By Reinhold Pregla
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Additional info for Analysis of Electromagnetic Fields and Waves: The Method of Lines (RSP)
16. 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.
4) with: εxx = ε11 cos2 φ◦ + ε22 sin2 φ◦ − (ε12 + ε21 ) sin φ0 cos φ◦ εxy = (ε11 − ε22 ) sin φ◦ cos φ◦ + (ε12 cos2 φ0 − ε21 sin2 φ◦ ) εyx = (ε11 − ε22 ) sin φ◦ cos φ◦ − (ε12 sin2 φ◦ − ε21 cos2 φ◦ ) εyy = ε11 sin2 φ◦ + ε22 cos2 φ◦ + (ε12 + ε21 ) sin φ0 cos φ◦ In case ε12 = −ε21 = −jκε , the last terms in εxx and εyy vanish and the terms in brackets of εxy and εyx result in −jκε . The calculation of the modiﬁed µ-tensor is done analogously. 2 Relations between transversal electric and magnetic ﬁelds – generalised transmission line (GTL) equations In this subsection we will develop the generalised transmission line (GTL) equations that are used for analysis.
A general model of a waveguide cross-section is shown in Fig. 13. In the ±z direction, the structure may be inﬁnite or a closed electric or magnetic wall. 5 70 60 0 6 50 4 40 2 30 0 20 2 10 4 0 6 Fig. 11 Independent propagation of two guided modes in a slab waveguide (see Fig. 55 µm 11 10 9 8 7 6 5 4 3 2 1 −5 0 5 10 15 20 Fig. 55 µm (Reproduced by permission of Elsevier) the ground metallisation acts as a boundary. In the x-direction, electric or magnetic walls may be introduced as well. The permittivity of the layers might be either a continuous or stepwise function of the lateral coordinate, and the layers may diﬀer in width.
Analysis of Electromagnetic Fields and Waves: The Method of Lines (RSP) by Reinhold Pregla