I

MEASUPEMENT TECHNIQUES FOR PEXIODIC STRUCTLTPES

BY W, J, Gallagher

In te ma 1 Memorandum

A,E.C. Contract AT(04-3)-21 (Project Agreement NO. 1)

M. L. Report No. 767 PrcJect M Report No. M-205

Novezber 1960

M i crorcve Iaboratory W. W. Hansen k b o r a t o r i e s of Physics

Stanford University Stanf ora, C a l S f ornia

I .

I1 D

Cavity Test Methods . . . . . . . . . . . . . . . . . . . . . . . . 1

A . Measurement of Group Velocity . . . . . . . . . . . . . . . . 4 B . Measurement of Q . . . . . . . . . . . . . . . . . . . . . . 6 C . Measurement of r/Q . . . . . . . . . . . . . . . . . . . . . 8 D . Fourier Analysis of Cavity Fields . . . . . . . . . . . . . . 11

Matching of Couplers . . . . . . . . . . . . . . . . . . . . . . 19 A . Resistive Match Method . . . . . . . . . . . . . . . . . . . 19 B. N o d a l Sh i f t Method . . . . . . . . . . . . . . . . . . . . . 21

I11 . Uniform Structure Measurements . . . . . . . . . . . . . A . Preliminary Getexmination of Running Frequency . . . B . Nodal Shif t ing of Periodic Structures . . . . . . . .

I V . Buncher Measurements . . . . . . . . . . . . . . . . . . Appendix A - I l l u s t r a t i o n of Metall ic Plunger Method f o r

Determining r/Q . . . . . . . . . . . . . . Appendix B . I l l u s t r a t i o n of Disc-Loaded Line Fourier

Analysis . . . . . . . . . . . . . . . . . . Appendix C . Construction of Resistive Termination . . .

. . . . 31

. . . . 31

. . . . 32

. . . . 42

. . . . 3 4

. . . . 55

. . . . 56

1

. ii .

T

Page LIST OF ILLUSTRATIONS

1 .

2 .

3 . 4 . 5 . 6 . 7 . 8 . 9 . 10 . 11 . 12 . 13 . 14 .

15 .

16 .

17 .

18 . 19 20 .

I l l u s t r a t i n g the n/2 mode of resonance w i t h shorting

planes a t e i t h e r re f lec t ion plane of symmetry . . . . . . . . . . 16 Arrangement of apparatus and idea l iza t ion of accelerator

section with couplers . . . . . . . . . . . . . . . . . . . . . . 22

Smith-chart explanation of coupler matching technique . . . . . . 23 .Example of transformation at first coupler . . . . . . . . . . . 26 Example of mismatched coupler data . . . . . . . . . . . . . . . 27 Determination of Iconocenter . . . . . . . . . . . . . . . . . . 28 Location of Iconocenter by a l t e rna te construction . . . . . . . . 29

Example of detuning plunger . . . . . . . . . . . . . . . . . . . 34 Nodal s h i f t pat tern f o r n/2 mode . . . . . . . . . . . . . . . . 35 Example of nodal-shift problem . . . . . . . . . . . . . . . . . 38 Example of nodal-shift problem . . . . . . . . . . . . . . . . . 39 Nodal-shift data of 41-cavity sect ion . . . . . . . . . . . . . . 40 Hinged bar f o r nodal-shifting buncher . . . . . . . . . . . . . . 45 Apparatus arrangement f o r f ie ld-s t rength measurement i n -

buncher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 Alternate apparatus arrangement f o r f ield-strength

measurement i n huncher . . . . . . . . . . . . . . . . . . . . . 48 Typical f ield-strength pa t te rn observed along ~ / 2 mode

waveguide section by setup of Fig . 14 . . . . . . . . . . . . . . 49

observed along 2n/3 waveguide sect ion by setup of Fig . 15 . . . . 50

Smith-chart p l o t of input impedance of typical pat tern

Example of r/Q measurement . . . . . . . . . . . . . . . . . . 51 Perturbation data produced by d i e l e c t r i c bead . . . . . . . . . . 52 Suggested match-termination f o r n/2 mode disc-loaded

waveguide a t 10.5 cm . . . . . . . . . . . . . . . . . . . . . . 57

1 T .

I. CAVITY TEST METHODS

The descriptive properties of slow-wave structures f o r accelerator

application are the voltage attenuation per uni t length (I), the group

velocity (v ) and the shunt impedance, o r resistance, per uni t length

( r ) . These parameters a r i s e natural ly from consideration of the per- formance of accelerators.

traversing an accelerator i s given by

g

Thus, the maximum energy gain by a par t ic le

4 V = f Ezdz

0

where the f i e l d strength EZ

defined by

is, by analogy t o cavity-resonator theory,

velocity a re taken

where w i s the l inear energy density, and the group velocity and energy

as ident ica l . That is, the power flow i s

By def in i t ion

tenuation i s given

P(2) = w(z)vg .

of the voltage attenuation coefficient the power a t -

by

or - = - e 21P . -212 P(Z) = P0e dz

By definit ion, the Q of a transmission system i s 231 times the

r a t i o of energy stored per un i t length t o energy dissipated per uni t

length per cycle, which i n view of (3) is given by

WW

Q=--

Therefore, from (j), (4) and ( 3 ) , ' 0 ) Q = -

Q 21v

In addition, from (2), ( 3 ) , (4) and

which i s sometimes used as

T . 1

a defini t ion.

- 1 -

(6) it i s evident t ha t

21Pr

(4)

( 5 )

(7)

1 .

The i n t e r e s t i n group veloci ty fur ther a r i s e s from the f i l l - t ime of

the s t ructure ,

i n view of (6). By defini t ion, of course,

where 2 2

n P2 = B, + B

a r e thepropagation constants.

dispersive charac te r i s t ic of the accelerator structure, and t.herefore

The group veloci ty i s descriptive of the

ind i r ec t ly a l so spec i f ies the dimensional tolerances which must be held

i n construction.

The voltage-attenuation coef f ic ien t i s of in te res t , since fcr ef-

f i c i e n t use of the input power it determines the length of the acceler-

a to r . For example, from Eq. (1) the constant-impedance s t ructure at. no

load has an energy gain

from which it i s seen tha t

av - =iq (214 + 1) e-''*- 1

(11 1 13 I

-1e - 1 , or 14 = 1.255. This, of which has a maximum when

course, means t h a t the maximum energy gain occurs fo r a par t icu lar length

of such a guide.

218 = e

It i s perhaps noticeable that these parameters a re not independent

of each other. Further, these a r e not conyenient parameters t o measure

experimentally, but it i s possible t o relate them (for a t ravel ing wave

i n the per iodic s t ruc ture) t o o ther parameters of a f i n i t e length of such a s t ruc ture when resonant, which can be measured conveniently.

It i s a well-known result that l i n e a r homogeneous d i f f e ren t i a l

- 2 -

T

equations whose coefficients a r e singly periodic f'unctions of the inde-

pendent variable have solutions which a re also periodic (Floquet ' s Theorem).

Thus, f o r a traveling w v e i n a loss less periodic c i rcu i t the axial elec-

t r i c f ie ld may be expressed

where p i s the periodic length of the structure. Then an axial par t -

i c l e synchronized wit.h one of these space-harmonics encounters an average

f i e l d over a periodic length,

= a (13) n

since

The ncnsynchronous space harmonics a r e orthogonal i n z over.the in te rva l

P -

t = (3 Z/CD fo r the p a r t i c l e synchronized with the n th harmonic. n

If a cavity i s formed by; placing e l ec t r i c ref lectors i n the t rans-

verse planes of symmetry, the propagating modes of the periodic c i rcu i t

satisfy theboundary conditions of the cavity; and the standing-wave f i e l d

may be expressed

E~ =l 2an cos pnz e ju t .

To resonate, the cavity must contain an integral number of half guide-

wavelengths, and, as the cavi ty i s a l so an integral number of half periods

i n length x P

q.;S-=L (q,s = 0,1,2 ...) 2 . 2

( - u) < n < + m )

1

- 3 - T

or, from Eq. (14)) the cavity f i e l d s may be written

This i s the bas i s of the re la t ion of the parameters of a travelicg-wave

f i e l d i n the periodic s t ructure t o the parameters of a f i n i t e length of

sucBa s t ructure when resonating.

a r e r / Q , v /c , and Q of the resonant cavity, from which r, v /c

and

l a t ions .

The properties which w i l l be measured

Q Q of the transmission system can be obtained by the foregoing r e - I

A. Measurement of Group Velocity.

A s mentioned above, a test cavity composed of a length of periodic

s t ruc ture terminated i n planes of re f lec t ion symmetry w i l l resonate

times i n each passband i f the cavity consists of r periodic lengths.

These resonances will occur when

r + 1

o r = ( q = 0,1,2...r) q 2 = L = r p x@;

that is , there a re e l ec t r i ca l phase shif ts from 0 t o n per period. If

the mode i s the number of resonances may be 2r + 1 i n each passband, thzt i s , elec-

t r i c a l phase-shifts from 0 t o 2n per period are exhibited by the s:ruc-

ture. This i s eas i ly determined by passing a bead through the cavity

(supported by a thread) and observing the number of wavelengths i n the

cavi ty a t each resonance.

@-varying, and per iodici ty i s defined by t ranslat ion alcne.

It is, further, possible that the structure has more than one re -

f lec t ion plane of symmetry per period. If the number.of periodic lengths

of t he s t ructure i s an odd number of half-periodic lengths, the n-mode of a l t e rna te passbands will not resonate.

Evidently these resonances f a l l on the dispersion diagram i n the

passband of the traveling-wave s t ructure .

often p lo t ted i n wave numbers, propagation constants, o r merely frequency

versus longitudinal phase-shift per period (or mode).

Such dispersion diagrams are

On the dispersion diagram, phase veloci t ies are indicated by the

coordinates

1

- 4 -

T '

or, i n general ( i f p i s the longitudinal phase in te rva l per period, p),

v 2rr p - E = - - . c

The group velocity is, by definition,

v ag d(l /1> A = - =

c wn d(l/Ag)

and i s therefore the slope of t he dispersion diagram.

diagram i s constructed from the discrete resonances of a f i n i t e length of

the structure, we may compute the slope over a frequency in te rva l

If the dispersion

A(l /h) = A f / c

and the prppagation number span

then

C c h c l

If f e w periods of the s t ruc ture a re available, it i s often assumed that

the dispersion diagram i s sinusoidal. Then the group velocity i s given

by

V T P A f A=- s i n (5- C C

where

at the veloci ty of l i gh t ,

i s the width of t h e passband. In par t icular f o r the sr/2-mode

v n p A=-@. C C

Another approximation fo r t he n/2-mode at the velocity of l i g h t depends

upon the f ac t that the dispersion diagram i s nearly l i nea r near the band

center. Then

- 5 -

'7 T

For the two adjaqent resonances hl and X2 , where the section consists

of n cavities,

and

where X i s the wavelength of t he n/2-mode.

B. Measurement of Q . r

By far the most d i f f i c u l t property t o nieasure accurately i s

With suf f ic ien t lengths of the s t ruc ture the e f fec ts of end-plates and

coupling loops i s considerably lessened, and reasonably accurate values

may be obtained.

Q

EI the case where two short lengths of the t e s t structure a re avai l -

able the following method may be employed,

of n and I J ~ periods, We may estimate the stored energy t o be propor-

t i ona l t o the npnber of periodic lengths

portional a l so t o the number of periodic lengths

fo r the end p l a t e s (Lp) . Then the Q of each secticn i s given by

Sugpose the two lengths consist

1 (nS> and the losses t o be pro-

(r?) plus a fixed lo s s

n, n A b L z

n2L + L Q2= nlL + L Ql=

P P

Eliminating L , and solving fo r the r a t i o of periodic storage t o loss, P

The measured &1 and &r are, of course, obtained by conventional tech-

niques. Generally, i n an experimental program the coupling apparatu.s i s

far from the c r i t i c a l l y coupled, so that d i rec t Q-circle methods a re im- pract ical , but i n such cases as this the half-pcwer method i s s t i l l prac-

t i c a l .

1 T

- 6 -

If the input standing-wave-ratio technique i s use& where the input

VSWR i s uo at resonance, the VSWR at the 3-db points i s

2 2 2 2 ' 0 bo + uo + 1) +Y(uo + a. + 1) - 0

u = 0 a

To obtain suff ic ient accuracy, the

method at each frequency, and the results plotted as a function of f r e -

quene>.* In any case it i s advisable t o obtain the c rys ta l l a w i f th i s

type of detector i s used.

VSWR should be measured by the 3-db

This en t i r e technique of obtaining the Q depends c r i t i c a l l y on the

equal adjustment of the end p la tes i n both cqses.

the case of two periodic lengths,

Consider, for example,

The Q i s calculated from n = 2n2 . 1 Eq. (19):

where i s the half-power bandwidths.

The e r ror may be expressed

or, eliminating squared terms of the fract ional error and rearranging,

- - 8Q - - 6Q2 + - - -8Ql . Q B Q l 2Q2 - Ql

When the e r rors a re of the Same sign and bf the same order (both measures

e i the r too high or too low) the fract ional e r ro r . i n the f i n a l value i s of

the same order as the individual measurements.

opposite signs and are of the same order, the result ing accuracy decreases

rapidly.

But i f the errors have

* The 3-db method of measuring high values of VSWR

soidal pa t te rn i n the s lo t t ed l i n e . If the carriage i s moved t o +3 db from'the n u l l on both sides and scale difference & noted, it may be shown that the VSWR i s u = Xg/& .

assumes a sinu-

- 7 -

T

C. Measurement of r / Q .

i n Eq. (2) the choice a r i s ing from analogy with cavity resonators, but

where by EZ we indicate the peak axial e l ec t r i c f i e l d of the t ravel ing

wave. Thus,

O f the various possible beam-interaction measures, we have expressed

-

r E,~ - = - ohms per unit length Q w

2 Hansen and Post have described a method of measuring r / Q which

involves the interact ion of a small metal l ic plunger of cross-section

A through the end p l a t e on the axis of the cavity.

the resonant frequency of the cavity i s then

The perturbation on

hf eOEc2( 0)AAz

f - - - -

C 4w

where w Az i s measured posi t ively in to the cavity. From Eq. (14), E (0) = C 2an,

and since the wave i s standing,

through the end-plate

i s the t o t a l energy stored i n the cavity, and the displacement C

w = 2wL ; so t ha t the plunger passes C

( z = 0) Eq. (20) may be re-written

where L i s the t o t a l length of the cavity. The nth harmonic peak shunt

impedance is, therefore, contained i n

Putting l/xc0 = qc/x = 120c t h i s becomes

%. W. Hansen and R. F. Post, "On the Measurement of Cavity Impedance, " J. Appl. Phys. 19, 1059, (1948).

- 8 -

'1 T

2 a 120 Lc Af (a>

P

The space-harmonic component with which the beam pa r t i c l e s a r e sychronized

i s the only one contributing t o the net energy of the beam and is, therefore,

the reason for the Fourier resolution i n Eq. (a). The above technique requires the determination of the slope of the -

frequency perturbation curve as the perturbation vanishes, which i s the

region of greatest curvature of the function, and consequently.of l ea s t

accuracy.

The accuracy of the perturbing plunger may be conveniently examined

by a preliminary measurement on a cylindrical cavity i n the !INolo mode.

The theore t ica l r / Q of such a cavity i s

The perturhation produced by a metal plunger on the ax is of a cylindrical

cavity propagating the !JNolo mode i s

Af /r\2 Az

where r i s the radius of the plunger, a i s the radius of the cylinder,

h i s the height of the cylinder, and aZ i s the plunger insertion. Thus,

which can be compared t o the exprimental determination.

Diff icul t ies arise i n this technique from plungers which a re too large

i n diameter. I n addition, plungers which do not f i t closely i n the end-

p l a t e produce a perturbation based on different cross-sections depending

on whether t he plunger i s protruding o r not.

the edges of the perturber hole not be bevelled.

It is a l so essent ia l t ha t

- 9 -

1

A numerical example of the use of the above technique i s presented

i n Appendix A.

An a l te rna t ive method f o r making this measurement has been described 3 which is, i n general, more rapid and produces more accurate r e su l t s If

a t h i n d i e l ec t r i c rod of cross-section A and d ie lec t r ic constant E i s

inser ted along the axis of the cavity, traversing

frequency perturbation on the resonance is

i t s en t i re length, the

C 4W

From Eq. (14) the in tegra l may be evaluated i n terms of an amplitude sum-

mation and the peak shunt impedance of the nth harmonic i s contained i n

the expression

Putting e ' = E / E ~ and l/xc0 = 120 c this becomes

The d i f f i cu l ty of this method i s the precise lmowledge needed of the

d i e l ec t r i c constant of the perturbing rod; but this, too, may be determined

by the perturbation produced i n a case where the theoret ical perturbation

is calculable. For example, an axial d i e l ec t r i c rod i n a cylindrical cavity

propagating the ~ o l o mode produces a frequency perturbation

1 2 Af aw E

f W

- - - - - = (y) I:) =:(Pol 1

from which, i f the diamater of the rod i s small enough, i t s d i e l ec t r i c '1 'W. R. e t al., M.L. Report No. 403, Stanford University, (1957).

- 10 -

7 T '

constant may be determined.

rection may be made f o r this, since the radial f i e l d decreases as

If the rod is too large i n diameter a cor-

Jo(kr).

Several additional methods have been employed t o measure shunt im-

pedance o r r/Q . For example, the Q before and a f t e r the inser t ion of

a r e s i s t i ve cblumn on the axis can be u s e d t o f ind the shunt impedance.

The r / Q may a lso be measured by means of a small metal bead, of radius

a and volume V , carried on a thread t o a specified location i n the

cavity.

This function i s projected t o a vanishing bead, at which point it may be

shown the shunt impedance i s given by

The frequency perturbation i s noted as the bead length i s decreased.

120 LZ lim M r = Q [ l - a ) A M iG

\ I

where L i s the length of t he cavity, and a i s a correction factor t o

account f o r dis tor t ion of the f i e l d due t o the bead, and. i s given approx- 2 imately by cx = 3/8 (kea) ,, where

D. Fourier Analysis of the Cavity Fields.

= k2 - k are propagation constants. kc %

If e i the r a small metal o r d i e l ec t r i c bead is carried on a thread

through the cavity along the axis, a frequency perturbation i s produced

of amount

where i s a shape factor accounting fo r L e d i s t o h i o n of the f i e l d due

t o the bead geometry and material, so that i n any case (where only uniform

e l e c t r i c fields a re considered) the perturbation is proportional t o the

square of the f i e l d in t ens i ty i.e.,

able be taken as the displacement along the axis of the cavity, it i s ap;-

parent from Eq. (16) that a Farrier analysis of the square-root of the

perturbation pat tern w i l l reveal the coefficients of the space- harmonics.

These coeff ic ients a re not normalized, of course, but since the energy

they represent i s approximately constant, the sum of the coefficients may

be used t o normalize these individual coefficients.

F

-nf a 1E I*. If the independent var i -

- 11 -

1 T '

The details of the procedure of a Fourier analysis for the per- turbation data are presented here for convenience.

Consider the function f(x) plotted for one guide wavelength in the structure, that is, electrically over the range -A < x < R .

and f2(x), by the definition The function is then decomposed into odd and even components, fl(x)

f2(x) = ; [f(x) + f(-x) L 1 .

Then the Fourier components of these sets are

bk sin kx fl(x) = b sin x + b2 sin 2x + b sin 3x + ... - 1 3 - 1 k = l

f2(x) = a + a cos x + a cos 2~ + a COS 3 + .... - - 1% cos kx . 2 3 0 1

k = O

To find the coefficients of the trigonometric terms, divide.$he range

at the midpoints of the intervals, or 0 < x < I[ into n intervals, corresponding to arguments xl, x2.. .xj...x n'

2 j - l n 2n x = J

at which the functions have the values (at the center of the intervals)

f ( ), f (x ), * . .f (x.) . . .f (x ); and f2(xl), f (x ), ..of (x ) . . .f (x ) 1x1 1 2 1 3 1 n 2 2 2 j 2 n

By the orthogonality condition over a fundamental period for integrally the series expansions can be shown to be reducible to related harmonics,

a = k

n

n 1 f2(Xj) cos px. J p = 1,2,3 ...

and

n

O n j = 1

- 12 - T '

2 n J bk = - 1 fl(xj) s in px .

j =1

There i s no point i n counting the harmonic-component analysis beyond

k = n, since the number of i n t e rva l s implies t h e degree of d e t a i l specified

about the function f (x) . I n Eq. (24) it i s indicated t h a t the index p

runs over the posit ive integers . However, from Eq. (12), which describes

the propagation modes, it i s evident tha t the propagation constants (repre-

sented by

i n the

Therefore 8, can only have the values x/2p, 3n/2p, 5n/2p.. . (2n + 1)1~./2p.

On.the other hand, when operating i n the 2n/3 mode, pn = (211/3p)(1 + 3n), and Bn has the values 21r/3p, 4n/3p, 81~/3p, 14n/3p ... The value of t h i s

f ac t i s apparent when ac tua l ly performing the work of determining the space-

harmonic amplitudes.

- 00 < n < + =) can be predicted. For example, when operating

n/2 mode, Bn = f3, + 2nn/p and Bop = n/2, o r B, = (n/2p) (1 -I- 4n).

As a matter of technique, it should be noted t h a t due t o the f i n i t e

s ize of the bead it i s desirable t o remove the bead, through the end-plates

without disassembling the apparatus set-up, i n order t o obtain the resonant

frequency without the bead i n the cavity.

where the f i e l d actual ly vanishes, a perturbation w i l l be indicated, and

the Fourier analysis w i l l be performed f o r an incorrect perturbation ampli-

tude.

If the bead bridges a region

Another source of e r ro r i n the bead perturbation measurements a r i s e s

when a minimum length of the s t ruc ture i s used. A s the bead approaches the

end-plates, the coupling of the bead with i t s image i n the end-plate causes

a l a rge r perturbation than i s j u s t i f i e d by the f i e l d strengbh a t the point.

It is, of course, possible t o ca l ib ra t e the bead by observing the pertur-

bat ion produced on the TM

end-plate.

a perturbation 1.1 times too large, and tha t t h i s e f f ec t vanishes when the

bead i s i t s own height away from the end-plate.

cy l indr ica l mode as the bead approaches the 010 It has been found tha t beads i n contact w i t h the end-plate give

A more subt le d i f f i c u l t y a r i s e s i n cases where a large number of per i -

ods of the s t ruc ture are used f o r the Fourier analysis.

resonances a re too close, these a l so contribute t o the f i e l d dis t r ibut ion.

This may be indicated by the dissymmetries i n the perturbation pattern.

I n addition, i f the bead is too large the range of the perturbation may

extend i n t o the adjacent mode. Further, i f the bead i s not s m a l l the

If the adjacent

- 13 -

1 T '

f i e l d w i l l be dis tor ted. The most corivenient manner of determining the

appropriate bead s i ze i s probably t o decrease the bead volume u n t i l the

perturbation pat tern no longer depends on the bead size.

upon the d i e l ec t r i c constant as well as the volume of the bead.

This size depends

I n general, the Fourier analysis i s the most inaccurate procedure

i n cavi ty analyses. Aside from the experimental error i n the data t o be

decomposed, t he approximation resu l t ing from a f i n i t e Fourier sum i s not

unique, but depends on the type of acceptable errcr .

analysis described above r e su l t s i n a minimum mean-square error, but the

coeff ic ients would be different from an analysis based on a uniform-toler-

ance error . Occasionally, when too few terms a r e used, it is obvious by

inspection that the f i t could be improved by adjusting the calculated

values.

For example, the

A numerical example of the technique i s given i n Appendix B.

There is, i n addition t o the above technique, mother method of

obtaining an approximation t o the space-harmonic coefficients when the

periodic s t ructure contains two o r more re f lec t ion planes of symmetry i n one period.

plane of the discs and half-way between the discs.

produced by an a x i a l plunger i n the end-plate i s measured i n cavi t ies termi-

nated i n e i the r of t he two ways, the results w i l l i n general be different .

The difference i s of course due t o the f a c t that the f i e l d in tens i ty o f t h e

space-harmonic sum i s d i f fe ren t i n the two cases.

For example, the disk-loaded l i n e mntains such planes i n the

If the perturbation

If the axial f i e l d i s expressed as a space-haknonic sum,

the f i e l d on the axis at the end plate , s i tuated i n the midplane bztwezn

discs ( i . e . , z = 0), where we consider t h e time average, i s

Ec(0) = 2kn .

- 14 -

T '

The fac tor of two ar i ses from the condition of resonance. meed by the plunger i s then

The perturbation

Af Aw E ~ A A Z E ~ ~ ( O ) - = - =

wc 2WL

and the slope of the perturbation curve as the plunger passes through the

end-plate i s

f “oA2 2 (:) =&an) *

0

If the end-plates are moved t o the plane of the discs ( z = p/2),

the f i e l d description i s the same as the previous case, except that the

t i m e i s advanced an amount Ir/k and the posit ion a half-period,

j(at + E x - B, E(z ) =. ane c

- 0 3

where k i s the number of d i scs per wavelength.

The time-average of the f i e l d on the axis a t the end-plate i s then

Ec($) = 2 E n cos (-m)

and the perturbation produced by the plunger i s

af Aw “ o m c 2 2 (2) - = - - -

2WL f w C

The slope of the perturbation curve as the plunger passes through the end-

p l a t e i s

A s an example, Fig 1 i l l u s t r a t e s two cavi t ies which w i l l resonate

- 15 -

1 T

Metallic Plunger

I t a l l .unge

.ic r

Fig. 1 - - I l lus t ra t ing the rr/2 m o d e of resonance

with shorting plane at e i t h e r re f lec t ion plane of symmetry.

- 16 - .

in the n/2 mode. The perturbation experiment for one case may be put in the form

a 0 + a -1 + a +1 + =j- [El 01 and in the other case

a - a - a + o o o

02 0 -1 +1. fe 2 A

neglecting higher space-harmonics. Adding these expressions gives a 0' the square of which may then be compared to (&n)2 to give the correction factor to the shunt-impedance measurement. That is,

which may then be used as shown in Eq. (21).

the accuracy of the procedures is obtained. When both techniques described earlier are employed, an estimme of

We use the notation

240 c ~f 120 Lc Af -

f2A (.E) z=o - (T/Q)~ - -

2 (T/Q)~ - (E' - l ) A f

for the dielectric rod and metallic plunger methods of measuring r/Q , respectively. Then for any space-harmonic component n

from which it is evident that

- 17 -

T

The dielectr ic-rod method measuring the r/Q is probably t o be

preferred t o the plunger method on the basis of the e f fec t of an e r ro r i n

the harmonic analysis.

the coefficient of the nth harmonic, the e r r o r i n the

space-harmonic is, closely, f o r the dielectr ic-rod method,

If an er ror of amount 6an occurs i n estimating r/Q of the zero-th

but f o r the plunger method the same e r r o r produces an e r ror in the

given closely by

r/Q

The l a t t e r i s la rger than the former when the power is carried preponderantly

by the zero-th space harmonic.

It is possible t o cmpute the normalized amplitude coefficients, con-

sidering the gap voltage t o be a rectangular pulse. In t h i s case, where

the gap is considered t o be between the edges of the disc hole (a length

6 = p - t , where t is the d isc thickness),

sin (PnS/2) Ib(7 r ) a = n n ( 8,6/2 ) 10 ( Yna)

and yn = Bn - . Ekperimental agreement with t h i s formula i s not

impress ive . 2 2

- 18 -

T

11. MATCHING OF COUPLEXS

The procedure i n matching couplers depends upon the d e t a i l s of the

whole section being constructed, and what components in the system can

be considered t o be without fabricakion errors.

section has both input and output couplers and a preliminary examination

(discussed i n Section 111-A) reveals no large ref lect ions i n the subsections,

then the method whose de ta i l s are described i n Section 11-B may be used.

If no output covpler is used the method of Section 11-A may be used.

For example, i f the

I n the matching of couplers par t icu lar note'should be taken t h a t

dsing 1000-cps square-wave modulation of a UHF' source w i l l present a d i f -

fe ren t matching picture from tha t f o r the microsecond pulses used i n the

accelerator operation. Suppose, f o r example, t ha t a ref lect ion termination

is inserted i n the disc-loaded guide and measurements are made with a

s lo t t ed section i n the rectangular-waveguide feed l ine. The match w i l l

be made t o the region a t which the load is located by adjustment of the

couplcing r a t i o and tuning of the mode converter o r coupler. The load

should be located a t such a distance down the disc-loaded guide that no

re f lec t ion from the front end of the pulse will arr ive back a t the klystron

before the pulse has finished. Otherwise, the output impedance of the

klystron w i l l change during the pulse, possibly resul t ing i n frequency

modulation o r a power-level change.

a short pulse w i l l not cancel, w i l l be-"washed out" by long square waves

used i n tes t ing . By means of a d i rec t iona l coupler a t the input, a short

pulse may be viewed on a fast sweep oscilloscope f o r the purpose of examining

the instantaneous ref l e c t ion.

Distributed reflections, which during

A. Resistive Match Method.

This method, described here fo r the x/2 mode of operatim, involves

The load, the use of the res i s t ive termination described in Appendix C.

which w i l l probably not be perfect (VSWR of - 1.2), is placed i n two successive

cav i t i e s and the impedance noted.

t h e m is taken as the match point.

of the load adds t o tha t of any previous ref lect ion i n opposite phase i n

the two cases, as they are a quarter-wave apart,. The only approximation

i s t h a t the coupler mismatch, as well as the load mismatch, is not too

large. ) I n

The half-way point on the l i ne joining

(This is quite val id since the ref lect ion

This match-point i s taken as the mismatch of the coupler.

- 19 -

T '

r e a l i t y t h i s is only the center of the impe@nce c i r c l e of the load, and

the "Smith center" is displaced s l i g h t l y more.

becomes increasingly be t t e r as a match is approached.

However, the approximation

The Brincipal d i f f icu l ty is i n deciding w h a t operation t o perform on

the coupler t o improve the match.

taken with respect t o the plane of the iris, by placing a detuning plunger

i n the coupler f o r a reference, since the observed impedance i s on the

&-circle of the coupler.

too small, the coupling m u s t be increased, and vice versa.

point l i e s above or below the r e a l axis the cavity is detuned.

(by Foster ' s theorem, extended t o lossy networks) the locus of impedance

passes through match i n a clockwise direct ion w i t h increasing frequency,

the direct ion i n which t o perturb the cavity t o tune it is evident.

t ha t when the operating frequency is below resonance the cavity i s resonant

a t a higher frequency and must be lowered.) But, as a check, before

deforming the cavity, it may be perturbed e i t h e r by dropping i n a slug t o

ra i se the frequency (magnetic or inductive decrease perturbation) or

placing a plunger i n the beam hole t o lower the fre'quency (e lec t r ic or

capacitive increase perturbation. ) the tuning operations, since f i l i n g a coupling hole detunes the cavity as

wel1,as increasing the coupling, and bending an iris changes the coupling

as well! as the tuning.-

The clue is provided when the data is

.

If the match-point implies that the c i r c l e is

If the match-

Since

(Note

Some care m u s t be exercised in performing

The above remarks apply pr incipal ly t o cavity couplers. However,

doorknob type couplers are t reated SimilarAy, but changing the coupling

r a t i o usually has a s ignif icant e f f ec t upon the coupling and cavity tuning,

and the two are d i f f i c u l t t o adjust separately.

The t ra jec tory of successive matching trials plotted on a Smith Chart

w i l l be curved, even f o r the same operation, so that the choice of what

operation t o perform is not always evident and the work should be done i n

short steps. If considerable d i f f i cu l ty i s encountered, suspicion of a junction

cavity ( i f one exists) is jus t i f ied .

resonant frequency of the coupling cavi ty o r junction alone, although they

are great ly overcoupled.

examined with the coupling cavity unexcited by placing a shorting plate

across the midplane of the cavity if it is bisected.

It is not d i f f i cu l t t o check the

If t h i s is necessary, the junction cavity can be

Otherwise, a shorting

- 20 -

1 T '

plunger i n the following cavi ty w i l l exci te the junction cavity only.

"he coupler with the junction opened o r detuned w i l l exci te only the

coupling cavity. I n e i t h e r case the input impedance is measured f o r

several frequencies.

resonance, but since the points a l l .have about the same VSWR near reasonance

a reference obtained by detuning the coupled cavity (or the n u l l of a

frequency far off resonance) w i l l allow one t o obtain a Q-circle.

The real-axis crossing or minimum VSWR marks the

B. Nodal S h i f t

The mathematical j u s t i f i c a t i o n of the following method is contained

"he system under discussion i s shown i n Fig. 2.

When the short i s s e t a t ' any a rb i t r a ry position, the admittance (or

i n references 4, 5 , and 6.

impedance) a t a single frequency looking into the system w i l l l i e on a

c i r c l e - o n the Smith Chart described i n the following anqlysis.

the second mismatch, the impedance of the short is always sane point on

the rim of the Smith Chart.

(Fig. 3 , P P P P ). 1 2 3 4 susceptances a r i s ing from the short .

of the eighth-wavelength points along the rim (PI P ' P ' PI ) .

a t tenuat ion of the l ine , viewed j u s t a f t e r the first mismatch, shrinks the

Viewed a f t e r

Consider points one-eighth wavelength apart

A t mismatch No. 2 a constant susceptance i s added t o This only red is t r ibu tes the posit ions

Now the 1, 2, 3 , 4

admittance locus in to what is ca l led an admittance c i r c l e , but note tha t

the r e l a t ive posi t ion angles of the eighth-wavelengkh points a re preserved and that the c i r c l e i s centered on the h i t h Chart (P" P" P" P"). ( In the

example of Fig. 3 the unspecified phase-shift occurring i n the subsection

i s ignored.) Next, a f t e r some a rb i t r a ry rotation of the admittance c i r c l e

due t o the phase-shift over which the attenuation occurred, the susceptance

of mismatch No. 1, due t o detuning of the coupler is added t o each point

of the c i r c l e . This is done along constant conductancq c i r c l e s on the chart

and, therefore, i n every case,. fu r the r shrinks the s ize of the admittance

1, 2, 3 , 4

c i r c l e , displaces the center and redist.ributes the points P;" P;' P;,' P1;" . J Y ,

Note t h a t the real-axis diameter of the c i r c l e is now along a constant

susceptance l i n e . The point SC on t h i s l i ne is the Smith center. The

geometric center of the c i r c l e 0 l i e s on the l i ne from the center of the

& i t h Chart t o the point SC. I n addition t o the susceptance of the

4J. Rybner, Journal IEE, Pt. 111, 93, 243 (1948). 'G. A. Deschamps, J. Appl. Phys., - 24, 1046 (1953). 6

- E. Feenberg, J. Appl. Phys., 17, 530 (1946).

- 21 -

1 T '

Input Coupling Cavity

I-.-+

Slot ted Line

1 1 1 1 1 1 1 1 1 1 1 I?

I EZf3

P

Power Source -I Input Output

Coupler Wave guide Coupler (Mismatch (Attenuation) ( M i s No.1 -

I Short

- - * e r

(Mismatch (Attenuation) ( M i s No.1 -

I

Slot ted u Line and Detector

Fig. 2--Arrangement of apparatus and idealization

of accelerator sect ion with couplers.

1

- 22 -

T

\ \

'\

I 5'

. Fig. 3--Smith Chart explanation of coupler-matching technique.

- 23 -

1 T

coupling cavi ty due t o detuning there is a renomalization of the admittance

c i r c l e due t o the coupling coeff ic ient between the two waveguides.

"he admittance c i r c l e , which is now centered on the critically coupled Q-circle

(uni t c i r c l e ) , is therefore transformed by the amount of the coupling ccef-

f i c i e n t k and the c i r c l e becomes, f o r example, P:: Pzy Pi; Pc'*.

is performed d i r e c t l y from P" P" P" P" and then the susceptive mismatch added,

or if the susceptive mismatch i s added t o P" 1, Pi, P;, PI: and the c i r c l e

p;" p;ll ,;I P1;" transformed.

l / k tance c i r c l e (shown as having a s ize

u = l / k and is of a diameter such that u The center of the c i r c l e therefore l i es on the bisection of the l i n e between

I n pr inc ip le it does not matter whether the coupling transformation

1, 2, 3, 4

) > , In the former case the c i r c l e Pi, Pl, P;, Pc is transformed by .& amount

(shown as k = 0.5 i n the i l l u s t r a t i o n of Fig. 4), such t h a t the admit-

2) then has i t s Smith Center a t "O=

= uo2 and u = 1 . 1 max m i n

the center of the Smith Chart and uo2 , indicated by the c i r c l e P P P P

"he susceptive mismatch is added t o each point , but now i n amount

(taken as B = 0.5 i n the example), producing the f i n a l c i r c l e P i "

mismatch i s added (B = 0.5 i n the example), producing the c i r c l e P"' Pi'' Pj" P1;".

This c i r c l e is then transformed by multiplying each point on it by

producing the f i n a l c i r c l e Pi': PGy P;; P;".

since the VSWR w a s i n f i n i t e .

A B C D ' B/k

1111 p"" p " " ~ 7 p2, 3 , 4

P" P" the susceptive pi, 3 , 4 In the l a t t e r case t o each point of Pi 2

1, 7 , l /k ,

Ear l i e r , i n the case of the output coupler, these d e t a i l s were omitted

Note tha t the s i ze of the admittance c i r c l e depended only on the

at tenuat ion and mismatch No. 1, and i t s location only on mismatch No. 1.

Consider now a case of data taken w i t h the shorting plane moved

successive one-eighth guide-wavelengths (Fig. 5 ) . four potnts . If the impedance c i r c l e contains the center of the Smith Chart

t h i s c i r c l e may now be centered on the Smith Chart (subtracting mismatch

No. 1) using a VSWR diameter u :

F i t a c i r c l e through the

=v'max U min'

where amax and u are the m a x i m u m and minimum VSWR of the c i r c l e . The min VSWR of mismatch No. 1 (ul) i s the transformation ra t io ,

m 1

min

1

- 24 -

T ' c

If the impedance c i r c l e

r e l a t ions are reversed.

uate the attenuation of

does not csntain the center of the Smith Chart these

The diameter of the new c i r c l e may be used t o eval-

the l i ne , which is given by

1 I& = a r c t a d - nepers a (29 )

The magnitude of mismatch No. 2 can be determined by means of the iconocenter

e i t h e r before o r a f t e r the t ransfer .

the intersect ion of orthogonal c i r c l e s passing through the alternate eighth-

wavelength points (see Fig. 6). by taking tangents to , say, points 1 and 3 and s t r ik ing an a rc passing through

1 and 3 centered on the intersect ion, and similarly f o r points 2 and 4. A

check on the accuracy of the construction is given by the f a c t that points

2 and 4 and the intersect ion of the tangents t o 1 and 3 l i e on a s t ra ight

l i ne . Obviously, 1 and 3 l i e on the l i ne passing through the intersection

of tangents t o 2 and 4 also.

point which l i e s on a l i ne through the iconcenter and the geometric center.)

When the eighth-wavelength points l i e nearly on diameters of the c i r c l e

This point is determined as being a t

The orthogonal c i r c l e s may be constructed

(Further, chords from 2 t o 4 in te rsec t a t a

the f igure i s inconvenient t o construct, and an a l te rna te geometric construc-

t i o n w i l l f i nd the iconcenter.

(see Fig. 7). Join the center of the c i r c l e (0) t o the chord intersection,

and e rec t perpendiculars t o both ends. Where these in te rsec t the impedance

c i r c l e (points a, b, c, d ) , j o in diagonal points. Their intersect ion w i l l

l i e on a l i n e from the center of the Smith Chart and w i l l be the iconcenter.

This value of VSWR or susceptance must now be renormalized t o the f u l l

h y off the chords 1 t o 3 and 2 t o 4

Smith Chart diameter and is then the VSWR of mismatch No. 2. The same r e su l t s

would be obtained i f the t ransferred impedance c i rc le , i .e. , centered on the

Smith Chart, were projected out t o the rim and the iconcenter then obtained.

This l a t t e r method gives an addi t iona l check on the experimental accuracy,

since the iconcenter must l i e on the unit-resistance c i rc le .

If the impedance c i r c l e were p lo t ted as suggested above, the d i f f i cu l ty

would be evident that, while the VSWR of both couplers and the attenuation

of the sect ion of periodic s t ruc ture was determinable, there would be no

clue as t o w h a t operation could be employed t o remove the mismatches. This

is remedied by p lo t t ing the impedance c i r c l e with respect t o a reference

obtained by detuning the input coupler. Then the Smith center of the im- (J from the center of the Smith pedance c i r c l e (a distance * = d u m = min'

- 25 -

1 T

Fig. 4--Example of transformation at first Coupler.

- 26 - T

c ,--

Fig. 5--Example of mismatched coupler data \ (P P P P ). The c i rc le , 1 2 3 4

centered at 0, f i t s the points so tha t the Smith Center SC may be ob-

tained from the m a x i m u m and minimum VSWR of the c i rc le . Since SC l i e s

on the impedance c i r c l e (or Q - c i r c l e ) . i t may be drawn immediately, from

which it i s seen tha t the system i s undercoupled (k = 0.5) and coupler

tuning is high in frequency.

- 27 -

T

Fig: 6--Detemination of Iconocenter

- 28 -

Fig. 7--Location of Iconocenter by a l te rna te construction.

- 29 -

T '

Chart on the l i n e passing thrb'ugh the geometrical center of the impedance

c i r c l e ) is a l s o on the Q-circle of the input coupler (see Fig. 5 ) . is evident since the displacement of the impedance c i rc le from the center

of the Smith Chart is caused by the reactance ar is ing from the input

coupler cavi ty only. The Q-circle i tself is eas i ly drawn since two points

on it (0 + j0 on the Smith Chart and the measured impedance) and i t s dia-

meter are known. Once the Q-circle is obtained, the operatiow tha t are

required t o tune the input coupler are obvious. When t h i s is accomplished,

the most p rac t i ca l procedure is t o reverse tk slot ted l i ne and the s l iding

short and adjust the output coupler, rather than observing through the ac-

ce l e rat or guide.

In pract ice one takes more than j u s t eighth-waveguide points t o deter-

This

mine the impedance c i rc le . If the periodic structure has any in te rna l

mismatches t h i s w i l l become evident, since the impedance c i r c l e w i l l not

then be a c i r c l e , within the experimental accuracy of the method.

As a f i n a l check on the matching operation a rectangular waveguide load

may be used in place of the slzding short .

- 30 - T

111. UNIFORM STRUCTURE MEASUREMENTS

A. Preliminary Determination of Running Frequency

A f i n i t e length of per iodical ly loaded waveguide w i l l , when frequency-

swept, resonate i n r + 1 m o d e s i n any passband i f the length is shorted

at re f lec t ion planes of symmetry, where

s t ruc tq re . As mentioned above th i s resu l t s from the condition f o r resonance,

which is t h a t there be an in t eg ra l number of half guide-wavelengths i n the

resonator:

r is the number of periods of the

A n d = ~ n = 0 , 1 , 2 , ... r .

2

In order t o determine the running frequency f o r cold-testing on periodic

s t ructures a length must be chosen such tha t the desired mode w i l l resonate

i n the cavity. In accordance with the above condition of resonance, the

e l e c t r i c a l phase-shifts per period which w i l l resonate i n N periods of

the s t ructure a re

nn I? - n = 0, 1, 2, _... N .

This presents a d i f f i cu l ty . I f , fo r example, the frequency of the

n/2 mode is desired, no length w i l l t e s t a l l the cavi t ies , as only al ternate

cavi t ies a re excited, and reversing the end being excited does not change

the cav i t i e s examined.

I n thds par t icular case, regardless of the number N of periodic

lengths of the structure, it may be resonated using only one shorting

plane, and then reversed and resonated frorn the opposite end. The correct

running frequency is t o be between the (N - 1)/2-th and (N + 1)/2-th reso-

nance.

a t each end.

Of course there are two such frequencies, given by the measurements

If these frequencies a re different it can be concluded tha t

there i s an e r r o r i n the construction.

difference it cannot be assumed there is no error . The foui. cases of

construction e r rors ( resul t ing i n shunt susceptances, or ref lect ions) can

be considered individually: (1) If two susceptances of opposite sign occur,

an odd number of quarter guide-wavelengths apart , then the two add.

(2) If the two susceptances of opposite sign are an even number of quarter

guide-wavelengths apart they cancel.

However, if there i s no frequency

( 3 ) If the susceptances have the same

- 31 -

1 T ’

sign and are an odd number of quarter guide-wavelengths apart they cancel.

(4) If the susceptances are the same and are an even nunber of quarter

guide -wave lengths apart they add.

The frequency differences are due t o the f ac t that the uncancelled re-

f lec t ions cause a rearrangement of the energy; and therefore the frequency,

in the resonant cavity.

Quantitatively, these constructional e r rors may be analyzed i n terms

of the reactances they introduce. Thus, a reactance j x causes a standing-

wave - rat i o Z + jx 0 a =

zO

or re f lec t ion coefficient

The f rac t iona l voltage transmitted through the reactance is 2

- - - 1 - - 8

+ ... (32) 2 + 7 - x

2 1 - lPl =d(l - P)(l - PI ' 4 + x

and has a phase-shift 8 , since 4 - j 2 x

- - j x 2 1 - p = l -

2 + j x 4 + x

of amount

( 3 3 ) X t an 8 - - - 2 *

Some idea of the magnitude of the phase-shift may be got from the

.-experimental data by means of the re la t ion , valid near the middle of the

passband ,

f 2

where n i s the number of cavi t ies i n the subsection, and k i s the number

of discs per wavelength.

cause of the swamping out of the e r r o r by other correctly constructed

cavi t ies , but t h i s weight must be divided by 2 because the frequency d i f -

ference was observed by the measurements a t both ends.

It is necessary t o weight the frequency data be-

N

Since 8 = x/2 , . - . x e2

PI = - = - a 2

- 32 - 1

and s imilar ly the VSWR may a l s o be computed.

The location of e r rors i n construction i s be t te r l e f t t o the nodal-

A t t h i s point the frequency s h i f t technique described in Section 111-B.

at which the nodal s h i f t ought t o be performed is determined.

B. Nodal Shift ing of Periodic Structure

Once the couplers have been matched it is convenient t o proceed with

a determination of the construction accuracy of the periodic structure.

The most p rac t i ca l method of obtaining. t h i s data is t o d r a w a shorting

(or detuning) plunger through the structure plot t ing the input impedance

on a Smith Chart. The pr inciple involved is the 'nodal s h i f t ' technique

described in re fs . 6 and 7. Briefly, if a shorting plane is moved a specified e l e c t r i c a l distance

i n a uniform transmission l i ne , the VSWR pat tern w i l l move the same e l ec t r i ca l distance i n the same direct ion if there are no reflections between the

shorting plane and the location where the VSWR pat tern is examined.

otherwise, the VSWR pat tern w i l l be displaced an unequal amount, depending If

upon the magnitude of the mismatch.

More specif ical ly , i n a lo s s l e s s system the driving-point admittance

f o r a shorted l i ne i s always susceptive (on the rim of the Smith Chart).

A t the shorting plane and a t every n u l l of the VSWR pat tern the susceptance

i s in f in i t e . The physical locat ion of these nulls from the shorting plane

is , i n general,

n n = 0, 1, 2, 3 ... V

but if a lumped susceptance j B exis ts i n the system, because of i t s associated phase-shift, the above oondition becomes more complicated,

since its effect-depends upon where it is located. In a uniform l ine the most p rac t i ca l thing t o do is t o move the shorting

plane by equal small increments (Ay ) and note the inequalit ies of the s h i f t

of the n u l l posit ion (&) on a s lo t t ed l ine .

9 against

of e l e c t r i c a l phase-shift w i l l have occurred, and it can be shown t h a t the

g From a histogram of Ax

, which is sinusoidal, every possible value of e l ec t r i ca l value Ayg

'E. Feenberg, J. Appl. Phys., 17, 530 (1946). 7E. Mallory, "A Comparison of the Predicted and Observed Performance of a

-

Billion-volt Electron Accelerator," HEPL Report No. 46, W. W. Hansen Laboratories of Physics, Stanford University (1955), p. 68 e t seq.

'E. L. Ginzton, Microwave Measurements, McGraw-Hill Book Co. (1957), p. 281.

1 T '

- 33 -

1/8 D. Groove Break Edges

Approx . 5/16 R.

.050 D

Fig. 8 - - ~ e t w i n g plunger f o r disc-loaded periodic structure where the disc hole is 0.822 in . and v = c a t 2856 Mc/s. The plunger grooves rest on

posi t ion a r e s i s t i ve termination. The 10-32 tapped hole is used f o r exten-

s ion rods t o in se r t the plunger in to the structure.

d i sc holes when correct ly P positioned. The 0.050 in. dim hole is used t o

- 34 - 1 T '

-z Towards Generator

Fig. g--Nodal-shift pat tern. lkpedance as detuning plunger i s moved equal

small increments toward the loeii, showing the "dwell points" when the plunger

of Fig. 8 is used.

- 35 -

T

amplitude of the pat tern D/2

the s ta t ionary obstacle by

i s re la ted t o the ref lect ion coefficient of

D p = s i n - 2 '

However, the problem i s somewhat d i f fe ren t in periodic structures.

When the detuning plunger (Fig. 8) is moved i n small increments away from

is observed f o r the n/2 mode.

s h i f t an equivalent e l e c t r i c a l distance. Furthermore, it moves i n the

opposite direction, that is, toward the generator. These e f fec ts are

both due t o the manner of detwing a cavi ty i n a periodic c i r cu i t . How-

ever, it is noticeable tha t when the plunger i s about half-way through a

cavity there i s a dwell-point of the driving admittance (or impedance).

It is t h i s property tha t is made use of i n nodal-shifting a periodic struc-

tu re .

is equivalent t o a shorting plane of re f lec t ion symmetry of the cavity

immediately ahead (toward the generator). Therefore the phase-shift in-

formation applies t o that cavity and not t o the one i n which the plunger

i s located.

- the generator, f o r example, a pat tern somewhat l i k e that shown i n Fig. 9 For equal displacements the n u l l does not

Note that when the plunger is detuning a par t icular cavity, t h i s

However, i f the plunger be placed i n successive cavi t ies and the icput

impedance (or admittance) be noted, it w i l l be evident tha t insuff ic ient

information e x i s t s t o determine the phase-shift due t o each cavity. The

impedance c i r c l e on which these points are located is displaced from the

center of the Smith Chart by an amount which depends on the sum of the

mismatches between the cavi t ies being measured and the location a t which

the measurements a re performed. The impedance c i rc le i t s e l f can be de-

termined by an addi t ional measurement, e i t h e r by some other locations of

the plunger or by a res i s t ive termination placed on the head of the plunger.

In the l a t t e r case we have then three points (two on the impedance c i r c l e ,

and the other i t s e l e c t r i c a l center).

impedance c i r c l e , intersecting it orthogonally, and are themselves on a

c i r c l e , from which construction of impedance c i r c l e can be determined.

(Tangents t o the 'diameter' a t the impedance points intersect at the

geometrical center of the impedance c i r c l e . )

These points are a 'diameter' of the

If the impedance c i rc le i s centered on the Smith Chart, the angular

phase-error of a par t icu lar cavity may be measured d i rec t ly off the rim of

- 36 -

1

the chart .

points a re diametrically opposite the center of the chart t ha t the c i rc le

is centered.

O f course it cannot be assumed t h a t because the impedance

If the impedance c i r c l e i s displaced, the tangents t o the diameters

at the match points intersect .

phase-error of the cavity being examined.

mation t h a t the mismatch in to the system is not greatly affected by the

mismatch associated with the phase-error, and therefore the two impedance

c i r c l e s may be approximated by the single c i rc le containing three impedance

points. The two diameters remain the same, and therefore if we assume tha t

angular displacements are uniform around the chart, we may take the half-

angle of the tangents t o the diameters at the m&ch points as the phase-error.

Figures 10, 11, and 12 i l l u s t r a t e several possible s i tuat ions when a

periodic s t ructure i s excited i n the n/2 mode (mid-band).

input impedances f o r the detuning plunger drawn from the generator are

a, b, and c. When the r e s i s t i ve termination is placed i n the same locations

on the plunger the measured input impedances are

Half the angle a t intersection is the

%is results from the approxi-

The measured

La I+, and Lc . I n Fig. 10 (a) the points a and c are coincident and the match is

perfect . Since cavi t ies a and c are not excited when the plunger is

located i n then;, the conclusion is tha t the cavity b

No conclusion about cavi t ies a or c can as yet be formed. The impedance

t ra jec tory as the plunger is moved incrementally is ambnc, but conclusions

about a, b and c are drawn as i f the t ra jectory were anbmc.

In Fig. 10 (b) a mismatch i s indicated ahead of cavity

phase-shift through b is 90'. The diameter of the impedance c i rc le is

the sement aMb (o r bMc).

has a 90' phase-shift.

b , since the

In Fig. 11 (a) is shown am example of a mismatch i n cavity b where

there is no previous mismatch.

matched cavity where a previous mismatch exis ts .

Figure 11 (b) shows an example of a m i s -

Figure 12 i l l u s t r a t e s a typ ica l pat tern obtained f o r the n/2mode by 1.

moving the plunger away from the generator.

move incrementally as if the plunger were being moved tcvard the generator,

only the 'dwell points ' are measured, and these are treated as if the i m -

pedance t ra jec tory was ac tua l ly traced clockw$se, t h a t is, the plunger was

drawn away from the generator. Thus, in the i l lus t ra t ion , every cavity

exhibi ts an insufficient phase-shift (less than lr/2 radians per cavity) a t

- 37.-

Although the impedance points

1 T '

Fig. lO(a)--Ekample of nodal-shift problem.

Fig. lO(b)--Example of nodal-shift problem.

- 38 -

1 T '

Fig. Il(a)--Example of nodal-shift problem.

Fig. ll(b)-'-Example of nodal-shift problem.

- 39 -

I I 9 .'7 2'

Fig. 12--Nodal s h i f t of 43-cavity section. Rotation shows frequency e r ro r (driving frequency 86 kcs too, low). I n

addition, the input coupler i s s l i g h t l y mismatched.

Numbers r e fe r t o discs .

1

- 40 -

T

apr

O f

In

is

the dr ive frequency.

running frequency.

from the dispersion diagram f o r the structure.

wave s t ruc tures increasing the frequency decreases the guide-wavelength,

o r increases the e l e c t r i c a l phase-shift per period. An es$imate of how much t o change the drive frequency can be got from the excess phase,

The conclusion ought then t o be t o increase the The requirement i n each par t icu lar case is e v i d p t

For example, f o r forward-

, obtained per cavity:

# vg f

course the phase excess must be computed from the match-point as center.

general, from inspection of the impedance t race it is evident tha t it

not necessary t o compute the phase-error of every cavity by finding the

impedance c i r c l e , but only those cavi t ies displaying excessively large

phase excursions. Note t h a t phase-error cancellations are not ref lect ion

cancellations. It is possible t o correct the tuning of such cavi t ies de- pending on the method of construction.

(dropping i n penny-size copper d iscs ) , slumping the cylinder w a l l in with

a punch, p la t ing or deplating, etching, e t c .

Such methods include slugging

An in te res t ing problem arises i n the case where the mode of operation

is not one of the two elementary cases, that is, the n/2 or

Two simple poss ib i l i t i e s exist: ac tua l ly test in the desired mode, involv-

ing the more complicated phase-shift t race on the Smith Chart; or t e s t i n the n/2 m o d e a t another frequency, ac tua l ly using the s t ructure in the way

planned.

t o examine f o r uniformity of construction.

J[ mode.

The l a t t e r i s reasonable since the primary object of tests is

After the nodal-shift da ta has been obtained it is possible t o obtain

an estimate of the Q of the s t ructure , or the voltage-attenuation con-

s t an t I , since the VSWR u of a shorted l ine depends only upon i t s

losses i n the manner

nepers . 1 I& = arctanh - U

From Eq. (6) the Q may be obtained. As a matter of practice it i s

somewhat more accurate t o perform t h i s measurement over a length t o t a l l y

within the periodic s t ructure , t h a t is , t o eliminate the coupler.

1 T

I V . BUNCHERMEASUREMENTS

The information which is required i n the case of bunching sections

isof'tkesame s o r t as required i n uniform sections, w i t h the additional re-

quirements t ha t these parameters be given as a function of location i n the

section. O f course, in addition it is necessary t o determine the phase

veloci ty as a function of location.

In the design of the buncher the pa r t i c l e t ra jec tor ies have been

computed on the basis of a specified schedule of phase velocity and f i e l d

strength. &om the longitudinal equations of motion, an electron gains a

d i f f e r e n t i a l energy

joules per meter ( 3 5 )

or, since rn = m0y , and expressing distance i n free-space wavelengths

5 = z/x >

d7 e l

m c 0 dE - - - - 7 E ( S ) s i n 6 = - a ( 5 ) s in 6 .

From Eqs. (36) and (7),

e l eh a(() = 7 E ( E ) = ~ 7 / - = - 2 m c 0 m c 0 m c 0

Thus the specif icat ioq of the accelerat ion parameter depends upon a com-

bination of the propagation propert ies of the structure.

Bunchers may be constructed on a number of different principles of

operation.

and phase velocity a re t o vary according t o a specified rule, and each

cavi ty is t o operate i n the n/2 mode.

presumed parameters w i l l have been previously obtained from individual

tes t -cavi t ies .

formed has been determined by the frequency demanded by the uniform sections.

(Care must be taken tha t these measurements are made a t suf f ic ien t ly close

temperatures.

t h i s amounts t o about + l0C. )

As an example, we consider the desigq where the f i e l d strength

Construction dimensions and the .

!Ihe frequency a t which the buncher tes t ing w i l l be pre-

For S-band, and measurement accuracies presently obtainable,

- Before matching the buncher coupler it is interesting t o examine the

d iscre te resonances of the buncher with shorting-plane terminations. There

is no resemblance t o the ordinary dispersion diagram of periodic structures.

- 42 - 1 T '

The observed resonances, of course, occur fo r the condition

Note t h a t the resonance spectrum of the buncher depends on which end-

p la te contains the exci ta t ion probes, as the width of the passband i s

d i f fe ren t f o r t he two end regions.

Eq. (37) are not zero and the length of the buncher,

quency at which the buncher i s designed t o be used the l a t t e r l imi t s apply, and i f the buncher i s ideal ly constructed, one of i t s discrete resonances

w i l l be the design frequency, providing the number of cavi t ies permits

t h i s resonance. Otherwise, the two adjacent resonances w i l l bracket the

n/2 mode.

This i s the reason tha t the l imi t s i n

L . Near the f r e -

That is, the resonances w i l l be (N/2 F l)n/N . I f it i s of in te res t , the transit-t ime of the rf energy through the

buncher may be estimated from the resonance spectrum i n the design region.

Although, precisely,

i f we consider the buncher as 2-port network the dispersiveness i n the

useful region may be taken as the group velocity through the network, so

t ha t

L d(l/A)/d(l/A ) '

T =

g

A more tedious but exact method of obtaining the transit-t ime of the rf

energy through the buncher i s based on the phase e r ror introduced by a

frequency error .

a t e i n the 2z/k mode, where k i s the number of disc per guide wavelength,

the excess e l e c t r i c a l phase A$/@ introduced by a frequency error Af/f i s

As noted e a r l i e r , when a structure i s designed t o oper-

Thus a nodal-shift performed o f f the correct running frequency can be re-

duced t o obtain the group veloci ty of each period of the structure, i f

the phase velocity i s assumed t o be known. Knowing the disc spacing of

- 43 -

-1

each period and the group veloci ty between them, the transit-t ime can be

determined. While the above is somewhat tiresome, i n the case of uniform

sections an accurate value of the group veloci ty i s obtainable, as a

ver i f ica t ion of the cavi ty- tes t method, w i t h l i t t l e trouble

It i s possible t o determine the shunt impedance of the e n t i r e buncher

by a slmple perturbation measurement, using an axial d i e l e c t r i c rod inserted

through the buncher as a resonator. With t h i s method the perturbation is

2 ( E - E o k 0

.e Af' (E - cO)A - - - f J z

C 0 4w 4w

or 240c "0 - = - =

Q UW (E' - l ) A

ignoring space-harmonics. I n a physical sense, i f the power i s converted

preponderantly by the fundamental space-harmonic, then since the d i e l e c t r i c

rod is performing i t s own Fourier ana,lysis, any correction w i l l be small.

The t o t a l R/Q can be cmpared t o t ha t computed by numerical integrat ion

from the specified values i n the design. The above does not check if the

specif ied f ie ld-s t rength parameter was ac tua l ly achieved but only if the

in tegra ls a r e the same.

calculated i n any case, since the correction fac tor i s always l e s s than

unity.

The measured value ought t o be greater than the

While the input coupler may be matched using e i the r of the methods

of Section 11, it is perhaps advisable t o perform a preliminary nodal-

s h i f t from the output end if the system permits.

be done through the following uniform sect ion if necessary.

cedure more o r l e s s guarantees t h a t the input coupler matching w i l l not

be affected by in t e rna l mismatches i n the buncher which were not known.

This can cwveniently

Such a pro-

Once the coupler matching has been completed a nodal-shifting of the

buncher, using a hinged bar, as shown i n Fig. 13, may be performed. The

bar is used, of course, since the customary detuning plunger cannot be

accommodated t o the varying conditions i n the section. This bar i s

hinged so t h a t it may be folded and drawn back through the d i sc holes.

I n use the bar i s positioned across the h c k of the disc. The n u l l

posi t ions of the s l o t t e d l i ne measurements do not agree with those which

- 44 -

1 T '

Lap t h i s

1/4 surf ace

. loo

Fig. 13-ainged bar su i tab le f o r nodal-shifting buncher. Dimensions a re f o r 3 kMcs.

- 45 -

A (inches)

1 318 2

2 1/4

1 118

1 T '

would have been obtained employing a conventional detuning plunger; how-

ever, the measurements may be used t o indicate the e l e c t r i c a l length of successive cavi t ies i n the same manner. As discussed i n Section 111-B,

it is also necessary, i n order t o analyze the construction errors , t o

determine the location of the match-point of each cavity on the Smith

Chart. This is accomplished using a termination of 377-ohm res i s t ive

' card, approximately a half-inch wide, placed across the d isc hole, or

fram modified versions of the paddle termination described i n Appendix C.

In general, the required accuracy is not great.

By means of the geometrical methods indicated i n Section 111-B the

phase-error of each incorrect cavity i s cmputed. "here are various

methods of correcting the excess phase, or phase-velocity e r ror , such as

deforming the cavity, slugging, e t c . Whichever method is applicable, the

work may be conveniently checked by t rying slugs (with an attached thread

f o r recovery) before actual ly making permanent corrections.

slugs f o r permanent correction is inadvisable a t high power levels and

obviously cannot be used i n moveable acceleratcrs.

The use of

The importance of correcting excess phase-errors is indicated by a

p l o t where the accumulated phase-error is recorded.

phase-velocity m a y b e

the accumulated phase-error may become important, depending on de ta i l s

of the t r ans i t i on in to the uniform section.

Although the actual

very l i t t l e different from the design value,

In addition t o the above examinations it is possible t o estimate the

f ie ld-s t rength parameter of the buncher section by means of a traveling-

wave perturbation. An arrangement of the apparatus is shown schematically

i n Fig. 14. magic-T. The calibrated phase-shifter is adjusted f o r a nu l l indication

on the detector when the system i s excited a t the desired frequency. A bead (preferably d ie lec t r ic ) i s then drawn through the section, and the

phase-shift introduced by its presence at each location i s measured by

rese t t ing the n u l l on the detector.

Alternately, a s lo t t ed l i ne could be used i n place of the

The introduction of the bead in to a cavity is equivalent t o a reac-

tance e r ro r i n the construction of the cavity, and causes a ref lect ion

and a phase-shift of the transmitted wave. The phase-shift due t o the

bead may be re la ted t o the detuning of the cavity, which is a frequency per-

turbation, and therefore t o the f i e l d strength of the wave a t the location

- 46 -

1 T '

of the bead.

therefore an analysis of the space-harmonic content ought t o be performed

t o obtain an accurate estimate of the acceleration f i e ld .

of sane d i f f i cu l ty , since now the space-harmonic coeff ic ients are a func- t i o n of location along the buncher.

The perturbation value is, of course, a time-average, and

This is a matter

An a l te rna te and somewhat simpler means of observing the traveling-

wave perturbation produced by a bead located i n a periodic structure is

shown i n Fig. 15. VSWR is noted f o r each posit ion, a repe t i t ive pat tern similar t o Fig. 16 or Fig. 17 is observed.

weak i n the d i sc holes.

consideration it is c lear t h a t the space-harmonics ex i s t physically fo r

the purpose of sat isfying the boundary conditions and t h i s necessity

arises at the discs.

If the bead is drawn along the axis, and the input

The f i e l d s appear strong i n the cavi t ies and

This is perhaps not in tu i t ive ly expected, but on

An e r r o r i n the construction of a cavity i n an otherwise perfect

l i n e may be interpreted as a shunt susceptance i n a uniform transmission system. The re f lec t ion coeff ic ient exhibited i n a lossless , i n f in i t e

l i ne by a normalized susceptance j B is

Yo - Y Yo - G - j B - j B

Y , + Y Y o + G + j B 2 + j B - - - p ’ -

The magnitude of the re f lec t ion coeff ic ient is approximately, when

small B i s

B B

To relate the susceptance of the bead t o the f i e l d strength, we

observe that the transmission phase-shift through such susceptance would

be given by the phase angJe of 1 - p :

4 - j2B

4 + B 2 1 - p =

or

. B N t m e = e = - 2 -

We may relate t h i s phase-shift t o the variation of the electrical . length

of the cavi ty due t o detuning,

1

- 47 -

T

Coupler Buncher Coupler

-.- Slot ted

-.- Line Matched 4 Brmination

Fig. lh--Apparatus arrangement f o r f ie ld-s t rength measurement i n buncher.

Detector

Fig. 15--Alternate apparatus arrangement fo r field-strength measurement

i n buncher.

- 4 8 -

T

Iength Along Waveguide

Fig. 16--Ty-pical pa t t e rn of f i e l d strength observed along

x/2 mode waveguide section by s e t up of Fig. 14.

- 49 -

1 T '

Fig. 17--Smith Chart p lo t of input impedance of typical pattern observed

along 2n/3 waveguide section by s e t up of Fig. 15.

1

r'.

4248

4244

4240

4236

Plunger Micrometer Reading (mm)

- 51 -

4214

4222

4230

4238

4246

4254

- - - - - - - _

I 1.125 2.125 . ~ 3.125 4.125 I

Posi t ion of bead, i n inches

Unperturbed

Fig. 19--Perturbation da ta produced by d i e l ec t r i c bead.

k v f g

c

.~

where k is the number of d i scs per guide wavelength. Since the frequency

perturbation may be taken as proportional t o the energy perturbation, 8 i s proportianal t o the square of the e l e c t r i c a l f ie ld-s t rength a t the

locat ion of the bead. A similar argument may be adduced f o r p . I n the uniform section it is similarly possible t o verif'y the r a t i o

of the f ie ld strength i n the cavi ty t o t h a t i n the d i sc hole.

is already known from the space-harmonic sum, as shown i n Section I-D. This r a t i o

The application t o the buncher t o determine the r e l a t ive f i e ld

s t rengths along the length of the s t ructure is obvious. It is only neces- sa ry t o es tab l i sh the scale t o obtain absolute values of the accelertition

parameter, since both at tenuat ion and f i e l d d is t r ibu t ion are taken in to account by the phase-shift method (Fig. 14).

The above techniques a l s o offer the poss ib i l i ty of examining the

transverse symmetry of the beam-interaction region; as f o r example i n the

coupling cavity.

The transmission-technique measurements do not depend upon the et ten-

uation since, t o f irst order, a l l the measurements are observed with the

same attenuation.

do depend upon the attenuation, and therefore the r a t i o of f ield-strength

measures a t the same location may be used t o estimate the attenuation of

the s t ructure .

On the other hand, the reflection-technique measurements

- 53 -

APPEM>uC A

ILLUSTRATION OF METALLIC PLUNGER METHOD FOR DETERMDIING r/Q .

Figure 18 is an i l l u s t r a t i o n of the resonant frequency of the cavi ty

i n a rb i t r a ry wavemeter divisions, p lo t t ed as a function of the plunger

inser t ion, i n millimeters. &ro in se r t ion is indicated. The slope at

t h i s point is 6.86 divis ions per mm. 0.081 Mcs/div.

From Eq. (21) the r/Q , ignoring the d i s t r ibu t ion of power in to the space-

harmonics, is 77.6 ohms/cm. '

From the d ie lec t r ic - rod method, the perturbation on the resonant

The wavemeter calibrat.ion is

Therefore the per turbat ion is 5.55 Mcs per cm insertion.

frequency of a length of the s t ruc ture resonating i n the

produced by a ' 25 mil rod (E = 9.56) is 9.784 Mcs.

r/Q , ignoring the space-harmonic d i s t r ibu t ion , is 48.9 ohms/cm.

for the fundamental space-harmonic by the method of the metal l ic plunger

is then

,21113 mode

From Eq. ( 2 3 ) the

From the Fourier analysis i n Appendix B the r/Q of the s t ructure

and by the method of the d i e l e c t r i c rod,

A s a camparison, from the Fourier analysis ,

and from the two r/Q meas&ements,

1

- 54 -

T

APPENDIX B ILLUSTRATION OF DISC-LOADED LINE FOURIER ANALYSIS

I n Fig. 19 the perturbation produced by a l/&-inch-long by l/&-inch-

diameter polystyrene bead drawn along the axis i s illustrated.

perturbation is given i n arbitrary wavemeter divisions, and the location

along the axis of the structure is i n inches. The s t ructure is operated i n the 2n/3 mode, so that the guide-wave-

The

lengths of space-harmonics which exis t i n structure are A = 3p, 3p/2, 3~34, 3p/5, 3p/7 ..., calculated from Eq. (12 ) f o r the propagation con-

s tants ; tha t is,

€5

Po = 2n/3p , 63, = - 2(277/3p) J B+l = 4(2n/3p) , . . . .

and noting the departures from the unperturbed wavemeter reading, we may

perform the Fourier analysis:

Dividing the int*erval of per iodici ty into, fo r example, e ight intervals,

e

11.25

33.75 56.25 78.75

101.25 123.75 168.75

+ 31 + 22

+ 3 - 0.5 - 8 - 12

- 9

Coefficients a CI

Therefore, 2

a n

@ COS e -

+ 5.35 + 3.89 + 0.96

- 0.14

-+ 0.55 + 2.07 + 2.94

+ 4.61

@?? COS 28

+ 4.54 + 1.79 - 0.66 + 0.66 + 2.61. + 1.32 - 2.78

+ 1.55

2 a, v

(Ca)2 = 0.57 - = " 0.90 C a2

@'cos 48

+ 3.86 - 3-30 - 1-22

+ 0.50 - 2.00

+ 2.44 - 2.12

- 0.11

@ cos 58

+ 3.03 - 4.57 + 0.33 - 0.60 + 2.32

- 3.34 + 2.12

= 0.63 ( CaI2

+ 0.05

- 55 -

1

APPENDIX c CONSTRUCTION OF A RESISTIVE LOAD

The specif icat ion of the requirements of a res i s t ive termination

fo r disc-loaded s t ructures is not d i f f i c u l t i n itseLf. What is d i f f i c u l t

is t o ac tua l ly construct a device with such properties.

more or less fabricated by a c u t - a d - t r y method.

The model must be

To simplify the in te r -

~ pretat ion of the measurements on the load a subsection with matched

coupler should be used.

A p rac t i ca l method of terminating the l i n e without ref lect ion would

be the inser t ion of a lossy material j u s t suf f ic ien t t o absorb a l l the

energy i n the axial e l e c t r i c f i e ld , f o r example a lossy s t r i p of some.

specified dc resistance per uni t length. However the lossy material m u s t

be deposited on some substratum t o support it i n the accelerator. This,

i n turn, is a d i e l e c t r i c and therefore detunes the cavity, causing a re-

f lect ion. Therefore i n the next previous cavity another ident ical mismatch

must be introduced t o cancel tha t of the load. The test of the success

of the cancellation is whether the d i e l e c t r i c s without coating of lossy

material mounted on a shorting (or detuning) plunger give the same nul l s

(on a s lo t t ed l i ne i n the feed l i n e ) as when they are removed.

i s so, the lossy material may be painted on t o match the l ine .

If this

The data may conveniently be taken by inserting the detuning, or

shorting, plunger in to two specified adjacent cavi t ies .

on the s lo t t ed l i n e is used as a reference plane.

with compensating paddle is then inser ted on the head of the plunger.

the nul l , when the assembly is placed i n the same cavity, is not identical ,

then the nature of the reactance e r r o r w i l l be made apparent, by p lo t t ing

on the Smith Chart.

When the n u l l has the same location whether or not the paddle and s t r i p

are mounted on the plunger, the canpensation is acceptable.

such as Aquadag may then be painted on the strip. If successive tr ials

are a l s o p lo t ted on the Smith Chart it i s obvious whether or not enough

lossy material is being applied. Note tha t more lossy material is l e s s

resistance. The indications should be taken from several successive

cavi t ies , since these are spaced e l e c t r i c a l l y and w i l l indicate if the

load is genuinely good.

The nu l l posit ion

The d ie lec t r ic s t r i p

If

Either the s t r i p o r the paddle may be corrected.

Lossy material

A suggested design f o r a load is given i n Fig. 20.

- 56 -

1

t

1 Scotch Tape

Resistance Material Eg. I R C card 300 ohms/sq Mi carta

.080

1.650

.030 f 1

Fig. 2O--Suggested match-termination for rr/2 mode

disc-loaded waveguide at 10.5 cm.

57 -

REFERENCES AND LITERATURE CITED

1.

2.

. 3. 4. 5. 6. 7.

0.

9.

A. W. Aikin, "Measurements i n Traveling-Wave Structures, '' Wireless

Engineer - 32, 230 (1B5). W. W. Hansen and R. F. Post, "On the Measurement of Cavity Impedance, It

J. Appl. Phys. - 19, 1059 (1948). W. R. Ayers e t al. , M. L. Report No. 403, Stanford University, (1957). J. Rybner, Journal IEE, PT. 111, 95, 243 (1948). G. A. Deschamps, J. Appl. Phys. 24, 1046 (1953). E. Feenberg, J. Appl. Phys. 17, 530 (1946). K. Mallory, "A Comparison of the Predicted and Observed Performance of

a Billion-Volt Electrpn Accelerator," HEPL Report No. 46, W. W. Haasen bbora to r i e s , Stanford University (1955), p. 68 e t seq.

E. Japes, "The concept and Measurement of Impedance in Periodically

Loaded Waveguides," J. Appl. Phys. 23, 1077 (1932). E. L. Ginzton, Microwave Measurements, McGraw-Hill Book Co., (1957) p. 281,

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