RL 741 A STUDY OF SPHERICAL CAP ANTENNAS H. Kimura, C. T. Tai and V. V. Liepa Radiation Laboratory Department of Electrical and Computer Engineering The University of Michigan Ann Arbor, Michigan 48109 Introduction Where space is limited, or at low frequencies where large antennas are required, reduced size antennas are often used. One method of achieving the necessary size reduction is through socalled capacitive or top loading [1]. Here we present analysis, and computed and measured impedance characteristics of one such antenna, specifically the spherical cap antenna. A variational technique [2] is used in the analysis. Analytical Procedure The geometry of the antenna is shown in Fig. 1. The antenna consists of a biconical section terminated by a pair of spherical caps. For analysis, two regions are defined, one inside the spherical shells (Region I), and the other outside (Region II). In the two regions electric fields are represented by equations (1) and (2), where Yt is the effective outward admittance defined at the interior surface of the caps [3]. The Ln(e) is a class of Legendre functions which go to zero at eo, 7r/2 and 7r-0o. In general, for arbitrary eo, n are not integers. The K is the characteristic impedance of the biconical transmission line and an and bk are constants. to be determined. EI () ZR sin ( - R) - a s [KYt sin ~(k - R) - j 2-aR sin e cos 6 (Q - R)] j0z 27rR a n(n + 1) S' (BR) n (ZR) n aL ( ) n 29 (n = n,n2,n3...; R < z) (1) (k = 1,3,5..; R > ), (2) - IIA jZ EII() = -G Zo 27TR bk k(k + 1) Rk(BR) aPk(e) Rk (fZ) 3e k RL-741 = RL-741

-2 - Z 9( where I is the driving point current, K = - In cot -, ' 0., S ( R) = (gR)1/2 Jn/2 (6R) Rk(SR) = (BR)1/2 H(/2( R) dS (SR) Sn (R) = d(RdRk (R) R)= - d(R) R~(~I\ d ~RT Introducing boundary aperture field Ea(e ), which is still unknown, variational expression for Yt is obtained from equations (1) and (2). Thus, Yt -_ 2j2T *1 Z 2 zo ~-01 L Ea(e)de - 1 - n n+ Nnnn U "("n nn *0 [ - 1 1 w-~ Ea ()L' n ()sin - 2 e de I - k k+l M)IkIkk Ea (e)Pk(e)sin e de ) -.0 (3) where N - n ) n S (ZQ) M n Mk = Rk(3) ' Jnn = f [L (9)]2 sin e de 0 Ikk Jo [Pk(e)] sin e d (n = nl,n2,n3,...; k = 1,3,5,....) To avoid non-integer Legendre functions in computation of interior modes, %o is assumed to be small, and in such cases n are close to integers and integration pertaining to eigenfunctions Ln(tE) can be approximated by integration of selected Legendre functions Pk(e) of odd integer order.

-3 - When taking the dominant mode, or the dominant plus one complimentary mode for the aperture field Ea(e), the stationary property of equation (3) yields zeroth or first order solutions for Yt, respectively. We found that for a wide-angle cap (small gap) the Re[Yt] is almost the same as that obtained by Stratton and Chu 14] in their study of spherical antennas with infinitesimal gap using spherical harmonic expansion. The driving point impedance or admittance of the antenna is obtained by applying transmission line transformation to Yt. An example of such calculations is shown in Fig. 2. Experimental Verification Two monopole version antennas with e1 = 45~ and 89~ were constructed and impedance measurements were performed. In each case a good agreement with prediction was found. Figure 3 shows an example of the results. Conclusions Variational technique was used to solve for impedance of the spherical cap antenna and the results agree well with the measurements. As much as 88 percent reduction in the physical height of the antenna can be achieved, but bandwidth and the radiation resistance is lost in the trade-off. If needed, more precise results can be obtained by using higher order modes when prescribing the aperture field. References [1] Schwering, F., Puri, N. N., and Stavridis, A., "The modal expansion solution of a top loaded dipole," AP-S Symposium at the University of New Mexico, Vol. 2, 584-587, 1982. [2] Tai, C. T., "Applications of a variational principle to biconical antennas," J. Appl. Phys., 20, 1076-1084, 1949. [3] Schelkunoff, S. A., "Theory of antennas of arbitrary size and shape," Proc. I.R.E., 29, 493-521, 1946. [4] Stratton, J. A., and Chu, L. J., "Steady-state solutions of electromagnetic field problems," J. Appl. Phys., 12, 236-340, 1941.

-4 - Region II / /R [ l II I Fig. 1: 'Geometry of the antenna. Fig. 3: Experimental results. Freq.; MHz ohms 2000 ohms,1 I 200 100 '000 0 -1000 -2000 I 0 1.0 2.0 0.0 2.0 Bp' Fig. 2: Example of numerical o = 1.77 deg. Real part; Left calculation (1st order) 0e = 45 deg. Imag. part; Right