PROCEEDINGS OF THE I.R.E.

Closed- and Open-Ridge Waveguide*T. G. MIHRANt, STUDENT MEMBER, IRE

Summary-Expressions are developed for the voltage-currentand voltage-power impedance of closed-ridge and open-ridge wave-guide, with the discontinuity capacitance taken into account. Ap-proximate expressions for impedance are derived which are validunder given typical conditions.

I. CLOSED-RIDGE WAVEGUIDE

N A RECENT PAPER, Cohn presented much in-formation on the cutoff frequency and impedanceof closed-ridge waveguide.' He noted excellent ex-

perimental checks of theoretical calculations for cutofffrequency; also for impedance, providing the ratio ofguide height to width was small. However, when thisratio was 0.5, calculated impedances were approximately25 per cent too high. He stated, correctly, that the rea-son for this discrepancy was the partial neglect of thediscontinuity capacitance at the edges of the ridge in thederivation of the impedance equation. In the followingcalculation, the effect of this susceptance will be takeninto account, giving expressions which are only slightlymore complex than those derived by Cohn. In fact,Cohn's curves may be used to obtain a preliminary an-swer, which is then modified to include the total effectof the discontinuity.

Cohn's derivation of impedance was based upon thesimplifying assumption that E lines within the guiderun vertically from top to bottom. At the edges of theridge, voltage was made continuous by assuming a dis-continuity in E at that point. Impedance was defined asthe ratio of voltage across the center of the guide to to-tal longitudinal current on the top face. This currentwas calculated on the basis of the assumed distributionof E, thus neglecting the effect of the higher order modes

Fig. 1-E lines terminating on ridge side.

which are necessary to satisfy the boundary conditionsat the edges of the ridge. As a result, when the ridge step

* Decimal classification: Ri118.1. Original manuscript received

by the Institute, August 30, 1948; revised manuscript received, De-cember 14, 1948.

t Microwave Laboratory, Stanford University, Stanford, Calif.1 S. B. Cohn, "Properties of ridge waveguide," PROC. I.R.E., vol.

35, pp. 783-788; August, 1947.

was of appreciable size, the omission of the contributionof current due to the capacitance of the ridge edges re-sulted in a theoretical impedance which was too high.

This discontinuity current is readily evaluated. Con-sider two banks of E lines terminating on the side of theridge in a ridge waveguide, as in Fig. 1. If their spacingis dn, the capacitance per unit length of the shaded con-ductors contained between the terminations of the Elines may be written as

EOdQ eoEdndC = =r v

J Edp

The longitudinal current in either conductor is given bythe tangential component of transverse H, i.e.,

1dIz = Hndn= -- Edn

ZTE

where ZTE is the ratio of transverse E to transverse Hfor a TE mode,

Et- 120wr= ZTE = ( )

I/ 1-{

Hence,1

dlz = VdC.E0OZTE

(1)

Equation (1) states that current flow down the guidemay be obtained by considering the product of voltageand capacitance. It is necessary to apply this equationonly at the discontinuity, since the current in the restof the guide walls has been determined by Cohn. Inridge guide there are two discontinuity capacitances atthe edges of the ridge. Their presence, therefore, in-creases the total longitudinal current by the amount2 VCd/EoZTE, where Cd is the familiar Whinnery andJamieson discontinuity capacitance2 given in Fig. 2 ofthis paper. Strictly speaking, this value of Cd must bemodified if the ridge edges are not far enough from thesides of the guide that reflection of higher-order modes isappreciable. However, with Cohn, these proximity ef-fects are neglected in the determination of Cd.The derivation of an expression for impedance may

now be carried through in a manner similar to that ofCohn, except that his expression for total longitudinal

2 J. R. Whinnery and H. W. Jamieson, "Equivalent circuits fordiscontinuities in transmission lines," PROC. I.R.E., vol. 32, pp. 98-114; February, 1944.

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Mihran: Closed- and Open-Ridge Waveguide

current must be augmented by the discontinuity cur-rent, namely, 2CdEob2 cos 02/EOZTE. Since his imped-ance has been based on the ratio of voltage to current,

120r

2Cd a2 1 Xe, 01- +- +- -tan

CO b2 X bi 2

atuiI.-w2

CcwCL

CsU.2-2

z

I?

14 It I _

12 _ X _ _

0 0.2 0.4 0.6 0.8I.b2

b,.0

Fig. 2-Discontinuity capacitance as a function of ridge step.

it will be termed Z,i. Accordingly, his equation (3) be-comes

1207r~~~~~~~~~~~~~(2)

2Cd cos 02 Xc b2 0il-+ sin 02 +-COS 02 tan-

E0 b2 21jwhere

Zvi = Zvi{I (&)

a2\ Xc ir

01= 1- 2a1 Xc' 2

02 =a2 Xc X

a1Xi' 2

.i

bt

bi'(a)

(b)Fig. 3-Closed-ridge waveguide.

If 'XC'/X, >3, little error results in replacing tan 01/2 by01/2, giving a very useful approximation which is inde-pendent of cutoff frequency

120rzvioo =

2Cd a2 1 a, a2

so b2 2 b1 al/

(4)

While the voltage-current definition of impedance isuseful in problems dealing with the matching of guides,it should be remembered that the definition of imped-ance is not unique when circuit dimensions are an ap-

preciable fraction of a wavelength. If it is desired to findthe voltage developed across a guide for a given power

propagating down it, a power-voltage impedance mustbe defined and calculated. Let this impedance be termedZpv. Then

X, = cutoff wavelength without ridge = 2a,

X,' = cutoff wavelength with ridge.

Equation (2) gives the impedance of single-closed-ridgeguide as illustrated in Fig. 3(a). The impedance of thedouble-ridge structure is just twice that of the corre-

sponding single-ridge guide.If a narrow ridge is employed, 02<<1; COS 02 may be re-

placed by unity and sin 02 by 02, giving the approximateexpression

V02zpv =

10.8.

(5)E X Hda

The effect of the discontinuity capacitance may againbe taken into account, with the aid of Fig. 2, by notingthat

1 1d2P= E X Hda = E2dndp = (Edn)(Edp) -

ZTE ZTE

(3)

--

L i

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PROCEEDINGS OF THE I.R.E. Jun

Since E dn is constant between the two E lines, the in-tegration over p may be carried out, giving

From which

VdP =- Edn

ZTE

1dP = -- V2dC.

60ZTEHence, the additional power flow due to 2 Cd is2 V2Cd/I oZTE. Writing (5) as

V02zpv =

1

+-- J E2da6OZTE ZTE C.

Cohn's expression for E may be inserted and the inte-gration carried out, giving

of Fig. 4, an advantageous structure for certain applica-tions. In this guide, the discontinuity susceptance is amajor factor governing guide impedance, since the shunt-capacitance has been greatly reduced by removal of theridge top. So far, the equations have been corrected forthe effect of Cd. They must be modified further to in-clude the change of geometry and the resulting loss ofcapacitance. Henceforth, it will be assumed that theridge is narrow and represents a lumped capacitancesubject to a voltage Vo. The discontinutiy capacitance isalso lumped with the ridge-top capacitance. A majorproblem is to determine how much the ridge-top capaci-tance changes when the bridging conductor is elimi-nated. The simplest approach to this problem is the ex-perimental approach. Measurements have been made todetermine the difference in capacitance between a ridge

1207r2b2

Cd c 2 ±02 sin 202 b2 cos2 02 -o sin 201- '27rb2- COS2 02 + A 2 + 4 + b1 sin2 0i 2 4 f

This simplifies, if a narrow ridge is assumed, to

120r

2Cd a2 1 Xc' 1 r0o sin 20,-+--F-+ -- --

CO b2 7r b1i sin2 0i L 2 4

If 0, <300, an excellent approximation is

zpv001207r

2Cd a2 1 /al-a2\

6o b2 3 bi/II. OPEN-RIDGE WAVEGUIDE

The preceding expressions were developed primarilyfor determiningthe behavior of the open-ridge waveguide

b2

bt

(a)

2b2 2b,

(b)Fig. 4-Open-ridge waveguide.

with and without a top. It was found that this changeof capacitance AC depends chiefly upon the ratio of theinside spacing of the fins s to the ridge spacing b2, pro-

30

AC.Cp C,

25

la5_ * 32

o 4

0 1 2 3 4 5

b2

Fig. 5-Experimental determination of AC as a function of s/b2.

viding this ratio is greater than unity. Experimental re-sults for several fin thicknesses are presented in Fig. 5,from which it is evident that varying this parameter haslittle effect on AC. This handily eliminates an unwantedvariable.

Sufficient information is now available to calculatethe cutoff wavelength and impedance of open-ridgewaveguide. The expressions will again be developed forsingle-ridge guide, but apply equally well to double-ridge guide, providing the impedance is doubled, The

zpv -=

zpv00

11

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a 01..

i- -

I(It

1AliMran: Closed- and Open-Ridge Waveguide

shunt admittance of Cd+ 2 C, must equal the admittanceof a transmission line of length 01 at cutoff, hence

CO(Cd + QCr) = Y,l cot 01.

Noting that Yol = 1/120irbl, this may be written

bl(2Cd+ Cr)

a, fo cot 017

____________- -- (7)a2 61

1--al

Fig. 6-Graphical solution of equation (7).

The solution of this equation may be found from theplot of Fig. 6, and \,,'/X, obtained from the definition of01,

XcI / a2\ 7r 1-= 1 - - -

c < a1, 2 01(8)

Impedance may now be obtained from (3) or (6), exceptthat a2/b2 must be replaced by C7leo as determined by

Cr a2 AC

so b2 so(9)

Hence, for open-ridge guide,

Zvio =120r

2Cd + Cr 2 Xc' a, 6l- + ---tan-

to ir X, bi 2

and

120xrZpv~~~~ =2- (11)

2Cd+Cr 2 X.' a1 1 r01 sin 201-

EO -rX,1bsin2 0L 2 4 J

III. SAMPLE CALCULATIONSConsider a closed-ridge guide with b1/a, = 0.5, b2lb,

=0.133, and a2/al=0.352. Cohn's Fig. 3 gives V/Xe as2.57 and a Z,ti° of 65 ohms. From Fig. 2 of this paper, Cdis found to 9.5 ,u,uf/meter for a step of 0.133. Since eo is8.85 ,u,uf/meter in mks units, the correction term may beevaluated, 2Cd cos 02/Co-= 2.1. Cohn's denominator musthave been 377/65 =5.8, but this must be increased by2.1. Hence, the impedance is 377/(5.8+2.1) =47.5.Cohn reported experimental data on a similar guide,but with bl/a, scaled down by a factor 0.472/0.500; itsimpedance was measured as approximately 50 ohms.Scaling 47.5 down by this factor, the corrected theo-retical impedance is 45 ohms, a figure within 10 per centof the experimental value. The approximate formula(4) gives 47 ohms.Cohn gave a second example, a guide with b1/a, = 0.5,

a2/a =0.4, and b2/b= 0.1. At a frequency one and one-half times cutoff, the uncorrected calculation gives 60ohms, while including the discontinuity term reducesthis to 46 ohms. Cohn reported 35 to 40 ohms experi-mentally. The approximate formula gives 45 ohms. Un-fortunately, no further experimental data are available,but it is evident that the correction is in the right direc-tion and of the right order of magnitude.

I010

(a)

T

12

_5.55

* 7.5

25.5(10)

(b)Fig. 7-Open-ridge guide for sample calculation.

COT e,el w y W - ---

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L-I I I I

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PROCEEDINGS OF THE I.R.E.

As an example of open-ridge calculations, consider theguide of Fig. 7(a). Here b1/ai=0.392, b2/bi=0.30, anda2/a=0.294. The discontinuity capacity for a step of0.30 is given by Fig. 2 as 5.0 ,uf/meter. The AC for aridge with s/b2=5.5/3=1.83 is given by Fig. 5 as 11,u4f /meter. Thus, from (9), C, 11 ,u,uf/meter. Substitu-tion in (7) and use of Fig. 6 gives a 01 of 0.77. This allowsevaluation of X¢'/Xi from (8), giving the value 1.44. Im-pedance is calculated directly from (10) and (11), givingZ,j= 113 ohms, and ZpX=124 ohms.No direct experimental check of these figures has been

made; however, the above guide was set up in an equiv-alent form on a rectangular network analyzer designedand built at Stanford University under the direction of

Spangenberg.3 Measurement of X0'/Xc from the boardgave the value 1.65. This is within 15 per cent of thetheoretical value, the lack of agreement probably beingdue to the difficulty of representing thin reentrant finson the board, since it makes use of discrete lumped ele-ments.

ACKNOWLEDGMENTThe writer wishes to thank E. L. Ginzton and M.

Chodorow for their encouragement and guidance.I K. Spangenberg and G. Walters, "An electrical network for the

study of electromagnetic fields," Technical Report No. 1, ONR Con-tract N6-ori-106, Task III; May 15, 1947. This report describes aboard designed for use with fields having cylindrical symmetry. Re-cently, however, another board has been constructed, one which al-lows solution of the two-dimensional wave equation in rectangularco-ordinates.

Broad-Band Dissipative MAatchingStructures for Microwaves *

HERBERT J. CARLINt, MEMBER, IRE

Summary-Interpolation in the complex plane is employed tohandle microwave network functions. This yields an approximatingrational function over a specified bandwidth, and leads to a lumped-circuit approximation for the microwave structure, which is used asa basis for the synthesis of matching networks. In various problemsinvolving dissipative devices, the poles of the rational approximatingfunction may satisfy special conditions. In such cases the ideal lumpedmatching network has a simple realizable form, and may be trans-formed into a suitable microwave structure. Applications of thismethod and experimental results are given for the synthesis of a newtype broad-band coaxial "chimney" attenuator.

I. INTRODUCTIONI N ORDER TO insure uniformity of apparatus de-

sign in a composite transmission system, and, atthe same time, maintain conditions for maximum

power transfer, it has been the practice to match theterminal impedance of components to the character-istic impedance of the main transmission line. Thispaper considers methods of design of microwave struc-tures which match the input impedance of transmissioncomponents over very wide frequency bands. Thematching structures are analogous to the constant-resistance networks of low-frequency circuit theory,and hence have loss. Thus the methods are most di-rectly applicable to devices in which dissipation can betolerated, such as attenuators, terminating loads, cer-tain types of broad-band probes, etc. Later in the paper,

* Decimal classification: R117.121 XR310. Original manuscriptreceived by the Institute, July 8, 1948; revised manuscript received,October 5, 1948. Presented, 1948 IRE National Convention, NewYork, N. Y., March 25, 1948. This paper was prepared in connectionwith Navy Bureau of Ships Contract Nobs-28376, and is based onpart of a doctoral thesis submitted to the Polytechnic Institute ofBrooklyn.

t Microwave Research Institute, Polytechnic Institute of Brook-lyn, Brooklyn, N. Y.

applications of the method to the design of fixed padcoaxial "chimney" attenuators will be given.The design procedure consists of essentially three

steps:Step 1: Determine the input driving point imped-

ance function of the microwave device when properlyterminated, and approximate this over a specified fre-quency band by a suitable rational function and itsequivalent lumped-parameter circuit representation.

Step 2: Determine an ideal lumped-parameter match-ing network which, in combination with the lumped-circuit approximation of Step 1, produces a constant-resistance input impedance.

Step 3: Transform the ideal matching network of Step2 into the final microwave matching structure.

II. REPRESENTATION OF MICROWAVEIMPEDANCE FUNCTIONS

The method employed here for approximating by arational function the nonrational function which repre-sents the input impedance of a specified microwave de-vice, is that of interpolation in the complex plane. Alge-braic methods of interpolation for the approximation ofimpedance functions have been used heretofore byZobel,' but a more general method employing contourintegration, described by Walsh,2 appears to be littleknown to engineers. The basic theorem for interpola-tion in the complex plane may be stated as follows:

Let the function f(s) be analytic in a region of the

X Otto J. Zobel, U.S. Patent No. 1,720,777.2J. L. Walsh, "Interpolation and Approximation by Rational

Functions in the Complex Plane," Amer. Math. Soc. Colloq. PublicVol. XX, 1935, chap. VIII.

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