Why do the odd and even modes in a coupled microstrip waveguide have different phase velocities? Is this also the case for a coplanar strips waveguide?
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Assuming that you are already familiar with propagation in a homogeneously filled waveguide then you should not be surprised that different modes may propagate with different velocities. A homogeneous waveguide can support both TE and TM modes, and if its cross-section is multiply connected a TEM wave can also propagate. In general, the phase velocities of the TE and TM modes, aside from some special symmetrical cases, are all different. The case of any TEM mode is special because it propagates without dispersion, just like a plane wave in free space or the voltage/current can propagate on a transmission line. All TEM modes propagate at the same phase velocity.
The situation changes the moment you fill the waveguide inhomogeneously with a dielectric. There are no true TE or TM modes that can propagate, and even if it is multiply-connected there is no TEM mode either. As you approach their cut-off frequencies the longitudinal components do get smaller and at cut-off become zero. when propagating all modes have longitudinal components, hence are called quasi-TE, quasi-TM, or quasi-TEM modes. They all have different propagating velocities. If one qTEM field is concentrated more in the dielectric than in the air then it will be slower moving than the one that is propagating more in the air than the dielectric. This is what happens in a coupled microstrip or coplanar line. Being multiply-connected, either one can support two types of qTEM mode, called even or odd. For the mode to exist its part in the air and its part in the dielectric must move together. Since the field distribution is different from mode to mode, the velocity should also be different.
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