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Chapter 3 Transmission lines and waveguides 3.1 General solutions for TEM, TE, and TM waves An arbitrary transmission line is characterized by conductor boundaries parallel to the z direction and is uniform along its axis. The conductor is at first assumed perfect. Assuming time-harmonic fields with an dependence and wave propagation along the z-axis. The fields can be written as (3.1a) (3.1b) where and represent the transverse field components, while and the longitudinal field components. The wave is supposed to be propagating in the +z direction. Assuming that the transmission line region is source free, Maxwell’s equation (1.27a) and (1.27b) can be written as （3.2a） （3.2b） With an z dependence, the three components of each of the above vector equations can be reduced to (3.5a) (3.5b) (3.5c) ， (3.5d) where (3.6) is defined as the cutoff wave-number. is the wave-number of the material filled transmission line. TEM waves To discuss the TEM wave, we need to discuss Helmoholtz Equation The Helmoholtz Equation In a source-free, linear, isotropic, homogeneous region, Maxwell’s equations in phasor form are, (3.6a) (3.6b) Taking the curl of (3.6a) and using (3.6b), we have, , Which is a equation for .The result can be simplified through the use of vector identity (B.14), , which is valid for the rectangular components of an arbitrary vector .Then, (3.7) Which is the Helmoholtz Equation. For TEM wave, , , must be zero, so we have . (3.8) From (3.7), we have (3.9) Considering z dependence, using (3.8) we have (3.10) The same expression holds for . So we have (3.11) It is also easy to show in the same way that the transverse magnetic field components also satisfy Laplace’s Equation: (3.12) The transverse distribution of the electric fields are the same as that of ele