1) Derive the solution for the fields for the TMn modes for a parallel plate waveguide using (3.23) - (3.25) and the appropriate boundary conditions.

2) You are given a parallel plate waveguide with a plate separation of 1 cm and filled with Alumina Nitride, which has a dielectric constant of 9.0.
a)Determine the cutoff frequencies for the first three TMn and TEn modes for the parallel plate waveguide.
b) Plot the real part of the wavenumbers for the first three TMn modes and first two TEn modes as a function of frequency from 0 to 15 GHz (use Mathcad, Excel, Maple, Matlab, or something equivalent to generate the plot).
b) Plot the imaginary part of the wavenumbers for the first three TMn modes and first two TEn modes as a function of frequency from 0 to 15 GHz (use Mathcad, Excel, Maple, Matlab, or something equivalent to generate the plot).
c) Plot the phase and group velocities of the first 3 TMn and the first 2 TEn modes versus frequency from 0 to 15 GHz.
d) Plot the characteristic impedance of the first 3 TMn and the first 2 TEn modes versus frequency from 0 to 15 GHz.
e) When designing a waveguide system, it is often desired to excite a single mode. Based on arguments of impedance, phase velocity, and wave number for the parallel plate waveguide, what reasoning would support single mode operation? Secondly, for the parallel plate waveguide, which mode is most desirable to excite? Why?

3) A parallel plate waveguide with separation of 1 mm and plate width of 1 cm is filled with 98 % Alumina with a dielectric constant of 9.8 and a loss tangent of 0.001. The plates are made with 1 mil thick copper, which has a conductivity of 5.8e+07 mhos/m. A TEM mode is propagating along this line.
a) Determine the guide wavelength for the TEM mode
b) Determine the attenuation over one guide wavelength due to dielectric loss at 1 GHz and 10 GHz.
c) Determine the attenuation over one guide wavelength due to conductor loss at 1 GHz and 10 GHz.


4) a) Using (3.197), determine the width of a microstrip line to be printed on a 10 mil thick Alumina substrate with relative permittivity of 9.8 such that the microstrip line has a characteristic impedance of 50 ohms.
b) Determine the effective dielectric constant of the microstrip line.
c) Determine the wave speed of the quasi-TEM mode supported by the line.
d) Determine the guide wavelength (or effective wavelength) for the quasi-TEM mode supported by the line.
e) If the loss tangent is 0.001 and the conductors are copper, determine the dielectric loss and the conductor loss at 10 GHz.
f) If the microstrip line were assumed to be a parallel plate waveguide, at what frequency will a higher order mode begin to propagate? How would this affect your chose of dimensions versus frequency for this substrate?

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