ECE 407 – ELECTROMAGNETIC COMPATIBILITYAdvanced Electromagnetic Fields and Waves I 1. (50 pts)...

Question

ECE 407 – ELECTROMAGNETIC COMPATIBILITYAdvanced Electromagnetic Fields and Waves I  1. (50 pts) Let’s consider a homogeneous Tellegen medium. A Tellegen medium isi-isotropic, and is usually described using the constitutive relations:�⃗⃗� = ?�⃗� + ?�⃗⃗� (1) �⃗� = ?�⃗� + ?�⃗⃗� (2) Here ?, ?, and ? are the constitutive parameters with ?2 do not depend on the position since the medium is homogenous. Assume there no sources in the region of interest.  a) (5 pts) Show that (1) and (2) may be written as:�⃗⃗� = ?�⃗� + ?�⃗� (3) �⃗⃗� = −?�⃗� + ?�⃗� (4) Here ?, ?, and ? are the new constitutive parameters. Determine these in terms of the old parameters ?, ?, and ?. ) (5 pts) Using (3) show that ∇ ∙ �⃗� = 0.c) (10 pts) Derive the following wave (Helmholtz) equation for �⃗� :∇2�⃗� + ?2�⃗� = 0where ?2 = ?2(?? − ?2).d) (10 pts) Verify that the following plane wave is a solution of the Helmholtzequation:�⃗� = �̂�?0?−???e) (15 pts) Use Maxwell’s equations to derive the following expression for �⃗⃗� :�⃗⃗� = (−�̂�??+ �̂�???)?0?−???Is this wave TEM? f) (5 pts) The phase fronts for this wave propagate in the z-direction. Does thepower flows in the z-direction as well?2. (50 pts) A parallel-plate waveguide is formed by placing infinitely large PECplates in the planes ? = 0 and ? = ?. The parallel-plate waveguide is filled by alossless material with permittivity ? and permeability ?. Let consider the TMzmodes propagating in the z-direction. The magnetic field can be expressed as a combination of the following two modes: �⃗⃗� + = �̂�?0???(−? sin ?−? cos?)�⃗⃗� − = �̂�?0???(? sin ?−? cos?)a) (5 pts) Show by adding �⃗⃗� + and �⃗⃗� − that the total magnetic field is equal to�⃗⃗� = �̂�?0 cos(???) ?−??) (10 pts) Derive �⃗�  and then ?? by using the boundary conditions at the twometallic plates.c) (15 pts) Derive ? and the cut-off frequencies of the modes, ??.d) (20 points) If the plates are not PEC, but instead have a surface resistance ℛderive the attenuation constant due to imperfect conductors, ??. (Hint: use thepertuation method).

Bhaumik · Accepted Answer

Advanced Electromagnetic Fields and Waves I
Solutions
Solution for Part 1:
a) Expressing the constitutive relations in terms of new parameters :
Given: 
                     (1)
                     (2)
The following must be expressed: 
                      (3)
                    (4)
The process is about identifying how to translate the parameters  into 
By comparing, we can set:
·   (from the  term in ),
· (from the  term in ),
· (from the term in ).
Thus, we have the relations: 
  , , and 
b) Showing 
Using the modified form: 
As there are no free charges in the region, Gauss’s law for the electric displacement field is:
Substituting  into this equation:
Since the medium is homogeneous, the parameters  and  are constants, so the divergence of this expression becomes:
Now, using the facts in the absence of magnetic monopoles, we have:
Thus, the equation reduces to:
Since    for a non-trivial medium, we conclude:
This shows that the electric field  is solenoidal (divergence-free), consistent with the problem’s assumptions.
c) Deriving the wave (Helmholtz) equation for 
From Maxwell’s equation, we know that:
Substituting  into this:
Since time-harmonic fields are assumed (), this becomes:
Similarly, for the curl of H:
Now, let’s derive the wave equation,
Taking the curl of 
Using the vector identity  and knowing  , this simplifies to:
To obtain Helmholtz equation for  using the constitutive relations  and  and assuming time-harmonic fields (), we substitute Maxwell’s curl equations. 
d) Verifying the plane wave solution:
Given: 
First, calculate Since, the wave is propagating in the z-direction,

Please solve the two problems provide. Make sure to include all steps and details leading to the answers

Solution

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