PROCEEDINGS OF THE I-R-E

The exponent of (18) can be put into the followingform:

b2 E ++b2±+s2d2] [c -+b21 1Fd2(+b +s1d2+

d2 [ +b 2 +4S2 b[+ 2 + s2d2] j

s2c2

4+ +s2d2J

(19)

Referring to the definitions of A, W, and B, and re-calling that in the Gaussian case the width of a curveis taken to the e-14 points, it is clear that

bA0 = , (20)

4 G 1/ i- +b2 + 4s2

1 /4B = b +b2+ s2d2,

and

sd d2sdd/(12+6b2)2+ 4s2 -W =-/=_b2 4- 2 + s2 2 b Ao2B

d2

(21)

(22)

The time of maximum response is given by

4

d2 1tm = C | -

L- + b2 + s2d2

Now g(t) can be written

g(t) I=AOexp { [ b[]B[E]}

(23)

(24)

For a cw input the signal is

exp j -+ at],

and the output can be obtained from (18) by taking thelimit as d approaches infinity.

6 b2s2Tlim Ig\,j = -exp 4s2fd- Ico ( ) I [b4 + 4S2]1[4 exp {-b-4 + 4] 2}

In the notation of (20) to (24),

b

[b4 + 4s2]1/41 1

W = b4+ 4S2 =--62 o

and

lim g(t) = Ao exp {- -2[]s}

(18a)

(20a)

(22a)

(24a)

Mutual Coupling Considerations in

Linear-Slot Array Design*M. J. EHRLICH,t SENIOR MEMBER, I.R.E., AND JOANN SHORTt

Summary--A study was made of the mutual coupling between tworesonant waveguide fed slots on a finite ground plane. The size of theground plane and the relative spacing of the slots were such that thegeometry corresponded to that of a pair of adjacent longitudinal shuntslots on the broad face of a rectangular waveguide. A null bridgemethod of measurement was used to determine accurately the rela-tive field strength excited in the second waveguide by mutualcoupling between the slots when the generator was applied to thefirst waveguide. The change in input slot admittance of the drivenslot arising from the presence of the parasitic slot was also measuredfor matched terminations of the ends of the parasitic slot waveguide.The changes in slot input admittance and excitation arising frommutual coupling were determined. It is shown that the changes maybe neglected in the design of the great majority of linear-slot arrays.

* Decimal classification: R1 18.l X>R142. Original maniuscript re-ceived by the IRE, May 18, 1953; revised manuscript received,October 15, 1953.

This technical memorandum is one of a series of reports which willprovide final information available on Project 532-A.

t Hughes Aircraft Co., Research & Development Labs., CulverCity, Calif.

INTRODUCTIONTVHE DESIGN of linear-slot arrays whose radiat-

ing elements are shunt slots cut on the broad faceof a rectangular waveguide is usually executed

without consideration of the effects of mutual couplingbetween the elements.' The mutual coupling is definedas that coupling which occurs in the free space exteriorto the waveguide. The magnitude of this coupling andsubsequently the evaluation of the validity of its neglectin design have been determined in this study.

NATURE OF THE EXPERIMENTThe particular external slot geometry that is con-

sidered in this experiment corresponds to two adjacentelements of a longitudinal shunt-slot array on the broad

I W. H. Watson, "Waveguide Transmission alid Antenna Sys-tems," Clarendon Press, Oxford, England, chap. 8; 1947.

956 June

Ehrlich and Short: Mutual Coupling Considerations in Linear-Slot Array Design

face of a rectangular waveguide. Each slot is centeredlongitudinally on the narrow face of a length of oneinch by one-half inch RG-52U waveguide as shown inFig. 1. One end of each of the guides is bent to provide

tance of the single slot in the finite ground plane.These data are then used to determine the changes in

slot excitation and admittance, normalized to the wave-guide, arising from the mutual coupling.

Fig. 3-Block diagram of experimental setup.

Fig. 1-Geometry of two slots in separate waveguides.

waveguide flange-input connections. The two guides are

then soldered with their broad faces together such as togive a spacing of X,/2 between the transverse center linesof the slots. Extension pieces one-half inch wide are

added on each curved end in order to locate the slotson the same one-inch wide ground plane as that of thebroad face of a single length of rectangular waveguide.The complete unit is shown in Fig. 2. The geometry ofthe experiment and the use of separate guides to feedeach slot was originally proposed by Dr. R. S. Wehner.

Fig. 2- Experimental two-slot array.

The experiment is simple in concept. A generator iscoupled to one waveguide and drives the slot cut in thenarrow wall of the guide. The slot conductance is suchthat the slot radiates approximately half of the energy

incident upon it, the remainder being absorbed in thematched termination. The field radiated by the drivenslot couples energy to the parasitic slot cut in the nar-

row face of the second guide. The field excited in thesecond guide by the parasitic slot is compared in phaseand magnitude by use of a directional coupler to thefield incident in the guide on the driven slot. Separatemeasurements are made of (1) the input admittance ofthe driven slot, normalized to the waveguide, when bothends of the waveguide excited by the parasitic slot are

terminated in matched loads; and (2) the self admit-

EXPERIMENTAL DETAILS

Fig. 3 is a block diagram of the bridge network usedfor the accurate measurement of the field excited by theparasitic slot in its feed guide.Each slot was cut to be resonant at 9,350 mcs in the

absence of the other slot and measurements were madeat this frequency.The generator was a stabilized X-band supply which

used a 2K39 klystron as the source of RF power. Theoutput of the supply was fed through an attenuator, adirectional coupler and then to the waveguide whichcontained the driven slot. A reference signal, which wasfurnished by the coupler and was -20 db with respectto the main signal, was fed through a calibrated preci-sion attenuator and phase-shifter to one of the arms ofthe magic-T bridge. The other input arm of the bridgewas connected to the waveguide fed by the parasiticslot. The signal from the difference arm of the bridgewas brought to a bolometer amplifier. The sum arm ofthe bridge was terminated in a matched load. A matchedload was seen looking into the symmetrical arms of theT. It is to be emphasized at this point that all measure-ments of the backward scattered waves in the wave-guide are measured relative to the field incident on thedriven slot and not to the total field.The self admittance of a single slot in the finite

ground plane, normalized to the waveguide, was meas-ured with the second slot filled with a metal plug sol-dered to the slot so that an unbroken metal surface waspresented on the exterior of the waveguide.The parasitic slot was then opened and the input ad-

mittance of the driven slot was measured as a functionof frequency when both ends of the parasitically excitedwaveguide were terminated in matched loads.

ANALYSIS OF EXPERIMENTAL RESULTS

The experiment is designed to measure the backwardscattered waves Aio and Ao1', (Fig. 4, p. 958), relative tothe unity amplitude TEjo wave incident in the wave-guide upon the driven slot.An analysis of the admittance and impedance proper-

ties of shunt and series slots cut in the broad face of arectangular waveguide has been formulated by Steven-

9571954

PROCEEDINGS OF TIE I-R-E

son.2 The backward scattered wave excited by a longi-tudinal shunt slot cut in the narrow face of a rectangularwaveguide for the TE1o dominant mode case is given as:A lo -jEo(2/irb) (k/l3io)2 cos (31oX/4) and for the shunt

slot= B1o the forward scattered wave as shown in Fig. 4.

I

~~~~~1 ---l-:.~~~~~~~~~~~- 0+ _lo ,

z0

(a)

z1

(b)

Fig. 4-(a) Waves scattered in waveguide by slot radiators.(b) Slot geometry.

where8=slot width

Eo = electric field across the center of the slota= width of broad face of the waveguideb =width of narrow face of the waveguidek = 27r/X =free space wavelength

i310= + [k2 (7r/a)2]1/2.The "voltage" across the slot is defined as the integralof the field across the slot at its center, i.e.,

Vo = 8Eo.

A1o is measured by determination of the reflectioncoefficient r in the guide containing the driven slotwhen the guide is terminated in its characteristicim-pedance, Zo. A lo' is measured in the parasitically excitedguide by the use of the null bridge method in which A lo'is compared directly to the unity amplitude incident,TE1o wave in the driven guide.The field excited in the parasitic slot Eo', relative to

the field excited in the driven slot Eo, is given by therelation

Eo' A 1o'

Eo A1oThis relation is approximate in that it neglects the

change in the admittance of the first slot arising from

2 S. Silver, "Microwave Antenna Theory and Design," M-97Radiation Lab. Series, McGraw-Hill Book Co., New York, N.Y.,vol. 12, pp. 286-300; 1949.

the presence of the parasitic slot. This change is, how-ever, very slight as it arises principally from the sec-ondary scattering of the parasitic slot.

Measurements were made of the driven slot admit-tance for two cases and the results are shown in Fig. 5.The first has the parasitic slot completely covered withmetal sheet. The second has the parasitic slot openedand with its waveguide terminated in matched loads.The ratio of the measured reflection coefficient for thetwo cases is IF'/F = 0.98. Thus the variations of thedriven slot admittance may be safely neglected.The measured values of A1o and A lo' give a value

Vol 8Eo' A iof-=-- = 0.087 exp-ji640

Vo 6Eo A10

for the ratio of the coupled field to the excited field.The total field at each slot when both are driven by

equal generators is

Vo,=Vo +±vi = vo[l + 0.087 exp-il64]

Vo"= 0.916 exp-ils'o

Vo

andIVovo

_0.916.

It is now desired to interpret the experimental resultsfor the two shunt slots on the narrow faces of the wave-guides so as to apply to two shunt slots cut in the broadface of the same rectangular waveguide.The shunt slot on the broad face of the waveguide is

excited by the transverse component of current, and isrepresented as a pure two-terminal element in shuntacross the equivalent two-wire line specified by thedominant TE1o mode. The shunt slot on the narrowface, when aligned parallel to the longitudinal axis ofthe waveguide, i.e., inclined at an angle of 90 degrees, isalso excited by the same current component. The con-ductance of the slot on the broad face is made equal tothat of the slot on the narrow face by increasing itstransverse displacement. When the conductances arethus made equal it is found that spacing between theslots on the broad face is very nearly equal that of thetwo slots on the narrow faces of 'the adjacent waveguideof the experiment for the specified dimensions and fre-quency. The external slot geometry and the groundplanes are then identical in the two cases. Both types ofslots are excited by the same transverse current com-ponent and are pure two-terminal shunt elements. Theinternal coupling arising from higher-order modes be-tween the two shunt slots cut on the broad face isnegligible. Therefore on a dom'inant mode representa-tion the two configurations are wholly equivalent as faras regards the admittance and excitation changes of theslots arising from external mutual coupling between theslots.

.I; i

Juene958

II

All o -..*- B/0

Ehrlich and Short: Mutual Coupling Considerations in Linear-Slot Array Design

(o Parasitic slot-completely coveredE1 Parasitic slot-open and waveguide terminated in matched loads

Fig. 5-Admittance of driven slot-normalized to waveguide.

VI

II Y2 I2

Yo I YI -Yi2 1YO IYo 2

0 --2o

Fig. 6-Equivalent circuit of two-shunt slots inseparate waveguides.

Consider equivalent circuit of configuration used in forexperiment, where the two slots are cut in the narrowfaces of separate waveguides, as shown in Fig. 6.

I1 = (Y11- Y12 + YO)Vl + Y12(Vl- V2).,. 11 = (Y11 + Yo)Vl -Y12V2

O = 12 = (V2 - Vl)Y12 + (Y22 - Y12 + 2Yo)V20 = - Y12V1 + (Y22 + 2Yo)V2

V2V, 12 =- (Y22+ 2Yo)V1

Y22= y11; Yo= 1

1954 9359

PROCEEDINGS OF THE I-R-E

V2Y12 =- (YV, + 2)

V1

and

Yi= V= (Yii + 1) )2(Y11 + 2)V1 V

and as the measured value of (V2/ VI)= 0.042 exp-i164Y,r~(Y,1+1) which confirms the results obtained bymeasurement as shown in Fig. 5 where Y11= 1.10and

Yii' = Y1,- (Yll + 2).

The case of interest is when both slots are cut in thebroad face of the same waveguide. The equivalent cir-cuit for this configuration is shown in Fig. 7.

Fig. 7-Equivalent circuit for two-shunt slots in same waveguide.

Vi' and V2' are the applied voltages and as Vl'= V2',due to the slots being spaced X,/2 apart along the line,the series arm of the II is removed. Thus,

Yin = (Y1l - Y12) + (Y22- Y12) + YO where, Yll = Y22Yin = 2(Y,1 - Y12) + 1.0 Y12 = Y21

Yo = 1.

Y1' is the admittance of one slot normalized to thewaveguide and is given by

Y1 = 2(Yin - 1.0) = (YV1 - Y12).Then Yi' for the measured case is given by

= 1.1 - 0.042(1.1 + 2) exp- j1640 1.22 expwl70

andYi'/Y,j = 1.11 a change of 11 per cent in the slot

admittance, as compared to the change when the slotsare in separate guides.

The 11 per cent represents an extreme change as thebulk of the slot arrays which have been designed havehad far smaller values of YV, for the most stronglycoupled slot.

EXTENSIONS OF THE RESULTS TO THE LIGHTLYCOUPLED CASE

A survey of the conductances of the slots in the greatmajority of the arrays previously built shows that the

maximum slot conductance is less than 0.1, normalizedto the waveguide. The experimental results for thestrongly coupled slot, i.e., g> 1.0 must now be extendedto the lightly coupled slot.The fraction of the power incident upon the slot in-

side the guide, that is radiated by the slot, is given bythe relation in Fig. 8, where ge = conductance of the restof the array on the load side of the slot position on theline.

Prad 9slot

Pinc - Prad g2e

I 4$+.$I

Fig. 8

The power that is radiated by the slot may be ex-pressed as:

Prad = 2VO'Yr

Vo=equivalent peak voltage of the slotYr=radiation admittance of the slot.As the driven slot conductance is reduced by the

factor A, the voltage Vo is reduced by the factor A"l2and the field excited by mutual coupling in the neighbor-ing slot is reduced by the factor A"l2. If, in addition,the neighboring slot's conductance is reduced by thefactor B, then by reciprocity the field excited in theneighboring slot will be reduced by the factor A"2XB 2.For the symmetrical case, i.e., A =B, the applied volt-ages are reduced by the factor All'2 while the coupledvoltages are reduced by the factor A. The slot voltagevariation due to nearest neighbor mutual coupling isreadily derived.

For the measured case of one adjacent slot

Vsot= [1.0 + 0.087 exp-i164].

The largest conductance likely to be encountered is 0.1

0.1A =--= 0.091;

1.1

accordingly

Vsl1t = [1.0(A)112 + (0.087 exp-i64±+)A]= 0.301 + 0.0079 exp-'(164+0°= 0.301 expil.'.

Vs lotThen | = 0.972,

V0

whereVo =voltage for zero mutual couplingO=change in the phase angle of the mutual admit-

tance due to the decreased path length betweenthe slots.

From the geometry of the array ck = 33 degrees at themeasurement frequency, and the change in Vo is neg-ligible.

960 Ju>ne

Mackay and Morris: Transient Response of Glow Discharges with Applications

When both nearest neighbors are considered

-lt 0.96.

voThis represents the greatest excitation change to be

expected and in itself may be safely neglected in design.It has been stated that as the conductances are reducedby the factor A, V2, the coupled field in the guide is re-duced by the same factor. In addition, the applied fieldat the driven slot V1= Vinc/(l +r). The new slot admit-tance Y2' for the lightly coupled case when both nearestneighbors are considered is readily derived from the re-lations

y2 = Y22- Y12- Y23,where

Y12 = Y23y2f = 22- 2Y12

Y12 = - (Y11 + 2)]

then

Y2'= Y2-2[-2 (1 + r)A] [Y1 + 2].

The numerical value, for g_0.1, is

Y2' = 0.1-2 [0.031(1.13) X 0.091] exp-il64 [2.1] expiOY2' = 0.108 exp150

y2,= 1.08.

Y2

The admittance change of 8 per cent is the largestto be expected and for the majority of the slots in thearray will be slightly less.

CONCLUSIONThe changes in the relative excitation and input ad-

mittance of the shunt slot radiator cut in the broad faceof a rectangular waveguide excited in the TEjo mode,arising from external mutual coupling between theslots, may be neglected in the design of most linear shuntslot arrays. Both the excitation and admittance changesare small and of the same order as those produced bydimensional variation within standard manufacturingtolerances.3

If the array is designed for extremely low side-lobelevels, say -30 or -40 db, and fabricated with greatcare such that dimensional variations are minute, thensome compensations for the effects of nearest neighbormutual coupling may be made in order to achieve mini-mal side lobes with the greatest aperture efficiency. Pre-cision fabrication is quite costly and it is far simpler toobtain low side-lobe levels by some over-design, whichwould introduce a slightly reduced aperture efficiency.One other case in which compensation may be re-

quired is for the short array of a few slots which arestrongly coupled to the feed and where some beam shap-ing is specified. In this instance, correction will haveto be made for the coupling arising from all the slotsrather than from the nearest neighbor only. Such cor-rection is feasible only in a relatively small array as itrequires a large amount of numerical computation.

ACKNOWLEDGMENTS

The authors wish to thank N. A. Begovich, S. Sen-siper, and R. S. Wehner, of Hughes Aircraft Company,for their helpful suggestions during the course of thestudy, and their critical reviews of this paper.

L. L. Bailin and M. J. Ehrlich, "Factors affecting performanceof linear arrays," PROC. I.R.E., vol. 41,Fpp.1235-241; February, 1953

Transient Response of Glow Dischargeswith Applications*

R. S. MACKAYt AND H. D. MORRISt

Summary-The transient responses of a number of glow dis-charges were observed. From the response time of several micro-seconds, and the characteristic overshoot, they can be described ashaving an inductive component approaching a henry. This is con-sistent with the observed frequency response.

It is shown how standard neon lamps can be used as fast andnoiseless coupling elements for voltage level changing in directcoupled circuits, in spite of the above.

* Decimal classification: 621.327. Original manuscript received bythe IRE, December 17, 1953; revised manuscript received, Jan-uary 13, 1954.

t Division of Electrical Enlginieering, University of California,Berkeley, Calif.

INTRODUCTIONT HERE FOLLOWS the description of some ex-

periments which measured the way in which aglow discharge responded to a sudden change in

conditions. This procedure yields a valuable techniquefor the study of fundamental discharge processes. Thepresent study was mainly concerned with rise time andbandwidth measurements. These measurements alsohave immediate application to practical electronic cir-cuits. Glow discharge tubes make useful voltage levelchanging devices (for taking in a signal at one level and

1954 961