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

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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.