By Cam Nguyen
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Additional resources for Analysis Methods for RF, Microwave, and Millimeter-Wave Planar Transmission Line Structures
Substituting Eq. 51b respectively. Note that, for degenerate modes, e D h . 52b Similarly, by expanding Eq. 53b Further expanding Eq. 54b jωεeš We can also determine the x and y components from the z components of the fields by using Maxwell’s equations directly. We can easily prove that both Ez and Hz cannot be even or odd with respect to x and y simultaneously. The results derived so far are very general and so are applicable to any possible modes 26 FUNDAMENTALS OF ELECTROMAGNETIC THEORY existing in any waveguide.
3 Orthogonality Relations Between Electric and Magnetic Fields Let’s consider again two different TMmn and TMkl modes. Assuming z is the direction of propagation, the fields of these modes can be expressed as a sum of the transverse and longitudinal fields. 88b 33 ORTHOGONALITY RELATIONS Note that we have introduced the notation e, h and ez , hz to signify the twodimensional transverse and longitudinal components, respectively. 89b where j D mn, kl. Making use of Eq. 90 Separating the W operator in Eq.
77) implies that there is always a unique solution for the scalar potentials and, hence, electromagnetic fields for waveguides having perfect conductors. 2 Orthogonality Relations Between Electric Fields and Between Magnetic Fields Let’s consider two different TMmn and TMkl modes. The surface integral of the dot product of the transverse electric fields is given, using Eq. 71b), we can rewrite Eq. 83 S This result indicates that the transverse electric fields of two different TM modes are always orthogonal to each other.
Analysis Methods for RF, Microwave, and Millimeter-Wave Planar Transmission Line Structures by Cam Nguyen