RL-63-J THE CURRENTS ON STRIP A!TENN]AS by T.B.A. Senior* An ex&ct expression is obtained for the longitudinal distribution of current excited on a perfectly conducting strip by a normally incident plane wave, and computations are carried out for quarter and half wave antennas. By using the known transverse variation of the current on strips of small width, the complete surface distribution is determined, leading to an expression for the total current carried by the antenna. This is compared with the current distribution for a thin wire, but little agreement is found. Some reasons for the differences are given. 1. Introduction It is frequently assumed that the currents excited on a strip antenna have a longitudinal distribution which is similar to that for a thin wire. As a result, the variation of the current as a function of position can be closely represented by a cosine term. This appears a reasonable assumption for narrow strips when the incident field is tedge-on', but at normal incidence the analogy with the wire is not quite so obvious, suggesting that further consideration be given to this case. In the following, attention is confined to an idealized antenna consisting of a perfectly conducting infinitely thin strip of arbitrary length and width. As such, the strip can be likened to some types of indocr TV aerials. The excitation is by means of a normally incident plane wave and no account is takan of antenna connections or impedance losses. *The University of Michigan, Radiation Laboratory RL-063 = RL-063

RL-63-J In Section 2 it is shown that the longitudinal distribution of the surface current on a strip of length 2a can be deduced from the transverse distribution on a strip of width 2a and infinite length. This last can be expressed as a series of Mathieu functions, and computations are carried out for a = A/e and: = i/ correspondcling to quarter and h.alf jwve antennas. For strips of relatively small width the total current distribution can be obtained using the known variation across the width of the strips and this is done in Section 4. The resulting distribution is compared with that for a thin wire. If the length of the strip is not large compared with the ravelength, the two distributions do not agree, and cannot be brought into agreement whatever the radius of the wire. Although the differences are not necessarily significant, it is clear that for practical strip antennas of small length, the new distribution will represent a better initial approximation in any iterative scheme for finding the true current. 2. The Method An expression for the current is most easily obtained by considering the current which would be induced in an infinite strip by an incident plane mave. In terms of Cartesian coordinates x, y, z the infinite strip will be assumed to occupy the region -a<x <a, -oo<z <oo of the plane y = 0. A plane wave is incident in the direction of the negative y axis and if its magnetic vector is entirely in the z direction H = (00, e-ik) (1) -2 -

RL-63-J where the affix tit is used to denote the incident field and k = 2 ' /; is the propagation constant. M.k.s. units are employed and a time factor e Ji t i suppressed throughout. [FIG. 1] From symmetry considerations it is clear that the magnetic vector in the scattered field must also be confined to the z direction and since the induced current is determined bar the component of the total magnetic field parallel to the surface of the strip, the current vector I must be entirely in the x direction. This implies that the only current flow is across the strip. Moreover, the fact that the whole problem has two-dimensional symmetry requires the current to be the same at all points along the length of the strip and hence I can only be a function of x. Since I = n A^ H, where n is a unit vector normal to the surface, we now have I -(I, O, 0) and on the upper surface (y =+ 0) I = I (x) with I+ (x) [Hz] Y= + 0 = 1 + jHj]. (2) yI= +O On the lower surface of the strip the sign of I is reversed. A strip antenna of the type under consideration can be obtained by chopping up an infinite strip with cuts parallel to the x axis, thereby producing a rectangular surface of length 2a and width 2d (say). In general d will be small compared with a. The new edges which are formed in this way cannot generate any -3 -

Ol~~4 - - -ll c.'ll-~L C 'C e c C- ft>Cs~ I, 0 ~ S~I (>5 > Z' C-s I J. c, IJ-. SI grte u ~.~ te~~ be Ce-crec Z )QheAnalIsat 8DWasple. matter ~ deteiuiine the atenna curre-',nre us thn cpes s on- f or the- cU~rrente,, excitedi on an i.nf initL-e strlir, by a no IIv ir-c'mn r Eas INave. 2b00 liel () 1 e4n +2() L...j 'e2nnl 00n m (2-1 se( e 17 (-i) (2,., 2r~+I =O 2mO Ne(1 ()(2n + l) 2(2n+-l) 2n+- 1M -0 m=O -4 -

:]er= ~ and Q are given Lby the Bessel function formiae is T- *- -1 m. = = (b.)Y (b) i c (b)Y,+(b) - b (b)~ ) (b I tI Mm m,r +1-1 T i- b m +1 m Hences 0o Ne (0) (2n+ 1) 1() n (x) + 2 2n+l Ne2n +1 For the lower surface (y = -0) the sign of I has to be reversed, but since rmust also be replaced by 2 - t7, I(x) = I (x) -2 (4) The coefficients B(2n+1) are, of course, functions of b and their values have 2m+1 been tabulated by the Computation Laboratory, National Bureau of Standards (1951). For large b the series for H5 is only slowly convergent, but when b is less than unity the convergence is sufficiently rapid for the series to be cut off after the first few terms. Two examples will be considered, corresponding to quarter and half wave antennas (a = A/8 and A/4 respectively). Using tabulated values of B ) and 2m +of the Bessel functions (the latter being supplemented, where necessary, by direct calculation of J (b) and Y (b) from their series expansions), the following results n n are obtained: -5 -

.. = 1 - (0 3.2212-i -1127S - 0035-. 4. - (0.00005-i O.00021) sin 5, (5) I4 (X) I+(1.80365 - i 0.88454) sin; - (0.14997 r i 0.03026) sin 37 +(0.00396 + i 0.00061) sin 57 - (0.00005 + i 0.00002) sin 77. (6) The coefficients of the higher trigonometrical functions are zero to the first 5 places of decimals. The amplitudes and phases of I+(x) are plotted as functions of x in Figs. 2 and 3 respectively. Taking first the quarter wave antenna, it is seen that the phase behaves in a perfectly regular manner, increasing monotonically as x increases from 0 to a, but the amplitude curve shows a minimumu at about x = 0.95a. This would be explained if there were an appreciable build up of charge at the ends, giving rise to a capacitive effect. The antenna would then resonate at a wavelength which is greater than that predicted by its physical dimensions. [FIGS. 2 & 3] No such minimum is found in the amplitude curve for a = A / and this can be attributed to the fact that the first harmonic (which is zero at the ends) now dominates the current distribution. Since the constant current is mainly responsible for any charge appearing at the ends, the capacitive effect is no longer important. -6 -

'-*,, yomipariscn wi h-i. Wire " Curen -le disc.ussin f ar has beer devoted to the curr.ent T (x) vi:.ic wouic be measured if a pirobe were placed on the upper surface of the antennae S:"measures r eteen e entirely feasible in ste o the experimental di:ficultie s involved and soe. Mr es ts c 1 t tained th r -ect gul 1a ar n. riangu l surf For the l ower surface of the antenna the expression for the current differs from that of equation (3) in the sign of the constant term. This discontinuitv between the currents on the illumirated and non-illuminated sides of the surface is to be expected and is the same as that occuring in the case of a half-plane (see, for example, Clemmow, 1951). To determine the total current carried by the antenna it is necessary to multiply each of the surface currents by a factor which takes into account 1the variation across the width of the strip. This dependence has been considered at length by Moullin & Phillips (1952) and they have shown that for narrow strips (kd <<l) a close approximation to the transverse distribution is provided by the function (1 - 2/d2)1/2 This is precisely the z dependence which would be arrived at by a study of the current near to the edge of a half-plane, and the singularities at z = d are those which are required in order to satisfy the edge conditions in cdiffraction theory (Jones, 1952). The distribution of current over the upper surface of the strip can now be represented as a function of x and z by I +(x) (1- z2/d )1/2 -7 -

and to obtain the total currentflo in to integrate, with respect to z fromz -d the current ca Lr ic ed by both the ~upper n and the msagnitude ofthe It will be observedta occured with I+ (x). The above expeso o fact suggest's that the di of suitably chosen rdiu wire has been ebtnsidere tion is a-noral i is where f Xx)is aca U where r is the d'

RL-63-J It is imlrediately apparent that the distributions for L strip and a wire are entirel ff'erent+ in character b-t ' th s does not rue c't the poss:bity: of finding an alternative expression for the strip current which wTil- brinE the,-wo int agreement. The natural expansion for I+(x), and hence I (x), is based tot since K X 1/2 sin = 1 - - i, \ 2 J 2 1/2 3 2 3/2 sin 3 1 x42 n 1/2 etc., the series can also be written in terms of the functions (1 - - / a2 n=0, 1l, 2,....0 Both of these expansions are rapidly convergent for values of ka of order ^unity and a reasonable approximation to the current distribution can be had by neglecting all harmonics above the first. Thus, for a = //, x2 1/2 I (x) "- 2rTd (0.32212-i 1.01278) (1 - ) (10) tot 2 a and for a = X/4, 2 1/2 I (x) > 2iTd (1.8035 + i 0.88454) (1 - X (11) tot. 2 a The function sin 7 can also be expressed in a Fourier series of the form sin 7= - J (m + ) cos (2m +)kx m -O 2mnl 2 2 J -9 -

RL-63-J nhere J is the second order Bessel function, and similarly sin =, -- -— C J 1(m+ — )TT cos (2r+l)kx. m=o 1( (2ml)rJ 21 2 ) S'c;. expansions iowever converge on3- slo-Sy and i' they are inserted i'nte the equation for the total current, a large number of terms are required in order to reproduce the accuracy represented by the single terms in, for example, equations (10) and (11). The first terr involving cos kx in no sense dominates the series when ka is small and hence, if an attempt is made to write the current distribution for a strip in a form analogous to that of equation (9), the correction term corresponding to f(x) will be at least as important as cos kx. As a result the differences betwen the distributions for a strip and a wire must be regarded as fundamental and, indeed, if a numerical comparison is made, no practicable value of L exists which will bring them into even approximate agreement. 5. Conclusions In the preceding section it has been shown that the longitudinal distribution of current on a strip of finite length a can be derived from the exact Mathieu function expansion for the current on an infinite strip. The resulting expression is valid for all a. If the width of the strip is small, the known variation of the current in the transverse direction may be used to predict the entire surface distribution, leading to an expression for the total current I t(x) carried by the strip. For small ka it is found that 2 1/2 I (x) 2T1 d o((l - - ) (12) tot a2 J

where Co is a complex numerical co depe a rapidly convergent Mathieu function series of equation (12) ist theit- r but for ka 1 the coei Although it is freely admitted that the most idealized type,of stri t ative scheme f'or.fi n e n 6.,

RL-63-J Legends for Figures Fig. 1 Fig. 2 Amplitude of surface current I (x) on quarter wve (a = X/8) and wive (a = A/4) strips. Fig. 3 Phase of surface current T (x) on quarter wave (a = /8) and half X (a= A /4) strips. Fig. 4 Longitudinal distribution of total current amplitude.

It 11, 9 't I. t i t Za I / 4f ~ / I 1.1, T/1 A Fig. I

+ o__ t~~~~~~~~~~~~~~~...................................... cl 0 i / II / Cr)I if( Nr9 c 0 / ~/ / /./,/ I I, / /1. iy /3 /

?/ J J /. ii 'i -~~I-r- =k/8 ti L~ 1i O0,1 bC ~/ C) 0.0 ~204 0. 08. '.dlstanc x/a1 i- ~ I -~ ~xt 10 -I — -_ ---~ — _ 0,0 0! 2 0.4 0.6 0.8 t distanbce x/a Fig. 3

2N itot 27rd 0. 0 0 0.2 0.4 distance x/a a, = 1 -Xj4 0"6 0. 8 Fig. 4