## 3 Parametric Analysis  
In this section, a parametric analysis devoted to gathering the limits of validity of the quasi-magnetic wall approximation is carried out. However, rather than considering a slot structure over a dielectric substrate (Figure 2a), the structure under study is a coplanar waveguide (CPW) with a central strip width 10 times larger than the width of the slots, i.e., \(W=10S\) (see Figure 2b). The reason is that some authors previously developed a software tool useful to calculate the quasi-static capacitance and the effective dielectric constant of a CPW. With such relation between the slot and strip width and symmetric (even-mode) excitation (the fundamental mode of a CPW), the electric field does not appreciably reach the bisecting symmetry plane (a magnetic wall). Thus, the system can be considered to be composed of two independent halves, where the behavior of each half is identical to the one of the slot structure of Figure 2a (actually, the same conclusion is deduced for odd-mode excitation of the CPW structure). Therefore, carrying out the parametric analysis on the CPW transmission line (with \(W=10S\)) is fully justified. The effective dielectric constants of the CPW structures (used to check the validation of the behavior of the slots) have been calculated by means of the highly efficient semi-numerical algorithm reported in [90], which is basically an enhanced version of the well-known method of moments (MoM) in the spectral domain. In order to obtain the per-unit-length capacitance (C) and inductance (\(L\)) matrices of a general multi-conductor CPW, an integral equation is posed for the tangential electrical field in the slots. The kernel of such integral equation in the spectral domain can be analytically determined for a very general class of multilayered substrates. In addition, the use of appropriate basis functions with edge singularities and analytical techniques to speed up the calculation of the entries of the MoM matrix results in a very fast and extremely accurate computer code, which has been the considered one (see [90] for further details, which are out of the scope of this paper). The parameters under consideration are the slot width \(S\), the substrate thickness \(h\) and the dielectric constant of the substrate \(\varepsilon_{r}\) (as mentioned \(W=10S\)). It is assumed that the conductivity of the metal layer is infinite and its thickness is null (perfect conductor approximation). Due to the scalability of the Maxwell equations, it follows that the relevant geometry parameter is the ratio between the slot width and substrate height, \(r=S/h\). Thus, the parametric analysis reduces the study of the effects to only two parameters, \(r\) and \(\varepsilon_{r}\). For any given combination of \(r\) and \(\varepsilon_{r}\), the tool provides the effective dielectric constant, \(\varepsilon_{eff}\), of the actual (CPW) structure (Figure 2b), as well as the effective dielectric constant of a CPW covered with a dielectric material identical to the substrate (i.e., with the same dielectric constant and thickness), see Figure 2c. Such an effective dielectric constant is indeed the equivalent dielectric constant, \(\varepsilon_{r,\alpha}\), as defined in the previous section. If the quasi-magnetic wall approximation assumed in the previous analysis is valid, the effective dielectric constant, \(\varepsilon_{eff}\), should coincide with the effective dielectric constant of the CPW calculated simply as \((1+\varepsilon_{r,\alpha})/2\equiv\varepsilon_{\alpha r}\), i.e., the average value of the dielectric constant of air and \(\varepsilon_{r,\alpha}\). Therefore, an indicative parameter of the validity of the quasi-magnetic wall approximation is the relative difference between \(\varepsilon_{eff}\) and \(\varepsilon_{\alpha r}\). When expressed in percentage, such a parameter or relative error is simply \(E_{rel}=100\cdot(\varepsilon_{eff}-\varepsilon_{\alpha r})/\varepsilon_{eff}\).