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Additional info for Analysis Methods for RF, Microwave, and Millimeter-Wave Planar Transmission Line Structures
We consider a medium that is source free ( D J D 0) and simple (homogeneous, isotropic, and linear) and assume that the fields are time harmonic. The medium is characterized by a dielectric constant ε and permeability . Taking the curl of Maxwell’s Eq. 5a) and making use of Maxwell’s Eq. 24 is the wave number. 25 where A is an arbitrary vector, we can then rewrite Eq. 26 where r2 denotes the Laplacian operator. This equation is called the wave equation for the electric field. 27 W2 H C k 2 H D 0 Both of these wave equations are also known as Helmholtz equations.
101). 109d D Hkl PROBLEMS 37 where the constant C is chosen to satisfy the power orthogonality. It should be noted that the power orthogonality is only approximately held for low-loss waveguides propagating nondegenerate modes. For degenerate modes, strong couplings between these modes occur. One remark needs to be made at this point. All the derived orthogonality relations are completely satisfied only if the modes were calculated exactly. Normally, the eigenmodes in planar transmission lines can only be determined approximately.
For instance, we consider the TE mode and let Hz in Eq. 55c) equal zero. This leads to kc,h D 0 and, consequently, Eq. 65 28 FUNDAMENTALS OF ELECTROMAGNETIC THEORY which is basically Laplace’s equation in the transverse plane. The transverse fields can be obtained from Eqs. 66b where is the TEM mode’s propagation constant. Proceeding with the TM mode also gives identical results. It should be noted that for the TEM case, both e x, y and h x, y are equal to the two-dimensional voltage or potential V x, y .