Department of Electrical Engineering and Computer Science
EEL6482 Electromagnetic Theory
Homework/Project Set No. 4
Problem 20.
For a standard X-band rectangular waveguide, the waveguide dimension is a = 0.9 in
(22.86 mm) and b = 0.4 in (10.16 mm). Assume the waveguide operates at its dominant
mode TE10. The waveguide is air filled.
(1) Find the frequency range so that only the dominant mode can propagate;
(2) Determine its length for 2 s group delay at 10GHz;
(3) We use this waveguide to send a signal using dominant mode with
E y  (2  cos 2f1t  cos 2f 2 t ) cos 2f 3t at z = 0 passing through 1000 meter long.
Assume f1  100 MHz, f 2  400 MHz, and f 3  10 GHz. Find the expression of the
signal at the output of the waveguide (z = 1000 m).
Problem 21.
For the X-band waveguide discussed in last problem
(1) Find the analytical expression for dominant mode’s cutoff frequency, guide
wavelength, phase velocity, group velocity, characteristic impedance if the waveguide
is filled with lossless polystyrene ( 𝜖𝑟 = 2.56 , nonmagnetic ).
(2) When we connect an empty waveguide (air filled) with a lossless polystyrene filled
waveguide, we can design a binomial transformer (waveguide sections fully filled
with appropriate dielectrics) like what we did for multi-layer slabs. The dimensions
keep the same. If we need the maximum allowable reflection coefficient to be less
than 0.005, find the order, dielectric constant and length of each waveguide section.
(3) If the waveguide is made of copper (𝜎 = 5.76 × 107 𝑆/𝑚) and loss tangent for a
lossy polystyrene is 3 × 10−4 , find the attenuation constant (considering both
conductor and dielectric losses).
Problem 22.
The following figure shows the cross section of a waveguide. All the walls are perfect
magnetic conductor (PMC). For the TM modes in the waveguide,
(1) write down the Helmholtz equation that Ez must satisfy;
(2) write down the boundary conditions for Ez;
(3) solve the equation from (1) together with boundary conditions in (2) to find kx and
ky;
(4) find the expression for the propagation constant kz;
(5) find the expression for Ez;
(6) find the other field components;
(7) find the expression of guide wavelength;
(8) find the expression of cutoff frequency;
(9) find the expression of phase velocity;
(10) find the expression of group velocity;
(11) sketch the field pattern for TE12 and TM21 mode.
y
PMC
PMC

b
PMC
PMC
a
x
z
Problem 23.
An air-filled circular waveguide of radius a has a perfect magnetic conductor baffle
placed along its length at =0 extending from =0 to a, as shown in the figure. For TE
modes and TM modes,
(1) Solve Helmholtz equations together with boundary conditions, find the expressions
for Ez and Hz;
(2) Find the cutoff frequencies for TE and TM modes;
(3) Find the guide wavelength for TE and TM modes;
(4) What are the three lowest TE modes and three lowest TM modes? Which mode is the
dominant mode?
(5) When radius a is fixed at 2cm, find the frequency range so that only the dominant
mode can propagate.
Hint: You need to find the zeros of J 1/ 2 ( x)  0 , J 3 / 2 ( x)  0 , J 5 / 2 ( x)  0 ,
J 1 / 2 ( x)  0 , J 3 / 2 ( x)  0 , J 5 / 2 ( x)  0 etc. using fsolve and compare.
Problem 24.
For perfect magnetic conductor grounded dielectric waveguide as shown in the figure
below,
(1) Find the characteristic equation for TE modes using both field analysis and transverse
resonance method;
(2) Find the characteristic equation for TM modes using both field analysis and
x
d
0, 0
r0, 0
z
Perfect Magnetic Conductor
(3)
(4)
(5)
(6)
(7)
(8)
transverse resonance method;
Find the cutoff frequency for TE modes;
Find the cutoff frequency for TM modes;
Find all the field components for TE modes;
Find all the field components for TM modes;
When r=2.55, plot the dispersion diagram for the first four modes. You need to plot
re versus (d/0).
When r=2.55 and (d/0)=0.3, plot the Ey distribution for TE0 mode and
Hy distribution for TM0 mode.
Problem 25.
For the non-uniform parallel plate dielectric waveguide as shown in the following figure,
the dielectric constant distribution inside the slab is given by
 r  2.6  0.1(2 x / d ) 2
Write Matlab programs to do the following:
(1) Find the cutoff (d/values for TE0, TE1, TE2, TM0, TM1 and TM2 modes
numerically;
(2) Plot re versus (d/from cutoff value to 2.0) for the above 6 modes;
(3) When (d/= 1.2, plot Ey(x), Hz(x) and Hx(x) versus x (from -5 to 5for the
first 3 TE modes and Hy(x), Ez(x) and Ex(x) versus x (from -5 to 5for the first
3 TM modes

0 0
x
0 r(x) 0
d
z
0 0
Problem 26.
Use the effective dielectric constant method to analyze the mode with electric field
direction shown below for ridge waveguide. Here all the dielectric materials are assumed
to be non-magnetic. r1=6.6, r2=2.56, r3=2.10, a=2b, b=c, and a=0.80. Find re. You are
required to use both approaches of EDC method to solve the problem and compare your
results of different approaches.
E
b
c
r10
a
0
r20
r3