By Hyo J. Eom
Electromagnetic wave idea is predicated on Maxwell's equations, and electromagnetic boundary-value difficulties has to be solved to appreciate electromagnetic scattering, propagation, and radiation. Electromagnetic idea unearths functional purposes in instant telecommunications and microwave engineering. This publication is written as a textual content for a two-semester graduate direction on electromagnetic wave thought. As such, Electromagnetic Wave concept for Boundary-Value difficulties is meant to assist scholars improve analytic talents by way of fixing pertinent boundary-value difficulties. particularly, the innovations of Fourier remodel, mode matching, and residue calculus are applied to unravel a few canonical scattering and radiation difficulties.
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6 Terminated Transmission Lines 43 I (z) Zo + Zs • V+(z) V(z) + Vs ~ Zl V- (z) z V(z), l(z) z=O z =l Fig. 12. Transmission line terminated with load Z/. 118) Since 10+ = 11,+ 0 Zo , =_ and 10- 11,0 Zo ' I(z) = 1 (Vo+e ikz _ Vo-e-ikz) . 120) Zl = I(l) VB = I(O)Zs + V(O) . 121) for Vo+ and Vo- gives 11,+ = o (Zo 11, o = r. 123) where rl Zl - Zo Zl + Zo is the voltage reflection coefficient at the load z = l. I. Ie 2ik(l-z) . 124) y FI The impedance Z(z) at z is defined as Z( ) = V(z) = Z 1 + F(z) 01-F(z) .
46) where the tip of the electric field traces a line on the ECaxis and the wave is said to have linear polarization. Its graphical representation is shown in Fig. 4. ~\-'--=-------. Ex \ \ \ \ \ Fig. 4. Linear polarization. ,/ , E~ ,/ ,/ ,/ ,/ \ \ \ ,/ ,/ \ ,/ 'If ----;---,/-7~\-'--=--i_----. Ex ,/ ,/ \ \ \ ,/ ,/ ,/ ,/ ,/ Fig. 5. Circular polarization. 47) which represents a circle on the ECEf) plane; thus, the wave has circular polarization. Its graphical representation is shown in Fig. 5. When Oy - Ox = - ;, the polarization is called right-hand circular polarization, since the electric field behaves as a right-handed screw that advances in the +z-direction.
1 . cos t cos i 1:2 1:1 Let us evaluate the time-average power densities entering and leaving the boundary at z = 0 for lossless media, where both Z1 and Z2 are real. 2 = angle is - 1- - • 2 () sm i. 1. 4 Reflection at Plane Boundary When the incident angle (}i is greater than the critical angle (}e = sin- 37 1 , 1"1 the term cos(h (thus Z2) becomes purely imaginary, thereby yielding IRTEI = 1, Sr = Si, and St = 0. This means that the total internal reflection occurs when a wave impinges on a less dense medium (1"1 > 1"2) at an incident angle greater than the critical angle ((}i > (}e).