24 IEEE TRANSACITONS ON MICROWAVE THEORY ANO TECHNIQUES, VOL. MTr-25, NO. 1, JANUARY 1977

instead of (3). Instead of being governed by the Mathieu

equation, the TM wave is governed by the general Hill’s

equation [10]. However, for the case of MA/d << 1, we can

neglect the third term because it varies as M/2d while the

first two terms vary as l/A2. We then have an equation

identical in form to (3) and the problem can be solved by a

parallel treatment.

It is worth pointing out that the modal theory provides

the most rigorous approach to the problem of spatially

periodic media. Retaining only two modes in the equations,

we have derived the popular coupled wave equations from

the postulate of small index modulation. The modal theory

is shown to reduce to various previously arrived theories

under simplified assumptions. Calculations are presented

for cases when other theories fail to apply. We stress that

the dispersion analysis techniques as discussed in this paper

are very useful tools in studying wave behavior pertaining

to periodic media. The paper is directed toward applications

to optical components in integrated optics systems. Applica-

tions of the theory to other fields are topics worthwhile

exploring.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

REFERENCES

R. S. Chu and T. Tamir, “Guided-wave theory of light diffractionby acoustic microwaves,” IEEE ,Trans. Microwave Theory Tech.,vol. MTT-18, no. 8, pp. 486-504; Aug. 1970.S. T. Peng, T. Tamir, and H. L. Bertoni, “Theory of periodicdielectric waveguides~’ IEEE Trans. Microwave Theory Tech.,vol. MTT-23, no. 1, pp. 123–133, Jan. 1975.L. Bernstein and D. Kermisch, “Image storage and reconstructionin volume holography,” Symposium on Modern Optics, Poly-technic Institute of Brooklyn, pp. 655-680P Mar. 22-24, 1967.H. Kogelnik, “Coupled wave theory for thick hologram gratingsj’Bell Sys. Tech. Journal, pp. 2909-2947, Nov. 1969.R. S. Chu and T. Tami~, “Wave propagation and dispersion inspace-time periodic media:’ Proc. ZEE, vol. 119, no. 7, pp. 797-806. .Iuh? 1972., .-.., -. .—.D. A. Pinnow, “Guided lines for the selection of acoustoopticmaterials:’ IEEE J. Quan. Elec., vol. QE-6, no. 4, pp. 223-238,Apr. 1970.W. R. Klein and B. D. Cook, “Unified approach to ultrasoniclight diffraction;’ IEEE Trans. Sonics and Supersonics, vol. SU-14,no. 3, pp. 123–134, July 1967.M. Born and E. Wolf, Principles of Optics, Pergamon Press,New York. 1972.H. Kogel~k and C. V. Shank, “Coupled-wave theory of dk-tributed feedback lasers,” J. Appl. Ph.vs., vol. 43, no. 5, pp. 2327-2335, May 1972.C. Yeh, K. F. Casey, and Z. A. Kaprelian, “Transverse magneticwave propagation m sinusoidally stratified dielectric media,”IEEE Trans. Microwave Theory Tech., vol. MTT-13, pp. 297-302,May 1965.

Generalized Analysis of Parallel Two-PostMounting Structures in Waveguide

OSMAN L. EL-SAYED

Abstract—An analytical expression is obtained for tbe reactance of

parallel two-post mounting structures having unsymmetrical strip and

gap positions and different strip and gap widths. An equivalent circuit isderived and analytical expressions for its components established giving aphysical insight into the problem of coupling between the two gaps. Theanalysis is based on deriving a variational expression for the structurereactance from the boundary conditions at the structure position. Thetheoretical results are experimentally verified for a wide range of cou-pling conditions.

Exploitation of this model for the design of wide-band varactor-tuned

negative resistance oscillators and multidiode parallel mounts in general is

discussed.

LIST OF SYMBOLS

J(r) Strip current density.

I(y) Strip current.

Zg Gap current.

V9 Voltage drop across gap.

Ein y-directed incident electric field intensity.

E,(r) y-directed scattered electric field intensity.

Manuscript received December 23, 1975; revised May 25,.1976.The author is with the Department of Electronics, Umversity of

Provence, Aix-Marseille, France, on leave from the Faculty of Engi-neering, Cairo University, Cairo, Egypt.

E,(r)

R

n

an

m,n

kx

k,

kPm

k e.

A

)

;

rky

Z.

u(x)

Gap field intensity.

Reflection coefficient at mount position.

Free-space wave impedance equals Juo/co.

Kronecker delta.

Mode number.

Equals mII/a.

Equals nII/b.

Post coupling coefficient equals sin kXS(sin (kxW/2)/

(kxW/2)).

Gap coupling coefficient equals cos kYh(sin (kYg/2)/(kYg/2)).

Free-space wavelength.

Guide wavelength (dominant mode).

Free-space wave number equals 211/L

Guide wave number equals JkX2 + kY2 – k2.

Dominant wave number equals (211/J.J = (rIO/j).

Guide characteristic impedance equals (2b/a)n(J.g/A).

Distribution function given by

(

1, for S–~<x<S+~u(x) =

[0, elsewhere.

EL-SAYED : TWO-POST MOUNTING STRUCTURES 25

/In general, mounting structures may be tackled either as

z

an antenna system radiating in the waveguide [1], [5] or

as an obstacle in the waveguide [4]. In th~e following

analysis the mount will be dealt with as an obstacle. A

b dominant mode (TEIO) wave is assumed to be incident

/on the mount, thus inducing, in general, two different

currents in the posts. These currents, which are assumed

to be uniformly distributed across the strips, will develop

—x a voltage drop across each gap. Resulting gap fields will bez assumed uniform all over the gaps.

1- SS JA. Determination of Gap and Scattered Fields in Terms

Fig. 1. Cross section of the rectangular guide showing the structureunder study and its parameters. of Strip Current$

Since the strips are not uniform along the y direction,

o(y) Distribution function given byboth strip currents will have a nonuniform distribution

along the y direction. These current distributions are,

(

forh–~<y<h+~a priori, unknown but will be expanded in a general orthog-

e(y) = 1’ onal set of harmonic functions of period 2b

(0, elsewhere.

I,II,N Superscripts indicating that the quantity is re-~ ‘(y) = ~ (2 – dw)~~NCOS kYy, N == 1,11. (1)

n=O

spectively relevant to strip I, strip 11[, or both

strips. The current densities are then given by

?“)~:COSkYy U1(X)~(z)I. INTRODUCTION ~l(r) = ~ (2 -

P

(2a)

ARALLEL two-post mounting structures are frequently“=0 WI

used to couple two active or passive devices to a ‘2–bnrectangular waveguide or cavity. In tlhis type of mount both J1l(r) = .~o ~ ~/* COS kyy U’*(X)C$(Z). (2b)

devices are mounted in separate gaps along two separate

posts positioned in the same cross section of the guide. Considering that the currents laI and 1,*1 flowing throughA typical circuit in which this mounting structure is used the elements (Z~l,Z~[[) present in the gaps are respectively

is the broad-band varactor-tuned Gunn oscillator. It maY equaltotheaveragecurrentineachgap, we can write

also be used for multiple Gunn diode oscillators, p-i-n

diode samplers, and commutators and varactor-tuned

filters.

For the investigation of the mount configuration yielding

the optimum coupling between the different elements, a

lumped equivalent circuit for the mount is needed. The

development of such a circuit will be our main concern

here.

In a previous publication [I] we have studied the special

case of the symmetrical mount where the posts are identical

and symmetrically disposed with respect to the center of the

guide, the gaps being at the same height. More recently,

Chang and Khan [2] analyzed the coupling between two

narrow transverse strips unsymmetrically located in the

same cross section of a rectangular waveguide by extending

to the two-post case the constant current density used by

Collin [3], In this paper, we treat the most general con-

figuration of parallel double-post mounting structures by

extending Lewin’s analysis of a single post [4] to the

double-post case.

II. THEORETICAL ANALYSIS

The structure under study is shown in Fig. 1. The strips

are assumed to be infinitesimally thin and perfectly con-

ducting. Two impedances Z~[ and Z~[! are placed in the

gaps of strips I and II, respectively.

(3)

The voltage drops and electric fields in the gaps are then

given byN=_

V9IgNZdN (4)

Egl(r) = 1= u1(X)O*(y)8(Z)91

~gllzdlI

Eg*l(r) = ~ Z41yx)eqy)a(z).

(5a)

(5b)

The determination of the scattered y-directed electrical field

E(r) follows by using the expression

E,(r) = –jcopoJ

G(r I r’)[J1(r’) + J*’(r’)] dV’v’

where G(r I r‘) is the $-j component of the dyadic Green’sfunction for a rectangular guide [1]:

. sin kXx sin kXx’ cos kYy cos kYy’’e-rmnlz-z’l.

26 IEEE TRANSACT1ONS ON M1cROwAw THEORY AND TscHNIQws, JANUARY 1977

This gives

o[l~k,~ + l~’kp~’] sin k.x cos k, ye-rmnlzl. (6)

Let the y-directed electrical field of the incident TEIO wave

(Ein) be given by

()~_sin ~X ~r,ozin —

a(7)

The reflection coefficient R is then obtained by taking the

ratio of the dominant mode back-scattered field to the

incident field at the mount position

~= E,(n=O, nz=l)

Ei~ Z=O

(8)

B, Boundary Conditions

In t~is section the boundary conditions are exploited,

together with the relations established in Section II-A to

derive a variational expression for the mount impedance

as viewed from the waveguide, in terms of the currents

spatial harmonics (which are still unknown).

The boundary conditions, at the mount, are such that the

sum of the y-directed incident and scattered electrical fields

vanishes all over the two strips except at the gaps where it is

respectively equal to the gap fields.

[.Ein + Er(~)]uN(x)8(z) = E,N(r), N = 1,11.

Integrating the preceding equality with respect to x, over

each post, we obtain, respectively,

kP,l _ j ~ ~ ~ (2 – ?.)(k2 – /’c,z)Ukn=Om=l r mn

11] COSky y = ; I;-Z;61(Y) (9)“ kPwl[InlkP~l + I.llkP~

1I I’zd*’eq y). (lo)“ kP~ll[l:kPml + I:lkP~ll] COS kYy = ; ~

It is to be noted that the integrations over x of the bound-

ary condition equations were carried in order to bring out

the dominant mode term in the form of the bracketed term

in the right-hand side of (8).

In order to obtain a variational expression for the mount

impedance, we multiply (9) and (10), respectively, by ZI(Y)

and 111(y), integrate the resulting equations with respect to

y from O to b, and add the two obtained equations. This

gives

b[Io*kpll + I:lkP:] - ; [1.’k,,’ + Io’’k,,”]’

_j~n~:~~1(2–6~2–kY2)

mn

. [~;kp; + ~:’kp:’]’ = I:’Z} + I:2Z;1 (11)

where the prime between the summation symbols indicates

that the term n = O, in = 1 is excluded from the summation.

Dividing (11) by Zo[loikPII + 10111CP,I[]2and substituting

from (8), we obtain

()

ll+R———2R

= j ~ ~$o ~~1 (2 _ 6J~ – kY2/k2)

mn

(. I;kpml + I;[k II 2pm_

Iolkp,l + lo[[kr,” )

I I’(z;/zo) + l:’(zdwo)< @)

+ g (I:k,, + I;’kP:)’

It can be readily shown that the left-hand side of (12) is

equal to the normalized obstacle impedance zO~= (ZJZO).

Moreover, if we refer all the quantities to the center of the

guide and normalize the impedances, (12) can be rewritten

as

where

InN’ = InNk Np, 9

n = 0,1,2,”” “,m, N = 1,11. (14)

I~N’ = IgNkplN (15]

N’ zdNz~ =

Zo(kP,N)2 “(16)

C. Evaluation of the Currents Spatial Harmonics and

Obstacle Impedance

In order to express the mount impedance only in terms of

frequency and the mount geometrical parameters, the dif-

ferent currents spatial harmonics should be evaluated and

substituted in (13). For this we proceed as follows. .

1) By exploiting the orthogonality of the different terms

of (9) and (10), we obtain expressions for the different

currents spatial harmonics (1#1) in terms of the voltage drops

across the gaps vgl and v~II.

2) Referring to (3) and (4), we can see that the gaps

voltage drops are themselves expressed in terms of the

currents spatial harmonics. Thus, substituting by the expres-

EL-SAYED : TWO-POST MOUNTING STRUCTURES 27

Fig. 2. Equivalent circuit for the structure (referred to the center ofthe guide) when the two gaps are short circuited.

sions obtained in step 1), in (4) we can express the gap

voltage drops in terms of the mount parameters.

3) Substituting again bytheexpressions obtained instep

2) for #1 into those found in 1), we obtain the required

expressions for the currents spatial harmonics and hence

the final expression for zO~.

The details of the aforementioned mathematical manip-

ulations can be found in the Appendix and car. be seen to

lead to the following expression:

(x: – Xm)(x:’ – x.)zo~ = Xm +

(x: – Xm) + (X:1 – Xm) + l/Y~

I 1

Fig. 3. Equivalent circuit for the structure under study (referred tothe center of the guide.)

coupling network corresponding to all n > 0 with the

coupling network corresponding to n = O (Fig. 3).

It can be readily seen that the obtained expression for zOb

[see (17)] fits the equivalent circuit just described. This

[~+—

(xeI - Xm) “ (1/Y~)

1[~+

(X:1 – ‘m) “ (l/yM)

+Y: (x: – q) + (xelI – Xm) + (l/YM) Y,II (xeI – Xm) + (Xell – Xm) + (l/Y,;) 1

[~+

(x,’ – Xm) “ (l/YM)

1[

1 (xeII

1

– ~rn) - (1/yJ.f) —

Y; (x=’ – Xm) + (x=” -- Xm) + (l/YM) + q+(x:– Xm) + (x.” – Xm) + (1/~)

D. Determination of Equivalent Circuit Configuration

In order to determine the configuration of the equivalent

circuit, we will first consider the special case where the two

gaps are short circuited (z~I’ = z~II’ = O). It can be readily

seen that in this case

lnl’ = ]“11’ = (), for all n >0

Ig’~ = 1.11’ = ; (x: – Xm)

Zob= Xm + (x: - %)(x:’ - %?!).

X=l + xe*I — 2xm

The preceding equations are seen to be satisfied by the

equivalent circuit of Fig. 2, which is the same as that found

by Chang and Khan [2]. It represents the coupling between

the posts due to all the modes n = O, m = 1,2!,. . . ,co, x~

being the coupling impedance and X,[,X,l[ the impedance

of each post, respectively, when the other is absent.

In the general case, beside the coupling due to the modesn = O, further coupling occurs due to all the modes n > 0.

This can be accounted for by an infinite number of II coup-

ling networks connected in parallel between the two gaps,

each one corresponding to a value of n [1]. The complete

equivalent circuit is obtained by combining the resultant H

(17)

enables us to give the analytical expressions for each of its

elements.

A further evidence of the validity of this equivalent cir-

cuit is brought by the fact that the same configuration was

found in [1] although the problem was dealt with as a

system of coupled antennas. We can also verify that the

expressions for the different components reduce to those

obtained in [I] for the special case of symmetrical mount.

III. EXPERIMENTAL VERIFICATION

Measurements of the mount impedance, as viewed from

the waveguide port, were carried out in conventional X-band

rectangular guide, over a wide frequency range for different

combinations of strip and gap positions. The results were

compared to theoretical values.

In Fig. 4(a)-(c) strip I was kept in the san.ie position

(S1/a = 0.5) while strips II were respectively positioned at

Sz/a = 0,65, 0.75, and 0.85. In the three cases the gaps

were positioned at the bottom of the gtiide and had the ,

same widths gl = gz = 0.8, 1.6, 2.4, and 3 mm. In Fig. 5

each of the two gaps was alternatively short circuited while

the other remained open (S1/a = 0.5, S2/a = 0.85). These

configurations correspond to the cases Z~* = 0, XdI* = coand vice versa. In Fig. 6, the gap of strip I was maintained

at the bottom of the guide while the height hz of the other

gap was varied between O.12b and 0,88b, the gap widths

being equal. On the whole, experimental results show a good

agreement with theoretical ones over a wide range of coup-

28

0.2

0.1

0

-0.1

-0.2

0.:

0.

-o

-o.

,b

S* .0.750

>b, p

/

(a)

0.2

0. f

o

-Of

/

A

&

-0.2

(b)

~ob

4.6

S~=0.e5aZ.@

A 3

f (Gtk).

II 12

A A

A

4

(c)

Fig. 4. Obstacle reactance as a function of frequency for different positions of strip II: SJa = 0.5; WI = Wz = 2 mm;91 = 92 = 9; hi = hz = 9/2.(a) &/a = 0.65. (b) &/a = 0.75.(c) SJa = 0.85.0 Experimental point. — Theoretical curve.

EL-SAYSD : TWO-POST MOUNTING STRUCTURES 29

0.4

O.a

0.2

0.1

0

-0.1

-0.2

<o~

$(GHz)

12

A

Fig. 5. Obstacle reactance as a function of frequency for differentvalues of gap impedances (Z~’I and Z#). O Experimental point.— Theoretical curve.

ling conditions as long as the gap widths (gl,gJ do not

exceed b/4.

It must be noted that the mount configurations tested

were chosen so that the high-order components of the

‘[ l/Y~) are not masked from theequivalent circuit (x~*,x~ ,

waveguide port by high impedance values for x,,I, X.*[, and

x~. This is specially the case in Fig. 6 where the anti-

resonance frequency is very sensitive to the values of the

high-order components.

IV. INVESTIGATION OF OPTIMUM MOUNT CONFI~URATIONS

Although the analysis has been conducted for two flat

strips, the results can be extended to the case of two

cylindrical posts through the introduction of an equivalence

relation between strip widths (W) and post diameters (d) of

the form W = cd where c is an experimentally determined

constant [1], [5]. In addition, two identical series reac-

tance should be introduced in series to account for the

phase difference between reflected and transmitted wavesdue to the finite dimensions of the posts in the z direction [I].

The large variety of possible configurations implies that

the experimental constants should be determined each time

for a definite configuration.

However, the value of the established equivalent circuit

lies in providing a good physical insight into tlhe problem

4.2

4

O.f

0.6

0. Y

0.2

%b

s, = o.5aS2 . 0.?S0W,.n=. 2h8mq,, 9*= 2.um” ,0.5

Ih,(b

Fig. 6. obstacle reactance as a function of a frequenqy for differentvalues of gap height. S1/a = 0.5; SZ/a = 0.75; WI = WZ = 2 mm;gl = az = 2.4 mm; Al/b = 0.12; h2/b = 0.25,0.33,0.5,0.88.

x; 11[ x:}, “1

Fig. 7. Simplified form of obstacle equivalent circuit.

of coupling two diodes in a parallel mount, thus permitting

the investigation of optimum mount configurations for

multidiode circuits.

Let us investigate, for instance, the optimum configura-

tion, for broad-band varactor-tuned Gunn oscillators. In

order to do so, the equivalent circuit of Fig. 3 is reduced tothat of Fig. 7 by a A/Y transformation, and tlhe values of

the components x~t, xsI*, and xs~ computed :for different

post and gap coupling conditions (Figs. 8-10).

In general, wide-band tuning is achieved when Xs[, xsll,

and xs~ are small and do not show large variations within

the frequency band of interest. Inspection of Figs. 8-10

30 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, JANUARY 1977

2.5

2.

4,5

4.

0.5

0

-0.5

-1.

-1,5

-2

-2.5

~

.2,. O.sa !k=o.lz 0.25

s?= 0.65a0.33

b

v#,=w, .2rn*

9,= 9,=2Qm”

__~:;

0.s8

2

1.

f(GHz)

9 10 n 12 0

h0.12

0.25

-1.

-2.

s,= 0.50S*. 0.850

01f (GHz)

8 Q *O T4 42

(b)

(c)

Fig. 8. Strip I series reactance (x,’) as a function of frequency for different values of gap II height (h,). S,/a = 0.5;WI = W2 = 2 mm; g, = g, = 2.4 mm; hi/b = 0.12; h,/b = 0.12 + 0.88. (a) S,/a = 0.65. (b) S,/a = 0.75.(C) S,/a = 0.85.

shows that these conditions are fulfilled over the whole X center. This confirms the experimental results obtained by

band when the posts are closely coupled around the center Joshi [6]. This same configuration can also be used forof the guide with one gap at the bottom of one post and coupling two identical Gunn diodes to the same cavity.the other gap at the top of second post. This holds whether The parallel double-post mount can also be exploited forthe posts are symmetrical or not with respect to the guide the design of mechanically and/or electronically tuned

EL-SAYSD : TWO-POST MOUNT2NG STRUCTURES 31

2.

<.

0

-1.

-2.

?!?2=0.12

#’

b 025 0.33

s,, o.5aS2z 0.6!kIw, ,w2s2mm

‘3 =%=19 mm

/0.5

~<=,,

——0.89

f(GH.)9 ?0

——n 42

(a)

2.

4,

0

fo

8

6

u

2

c

,&

~=o,f~ 0.25 0.33

~

,0.5

S,, 0.5aSt. O.wtl

/1:

f(GHz)

9 10 (4 i2 -

(b)

$(LHZ)-

8 9 ‘to 41 12

(c)

F]g. 9. Strip II series reactance (x.”) as a function of frequency for different values of’ gap II height (hz). ~1/a = 0.5;WI = WZ = 2 mm; gl == g~ = 2.4 mm; h,/b = 0.12; h./b = 0.12 + 0.88. (a) S,/a = 0.65. (b) Sz/a = lD.75.(c) S./a = 0.85.

bandpass filters. To obtain a bandpass, the two branches The preceding examples, among many others, demon-

(x.’ + X,l) and (x~’[ + x~II) of the equivalent circuit (Fig. strate the usefulness of the obtained model in the design

7) should resonate in parallel at the required frequency, the and the optimization of multidiode circuits in rectangular

impedance of the mount being small outside the passband. waveguide.

This performance can be obtained for any frequency in the

band with a configuration similar to that tested in Fig. 6 V. CONCLUSIONwhere we can see that by changing gap II height (or its

width) we can obtain coarse (or fine) mechanical tuning. A variational expression for the “obstacle” impedance

If in gap II a varactor is introduced, electronic tuning can of the most general configuration of parallel double-post

be achieved. mounting structures has been established. Starting from this

32 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, JANUARY 1977

0.4

0

-1

-2

t ‘JM

2.

f.

o

-4,

-2.

~.o,S1 = 0.5a

\

0.25

$1 = 0.65*

t

SMs, =o.5a52. o.75a

(a)

0.5

9 10 44088

~o, 0 89

10 11 1’2 ~cGHz)

~“? -o?<e

—05

hz~=ol 2 02 5 033

(c)(b)

Fig. 10. Guide coupling reactance (xs.J as a function of frequency for different values of gap II height (hZ). S Ja = 0.5;W1=W2=2mm; g1=g2 =2.4 mm; h, b = 0.12; h,/b = 0.12 + 0.88. (a) S,/a = 0.65. (b) Wa = 0.75.(c) Sz/a = 0.85.

Xs”s,. o.5as~. o.850

:’0”2’J,

EL-SAYSD : TWO-POST MOUNTING STRUCTURES 33

expression, a lumped equivalent circuit has been developed

and analytical expressions for its components derived.

Theoretical results were experinnefitally veriified over a

wide range of frequencies and coupling conditions., Optimum

mount configurations for wide-band varactor-tuned Gunn

oscillators, double-diode Gunn oscillators, and mechanic-

ally (or electronically) tuned bandpms filters were discussed,

demonstrating thereby the usefulness of the model as a tool

for the design and optimization of multidiode circuits in

rectangular waveguide.

APPENDIX

Let us define the following quantities:

k’ * (kPtiN/kP1N2~N_.

–.qmgz)

ermO

Xm = j ~ ~~z (k~mlkp~/kPjlkP1ll)

mO

(Al)

(A2)

(A3)‘l–

y’k2 (??)2XN=j~~~l r

9“

* 1 – kY2/k2 (kPml “ kPmll=j~m~l r — W,,’ ‘~) (A~)

x~n

mn k~nlk~nll

ZeN=&+xeN (A5)

Zm = $+xm (A6)

boo=—

Zo(A7)

N’V9 = [’ 1

-I,WZ,N’ = - $0(2 - d.)l.N’k,: Z,N’ (A8)

where N = 1,11.

1) Substituting by (14)-(16) and (A1)-(A8) into (9) and

(10), we obtain, respectively,

n=l

+ 2 ~ xMnI;’ k~nlkonll COS kYy~=1

= V. + ~ v; ’W(y). (A1O)92

Integrating (A9) and (A1O) over y from O to b and solving

the resultant equations for lor and 1011’,we obtain

Iol’ = ~ (Z=’I – Zm) + !“*’Z’ll–‘9W (All)Do Do

‘zeI — Vg”z 1’1.’1’ == 2!2(Z=I – ZJ + !91’ ~ (A12)

Do Do

where Do = z, Iz,II — z~2.

Similarly, if (A9) and (A1O) are multiplied by cos kYy,

integrated over y from O to b and the resultant equations

solved for 1.1’ and 1.11’, we obtain

(A14)

where D. = xgn*xgnll — xMn2.

2) Substituting by (Al1)-(A15) into (A8), we obtain two

simultaneous equations for ogl’ and 091[’ which when

together give

2)0 (Ze11 _1’ Zm) Y2 + (zel — -Q Y3

U9 = ‘-Do F

v 11’ 00 (zel — Zm)Y1 + (zel* – zm)Y3.— _

9 Do F

where

Y1 =4+$+52$ZJ “=1

z; x:yz=++~+njlz;

n

Y3=$+52~n=l ~

F = Y~Y~ – Y32.

solved

(A15)

(A16)

3) Substituting with the expressions obtained for v~I’ and

11’into (A8), (Al 1)–(A14), we obtain the rec~uired expres-

~ons for IJ’, Is’” , 10[’, ZoIi”, 1~1’, and InII’.Finally, ‘we substitute into (13) by the obtained expres-

sions for the gap and spatial harmonic currents, and (17)

is obtained in which

y;=l+2~xgn’1–~Mn (A17)Zdr’ *=1 D.

m X9.1 — x~ny:=A+2~ ~ (A18)Zjl’ ~=1 n

YM = 2*$13”.

n

(A19)

ACKNOWLEDGMENT

The author wishes to thank Prof. J. Herv6 fc)r his valuable

suggestions and for his continuous encouragement. He also

wishes to thank G. Mailhan for his technical assistance.

REFERENCES

[1] O. L. E1-Sayed, “Impedance characterization of a two postmounting structure for varactor-tuned Gunn oscillators,” IEEETrans. Microwave Theory Tech., vol. MTT-22, pp. 769-776, Aug.1974.

[2]

[31

[4]

[5]

[6]

K. Chang and P. J. Khan, “Coupling between narrow transverseinductive. strips in waveguide,” IEEE Trans. Microwave TheoryTedi., tcibe published.R. E. Collin, Field Theory of Guided Waves. New York, McGraw-Hill. 1960.L. ~ewin~””A contribution to the theory of probes in waveguides,”Proc. Inst. Elec. Eng.—Monogr: 259R, Oct. 1957.R. L. Eisenhart and P. J. Khan, “Theoretical and experimentalanalysis of a waveguide mounting structure,” IEEE Trans. Micro-waue Theory Tech., vol. MTT-19, pp. 706–719, Aug. 1971.J. S. Joshi, “Wide band varactor-tuned X-band Gunn oscillators infull-height waveguide cavity,” IEEE Trans. Microwave TheoryTech., vol. MTT-21, pp. 137–138.