 = X2op. For bigger values of X2 > X2o (and for all values
of X2 if g ^ gop) the output power decreases with increasing X2. This
is an important fact to be taken into consideration in the appli-
40
cation of a synchronised reflex klystron. It may be generally stated
that the amplitude of the synchronising signal should be as small
as possible in order to obtain an output power which is not much
smaller than the optimum.It is obvious from (24b) and (25b) that the amplitude and
phase curves with X2 as a parameter are symmetrical about the
8 = 0 axis. It is also obvious that the maximum frequency deviation
which satisfies the above equation is given by that value of 8 = ±8mwhich makes j3= ± 90°. At this value of |3 Eq. (24b) gives
2Jj^-9=l, (31)
where Xm is the amplitude at the boundaries of the synchronisation
region. We notice that Xm is independent of X2 and has a value
equal to its undisturbed amplitude (compare with (24a)). Thus if
an external signal is applied having some X2, the amplitude of
oscillation X is greater than the undisturbed amplitude over the
whole synchronisation region, it decreases with increasing 8 to
attain its undisturbed amplitude at S = 8m. The value of 8m is givenfrom (25b) and (31) by
\*-\ =—r-^rrTTr^ (32)
Xm(l+eg^)As Xm is independent of X2, |8m| is proportional to X2.
The above discussion enables us to predict the shape of the
curves describing the dependence of the amplitude and the output
power on the frequency deviation for different values of X2 and g.
These are shown in Figs. 2.4a and 2.4b. In Pig. 2.4a it is assumed
that g is smaller than gop and thus the undisturbed amplitude is
smaller than Xop. Curves 1 and 2 are drawn for 2Jf2<3,83, curve
3 for 2X2 = 3,83 and curve 4 for 2X2>3,83. It is obvious that the
best operation is obtained by the state represented by curve 2,
where the output power remains nearly constant over the greatest
part of the synchronisation region. It is interesting to notice that
for 2X2>3,83 the oscillation stops in the middle of the synchroni¬sation region up to a value of the frequency deviation at which
41
xm >x.m '
Aop
a) 9<9op b) 9 >9op
Fig. 2.4. Amplitude of oscillation and output power for a reflex-oscillator
synchronised by an external signal. PQ = Power output when oscillator is
undisturbed, P = Optimum power output.
JT = 3,83 where oscillation starts with zero output power. Fig. 2.4b
shows the amplitude and power output for g > gop; the curves are
numbered in accordance with the values of X2 taken in Fig. 2.4a.
Here the power output is always smaller than the undisturbed
value, which is less than the optimum power output. Operationunder such conditions are therefore less advantageous than those
represented by Fig. 2.4a.
Reflector Voltage Initially adjusted to some <p = y>0
Substituting the value of <p as given by (28) in (24)—(26) yields
^—g(cos<p0-dsm<p0) + -^~cos)8 = 1 (24 c)
Xgsaup0 + --g-Bmp = 8 1 +—~ egco$cp0\ (25c)
P = p X Jx (X) • (cos <p0— e S sin cp0) (26c)
42
These expressions are comphcated and their discussion is rather
laborious. The effect of <p0 can be best seen in the calculated examplein the following paragraph.
2.5. Calculation of an Example
The values used in the following calculation "may be taken to
represent some average values typical to the 2K 25 reflex klystron.Some of these values has been experimentally measured. (See the
experimental part.) The values assumed are:
oj0 = 2ttX 8,9 X 109 C = 1 pPQ == 250 Q0 = 700
0 = 220 ftv GR = 80
Ge = 440 p
The assumed values of G and Ge give a conductance parameter
(/ = 2<<7op = 2,31 in order to obtain such curves as those shown in
Fig. 2.4a.
The first family of curves Fig. 2.5 has been calculated from
(24b)—-(26b). The second family Fig. 2.6 has been calculated from
(24c)—(26c) for ^0 = 30.
Xi'0.5 x2-1 x2*1,5 x2=1,9S
1,5 6
a) b)
Fig. 2.5. Synchronisation of a reflex-oscillator by an external signal of nor¬
malised amplitude X2. a) relative phase, b) amplitude X and power output
(proportional to 2XJ1(X)) as functions of the detuning S, with relative
reflection angle adjusted to zero.
43
T
1.0 o 1.0 eo
Fig. 2.6. Same as Fig. 2.5 with relative reflection angle <j>0 = 30°.
Chapter 3. Mutual Synchronisation of two Klystrons
3.1. The Equivalent Circuit
Fig. 3.1 shows a circuit arrangement which enables the experi¬mental investigation of the mutual synchronisation of two klys¬trons. The directional couplers are supposed to couple a small
^_^ ^^
Kt (~) 1Direct.
3
Coupler
War. rnase
Shifter
var. LalD.
Atten.1
Direct.3
2,Coupler
4,
-fc)«<,
Receiver Det. Det. Receiver
•Jr /
To indicating Instruments
Fig. 3.1. Circuit arrangement for the experimental investigation of mutual
synchronisation of 2 klystron-oscillators.
44
portion of the power flowing in the main line 121' 2' to be used for
measuring the frequency and the power output as well as to indicate
whether synchronisation takes place or not. The coupling bracnh 21'
contains a variable calibrated attenuator and a phase shifter and
thus enables to adjust the magnitude and phase of the couplingbetween the two klystrons. The power output from one klystron is
partly absorbed in the attenuator; the rest is coupled into the other
klystron to affect synchronisation. The circuit arrangement does
not provide for obtaining useful power output to be supplied to
some external load. This is intentionally done so as to extend our
investigation to include any degree of coupling between the two
klystrons. A schematic diagram of a possible arrangement where
useful power output can be supplied to some external load is shown
in Fig. 3.2. Here the directional coupler provides a small fixed
Useful output
from Kj —
c,0- Var. Phase
Shifter
1Direct.
3
2Coupler
4
H0K>—"• Useful output
from Kt
Fig. 3.2. Simple circuit to operate two klystron-oscillators in synchronism;
useful power out is available for external loads.
coupling between the lines 12 and 34, so that the greatest part of
the power output may be obtained from arms 1 and 4. The variable
phase shifter serves to adjust the phase of the coupling to its
optimum value.
In either of these circuits the coupling branch may be represented
by a symmetrical 2-terminal-pair network having the following
characteristics:
1. Its characteristic admittance is equal :to that of the line con¬
necting the klystrons.2. It produces an attenuation equal to that of the attenuator
(Fig. 3.1) or of the directional coupler (Fig. 3.2).
3. It produces a phase shift equal to that produced by the phase
shifter and the line length between some arbitrarily chosen reference
planes.
45
This 2-terminal-pair network is either connected in series between
the klystron outputs to represent the circuit in Fig. 3.1, or in par¬
allel between the 2 output lines to represent the circuit in Fig. 3.2;
this is shown in Fig. 3.3. Without making any reference to the
14 < i
Hi, ,
. i2_Jk°
Coupling ~^[^)o— Nefworh —g—\_y
©
Coupling
Nehvork
&2i
Fig. 3.3. Circuits of Figs. 3.1 and 3.2 with coupling branch replaced by a
two-terminal-pair network.
circuit elements to be contained in this 2-terminal-pair box or to
their arrangement which may fulfil the prescribed requirements,we may define the electrical properties of the network completelyby any of the matrices used in the 2-terminal-pair network theory.
I, ro *—
v'L -30,1' V26
Fig. 3.4.
Thus with the directions of currents and voltages chosen in Fig. 3.4
we define an ||a||-matrix by the equations
so that
vi = anV2 + a12I2
\CL\\ ="11 *12
(1)
(2)
For a symmetrical 2-terminal-pair network with characteristic
impedance g0 and propagation constant y the a-matrix is given by
46
ft =
cosh y —
£0 sinh y
—sinhy
—coshy
So' '
(3)
Thus if j0 is taken equal to the characteristic impedance of the
output line from the klystron, we have only to define y in terms of
the attenuation and phase shift existing in the line between the
chosen reference planes.Let the reading of the attenuator (usually in decibels) correspond
to a power ratio a2 (ratio of input to output power), and let the
equivalent line length between the 2 reference planes (including the
phase shifter) be ifs electrical degrees. Thus closing the line at one
end by a reflectionless termination and supplying power from the
other end we get:
' in ' out
^ in -* out
ho
and
so that
P = \V-in
P=
\V T \* out I ' out *- out I
IF- /• I IF- I2\ '
iri m I rin I
P W T \ W I2:x out I r out * out\ I ' out I
with j0 taken to be real. As the phase of the output voltage is
retarded tp° behind that of the input, we have
£*- = «e->* (4)' out
Now closing terminals 2 — 2 of Fig. 3.4 by j0, we get from (1) and (3)
fr = C• (5)
^ out
Eqs. (4) and (5) define y of the equivalent 2-terminal-pair network
in terms of the attenuation taken as a power ratio and the equi¬valent line length tfi by the relation
ey = a.e-H. (6)
It is to be noted that due to the symmetry of the 2-terminal-
pair network together with the symmetry of the chosen directions
47
of the terminal voltages and currents, the a-matrix defined by (3)
is directly applicable to the network with either terminal pair
taken as the input terminals. Thus the 2 pairs of equations
and
Fj = V2 cosh y — 72 g0 sinhy
h So = V2 sinh y-lzho cosh Y
F2 = V1 cosh y — Ix j0 sinh y
h i o= Visinh Y-hbo sinh7
(7)
(7a)
(8)
(8a)
are both valid; the one pair being directly derivable from the other.
Therefore we may choose any 2 of the above 4 equation to describe
the relations between the terminal voltages and currents. For our
problem the currents are functions of the voltages, so that the
most convenient pair are (7) and (8) as they give 2 similar expres¬
sions containing the 2 unknowns V1 and F2.The reference planes are chosen such that the distance between
one plane and the gap of the adjecent klystron is an integral mul¬
tiple of half a guide wavelength. This choice reduces the transfor¬
mation of voltages and currents from any reference plane to the
adjecent gap and vice versa to a simple multiplication by a real
number representing the transformation ratio of the line coupling
the klystron cavity to the waveguide.
Ci 4 Q,
Fig. 3.5. Simplified equivalent circuit for two reflex-oscillators coupled
together by a two-terminal-pair network. (Coupling between each cavity
and the guide is assumed lossless.)
According to the above considerations our equivalent circuit is
as shown in Fig. 3.5. Here we have a system containing 2 resonant
circuits representing the cavities of the 2 klystrons; the resonant
circuits are coupled by a 2-terminal-pair network of known charac¬
teristics. The system is driven simultaneously at both ends by the
48
bunched beam currents M1^1 and M2$2 which are given by the
process of velocity modulation and bunching as nonlinear functions
of the gap voltages SSX and 232-
3.2. Steady State Equations under Mutual Synchronisation
The resonant circuits shown in Fig. 3.5 have large Q-factors
(also when loaded by the attenuator) so that in the steady state the
terminal voltages 23x and 932 are very nearly sinusoidal and the effect
of all higher harmonics may be neglected. It has been shown in the
preceding chapter that for such a system the relations obtained by
integrating the energy equation and its first derivative over one
period can be directly obtained by simple circuit theories under the
restriction that the law of superposition does not hold. If such an
analysis is applied the driving currents are replaced by their funda¬
mental components obtained by representing the former currents
by a Fourier's series for the fundamental frequency of oscillation a>
and its higher harmonics. The fundamental component of the
bunched beam current is itself a nonlinear function of the corres¬
ponding gap voltage and is given — as was shown in the last chap¬ter — by
MQ1 = -jM1Im2J1(X)er'*. (9)
The different terms in (9) have exactly the same meanings as
defined in the preceding chapter. We substitute again for the reflec¬
tion transit angle <9X the value
01=2ir(n + i)+<Pl, (10)so that with
»! = V1 (real),
Eq. (9) gives (H)
M1Ql = M1I01-2J1(X)e-^^.
Referring all phases to V1 we may write
232 = F2e^(12)
Jf,3, = lfsJM-2J1(Z)e-'<w-0>.
With reference to Fig. 3.5 and using Eqs. (3), (7) and (8) the steadystate values of the voltages 931 and 932 are given by
49
»! = 85,00^7-(8,-«,D,)i0sinhy
»! = *! cosh y - (St - »! $1) So sinh y
Where ^ and ^)2 are the admittance of the shunt (C, L, 0) at the
working frequency. If the cavities are tuned to the resonant fre¬
quencies
<**x = TTcand a'«a = L~c~ (14)
and if the working frequency is co, the admittances ^ and |)2 are
given by
Si^i + Jfoo + Wi. » = lor2, (15)with
B^QJ^L-^). (16)
QLi is the loaded-# of the cavity and is given by
«*-£& <">
i/0 = — is the characteristic admittance of the line.So
The bunching parameters are given by
M-V-&-Y l I l
(18)
so that
""* 2 V
v1 -k^X,'
with k = uyJ)1-ta
2 ^2
and the conductance parameter is
9t = with Gei =2^oi
Using (6) we get
and
coshy
sinhy
1 \ a
= -(ae^- -e-2 \ a
-iA
50
(19)
(20)
51
para-auxiliarytwoasconsideredbecan<p2and<ptanglestransit
reflectionrelativeThei/j.lengthlinetheandaratioattenuation
voltagetheparameters:twothewith81;oscillationofquency
fre¬theandj8anglephaserelativetheirX2,andXtoscillations
ofamplitudestheunknowns:fourcontain(15)Eqs.complexThe
variable.independenttheasco02chooseandconstantco01keepWe
Quq
Gi.a
with
(25)q281-q§2=82
giveswhich
(24)l^(^_Wo.)=3.
approximationtheuse
mayweoj0'stheofonearoundfrequenciesofbandnarrowaFor
(23)/^i;(^-^r)-«i.=
bycavitiesklystrontheof
detuningrelativethedefinewe(15)bygiven82and82oftermsIn
2/o
(22)
«,•and
(Xf)A1with2J1(Xi)
,-81)}(«^-i^),+,1.4l(Z1).e-M-^-/a2/i\yii
.A1.,
/1
e+JP2
&X
1
e+#
21,
(21)and
,,\1.,1/2kX1e~*P
finallygetwe(11)—(20)Using
meters, which depend on the initial adjustment of the reflector-
voltages. In addition all the three angles <pt, <p2 and <fi are functions
of the frequency of oscillation.
It is obvious that the implicit form of (21) lends it impossible to
express any one of the four unknowns as a function of the indepen¬dent variable 812-and the four other parameters. However, we have
two special cases of interest. The first is the case where one of the
klystrons is much more powerful than the other, so that synchroni¬sation of the weaker klystron can be attained, with the stronger one
practically unaffected, by introducing a very small coupling between
them. This leads us back to the case of synchronisation by an exter¬
nal signal of constant frequency and constant amplitude, which has
been considered in the preceding chapter. The second case of special
practical interest is that of the synchronisation of two identical
klystrons. For this case the number of constants contained in (21)is considerably reduced and the discussion is to some extent simplerand clearer than that of the general case with two different klystrons.The following treatment will therefore be mainly concerned with
the case of two identical klystrons.
3.3. Synchronisation of two Identical Klystrons
As was indicated in the preceding chapter, the main effect of
adjusting the relative reflection phase angle <p to some value other
than zero is to distort the amplitude versus detuning curves and
to destroy their symmetry about the 8 = 0 axis. It was also indi¬
cated that for q> =t=0 synchronisation takes place over a band of fre¬
quencies, the width of which is effectively the same as with <p = 0,
although its limiting frequencies are no more equally displacedfrom the undisturbed frequency. Comparison between (2.24) and
(21) shows that the effect of 93 4=0 in the case of mutual synchroni¬sation is approximately the same as in synchronisation by an
"independent" external signal. The present discussion may thus be
limited to the case where the reflector voltages are both preadjustedto give 9>1 = oj2 = 0; the preadjustment being carried out with the
klystrons connected to matched loads. Further we may neglect the
dependence of <p on the frequency. The phase angle ifi introduced by
52
the coupling branch may be assumed constant over the frequencyband of interest; an assumption which holds good, if the line lengthlying between the two cavities is not excessively long. The long-lineeffect will be considered later on in this chapter.
Under these assumptions (21) yields
2X
(26)
$T-iH+H-{h(i->-is)->MKHwhere, for two identical klystrons
k = i, yx = y% = y, G1 = G2 = G, q = \
so that from (25)
§2 = 3i-812 (27)
Equations (26) are still complicated due to the presence of the
coupling phase angle. Therefore, we may begin our discussion by
studying the effect of tfi on the behaviour of the system
3.3a. Effect of the Coupling Phase Angle
Using the abbreviation g' = and separating real and imagi¬
nary parts, Eqs. (26) yield
=r^cos/3 = -cosift — b(g J1(X2)— g') costfj — a sin if/S2V ^2 y
1 2 X a
=r-^cos/S = -cos^— b(gAl (XJ—g') cos t// — a sin tfi 8Xy ^i y
=r^sin,6 = - simfj — a (gA1(X2) — g') smtp + b cos^S2y ^2 y
—=r-isin/J = - sin ^4 — a(gA1(X1) — g')sraifi + bcosifih1y ^2 y
with a = <x + ~ and b = a.— (29)a a
(28)
53
Eliminating 8X between equations (28) we get
\ (J1- ~x) °OS^ = gb cos^^i(Z^ - Ji (Xa)} +a sin,/.812
-l^ + ^Jsin/S = -£asin«/,{J1(X1)-J2(X2)} + &cos</-S12
(30)
As we are dealing with two identical klystrons, we may expect
some identity in their behaviour for different values of the indepen¬dent variable 8la. A closer study of (28) and (30) shows that, if for
some value of 812 = 8[2 the four unknowns, as determined from these
equations, have the set of values:
Z1 = X', X2 = X", S1 = 8', 32 = 8", /? = /?'
then a second set of values, possessing some sort of "reflection
symmetry" with respect to the first set, namely the set
X2 = X", X2 = X', 81 = 8", 82 = 8', £ = —/?'
will satisfy our equations for 812 = — Sj2. Stated in other words this
behaviour can be described as follows: Taking the resonant fre¬
quency of cavity of klystron 1 as reference we adjust w02 of klystron 2
successively to two values each on either side of cu01 but equallydisplaced from it, i.e. oj02 = a>01 + Aw0. With a>02 = w01 + Aw0 the
system will oscillate at the frequency oj = oj01 + Aui, whereas if
<u02 = co01— Aco0 the oscillation frequency is cu = cu01— A cu. Or, in
other words, oscillation always takes place at a frequency lyingbetween wel and <o02; its position being a function of A co0. Moreover,the amplitude of oscillation of klystron 1 and its phase at its
reference plane for co02 = w01 + A oj0 are the same as those attained
by klystron 2 for tu02 = w01—Aco0 and vice versa. However, the
behaviour of either klystron is not symmetrical about the 812 = 0-
axis (X' +-3T"), neither is the frequency of oscillation of the system(8' +8"). Thus, although the limiting values of 812, between which
synchronisation is possible, are of equal magnitude but opposite
sign, the corresponding values of the frequency of oscillation are
not of equal magnitude and are not necessarily of opposite sign,depending on the magnitude of i/i. In addition, it is to be noted
54
that the frequency of oscillation of the system when both cavities
are tuned to the same resonant frequency i. e. when S12 = 0, is not
equal to that resonant frequency but deviates from it by an amount
which depends on the magnitude and sign of tp. Further, althoughthe power output from each klystron is not a symmetrical function
of 812, the sum of the 2 powers is symmetrical. However, remem¬
bering that the frequency of oscillation is not a symmetrical func¬
tion of 812, the total power output of the system is also not a sym¬
metrical function of the oscillation frequency.
Summarising, we may write
-^l(812) = ^2(~812)
»i (8U) = 82 (-M (31)
P (8M) =-j8(-8u)
which shows that the function X1 is simply the reflection of the
function Xa about the S12-axis. The same holds for 8X, and 82; but
j3 has a skew symmetry.
Now let us find the value of if/ by which the following will be
satisfied
%i (8i2) = ^2 (812) = x (812).
2.
81 (812) = - 82 (812) = 8 (812)
i. e. the amplitudes of oscillations are always equal for any value
of 812 and the frequency of oscillation lies always middle way
between the resonant frequencies of the two cavities. Since Eqs. (31)
are valid for any value of if/, thus combining (31) and (32) gives
X(S12) = X(-8lt)
8 (812) = -S(-S12)
which means that, with the auxiliary condition (32) satisfied, the
behaviours of the 2 klystrons are exactly identical, each being in
addition symmetrical about the S12-axis and consequently a sym¬
metrical behaviour of the system as a whole may be obtained.
Substituting the first of conditions (32) in the first of Eqs. (30), the
condition for symmetrical operation is given by
a-sin^-812= 0. - (33)
55
If this is to be satisfied for all a and S12, we get
xfi = nrr with n = 0, 1, 2,
or the equivalent line length between the chosen reference planesshould be an integral multiple of half a guide-wavelength. As the
reference planes were chosen at distances equal to —^ from the
respective control grids, the line length between the 2 grids should
thus be an integral multiple of ~.
Putting ^r = 0 in (28) and remembering (32) we get the two
simple expressions
gAx(X) = i + -^J-^(i- a cos^and (34)
The first gives the amplitude of oscillation and the second the fre¬
quency of oscillation and the relative phase angle as a function of
the relative detuning. As described by (34) the behaviour of the
system is quite clear and simple.Consider first the limits of the frequency band over which syn¬
chronisation can take place. This is directly given by putting
j8= ±90°
l812lm_
2oc/ok\
~^--y(«>-l)> (35)
giving an infinite value of |812|m at <x= 1, i.e. when no attenuation
exists between the two klystrons (maximum coupling). For very
small coupling (ajs-1) Eq. (35) reduces to
¥"-4' (36)
18 1' 1i.e. LMm varies nearly linearly with - for «> 1.
Consider next the amplitude parameter at the middle of the
synchronisation region i. e. at S12=j8 = 0. This is given as a function
of the coupling by
56
The undisturbed amplitude is given by putting a = oo (zero coupling)
g-A^XJ-1. (38)
It is to be noted that for (0^X^3,83) the function A1(X) is
(1^A1(X)^0). Thus, as the coupling is increased (a decreases),AX(X) decreases and the amplitude of oscillation increases. If
Xu < 2,4, the power output increases by increasing the coupling,reaching a maximum at X0 = 2,4 and then decreases to zero byX0 = 3,83 or A1(X0) = 0. Thus, the maximum coupling with which
the system can still oscillate at 812 = 0 is given from (37) by
2a, = 1 for y < 2
and by V (37 a)
a, = 1 for y > 2
For values of <x<a, there will exist a gap in the middle of the
synchronisation region where no oscillation takes place.At the boundaries of the synchronisation region, the amplitude
of oscillation is given by
gAl(XJ = l +^~-T) (39)
M>From (39) we notice that Xm is smaller than both X0 and X.t
whereas X0 is always bigger than Xu. Thus if Xu < 2,4, the power
output v s detuning will possess 2 maxima unless X0 < 2,4. Another
important fact is included in Eq. (39). We notice that as <x-» 1 the
RH8 of (39) tends to infinity. But the maximum value of the
function Ax (X) is unity and occurs at X = 0 where both the voltagedeveloped and the power output will fall to zero. Thus, for a certain
value of a = ac, A1(Xm) becomes unity and the power output will
be zero just at the boundaries of the synchronisation region. PuttingJ1(ZJ = lin(39)weget
i1 + -A-K (39a)2/(^-1)
57
X=S,8S
Fig. 3.6. Amplitude and power output
as functions of the detuning with a as
parameter and 0 = 0, in a system of 2
identical reflex-oscillators.
Fig. 3.7. Maximum magnitude of 812, de¬
fining the boundaries of the synchronisa¬
tion region, as function of — with </i = 0.
At values of <x<<x0 oscillation will stop before the boundary is
reached; this obviously occurs at a value of the detuning smaller
than that given by (35) for the value of a in question.The results of the above discussion is illustrated in Fig. 3.6 which
shows the amplitude of oscillation and the power output for a case
58
where Xu<Xop =2,4. The dotted curves denoted by Xm and Pm
give the amplitude and power output at the boundaries of the
synchronisation region. We notice that:
1. For large values of a the system oscillates in synchronism
up to the boundaries |S12|m given by (35). Beyond these boundaries
synchronisation stops but oscillation continues at more than one
frequency.2. For a = ac, oscillation can only occur at one frequency (syn¬
chronisation). Just at the boundaries oscillation stops.3. For values of a smaller than both a, and <xc, the system can
only oscillate in synchronism, but the region of oscillation is splitinto 2 parts separated by a gap of no oscillation. Moreover, the
limits of S12 defined by (35) cannot be reached, since oscillation
stops at a value of |S12j < |S12|m.
3.36. Case of Small Coupling
In a system of 2 reflex-oscillators, where mutual synchronisationis affected by introducing a small coupling between them, a great
percentage of the power output from each will be available for use
in an external load. This case is, therefore, of special interest for
practical applications and will be discussed in the present section
in more detail. The simplifications introduced by assuming <x>l
allows us to study more exactly the effect of the coupling phase
angle ift, as well as the long line effect.
Again we assume that the two klystrons are identical and that
the relative reflection phase angles are adjusted to zero. For «>1
we may neglect —er^ with respect to ae^ in (26). This yields the
following four expressions
A|iCos(«A + /3) = l-!7J1(Z2),ccy X2
_l|«oo8(^-j3)=I-!741(X1),«y Zl
(40)
ay X2yisin(>£ + jS)=82,
X,^sin(«/,-J8)=81.
<x.y li
59
Since the percentage change in the amplitudes over the frequencyband of interest is very small, we may express Xx and X2 in terms
of their value at the middle of the band by the approximations
X^Xoil + x), X2 = X0(l + x). (41)
For the functions A1(X1) and A2(X2) we use the Taylor expansion
up to the first order; thus
A1(X) = A1(X0) + (X-X0)A1'(X0),
which yields
A1(X)=A1(X0){l-xK(X0)},where
T /y \(^)
*(*o) =
J2 (Xq)
A(x0y
Substitution in the first equation of (40) we get
-^-(l+x1-x2)cosy + p) = {l-gA1(X0)} + x2gJ2(X0). (43)
Remembering that at B = 0, XX = X2 =X and x1 = x2 = 0, the value
of X0 is given by the expression
ACoS^ = l_<7Jl(X0) (44)
Since by assumption <x>l, the second term x2J2(X0) may not be
neglected with respect to \~gAx (Xg), whereas in the left hand side
of (43) (xx—x2) may be neglected with respect to unity. Thus, we
may put ~^ = 1 in all of the four expressions (43) to obtain
2— (cos (iff+B) - cos iff) = g x2 J2 (X0),
2— (cos (if,-B) - cos^r) = gx1J2(X0),
2
2sin (iff — B) = 8, .
ay
(45)
60
Eliminating §x between the last pair we get
4— sin /S cos i/j = S12. (46)
For any value of ifi the maximum values of 813 that define the
boundaries of the synchronisation region is obtained from (46) byputting j8 = +90, giving
8l2im = — |0OSi/r|.a y
(47)
Eq. (47) shows an important feature in the mutual synchronisationof two identical oscillators. By varying the phase of the coupling,the width of the band over which synchronisation is possible,
varies between zero and a maximum =
ocythe maximum takes
place at <p = mr with n = 0,1, 2,. . . Thus, the proper adjustment of
the phase of the coupling is quite important. With ifi = mr, two'
requirements are fulfilled:
1. Maximum width of the band over which synchronisation is
possible.2. Symmetrical amplitude and phase versus detuning charac¬
teristics for each oscillators; at the same time the behaviours of
both oscillators are identical.
X XFor any value of i/>, curves for r=~ and -— are shown in Fig. 3.8a.
XIt is to be noticed that the curve for ~ can be obtained from that
-90 *90° ft -90° *90' ft
Fig. 3.8. Relative amplitudes developed by either reflex oscillator as func¬
tions of the relative phase angle /! for the case a5>l, (a) </>#=0, (b) ^ = 0.
61
for ^ by reflecting the former on the vertical axis. Also the maximaXq
of the curves are displaced to the right or to the left from the
vertical axis instead of lying directly upon it as in the case where
i^ = 0. The corresponding curves with $ = 0 are shown in Fig. 3.8b.
Here both ^ and ^ coincide. In Fig. 3.9a and 3.9b the 8's areX, X,
shown.
Fig. 3.9. The different S's as functions of p for the case a*
(b) * = 0.
1, (a) ^#=0,
For the case of a small coupling but non-identical oscillators, the
following relations can be derived in a manner similar to that used
in deriving Eqs. (45):
*«/2 ^02(cos]8-l) = x2-g2J2(X0
Xn—-^(cos p - 1) = xx -9l J2 (X01)«2/i kX01
(48)
*2/i &^oi
2 fj_ JfeX,
sin/3
X,91+
1
Si.
X,02
sin/3 Jia •
Lo2 Vx "'-^oi;
In (48) ifi has been put equal to zero. No further discussion of (48)is going to made. It is only intended to use them in deriving a
simple equation which enables us to compare quantitatively the
theory with the results of the experiment. Since we are going to
use similar tubes we may put k = 1, for it depends only on the beam
voltages, the beam coupling coefficients and the d. c. transit angles,
62
which may be expected to be the nearly equal. Further, if we choose
the case of very small coupling where changes in the amplitudes
may be neglected, we may define X01 and X02 by the equations
?1J1(X01)=g2J1(Z02)=l, (49)
which hold for <x = co. Thus the initial inclination of the curve
relating |Sia|m to — will be given by
an expression which contains values that can be easily measured.
3.4. The Long Line Effects
If the equivalent length of the line lying between the two planes
containing the klystron grids is excessive the phase angle ip cannot
be assumed constant and its dependence on the frequency should
be taken into consideration. Due to the implicit form of the generalequations (26) we may limit our present consideration to the case
of small coupling described by the simpler expressions (45).
Mm.// /V ' /1 1 /
1 /1 1 /11/1 '/
l<Mm
/\ \
f \ \
\ \\l6'2'\ \ \\ \ \\ x \\ * \\\ \\\\
m-
*9~ Physicalline-length
•>
Fig. 3.10. Detunings at the boundaries of synchronisation region |Slajm as
functions of physical line length; showing the long line effect. Curve denoted
|812|Bl+ occurs when <o02>tu01 and that denoted |8la|nl_ when a>02<a)01. The
dotted curve shows both J812jm+ and |812|m_ coinciding on one another, if the
long line effect were conipletely absent. A = guide wavelength at the
frequency co01.
63
From Eq. (47) it is obvious that |§12|m reaches its maximum if
the physical line length makes tfi = 0 or n at the corresponding
frequency of oscillations. Since the 2 frequencies occuring at the
boundaries of the synchronisation region are different, different
physical lengths of the coupling line is necessary to obtain the
maximum values of |S12|m. Thus if we try to measure experimentally
|812|m as, a function of the physical line length, we get such curves
as those shown in Pig. 3.10.
Another long-line effect will be noticed, if the physical length of
the coupling line is adjusted to give if/ = 0 at the center of the syn¬
chronisation region i.e. at a> = w01. At the boundaries of the syn¬
chronisation region (^8 = 90°) will be different from zero, resultingin a corresponding reduction in the limiting value of S12 as is
directly seen from (46). If the value of ip at the boundaries is
denoted by tfim, where i/jm is a small angle, the reduction factor is
cosi/im= 1 — -~~.... Thus, this long-line effect will only be noticed
for an extensively long-line, where </^ becomes comparable to
unity. It is obvious that a slight deviation from symmetry will also
be noticed in the amplitude behaviour of the system. (See Eqs. (45).)
Chapter 4. Synchronous Parallel Operation of Reflex Klystrons
4.1. General Requirements
A combining network which allows two klystron oscillators to be
operated simultaneously into a common load must fulfil the
requirements that:
1. The two klystrons remain locked-in to the same frequency, in
spite of a possible error in tuning them and in spite of the random
variation of the oscillation frequency of each klystron.2. The total power supplied to the common load is optimum.3. Simultaneous modulation of one of the oscillators' parameters
(e. g. of the reflector voltages) does not throw the system out of
synchronism, provided that the frequency of the modulating signal
64
is small compared with the synchronisation time-constant. In other,
words simultaneous modulation of both reflector voltages should
result in a single instantaneous frequency of oscillation -of the
system as a whole. In addition the resulting frequency deviation
should be proportional to the amplitude of the a. f. signal. Both
conditions should be fulfilled under the assumption that the static
characteristics of the oscillators relevant to modulation are not
exactly identical.
Thus, the combining network, should allow a certain fraction of
the power output from either klystron to couple into the other, in
order that mutual synchronisation may take place. Further, the
phase of this coupling should correspond to optimum width of the
band over which synchronisation is possible. Under such circums^
tances both the first and third requirements mentioned above are
fulfilled.
Concerning the second requirement some features of the available
tubes should be mentioned. Attention is concentrated on the
723 A/B family of reflex klystron, for all the experiments carried
out in the present work were performed on the 2 K 25 tube. These
tubes have integral resonators with built in output circuits that
consist essentially of a coupling device and an output transmission
line. The coupling device is an inductive pickup loop formed on
the end of a coaxial line and inserted in a region of the resonator
where the magnetic field is high. The output lines are small coaxial
lines provided with beads, which are also vacuum seals, and carry
an antenna that feeds a waveguide. The design of the coupling looparid output line together with the position of the antenna in the
waveguide are so chosen that the tube may be correctly loaded bya matched guide. Yet most of the tubes of the above family require
individually adjusted transformers .of some sort dn order to deliver
full power to a resistive load that is matched to the "waveguide.This is because of the relatively 'big tolerences, which must be-
allowed in manufacturing the small coupling loops and the coa,xiaJ.
line bead seals. However, the fact that the tube used will deliver
its optimum power to a load which is not matched to the wave¬
guide, may be used with advantage in the case of synchronous
parallel operation of two klystron oscillators. This may be explained
65
as follows. Consider the combining network with the two klystronsconnected to two different arms. Suppose that, if one of the klys¬trons is replaced by a reflectionless termination, the admittance
seen by the other klystron is a perfect match i. e. no reflected wave
exists in the guide connecting this klystron to the network. Now,
with both klystrons connected to their respective arms and due to
the intentionally introduced coupling between them, a wave
travelling towards either of them will exist in the guide connectingit to the network. In a steady state of mutual synchronisation such
a wave has a frequency which is exactly equal to the frequency of
oscillation of the klystron. For the klystron itself it does not matter
at all whether such a "reflected" wave does originate from a non-
matched line or from another oscillator connected elsewhere in the
network. The only thing that really matters is the fact that there
exists a standing wave in the connecting guide. Now, if the
"standing wave ratio" is adjusted to the correct value optimum
power will be supplied from the klystron to the network. In this
event the synchronising signal coupled from either klystron to the
other serves to affect synchronisation as well as to optimise the
power output. The required "SWR" may be achieved by proper
choice of the strength of the coupling introduced between the two
oscillators. Here the question arises whether the magnitude of the
coupling required for optimum power output will be associated
with the required width of the frequency band over which synchro¬nisation is possible. Fortunately, by the 2K25 tubes used in the
coarse of our experiments optimum power was obtained by a
coupling whose magnitude is about 10 db. The bandwidth associ¬
ated with this .coupling is about 10Mc/s°or more; a width which
may be considered adequate for most applications.For the experimental investigation of the synchronous parallel
operation the most suitable network element which enables an easy
fulfillment of the above requirements is the magic-T. If the two
oscillators are connected to two adjacent arms of the magic-T with
the two remaining arms connected to reflectionless terminations,
the two oscillators are completely decoupled from one another. The
required magnitude and phase of the coupling may then be adjustedat will by using, for example, reactive screws to introduce reflection.
r'
-
66
Before going into the details of the combining circuit, the magic-Tused in this work may be described and its scattering matrix may
be derived. This is done in the next section.
4.2. Magic-T; Scattering Matrix
The magic-T used here consists oftwo parallel waveguides havinga common wall in the broad side. The common wall contains a
number of small coupling slots, which are dimensioned and arrangedin such a manner that power incident on the junction from one arm
splits equally between the two opposite arms with no power coupledinto the adjacent arm. Also, no power will be reflected back to the
source if the two opposite arms are connected to matched termina¬
tions. The measured characteristics of these magic-T's gave the
following results:
1. Voltage standing-wave ratio (VSWR) in the feed arm <1,1.
2. Power outputs from the two opposite arms are equal to within
1 db or less.
3. Isolation between the two adjecent arms is better than 40 db.
(The values given up represent the limits up to which our measuringdevices could give reliable indications rather than the actual values
of matching, balance and isolation.) In our subsequent calculations
we may, therefore, suppose that the magic-T used is perfectlymatched with equal power division between the two opposite arms
and perfect isolation from the adjacent arm.
The scattering matrix is a useful tool which is often used in cal¬
culating microwave circuits containing waveguide junctions with
several arms. This matrix is a simple extension of the wave forma-
D
1
|E0, Eos !U— —-i
1
»j
1
E04 '
D
Fig. 4.1.
67
lism generally employed in transmission line theory. Consider for
example the four terminal pair junction shown in Fig. 4.1. Let
Ein be a complex number representing the amplitude and phase of
the transverse electric field of the incident wave at the n-th ter¬
minal pair. Let Eon be the corresponding measure of the emergent
wave. It is assumed that Ein is normalised in such a way that
\EinE*in is the average incident power, and correspondingly for
E^. (E*n is the complex conjugate of Ein.) It is obvious that the
amplitude of any emergent wave Eon may be related to the ampli¬tudes of all incident waves by a simple linear combination of these
amplitudes, each being multiplied by an appropriate proportionality
(complex) constant. For example
Eon =snlEil + Sn2^o2+ +SnnEon+ (l)
The meaning of the coefficients snk is rather important. Consider
the case where power is supplied to the junction from the n-th
terminal pair with all other terminal pairs connected to matched
loads. In this case all Enk — 0 for all k #=n, and (1) reduces to
E =s E-
Thus, although all other arms are perfectly terminated, there may
exists an emergent wave in the feed arm, which is reflected by the
discontinuities at the junction. Thus, snn describes a property of the
junction itself independent of the manner in which its terminals
are terminated. If snn — 0 the junction is said to be matched lookinginto the n-th. arm, meaning that all power incident on the junctionfrom this arm will couple into the other arms, no direct reflection
at the junction back into the n-th arm takes place. If all snn = 0 the
junction is then matched looking into all arms. Again, for the case
considered — power fed through the n-th arm only — the wave
coupled into the k-th arm is given by
E0k ~ snk Ein
Thus, while snn represents a reflection coefficient, snk indicates the
coupling coefficient between the n-th and k-th arm. Further it is
obvious that, due to the reciprocity theorem,
snk~
skn
68
These coefficients may be written in the form of a matrix that is
known as the scattering matrix of the junction. Thus for the four-
terminal pair junction shown is Fig. 1, the scattering matrix has
the form
l-SII =
*11 S1Z S13 *H
S12 522 S23 S24
513 S23 S33 *34
su 524 S34 S44
(2)
Further, it may be shown that in addition to its symmetry the
scattering matrix is also unitary i.e. <
Zj \snk\ ~ 1 anC* Zj snk8mk ~ 0 (3)
Now, in the magic-T described at the beginning of this section,the junction is matched looking into all arms i. e.
snn = ">
the adjacent arms are decoupled i. e.
S12 = 534 = "'
thus the scattering matrix reduces to
1-81
0 0 S13 su
0 0 *23 S2i
S13 S23 0 0
«14 S24 0 0
(4)
Remembering that power fed from one arm divides equally between
the two opposite arms, the magnitudes of the four unknowns con¬
tained in (4) are equal. Using the first of (3) we get
+ 4-U I2 — 1
+ *24 — *
Since all magnitudes are equal, we have
312 I '14 I~ \S?a\ — lS24l —
1
71(5)
and 11$|| may be written
69
151
0 0 c f
1 0
c
0 9 h
a 9 0 0
/ 9 h h
(6)
with the new complex unknowns having all unity magnitudes. The
determination of these unknowns will obviously give the phases of
the coupling coefficients. This can be done by considering the sym¬
metry of the magic-T. Choosing the reference planes at CC and DD
(see Pig. 4.1), the junction has a reflection symmetry about the
plane AA. This reflection symmetry may be described in the form
of the matrix (see [10], chap. 12).
0 1 0 0
1 0 0 0
0 0 0 1
0 0 1 0
\F\\ =
which commutes with the scattering matrix \\8\\ so that
||£|j.||.F|j = plJ-H^II.
Performing the multiplication, (8) gives
c — h and / = g .
Using the second of (3) and (9) gives
hg* + gh* = 0.
Remembering that \g\ = \h\ = 1, we may write
g = ei@l, h = ei0'i.
Substitute (11) in (10) gives
e i(Si-02) + e->«9i-<92) = 2 cos (©i-©2) = 0
or g and h are in quadrature. Putting @1-&2=^ and ®2 = *
(7)
= ioiQg = je
h = eie,
(8)
(9)
(10)
(11)
(12)
we get
(13)
70
and the scattering matrix assumes the simple form
1511 =
0 0 1 i
e;0 0 0 i i
12 1 j 0 0
1 1 0 0
(14)
The incident and emergent waves shown in Pig. 4.1 are now related
by the expressions:
aE01 = Ei3 + jEti
a^oi = i®iz+ En
aE03 = Etl +jEi2
aEn = jEtl + Ei2
(15)
with the abbreviation
= ]/2 e-'e (16)
It is obvious that with the proper choice of the reference planes the
phase angle @ may be made zero with a subsequent simplificationof the expressions (15). However, if for any practical consideration
the reference planes are preferably chosen to coincide with the
physical terminal planes of the magic-T, the angle © may be
measured experimentally as follows: If arms 2 and 4 are connected
to reflectionless termination and arm 3 is short circuited while
power is supplied through arm 1, we have
E*
so that (15) gives
Eu = 0 and Ei3 En
E<
01g-j'2® ej(«r-»8>
(17)
Using a standing wave indicator to measure the distance of the
first standing wave minimum from the terminal plane, this distance
expressed in electrical degrees will give directly (n — 2®), as indi¬
cated by (17).A special case of loading the magic-T is going to be often met in
the subsequent sections and may, therefore, be derived here. Let a
71
matched generator be connected to arm 1 with arm 2 perfectlyterminated and arms 3 and 4 connected to loads producing at the
respective reference planes the reflection coefficients rz and /\.'Jhis gives
^£3
p _
^u
^03 "^04^2 = 0, r, = -=**, A = -=i* (is)
Substitution in (15) yields
Em 101= f.(A--T
a
02
E
E02_
i(19)
= Mrs+ri)H2
Placing the generator on arm 2 with arm 1 perfectly terminated
and arms 3 and 4 loaded as before, similar expressions are obtained:
Ei2~ a2( 3 4>
(20)
| = -^(A+r4)We notice that with rs — ri = r the generator sees a match in
either of the above cases. The relative voltage coupled into the arm
adjacent to that containing the generator is then given by
^f (21)
which is the coupling coefficient between the two adjacent arms 1
and 2 produced by two identical loads on arms 3 and 4.
4.3. Alternative Combining Networks for SynchronousParallel Operation
In this section some combining networks which allow the
synchondtfs parallel -operation of 2 or 2n klystrons, are suggestedand their main features calculated. These networks represent by no
meafis all the possible circuit arrangements, which may be used
<f»t this application. They rather serve to indicate the important
72
points of view which should be taken into consideration to attain
the proper behaviour of the system.In the following discussion it is assumed that all generators are
matched to the line, so that any signal incident on the generator is
absorbed there. Each generator is mounted on a length of wave¬
guide, whose terminal plane is assumed to be at a distance <p°(electrical) from the generator grids. All generators are assumed to
be tuned to same frequency with the reflector voltages adjusted to
centre of the mode. The reference planes of the magic-T are taken
to coincide with its physical terminal planes and its scatteringmatrix is therefore given by (14) with a certain angle 0, that can
be determined by measurement as mentioned above.
4.3.1. Parallel Operation with an External Synchronising Signal
Here, a simple combining network is described which enables
2 or 2n klystrons to operate in parallel and supply their output
power to a common load. No direct coupling exists between the
different generators and synchronisation as well as the requiredphase relationships are affected by an external signal supplied from
a matched generator that is assumed to be completely uninfluenced
by any signal coupled into it from the system.
1
MT
3
4
bignaiSource
2 Matched
Load
Fig. 4.2. Parallel operation with an external synchronising signal. If Kxand if2 are identical no power couples from them to the signal source; all
! power output is supplied to the load.
The circuit suggested is shown in Fig. 2. The four terminal pairnetwork denoted MT is the magic-T. The planes denoted by 1, 2, 3
and 4 are the terminals of the MT and are taken to be our reference
planes as mentioned above. K1 and K% are the two klystron oscil-
latprs to be synchronised.
73
Consider first the case where Kx and K2 are replaced by matched
loads and power is supplied from the external source through arm 3.
Let the incident wave at plane 3 be denoted by Es — real i. e. we
refer all phases to the phase of Es at plane 3. Using (15) the emergentwaves at planes 1 and 2 are given respectively by
E01 = ~ES and EM = ?-E8 (22)
which are in quadrature and of equal magnitudes. Taking plane 1'
on line 1 and 2' on line 2 such that the distances ll' = 22' = <p the
emergent waves at 1' and 2' lag by an angle <p behind those given
by (22) and we get
EW = E,— and EW = E„!—— (23)a n
Under the assumption that the external source is uninfluenced by
any signal coupled to it from the system the emergent waves Eorand E02. will hold their phase and magnitude if Kx and K2 are
reconnected to arms 1 and 2. Remembering that synchronisation
by an external signal, which is in tune with the resonant frequencyof the oscillator cavity, results in phase coincidence between exter¬
nal signal and self-exciting signal, the klystrons will deliver 2 waves
given at planes 1 and 2 by
Eix = £e-'(»-8) and E = jEe-J<f-&) (24)
With E real; these waves are of equal amplitudes and have the
phases of Eor and E02,, given by (23). At planes 1 and 2 these
become
Eil^Ee~J^v-&) and Ei2 = jEeri<*<p-*> (25)
Now considering that both arms 3 and 4 are perfectly terminated
and the two waves given by (25) are incident on planes 1 and 2,
substitution in (15) gives
#03 = 0 and EQi = ji~2Ee-J2<i'-» (26)
i. e. the whole power comes out of arm 4, the 2 waves coupled in
arm 3 cancel. It is now obvious that in the system shown in Fig. 2
74
an external signal fed in arm 3 divides equally between arms 1 and
2 and compels the oscillators connected there to adjust their phasesto be in quadrature so that the total power output from Kx and K2comes out of arm 4. The common load can thus be connected to
arm 4.
From the above results it can be easily shown that, if the line
connecting terminal 1 to Kt is longer by a quarter guide wavelengththan that connecting terminal 2 to K2, the total power outputcomes out of arm 3. This fact can be used to design a simple net¬
work for the parallel operation of four klystrons. This network is
shown in Fig. 4.3. The signal source is shown connected to arm 3
of M Tz. Kx and Kt are displaced to the left a distance-j-
from the
plane containing K2 and Kz, so that power output from K1 and K2comes out of arm 4 of M Tl and that from K3 and Kt comes out of
K,Q
K2QM2MT,
_Matched
J Termination
ML
MT3
SignalSource
Load
Matched
Termination
Fig. 4.3. Parallel operation of 4 reflex-oscillators synchronised by an exter¬
nal signal. If all oscillators are identical all power output is supplied to the
load; no power couples into signal source.
arm 3 of M T2. These output waves are in quadrature, with the
latter wave leading the former one, so that the total power output
comes out of arm 4 of M T3 to the load. No power couples from the
system into the signal source.
The main features of this network are: 1. that the total power
output is directly available from one arm without the necessity of
using any phase shifter and 2. that no power originating from the
synchronised oscillators couples into the arm containing the signalsource. This latter feature is quite important, if the signal source is
76
a harmonic generator using some crystal rectifier as a multiplierelement. In such a case the crystal rectifier is generally fully loaded
from the source of the fundamental wave so as to obtain high har¬
monic power and thus no additional loading can be allowed fpr.
4.3.2. Symmetrical Combining Network; Coupling through Reflection
This combining network is especially convenient for laboratorywork to investigate the parallel operation of two klystron oscillators
wh'ch mutually synchronise. As shown in Fig. 4.4 coupling between
K2gh
Movable
Sifrew
Tuners
MT,
4 PhaseShifter
s t=p j .miner
Matched
Termination
MT2
Load
Fig. 4.4. Symmetrical combining network appropriate for the experimental
investigation of synchronous parallel operation of 2 reflex-oscillators.
Kx and K2 is achieved by two movable screw-tuners connected to
arms 3 and 4 of M Tx. The phase shifter is used to adjust the phaseof the incident wave in line 3—V to a value such that the two
waves incident on terminals 1 and 2 of M T2 are exactly in quadra¬ture. If these two waves are of equal amplitudes all the power will
couple either in 3 or 4 of M T2, where the common load may be
connected. Thus the part of the network to right of the plane AA
serves only to convey the total power output to the common load.
To simplify the following discussion we may suppose that lookinginto either terminal pair to the right of the plane AA the admittance
seen is a match. With this assumption the reflection coefficients at
terminals 3 and 4 of M Tx are solely determined by the movable
screw-tuners. The magnitude of the reflection from the screw
depends only on the insertion and does not vary with changes in the
position of the tuner. Thus by the help of the movable screw-
76
tuners used the required magnitude of the reflection coefficients at
terminals 3 and 4 can be obtained by choosing the proper insertion
of the screw, while the required phase can then be independentlyobtained by adjusting the position of the screw along the guide.Choosing equal insertions and adjusting the positions at equaldistances from the terminals, the coupling coefficient between Kxand K2 is given by (21):
jTej2@. (21)
As derived in section 3.2 this coupling coefficient is valid for the
reference planes at terminals 1 and 2. If the length of the line
between either plane and the grids of the klystron oscillator con¬
nected to it is given by <p, the coupling coefficient referred to the
planes containing the grids is
yjV2*®-**). (27)
r=yeJoi,
yeV
* i) (28)'
giving a magnitude of the coupling coefficient between Kx and K2equal to the magnitude of the reflection coefficient introduced byeither screw and a phase given by
^ = 2©-2^. + a + ~ (29)
It has been shown in the preceding chapter that for symmetrical
operation of either oscillator and identical behaviour of both as well
as for maximum width of the frequency band over which synchroni¬sation is possible, the phase of the coupling coefficient should be
zero. The necessary phase of the reflection coefficients is then givenfrom (29) by
a = 2<p-2©-~ (30)
It is seen that this combining network allows in a simple manner
the adjustment of each parameter completely independent from
If we put
(22) becomes
77
the others and thus enables the fulfilment of all the requirements
for proper synchronous parallel operation as well as the study of
the effect of each parameter separately.An alternative of this circuit which allows the omission of that
part to right of the plane AA is shown in Fig. 4.5. It can be easily
shown that the requirements imposed on the coupling magnitude
C0
<>0MT,
-jfj-|wI- 2 Load
Fig. 4.5. Alternative network of that in Fig. 4.4.
and phase are also fulfilled by this combining network, if K1 as well
as both screws are displaced to the right a distance = ~. In
addition to its simplicity this network has also all the advantagesof that shown in Fig. 4.4.
4.3.3. Combining Network Composed of a Single Magic-T with
Complimentary Holes for Coupling
The circuit described above suggests another simpler one, where
neither screw tunes nor phase shifters are necessary. If the common
wall of the magic-T is designed in the usual manner to fulfil the
requirements of equal power division and directivity, a Bethe-hole
78
coupler may then be bored at one end of the common wall to intro¬
duce the required coupling between the oscillators. Thus as shown
in Pig. 4.5 the Bethe-hole perovides for the coupling while the
magic-T serves to convey the total power output to the load. For
two parallel, equal waveguides coupled through a circular hole in
the centre of the wide side, a wave travelling in one guide in a
certain direction, say to the right, excites two waves in the other
guide, one travelling to the right (termed forward wave) and the
other to the left (backward wave). With unit amplitude of the wave
in the exciting guide, the squares of the amplitudes of the excited
waves are given by [12]
16 tt2 r6 ( • A 2\for forward wave A* =—
^^(2-^)
-2-^(2+V\(31)
and for backward wave B =
r — radius of the whole
a, b = dimensions of guide cross section
A0 = free space wavelength
\g = guide wavelength.
The amplitude of the forward wave is zero at a wavelength
satisfying the relation
A9=V2A0 = 2a. (32)
For a standard 1-in by J-in waveguide (32) is satisfied at a free
space wavelength of 3,2 cm. At this wavelength the directivity of
a Bethe-hole coupler is infinite and a wave travelling in one guidein the forward direction excites only a backward wave in the other
guide. Such a coupler is thus suitable to produce the required
coupling between the two klystron oscillators.
If planes AA and BB are taken to represent the planes con¬
taining the grids of the oscillators, the best phase angle of the
coupling is obtained with the length of the dotted line equal to
-~-; with the phase shift introduced by the Bethe-hole being taken
into consideration in calculating this length. If, in addition, the
distance between AA and terminal 1 is ~ shorter than that between4
79
BB and 2, the incident waves at 1 and 2 are in quadrature and the
total power couples into arm 3 where the common load is connected.
If the system is to operate at a wavelength other than that
where the directivity of the Bethe-coupler is infinite, forward waves
are excited in both guides. If the main waves are in quadrature at
the plane of the whole with the wave in arm 2 lagging, the excited
waves are also in quadrature with the wave in arm 2 leading. Thus
the excited waves will couple into arm 4 instead into the load,
resulting in a reduction of the net power supphed to the load. But
if the Bethe-hole is designed for coupling of about 20 db at A0 = 3,2,
thus in a system working at A0 = 3,4 the power in the excited forward
waves will be 4<? db beneath the power in the main waves and the
resulting reduction is negligible. Remembering that the working
range of the 2K 25 reflex klystron lies between the wavelengthes
3,1 to 3,5 mm, it becomes obvious that the reduction in the power
delivered to the load is negligibly small all over that range. Thus
the Bethe-hole coupler is quite suitable for such a purpose.
Chapter 5. Experimental Results
5.1. Introduction
A number of experiments were carried out to investigate the
behaviour of the synchronised reflex-klystron oscillator and the
possibility of synchronous parallel operation. Nearly all measure¬
ments were performed at a wavelength of 3,4 cm using the 2K25
reflex-tube. The choice of this working wavelength was primarilydetermined by the available cavity-wavemeters which have a range
extending between 3,37 and 3,43 cms. The experiments are going to
be described here in the sequence of their development rather than
the sequence of the theoretical treatment of the preceding chapters.The present investigation necessitates the knowledge of the pro¬
perties of the tubes under test as seen from the waveguide, as well
as their performance under different loading conditions. Prom the
properties of the tube we have to measure the different (J-factors
80
(unloaded- and radiation-Q) of the cavity resonators and the appa¬
rent position of the cavity grids relative to the terminal plane of the
waveguide, over which the tube is mounted. The knowledge of the
Q-factors are necessary to calculate the expected width of the band
over which synchronisation is possible, as well as to compare the
experimental results with those predicted by the theory. The
knowledge of the tube performance with different loads enables to
determine the conditions most appropriate for synchronous parallel
operation.Measurement of impedance involved in the experimental proce¬
dures to obtain the informations mentioned above is carried out by
using a standing-wave indicator. This consists of a section of wave¬
guide into which a small, movable probe can be introduced througha slot. The probe extracts a small fraction of the power flowing in
Square Wave
Modulator
a-f Calib.
Attenuat.
Selective
Amplifier
Power
SupplyPreamplif.
Klystron — 15 db Atten.Slotted
SectionTest Piece
Fig. 5.1. Arrangement of circuit for standing-wave measurements.
the guide to deliver it to a crystal rectifier, the output of which is
then supplied to some indicating device. In order to achieve the
required sensitivity and in the same time to allow the signal source
to be isolated from the line by attenuating pads, the system shown
in Fig. 5.1 is used. A square-wave modulation voltage is applied in
series with the reflector voltage of the klystron oscillator to produce
an "on-off" modulation. Power from the klystron is fed through
a 15 db attenuator to the measuring device. As the 2K25 reflex-
klystron supplies about 20 mW at cw operation, the average power
output when square-wave modulated will be about 10 mW. Power
in the incident wave in the slotted line is therefore about 0,3 mW.
81
Suppose it is required to measure VSWR's up to about 35 which
corresponds to a PSWR about 1000. At this value the power at
the standing-wave maximum is about 4 times the power in the
incident wave i.e. about 1,2 mW while that at the standing-waveminimum is 1,2 x 10-3 mW. To reduce the errors attributable to the
presence of the probe in the line, we may assume a small probe-
coupling of the order 0,5%. Thus the power supplied to the crystaldetector by the probe is 6 /xW at the standing-wave maximum and
6x 10~3 /xW at the minimum. The sensitivity that can be achieved
with standard microwave crystals is of the order of 1 mV of rectified
open-circuit voltage for 1 /xW of r-f power absorbed in the crystal.The rectified voltage is in most cases very nearly proportional to the
power absorbed for powers not greater than a few microwatt. Thus
the 6 /xW power absorbed at the standing-wave maximum in the
above example may be considered as the maximum allowable for
the assumption of proportionality between the output voltage and
the absorbed power to be valid. Now, at the standing-wave mini¬
mum the voltage output from the crystal is of the order of 6 /xV.Thus, if the full-scale meter deflection is obtained by a 6 V outputfrom the amplifier, the total amplification required is about 106 or
120 db.
To achieve the high sensitivity imposed on the indicating device
we designed and constructed a preamplifier unit and an amplifiermodulator unit shown in the block diagram of Fig. 5.2. The sensiti¬
vity of an amplifier, working in the audio range of frequencies, is
usually limited by microphonics, nicker effect, and harmonics of
power-line frequency rather than by thermal noise. These effects
are considerably reduced if the recurrence frequency is increased to
15 to 20 kc/sec. Therefore, we chose our modulation frequency at
17 kc/sec. The first stage of the preamplifier uses a 6F5 triode,that has low microphonics, little hum modulation and low equi¬valent-noise resistance, while the last stage is a cathode-follower
with an output resistance of 75 Q equal to the characteristic
resistance of the a.f. calibrated attenuator. The amplifier-modu¬lator unit consists of an R-C oscillator which drives through a
buffer stage a slicer circuit (used to produce the square wave
modulating voltage), and supplies the reference signal through a
82
Preamplifier Jnit
r
From i
Crystal 1V Amp. 2nd Amp.
Cath.
FollowerL
4-» a-f Calib. Attenuator
r
x
Coherent
SignalDetector
1
Amp.
<5 -
Amp. H
Phase
Inverter
Selective
Amp.
Butter
Amp.
Buffer
Amp.
Phase
Shifler
RC
Oscillator
Buffer
Amp
Slicer
Circuit
Amplifier-modulator
Unit.
To KlystronReflector
Fig. 5.2. Block diagram showing the preamplifier unit and the amplifier-modulator unit.
phase-shifter and a buffer to the coherent-signal detector. The
coherent-signal detector is preferred to the conventional detector
circuit because it reduces the pass band to about 1 cps and thus
allows to measure the amplitude of the fundamental component of
the signal with greater accuracy. The output from the a-f atte¬
nuator is fed to a matched buffer amplifier followed by a selective
amplifier stage, using a twin-T feedback network. The output from
this selective stage is fed to the coherent-signal detector througha phase invertor and two buffer amplifiers. In spite of the high
selectivity of the coherent-signal detector the twin-T selective stageis not omitted since it removes a great deal of noise and inter¬
ference prior to the detector.
Measurements on the constructed units at the working frequency
gave the following results:
1. The preamplifier gives a maximum undistorted output of
about 300 mV to a load resistance of 75 Q with a voltage gain of
about 600.
2. The slicer circuit gives a square-wave of amplitude 40 V max.
with a time of rise of about 1 % and a smaller time of fall.
83
3. The selective stage has a pass band of about 1,5 kc/sec cen¬
tered at the working frequency.4. The coherent-signal detector has a linear characteristic up to
full-scale deflection of the indicating meter (an ampermeter of range
1 mA). The detector has a pass band which is certainly less than
1 cps.
5. The voltage gain of the amplifier up to the detector input is
about 7000. Thus total gain of both units is about 130 db.
The system allows to measure input voltage ratios up to 60 to
70 db corresponding to VSWR's 30 to 35 db with an accuracy
< 1 db. Small VSWR's (as low as 0,1 db) can be measured by usingthe output meter.
Apart from standing-wave measurements we still requiremethods to enable the measurement of absolute and relative power,
attenuation, wavelength and frequency differences of the order of
Slotted
Section—to Atten. 1 m Atten. II
Matched
Detector
'
InicatingDevice
Fig. 5.3. Circuit arrangement used for calibrating the attenuators.
magnitude of some Mc/sec. At the time these experiments were
carried out no thermistor mount for 3-cm band was available, also
no attenuation standards. The only attenuators available were two
variable waveguide attenuators of the flap type using IRC resistive
strips as dissipative elements. To use these in measuring attenuation
and comparing power levels, they were calibrated in the followingsimple manner. A crystal detector, matched to a VSWR < 1 db
receives power from a modulated source through the 2 attenuators
under test as shown in Pig. 5.3. The indicating device is that shown
in Fig. 5.2. With the dissipative strip of attenuator I fully inserted
in the guide (maximum attenuation) and that of II fully withdrawn
(zero attenuation) the reading of the output meter was adjusted
by the a-f attenuator to 0,7 its full deflection. A magic-T, also
84
matched to a VSWR < 1 db is then inserted between the detector
and the attenuator II. Power division between the output arms is
then checked and found to be better than 0,2 db. With the crystaldetector connected to one output arm and the 2 other arms con¬
nected to reflectionless terminations, the reading of the outputmeter is observed. In this manner we have obtained 2 values on the
output meter with 3 db seperation between them. Now, using the
circuit of Fig. 5.3 the method of calibration is quite obvious.
Increasing the attenuation of II till the meter indicates the half-
power point and then decreasing the attenuation of I to the full-
power point enables to determine the attenuation curve against
40
-Q
~°30c
c
o
1520
c
tt 10
60 SO 40 30 20 10
Sceale Reading
Fig. 5.4. Measured attenuation characteristics for the attenuators used.
scale reading by a succession of points with any 2 successive points
separated by 3 db. The slotted line was used to check the VSWR
during the coarse of measurement. An example of the curves thus
obtained is shown in Fig. 5.4. It is not worth while to discuss the
sources of error in the method used nor to mention the disadvan¬
tages of such an attenuator, which is generally used as a buffer
attenuator only. The important thing is that with the help of this
attenuator we could, with a good enough approximation, estimate
relative power levels and measure attenuation.
85
For the measurement of frequency differences of the order of
some Mc/sec an obvious method is to use a mixer crystal connected
to a superheterodyne receiver that produces a tone when tuned to
the frequency of the signal received from the crystal. In some cases
it was found more appropriate to use the cavity wavemeter for this
aim in order to avoid undue complexity of the circuit. The circuit
shown in Fig. 5.5 was used to measure the difference between any
2 resonant frequencies corresponding to 2 different settings of the
wavemeter tuning plunger. Thence, the frequency difference pro
scale division near a particular resonance was calculated. This was
done by tuning the wavemeter to the frequency of Kz and mea¬
suring the frequency difference between the frequency of K2 and
that of Kx by using a superheterodyne receiver connected to a
crystal mixer (Det. (2) in Fig. 5.5). The frequency of K2 is then
*-&i20db
Term H2Direct.
4H1
(~)-20db-Coupler
2_. ,4 — Term.Direct.
Coupler3 — Det 2 _
Superhet.Receiver
C.W. — Det.1 -/?)
Fig. 5.5. Block diagram for the circuit used to measure the frequencydifference in Mc/sec pro scale division for a cavity wavemeter.
changed and the same steps are repeated. The 2 sets of readingsthus obtained (2 readings on the wavemeter scale and the cor¬
responding 2 readings on the receiver scale) enable to calculate the
frequency difference in Mc/s pro scale division. This was done for
a number of resonant frequencies covering the whole tuning rangeof the wavemeter. The measurement gave a value of 10 Mc/s pro
scale division at the resonant frequency lying in the middle of the
tuning range. A slightly higher value was observed at higherresonant frequencies towards one end, and a slightly lower one
towards the other end of the tuning range. It is to be noted that
the scale of the wavemeter used has 18 divisions, each containing10 subdivisions and that the resonance wavelength at the middle
of the tuning range was estimated to be about 3,39 cm.
86
5.2. Cold Test to estimate the Q-Factors °f tne Klystron Cavity
Near a particular resonant frequency, the impedance lookinginto the cavity across an appropriately chosen reference plane d0can be written in the form (see [8], chap. 12)
Zw = ±^tl^\ + " (1)
with Z(do) and r expressed in terms of the characteristic impedance
30 of the measuring hne. Thus, at that reference plane d0 the cavity
Rvwwvvv o
c it to
Fig. 5.6. .Equivalent circuit of a klystron cavity at particular reference
planes and near a particular resonance.
can be represented by the equivalent circuit Fig. 5.6, if
Q0 = —^j— = the unloaded-Q of the cavity,
and QR = oj0Cj0 = the radiation or external Q of the cavity.
Using the abbreviations
a = ^ and S = QB(^-^)=2Q^,Eq. (1) becomes
Z = ~^ + r. (la)
From Eq. (1) it is obvious that the reference plane to be chosen
should coincide with one of the standing-wave minima planes, when
the cavity is detuned far enough from resonance to make
87
10 and Z<d.) = r = real. (2)
a+ j8
It is to be noted that Q0 is a measure of the internal losses of the
cavity whereas the series resistance R is included to account for
the losses in the line coupling the cavity to the waveguide.To estimate the parameters contained in Eq. (1) it is customary
to plot the VSWR as a function of the frequency throughout the
resonance curve of the cavity. The circuit we used to perform this
measurement is shown in Fig. 5.7. The procedure of measurement
Square-wavemodulated
Source
0- 15 db Atten. 'Direct.
3
I . Coupler
IndicatingDevice
ISlotted
Line
:.w
Klystron under
Test. (Cold.)
O
Det.D-0
Fig. 5.7. Block diagram of apparatus used for making the cold test.
is as follows. The signal source is tuned to the frequency of interest
and the klystron under test (not oscillating) is sufficiently detuned
away from this frequency. In this case the position of the minimum
as indicated by the slotted line defines the position of the reference
plane d0 and the measured VSWR is equal to reciprocal of the
series resistance r. With the probe of the slotted line at the plane
d0 the cavity is then tuned to the frequency of the signal. This is
indicated by the fact that either a standing-wave maximum or
minimum will be found at the plane d0, if the cavity is in tune with
the frequency of the incident signal. With the cavity thus tuned
the resonance curve is plotted by changing the frequency of the
signal source and measuring the corresponding VSWR and positionof the minimum. The frequency of the signal is indicated by the
wavemeter.
The data obtained from the above measurement is shown in the
curves Fig. 5.8 for the 4 reflex-klystrons to be used in the coarse of
our experiments. In Fig. 5.8 the VSWR is given in db and the
frequency axis indicates frequency differences from the resonance
88
-60 -40 -20 20 40 60 SO
i 1 r
-60 -40 -20 0 20 40 60 80
-60 -40 20 40 -60 -40 20 40
Fig. 5.8. Measured resonance curves of the four reflex-oscillators under test.
AF = Frequency deviation off resonanbe. r = VSWR in db
frequency. These are estimated from the results of the calibration
of the wavemeter that was mentioned in the introduction.
The determination of the cavity parameters from the measured
data is carried out in the following manner. Consider first the input
impedance Z(do)o when the cavity is in tune with signal frequency.
This is given from Eq. (la) by
89
A*,),, = -~ + r (3)a
Let the corresponding VSWR be r0. Since the quantity Z^ may
be greater or less than unity, two cases must be distinguished. In
Case 1, the standing-wave maximum at resonance is found at d0and Z(do) is greater than unity; in Case 2, Z^ is less than unityand the standing-wave minimum occurs at d0. This gives
r0=I + r, Casel,
= h r, Case 2.
r0 a
(4)
For the 4 tubes under test it was found that Case 2 holds. In what
follows only Case 2 will be considered. Let roii denote the VSWR
with the cavity sufficiently detuned. From Eq. (2) we have
~y=Z{di))-r. (2a)
From Eqs. (2a) and (4) — —
rr can De determined. To separate
Q0 and QB it is customary to determine the width of the resonance
curve at some properly chosen value of the VSWR. To find the
relation between 8 and the measured VSWR, let us denote the
value of the latter for a certain 8 by rg and let us introduce the
complex reflection coefficient r defined by the equation
Z(dn\ — 1
Wo)Ado) +l
Since only the phase of r will vary by varying the position of the
reference plane, we may calculate the magnitude y = \r\ for any 8
from (la) and (5), thus giving
2=
Mr-l) + l}2 + 82(r-l)27
{ff(r+l) + l]2 + 82(r+l)2-( '
Since a and r are known, the value of y for any 8 can be calculated
from (6). Substituting this y in the relation
90
gives rs at the assumed value of 8. Let the width of the resonance
curve at the calculated value of rg be A F Mc/s. Since from defi¬
nition
QR can be obtained. If QR is thus calculated for a number values
of 8 and the average is taken, this average will represent a better
approximation to the value of QR.This calculation was carried out for the 4 tubes under test. Let
us denote these by K1, K2, K3 and K±\ these symbols are goingto be used to refer to any of these tubes when used later on in the
following experiments. The results obtained are:
Klystron r0 r Qr Qo
*i 1,26 0,05 885 650
K2 1,42 0,10 885 535
K3 1,42 0,03 650 440
K, 2,82 0,03 950 310
r0 = VSWR at resonance. „
r = normalised series resistance — giving lossesho
in the line coupling cavity to waveguide.
QR= radiation Q.
Q0 = unloaded Q.
5.3. Mutual Synchronisation
An arrangement of a circuit appropriate for the experimental
investigation of the mutual synchronisation of 2 reflex-oscillators
has been described in section 3.1. The arrangement actually used
for the following measurements is shown in Fig. 5.9.
The directional coupler used here couples about 3% of the power
flowing in the main line 13 into the auxiliary line 24. Signals coupledinto this line are used for indication and measurement. The cali¬
brated attenuator in series with the phase shifter enable to vary the
magnitude and phase of the coupling between the 2 oscillators. In
the phase shifter used the change in guide wavelength is brought
91
K,0-
Phase
Shifter
Calib.
Atten.
Det.2
Receiv.
1Direct.
3
Coupler4
Oor£
C.W.
Det.1
&
CRT
Fig. 5.9. Circuit arrangement used to measure the width of the synchroni¬sation region for mutual synchronisation of 2 reflex-oscillators.
about by moving a long polystyrene slab laterally across the interior
of the waveguide. The position of the slab is indicated by a micro¬
meter scale; with the slab near the side wall of the waveguide the
reading on the scale is zero. Thus, this reading corresponds to the
longest equivalent length introduced by the phase shifter.
The aim of the measurement is to determine the width of the
synchronisation region as a function of the phase shifter setting for
different values of the attenuation (i.e. |812|OT as a function of ifiwith a as parameter). The same measurement was carried out twice;
once for the pair of klystrons K1 — K3 and the other for the pair
Ki — Ki. The following description is going to be referred to the
pair K1 — K3.Consider first the case where the attenuator is set to give maxi¬
mum attenuation (about 45 db for the attenuator used). Since the
power coupled from one oscillator to the other is now very small,
each can be assumed to be oscillating at its "free-running" fre¬
quency. Signals arriving at detector (2) will be mixed by the crystalmixer and an i-f signal, of frequency equal to the difference between
the "free-running" frequencies of Kx and K3, is supplied to the
receiver. The latter is adjusted to receive unmodulated signals.Thus an audio tone will be heard if the receiver is tuned to the
frequency of the i-f signal. Now if the reflector voltages were pread-justed such that the relative reflection phase angles are both zero,
the "free-running" frequency of each will be equal to the resonant
92
frequency of its cavity. Thus, the frequency measured by the
receiver gives directly the difference between the resonant frequen¬cies of the cavities.
Let us now consider the case where the attenuation is adjusted
to give a certain coupling of magnitude a. We keep the resonant
frequency oj01 of iC1 constant and vary that of K3 (co03). If oj03 is
near enough to ioQ1 but no synchronisation takes place, we observe
beats — each oscillation undergoes periodic variations of frequency
and amplitude. The period of these variations becomes longer, as
a)03 is brought nearer to one of the limiting frequencies at the
boundaries of the synchronisation region. At the same time the
average frequency of each beat is "pulled" away from the resonant
frequency of the corresponding cavity and is brought nearer to
that limiting frequency. The signals reaching the detectors are com¬
posed of a mixture of the 2 beats. Hence, the output from the i-f
terminals of the crystal rectifiers is again a beat of average fre¬
quency equal to the difference between the average frequencies of
the r-f beats. Thus, if the receiver is tuned to this average inter¬
mediate frequency, a tone will be heard. Due to the complex
character of the i-f beat received, it is obvious that the tone will be
heard, if the receiver is detuned to either side of the average.
However, the middle frequency of the band, over which the tone
can be heard, may be taken equal to the average frequency of the
i-f beat. Also, if this average frequency is low enough, the signalreceived by the cathode ray tube (CRT) will produce a wave of
some particular form. Now, if co03 is brought slowly towards the
boundary, a reduction in the frequency of the i-f beat will be
indicated by both the CRT and the receiver. The disappearance of
the wave seen on the CRT accompanied by the vanishing of the
tone heard from the receiver will indicate that synchronisation has
taken place. Due to the lack of a fine adjustment in the tuning
mechanisms of the existing reflex-tubes great care should be taken
to adjust w03 so that the system oscillates in synchronism just at
one of the boundaries. If this is done, then by introducing again full
attenuation the difference between the resonant frequencies of the
cavities can be measured as explained in the preceding paragraph.
In this manner we obtain for each setting of the attenuator and
93
6 8 2 4 6 8
Phase-shifter Scale reading
Fig. 5.10a. Measured width of the synchronisation region as a function
of the phase shifter setting for different values of the magnitude of the
coupling a (expressed in db), for the Klystrons Kx and K3.The circles o give AFly and the crosses + AF2.
94
^ 7
(24 db) (21 db)
f\lr~—i 1
(18 db) (16 db)
T I I I
(12 db) (Wdb)
"i 1 r—i ~i 1 r
16-
14-
12-
10
^8^%6A
4-\
2
0
(8db) (6db) (4db)
—I 1 1 —i 1 1 r
24632463246
(2db)
i 1 1 —I r
2 4 6
Phase-shifter Scale reading
Fig. 5.10b. Same as Fig. 5.10a for the Klystrons Kt and Kt.
95
phase shifter the 2 frequency differences defining the synchroni¬sation region. Denoting by A F± the measured frequency difference
at the upper boundary i.e. for «j03>aj01 and by A F2 that at the
lower boundary i.e. for oj03<oj01, then the sum (A Fx-\-A F2) givesthe width of the band, over which synchronisation is possible.
Following the procedure just described, we obtained the results
shown in Fig. 5.10a and Fig. 5.10b. Here each pair of curves givesA F± and A F2 as functions of the phase of the coupling (expressedin terms of the phase shifter reading), with the magnitude of the
coupling kept constant. It is obvious that the form of these curves
is in good agreement with the prediction of Eq. (3.47) which showed
that [812|,„ should vary as the cosine of the coupling phase angle. It
is also noticed that the maximum of the A _Fx-curves are displacedto the right (towards shorter equivalent length of the line) with
respect to those of the A F2-curvea. This is due to the long-lineeffect described in section 3.4 and in accordance with the results
deduced there. The length of the line between the grids of the
2 klystrons under test was about 25 Xgi). A simple calculation will
show that for a band 20 Mc/s wide with a middle frequency at
8850 Mc/s (our working frequency) the difference between the elec¬
trical lengths at the sides of the band of a line of physical length25 Xg is about 40°. Since the phase shifter used is not calibrated, it
is not possible to check this result.
In Fig. 5.11 the maximum values A Flmax and A F2max are plot¬
ted as functions of —. Comparisons between this figure and Fig. 3.7
shows the qualitative agreement between theory and experiment.To make a quantitative check we use the expression (3.50) giving
the initial inclination of the curve relating |812|m to — (with </< = 0)
together with the measured Q-factors given in section 5.2. Direct
use of expression (3.50) necessitates the determination of the ratio
-^~from Eq. (3.49). This contains the conductance parameters
which, as given by Eq. (3.19), are functions of the small signaltransconductance Ge as well as G and Q; Ge and G have not been
measured. Thus we may either assume values for Ge and C or pur
y^- = 1. This latter assumption is rather justified since the first
96
K4.
K2,
pair
the
for
b)Ka.
K1,
pair
the
for
a)
of
funktion
as
region
synchronisation
of
width
Maximum
5.11.
Fig.
a0db'20logw
b)
«±1
0,9
0,8
0,7
0,6
0,5
0,6
0,3
0,2
0,1
0
II
II
1I
II
II
I
68
10
12
1624
i—r
i>
i11
a.
11
0,9
O.H
0,7
0,6
0,5
0,k
0,3
0,2
01
0
II
II
II
II
II
I
Odb
2«
68
10
1216
2t
~
11
11
i—i—i—i
ii
111
term of (3.50) is multiplied whereas the second term is divided bythis ratio. Thus, putting this ratio equal to 1 or 0,8 wjll give two
results which differ only by a few per cent. Taking ^~ = 1 and
substituting (3.22)—(3.25) in (3.50) yields for the initial inclination
the simple expression:
Mi+i)Mc'8'^ <8)
with F0 = 8850 Mc/s = working frequency. For the first pair Kxand K3 the initial inclinations as determined from Fig. 5.11 are:
for A Flmax 27 Mc/s and for A F2max 22,8 Mc/s while the value cal¬
culated from (8) is 23,8 Mc/s. For the second pair K2 and K4 the
experimental values are: for A Flmax 12,7 Mc/s and for A F2max11,3 Mc/s, while that calculated is 19,3 Mc/s. Reference to the table
in section 5.2 shows that the klystron K2 has a series resistance r
which is rather big. As shown in Fig. 5.6 this series resistance reduces
the magnitude of the signal coupled into the cavity and hence will
result in a reduction in the width of the synchronisation region.
Taking this fact into consideration and remembering that Eq. (3.50)
was deduced under the assumption of a lossless line coupling the
cavity to the waveguide (i.e. R = r = 0), we notice that the theoryalso agrees quantitatively with the experiment.
From Figs. 5.10 and 5.11 we notice that A Flmax is always biggerthan A F2max. It can be shown that inequality between the 2
limiting frequencies can be the result of an error in the preadjust-ment of the reflector voltages to some other values than those
making the relative reflection phase angles equal to zero. To show
this we use Eqs. (3.21) under the assumptions:
1. identical klystron with a small coupling,2. cPl = ^ = 0,
and 3. cp2is small so that e~J'<*'2 = l—j<p2, so that
Xx = X2 = X,
2
xyoosi8 = l-j/J1(Z),
(9)
-\ sin 8 = 8, .
y
98
Since 82 = 8t — 812 we have
S12 =-*'
amP + MA^X), (10)a y
which shows that the maximum values of 812 obtained by putting
(8= +90° are no more equal and are given by
4
S12 = S12, = —+ <Pf9Al (X)£= 90°
+ *?/
4and S12 = 8U_= --tpa-gA^X).
(3=-9o° <xy
Thus, if one of the relative reflection phase angles has a value other
than the zero, the values of A Flmax and A F2max are not equal.
Eqs. (11) show that
A FlmaT > A F2mar ^T <?% > 0
and A Flmax < A F2max for <p2 < 0.
Subtracting Eqs. (11) yields
|§12+-Sl2_| = 2l<P2|-^lP0- (12)
Assuming that (12) remains valid for large values of the couplingas well, the value of gAx(X) in (12) is then given by Eq. (3.39),
so that
IV-^h^Wl + ^t!,-) d3)
which shows that the difference increases by increasing the coupling
(i.e. reducing a). Comparison between (13) and Fig. 5.11 shows
again the agreement between theory and experiment.Further Eqs. (9) enables us to make some remarks on the
behaviour of a system of 2 mutually synchronised reflex-oscillators,
if either or both of the reflector voltages are modulated. Under the
assumption that the period of the modulating signal is small com¬
pared with the synchronisation time constant, the system can be
assumed in equilibrium at each instant of time and a succession of
steady state solutions give a good enough approximation. Hence,
Eqs. (9) give the solution when only the reflector voltage of K2 is
99
modulated; <p2 being thus the result of the application of a small
modulating signal. Assuming that both cavities are pretuned to the
same frequency we get
8ia = 0
8i = 82 = - |?a^iffl
which shows that the frequency at which the system oscillates is
directly proportional to q>2 and hence also proportional to the ampli¬tude of the modulating signal, if the latter is small. The factor \arises naturally from the fact that only one reflector voltage is
modulated. This enables us to conclude that if the static charac¬
teristics (relative reflection angle as a function of reflector voltage)of the klystrons considered are different from one another, the fre¬
quency of oscillation of the system remains proportional to the
amplitude of the modulating signal, when both reflector voltagesare simultaneously modulated. The equivalent static characteristic
relevant to the modulating signal, for the system as whole, will be
somewhere between the characteristics of the individual oscillators.
The behaviour of a "disturbed" oscillator in the region of beats
(a) (b)
Fig. 5.12. Frequency "Pulling" in the region of beats.
AF0= "undisturbed" frequncy difference.
Af = "disturbed,, frequency difference.
(a) for the'pair Ky, K3, (b) for the pair K2, K^.
WO
has been fully discussed in Chap. 1. A similar behaviour may be
expected in the case of two oscillators coupled together and gene¬
rating frequencies that are not widely different. If the two frequen¬cies differ by only a small percentage, they are both shifted from
their normal values in such a way as to reduce the difference. This
attraction of the two frequencies becomes more pronounced as the
difference between the normal oscillating frequencies is reduced
and finally becomes so great that the oscillators pull into synchro¬nism. This behaviour is illustrated in the experimental curves
Fig. 5.12a and b for the 2 pairs of klystrons under test. Here, the
average frequency of the i-f beat is plotted against the difference
between the undisturbed frequencies.
5.4. Synchronous Parallel Operation of two Reflex Oscillators
This problem was theoretically discussed in chapter 4, and the
condition for identity and symmetry of the operation was given in
Eq. (4.30) which defines the phase of the coupling coefficient. It
was also mentioned, that the magnitude of the coupling should be
chosen in such a manner that optimum power may be supplied to
the common load. The choice of the magnitude of the couplingnecessitates the knowledge of the behaviour of the particular kly¬stron under various loading conditions, i. e. the knowledge of its
rlieke diagram (see [8], chap. 12). However, for the particularinformations required for our present application it is not necessary
to plot the whole of the Rieke diagram. It is only necessary to
locate that region of the diagram where optimum power is suppliedto the load. This is done as follows. A movable screw tuner is
calibrated when connected to a matched termination and the com¬
bination is used as a standing-wave introducer. Here, the VSWR
depends solely on the screw introduction, and the phase on its
position. If this is used as a load for the klystron a quick deter¬
mination of the required data is possible. With a constant screw
introduction and variable position the point on the Smith chart
representing the load seen by the klystron traces a constant-y
circle (y = magnitude of the reflection coefficient). On this circle
we have to determine the position of the two points at which maxi-
101
mum and minimum power is supplied from the klystron. If this
is repeated for a number of values of y and the results plotted on
a Smith chart, the region of maximum power is readily determined.
The circuit we used for this purpose is shown in Fig. 5.13. The
reference klystron together with the mixer and receiver are used to
measure the change in the frequency of oscillation of the klystronunder test when the particular load of interest is presented to it
from its frequency when acting into a matched load. The cavity-
wavemeter (C. W.) is used to check the frequency of the reference
klystron during the coarse of the experiment. The calibrated
9 n A
s~\ Slotted
\y~ Line HMov.TuiScrew
ner
Calib.
Alien.
Klystronunder test
Term
/•AMput faDe,\^_
Output
Direct.
Coupler
Receiv.
Reference
Klystron
Direct.
Coupler
— Mixer
20 db
C.W.
/M0- Det.
Fig. 5.13. Arrangement of circuit used to determine the behaviour of the
klystron oscillator under different loading conditions.
attenuator together with the output detector and the micro-ampere¬meter are used to measure the level of the power output from the
klystron under test. The slotted line is used to determine the positionof the standing-wave minimum of any particular load of interest.
The data obtained for the 4 klystrons Kl, K2, K3 and Kt is plottedon the Smith chart Fig. 5.14; the reference plane used here is the
terminal.plane of the waveguide on which the klystron is mounted.
Curves in Fig. 5.15 give the level of the maximum and minimum
power outputs as functions of the VSWR of the load. It is now
obvious that with the help of Fig. 5.15 the location of the regionsof maximum power output can be easily determined on Fig. 5.14.
These are shown dotted in Fig. 5.14.
102
Fig. 5.14. Loci of maximum and minimum power output obteined by
fixing the magnitude of the reflection coeff. of the load and varying its
phase. For the four Klystrons under test the points determined are shawn:
x for Ky, o for K2, • for K3 and + for Ki. The regions enclosed within
the dotted curves denote those for optimum power outpus.
103
max.
5 rswR
1.0-
Ki
0.9 -
\ \^max.0.8 - \° ^^
0.7-
0.6 -
0.S-
\ min.
0.4 i
0.3-
0.2-• \>
0.H
1 1 1 1
srswR
Fig. 5.15. Plat of ^E and ^
6 snwR
minas functions of the VSWR.
104
Choosing K% and K3 to be operated in parallel by using the
combining circuit Pig. 4.4, the "operating point" should lie within
the common part between their respective dotted regions in Fig.5.15. At this operating point the necessary magnitude of the cou¬
pling coefficient is about 0,262 (corresponding to a VSWR 1,75)and the necessary phase is read directly on the "Wavelengthtowards generator" scale. Adjusting the screw tuners in Fig. 4.4 to
give this value of coupling, the adjustment of their positions to
give the required phase at terminals 1 and 2 of M T1 (see Fig. 4.4)can be carried out by a simple standing-wave measurement. With
everything properly adjusted the level of the maximum power
supplied to the load as well as that coupled into arm 3 of M T2were measured. The results of the measurement are: level of maxi¬
mum power supplied to the load is 3 db over the optimum power
level from K3 alone and 2 db over that from K2 alone; the level of
power coupled into arm 3 is 18 db beneath that supplied to the load.
Reference to Fig. 5.15 shows that the power supplied to the load
is at least equal to the sum of the powers supplied from each
klystron separately to a matched load. That complete cancellation
of the waves coupled into arm 3 does not take place, is obviouslydue to the unequal power outputs from the klystrons. To investi¬
gate the stability and reliability of the operation of the system, it
was left running for a couple of hours, switched on and off; the
system remained always in synchronism with practically constant
power supplied to the load.
With the appropriate adjustments performed the alternative
combining circuit Fig. 4.5 was then examined. The same results
were obtained here as for the original circuit Fig. 4.4. Further, it
was found possible, by slight readjustment of the resonant fre¬
quency of one the klystrons, that complete cancellation of the
waves coupled into arm 3 may be attained, accompanied by a
slight increase of the power coupled into the load. It was also found
that the adjustment of the positions of the screws is not critical:
the power supplied to the load varies slowly with varying the
positions of the screws, whereas their insertions produced marked
effects. This enables the proper adjustments of this circuit for the
synchronous parallel operation of 2 klystrons without having any
105
knowledge of their active performance. This is done by tuning each
separately to the same frequency with its reflector voltage adjusted
to the center of the mode. When connected to the circuit of Fig. 4.5,
a simple trial enables to find the proper insertions of the screws;
then maximisation of the power delivered to the load is performed
by adjusting the positions. It is advisable in this case to take
insertions of the screw which produce a VSWR 1,5 in order to
insure stability of operation. A circuit adjusted in this manner may
be useful for application as a signal source of a higher power output.
5.5. Synchronisation by a Signal from a Harmonic Generator
Frequency standards available in the microwave region are
either the frequencies of the absorption lines in the spectrum of an
absorbing gas (mostly NH3); these being found in the 1 cm region,or a signal of microwave frequency produced by multiplying the
frequency of a crystal oscillator. The multiplication into the vhf-
region is made by vacuum tube multipliers. The vhf multiple of
the crystal frequency is raised to about 900 Mc/s by using a light¬house triode multiplier. The last step in multiplication into the
microwave region is accomplished by harmonic generation in a
silicon crystal or by using klystron multipliers. Although the
klystron multiplier provides a larger power than does the crystal
rectifier, the alignment of the 2 cavities of the former is critical,
and the multiplier is difficult to use.
However, the possibility to synchronise a reflex klystron oscil¬
lator gives another alternative which provides a large power
without having the disadvantages of the klystron multiplier.To investigate this alternative we constructed the harmonic
generator shown in Fig. 5.16. The signal of fundamental frequencyis applied to the crystal through the i-f terminals and the 3-cm
harmonic is propagated down the waveguide. The transmission
characteristics of the waveguide provides adequate filtering of the
fundamental and thus no power of fundamental frequency exists
in the output. The input line is fitted with chokes which prevent
any harmonic power from flowing into this line. For best harmonic
generation one adjustable short circuit and 2 screw tuners are
106
3
band.
em
3the
for
Generator
Harmonic
5.16.
Fig.
Rings
Auxi
liary
Screws
Tuning
included to enable matching; in addition the crystal can be moved
across the guide by use of the auxiliary rings.Fundamental power at a wavelength of about 10,3 is supplied
by a lighthouse triode oscillator. Proper matching was attainable byusing a three stub tuner so that the VSWR seen by the 10,3 oscil¬
lator was about 1,22. The power supplied to the crystal was mea¬
sured and found to be about 45 mW. The harmonic power outputmeasured by comparison, using our calibrated attenuator, was
found to be at level 15 db beneath the power level of the 2K25
reflex-klystron; thus estimated to be about 0,8 mW. Thus the
conversion loss of the harmonic generator is
logifundamental power input
310 harmonic power output
in producing the third harmonic.
= 17,5 db
1 = 3,43 cmI
X -= 10,3 cm
ri
^
Harmonic
Generator
3-Stub
Tuner
Slotted
Line
Light-houseOscillator
Synchronised
Klystron^ 1 3
MT
2 4
o>
Term. Atten. Det.~l
To CRT.
Fig. 5.17. Synchronisation of one reflex oscillator by signal from harmonic
generator.
The signal thus obtained was used to synchronise a 2K25
reflex-klystron. The circuit used is shown in Fig. 5.17. The signalreaching K is thus \ that generated by harmonic generator, i. e. of
about 0,4 mW. An external signal of this magnitude will affect
synchronisation over a band of frequency, the total width of which
is about 1,5 Mc, which is rather narrow. However, with proper
adjustment, the system remained in stable synchronism for a
couple of hours giving a power output of about 10 mW or more.
Thus the total conversion loss of the system as a whole is given by
logfundamental power input
synchronised harmonic power output= 6db
108
with an increase of about 11,5 db over that obtained by the har¬
monic generator alone. When the system was switched off and left
to cool, to be switched on afterwards, no synchronisation occured.
This is due mainly to the fluctuation in frequency of the lighthousetriode oscillator, which, even when small, will result through multi¬
plication in a rather high frequency difference.
Next, the signal from the harmonic generator was applied to
synchronise simultaneously a system of 2 reflex oscillators by usingthe circuit shown in Fig. 4.2. The stability of the system was bad;but as it once happened that the system remained in synchronismfor a few minutes the power output was double that obtained from
a single klystron with a corresponding conversion gain. However,
to attain a good stability of such a system the signal provided bythe harmonic generator should be of about 3 mW to affect syn¬
chronisation over a band of 3 Mc/s. With the same conversion loss
of the harmonic generator alone the fundamental signal power
should be about 150 mW. However, if the fundamental signalsource has a stable frequency, only about 100 mW fundamental
power is necessary to obtain good stability and steady locking.Since the harmonic power output in such a system is about 50mW
the conversion loss of the system as whole will be about 3 to 5 db.
I wish to express my deep gratitude to my Professor Dr. F. Tank
for his valuable advice and superior guidance. It is also to acknow¬
ledge that some of the apparatus used in carrying out the above
experiments has been paid through the fund of the Jubilee 1930
of the Swiss Federal Institute of Technology. I wish here to express
my best thanks and appreciation.
109
Literature
1. Balth. van der Pol, "The nonlinear Theory of Electric Oscillations".
Proc. I.R.E. vol. 22, pp. 1051—1086. Sept. 1934.
2. Heinz Sarnulon, ,,tjber die Synchronisierung von Rohrengeneratoren".Helvetica Physica Acta, vol. 14, pp. 281—306, 1941.
3. Fritz Diemer, „YJber Synchronisierung von Rohrengeneratoren durch
modulierte Signale". Mitteilungen aus dem Institut fur Hochfrequenz-technik an der ETH in Zurich, Nr. 7, Verlag Leemann Zurich.
4. Robert Adler, "A Study of Locking Phenomena in Oscillators". Proc.
I.R.E., vol. 34, pp. 351—357, June 1946.
5. Robert D. Huntoon and A. Weiss, "Synchronisation of Oscillators". Proc.
I.R.E., vol. 35, pp. 1415—1423, December 1947.
6. J. R. Pierce and W. O. Shepherd, "Reflex Oscillators". Bell SystemTechn. Journal, vol. 26, pp. 97—113, March 1946.
7. E. L. Ginzton and A. E. Harrison, "Reflex-Klystron Oscillators". Proc.
I.R.E., vol. 34, pp. 97—113, March 1946.
8. D. R. Hamilton, J. K. Knipp and J. B. H. Kuper, "Klystrons and
Microwave Triodes". McGraw-Hill Book Co. Inc., New York, N. Y., 1948.
9. A. H. W. Beck, "Velocity-Modulated Thermionic Tubes". The Macmillan
Co., New York, N. Y., 1948.
10. Montgomery, Dicke and Purcell, "Principles of Microwave Circuits".
McGraw-Hill Book Co. Inc., New York, N. Y., 1948.
11. Montgomery, "Technique of Microwave Measurements". McGraw-Hill
Book Co. Inc., New York, N. Y., 1948.
12. M. Surdin, "Directive Couplers in Waveguides". Journal I.E.E., vol. 34,
pp. 725—745, Sept. 1946.
110
Course of Life
I was born on 18th November 1921 in Alexandria (Egypt). After
finishing the primary and secondary schools I obtained my certi¬
ficate of maturity in 1939. Then I joined the Military College in
Cairo and studied there as a student-officer for 2 years. In 1941
I entered the Faculty of Engineering in the Farouk I Universityin Alexandria. After 5 years study I obtained in 1946 my B. Sc.
degree in Electrical Engineering. I worked in the same faculty as
assistant for one year and then came to Switzerland. Since the
summer term 1948 I studied under the guidance of Prof. Dr.
F. Tank in the Institute of High-Frequency Techniques at the
Swiss Federal Institute of Technology.