5. Poles and zeros

An impedance z(k) as defined by (15) satisfies

z(0) = 1

(29)

and is an RL impedance, so that its zeros k1, k2, ... and its poles k1', k2', ... are negative real and separate each other:

0 < -k1 < -k1' < -k2 < -k2' < ...133;

(30)

The present section is devoted to an analytical and numerical study of the distribution of the poles and zeros, both for the ellipse and for the rectangle, with the idea of obtaining additional information on the impedance behaviour of the rectangle and, more particularly, on its asymptotic behaviour.

Physically, poles and zeros characterize transient modes of decay (with different boundary conditions) and correspond to free solutions of the skin-effect equation, hence to eigenvalues of the Laplace operator in (13). The asymptotic distribution of these eigenvalues has been studied in mathematical physics in connection with various problems of statistical thermodynamics; in particular, the nth eigenvalue of (13), for a two-dimensional domain of area S, and for quite general boundary conditions, is known to be asymptotic to . From (14), (11) and , we thus obtain -n as the asymptotic value of the nth pole and zero, which means:

(31)

with and tending to zero for large n. Moreover, the alternation (30) of poles and zeros restricts the deviations of (31) to

(32)

Although the asymptotic distribution (31) is independent of the shape of the section so that it also holds for a circular cross-section of radius a, not all natural frequencies are excited by the forced current in this case (by circular symmetry), so that only a small subset of the eigenvalues (corresponding to a one-dimensional problem where is asymptotic to ) appear as poles and zeros of the impedance.

Since and certainly tend to zero if and tend to constant values, it is interesting to discuss the case where and are rigorously constant in (31). The function satisfying (29) is then

(33)

and is asymptotic to

(34)

for large k, by Stirling's approximation for the gamma function.

If and are not constant but tend sufficiently quickly to constant values and , the asymptotic expression of the impedance is still of the form , but with a coefficient C different from that of (34), because the latter was imposed on (33) by condition (29). The asymptotic law may, however, become quite different when the variation and is very slow, as shown by the example of the logarithmic derivative of the gamma function. For the function we have , whereas tends to zero as . Although (34) gives the value 1 for, the function tends to infinity as .

For the ellipse impedance (17), it is known that the nth zero of is asymptotic to . Since the exponent of (34) is known to be in (24), one thus expects the nth pole to be asymptotic to . This is confirmed by Table I based on a numerical computation for the equivalent circuit of fig. 4 with 20 and 30 sections.

Table I. Zeros (kn) and poles (kn') of the function .

N

kn

k'n

1

-0.83752

-1.11712

2

-1.83490

-2.13294

3

-2.83422

-3.14014

4

-3.83393

-4.14441

5

-4.83377

-5.1471-8

6

-5.83368

-6.14937

For the rectangle, the zeros and poles have been computed from (27)-(28) on matrices truncated at orders 30 and 40, and are given in Table II which confirms the asymptotic behaviour (31). This behaviour was also apparent in Silvester's results5 in spite of differences in normalization in (11) and (15): Silvester's rough tabulation gives the zeros (but not the poles) of z + k ln 2 in terms of the frequency variable 4k, but is coherent with our results.

Table II. Zeros (kn) and poles (k'n) of the impedance of a thin rectangular conductor.

N

kn

k'n

1

- 0.87914

- 0.9716

2

- 2.06221

- 2.1591

3

- 3.10107

- 3.2156

4

- 4.12081

- 4.2496

5

- 5.13360

- 5.2742

6

- 6.14298

- 6.1935

7

- 7.1504

- 7.3093

8

- 8.1565

- 8.3227

9

- 9.1618

- 9.3341

10

-10.1664

-10.344

11

-11.1705

-11.352

12

-12.1743

-12.358

13

-13.1777

-13.360

14

-14.1809

 

15

-15.1838

 

Although the natural frequencies have similar asymptotic distributions for the rectangle and the ellipse, the deviations and show a markedly different small-scale behaviour, as it appears in fig. 6. In contrast with the rapid convergence to the asymptotes for the ellipse, the deviations show a logarithmic drift for the rectangle, with an increasing difference . Since (32) prevents such an increase from continuing indefinitely, we are far from having reached the asymptotic behaviour of the deviations. Since, however, a continuation of the logarithmic drifts with the slopes resulting from fig. 6 is compatible with (32) up to about n = 108, there is little hope of obtaining the true asymptotic behaviour by numerical computations.


Fig. 6. and , the deviations from n for the zeros and poles of the normalized impedance of a conductor with elliptical and rectangular cross-section (subscripts e and r). The dashed lines give the asymptotic values and .

Our success in obtaining the asymptotic expression (24) of the ellipse impedance is due to the existence of the closed-form expression (17) and to the availability of relatively advanced Bessel-function data. If it were possible to establish (24) directly, without using the closed form (17), a similar approach might succeed for the rectangle where such a form is not available. Such a direct method is described in the next paragraphs for the ellipse. For the rectangle, the relevant mathematics are much more complicated and the treatment is given in the Appendix.

The asymptotic impedance of the ellipse will now be obtained by simple physical considerations of the network of fig. 4, which are, of course, equivalent to mathematical. considerations of the corresponding continued fraction (22). A finite approximation of degree n of the network is obtained either by short-circuiting the (n + 1)th shunt inductance or by opening the (n + 1)th series resistance, and the corresponding impedances will be called zs and z0. At high frequencies, the network reduces to n unit resistances in series in the first case, so that the approximate impedance is

zs = n

(35)

In the second case, however, the impedance of the last inductance dominates the last resistance at high frequencies, and the resistance can be neglected: the last inductance thus combines in parallel with the preceding one, and the reasoning applies again. Ultimately, the network reduces to the parallel combination of the first n inductances. Since the total susceptance is 2(1 +2+...+ n), which is approximately n2 for large n, we obtain the impedance

(36)

In classical network and line theory, the input impedances z0 and zs, of a dissipative 2-port opened or shorted at its output converge to a common value z (the characteristic impedance) when the network attenuation, or the line length, becomes infinite. By contrast, the divergent behaviour of (35)-(36) is due to the essential singularity at infinity of the function (17), resulting from its asymptotic behaviour (24). Although all three impedances z0, zs and z thus diverge for large k, there is some hope of obtaining the asymptotic expression of z by imposing a common asymptotic behaviour on z0 and zs. As first attempt, if is forced into (35)-(36), the resulting constraint

(37)

imposes the common value on z0 and zs so that the asymptotic law (24) is confirmed, except for a small difference in the coefficient C, whose correct value (25) is replaced by 1. The discrepancy is due to the fact that the principal values (35)-(36) of zs and z0 have been computed by making k large for a fixed value of n, without considering the constraint (37) which was only found a posteriori, and as a first approximation. This suggests that an iterative process might lead to an improvement both of the constraint (37) and of the resulting value of the coefficient C of (24). The mathematical justification of this process is based on the inequalities zsz < z0, holding for any positive n and k because of potentiometric effects, which impose a common asymptotic behaviour on all three impedances if is forced to tend to 1. Owing to the divergent values (35)-(36) of z0 and zs for large k, fixed n, the ratio can only tend to 1 if n and k tend simultaneously to infinity, in accordance with some (as yet unknown) constraint. Because the constraint is unknown, it can only be reached by successive approximations, which lead to improved estimates for the asymptotic expression of z because the margins resulting from the potentiometric inequalities are decreased at every step. Although the process has not been proved convergent, the margins obtained in one or two steps are already sufficiently small for all practical purposes.

The second approximation replacing (35)-(36)-(37) is obtained as follows. When a unit current is injected in the network of impedance (35), the input voltage is n and the voltage at the ith mode is proportional to -n - i. The total magnetic energy in the shunt inductances is thus

,

and must be equated to the energy in an equivalent inductance L shunting the resistance (35). Since v = n, we obtain , and (35) is replaced more accurately by

(38)

By a similar reasoning we are led to evaluate the total dissipation in the network of the initially reactive impedance (36) and to represent it as a series resistance, which is found to be , so that (36) is replaced by

(39)

It is not legitimate to equate (38) and (39) rigorously, for the resulting second-degree equation contains terms in that are still neglected in (38)-(39). One must therefore linearize the equation around the first-order approximation (37), by replacing by 1 in the correction factors. This reduces (38) to and (39) to . By equating the last two expressions we obtain for both the value (24) with C = 1.044....

[ Back to index ]