## 5 Single DIE for Conductive Disks  
An approximate boundary condition is derived here for purely conductive disks. The Mitzner GBCs allow for exact representation of the screen of infinite extent in a spectral-domain equivalent transmission line (TL) as a T network, as shown in Figure 2[35]. The expressions of the longitudinal impedance \(Z_{\rm L}\) and the transverse impedance \(Z_{\rm T}\) are respectively  
\[ \begin{split} Z_{\rm L}&=\eta_{0}\hat{Z}_{\rm L}= \mathrm{j}\eta_{0}\xi_{\rm cr}\tan\left(\frac{k_{\rm c}d}{2}\right),\\ Z_{\rm T}&=\eta_{0}\hat{Z}_{\rm T}=-\mathrm{j} \eta_{0}\frac{\zeta_{\rm cr}}{\sin(k_{\rm c}d)}\,.\end{split}  \tag{31}\]  
When a loop source radiates in the presence of the infinite screen, the equivalent TL of the problem in a Schelkunoff approach [34; 36] is as reported in Figure 3, where \(Z_{\rm loop}\) is the impedance of the current loop and \(Z_{0}\) is the free-space impedance \(\eta_{0}\). For a magneto-conductive screen, this always results in \(|Z_{\rm L}|\ll Z_{0}\), and the presence of the longitudinal impedances \(Z_{\rm L}\) in the TL network in Figure 3 can be neglected if the result is \(|Z_{\rm L}|\ll\left|Z_{\rm loop}\right|\). The impedance of a parallel loop is a function of the distance; in particular, at the air-screen interface for a small loop, it results in [34; 37]  
\[Z_{\rm loop}\simeq j\eta_{0}\frac{k_{0}h}{3}\,. \tag{32}\]  
Therefore, the effects of \(Z_{\rm L}\) can be neglected if  
\[\left|\xi_{\rm cr}\tan\left(\frac{k_{\rm c}d}{2}\right)\right|\ll\frac{k_{0}h }{3}\,. \tag{33}\]  
Figure 3: Equivalent TL for a loop radiating in the presence of an infinite screen of finite thickness. Figure 2: T network representing a thick infinite screen in a spectral-domain TL. Thus, it should be noted that for purely conducting screens (33) is always satisfied provided that \(d\ll h\), which always occurs in practice. If the presence of the longitudinal impedances \(Z_{\text{L}}\) can be neglected, then the screen is represented only by the transverse impedance \(Z_{\text{T}}\). This results in only one boundary condition,  
\[\frac{1}{2}\left(\mathbf{E}_{\text{t+}}+\mathbf{E}_{\text{t-}}\right)=Z_{ \text{T}}\mathbf{u}_{\text{z}}\times\left(\mathbf{H}_{\text{t+}}-\mathbf{H}_{ \text{t-}}\right)=Z_{\text{T}}\mathbf{J}_{\text{S}}\,, \tag{34}\]  
which is the same as in (1). Thus, we can derive a single set of DIEs which solves the problem, which consists of (5) and (6) with \(i=1\) and \(\hat{Z}_{\text{T}}\) instead of \(\hat{Z}_{S}\). Therefore, under assumption (33), the solution of the original problem requires the introduction of only one unknown (i.e., an equivalent surface electric current density) and the solution of _only one_ set of DIEs; this is exactly the same formulation adopted to solve the problem under the assumption of thin screens [16; 23]. In such a case, the scattered magnetic field is provided by (24), with coefficients \(\psi_{n}^{(1)}\) provided by the solution of (13) with \(i=1\). Thus, it is clear that with respect to the full formulation based on Mitzner GBCs, the savings in terms of computation time are about 75%. Another important advantage is that for \(k_{0}a\ll 1\) (a condition which in practice is always met in EMC) the elements of the matrix (18) have an exceptionally simple expression. In fact, using the series expression derived in [38] to express the integral term in (18), we have  
\[A_{mn}^{(1)}\simeq\frac{\hat{Z}_{\text{T}}}{4m}\,\delta_{mn}+\frac{k_{0}a}{8} \left(-1\right)^{1-p}\sum_{l=1}^{\infty}\frac{\left(-\text{j}\right)^{l}\Gamma \left(p-\frac{l}{2}\right)\Gamma\left(q-\frac{l}{2}\right)\left(k_{0}a\right) ^{l}}{\Gamma\left(p+1+\frac{l}{2}\right)\Gamma\left(q+1+\frac{l}{2}\right) \Gamma\left(-\frac{l}{2}\right)\Gamma\left(1-\frac{l}{2}\right)} \tag{35}\]  
where \(\Gamma(\cdot)\) is the Gamma function [33], \(p=m-n\), and \(q=m+n\).