By Reinhold Pregla
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Extra info for Analysis of Electromagnetic Fields and Waves: The Method of Lines (RSP)
17). 32) Therefore the eqs. 19) show that the matrices QE and QH are determined as products of RE and RH . g. ), a product of two square matrices results in identical eigenvalues if the order of the product is reversed. Also, the corresponding 24 Analysis of Electromagnetic Fields and Waves eigenvectors are related. We will use this characteristic later. 35) In what follows, we show the steps of the analysis for the cases where we start with the electric or the magnetic ﬁeld in parallel. However, when applying the MoL, we will only solve one of the eigenvalue/eigenvector problems in eq.
We start with a complete metallisation in plane A (z = 0), leading to EA = 0. 74) The subscript M (instead of B) is used for the plane of metallisation (lower side). The superscript d denotes the (inhomogeneous) dielectric layer. 34 Analysis of Electromagnetic Fields and Waves z h pm h s pd h s w BC BC ε ra M ε rd y d a e -lines: E z , E y , H x , S y h -lines: H z, H y, E x , S x x MMMS2011 Fig. 16 Cross-section of a microstrip with ﬁnite substrate width (Reproduced by permission of Taylor & Francis) The layer above the strip (indicated with the superscript a) may be of ﬁnite or inﬁnite thickness.
With the relations given in eq. 71) Now, by introducing these expressions into an expression analogous to eq. 73) (k) = (k) z 2 z 1 |k −H z 2 = Z 0 / sinh(Γd) E B B As mentioned before, Y 0 Z 0 are unit matrices here. 73) can also be used in cases of diﬀerent normalisations leading to other expressions. Therefore, we kept Y 0 and Z 0 here. The formulas for the relation of the ﬁelds in planes A and B are independent of the coordinate system. 3 Examples of system equations Let us show the procedure for the analysis of the structure shown in Fig.