Answer Happy

In this problem we will consider a waveguide made of two parallel, perfectly con- ducting plates. Between the plates is a nonmagnetic dielectric, and as in lecture, let's say that the bottom plate lies in the yz-plane at the position x = 0, and the top plate lies parallel to the yz-plane at the position x = d = 3.3 mm. The dielectric has a relative permittivity of 2.25. Modes in this waveguide propagate in the z-direction. (a) Sketch the dispersion relation for the TE modes associated with this waveguide for frequencies from 0 to 100 GHz. Calculate and label the cutoff frequencies in this range. Briefly explain the meaning of the dispersion curves you have sketched. For parts (b)-(e) of this problem, let's say that the waveguide is being used for a high-speed microwave data link, and an antenna radiating an electromagnetic wave with frequency 40 GHz excites TE modes in the waveguide. (b) One of the modes that this 40-GHz wave excites will propagate without loss. Find an expression for the electric field associated with this mode as a function of x, z, and t. (Assume that the electric field of this mode has a peak amplitude of Eo.) (c) Consider the time-average power density carried by the modes of this waveg- uide. What is the time-average power density carried by the TE₁ mode? In this part of the problem, you may assume that the electric field in the TE₁ mode has a peak amplitude of Eo. (HINT: You can calculate the time-average power density as we did for uniform plane electromagnetic waves, that is, with the time-average Poynting vector, Sav=Re{Ẽ × H*}. Note that unlike the uniform plane wave, the TE₁ mode has a spatially varying time-average power density!) (d) Lastly, let's say that we want to use our waveguide as a "mode filter". The 40-GHz electromagnetic wave coupled into our waveguide excites a number of TEm modes. After propagation through some length L of the waveguide, we want the power in any of the TEm modes, where m > 1, to be attenuated by at least 60 dB. What should I be? (Note: an attenuation of 60 dB corresponds to a reduction in the amplitude by a factor or 1/1000.)