the design of a wave-guide-fed array of slots to give a specified radiation pattern

THE DESIGN OF A WAVE-GUIDE-FED ARRAY OF SLOTS TO GIVE A SPECIFIEDRADIATION PATTERN*

By A. L. CULLEN,f B.Sc, Graduate, and F. K. GOWARD, B.A.J

{The paper was first received 8th March, and in revised form 27 th May, 1946.)

SUMMARYIt is well known that the distribution of amplitude and phase which

is necessary across an aerial aperture to obtain a specified radiationpattern can be calculated by using the Fourier transformation. Thispaper describes a method by which any specified aperture distributioncan be obtained from a wave-guide-fed array of slots. Thus theproblem of obtaining a specified radiation pattern is solved.

Experimental results are given to confirm the theory.

(1) INTRODUCTIONWave-guide-fed arrays of slots have been used extensively by

W. H. Watson1 and others2 to produce narrow beams of radia-tion. For many applications, however, it is necessary to have aradiation pattern which is a specified function of angle, either inbearing or in elevation.

The general problem of designing an aerial system to producea specified polar diagram was first considered by I. Wolff.3 Hepointed out that a linear array of dipoles had: a radiation patternwhich could be expressed as a Fourier series in which the co-efficients of the sines and cosines were proportional to the cur-ents in the dipoles. It followed from this observation that itwas only necessary to Fourier analyse a given radiation patternin order to calculate the currents in the dipoles necessary to pro-duce an approximation to it. The accuracy of the approximationwas limited only by the number of dipoles and the total apertureavailable. Each dipole added, corresponded to an additionalterm in the Fourier series. The method has been discussed anddeveloped considerably by S. A. Schelkunoff.*

When a large number of dipoles is used it is convenient to usean equivalent continuous distribution of current. This is calledfor convenience the "aperture distribution." Just as the radia-tion pattern of a linear array of dipoles can be expressed as aFourier series, so the radiation pattern of a continuous aperturedistribution can be expressed as a Fourier integral. It has beenpointed out independently by several authors following Stratton5

that the Fourier transform formulae enable the appropriateaperture distribution to be calculated if the required radiationpattern is specified. A further method of evaluating the aperturedistribution in cases where integration in closed form is notpossible has been devised by Woodward.6

This paper describes how the slotted linear-array techniquedeveloped by Watson can be combined with the Fourier analysismethods described. Calculations are made of the required offsetand spacing of longitudinal slots in the broad face of a waveguide which will produce a specified pattern.

(2) CALCULATION OF REQUIRED APERTURE DISTRIBUTIONTo produce a specified radiation pattern exactly would require

an infinite aperture. Given a finite aperture, we can makevarious approximations to the Fourier transform of which weshall use the following.

1. We can use a portion of the infinite aperture dictated byxvauiu isecuon paper.

t University College, London, formerly at the Royal Aircraft Establishment.X Telecommunications Research Establishment.

the Fourier transform. The finite aperture must include thegreater part of the energy in the infinite aperture. This methodis the one described by Stratton5 and is treated in detail in theAppendix (Section 12.1).

2. A finite aperture can be assumed initially and the radiationpattern made to coincide with that desired at a finite number ofangles. This is the method described by Woodward6 and canbe approached quite independently of the Fourier transform.

The difference between the two methods can best be seen byapplying both to the same problem. Let us assume that thespecified radiation pattern is that shown in Fig, 1, and the

10

0-0Sin 0

Fig. 1.—Specified radiation pattern.

aperture available is 24A. The radiation pattern obtained by theFourier transform method is shown in Fig. 2A. The patternobtained by Woodward's method is shown for a value at 0 = 0of both i and 1 (Fig. 2B). The close resemblance between

--o-i

1// *̂̂

Fig.

10°Angle 0

2A.—Fourier transform approximationpattern.

20°

to specified radiation

l-2

-c 10f 08'£ 0-61 0-4t 02"I 0

Angle 010° 20

Fig. 2B.—Woodward approximations to specified radiation patterns.

curves a and b is apparent. The Fourier method gives approxi-mately the value $[E(0 + 0) + E{0 — 0)] at a discontinuity atE0 provided the aperture is large compared with a wavelength.The Woodward method with the value 0-5 at 0 = 0 gives a verysimilar approximation to the pattern, but other quite differentappiroximations such as curve c can be made by the Woodwardmethod.

[683 ]

684 CULLEN AND GOWARD: THE DESIGN OF A WAVE-GUIDE-FED ARRAY OF

The essential difference between the two methods should benoticed. The Fourier transform method finds the best approxi-mation, in the sense of least mean-square deviation from therequired pattern, which can be achieved with a given aperture.Woodward's method ensures that the approximate diagram isequal to the ideal diagram at certain specified points but saysnothing about the curve between these points. It is possible,however, as Woodward has shown, to control the deviations onthe sloping tail at the expense of the initial rise or vice versa.The control can be exercised before the aperture distribution iscalculated. This flexibility is a great advantage over the Fouriertransform method.

Which method is used for a particular problem is partly amatter of mathematical convenience and partly depends on thetype of approximation to the specified diagram which is required.Mathematically, the Fourier method lends itself particularly toanalytical treatment of simple diagrams, the Woodward methodto graphical treatment of more complicated patterns.

With either method the aperture distribution is complex andcan be written

A(z) = [/(*)]* exp [#(*)] . . . - . (1)where/(z) is proportional to the power radiated per unit lengthand 0(z)is4he phase distribution, z being the distance along theaperture in wavelengths, measured from the centre.

(3) CALCULATION OF SLOT SPACINGTo make a simple calculation of the slot spacing required to

produce the phase distribution, ifj(z), it is necessary that thephase shall be controlled independently of amplitude. This willonly hold provided

1. The guide is matched down its whole length.2. There are no mutual interactions between slots.The first condition requires that the conductance of the slots,

should not be very high. This is arranged for slots near theload end of the array by dissipating a suitable amount of powerin a matched load. The conductance of other slots will be fixedby the required amplitude distribution. The conductance of theslots then fixes how closely we can bring the slots to a halfwavelength (in the guide) apart. The necessary condition is that

g2cosec2j8rf<l (2)

where j3 = 2TTIX8, d is the distance apart in cm, and gs is theslot conductance. See Section 12.2, equation (45A).

The second condition requires the use of longitudinal slots inthe broad face (Fig. 3). The slots shown will radiate alternately

Fig. 3.—Longitudinal slots in the broad face of a wave guide.

in phase and out-of-phase with the electric field in the guide attheir centre. The conductance of this type of slot is given bythe formula,

gs = A sin2 (vxja) (3)where .v is the off-set, a the internal broad dimension of the

guide and A a constant whose value depends on the guide dimen-sions and the wavelength. Its value is usually of the order ofunity and can be determined by experiment. The resonant slotlength would be measured at the same time.

If the above conditions are fulfilled, the phase of the voltagedown the guide relative to the phase at a fixed point O will havethe form shown by the full sloping lines OA, BC of Fig. 4. The

Direction of power flow down guide

^Phase relative to(0+180°)

Required phase f (z)Phase relative to 0

-360'

Fig. 4.—Construction for finding slot positions.O Shows position of slot off-set on one side of centre line.• Shows positions of slot off-set on the other side of centre line.

dotted sloping lines O'A', etc., give the phase + 1803. The re-quired phase, «/r(z), is now plotted on the same graph and theintersection of this line with the sloping lines gives the slotpositions. Intersections with the full* lines give the position ofslots offset from the centre-line in one direction; intersection withthe dotted lines correspond to slots offset in the other direction.

In order to preserve the matching down the guide it is usuallynecessary to add a linear phase distribution to the required phasedistribution. The effect of this is to swing the polar diagramthrough a certain angle leaving its shape unchanged whenplotted against sin 0. The linear phase distribution added mustbe such as to make the slots at no point sufficiently close to A /2to violate the match condition of equation (2). The slots must notbe allowed to become so close together that they run into oneanother, nor so far apart that second-order beams are trouble-some. Better match may be obtained in some cases by reversingthe direction of the travelling wave in the guide. This changesthe sign of the slope of lines OA, BC, etc., in Fig. 4.

It will usually be possible by one of the devices mentioned toavoid mismatch. If, however, mismatch is unavoidable, cor-rection can be made without much difficulty. The method isoutlined in Section 12.3. Briefly the phase in a mismatchedguide does not vary linearly with distance but follows a cyclicalcurve, which must therefore be plotted to replace the lines OA,BC, etc., of Fig. 4. Correction for mismatch will normally benecessary only near the centre slots which take a large amountof power, and near the load when it is desired to increase theefficiency by absorbing only a small fraction of power in the load.

(4) CALCULATION OF SLOT CONDUCTANCEThe required distribution of power radiated per unit length is

f(z) in equation (1). With the slot spacing determined, the indi-vidual slot conductances can be found either graphically ornumerically depending upon whether f(z) is integrable in closedform or not.

The analytical method relies on the fact that an array of dis-crete radiators can be replaced, for the purpose of making trans-mission-line calculations, by a continuous leakage from the array.This leakage conductance is defined as the "equivalent distributedconductance" g(z). The equivalent distributed conductance re-quired to produce the power distribution f(z) is given by

-\f(z)dz~\Az)dzlJ—s

SLOTS TO GIVE A SPECIFIED RADIATION PATTERN 685

where 2s is the aperture width in wavelengths and 77 is the effi-ciency of the array. The conductance of each slot is obtainedfrom this equation by the equation:

ga = d{z)g(z) (5)

where ga is the normalized conductance of a slot placed at zand d(z) is the distance in wavelengths from an adjacent slot.It is not important on which side of the slot d(z) is measuredsince d(z) does not vary rapidly with z. A full statement of theanalytical method and the theory supporting it will be found inSection 12.2. Equations (4) and (5) correspond to equations(68) and (45B) of that Appendix.

The graphical method works out the conductances slot by slot.The curve of the power to be radiated, f(z), is plotted against zand the position of the slots marked. From the distance apartof the slots the relative power, fr radiated by the rth slot is foundand the ordinate drawn (Fig. 5) using the equation:

fr=f(z)d(z) (6)

where/(z) and d(z) are measured at the value of z correspondingto the rth slot. The slots are most conveniently numbered from

. power flow

Fig. 5.—Construction for finding slot conductance.

the load end. The total power radiated, PR, is now found bysumming the ordinates

The total input power is PR[r) where 77 is the efficiency of thearray and the power absorbed in the load is PL where

PL = PR^-l)h (8)

If the first slot, counting from the load, is to radiate power/j itsconductance gt must be

SI-AIPL (9)

For the second slot

S2-fiKPL+fi) (10)

and so on down the array until for the nth slot:

w - 1

It will be noticed that equation (9) implies a relationshipbetween the efficiency and the allowable conductance of the firstslot. Normally the greater we can make this conductance the

greater can we make the efficiency. For long arrays an efficiencyof 95 % is usually possible without introducing mismatch near theload. If the band-width required is not too great, it may beimproved to practically 100% by using a short-circuiting plateas described in Section 12.3. However, loss in the guide wallsmay reduce the efficiency in a long array, by a small percentage.

The conductance of each slot having been determined it isnow only necessary to determine the offset. This is done fromequation (3) of Section 3.

(5) CENTRE-FEEDING AGALNST END-FEEDINGAn array will normally be fed from one end and the necessary

waste power absorbed in a matched load at the other end. Withsuch an arrangement the tolerances on the offset of the slotswhich are fed first may be unduly tight.

In such cases, it is possible (o feed the array from the centreand have a matched load at each end. The tolerances on theend slots are now eased. Such a method of feeding does notaffect either the efficiency of the array nor the high conductancerequired in the centre slots. Attenuation in the guide walls willbe less in centre-feeding since most of the power is removed aftertravelling only a short distance down the guides.

The effect of frequency will be different in the two cases. Withend-feeding the radiation pattern is swung in angle with littlechange of shape. With centre-feeding the effect is a broadeningof the pattern without angular movement. This is because thetwo halves of the array swing their contributions in oppositedirections as the frequency changes. The change of impedancewith frequency will in general be greater for a centre-fed array,because of the high conductance near the feed point.

Each method has its particular attributes and either methodmay be preferable according to circumstances.

(6) POWER HANDLINGThe form of the function for power radiated normally requires

that a large amount of power be removed from the centre slotsof an array. This may be a limitation in some cases and callfor widening or special design of the centre slots. Watson hasmeasured the power-handling capacity of a £-in wide slot at awavelength of about 10 cm for 1-microsec pulses at 150 kW peakpower.

(7) APPLICATIONSThe linear arrays described are normally used to feed parabolic

cylinders in order to obtain directivity in a plane at right anglesto the array axis. The shaping of the radiation pattern occursin a plane through the array axis. The method is therefore ap-plicable only to apertures where the shaping is produced by thelonger dimension. Square apertures have been used with successbut in cases where beaming is to be effected by the long dimensionand shaping by the short dimension another method must beused. Such a method has been suggested by L. J. Chu; in thismethod a line source is used to feed a shaped reflector.

A special advantage of the present method is that the radiationpattern may be altered by changing, electrically or mechanically,the feeds to the reflector.

(8) EXPERIMENTAL RESULTSThe performance of two specific arrays is described.(a) An end-fed array 24A long designed to produce a triangular

radiation pattern. This array was designed by the methoddescribed in Sections 2 and 4.

(b) A centre-fed array 55A long designed to produce a "solid-cover" pattern, i.e. a pattern which illuminates equally all pointsalong the edges of a rectangle of which the aerial is a corner(see Fig. 6). This array was designed by the method describedin Sections 2.1.2 and 4.3.


1-25

Horizontal

(a) Polar diagram.

,Sec0 curve

JB>Cosec 9 curve

(b) Cartesian diagram.Fig. 6.—Solid cover pattern.

The end-fed array was designed to work at 9.1 cm giving atriangular radiation pattern. The required aperture distributionis shown in Fig. 7 whilst Fig. 8 gives the experimental radiation

90*

-100*. — - " '

z)V

10

0-5

00-12 -6

z, distance along array in wavelengths.

Fig. 7.—Aperture distribution to produce triangular radiation pattern.

10

I 0-0-5

0-0Jj

I

-5° 0° 10° 20°Angle: 0

Fig. 8.—Experimental triangular radiation pattern.; Ideal pattern.* Theoretical pattern.

Experimental result.

pattern obtained. The efficiency of the array was measured overthe 8-9-9-3 cm band, and varied from 90 to 98%. This com-pares with the theoretical 90% for the mid-band wavelength.The input s.w.r. was measured and remained between 0-83 and0-90 over the same band.

The centre-fed array was designed to work at 3-20 cm. Therequired radiation pattern and aperture distribution are shown inFigs. 9 and 10 together with the experimental results. The fieldstrength was measured at angles up to 90°, but surfers only asmall reduction between 40 and 90°. The radiation patternobtained when the array is placed in a parabolic cylinder is shownin Fig. 11, together with the effect of a frequency change. Theparabola was cut in its focal plane to give a square section ofside equal to the array length.

(9) CONCLUSIONThe technique described is seen to give reliable and useful

results. The complete design may be accomplished theoreticallyand an accurate and rapid estimate made of the degree of ap-proximation to a specified polar diagram which will be obtained.

1-0

fo-75

210-25

0 0

V

-5* 0* 10° 20°Angle :tf

30*

Fig. 9.—Experimental solid-cover diagram.

Theoretical pattern at X •= 3 • 2 cm.Experimental results at X 3 -2 cm.

100*

50°

Phase<P(z)

°V

-100*

if1

-22-5 -11-25 22-5

•00

0-75

Amplitude

0-50 /

0-25 / .

-22-5 -11-25 1105

Fig. 10.—Experimental aperture distribution.Theoretical, at X = 3 -2 cm.Practical, at X -•= 3 -2cm.

z — distance along array in wavelengths.

1-2S

1-0

£075

0-0

\V, ....

10°Angle:

20° 30° 403

Fig. 11.—Effect of parabolic cylinder on solid-cover pattern overfrequency band.

x = 3-2 cm. /. = 3 -22 cm.


There is no recourse to the trial and error adjustments which arefrequently necessary in the shaped-reflector method.

(10) REFERENCES(1) WATSON, W. H.: "Resonant Slots," Journal I.E.E., 1946,

93, Part IIIA, p. 747.(2) FRY, D. W.: "Slotted Linear Arrays," ibid., p. 43.(3) WOLFF, I.: "Determination of the Radiating System which

will Produce a Specified Directional Characteristic,"Proceedings of the Institute of Radio Engineers, 1937, 25,p. 630.

(4) SCHELKUNOFF, S. A.: "A Mathematical Theory of LinearArrays," Bell System Technical Journal, New York, 1943,22, p. 80.

(5) STRATTON, J. A.: "Electro-Magnetic Theory," Section 6.7(McGraw-Hill Book Co., Inc., 1941).

(6) WOODWARD, P. M.: "A Theory of Approximating SpecifiedPolar Diagrams with Aerial Systems of given ApertureWidth." Paper submitted for publication in the Journal.

(7) TITCHMARSH, A. C : "Introduction to the Theory of FourierIntegrals" (Oxford: Clarendon Press, 1937).

(8) SLATER, J. C : "Microwave Transmission" (New York:McGraw-Hill, 1942). Equation (2.18).

(9) JACKSON, W., and HUXLEY, L. G. H.: "The Solution of

Transmission-Line Problems by use of the Circle Diagramof Impedance," Journal I.E.E., 1944, 91, Part III, p. 105.

(11) ACKNOWLEDGMENTSAcknowledgment is made for assistance and encouragement

from our colleagues at R.A.E., Farnborough, and T.R.E., GreatMalvern. The copyright of the illustrations in this paper isvested in the Crown.

(12) APPENDIX(12.1) The Fourier Transform Method

Consider an aperture distribution A(y) where y is the distancemeasured from a convenient origin to which phase is referred

Time factor = tjat

Advance in phase of contribution from element P ata distant point in 6-direction is:—

(2Tt/X)>> sin 6 = 2rcz sin 6Therefore contribution from element at P is:—

Time factor - e Jat

Advance in phase of contribution within angle SOarriving at P from a distant point is:—

2rrc sin 6Therefore contribution from elementary angle SO is

For small angles this is

Fig. 12.—Fourier transform regarded as a wave retracing its path.

(see Fig. 12). Then the field produced by this aperture distri-bution at a great distance can be written, apart from a constantfactor, as

where k — 2TT/A. This formula can be expressed in a more con-venient form if distance across the aperture is measured in termsof wavelength by the substitution z = v/A. Then equation (12)becomes, dropping the constant factor of 1/A,

J.+co

A(z) exp QITTZ sin 0)dz

—00

The Fourier transform theorem7 enables us to write

J.+oo

£"(sin 0) exp ( - JITTZ sin 0)tf(sin 0)

(13)

(14)

Thus if the radiation pattern is expressed as a function of sin 0,i.e. as .E(sin 0) evaluation of equation (14) gives the requiredaperture distribution.

In general the required aperture will extend from - c o to+ co, so that only a part of it can be used. The larger theaperture the more accurately will the required radiation patternbe reproduced. Let £'(sin0O be the approximation to thedesired radiation pattern produced by that part of the aperturedistribution given by eqn. (14) for which — s < z < + s. Thiscorresponds to the use of an aerial with an aperture of 2swavelengths. Then

= \AJ—s

E'(sin 00 = \A(z) exp (J2ITZ sin (15)

By substituting the result of eqn. (14) in eqn. (15), the approxi-mate pattern can be expressed in terms of the desired pattern thus:

f r J . /vSjSjpm ft' — sin0)1

_ s in

£(sin 0) = A(y) exp (jky sin B)dy . (12)

apart from a constant factor Is.When s is very large the (sin u)/u term is practically zero for all

values of sin 0 except sin 0', when it has the value unity. Thusit is clear that E' can be made as good an approximation to E aswe choose if s is chosen sufficiently large.

The effect of a linear phase distribution added to the aperturedistribution A(z), is worth studying at this point. Let us sup-pose that the shift in phase is ip radians/wavelength. Theneqn. (15) is modified to

r+I

E'(sin 00 = A(z) exp [J(2TT sin 0' + ifi)z]dz . (17)J— s

It is clear from this equation that if a new variable sin 0" is used,where

sin 0" = sin 0' + I/,J2TT . . . . (18)

then equations (15) and (17) are formally identical. Thus theonly change is a shift of the origin of sin 0' from 0 to — 0/2TT,

or a shift of the whole radiation pattern of tfjl2v, its shape beingunchanged when plotted on a sin 0 base. For small angles theshape is unchanged when plotted against 0.

A better understanding of the significance of expressions (12)to (16) is given by two physical pictures which will now bedescribed.

First consider eqn. (12) as building up field at a distant point bysumming the contributions from all elements dy of the aperture.The contribution from the typical element P (see Fig. 12) isA(y)dy exp (jX-1 liry sin 0); summing all contributions leads tothe integral of eqn. (12) or eqn. (13) with the change of variablez = y/X.

Now the wave equation y2£ = c-27)2£p)t2 is not affected bya change in the sign of t, so that if it is satisfied by a function


£(.v, y, t) exp (Juit) it is also satisfied by g(x, y, t) exp (— joit).This last wave can be interpreted as the first wave retracing itspath. We can therefore consider equation (14) as finding thefield produced at the aperture by the specified distant field, atypical contribution being £"(sin 0)</(sin 0) exp (— JITTZ sin 0).The change in sign of the index of the exponential is necessarybecause of the change in sign of t.

The second picture of the Fourier transform is due to Wood-ward. Consider an infinite plane wave passing the yz-plane, asshown in Fig. 12, the wave-front making an angle 0 with they-axis. This plane wave represents the sole contribution to thedistant field in the 0 direction and must therefore have theamplitude £"(sin 0). Its propagation factors are k cos 0 andk sin 0 in the x and y directions respectively, so the plane waveis represented by

£-£(sin0)exp[jA:(;ccos0 + ;>sin0)] . . (19)The field across the aperture due to this single plane wave isgiven by putting x = 0.

A(y) = £(sin 0) exp (jky sin 0) . . . (20)

The aperture distribution is given by summing all the waves atthe aperture plane, thus:

/-length Iff

A{y) =-- £(sin 0) exp (jky sin 0)</(sin 0) (21)

The effect of restricting the aperture is brought out very clearlyby this argument. Thus consider the plane OYZ in Fig. 12 tobe an infinite screen open only from y = — S to + S. Let aplane wave be incident on this aperture at an angle 0. Theradiation pattern at a great distance to the right of the screen isgiven by

E'(sin 9Q - £(sin ftA:£(sin d - sin

(22)

Summing all contributions leads directly to the integral (16).The physical significance of the summation will be realized ifterms of the type (sin u)fu are thought of as radiation patterns ofa uniformly illuminated aperture with a linear phase shift.Clearly the larger the aperture the narrower will be the beam ofthe fundamental radiation patterns, and the more accurately canthe fine detail of the specified pattern be reproduced.

(12.2) The Equivalent Transmission-Line Problem(12.2.1) Uniform Line.

Watson has shown elsewhere1 that slots in a wave guide, ofthe type shown in Fig. 3, behave like shunt conductances on aloss-free transmission line if terminal conditions only arerequired.

It will now be shown that if the slots are all separated by adistance d and have the same conductance Gs, an equivalentuniform "lossy" line can be found by a method analogous toCampbell's method for a coil-loaded telephone line.

Consider the relations between the input and output currentsand voltages of a uniform transmission line of characteristicimpedance Zo and propagation constant P. Referring toFig. 13 for the notation, we have

VR = Vs cosh PI - fsZ0 sinh PI

^- VR cosh PI JRZQ sinh Pi

Now consider a symmetrical II network.with the notation of Fig. 14 that

K =

• • (23)

• • (24)

It is easy to show,

. . . . (25)

. . . . (26)

s I Zo,P= Propagation constant ; \x'O • O '

Fig. 13.—Transmission-line notation.

Fig. 14.—Symmetrical II-network notation.

By equating coefficients of VR, Vs, IR and Is, the line and thenetwork can be made equivalent as regards terminal conditions;this gives

cosh PI = 1 + ZX\Z2 (27)

ZosinhP/ = Z! (28)

From equations (27) and (28) we can readily deduce that theseries and shunt arms of the equivalent II network are given by

Z1 = Zo sinh PI (29)

Z2 = Zo coth \Pl (30)

In the case of a loss-free line, Zo is real, and P = + y'jS, and ispurely imaginary.

For this case:Z1 = +7Zosin)3/ (31)

Z 2 = - y 7 0 cot tf/ (32)The section of guide, length d, between two slots can thus bereplaced by a II-network of pure reactances. The effect of theslots can be included by adding half the slot conductance toeach shunt arm of the II-network. Thus the admittance of themodified shunt arm is:—

where Yo = 1/ZO.We can now find a lossy line which is equivalent to the modified

II-network. Let its propagation constant be P' = a' + y0'.By eqn. (27)

cosh (P'd) = 1 + Z{Y'2= cos 0rf + yiG,ZosiQ0</ . . (34)

It is convenient to normalize the shunt conductance of the slotsto the line admittance, writing—

g,= G^ (35)

and thus, expanding cosh (P'd), we get

cosh a.'d cos j3W — cos fid . . . . (36)

sinh add sin j3W = \gs sin fid . . . (37)

Eliminating a! between equations (36) and (37) we get—

. . . (38)Note that if 0</ = (2n + 1)TT/2, 0 ' - 0.

If the condition (gJ2)2 < 1 is fulfilled, equation (38) simplifiesto

(39)


From eqn. (37) we have

sinhaW - ±g, sin jfc//sin jS'rf . . . (40)

so that a' can be calculated from equations (39) and (40).However, we are more interested in the value of the equivalent

uniformly distributed conductance per unit length, G, whichwould give the same value of a'. Let us again normalize to theline admittance, writing:

g=GZ0 (41)

Now if gXg < 1 the normal transmission line formulae8 givedirectly to within 0-3%:

*' = \g (42)

(43)

(44)

(45A)

and if gd < \ we have, from equations (40) and (42),

g d = g s sin fid/sin frd . . .

and thus from eqn. (39):

sd = gs2i[ 1 + (1 + g} cosec2 j&O*]-* .

within 1 /,;. It is clear from eqn. (44) that if

#2 cosec2 £</< 1we can write

gd=gs (45B)a formula which is very plausible on purely physical arguments.The equivalent characteristic admittance can be found fromequations (28) and (31).

Thus Y'0IY0 = sirihP'dlJ sin fid . . . . ( 4 6 )

Expanding sinh P'd, and using the same approximations asbefore, we find

(47)! [ l v (1 + g} cosec2

- j\gs cot 041 + (1 + g} cosec2 j8^)i]-*

The practical significance of these results is best seen from thecurves of Figs. 15 and 16. It will be seen from Fig. 15 that for

1-2

gd 1

a

10 (H 0-2 0-3 0-4 0-5

oJ/Ay Slot separation in guide wavelengths

Fig. 25.—Limitations of equation, gd = gsCurve c/: for slot conductance g t= 0-1.Curve b: for slot conductance gt = 0-2.

slot conductances as high as 0-2, equation (45) is very accurateunless d is within 10 % of 0 • S\g. With the same limitation on d,the variation of phase constant and characteristic admittance(Fig. 16) is negligible.

(12.2.2) Non-Uniform Line.Now suppose that the slot conductances are not all the same,

but that the difference in conductance of any two adjacent slotsis small compared with the conductance of either. Moreover,suppose that d is no longer constant, but that the conditiong} cosec2 fid < 1 is everywhere fulfilled. It is then reasonableto assume that the slotted guide can be represented by anequivalent non-uniform transmission line, whose parameters arecalculated at each point as though all slots have the same con-ductance and separation as the slots nearest the point considered.

•a o

JJ

a 0

0 01 0-2 0-3 0-4 0-5u/Ay Slot separation in guide wavelengths

Fig. 16.—Additional phase shift per section <5i, and modified charac-teristic admittance, YQ = G'Q + jB'o caused by purely conductiveshunts. Conductances = 0-1.

We have already shown that the initial characteristic admittanceand phase constant are only slightly changed by the presence ofthe shunt conductances, and the addition of a distributed shuntconductance to the initial primary constants of the line is asufficiently good approximation to the equivalent line for ourpurpose.

We have therefore to solve the problem of a transmission linewith a non-uniform distribution of leakage conductance. Theline equations are

dV/dy=-ju)LI (48)d I / d y = - [ G ( y ) + j a ) C ] V . . . . ( 4 9 )

where L, G(y) and C are the distributed inductance, leakage,and capacitance per unit length. Eliminating / from these twoequations, we get—•

d2V\dy2 = ja)L[G(y) + jcoC] VNow let

Zo = (LIC)±P - 2n/\ = co(LC)i

Thus d2 V/dy2 - jpzQG(y)V + jS 2 V = 0The power dissipated per unit length of line,

p(y) = V2G(y) . .The problem can be stated as follows; given P(y), rind G(y)

such that equations (52) and (53) are satisfied.Although an exact solution of this problem is difficult, it is

possible to get an exact solution of a closely related problem.Instead of a distributed conductance let us take a distributedadmittance Y(y) per unit length. Then eqn. (52) becomes

d2V/dy2 - jPZ0Y(y)V + pV = 0 . . (54)

We can now put:. . . . (55)

(50)

(51)

(52)

(53)

and specify that U is real. Then, from equations (54) and (55),

(56)

Write

Then

and

Y°(jU ~dyr)

(57)

6 9 0 CULLEN AND GOWARD: THE DESIGN OF A WAVE-GUIDE-FED ARRAY OFFrom equations (48) and (55),

Therefore / = YQ(u + J-*g) exp ( - ffiy)

The modified characteristic admittance Y'o is:—

pu ~dj)

(58)

• • (59)

of the array, expressed as the ratio of power radiated to totapower input, and denoted by -q, so that

(66)

where 25 is the total aperture of the array (i.e. the total length ofthe equivalent line). If we put:

and we find that the real part of the modified characteristic ad-mittance is equal to the original characteristic admittance, whichwas a pure conductance, but that a susceptive component hasbeen added. This susceptive component can be expressed interms of G(y) by using eqn. (57);

, (67)s = SIX J

where Po is a constant, the formulae (65) can be expressed in amore convenient form:

f(z)

- O/2j3)GO0 (60)L\f(z)dz

It is interesting to note that this formula is obtained approxi-mately by the non-rigorous method of applying the usual formulafor characteristic admittance in the case of uniform distributionof conductance, thus:

\f(z)dz

g(z) dj*\

(68)

. • (61)

Thus the characteristic admittance at any point is very nearlythe value it would have if the leakage were uniformly distributedand had the value at the point considered.

Now consider the power flowing past any cross-section of theline. For simplicity we assume that r 0 , the original characteristicadmittance of the line, is unity, so that

W(y) = C/2 (62)

There is an integral relationship between W(y), the power passing

j /fe)Note that the function f(z) can have any arbitrary constantmultiplier, so that it can always be written in its simplest form

The design procedure is to find/(z) from A(y), the amplitudedistribution, and then calculate g(z) and b(z) from eqn. (68). Pro-vided that b(z) is very small compared with g(z), as will usuallybe the case, g(z) alone will be a sufficiently good approximationto the desired admittance distribution.

An alternative method of deriving g(z) will now be describedWe have

P(y) - U*g(y) ( 6 9)which with eqn. (62)* gives

P(y) = W(y)g(y)From eqn. (63) we then get directly

(71)

1 >.

•

wty)

+

y

Fig. 17.—Integral relationship between W(y) and P(y).

any cross-section of the line, and P(y) the power dissipated perunit length (Fig. 17). If the input power is unity:

(63)

So that u =-s

(64)

where S is the half-aperture. By substituting this expressioneqn. (57) we get: in

siy)=P{y)j \\~\p{y)dy~\

(65)

where g(y) and b(y) are the normalized values of GO) and B(y)It is convenient in practice to work in terms of the efficienc

)terms of the efficiency

in agreement with equation (65).It is interesting that this simple method of approach leads to

the same results for the shunt conductance distribution.

(12.3) Effect of MismatchIn the main report the design of arrays which are matched

along their whole length has been described. If mismatch ispresent design can still be effected quite simply by a graphicalmethod.

«e T ^ e u m e t h ° d i s b a s e d o n t h e p o l a r f o r m o f c i r c l e diagram orSmith chart."* We use the admittance diagram since we are

considering shunt slots.On this representation if we move down a length of mismatched

line from an admittance Ax to A2> as shown in Fig. 18, the phase-change is not \S as it would be with the same length of matchedline but %Q + <f>. This simple relation means that we can re-place the straight phase lines OA, BC, etc., of Fig. 4, Section 3,by appropriate curves, the curve shape being calculated from anadmittance plot on the Smith chart. The intersection of thesephase curves with the required phase curve now give the slotpositions.

The amount of power radiated by a slot of a certain con-ductance is also affected by mismatch. Since the slots are shuntconductances we have the following relationship:

(Voltage across Slot)2

= (Power Radiated) (Slot Conductance)= (Power Transmitted) (Conductance at load side of slot)

^ d ° n a b a s i s °f,ne8ligible change of guide admittance,e more generally true.


Fig. 18.—Phase change down mismatched line.Phase change passing from admittance, A-^ to Ao =• \Q -J- q>.

Q2

1-43-95 i•33

M--95J 1-3+1-Oj0-3

1-0+1-Oj

As an example let us assume that it is desired to produce apower distribution along the array, fly), and a phase distributiontfi(y) as shown in Fig. 19. The wavelength in the guide, Xg, istaken as 10 cm. The units of/O) are arbitrary. Further, let usassume that the load into which we work the array is not matchedbut has an admittance, Lv = 1 0 + 1 Q/. We now require tocalculate the position and conductances of the slots. Slot 1,counting from the load, can be fixed arbitrarily in position andconductance. Let us decide on conductance = 0-3 and positiony = 0. This gives the admittance looking into the slot and loadasGt = 1-3 + 10/.

The position of slot 2 can now be found. If there were nomismatch in the guide [see Section 3], the phase relative to0 would follow the straight line P ^ and the (phase + 180°) theline BC in Fig. 19. This would give the second slot at y = 7 • 4 cmthe point Pj. In fact, however, the phase follows the cycliccurve Pxabc and the (phase + 180°) curve Bd. This curve isobtained by plotting the admittance relations on a Smith chartas shown in Fig. 20 and measuring the angle (f>12 as we move

y distance along a r ray in cm

Fig. 19.—Mismatch correction example.

Fig. 20.—Mismatch correction example.

down the line, <f>l2 being added to the phase change expectedassuming matched conditions. The position of the slot is theintersection of the cyclic curve with the iff(y) curve, i.e. point P2.The separation of slots 1 and 2 is therefore 6-55 cm.

The power relations at slot 1 can now be fixed. Since thereis no slot to the right of slot 1 we must take as the slot separationat slot 1 the value 6 • 55 cm. The power required to be radiatedper centimetre at slot 1 is 0-6, from the graph for fly), so thepower to be radiated by the slot is 0-6 x 6-55 = 3-93. Thetransmitted power is given by:

transmitted power =(radiatedpower) (conductance at point Lj)/(slot conductance)

therefore transmitted power = 3-93 x 1-0./0-3 = 13-1

Thus the incident power is 13-1 + 3-93 = 17-03.We can now consider the power relations at slot 2.

The value of/O) at y = 6-55 cm is 0-78 power units/cm. Theslot separation at slot 2 is taken as that to the right of slot 2and is 6-55 cm.

692 CULLEN AND COWARD: DESIGN OF WAVE-GUIDE-FED ARRAY OF SLOTS TO GIVE SPECIFIED RADIATION PATTERN

Therefore the power required to be radiated by slot 2 = 0-78x 6-55 = 5-11. But the conductance at the load side of slot 2is 1-10 and the power there is 17-03.

Therefore the slot conductance required = 5• 11 x 1-10/17-03- 0 - 3 3

It is now possible to plot the point G2 on the Smith chart ofFig. 20 and thus moving back down the line (i.e. on the circlethrough G2) we can plot on Fig. 19 the cyclic curve P2e bymeasuring a new angle <f>2ty The line P2D shown on Fig. 19 isthe new phase line we should get if the line between slots 2 and 3had been matched. Continuing the curves P2e and fg untilintersection with the tft(y) curve occurs, gives the next slotposition, P3, and so on.

The calculation technique described can be simplified in

practice. The full curves Pjabc, Bd, etc., of Fig. 19 need notbe plotted. It is easily possible to estimate two points on Bd,say, which are near enough to P2 to give an intersection to therequired accuracy by interpolation.

The method described is applicable not only to the high con-ductance slots which normally occur near the centre of an array,but also to the load end slots. To obtain high array efficienciesit is necessary to use high-conductance slots near the end whichwould normally give mismatch and poor phasing. Utilizing thetechnique described above, however, it is possible to terminatethe array by a short-circuiting plate, and still maintain the correctphase and amplitude relationships down the array. Judiciouschoice of the short-circuit position will make the admittancelocus plotted on the Smith chart spiral close to the (1 -+- Oy)point after a few slots only and thereafter match can be assumed.

the design of a wave-guide-fed array of slots to give a specified radiation pattern

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