31
RESONANT SLOTS* By W. H. WATSON, M.A., Ph.D.f {The paper was received 10th April, 1946.) SUMMARY This paper describes how the coupling of a resonant half-wave slot to a rectangular wave guide in the wall of which the slot is cut, came to be studied in order to solve the problem of linear microwave radiators fed from wave guides. The methods of experi- mental investigation* are described and the results are presented in terms of a method of representing the loading of the dominant wave in the guide. The important conception is the transformation of the circle-diagram variable (w) representing the dominant wave-system in the guide. It is shown that wave guides may be coupled by resonant slots. If such a slot is cut in the wall of a wave guide and lies opposite a register- ing slot in a second guide in contact with the first, the wave guides are coupled if the slot can be excited by the dominant wave in both guides. The type of coupling depends on the aspect of the slot in each guide. The laws of guide coupling are explained in terms of the manner in which impedance is transferred from the position of the slot centre in guide 2 into guide 1 at the same position. The coupling of variable reactances to the guide by resonant slots to pro- duce a T-section load is described, with experimental confirmation of the transformation of impedance and phase by the load. The method of radi- ation coefficients is applied to deduce the law of guide-coupling in the general case; it may be applied to treat loading and coupling of two waves in the same guide. Finally, directive aerial coupling by a pair of slots is discussed. Finally, the elements of the design problem for a linear microwave array and the theory of the wave-guide feed are discussed. Both transverse and longitudinal polarization are considered, together with the effects of mutual interaction between the inclined slots cut in the narrow face of the guide in the longitudinally-polarized array. The band- width of arrays is treated and a broad-band array of inclined-displaced slotsinthebroadfaceis described with measurements of its performance. The principle of the microwave Yagi aerial is briefly presented. Part I. THE COUPLING OF A RESONANT SLOT TO A WAVE GUIDE (1) INTRODUCTION In the early years of the recent war several attempts were made, for example by Southworth and Hansen to construct microwave antennas of large aperture by allowing energy to leak from the wave guide through a long slot parallel to the axis of the rectangular guide. It is easy to demonstrate that such a slot behaves as a parallel-wire transmission line which is continuously coupled to the H 10 -wave in the guide. Accordingly, standing waves are set up on the slot due to its finite length and radiation from the slot is end-fire instead of broadside to the guide in the way desired. Linear arrays of dipoles fed from the guide through probes connecting the coaxial-line feed-supports of the radiators to the wave in the guide, were then proposed by Alvarez. Since the probes were introduced through the centre of the broad face of the guide, they had to be quite short to allow the energy in the guide to be distributed properly over the array. It was found that in order to feed such an array the spacing of the elements had to be different from A /2 and the guide had to be terminated in a matched load to avoid intolerable reflections in the main guide. The dipoles must, of course, be coupled with alternately reversed phases in order to produce a beam nearly normal to the array. The work to be described in this paper was the natural outcome of experiments at McGill University which were intended to reveal the proper principles on which an array fed from a wave guide should be designed. First it was shown that the impedance presented as a shunt to the guide by a single probe-coupled dipole could be measured by standing-wave technique and that the standing-wave ratio at the input to an array of such dipoles could be calculated according to well-known electrical principles using either the circle diagram or matrix methods. It was further pointed out that only if the radiators were spaced half of the wavelength for propagation in the guide, is it possible, in general, to impose arbitrarily chosen distributions of phase and amplitude along the array by independent adjustment of the couplings. It was * Radio Section paper, based on reports to the National Research Council of Canada by W. H. WATSON in collaboration with J. W. DOUDS. E. W. GUPTILL, R. H. JOHNSTON and F. R. TERROUX. t University of Saskatchewan, Canada. readily seen that in order to produce a matched input to an array of many elements, it must be possible to couple the radiating loads so that the conductances presented to the H 10 -wave in the guide are many times smaller than the characteristic admittance of the transmission line in terms of which the propagation can be represented. Once it had become clear that the impedance of a load in a wave guide is a well-defined conception, it was obvious that to design a satisfactory array at resonant (guide) spacing, all that was required was to find suitably coupled radiators. To achieve loose electrical coupling, the probes should be introduced near the narrow face or edge of the guide where the electric force in the H 10 -wave is small. The first attempt to exploit this idea was to mount the dipoles on the narrow face and to couple them to the electric field by bending the central conductor of the coaxial-line feed to a dipole, into a plane parallel to the narrow face of the guide. The bent probe so formed could be turned round in this plane and the coupling weakened by increasing the angle between the bent probe and the electric field in the wave guide. As a practical device this was not very attractive. To remove the need for insulating sup- ports, slot radiators on the ends of small pieces of wave guide which were coupled by means of the bent probes were developed. The slots were suitably "choked" to prevent radiation over the surface of the guides (see Fig. 1). An array of 50 of these radiators was constructed and tested in April, 1943. Both the input impedance and radiation pattern were in accord with theoretical expectation. The spacing of the couplings was A^/2, and the feed-guide was terminated by a reflecting plunger placed X g /4 from the nearest coupling point. Fortunately the com- plexity and weight of this array prevented it from being adopted, leaving essentially the original problem but in quite different circumstances. The ideas concerning the loading of an H 10 -wave were clearly grasped, and secondly, knowing from the work of Booker that a resonant slot in a sheet of conductor is essentially a high-impedance device, it was decided to persevere with slot radiators. The problem then became the following. A resonant slot (i.e. about half a wavelength long) is cut in the wall of the [747]

Resonant Slots Watson

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Page 1: Resonant Slots Watson

RESONANT SLOTS*

By W. H. WATSON, M.A., Ph.D.f

{The paper was received 10th April, 1946.)

SUMMARYThis paper describes how the coupling of a resonant half-wave

slot to a rectangular wave guide in the wall of which the slot iscut, came to be studied in order to solve the problem of linearmicrowave radiators fed from wave guides. The methods of experi-mental investigation* are described and the results are presented interms of a method of representing the loading of the dominant wavein the guide. The important conception is the transformation of thecircle-diagram variable (w) representing the dominant wave-system inthe guide.

It is shown that wave guides may be coupled by resonant slots. Ifsuch a slot is cut in the wall of a wave guide and lies opposite a register-ing slot in a second guide in contact with the first, the wave guides arecoupled if the slot can be excited by the dominant wave in both guides.The type of coupling depends on the aspect of the slot in eachguide.

The laws of guide coupling are explained in terms of the manner in

which impedance is transferred from the position of the slot centre inguide 2 into guide 1 at the same position.

The coupling of variable reactances to the guide by resonant slots to pro-duce a T-section load is described, with experimental confirmation of thetransformation of impedance and phase by the load. The method of radi-ation coefficients is applied to deduce the law of guide-coupling in thegeneral case; it may be applied to treat loading and coupling of two wavesin the same guide. Finally, directive aerial coupling by a pair of slotsis discussed.

Finally, the elements of the design problem for a linear microwavearray and the theory of the wave-guide feed are discussed. Bothtransverse and longitudinal polarization are considered, together withthe effects of mutual interaction between the inclined slots cut in thenarrow face of the guide in the longitudinally-polarized array. The band-width of arrays is treated and a broad-band array of inclined-displacedslotsinthebroadfaceis described with measurements of its performance.The principle of the microwave Yagi aerial is briefly presented.

Part I. THE COUPLING OF A RESONANT SLOT TO A WAVE GUIDE(1) INTRODUCTION

In the early years of the recent war several attempts weremade, for example by Southworth and Hansen to constructmicrowave antennas of large aperture by allowing energy to leakfrom the wave guide through a long slot parallel to the axis ofthe rectangular guide. It is easy to demonstrate that such a slotbehaves as a parallel-wire transmission line which is continuouslycoupled to the H10-wave in the guide. Accordingly, standingwaves are set up on the slot due to its finite length and radiationfrom the slot is end-fire instead of broadside to the guide in theway desired. Linear arrays of dipoles fed from the guidethrough probes connecting the coaxial-line feed-supports of theradiators to the wave in the guide, were then proposed byAlvarez. Since the probes were introduced through the centreof the broad face of the guide, they had to be quite short toallow the energy in the guide to be distributed properly over thearray. It was found that in order to feed such an array thespacing of the elements had to be different from A /2 and theguide had to be terminated in a matched load to avoid intolerablereflections in the main guide. The dipoles must, of course, becoupled with alternately reversed phases in order to produce abeam nearly normal to the array. The work to be described inthis paper was the natural outcome of experiments at McGillUniversity which were intended to reveal the proper principleson which an array fed from a wave guide should be designed.

First it was shown that the impedance presented as a shunt tothe guide by a single probe-coupled dipole could be measuredby standing-wave technique and that the standing-wave ratioat the input to an array of such dipoles could be calculatedaccording to well-known electrical principles using either thecircle diagram or matrix methods. It was further pointedout that only if the radiators were spaced half of the wavelengthfor propagation in the guide, is it possible, in general, to imposearbitrarily chosen distributions of phase and amplitude alongthe array by independent adjustment of the couplings. It was

* Radio Section paper, based on reports to the National Research Council ofCanada by W. H. WATSON in collaboration with J. W. DOUDS. E. W. GUPTILL,R. H. JOHNSTON and F. R. TERROUX. t University of Saskatchewan, Canada.

readily seen that in order to produce a matched input to an arrayof many elements, it must be possible to couple the radiatingloads so that the conductances presented to the H10-wave in theguide are many times smaller than the characteristic admittanceof the transmission line in terms of which the propagation can berepresented. Once it had become clear that the impedance of aload in a wave guide is a well-defined conception, it was obviousthat to design a satisfactory array at resonant (guide) spacing, allthat was required was to find suitably coupled radiators.

To achieve loose electrical coupling, the probes should beintroduced near the narrow face or edge of the guide where theelectric force in the H10-wave is small. The first attempt toexploit this idea was to mount the dipoles on the narrow faceand to couple them to the electric field by bending the centralconductor of the coaxial-line feed to a dipole, into a planeparallel to the narrow face of the guide. The bent probe soformed could be turned round in this plane and the couplingweakened by increasing the angle between the bent probe andthe electric field in the wave guide. As a practical device thiswas not very attractive. To remove the need for insulating sup-ports, slot radiators on the ends of small pieces of wave guidewhich were coupled by means of the bent probes were developed.The slots were suitably "choked" to prevent radiation over thesurface of the guides (see Fig. 1). An array of 50 of theseradiators was constructed and tested in April, 1943. Both theinput impedance and radiation pattern were in accord withtheoretical expectation. The spacing of the couplings was A /2,and the feed-guide was terminated by a reflecting plunger placedXg/4 from the nearest coupling point. Fortunately the com-plexity and weight of this array prevented it from being adopted,leaving essentially the original problem but in quite differentcircumstances. The ideas concerning the loading of an H10-wavewere clearly grasped, and secondly, knowing from the work ofBooker that a resonant slot in a sheet of conductor is essentiallya high-impedance device, it was decided to persevere with slotradiators. The problem then became the following. A resonantslot (i.e. about half a wavelength long) is cut in the wall of the

[747]

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748 WATSON: RESONANT SLOTS

Field

Fig. 1.—Slot radiator (with chokes) terminating auxiliary guidecoupled by rotatable probe.

wave guide; how can it be coupled to the wave in the guide?Two answers were soon found for a longitudinal slot, that is,one whose length is parallel to the axis of propagation in theguide.

A slot is excited by current flowing through it, so a longi-tudinal slot cut in the centre of the broad face is not excited bythe H10-wave. If, however, the flow of current in the guide wallis distorted in such a way that the slot axis is no longer a nodalline of transverse current, the slot will be "excited" and radiationto space take place. A bent probe was therefore introducedthrough the narrow face of the guide in the transverse sectionthrough the centre of the slot. The probe was in conductingconnection with the guide wall. By proper choice of probelength and slot length, it was found possible to produce a non-reactive load which could be varied in magnitude merely byturning the probe. Further, the phase of the radiation from theslot could be reversed by rotating the probe through 180n. Thesecond method which immediately suggested itself was to dis-place the slot from the centre of the broad face parallel to itself.By properly proportioning the slot, a pure conductance wasobtained which increased in magnitude as the lateral displace-ment of the slot was increased. Phase reversal was achieved byreversing the sense of lateral displacement from the centre lineof the broad face of the guide. The same design data as beforewere applied to the construction of the array and on test it provedsuccessful.

It would be out of place to repeat in historical order thevarious steps by which the whole picture of the coupling of slotswas elucidated. In what follows the experimental method willbe explained and then the results will be presented as succinctlyas possible after a discussion of the method of representing theloading of the H10-wave in the guide.

(2) EXPERIMENTALOn account of the less stringent demands for precision of work-

manship, it is convenient in treating single slots to work withmicrowaves in the 10-cm band rather than with shorter waves.If arrays of slots are to be studied in the laboratory, however,it is advisable to choose waves in the 3-cm band. Standardguide sizes were used. The source of microwaves was a klystronor other suitable microwave oscillator fed from a power supplyof good regulation, and amplitude-modulated in a square waveso as to permit l.f. amplification of the detected signal and toeliminate frequency modulation.

Since dispersion is an important factor in wave-guide propa-gation it is necessary to arrange to monitor the frequency. Thisis done by sampling the output of the generator and applyingthe amplified output of the detector of a secondary-standardwavemeter to a cathode-ray oscillograph. Incipient frequencydrift is then made evident by distortion of the square-wavepattern.

The electric force of the H-wave in the guide was measured bymeans of a travelling probe moving in a central slot 0-06 in

wide in the broad face by means of a rack and pinion withVernier or micrometer attachment. Obvious precautions weretaken to eliminate errors due to transverse displacement of theprobe, which was kept as short as possible. The signal wasdetected by a calibrated crystal, amplified in a "noise-free"amplifier and indicated on a multi-range vacuum-tube voltmeter.In much of this work high s.w.r.'s were encountered; to avoiderror, great care was taken.

Auxiliary equipment consisted of reflecting plungers, guidecouplers, attenuators, double-stub tuners, matching loads, aphasemeter of simple design, and antennae coupled by coaxialline of specially small diameter for the exploration of rad iationfields near their source. High-power testing equipment was alsoavailable.

Because it was necessary to measure the impedance and admit-tance of essentially weak loads, it was obviously advantageous tomeasure the property of the slot by itself, free from thedominating effect of the guide beyond. Shunt loads were there-fore measured with the guide terminated in an open-circuitreckoned at the position of the load. This was achieved byplacing a reflecting plunger A /4 from the point of loading.Similarly series loads were measured with a short-circuit termina-tion secured by placing the plunger A /2 from the load. Theimpedance or admittance in the wave system between the loadand the generator was determined with the aid of the circlediagram from the s.w.r. (corrected for noise) and the distance ofthe minimum in the standing wave from the point of loading.

The travelling detector was also used in another type ofmeasurement designed to elucidate the type of loading by a newcoupling. At first, we measured the impedance in the standing-wave system on both sides of the slot load when the guide beyondthe slot was terminated in a matched load or variable reactance.Later it was found much more effective to regard the loadas transforming the complex plane of the circle-diagram variablew = (Z — 1)/(Z + 1), where Z is the impedance. This methodwill be explained in connection with the general slot.

When slot radiators which were very loosely coupled to theguide had to be measured, standing-wave measurements of singleslots became unreliable. The conductance of a non-reactiveshunt load or the resistance of a non-reactive series load wasdetermined from the square of the ratio of the field strengths inthe radiation from two similar slots, one of which could bemeasured by standing-wave technique. Another method was tomultiply the effect to be measured by placing a number of similarslots A /2 apart, when the mutual interaction between them wasknown to be negligible.

(3) IMPEDANCE AND OTHER REPRESENTATIONSOF LOADING

The. wave guide has been treated as a transmission line andreference has been made to the impedance of a load. It shouldbe understood that impedance as conventionally used is a com-posite notion.1 On a transmission line, treated by the laws ofelectric circuits, impedance represents a function of the relativeamplitudes of the principal waves travelling in opposite directionson the line at the place where it is measured. To this extent,impedance at any point of the line depends only on propagationand discontinuities therein. The scale factor or characteristicimpedance of the line, which determines the actual numericalvalue of the impedance, enters through the relation of the par-ticular vector representing the wave to the energy flux on theline. So long as this relation remains the same, we may ignorethe scale factor and define impedance in terms only of waveamplitudes. Thus for a transmission line parallel to z on whichare propagated waves proportional to ۥ>"' the voltage is given by

(1)

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WATSON: RESONANT SLOTS 749

where k 2TT/X and the impedance in our sense at the originz 0, is Z (A } B)/(A-B). The admittance y-(A-B)/(A-| B).A is the complex amplitude of the wave travelling in thedirection of ^-increasing (right) and B is the complex amplitudeof the wave travelling in the opposite direction. Tf the generatoris on the left, we denote B/A by w, the circle-diagram variable,

2 . ]and \w\ < 1. Evidently w - •

The equations which represent the field components in thedominant T.E.-wave in a rectangular guide are well known.They show explicitly the dependence of these components on apropagation factor common to all and a distribution factorspecifying how they vary over the guide cross-section. If thewaves are detected by a method which does not vary the aspectof the detecting antenna with respect to the guide cross-section,the only variations in the measured fields arise from propagation.Accordingly the dominant wave may be treated in this respectlike the principal wave on a conventional transmission line andmatrix methods may be applied.2

Localized loading of the wave by an antenna or obstruction inthe guide causes the radiation of waves which at a sufficient dis-tance to allow the disappearance of the evanescent waves ofhigher order in the vicinity of the land, consist of dominantwaves in a special phase and amplitude relation to the incidentdominant wave. If the secondary waves of electric force on thetwo sides of the load are of equal amplitude and in phase atequal distances from it, the load behaves as a shunt presentingan admittance which can be measured by standing-wave pro-cedure. If the secondary waves of electric force are of equalamplitude and in opposed phase at the same distance, the loadbehaves as a series one presenting to the wave an impedancewhich may also be measured. Shunt loading involves a discon-tinuity in the transverse component of magnetic force at thepoint of loading; series loading involves a discontinuity inelectric force.

For the convenience of the reader, the example of a shunt loadwill be considered. [See Reference (2) for further details]. Letthe load on which is incident from the left a wave of complexamplitude A' (reckoned at the position of the load) radiate wavesof amplitude —/A' and —/A' in each direction. All amplitudesare used with the same convention as to the positive direction ofthe vector representing the waves. Primed and unprimed lettersare used to denote waves on the left and right, respectively, ofthe point of loading. Further, A denotes a wave travelling to theright, and B a wave travelling to the left. From the waveprinciple of superposition, we have on equating outgoing waveamplitudes with the ingoing plus scattered waves,

A=(1-/)A'-/B 1

HenceJ . . (2)

A'1 - / ' I - /

B

The transformation from the complex number pair (A, B), ar-ranged as a column vector, to (A', B') is effected by the matrix

1 /1-/ 1-/

- / 1-2/1-/ 1-/

An cn\\C21 C2l)

M

and w =_ C21 -I- c22w

cl2w

(3)

(4)

If w represents an open-circuit, i.e. is -f 1, with our choice ofthe meaning of A and B,

C22 _ * - 3 /" C2W

f ^1 ! /

Hence2 /

This is the admittance presented by the load to the wave. Solong as we have to deal with simple series and shunt loads, con-ventional circuit methods can be applied with advantage, but inmore general loading of the guide it is much more advantageousto treat the loading of the wave directly by means of the trans-formation (4) of the circle-diagram variable w. Thus themeasurements of |»v| and arg w are applied directly.

It should be obvious that for a shunt load, if w = — 1 (short-circuit termination) w' = — 1; i.e. a short-circuit is invariantunder the transformation. Similarly for a series load, an open-circuit is unchanged by the loading, and, of course, in neithercase can the load draw energy from the guide. This affords thebest test of these types of loading the wave. For greater sensi-tivity when the loads are weak, a number of them spaced XJ2apart may be used in making the test.

Since a slot is excited, by current flowing through it, moststrongly at its centre, we can infer the simple types of loadingor coupling from the position and disposition of the slot on theguide wall. If the slot is excited only by the longitudinal com-ponent of the surface current, it must radiate equal waves inopposed phase in opposite directions and is therefore a seriesload. If it is excited only by the transverse current, it is a shuntload. If the slot is resonant on the guide wall, then for seriesexcitation at a certain place in the guide, the radiation from itwill be in phase quadrature with that from the resonant slotwhich is shunt excited with its centre in the same guide cross-section. The reason for this is that the transverse and longi-tudinal components of the surface current on the guide wall arein phase quadrature in a travelling wave.

(4) THE SIMPLE TYPES OF SLOT COUPLINGIn presenting the results, we take the simple rectangular shape

as fundamental, for only two parameters are required to specifyit; nevertheless, it must be pointed out that in actual practice itmay be desirable to depart from this shape in order to facilitateaccurate but inexpensive cutting.

(4.1) Displaced Longitudinal Shunt SlotsLet a slot be cut in the guide wall with its long axis parallel

to the axis of the guide: such a slot is found to present a shuntload to the dominant wave in the guide. In spite of the lengthof the slot antenna, this load can be treated as lumped at theposition of the centre of the slot.

It has been found appropriate in every case of loading thedominant wave in rectangular guide by means of a half-waveslot, to treat the load as applied in the section of the guide inwhich lies the centre of the slot.

When under the chosen conditions Outside the guide, the slothas been tuned by cutting to the proper length, which is closeto the half free-space wavelength, the slot presents a pure con-ductance G to the wave. Let xl denote the distance of the centreof the slot from the centre line of the broad face of the guide(Fig. 2). Expressed as a fraction of the characteristic admittancefor the dominant wave,

° = K™2^ «where a is the internal width of the guide, perpendicular to theelectric force in the wave, and K is a constant dependent on the

Page 4: Resonant Slots Watson

750 WATSON: RESONANT SLOTS

I .Longitudinal slot

Fig. 2.—Transverse section of rectangular guide through centre ofslot cut in broad face.

guide dimensions and on the wavelength, A. This result isreadily understood as it shows that G is proportional to thesquare of the transverse current in the guide wall.

The conductance may be measured by the standing-wavedetector provided that a reflecting plunger is placed A /4 fromthe centre of the slot, at which point the line is regarded as loaded.But this method is quite unsuited for the measurement of smallconductances OO^o) which result from placing the slot close tothe centre line. In that case, the difficulties arising from theneed to draw energy from the wave into the detector and thegreat disparity of the field strengths at the maxima and minimaof the standing-wave system can be avoided by comparing theradiation fields of two slots one of which can be measured bystanding-wave technique. On the assumption that the slotsradiate with similar directive patterns, the slot conductances arein the ratio of the squares of the field strengths measured inspace outside the guide at the same position with respect to theslots. Experimental results in Fig. 3 show the dependence ofG on xx for standard S- and X-band guides.

The length of slot required for resonance, that is, to present apure conductance, is dependent on the position of the slot onthe guide and on its width. Measurements of both conductanceG and susceptance B are shown as functions of slot length forseveral widths in Figs. 4 (a)-(/). The values of conductance andsusceptance in Figs. 4 (a)-(f), 11, 12, 13 are normalized. In

10080

60

40

20

1

0-8

0-6

0-4-

4—\\\ I

A—V

VN

^V|

0-E 1-2 1-40-4- 0-6 08 10Slot displacement from centre oF guide,in.

Fig. 3(o).—Longitudinal slot: width & in, resonant length 2 00 in fordisplacement x less than 0 • 4 in. A = 10 • 70 cm.

Zo/R = 1 • 73 sin* (nx(a). a = 2 • 80 in.Guide dimensions: 3 in X l i in .Wall thickness: 0-081 in.

1-0

0-8

0-6

0 4

0-2

0-1

0-08

0 06

004

002-

0-01

. /

/

/H-58cr

//

i

/

/

59 cm

^T62cm

0 0-300-05 0-10 0-15 0-20 0-25Slot displacement from centre of guidejin.

Fig. 3(6).—Longitudinal slot: width ^ in, resonant length indicatedopposite o on graph. A — 3-20 cm.

G/y0 =1 -40 sin* Inx/a). a = 0 • 90 in.Guide dimensions: 1 in X -Jin.Wall thickness: 0-050 in.

Fig. 5 it is seen that for a given slot position the resonant slotlength is linearly dependent on the wavelength of the radiationover a 10% band. At S-band the Q of a i in or wider slot ofthe type considered is about 9, which is much higher than for aslot of the same width cut in an infinite plane sheet. This isundoubtedly due to waves spreading round the back of the guideand points to the need for experimental precautions in determin-ing resonant slot lengths.

It follows from equation (5) and it has already been pointedout that when the longitudinal slot is aligned with the centre ofthe broad face, the slot is not coupled to the dominant wave.This is because the transverse current in the inside wall of theguide vanishes there. This current reverses phase across thecentre line, consequently phase reversal without change in loadis effected by transferring the slot to the image position in thebroad face with respect to the central line.

(4.2) Series-Coupled SlotsA transverse slot with its centre in the middle of the broad

face is excited by the longitudinal current and presents a seriesload to the guide. The magnitude of the resistance is decreasedif the centre of the slot is displaced laterally; the law is that the

Page 5: Resonant Slots Watson

WATSON: RESONANT SLOTS 751

050 8 5

2-75 3-75Frequency, c/s «10

0-4

o-i

f-0-1O

z- 0 2

-0-3

Fig. 4(a).—Frequency dependence of G/Yo for longitudinal slots ofdifferent widths.

O a in; x I in; El iin'< A ijn.Centre of slot 0- 78 in from centre of guide in each case.

- 0 -

3-75

\ \

•w11

—t

• 3-0 Frequency,c/s-109 3 ' 2 5

Fig. 4(b).—Frequency dependence of b/Y0 for the slots described inFig. 4(«).

03axnin

0-6

0-4

0-2

O2-75

1o2 -0 -2

-0-4

§0-6

0-5

Frequency,c/s*109 3 2 5 fc Q-4

0-3

11

IIIf

JJ16It

ItII

IIII

i i

/

/

f

U\ \N\

\l} Resonant, lengthY 4-78 cm\\

\\Resonant length-V

4-88cm \

\ .

\

2-75Frequency, c/s

Figs. 4 (c) and (d).—Longitudinal slots.O i in dumb-bell slot (holes j in diameter). H i in rectangular slot.

In each case the centre of the slot is 0-78 in from the centre of the guide.

3-25

Page 6: Resonant Slots Watson

752 WATSON: RESONANT SLOTS

3 Frequency, c/s« 10 s 3 ' 2 5

1 0

Fig. 4(cj.—Frequency dependence of arc tan b/G for the slots describedin Fig. 4(a).

014

0-12

010

£008

S•d 006.6

1004

002

\

>

\

\

<

\

\

0-28

0 Z 4

0-20

016

012

008

0 0 4

V50 1-52 1-54 1-56Slot length,cm.

1-58 1-60

Fig« 4(/).—Longitudinal slot.G/y0 and 6/yb plotted against slot length. Width ^ in; centre of slot 0 • 133 in from

centre of standard X-band guide (1 in X } in, o.d. wall 0-050 in). X = 3 -20 cm.

resistance is proportional to the square of the longitudinal surfacecurrent at the centre of the slot. Likewise if the symmetricaltransverse slot is rotated about its centre, the resistance is de-creased, and when close to the central line it is approximatelyproportional to the square of the small angle from the unexcited

0-2

\

0t>

0 +

0-3 t.

0 2 .21z

01

1-50 1-52 V54 156 156 1M) \£,2Slot, length) cm

Fig« 4(g).—Longitudinal slot.G/y0 and 6/K0 plotted against slot length. Width ,',, in; centre of slot 0*268 in

from centre of standard X-band guide. ?. -•- 3-20 cm.

5-2

5-1

/

/

a//

/

7

4-9

4-8

4-7

4-6

4-5

3-50 975 1000 10-23 1050 r 10-75Wavelength, Xa|cm.

Fig. 5.—Longitudinal slot.Variation of resonant length with wavelength; width i in; centre of slot 0-648 in

from centre of standard S-band guide.

position (see Fig. 6). For these inclined series-coupled slots,phase-reversal is achieved by changing the sense of rotation.This type of slot is suitable for arrays but is not so convenientto cut as the longitudinal shunt slots. However, it affords theelectrical designer the chance to introduce series as well as shuntloading of the guide.

(4.3) Inclined Transverse-Shunt SlotsSince the longitudinal component of surface current vanishes

on the narrow face of the guide, a transverse slot perpendicularto the axis of the guide and cut in the narrow face will be parallelto the lines of current flow and hence unexcited by the dominant

Page 7: Resonant Slots Watson

WATSON: RESONANT SLOTS 753

Zo

1 •

0-8

0-6

0-4

02

01-0-08

006

0O4

00?.

I//

//

/

/

/

//

.—

V V

0° 15" 75* 90*

6.

30° 45° 60°Inclination to guide axis > 8

Inclined series slot: centre of slot on centre of broad face(S-band).

wave in the guide. Since the depth of the guide is less than A/2,in order to achieve resonant length it will be necessary to cut theslot partly in the two broad faces, but even so, the slot if cutsymmetrically, its centre on the centre of the narrow face, willstill be unexcited by the dominant T.E.-wave in the guide. Itmay, however, be excited by other types of waves inside andoutside the guide.

Suppose now that the slot is turned about its centre as shownin Fig. 7(a). Such a slot could be cut with an end-milling cutter,

Fig. 7(o).—Inclined edge slot (X-band).

Fig. l{b).—Edge slots with alternate phase reversal.

its axis parallel to the broad face and perpendicular to the axisof the guide, and the cut made in the plane making <f> with thetransverse section of the guide through the centre of the slot.The latter is again excited by the transverse current and loadsthe dominant H10-wave as a shunt.

In order to establish the pure shunt nature of the coupling ofthese slots, and to obtain precise values of the conductance pre-sented by one of them as a function of <f>, when <f> is less than

303, it is necessary to work with a number of slots cut XJ2 apart,alternately reversed in phase by alternate reversal of the senseof <f> [see Fig. 7(6)]. It was found that a terminating reactancewas transformed unchanged by their presence, hence there is noeffective series element in the coupling of the slots. It was alsofound that with 0 = 0, the conductance/transverse slot is lessthan 0-0004.

It was soon established that mutual interaction occurs betweenmembers of the array of these slots, i.e. the excitation of theradiators fed from the guide is in part determined by wavestravelling outside the guide with approximately free-spacevelocity. The presence of this interaction is shown by measure-ments of the input admittance of an array of similar slots. Ifthere were no interaction the graph shown in Fig. 8 would be a

0-3

0-2

0-1

A3/4

t^rlo"—0——0

u \z

0-2 0-4 0-6 08 10

Fig. 8.—Admittance of 1, 2, 3, . . . edge slots spaced \\g, inclination15°, alternate phase reversal. A = 3-20cm, standard X-bandguide.

straight line through the origin. In practice a somewhat irregularcurve is obtained for the first six or seven slots; thereafter itsettles down and by proper choice of the common length of theslots, the later slots are made to contribute pure conductance tothe input admittance.

Experiments were made which established the presence ofwaves (especially the principal T.M.-mode) on the outside of along guide when an inclined transverse slot, shunt-coupled tothe inside of the guide, was cut near its centre and reflectionfrom the ends of the outside of the guide was minimized. Figs.9(a) and 9(b) respectively show what may be called the principalpattern of the slot and the way in which the electric force per-pendicular to the guide wall was distributed round the guide upto a distance of 30 wavelengths from the slot. In Fig. 10 theprincipal pattern for an array of two slots at A /2 spacing andcoupled in reversed phase to the wave in the guide is shown forcomparison. This indicates the beginning of the process bywhich an array of slots "lifts" the energy off the guide andthrows it normal to the array.

Because slots are intended to be used in arrays, the fact ofmutual interaction renders of little use the methods adopted forthe study of other types without significant mutual effects.Nevertheless it is instructive to note the reduction of the singleslot Q because the radiation from the slot can escape along theguide. Measurements of conductance and susceptance for a15° slot at S-band are shown in Fig. 11.

As a step towards the elucidation of the mutual effects betweenthe inclined shunt slots (sometimes called edge slots) experimentson transverse slots on the broad face were undertaken becausethe standing-wave measurements are much easier. Seven series-coupled transverse slots were cut on the broad face of the guideXg/2 apart. The mutual impedance between the first of theseand each of the other six was measured in the following way.The series impedance of each slot was measured singly, then theimpedances of the pairs. Twice the mutual impedance of any

Page 8: Resonant Slots Watson

754 WATSON: RESONANT SLOTS

Normal to arrayFig. 9(«).—Radiation from single edge slot.

Amplitude distribution at about 1 metre in the plane half-way between the parallelto the broad faces of the guide (S-band).

10

•075

050

0-2!

\

0 50 300 350100 150 200 250Distance From slot, cm

Fig. 9(6).—Amplitude decay of waves launched from a single edgeslot. (A =10-7 cm.)

O wave on narrow face containing the slot.[3 wave on a broad face.A wave on edge opposite slot.

Normal to array

Fig. 10.—Amplitude distribution at about 1 metre from pair of edgeslots, iA^ apart and radiating in phase (slots antiparallel).

pair is the impedance of the two slots acting conjointly minusthe sum of their separate impedances. Fig. 12 shows the valuesof the mutual impedances between 1, 2; 1, 3; . . . and 1, 7;the points lie on a spiral. The angle between successive pointsis 75° which is the difference between 255J and 180°, the respectiveelectrical spacings of the slots at free-space and guide phase-velocities. The limiting point of the spiral is not exactly at theorigin. This displacement is completely explicable by a con-sistent error in the setting of the plunger terminating the guide.The observations quoted indicate that the mutual impedance

005

.8

004

003

002

0010 3 0-9 10 11 \-Z 13

Depth of cut,cm

Fig. ll(o).—Variation of GfYo with depth of cut.15° edge slot d i n wide). S-band, >. - 10-7 cm.

0-8 09 1-0 H W 2 1.3

--0002

Fig. ll(b).—Variation of b/Yo with depth of cut.15° edge slot ($ in wide) S-band, X = 10-7 cm.

0 2

Fig. 12.—Mutual impedances in array of series slots spaced $\g(2 • 27 cm) in standard X-band guide. A = 3 • 20 cm.

falls off approximately as the reciprocal of the distance betweenthe slots. Thus the main factor in determining mutual impedancecorresponds to the propagation of a nearly spherical wave fromone slot to the other, and on this basis it is not difficult to estab-lish the result exhibited in Fig. 12. It is of some practical im-portance to note that the waves on the outside of the guide maybe reflected by obstacles mounted on the guide such as thestanding-wave detector: care should be taken to minimize suchreflections.

It is fairly evident that when an array of slots is excited fromthe guide, the mutual effect between any two members of thearray will be reduced by the operation of the intervening slotswhich will radiate into space, and into the guide, part of thewaves by which the mutual interaction is caused.

In a long array of shunt-coupled slot radiators at A /2 spacing,it would be anticipated, on the basis of the foregoing paragraphs,that the input admittance to r slots of the array, terminated by areflecting plunger at the centre of the space before the (r + l)thslot from the input end, will vary with r according to quite adifferent law when r is small, < 6 say, compared with the lawwhen r is considerably greater.

Page 9: Resonant Slots Watson

WATSON: RESONANT SLOTS 755

Admittance measurements were made on arrays of inclinedslots cut in the narrow face of the guide (1 in x | in outsidediameter) at A = 3 • 2 cm. The slots were coupled alternatelyin reversed phase. In any one array all the slots were cut withthe same or an equal reversed inclination and the followingvalues of cf> were used: 30°, 20D, 15°, 10°. Provided that thelength of the slots is properly chosen, it is seen that the inputadmittance shows increasing conductance and susceptance as r isincreased from 1 to 6 or 7; thereafter, for larger values of r, theinput susceptance remains constant and can be tuned out by re-setting the terminating plunger. This behaviour is shown inFig. 13, where the results are plotted on a G-b diagram. The

0-8 10 1-2Conductance

-0-4Fig. 13.—Admittance of 1, 2, 3 , . . . edge slots, for three lengths of slot.

graph tends to a straight line parallel to the G-axis. If the slotsare not of the proper length, the straight line will no longer beparallel to the G-axis but will slope up or down according asthe slots are too short or too long.

If the conductance (see Fig. 14) per slot is plotted as a function

0-t2

o-io

5 009

sO-08

0071—1

)

y

//

y

t _

0 1 2 3 4 - 5 6 7Number oF slots

8 9 10 11

Fig. 14.—Edge slots {\\g spacing): Conductance per slot plotted againstnumber of slots. (X-band.)

of the number of slots in the array when the input admittance ismeasured, we find it tends to a limiting value which will becalled the incremental conductance of the slot in a long arrayof similar slots. In Fig. 15 the incremental conductance isshown as a function of <f>. To a first approximation for anglesless than 15°, it is proportional to sin2^. The measurementsare sufficiently precise, however, to show that this law is notexact for angles greater than 15°.

In order to extend the graph of conductance against slot in-clination to smaller angles, the intensities of radiation from singleslots at 3,6, 10 and 15° were compared. On the assumption thatthe values for the larger angles were approximately proportionalto the incremental conductances in arrays of slots, these valueswere used to interpolate in the graph of Fig. 15 between 0° and10°. As a final check, the equiphase surface in the radiationfrom the array of 10° slots was plotted in order to see to what

02

<u 0-1g 008| 0-06

| 004

1I 002£3

* 0-010008

0-006

0-004

-—1— / •

1

//

//

10 15 IK)Inclination oF slot 0,degreec

Fig. 15.—Edge slots: Variation of incremental conductance withinclination.

[X = 3 -20 cm, standard X-band guide, see Fig. 7(a).]

extent the waves outside the guide causing mutual interaction inthe excitation of the slots affect the phase distribution along thearray. The equiphase plot is shown in Fig. 16. It is flat towell within 1 mm over the whole length of the array. This factseems of outstanding significance in understanding the operationof an array of elements with mutual interaction at a resonantspacing in the feeder guide.

il: > o w ( o o o c O O o O o o 3 O O > O O Q

8 1 2 3 . 4 5 G 7| Array

Fig. 16.—Equiphase plot opposite first seven slots of 15-elementedge-slot array (inclination 10°).

The physical facts which must be taken into account in thinkingof an array of edge slots may be summarized thus:

(a) the slots are shunt-coupled to the wave in the guide insidewhich they may be treated like any other shunt load;

(b) outside the guide the slots may be regarded, very approxi-mately, as series-coupled to the principal wave on the outside ofthe long guide in which they are cut. Slots which are not coupledto the waves in the guide (<f> — 0) can be excited by the waveson the outside of the guide. Of course, the conception of theprincipal wave is very crude as a representation of the mutualinteraction of two slots close together.

(5) THE HALF-WAVE SLOT: GENERALAs soon as it became obvious that a half-wave slot cut in the

broad face of the rectangular guide with the centre of the slotdisplaced from the central line, and the axis of the slot inclinedto the guide axis, is not a simple series or shunt load, it was anatural first step to try to find an equivalent circuit to representthe loading. In Fig. 17 and 18 the behaviour of the slot isshown when its length is varied, the centre of the slot and itsinclination being kept fixed. Four types of measurement weremade; the input impedance was measured when the guide bear-ing the slot was terminated in (a) an open-circuit, (b) a short-circuit (both reckoned at the position of the slot centre) and(c) a match, the slot length being varied in each of (a), (b) and(c); in (d) the slot remained unaltered, while the reactanceterminating the guide was varied. In each case, as the length of

Page 10: Resonant Slots Watson

756 WATSON: RESONANT SLOTS

Pure Series

90°Pure Series

Fig. 17.—Inclined and displaced slot (& in wide, S-band).Above: 0 = 30°; each circle corresponds to the indicated value of x\.Mow: x\ - 1 -0 cm; each circle corresponds to the indicated value of 6.The arrows show the sense in which the circles are described as the slot length is

increased. The guide had a matched termination.

Fig. 18.—Circle-diagram plot of input impedance to slot (i in wide,displaced 0-536 in and inclined at 45° in S-band guide), forincreasing lengths of slot, and for both short- and open-circuitterminations.

Note that the two circles do not cut the real axis for the same slot length.

the slot was increased from a small value, the point representingthe input impedance on the circle diagram, or the Z-plane, wasobserved to describe a circle in the clockwise sense. With amatched termination, one can see in what respect the inclined-displaced slot is intermediate between a pure series and a pureshunt load (see Fig. 17). As the slot is turned from parallelismwith axis of the guide, the circle turns from the characteristicshunt position to that for the series slot which is perpendicularto the axis of the guide. The diameter of the circle for the in-clined-displaced slot is intermediate between the diameters forthe pure series and pure shunt slots with centres at the samepoint on the guide wall. Similar behaviour is observed whenthe inclined slot is moved towards the middle of the guide facecarrying the longitudinal surface current.

The key to understanding the behaviour of these slots was toplot the results of the method (d) above on the circle diagram.In this way the transform of the unit circle was obtained on thew-plane. These transformations are bilinear since circles aretransformed into circles and the striking experimental factemerged that the self-corresponding points of each transformationare coincident and lie on the unit circle. This type of loadinghas been discussed in Ref. 2, Section 8. If e-'8 is the position ofthe double self-corresponding point, the transformation is

g€js + (i _ 2g)ww = — - . j» (o)

1 — ge~JSWwhere geis is the transform of the matched termination w 0.

Thus by finding first the reactive termination tan 8/2 whichwill put the slot out of action as a radiator, and then measuringthe transform of w ~ 0, we can determine the bilinear trans-formation completely. The radiation coefficients of the slotantenna in the guide are as follows: when the dominant wave ofunit amplitude is incident from the left, the slot antenna radiateswaves (of the same type) of amplitudes geJs and — g to the leftand to the right respectively in the guide; when the dominantwave of unit amplitude is incident from the right, the slot pro-duces secondary waves of amplitudes — g and ge~J8 to the leftand right respectively. On account of the use for which theseslots were intended, it was found to be convenient to use the>v-plane admittance-wise; tan 8/2 then represents the invariantsusceptance Yv

It is found that 8 does not depend on the length of the slotbut depends on the position of its centre. The dependence ofY1 on the lateral displacement and inclination of the slot can berepresented approximately in the following way. Let a be theconductance of the pure shunt slot with its centre coincidentwith that of the displaced-inclined slot, and let y be the re-sistance of the pure series, inclined at the same angle, its centrebeing on the centre of the broad face. Experiment shows (seeFig. 19) that when the inclination and displacement are not too

large, Yl = ± f~\ The sign to be given to Yl depends on

the inclination and displacement chosen according to the rulespresented in Fig. 20.

The dependence of g on the length of the slot can be explainedwith the aid of Fig. 21. The self-corresponding point S is thepoint of contact of the unit circle |>v'| = 1 and the circle 1 whichis the transform of the unit circle. The point P is the transformof w = 0 when the length of the slot is /0. As the length of theslot is changed, the displacement and inclination being keptfixed, w — 0 transforms as we have already seen, into a pointof the circle 2. It will be noticed that the centres of the circles 1and 2 lie on the same radius OS making 8 with the positive realaxis of w'. On the circle 2, the arrow indicates the directionof the displacement of P when the length of the slot is increased;tp is therefore a function of the length of the slot and it is found

Page 11: Resonant Slots Watson

WATSON: RESONANT SLOTS 757

0

-01

§.g-0-3•goo.

sSi -n-r)U)

•s8-Of,

-08

-09

C

r_\\

\

\

\

\

\

\

\) 0-2 04 06 0-8 10

Fig. 19.—Susceptance of self-corresponding point for general slotcut in the broad face.

Generator Generator

/aJy

Fig. 20.—Rule for sign of invariant susceptance.

W'-pltne

Fig. 21.—Circle diagram showing transformation of the unit circleby the general slot.

also to depend on its width. When the length of the slot is /0,g is real and we shall call this the resonant slot corresponding tothe chosen position of the slot centre and inclination of its axis.Evidently in general

ifj = - arg gThe facts just presented are illuminated by the results of field

representation applied to the problem of the general near-resonant slot in a rectangular wave guide. Without enteringfully into the details of the field calculation due to Stevenson, wemay profit by a brief reference to the principle of it. Let p'denote the amplitude of the voltage across the centre of the slotwhen excited by the dominant T.E.-wave of unit amplitude inci-dent from the left. The solution of the electromagnetic boundaryvalue problem under the assumption that the slot is narrow, thatthe thickness of the guide wall may be ignored, and that dif-

fraction round the outside of the guide may be left out of con-sideration, leads to the result

P !K

where the complex number % depends on the displacement andinclination of the slot only, and not on its length and width, andwhere the real part of K, like %, can be evaluated without diffi-culty, but the imaginary part of A'involves summation of doubleseries.

When the slot is similarly excited only by a wave of unitamplitude from the right, the voltage across the slot has theamplitude

• • "iwhere £ is the conjugate complex of £.

Now when the slot radiates with unit voltage amplitude at itscentre, there is radiated to the left a wave of amplitude Z.£ andto the right a wave of amplitude L%, where L is real and dependsonly on the dimensions of the guide cross-section, the frequencyof the radiation and the field vector used in the representation.It follows at once that

r \Y\->

and 8 2 arg £ '•• nK

The radiation coefficients have therefore the form indicated bythe experimental results.

(6) PROBE-COMPENSATED SLOTSThere is a useful type of inclined-displaced slot which trans-

forms a terminating match into an admittance whose real part isunity and whose susceptance is negative, and therefore inductive,

•" when 8 = — -• The transform is represented by Q in Fig. 22.

W- piano

Fig. 22.—Circle diagram for probe-compensated slot

It is not difficult to show that in this case K must have the formr(a + 1 — j) where r is a real negative number. The diameter

of the circle 2 in Fig. 21 is ——- and the variation of the phasea + 1

of the radiation from the slot when the guide beyond it is termi-nated in a match is determined entirely by — \\i. The matrixrepresenting the loading by this type of slot is

a ~ j

a - l - ja - j a~J

If a short probe of the proper length is introduced through theopposite broad face so as to keep as small as possible the directmutual interaction of the slot and probe antennae, the inductive

Page 12: Resonant Slots Watson

758 WATSON: RESONANT SLOTS

input susceptance of the slot when alone may be compensatedby the capacitive susceptance of the probe. The latter should be2/(a -f 1). The slot and probe combination allows the extractionof energy from the guide without reflection when the terminationis matched. The fact that the two antennae are in the same cross-section of the guide gives a measure of broad-band behaviour tothe compensation.

(7) PAIRS OF SLOTSIf two half-wave slots are cut at right angles to each other

intersecting at their common centre (Fig. 23) the following factscan be established.

Fig. 23.—Crossed half-wave slots.

Fig. 24.—7r-sectionslot-pair.

(a) When one of the slots lies along the centre line on thebroad face, only the other slot is effective as a radiator. Theelectric field is polarized parallel to the length of the guide.This shows that the mutual coupling of these slots is negligiblysmall.

(b) As the cross is turned about its centre both slots are ex-cited and consequently the direction of polarization is not sub-stantially altered. The cross presents a series load to the guide.

(c) Suppose now that the centre of the cross is displaced fromthe centre of the broad face, one slot being kept parallel to theguide axis. The transverse slot is a series element and thereforeexcited in quadrature with the other which is a shunt element.One expects that this device will radiate elliptically-polarizedwaves normal to the guide. This is found to be the case.Circular polarization is possible as a special case. If the cross ismoved to its image with respect to the centre line on the broadface, the sense of circular polarization is reversed. The effect ofterminating the guide by a movable reflecting plunger can easilybe deduced on the basis of the principles already adduced.Radiation from the longitudinal slot is suppressed when theplunger is A /2 from the centre of the cross, and from the trans-verse when the plunger is A /4 from the centre.

Another combination pair of slots, one longitudinal shunt, theother transverse series, with their centres in the same transversesection of the guide is shown in Fig. 24. If the conductance ofthe shunt slot equals the resistance of the series one and bothequal 2, then the pair of slots presents a match to the generatorand all the energy is radiated, none passing the slots in theguide, irrespective of the termination of the latter.

(8) GENERAL REMARKS ON APERTURES IN THEGUIDE WALL

The rectangular slot is a particular type of aperture in the wallof the guide. Imagine the slot deformed but still retaining a

major and minor axis of symmetry, the former associated withthe length and the latter with the width. So long as the deforma-tion is continuous without changing the connectivity of theaperture, we obtain essentially the same behaviour as the half-wave slot near its first resonance. Dimensions for the aperturecan be found to produce resonance. Such an aperture will loadthe guide just like a resonant rectangular slot. For instancedumb-bell slots have been used; the dimensions of such slots forS-band are shown in Fig. 25 and the accompanying table. The

I c 1

Dumb-bell slot resonant at 10-7cm on end of standard guide:Dimensions (inches):

a

iii

C

1-901-771-691-592-00

d

i

Fig. 25.—Dumb-bell slot.

larger the circles at the ends, the narrower the gap required inthe centre. The deformed slot now resembles a conventionalresonant circuit in that in the circular ends of the slot, the energyis mainly stored in the magnetic field as in an inductance, whilein the gap, the storage is electrical as in a condenser.

Slots may be covered with dielectric so as to close the guide inwhich they are cut. The main effect of this is to increase thecapacitance and thus reduce the length of the slot for resonance.

If the aperture is a sufficiently large circular or square hole cutin the broad face off centre, there are two natural modes ofoscillation of the aperture with the resonant frequencies suffi-ciently close so that either or both modes may be excited. It iseasy to demonstrate experimentally that the radiation from thehole corresponds to different types of oscillation of the currentsystem about it, depending on the mode of excitation of the hole.Thus, suppose a reflecting plunger is used to terminate the guide.Let the plunger be placed A /2 beyond the centre of the holeso that the transverse current near the centre of the longitudinalside of the hole would, in its absence, be reduced to zero. Thehole oscillates with electric force parallel to the guide axis like atransverse series slot. If the plunger is moved Aff/4, the othersimple mode with transverse electric force predominates as itdoes in the longitudinal shunt slots. Thus the polarization ofthe radiation from the hole, and its radiation pattern also,depends on the position of the plunger inside the guide. More-over, these two oscillations have different phases with respect tothe waves inside the guide. Corresponding effects can be shownfor a circular hole: the two simplest modes correspond to theHn-modes of the transverse field-distribution in guides of circularcross-section. If the hole is large enough, it is also possible toexcite the H2i-mode of oscillation, the notation again being basedon analogy with the names of the circular guide patterns. Theforegoing observations are founded on quite rough experiments,so the question should be investigated further.

Part II. THE COUPLING OF WAVE GUIDES BY RESONANT SLOTS(9) INTRODUCTION

The study of the coupling of guides by means of slots arosequite naturally as an essential element in the design of two-dimensional arrays. In a two-dimensional array in the xy-planemade up of a series of linear arrays parallel to x, each of thedistributing guides supporting them must be fed from a trans-verse guide, for example, parallel to the .y-axis, in such a waythat energy is properly distributed to secure the desired patternin planes containing the .y-axis. An example of such a couplingis shown in Fig. 26. The slot is cut longitudinally in guide T

and a larger registering slot is cut in guide L. The slot A pre-sents a shunt load to T and is coupled series-wise to L. Ifseveral guides L p L2 . . . are to be coupled to T so as to presentdifferent admittances in conformity with a chosen gabling dis-tribution parallel to y, the corresponding slots A p A2, . . .must be displaced by different amounts from the centre of thebroad face of T.

The length of the slot for non-reactive coupling is different inthe presence of L from its value for an exposed slot radiator.The impedance of a slot (such as A), was measured when L was

Page 13: Resonant Slots Watson

WATSON: RESONANT SLOTS 759

-ouplitij' slolin guide T

lot Hole inguide L

Fig. 26.—Shunt-series coupling of rectangular guides.

terminated so as to maintain a standing-wave ratio of 2 : 1 inthe guide T on each side of the slot. The variation in impedanceas the slot length is increased is shown in Fig. 27 and for com-

2 0

1 6

12

x_R

Oft

O47

Slot --coupled tfree spac

\-*-Sl

\

kot when c"ansverse

(see Fij

\\

\

\

I

oupled toguide

i 26)

\\

\/ /

48 49 50 51 5-2Length of slot, cm.

Fig. 27.—X/R plotted against length of slot.

parison the corresponding curve for a slot uncovered by theguide is also shown.

The importance of the laws of coupling of guides by means ofslots or holes cut in the metal sheet which forms part of the wallor termination of both guides arises from the fact that in micro-wave practice it is often required to divide wave-guide paths sothat energy may be distributed according to a definitely pre-scribed law, and it may further be desirable to introduce phasechanges in the coupling. The slot once excited, radiates intoboth guides. Depending on the aspect of the slot in each guide,the type of coupling and hence the law of impedance or admit-tance transfer from one guide to the other will change from onedisposition of slot and guides to another.

(10) THE SIMPLE LAWSThe simplest types of guide coupling by a single slot are

classified by the modes of coupling of the corresponding slotradiator cut in the same aspect with respect to each of thecoupled guides. These are shown in Fig. 28 as follows:

(a) Series-series coupling: the coupling slot is transverse inboth guides which must therefore be parallel;

(b) Shunt-shunt coupling: the slot is longitudinal in the twoparallel guides; and

(c) Series-shunt coupling: the slot is transverse to guide (1)and longitudinal in guide (2). In this case the guides must beat right angles. This is the type of coupling introduced inSection 9.

VOL. 93, PART IIIA.

(a) Series-series (b) Shunt-shunt

(c) Series - shunt

Fig. 28.—The simple types of coupling.

It is found that the coupling depends on the length and widthof the slot and on the thickness of the metal between the adjacentinside surfaces of the two guides. It is likewise determined inpart by the disposition of the slot with respect to each of thecoupled guides. The most striking result obtained when thelength of the coupling slot has been properly adjusted, concernsthe manner in which impedance is transferred from the positionof the slot in guide (2), say, into guide (1) at the same position.

Assume that guide (2) is terminated on each side of thecoupling slot, and let guide (1) be fed from an oscillator on oneside and be terminated on the other side so as to produce either(i) a short-circuit at the centre of the slot (plunger distant A /2)if it is series to guide (1) or (ii) an open-circuit at the centre ofthe slot (plunger distant A /4) if it is shunt to guide (1).

The two impedances seen from the slot in guide (2) will addby the law of series or shunt combination if the slot is trans-verse or longitudinal, respectively, to guide (2).

The laws of impedance transformation for these three simplesttypes of guide coupling, when the guides have the same trans-verse dimensions, are shown in Fig. 29, and experimental datasupporting these conclusions are presented in Tables 1, 2 and 3.

More generally, let Z', Y' be respectively the total impedanceor admittance in guide (2) as seen from the point of coupling,which is the centre of the slot, and let Z, Y be the correspondinginput impedance or admittance to guide (1) terminated in theway indicated above. The input impedance is reckoned at thepoint of coupling. Provided that the length of the coupling slotis properly adjusted for the frequency used, for the aspect andwidth of the slot, and for the dimensions of the guide cross-sections, we have the following simple laws of impedance andadmittance transfer through the slot at its centre when the wallthickness is negligible.

(a) Series-series

(6) Shunt-shunt

(c) Series-shunt

(7)

(8)

(9)

49

Page 14: Resonant Slots Watson

760 WATSON: RESONANT SLOTS

(a) Series-series (b) Shunt-shunt

Z't \

' i n

\

hi.

m

(c) Series - shunt

7*

J

Zm~*"

/7Ad

2+ 7l

'in I4

77

K111 /-r n 111 ,-r , ry I

Fig. 29.—The simple laws of impedance transformation.

Table 1

(i) Zin =

Z'20002-3

- 2 - 3

z20-512-9

- 2 - 0 52-32-3

Zin0-542-9

- 1-94-50-12

Table 2

K") *-in —Z2 -\-Z2

Z'I00oo00

2-30-96

z2- 0 1 7- 0 - 9 0+ 0-21- 2 - 3

2-3

Zm- 0 1 9- 0 - 9 7

0-23110-71

Table 3

(iii) Zin = > T , - 7Z\ + Z\

Z[00

-0-82-0-315- 3-3

Zx0000

Zin< 0 1

0-902-330-23

1*10-740-730-75

7?j and n2 are numerical constants and are equal to unity foridentical guides coupled in similar aspect; the numerical con-stant «3 may be varied by changing the displacement of thecoupling slot from the central line in guide (2).

Not only may series coupling be achieved by means of a slottransverse to one of the guides, it may also be achieved forguide (2), for example, when one end of this guide abuts on toguide (1) as shown in Fig. 30. In this case the law of impedance

Fig. 30.—Series-series coupling.

transfer (1) holds, but now Z' stands for the single terminatingimpedance in guide (2).

Inclined slots may also be used to achieve these three differenttypes of coupling, but the possibilities are somewhat restrictedin practice by the difficulty of accommodating slots of sufficientwidth to transfer large power. (Under the most favourableconditions, at a wavelength of 10 cm a i-in resonant slot willbreak down at powers exceeding 150 kW. This performancemay be improved by covering the edges of the slot to increasetheir radius of curvature and thus reduce the electric fieldstrength at the metal surface.) In series-coupling, the slot centremust lie on the centre line of the broad faces of the guides whichtouch each other as in Fig. 31, or it may lie on the centre line

Fig. 31.—Series-series coupling.Slot centred on the broad face of both guides, may be inclined in one or both.

Fig. 32.—Series-series coupling.The strength of coupling may be altered by rotating the slot about its centre.

of one guide and in one of the ends of the other as in Fig. 32.Similar cases are shown in Fig. 33. An important case of shunt-series coupling is shown in Fig. 34. The axes of the guides areperpendicular to each other. Guide (1), which carries the in-clined shunt slot, fits into the recessed broad faces of guide (2)so that a mechanical as well as an electrical junction is effectedat the coupling. It is unnecessary to place a conducting termina-tion in the series-coupled guide (2) behind the coupling slot. Instandard S-band guide, the maximum inclination allowable witha slot |-in wide is 40°. The dependence of the coupling coeffi-cient on the inclination of the slot is shown for this junction inFig. 35.

Page 15: Resonant Slots Watson

WATSON: RESONANT SLOTS 761

When the slot is not of resonant length, the coupling laws are

Fig. 33(a).—Series-shunt coupling.Slot inclined in broad face of one guide and in the narrow face of the other.

Fig. 33(b).—Shunt-shunt coupling.

YT • O (at slot)

Fig. 34.TTi

Shunt-series coupling.

14

•oo;§3a.

\

\

\

\

20 25 35 40

Fig.

30&, degrees

35.—Coupling coefficient m plotted against 6 for the slot ofFig. 34.

(a) Series-series

(b) Shunt-shunt

(c) Series-shunt

1 //, , .

z Y- /a'

1

z"3

T

(10)

(11)

(12)

where oq, a2, a3, are real and dependent on the length of the slot.It is found experimentally that the form (10) applies to the E-typecoupling of guides, and (12) to the H-type coupling.

(11) SOME SIMPLE CONSEQUENCES OF THE LAWSThe reactance introduced by the non-resonant slot is in shunt

with the impedance in a series-coupled guide, and in series withthe admittance in a shunt-coupled guide. In the series-shuntcase the coupling acts as a transformer.

The resonant-slot couplings may be conveniently applied inthe design and construction of wave-guide circuits. Since thecoupling of similar guides by a resonant slot in the same aspectto both guides allows the transfer of impedance in the seriescase, and admittance in the shunt case, without change, a meansis provided for coupling different loads to a wave guide withoutrequiring at each coupling operation mechanical work on theguide in question. In particular, short-circuits and open-circuits may be introduced without any plunger in the mainguide. It should be noted, however, that in the coupling of re-actances and susceptances, particularly when high ratio trans-formations are carried out, the resistive losses at the slot may notbe negligible due to the very high currents flowing to it.

In practice the series-shunt coupling is of special importance.It has been found that the thickness of guide wall, which isrelatively greater in guides intended for higher frequency micro-waves, introduces a departure from the simple law (12). Theform which fitted the experimental facts with standard S-bandguide is

1 m

where a may be reduced to zero by tuning the slot length, Kmay not: it depends on the thickness of metal between theadjacent inside surfaces of the guides being coupled by the slotas shown in Fig. 36. K may be tuned out by the appropriateseries reactance in guide (2). It may be remarked that theform (13) was obtained with the guide (1) terminated on one sideby an open-circuit, and guide (2) terminated on one side by ashort-circuit at the position of the slot centre.

(12) COUPLING OF VARIABLE REACTANCES TO THEGUIDE BY IT- (OR T-) SECTION COMBINATION OFSLOTS

The guide couplings used to introduce variable reactances areshown in Fig. 37, first the series reactance and then the shuntreactance. In the former, the reactance presented to guide (1)is that seen at the slot looking into guide (2). In Fig. 370), thereactance presented to guide (1) is the transformed value. Inboth cases it will be assumed that the slot has the proper length.If a shunt reactance without transformation was required itwould be necessary to turn guide (2) into parallelism with (1)and provide it with two terminations.

A series-coupled and a shunt-coupled reactance with thecentres of the coupling slots in the same transverse section ofthe guide behave as a IT-section circuit, to which, of course, thereis an equivalent T-section. In proof of this, data will be pre-sented to show how the section, which is shown in Fig. 38, trans-forms impedance. In Fig. 39 the input reactance in guide (1) is

Page 16: Resonant Slots Watson

762 WATSON: RESONANT SLOTS

vat

0 25

O20

K 0 15

010

005

n

/

/

/

0 05 010 015 020Wall thickness, in.

0 25 030

Fig. 36.—The parameter A: [see eqn. (13)] in terms of wall thickness.

The wall thickness is measured as the thickness of the metal in which the slot is cutand does not include the thickness of the wall of the second guide, which is standard.

(a)

Fig. 37.—Slot-coupled reactances.(«) Series-coupled. (6) Shunt-coupled.

Variableplunger

Fig. 38.—II-section arrangement.

plotted against the terminating reactance in guide (1), both beingreferred to the characteristic impedance of the line equivalent tothe dominant-wave circuit in the guide. Fig. 40 shows theequivalent II- and T-sections. It is not difficult to show withthe aid of the matrix methods described in a recent paper2 thatif a is the common admittance of the shunt elements of the

-3 0 -20

-30

2 0

-1-0

-1-0

-2 -0

-30

-4 -Ql

0 2 0 30

Fig. 39.—Input reactance vs. terminating one with 77-section.

2s Zs Zs

Zo ZP •p 'O

7T- Section T- SectionFig. 40.—Equivalent sections.

II-section and y the common impedance of the series elements ofthe T-section equivalent to it, then

The graph of Z.n against Z when the latter is a pure reactanceis a rectargular hyperbola with centre at [— Ky + I/a),Ky+l/a)]. The constant of the hyperbola is [i(y—I/a)]2. Fromthese facts a and y have been determined for a number of pairsof slots, and Fig. 41 shows these values against the measuredreactances introduced by the shunt and series slots separately.It will be seen that a very fair approximation is obtained byputting

a = One half of the shunt admittance of the longitudinal slotand

y = One half of the series impedance of the transverse slot.A wave explanation of this result using the methods of Part Iis presented later; the immediate deduction is that the mutualcoupling of the two reactances is zero.

Before leaving consideration of the circuit shown in Fig. 38 itwill be indicated how it may be used as a phase-changer in amatched guide. Suppose that we make a = y = j tan \d, thenwe may easily deduce that when the guide termination is a match,the input is likewise matched and there occurs the phase change 9.The theoretical and observed phase shifts are shown in Fig. 42and the equivalence of a and y is also established. They mustboth be pure reactances and the condition for matched trans-mission is that the product of the series and shunt reactancesshall be — 1. This is shown in Fig. 43.

Page 17: Resonant Slots Watson

WATSON: RESONANT SLOTS 763

5 1-05 -1 --5Observed reaof shunt slot

Fig. 41(o).—Observed vs. theoretical reactance of shunt slot

rlO

(Theoretical

Observed reactance of series slot-30 -2*5 -20 -1-5 -1-0 0-5 1

- 5 -

-1O-

•1-5 •

0 l

Fig. 41(6).—Observed vs. theoretical reactance of series slot.

By the method previously used by the author, we can explainthese results for the wave guide in terms of the waves radiatedfrom the slots, instead of relying on the somewhat artificialcircuit analogue. The slot coupling the shunt reactance actsas an aerial in the guide and radiates on each side H10-wavesof equal amplitude and in phase at equal distances from thecentre of the slot. The series slot radiates waves of equalamplitude and in opposed phase. Let / be the rad:ationcoefficient of the former and/ ' that of the latter. Then applyingthe principle of superposition to the complex amplitudes of theingoing and outgoing waves, we have

A - [I -(/+/')]A' | (/'-/)B

Thus

A' -

B'

B' = (/' - /)A' I [1 - (/ + /')] B

1 r-f1 ( / I / O

-:- (/i /o

-B

(15)

L w ' ' ' l~(frJJJ jThe matrix M representing the transformation due to this doubleloading can be read from these equations. For matched trans-mission it must be diagonal. Thus/ ' — /, then M becomes

~ 2/)~i

0

0 11-2/J

But since If - the phase shift is given by arg1 ( \OL

which immediately leads to the result stated above.

2-5

20

1-5

10

05

W3-O5

-10

-1-5

- 20

60 120 180 240 300

Experiinentolucurve

V Resulting change of phase, degrees.

\ Exponmenta:^ Elrve

-25

Fig. 42.—Phase change due to matching 7r-section slot combination

(13) RADIATION COEFFICIENTS IN GUIDE COUPLINGIn order to explain the method of radiation coefficients by

which the laws of guide coupling may be deduced, we shall firstconsider the simple case of series-series coupling of similar guidesin similar aspect to the coupling slot. When a wave of amplitudeAj is incident from the left in guide (1), the coupling slot radiatesa wave gA\ to the left and — gA\ to the right in guide (1) and— gA\ to the left and gA\ to the right in guide (2). These arerepresented in Fig. 44. The senses assumed correspond to therelative voltage due to the series slot cut in both guides. Theequations expressing the outgoing wave amplitudes in terms ofthe incoming waves are given in equations (16) where subscriptsrefer to the guides, primed letters to the left of the slot, unprimedto the right, and as usual A denotes a wave travelling to theright, B a wave to the left; thus

A, = (1 - g)A\ + gA'2 + gBl - gB2

A2 = gA[ + (1 - g)A'2 - gB{ + #B2

B', = gA[ - gA2 + (1 - s)B, + gB2

B'2 = - sA; + gA'2 + gB{ + (1 - g)B2

In order to determine the impedance relations we put B — An;thus reducing (16) to a set of four equations homogeneous in theincoming wave amplitudes Bv B2, A'v A'2. The determinant ofthe coefficients must therefore vanish; i.e.

(16)

1 - \OL

1 ' _1»/

1

g

— g

g

— g

I

g

g

- g— g-w'

g - 1/H',

~g1 — g

g

g~ g

— 1/

g1 — I

= 0 (17)

Page 18: Resonant Slots Watson

764 WATSON: RESONANT SLOTS

- 2

-3

Shunt reactance, -?fa,ohnis

Fig. 43.—Series reactance plotted against shunt reactance for noreflection from the section.

Guide 2

\ •*-\— I | »• -gA'i Guide 1

ICoupling

point

Fig. 44.—Wave amplitudes in series-series coupling of similar guides.

From this we obtain

1 - 2g (w'2 - 1)(1 - wv> , (w; - 1)(1 - w,)= " I "

Z — 1We now substitute M1 -- —-—-. all impedances being reckoned

positive looking to the right, and we have

1 - 2g 1 , 1

g Z2 Z2 Zj — Zj

Since Z, - 0 and Z2 — Z2 ---- Z', we obtain

1 1 l-2g

z; 2gWhen the slot is properly tuned and g is real, it follows from theequation of the input and outgoing energy in both guides that

1 - 3g2 r (1 - g)2

and g -J. Thus the simple form (1) for guides of the samecross-section is deduced.

tan

N

N

C5.

8181

NT(N

11(M8

-{-- fS

N

+-

II

1

v r-i

N1<N

N

1

N

N

N

4-j

N

o N

N

N

g ^

a *°2 *c3

Jit

'—I ' '

N I f

is

| !

+

8

+

+

+

8

+

* k

00

'Sioo

"a3OU

(5

1

lot

I / ]

unt

x:T3CC3on

.u'u.

Page 19: Resonant Slots Watson

WATSON: RESONANT SLOTS 765

The foregoing method has been applied to a variety of cases,By making suitable assumptions regarding the phases of theradiation coefficients, it is possible to represent any guidecoupling by slots The results for a number of cases are givenin Table 4; Fig. 45 shows the meaning of the coefficients intro-

'4 ,Z9

(a) (b)

-fet

z;+ X(c) (d)

Fig. 45.—Radiation coefficients (see Table 4).

duced. In the treatment of the general case of single slotcoupling, the principle of the calculation based on the ideasintroduced in Section 5 will be used.

Let the slot be excited by a wave of unit amplitude from theleft in guide (1), all the other incoming amplitudes being zero,then in place of the equation p' = XJK'm Section 5, which appliesto the slot radiating to space, we find for the voltage amplitudeat the centre of the slot

where C plays the part of K in the previous discussion, and £tis the value of £ for guide (1). Similarly for excitation by unitamplitude from the right

When the incoming wave is of unit amplitude in guide (2), thecorresponding voltage amplitudes at the slot are

P2kC

and P2-

Now when the voltage amplitude at the centre of the slot is

unity, there are radiated waves of amplitude L&{ and L£2 tothe left in the respective guides and to the right L ^ , and L ^ , .Thus if the wave A', is incident on the coupling in guide (1) from

^ ^ ^ . ^ t Q t h e , c f t J n .g L ^C l

and to the right is ~^~AJ. and so on. We can set up the

equation system analogous to eqn. (16).the equations reads

For example, one of

Proceeding in the same way as in the simpler case, using theequation of energy and writing <j>l ---2 arg $v <f>2 -= 2 arg £2, wereach the result

H'. — W.

ImC— W-r»2 n 2

from which the law of impedance transformation can be deducedin any particular case.

(14) DIRECTIVE ANTENNA COUPLING OF GUIDESSuppose that the second guide is coupled, by a pair of antennae,

to the first. If one of the antennae is shunt-coupled and theother series-coupled to guide (2) at the same position in thatguide, then the radiation from the two antennae will be greaterin that direction along guide (2) in which the fields are in phase.When complete cancellation occurs in the direction which wasformerly that of weak radiation, all the energy is transmitted inthe other. If two similar guides are coupled by a pair of slotseach in the same aspect to the guides, one being shunt-shunt,the other series-series, a wave entering one guide is switchedcompletely at the slots to emerge from the other guide, therebeing no reflection from the junction. This result is in accordwith theory using the method indicated above.

Directive coupling may be achieved by a pair of couplingantennae not in the same cross-section of either or both guides.The pair of antennae may be regarded as a directive array. If,however, they are separated in both guides and both tuned toresonance, coupling will only occur if the electrical spacings inthe two guides differ by an integral multiple of 2TT.

In order to deal with the multiple coupling of guides, it isappropriate to use matrix methods.2

Part III. SLOTTED WAVE-GUIDE ARRAYS(15) INTRODUCTION

The advantage of microwaves is that they permit, in practice,aerial apertures to be many times the wavelength in width andhence they produce very narrow beams. To spread the energyover this aperture, an array may be fed from a wave guide. Thephysical considerations entering the design of such an array aretwofold. First, well-known principles are applied to determinethe amplitude and phase distribution over the aperture requiredto secure the desired radiation pattern. There remains then theelectrical problem of imposing these distributions and at thesame time ensuring maximum power transfer from the generator.Jt is with this problem that the present paper has to deal.

A linear array from which the electric force in the main lobeis at right angles to the axis of the array, will be called transverselypolarized. If the electric force is parallel to the plane throughthe point where it is measured and containing the axis of thearray, it will be called longitudinally polarized. These termswill also be applied even when they are only substantially correct.

(16) LINEAR RADIATORSIn long arrays with many elements among which the energy

is to be divided, each element must be comparatively weaklycoupled to the guide. It has already been pointed out howwell suited, for this purpose are slots cut in the guide wali,for they can be cut in somewhat different ways to control thedegree of coupling, without at the same time introducing intothe radiation pattern serious effects due to the varying aspect ordisposition of the slots, either on account of their displacementfrom the central line on the broad face of the guide or on accountof their inclination to the guide axis, or, in fact, due to a com-bination of these. Further, since the slot is placed close to aposition where the exciting current on the guide wall vanishesand changes sign, the reversals of phase which are essential inthe arrangement of linear arrays intended to radiate substantiallybroadside in a single lobe, can be introduced just as easily as thevariations in amplitude of excitation.

It is convenient to classify linear radiators in two types:

Page 20: Resonant Slots Watson

766 WATSON: RESONANT SLOTS

(i) fixed, (ii) variable azimuth for the main lobe, as the frequencyof the radiation is altered. The former is essentially a narrowband device, the latter is generally much less frequency-sensitive.

From the point of view of theory, it should be possible to cal-culate the phase and amplitude of excitation of the elements ofan array, each element being coupled in a given way to thesystem of waves in the guide, no matter what that system is onaccount of loading. Conversely, in principle, it should bepossible to deduce the couplings necessary to produce the re-quired amplitude-phase distribution. In practical arrays, thereare only two alternative conditions which can be adopted as thebasis of design because of the required stability of the phasedistribution for a good main lobe. These are: (1) the guide isterminated by a reflecting plunger and a standing-wave systemis established in the guide. Alternate couplings reverse phaseand the elements, being spaced Ag/2 apart, radiate in phase. Themain lobe from the array is normal to the guide. This is thenarrow band type of array, often referred to as resonant spacing.The band-width of this arrangement is proportionately reducedthe longer the array, for the limiting condition is that for N ele-ments, the fractional change l/7Vin the frequency will put one ofthe elements in opposed phase. For the fractional change n/N,(N — n) elements will radiate in phase and n elements in anti-phase, arid the beam will split. For this reason it is of thegreatest practical importance in constructing long arrays of thistype, to measure the wavelength in the actual guide to be usedfor the mean frequency of the band, otherwise the array will notfunction over the desired band. It may, however, functionsatisfactorily over a displaced band. (2) The alternative condi-tion which allows a good approximation to a constant phase-gradient along the array, consists of a travelling wave in theguide. Since the velocity of propagation in the latter is usuallyabout one and a half times that in free space, the main lobe willbe radiated at an inconvenient angle to the normal of the array,when the elements radiate in phase with the wave in the guidewhere they are coupled. Accordingly, alternate elements arecoupled in reversed phase, and, unless the coupling device doesnot reflect in the guide, it is necessary to space the elementssufficiently differently from A /2 in order to present a satisfactoryinput impedance over a band of frequencies, the guide beingterminated by a matched sand-load. Two spacings have beenused, viz. 160 and 200~ (guide), the latter being preferable.

It is possible to maintain a sufficiently good approximation toa travelling wave in the guide and to obtain the proper amplitudedistribution, only if all the elements are weakly coupled to thewave in the guide, that is, if each element produces negligible dis-turbance of the feed. This requires that a considerable fraction(from 5 to 20%) of the input energy may have to be dissipatedin the sand load terminating the guide. The waste of energyvaries with the frequency of the waves, and is one of the mainfactors limiting the band-width of arrays of this type.

The fundamental distinction between the standing wave andthe travelling-wave aerials is that in the former the loads areeffectively at the same place on the transmission line and com-pete for power exactly like the parts of a d.c. circuit, whereas inthe latter, the exciting wave is attenuated by the absorption ofpower in the series of loads; hence for the equal excitation oftwo elements, that farther from the generator must be morestrongly coupled than the other. With half-wave spacing, if theelements are all shunt coupled, the wave of electric force in theguide is not attenuated; the wave of transverse magnetic force isattenuated, but since the elements are voltage-excited, they areunaffected by this attenuation when the frequency correspondsexactly with the spacing. If the elements were series coupled atthe resonant spacing, the current (magnetic force) wave wouldnot be attenuated, whereas the voltage wave would. In order

to produce attenuation of both waves, both series and shuntloading would be required at A /2 spacing. The matter may bestated somewhat differently as follows. At a non-resonantspacing, with weak coupling, the standing-wave ratio neverdeparts seriously from unity anywhere along the guide when it isterminated in a match, whereas with resonant spacing, thestanding-wave ratio varies from infinity to unity if the guide isfed from one end and has matched input. It should be notedthat a travelling-wave aerial must be end-fed, otherwise a dis-continuity in phase-gradient will occur at the feed-point. In astanding-wave aerial, the feed-point may be at one end or itmay be near the middle of the array, and in the latter case byskilful design it may be possible to secure somewhat greaterband-width.

In order to prevent instability of phase when resonant spacingis used, the elements must be coupled at maxima in the standing-wave of mean frequency which excites them, i.e. shunt loadsshould be placed at voltage maxima, series loads at currentmaxima. The amount of power drawn from the guide shouldbe determined by presenting to the wave in the guide the properadmittance or impedance. Attempts to control the power takenfrom the guide by any other method are likely to lead to in-stability of the phase distribution.

(17) THEORY OF THE WAVE-GUIDE FEED TO A LINEARARRAY

In the theory of the feeding of an array from a wave guide, itis assumed, in the first place, that the mutual effects betweenelements of the array, apart from their coupling to the H-wave inthe guide, may be neglected. This method will therefore applyfairly closely to arrays of longitudinal shunt and series slots, butnot to transverse slots. For the purpose of the argument, onlyshunt loading will be discussed, but the principle of the methodis applicable to any type of loading.

Let yr be the admittance of the r-th shunt load in the array ofJV elements, the AT-th load being most distant from the generator.Let vr be the voltage in the equivalent transmission line at theposition of the r-th load. If the loads are spaced equidistant dapart, or 9 radians reckoned on the unloaded line, the differenceequation satisfied by the voltage* is

vf+l — 2 cos 6 vr + iv_! = — JY,-i sin 6 vr . (19)When 6 is an integral mutliple of 77, say mrr,

vr+l=-(-l)mvr (20)showing that the voltage wave is not attenuated. If ir denotesthe current immediately to the right of the r-th load

(21)

where Kis the constant voltage, and /0 is the current A /2 in frontof the first load. This shows the decreasing current wave in theshunt loaded line at resonant spacing of the loads.

It remains now to discuss non-resonant spacing. In a longarray the yr should be small compared with unity for weakcoupling in the proper physical sense of negligible local dis-turbance of the feed, and further, yr is not a rapidly changingfunction of r.

Let us treat equation (19) by the analogue of the method ofvariation of parameters writing

vr = Artor + Broj~r, co = e# . . . (22)

with the auxiliary condition(A r + 1 -A>H- i 1 ( B , + 1 - B > - ( ' + ' ) - 0 . (23)

so thatVf+l _ Vf == Ar(a> - l)a/ ! B,«-'(w-i 1) . (24)

* The time factor e~Jat is suppressed.

Page 21: Resonant Slots Watson

WATSON: RESONANT SLOTS 767

Thus the second-order difference equation (19) in vr can be re-placed by a pair of first-order equations in Ar and Br. In con-sequence of the smallness of yr we proceed on the assumptionthat Ar and Br do not vary rapidly with r and hence the pair ofdifference equations may be replaced by differential equationsin which the variable x\d replaces the index r. These equationsare

JA ,.dB n

dx dx(25)

(1,/A

dx dx

-JsinOyQ

Comparison of these equations shows that

Ay(u) — to-1)L =

dA}dx Id

(1.dB , By(co -

o>)— + — —jdx 2«

dAdx

dBdx

. (26)

(27)

(28)

where /x is a multiplier still to be determined.

If fix) = Yydx and f(L) = Q where L = Nd, then

B

. (29)

The value of /x is deduced in terms of the ratio of the input

and terminal values of the circle-diagram variable w = —A

reckoned at points distant an integral number of half-wavelengthsfrom the origin of x. We find

2(2= cos 0 + JXP -!- 1)* sin 0, where/? = (30)d **(£)

If H>r is near zero (matched termination) p will be small, andtherefore 1 — ii tends to cu, so we obtain the following approxi-mate formula

(31)

or reverting to the discrete loading

Ap= Ao exp (32)

while . . (33)

The expression for Ar shows the attenuation of the wavetravelling from the generator, that for Br the growth of the re-flected wave. It has already been shown how to calculate theratio w>i/wr by matrix methods in the case of weak loading.

In general ys is complex, ys — €s— jbs. The presence ofsusceptance modifies the velocity of phase propagation, but theuniformity of the phase gradient along the radiators of the arraywill be preserved if the gradient of the susceptance along thearray is small. The attenuation of the wave is determined by

the conductance gs. The fraction of energy reaching theterminating load is represented by

e X P

N

(18) AMPLITUDE DISTRIBUTION AND INPUT IMPEDANCEIn the case of half-wave spacing the radiators may be co-phased

by making them pure conductances gr, so that the input con-ductance to the array can be made unity or any desired value G,and the amplitude distribution along the array made to followthe desired law./, simultaneously, by means of the relation

(35)

To secure the desired amplitude distribution in a travelling-wave aerial, one must take into account the attenuation alongthe wave guide due to the loads. Let er be the fraction of energyto be extracted by the r-th radiator from the generator end, andlet fr be the amplitude of excitation for that radiator, then werequire

or . . . . (36)

Since the fraction of energy taken from the guide by a radiatorin a travelling wave is equal to the conductance presented by itto the waves, we have

. . . . (37)

g ,

In order to fix the values of #r, the conductances, the value ofone of them must be known. This will generally be decided byone or other of the following considerations: (a) there is amaximum value of conductance that can be tolerated consistentwith the hypothesis of weak coupling and satisfactory phasing,or in the case of slots, consistent with satisfactory radiating pro-perties with respect to polarization and side lobes from the array,and (b) the amount of energy to be dissipated in the terminatingload must be sufficiently small so as not to reduce unduly theefficiency of the array. Except for long arrays, these two con-siderations usually conflict, and in practice it is necessary tostrike a compromise.

Computation of the conductances is greatly facilitated bynoting that if

the simple formula

is obtained with

yr - yr+1 -f f?

(38)

(39)

If attenuation in the guide takes place in significant amountdue to the finite conductivity of the walls, let 8 be the fractionof energy lost due to this cause between two successive radiators,then equation (39) must be replaced by

Page 22: Resonant Slots Watson

768 WATSON: RESONANT SLOTS

To study the variation of the input impedance, we make useof the result already referred to, viz.

1 . (41)

When the termination is matched, wT 0, and

H.i J ? ^ I 2 S (42)

As 0 is varied, the sum which is the numerator of this fractionundergoes rapid fluctuations but its modulus remains smallcompared with }£ys except when 6 ^ mrr (m integral). Indeeda graph of the modulus of the sum against 6 will resemble thediffraction pattern of a grating with principal and subsidiarymaxima. Thus when the terminating load is non-reflecting,|M>J| and hence the standing-wave ratio will exhibit fluctuationslike the sum in question when the electrical spacing of the ele-ments is altered by varying the velocity of propagation in theguide, either as the result of changing the frequency or changingthe width of the guide cross-section. That is, it is not possibleto feed the same array when 6 — mv and in the vicinity of thesevalues. The fluctuations in standing-wave ratio to be expectedare illustrated in Fig. 46.

\ .10 cqu»l conductances

Y •i ,75 equal conductances

S\ I I

-0-1

O-0133/f

160" 165 190° 195° 2O5°170° 175° 180° 185°Electrical spacing of loads

Fig. 46.—Input standing-wave ratio for two arrays.Dotted curve: 10 equal conductances.Full curve: 75 equal conductances.In each case the total conductance is 3C0.

On the other hand, if wT ----- 1 (open-circuit termination) andN

>ys 1, then when B^nm, w, - 0, i.e. the input is matched.5 = 1

Even when Xys g < 1, we can secure a matched input bychoosing wT •=--—, but of course, if g > 1 there is no possible

2 - gtermination of the feed guide yielding a matched input. Fromequation (41) it is easily seen that for a system of equal loadsintended to match at half-wave spacing, the input admittance isreduced to a half when 9 is changed from IT to the nearest zeroof sin Nd, that is, the wavelength is changed by the fraction 1/2N.These considerations explain the sensitivity of the input admit-tance of a long array to frequency at A /2 spacing.(19) TRANSVERSELY-POLARIZED ARRAYS OF SLOTSIn Sections 1-8, it was related how the invention of slot

arrays was the natural outcome of the search for radiating ele-ments which can be coupled so as to present a low, pure con-ductance to the wave in the guide.

The first array of slots cut in the guide wall consisted of49 longitudinal laterally-displaced slots in the broad face, A /2being the spacing between the centres of successive slots, withthe guide terminated by a reflecting plunger A /4 from the lastslot centre. The conductance presented to the dominant wavein the guide by each slot was determined by its distance fromthe centre of the guide in accordance with the law G •— Ksir^irxja[see equation (5)], the constant K being found by measurement.Phase reversal was secured by placing alternate slots on oppositesides of the centre line as shown in Fig. 47(a). This particular

(a)

Fig. 47.—(a) Transversely- and (ft) longitudinally-polarkedslot arrays.

array was centre-fed: in such a case it is necessary to take intoaccount the phase relationship of the waves to the right and leftof the point of coupling. If the coupling of the generator feedin the guide array is series, the two waves are in anti-phase, ifshunt they are in phase. Consequently for a centre-fed array,it is necessary to know the method by which the transmitter iscoupled to the guide in order to cut the slots correctly.

This array had a sufficiently large number of elements so thateven the most strongly coupled slots in the centre of the arraydid not have to be displaced from the centre line by a distancegreat enough to distort sensibly the radiation pattern from thearray. The polarization was of course transverse. The stagger-ing of the slots with respect to the centre line introduced noobservable spurious radiation.

The following refinements and simplification arise in thepractical development of this device:

(i) choice of adequate slot-width;(ii) since the slots must be covered with weather-proof

dielectric the length of slot for resonance must be measuredfor the covered slots;

(iii) the length of slot for resonance varies slightly with dis-placement of the slot centre on the broad face of the guide;

(iv) it is not necessary to cut the slots with displacementsexactly as calculated, the displacements can be forced to thenearest figure found suitable for the machine work of cutting(usually with an end-mill). These same considerations apply inthe construction of travelling-wave antennae using these slots.

An array having the same polarization but not so convenientto cut could be made with inclined series-coupled slots; the in-clination of the slots to the guide axis would be small enough togive a good radiation pattern, provided that a sufficient numberof elements comprised the a/ray.

(20) LONGITUDINALLY-POLARIZED SLOT ARRAYSIn many practical antennae, longitudinal polarization is

desired because of the increased contrast possible in the radar

Page 23: Resonant Slots Watson

WATSON: RESONANT SLOTS 769

pictures resulting from the use of this polarization. It may beprovided by means of the inclined shunt-coupled slots cut in thenarrow face of the guide. These slots make only a smallangle (<j>) with the plane perpendicular to the guide axis. Becauseof the mutual interaction between these slots in virtue of wavespropagated on the outside of the guide the exact design problemis quite difficult. Firstly, it is not easy to obtain precise measure-ments of the parameters which represent the mutual effects,secondly, the conditions governing propagation outside arecomplicated, and thirdly, even if the foregoing information wereobtained with the necessary accuracy, special computing methodswould be required to give, in practice, the required result.

Arrays of inclined shunt slots may be constructed with resonantor non-resonant spacing [usually 200° (guide)]. The latteraffords a somewhat easier crude approximation in design whichwe now consider. With a matched termination, the attenuationof the wave in .the slotted guide, due to radiation by the slots,was measured as a function of depth of cut; the slots were cutat the same inclination but in alternately reversed senses [seeFig. 47(6)]. For a sufficiently great number of slots, there wasfound a particular depth of cut for each single angle of inclina-tion ((f>) of the slots which makes the power radiated by the arraya maximum (see Fig. 48). For different § « 15 ) the overall

36

32

:zo

\

\

085 090 095 100 105Depth of cut,cm

110 115

Fig. 48.-10° edge slots at 200° (guide) spacing. A = 10-7 cm.

length of the slot measured on the outside of the guide is thesame for maximum radiation. Consequently, to apply theinformation presented below one should approximate to con-stancy of length of slot rather than constancy of depth of cut.This attenuation measurement with a large number of similarslots gives a useful estimate of the contribution of the individualslots in a gabled array at the same non-resonant spacing, pro-vided that the array is long enough so that the gradient of slotinclination along the array is nowhere rapid.

Measurements showed that it is necessary to use more than10 slots in order to attain a steady value of the logarithmicdecrement (k) per slot. In Fig. 49, k is seen to depend con-siderably on n, the number of slots. The limiting value of k forw large is 0-049 when <f> = 10°; this represents ihe effective con-ductance G/Yo of each slot in the array. The admittance of oneof these slots was measured and found to be 0 017 + 0 015/.It is obvious that the particular depth of cut (slot length) yieldingmaximum radiation by the array at a given wavelength is deter-mined by the mutual interaction of a large number of slots atthe particular spacing adopted; so far as the wave in the guideis concerned, mutual susceptance is approximately cancelled byself-susceptance. Further, a different result would have beenobtained if the slots had been cut parallel (without phase reversal).

Fig. 50 shows the dependence of G/Yo on the inclination of

0 05

0 0 4

f 0 03

3f

0 0 2

I1

>/

8 12Number of slots

16 20

Fig. 49.—Array of 10° edge slots (width ft in) spaced at 200J (guide)./„ = fraction of input power radiated by n slots.k = C/Ko for n > 20.

O2O

0 10

0 08

OO6

O-O4

0 0 2

001

71

7

8 120, degrees

16 20

Fig. 50.—Effective conductance in terms of inclination of edge slotsin long array with alternate phase reversal.

Spacing 200c (guide). Standard S-band. G/Ko = 1-66 sin* <f>.

the slots. It is seen to follow the sin2</> law. It is fortunatethat at 200° spacing, the effect of mutual interaction is to in-crease the power radiated per slot at a given inclination, so thatthe amount of unwanted polarization to be tolerated for a givenradiation per slot is less than would be expected on the basis ofsingle-slot measurements.

Fig. 51 gives the depth of cut as a function of <j> for maximumradiation by an array of similar slots cut alternately in reversedphase at 200° spacing. This graph is consistent with the hypo-thesis that the overall length for maximum radiation at the givenfrequency is independent of the inclination (<f> < 15 ). As apractical compromise at A — 10-7 cm, it seems reasonable topropose that slots for which 10° < <f> < 15° should be cut0-95 cm in depth, also slots of smaller inclination than 101 shouldbe cut to a depth of 0 • 99 cm in standard guide for the band.

Since arrays of these slots are desired to give a certain measureof broad-band behaviour, it is interesting to compare the effectof varying the frequency with that of varying the slot length.Since the former changes the relative phases of the contributions

Page 24: Resonant Slots Watson

770

too

9 2

9 0

\

WATSON: RESONANT SLOTS

100

80

60

4 0

20

Q

/

Gener ator ei id

——0 "

Distan

iAr

reen s

\

ots. 9C

' * " " • • * " • .

6ctnN

9<(>, degrees

Fig. 51.—Depth of cut vs .inclination of S-band edge slots, ^ in wide.200° (guide) spacing, for maximum radiation from array of slots (see Fig. 48).

* „ = 107 cm.

to the mutual effect at each slot as well as changing the admit-tance of the slot itself, it is expected (and confirmed in Table 5)that to produce a given decrease from the maximum value ofthe power radiated by the array of similar slots, the fractionalchange in frequency required is about half of the fractionalchange in slot length.

Table 5

10° SLOTS

Change inradiated power

from themaximum

1025

Change in totalslot length

%314-9

Change infrequency

0/

l'°52-2

The simplest procedure in designing a broadside array ofinclined-shunt slots cut in the narrow face of the guide of alongitudinally-polarized array is to make all the slots of the sameoverall length corresponding to maximum radiation in a longarray of similar slots at the same spacing, equally inclined, butalternately reversed in phase. The inclination of the individualslots is then chosen so as to approach the desired amplitude dis-tribution and to yield a suitable dissipation of power in the load.The guide is assumed to be weakly loaded by each slot so thatthe phase of each radiator is nearly in constant relation to thephase of the wave in the guide at the position of the slot centre.An array constructed according to this plan will not give thedesired amplitude distribution. Whenever the mutual effect in-creases radiation, one may expect that the waves on the outsideof the guide travelling toward the generator have larger amplitudethan those travelling in the opposite direction.

An array of 20 slots of the same overall length and 200°spacing (guide) was cut with inclination varying from 6-3° at theinput end to 8 • 1J at the other. It was designed to extract 40%of the energy in the guide on the assumption that mutual inter-action could be taken into account by assigning to each slot theconductance corresponding to the limiting decrement of energyper slot in a long array of similar slots. It was found that 45 %of the energy was abstracted. The amplitude distribution wasintended to be uniform; the measured distribution is shown byFig. 52. The hump in the distribution at the input end of thearray corresponds with the observation that the wave energybeyond the array on the outside of the guide was twice as much

0 2 4 6 8 10 12 14 16 18 20Number of slots in array

Fig. 52.—Amplitude distribution and equiphase plot opposite S-bandedge-slot array.

at the input end as at the load end. The actual amplitude dis-tribution is only a fair approximation to the intended one;examination of the equiphase distribution about 20 cm awayfrom the array showed that the equiphase lines are very nearlystraight as is required for a good directive pattern from a lineararray. They were inclined at 4-2° to the guide: if the radiationwere phased according to the travelling wave in the guide, theangle would be about 3 • 8°. For many practical purposes, there-fore, this method may be adequate.

If higher accuracy is required, it seems necessary to prescribethe length and inclination of each slot in order to secure thedesired amplitude and phase distributions. A design procedurebased on approximate theory has been worked out by the author.

An approximate design for arrays of inclined slots in thenarrow face, spaced A /2 apart, with a plunger terminating theguide can be approached through the conception of incrementalconductance which was explained in Section 4.3. The incre-mental conductance having been found as a function of inclina-tion, arrays are then cut according to the law for shunt slots atresonant spacing without mutual admittance. The terminatingplunger must be adjusted by trial. While this procedure yieldsusable arrays with beams radiated normal to the guide, the pro-cedure does not enable the desired input impedance and amplitudedistribution to be obtained. Any attempt to improve the designby explicit representation of the mutual effects leads to formid-able numerical work. The experimental determination of thenecessary parameters is also difficult. As examples, we maymention the following measurements.

Suppose that two slots of the same length and inclination arecut at half-wave spacing in the guide, first, antiparallel (reversedphase coupling) and second, parallel to each other. Let theinput admittances be measured in each case and be denoted byY+ and Y_ respectively. The ratio of the mutual admittance Mto the self-admittance a for slots at the chosen inclination and

spacing is + Yj—• Results for f-in slots in standard guide at

a wavelength of 10-7 cm are presented in Tables 6 and 7.Thus Af/a does not vary much with the depth of cut for a given

fixed inclination of slot. Also Re(M/a), as one should expect,does not vary much with <j> for cf> < 15C. It should be noted,however, that if the depth of cut is maintained constant as theinclination of the slot is increased, the overall length of the slotis increased sufficiently rapidly to produce considerable changein the arguments of M, a and Af/a.

The foregoing is based on the assumption that the mutualadmittance remains unchanged in numerical value when one ofthe slots is turned from parallelism to antiparallelism at thesame inclination. This is not strictly true according to field

Page 25: Resonant Slots Watson

WATSON: RESONANT SLOTS 771

Table 6

15° SLOTS

Depth, cm

10761-1281-180

M/a

0-43e-v48-2°0-46e-y52-3°0-44€-y"52-7°

Table 7

TOTAL LENGTH OF SLOTS CONSTANT

Inclination

3°5°

15°

0-290-270-280-29

theory representation of the radiation on the outside of the guide,but if the inclination is small enough the error should not belarge.

A second measurement intended to estimate how the slots dis-turb propagation on the outside of the guide was suggested fromtheoretical considerations. A slot at zero inclination and there-fore unexcited by the dominant wave inside the guide will becalled a parasitic (or Yagi) slot. If the input admittance of aninclined shunt slot is measured alone and then in the presenceof a parasitic slot distant Xg(2, the fractional increase in admit-tance will be denoted by MJx; the ratio —-I— yields the

1 a / aapproximate value of the amplitude of the secondary sphericalwave reckoned at the distance X /2, when the exciting wave onthe outside of the guide has unit amplitude at the position ofthe slot. For example, the following results were obtained for12° slots:

M

leading to the amplitude of the secondary wave, 0-49e-v'18>2°.In order to check this, an attempt was made to measure im-

pedance on the outside of the guide using the device shown inFig. 53. A wave was launched on the outside of one end of

g-in.stot for detector Yagi slot

Fig. 53.—Standing-wave-detector intended to measure impedancespresented to waves on the outside of the guide.

the long guide. A Yagi slot at a distance of about one metrefrom the exciting slot was backed by a large reflecting coppersheet which was adjusted to place a short-circuit at the positionof the Yagi slot, the latter being series-coupled to the waves onthe outside of the guide. The longitudinal slot in the guidethrough which the detecting probe explored the standing-wavesystem in front of the Yagi slot was about 30 cm away. Themeasured series impedance was found to be pure reactance ofmagnitude 0 • 364. Taking into account the law of attenuationof the amplitude of waves propagated outside the narrow faceof the guide (see Section 4.3) the value of the secondary waveamplitude was found to be 0-25e^80°. This result is so much atvariance with our physical expectation as regards phase that weconclude that the current system on the guide face in this ex-periment was quite different from that when the Yagi slot wasnear the slot coupled to the guide. It was thought worth whileto mention these results for unless experimental methods of thistype can be satisfactorily developed, the exact design of thesearrays will have to depend on the mathematical theory alone.

(21) A BROAD-BAND ARRAY OF SLOTS (TRANSVERSEPOLARIZATION)

Because of the mutual interaction between the slots of alongitudinally-polarized array, and the sensitivity of the interactionto frequency so that the fraction of energy radiated by the slotsis fairly strongly frequency-sensitive, it is clear that the band-width of these arrays is limited by the loss of energy to theterminating load. Mutual interaction by waves outside theguide operates to reduce band-width. For a broad-band slotarray, it is necessary to use transverse polarization. A methodis now presented for achieving this by means of the displacedand inclined slots described in Section 6. The principle involvedis that each radiator is so coupled to the guide that it permits apure travelling-wave from the generator to the matched termina-tion of the guide, while abstracting a known fraction of theenergy from it.

The slots have small displacement x and inclination 9 measuredfrom the unexcited lontitudinal position on the broad face, andso chosen that the self-corresponding point of the w-plane trans-formed by the slot lies at — j . The length of the slot is such thata match is transformed into unit conductance with a small nega-tive susceptance. The latter is compensated by the positivesusceptance due to the silver probe (of diameter £ in for S-band)placed in the same guide cross-section as the centre of the slotand opposite to it in the other broad face. This disposition isrequired for proper compensation when a considerable fractionof energy is drawn from the guide by the end slots of the array.

Design of the array depends on the measurement of (i) thevalue of x and 9 to yield the proper type of displaced-inclinedslots (for practical purposes it is sufficient to determine the slotlength for only one value of 9, so that it allows perfect com-pensation by a probe), (ii) the compensating susceptances and thecorresponding probe lengths, (iii) the fraction of power ab-stracted by the compensated slot from a travelling wave as afunction of 9, and (iv) the phase shift (retardation) produced inthe travelling wave as it passes the slot, as a function of 9.

At S-band wavelengths, a slot displaced 0-4 in and inclinedat 25° abstracts 40% of the energy from a travelling wave in theguide; the corresponding phase shift is 14-4° in passing the com-pensated slot. In an array with its amplitude distributiontapered symmetrically about its centre most of the radiators willbe required to radiate very much smaller fractions of power than40%, and will produce a phase shift much less than 14°. There-fore we may ignore the longitudinal displacement required tocompensate for the phase shift when we are concerned withamplitude distribution, and choose as the most convenient

Page 26: Resonant Slots Watson

010

008

0-06

004

002

Iu(0

001

0-008

0006

0004

0002

/ /

y '•

7

z ,>0-6

0-4

-8To^ 0 2

a

1b 01

004

0020 004 008 012 016 020

Displacement, ins.0 2 4 6 8 10

Angle, degrees,and probe length, mm.

(a)displacement of slot.inclination of slot.length of compensating probe (0 125 indiam,), including the

wall thickness (2 • 1 mm).

01

0-6

0-4

0-2

1 01t.008

g.0-06

I 004|

(0

u.002

0 0 1

//

f // // // /

I 1\ 1\ 1I 1

11I

//

0-502 03 0 4Displacement, ins.

5 10 15 20 25Angle, degrees,and probe length,mm.

(b)

4i 2 3Retardation, mm.

0 4 8 12 16Retardation, degrees

(c)Phase shift produced in travelling wave on passing a slot.

degrees. mm (Xg - 16-2 cm).

20

i

Fig. 54.—Design data for broad-band inclined-displaced slot (S-band). Slot dimensions -& in x 2 in.

Page 27: Resonant Slots Watson

WATSON: RESONANT SLOTS

0-6

773

010

008

006

004

002

001

0008

0006

0-004

0 002

0001

11

n

//

i }

11

i

11f

/ , .

//

/1

1

/

//

//

/

/

//

//

s

ss

s

0 02 0-4 06 0-8 010 ' 012 014Displacement, mm.

0 1 2 3 4 5 6 7Angle, degrees

005 006 007 008 009 010 Oil 012Probe length, ins.

(a)displacement of slot centre.inclination of slot centre.

— — — length of compensating probe (0-125 in diam.), including wall thickness(0-050 in).

Fig. 55.—Design data for broad-band array of inclined-displaced slots

(X-band).

001

0

005

0-75

5

1-25

10

0-iO

175 2-25Displacement, mm.

15 20Angle, degrees

015Probe length, in

275

25

3-25

30

020

375

3b

Slot lengths: 10": l-495cm, 203: J-505 cm; 3(V: 1-510 cm.

0 2

I<J0L.Jj• 01

"008£V

a

ion

V(0t-i.

002

0-01

/

111 /

1llll

ll1

1

Ififll1ifM

/

' // /' /1 /

71

/

• —

0-2 0-4 0« 08Retardation, mm.

6 8

10

10

1-2

12

14

14Retardation, degrees

Phase shift produced in travelling wave on passing a slot.degrees; mm.

Page 28: Resonant Slots Watson

774

10

.6

WATSON: RESONANT SLOTS

i

\

\ XX .45 elen

nents

tents

y//

10-8 10-9 HO"10-3 10-4 10-5 10-6 10-7Wavelength Aa,cm.

Fig. 56.—Frequency dependence of power absorbed in terminatingload for 45-element array and for the latter half of it.

variable to characterize a radiator, the fraction of energy to beradiated by it. The array is then designed with the aid both ofequation (36) and of graphs in which are plotted against the

variable e (denoting the fraction of energy radiated by the slot)(i) the displacement x, (ii) the inclination 9, (iii) the length p ofthe compensating probe, and (iv) the phase shift converted intoequivalent displacements along the guide at mean frequency inthe band. The data for a wavelength of 10-7 cm in standardguide are shown in Fig. 54 (a), (b) and (c), and for a wavelengthof 3 • 20 cm in standard guide in Fig. 55 (a), (6) and (c). Item (iv)is used to correct the position of the centres of the slots withrespect to the basic spacing A /2 calculated for the mean fre-quency. The correction is cumulative. For a range of fre-quencies off the centre of the band, the maintenance of a suffi-ciently good approximation to a travelling wave in the guide isassured even if the individual slot-probe combination fails totransform a match into a perfect match off-frequejicy. Further,provided that the phase shift is taken into account, the beam fromthe array should be radiated normal to the guide at the meanfrequency.

In order to secure the alternation of phase necessary to pro-duce the beam, normal to the array, the slot centres will bealternately staggered on each side of the centre of the broad faceof the guide. In order that the proper slot shall be used, the endof the slot nearer to the generator will be nearer to the centre ofthe guide in every case. The phase correction is applied bybringing the slots nearer to the generator.

100

80

"a.

"40

20

-4 -2 0Degrees

X = 10-50cm; ZT = 1.

(c)

V

/\\

1V

-6 -4 -2 0Degrees

X 10-70cm; ZT ^ 1.

100

80

160

40

20

(e)

s/

//

/

/

\

\

-2 0 2Degrees

X - 10-90cm; Zr = 1.

"-4 -2 0 2Degrees

X = l l -0cm; ZT = 1.

Fig. 57.—Radiation patterns for 45-element array of inclined-displaced slots (S-band).

Page 29: Resonant Slots Watson

100

80

160

I,20

(f)

Ay A-6 -4 -2 0

DegreesX = 10-50 cm; Yf —

WATSON: RESONANT SLOTS

100

775

80

60

40

20

-6 -4 -2 0Degrees

X = 10-70cm; YT = 4 / .

-6 -4 -2 0Degrees

X = 10-81 cm; YT = +y.

100

80

60

40

20

(i)

\/J

\

IV-4 -2 0 2

DegreesX = 10-90 cm; YT =

J00

80

I"5-

20

f

11I

/

\

K-4 -2 0 2

Degrees

X = 11 00 cm; YT = +y.

Fig. ST.—Continued.

Page 30: Resonant Slots Watson

776 WATSON: RESONANT SLOTS

As an example of the application of these principles, a 45-element array was cut for S-band. The slots were -^-'m wide,2-in long, with a basic slot spacing of XJ2 at a wavelength of10-7 cm. The correction for phase shift was taken into account.The energy reaching the matching load at the end of the guidewas measured as a function of frequency and the result is shownin Fig. 56 both for the whole array and for the last half of it.On the assumption that not more than 6% of the energy wasdissipated in the terminating load, this array had a band-widthof 8% of the operating frequency. It may confidently be pre-dicted that a longer array would have greater band-width forsatisfactory power absorption. Throughout the band, thevoltage standing-wave ratio never exceeded 1 1 2 : 1 . Inpractice, however, it is possible to dispense with the terminatingload and to terminate the guide with a plunger %Ag from thecentre of the nearest slot. This particular reactive terminationmay be used because the last few slots of the array transform itinto a near match, and it is found that the disturbance of phaseis not serious. It should be noted that transformation bythese slots reduces the radius of a small circle near the originon the H'-plane, thus the matching arrangement is essentiallystable.

The results of the Held measurements are shown in Fig. 57(a)-(j). The average observed beam-width of the main lobe was2-35 compared to the theoretical value of 2 1° for a uniformlyilluminated array of the same length. When the guide was cor-rectly terminated in the maximum side-lobe amplitude was 15%of the maximum in the main lobe, in the wavelength range 10-4-11 -0 cm. When the guide was terminated by a reflecting plug,the maximum side-lobe amplitude varied from 19 • 5 % at 10- 5 cmto 12% at 10-8 cm and 16% at 11 0cm. The lobe of unwantedpolarization, due mainly to the inclination of the slots, wasobserved at ± 45J to the normal to the array and did not exceed11 % in amplitude. The band-width of this array is sufficientlygreat to allow control of the direction of the beam over four orfive degrees by variation of the frequency by ± 4%.

(22) TWO-DIMENSIONAL ARRAYS OF SLOTSThe linear arrays which have been described in the previous

Sections can be used as elements in a two-dimensional array ofslots, thus replacing the combination of a linear radiator with areflector. The linear arrays may be either standing-wave ortravelling-wave radiators. Mutual effect between them will be

important only for transverse polarization, but it may be sup-pressed by introducing a choke between each pair of adjacentguides. This choke is a narrow channel, a quarter-wave deep,which prevents waves from travelling from the surface of oneguide to the next. Care should be taken to prevent the trans-mission line formed by the channel from resonating and thusbecoming the source of unwanted radiation..

In the design of a two-dimensional array, the problem is toarrange that the linear arrays load the transverse feed-guide so asto give the desired transverse amplitude and phase distributionsand a suitable input impedance for the array as a whole. Toachieve this end, the coupling of guides by means of slots offersa wide variety of circuit arrangement. The spacing of the lineararrays, if transversely-polarized radiation is used, is practicallyfixed because of the width of the guide, but longitudinally-polarized arrays may be spaced more closely. If a feed-guide ofspecially chosen width is used, or if the guide may be turned soas not to be perpendicular to the linear radiators, the spacing ofthe loads on the transverse feed-guide may be made any desiredfigures resonant or not.

There are two basic types of coupling of shunt-series type thatpermit the choice of impedance transfer. For slots cut in thebroad face the couplings have already been explained in Fig. 28(o),while for slots cut in the narrow face of the feed guide, the ar-rangement is depicted in Fig. 34. With both types of coupling,the system of wave guides forming the whole aerial can easilybe rigidly bound together.

Inclined series-coupling slots in the broad face of the feedguide have been used to permit the turning of the componentlinear radiators of a longitudinally-polarized cosecant aerial.The required cut-off in the vertical radiation pattern was obtainedby mounting the two-dimensional array below (or above) aplane sheet of metal projecting about 10A in front of the array.By turning the linear arrays about their longitudinal axes somefreedom to adjust the shape of the pattern, was obtained, forthe degree of coupling to the feed-guide could be varied. Anantenna of this type was constructed at McGill University byGuptill in 1944.

(23) MICROWAVE YAGI AERIALIn discussing longitudinally-polarized arrays reference has been

made to the excitation of resonant slots cut in the narrow facewhen waves travel on the outside of the guide. If the slot is

20

/

to 03

1X

h

n%y

Elevation

rI

Plan Width of all slots = l-60mm

Fig. 58.—Microwave Yagi slot aerial (X-band).

Page 31: Resonant Slots Watson

WATSON: RESONANT SLOTS 777

symmetric; ,ily cut in the edge and with its axis perpendicular tothe axis of the guide, it is not excited by the dominant wave inthe guide. Tt may therefore be used as a parasitic radiator onthe outside of the guide excited by waves from a single inclinedslot which is coupled to the guide, and from other parasitic slotsin the same guide face.

An example of this type of aerial was constructed as follows.An array of slots was cut in each of the narrow faces of a pieceof X-band wave guide. Each array consisted of one 20° inclinedslot coupled to the guide with a reflecting plunger A /4 behindit inside the guide, a single parasitic-reflector slot and 15 director-parasitic slots at 0- 31A spacing. The two arrays were excited inanti-phase and their fields allowed to join in front at the edge ofthe wedge in which the guide carrying the slots was terminated.The details are shown in Fig. 58. The radiation is polarizedparallel to the broad face of the guide.

The microwave Yagi array has all the advantages of the wireYagi arrays for longer waves. These advantages are the smallcross-section presented to the direction of the main lobe, whichis approximately axially symmetric, and the extreme portabilityand simplicity of the array. In addition it is very easily made.Its directivity is limited, however, by the fact that it is an end-firearray.

(24) REFERENCES(1) WATSON, W. H.: "Wave Impedances and the Effective Cross-

Sections of Antennas," Transactions of the Royal Societyof Canada, Ottawa, Section 3, 1945, p. 33.

(2) WATSON, W. H.: "Matrix Methods in Transmission Lineand Impedance Calculation," Journal I.E.E., 1946, 93,Part IIIA, p. 737.

ACKNOWLEDGMENTThe investigations were carried out during 1942-4 in a special

laboratory of the Macdonald Physics Building, McGill Universitywhich assisted the work in every way possible. The majorfinancial support came from the National Research Council ofCanada and the facilities of the Field Station of its Radio Branchwere used in field measurements.

The fundamental work on shunt- and series-coupled slotradiators and guide couplings was reported by Mr. E. W.Guptill and the writer. Owing to the impossibility of writingseparate accounts of the original war-time reports in time for theRadiolocation Convention, the present consolidated report ofthe work at McGill University has been given. The author hasmuch pleasure in acknowledging the collaboration of the othermembers of the laboratory.