## ABSTRACT

The numerical method best suited for use with SIBCs is the boundary element method (BEM) because in both the BEM and the SIBCs the functions are approximated at the same points on the interface between the media. When applied to an eddy current problem consisting of conducting and nonconducting regions, the BEM yields a system of two integral equations over the conductor’s surface with respect to two unknowns: the required function and its normal derivative at the conductor=dielectric interface [1]. Under the conditions of skin effect, the electromagnetic ﬁeld behavior in the conductor is known, the surface impedance concept may be applied, and the formulation can be reduced to a single integral equation employing the fundamental solution of the Laplace equation. The extra unknown is eliminated using the SIBCs relating the function and its normal derivative at the conductor’s surface. This approach is almost ideal for solving timeharmonic problems. Because of this, BEM-SIBC formulations have been widely used for the analysis of skin and proximity effect problems of multiconductor systems [2-7]. In the general case of transient excitation of the surface integral equation,

including the time-domain SIBCs in the form Equation 3.81, 3.118, or 3.132, must be solved at every time step due to the time convolution terms. This leads to a signiﬁcant increase in the computational cost required for the solution and, for all practical purposes, renders its numerical implementation impractical. The best way to avoid this difﬁculty is the separation of variables in the formulation into space and time components. In this case, the integral equation for the space component needs to be solved only once for a given system of conductors, and the result multiplied by the time component to obtain the solution of the problem for any time dependence of the source. It is easy to see that the variables in the surface impedance function given

in Equations 3.82 through 3.84 cannot be separated because the functions Tm are different. However, in the low-frequency case (the so-called quasistatic approximation, in which the displacement current can be neglected) each

term in Equations 3.82 through 3.84, when considered independently, admits separations of variables. This circumstance and the fact that the SIBC in Equations 3.82 through 3.84 can be represented as power series in the small parameter are keys to the development of the formulation desired. Indeed, it is natural to suppose that use of the perturbation technique as described in Chapter 2 will lead to a set of integral equations so that every equation includes only one term of Equations 3.82 through 3.84 and, consequently, admits separation of variables. Such a formulation has a clear physical meaning: the zero-order integral equation gives the solution in the wellknown perfect electrical conductor limit and the other equations contribute corrections of the order of Leontovich, Mitzner, and Rytov approximations. Thus the total number of integral equations in the formulation will not exceed four. In fact, it may be even less depending on the problem as will be shown in Chapter 9. Derivation of this type of formulation is the aim of this chapter. Consider a system of N cylindrical conductors of arbitrary cross sections

surrounded by a homogeneous nonconducting space as shown in Figure 6.1. The parameters of the conducting and nonconducting media are assumed to be constants. To simplify derivations and without loss of generality, let us assume that the magnetic permeabilities of the conducting and dielectric regions are the same. Let an external source produce quasisteady current pulses ﬂowing through the conductors so that the condition of applicability of the surface impedance concept in Equation 2.1 is satisﬁed. In low-frequency problems, the electromagnetic ﬁeld distribution in both the dielectric and conducting regions can be described by the following equations:

Conducting domain:

r~E ¼ m @ ~H @t

(6:1)

r ~H ¼ s~E (6:2)

A

r ~H ¼ 0 (6:3) r ~E ¼ 0 (6:4)

Nonconducting domain:

r~E ¼ m @ ~H @t

(6:5)

r ~H ¼ 0 (6:6) r ~H ¼ 0 (6:7) r ~E ¼ 0 (6:8)

Consider the particular case of long parallel conductors of constant cross section as shown in Figure 6.1. We direct the tangential coordinate, j1, along the conductors; set d1 !1 and assume that no variation of the electromagnetic ﬁeld takes place in this direction

@f @j1

¼ 0 (6:9)

where f denotes any function. Under these conditions, the problem of the electromagnetic ﬁeld distribution can be considered as two-dimensional in the plane of cross sections of the conductors, and the vectors~Ii, ~E, ~H, and ~A1 can be represented in the following form:

~Ii ¼ Iið Þj1~e1; ~E ¼ Ej1~e1; ~H ¼ Hj2~e2 þHh~e3; ~A ¼ Aj1~e1 (6:10)

Further derivations for the two-dimensional case will be performed in terms of electric-magnetic ﬁelds and vector potential formalisms.