4. Summary of results

4.1 Thin elliptic strip

For the thin elliptic strip, the normalized impedance (l5) is

(17)

where Jk(k) is the Bessel function of the first kind of order and argument k given by (11). In the denominator of (17), J' denotes the derivative with respect to the argument (and not to the order), and hence the value of d Jk (s)/ds at s = k.

In terms of the auxiliary variable

u = arc cos x/a

(18)

the linear current density is given by any of the three following equivalent expressions:

(19)

(20)

(21)

where all sums are for n = 1, 2, ... and where i0 is (8). For k infinite, (19) reduces to its first term and gives (7), by (18). The series (19) is, however, divergent at the end-points x =  a corresponding to u = 0 or . By contrast, (20) and (21) are convergent and (20) reduces to (10) for k = 0.

A continued-fraction expansion of (17) is

(22)

where the successive denominators are alternately 1 and 2n/k (n = 1, 2, ...). The impedance (17) is thus represented by the equivalent circuit of fig. 4. The approximation of (22) limited to order k2 is

(23)


Fig. 4. Equivalent circuit for the impedance of eq. (17) based on the expansion (22).

At high frequencies, the known asymptotic expression of (17) is

(24)

where the numerical factor C is given by:

(25)

With the definition (11) of , we thus obtain:

(26)

The real and imaginary parts of the function (17) are shown in fig. 5 and compared with the tangents resulting from (23) and with the asymptotic expression (26). The numerical computation of (17) was based on the expansion (22) with truncations corresponding to 20 or 30 RL sections in the equivalent circuit of fig. 4; this produced no significant difference in the results, in the range .


Fig. 5. Real and imaginary parts of the function with k = j. The dashed lines show the tangents for  = 0 and the asymptotes for large .

4.2 Thin rectangular strip

In comparison with the full analytical results just summarized for the thin ellipse, very little is known for the thin rectangle. From a numerical study of the integral equation for the linear current density, V. Belevitch et al. have obtained low-frequency approximations for the impedance. On the other hand, a purely numerical treatment of the problem (by different methods) has led P. Silvester and C. Beccari and C. Ronca to a resistance law confirming the earlier measurements of A. E. Kennelly and H. A. Affel. From all these results it appears that the relative resistance increase Re(z-1) for the ellipse is about twice that for the rectangle. The only new result obtained in this article is an analytic expression of the rectangle resistance as the ratio of two infinite determinants:

(27)

where is the determinant of the symmetric matrix

(28)

and the principal minor separated by dotted lines. In the matrix elements, the first term is constant on a parallel to the main diagonal and the second on a parallel to the second diagonal. Results (27)-(28) are discussed further in the next section and in the Appendix.

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