1
ECE 6341
Spring 2009
HW 2
1) A grounded dielectric slab of Teflon has a relative permittivity of 2.2 and a thickness of
50 mils (1.534 mm). Use a numerical search to find the normalized phase constant 0
/k
β
of the TM0 surface-wave mode at the following frequencies: 1 GHz, 10 GHz, 100 GHz.
2) Assume that we have the same grounded slab as in the previous problem, but now we are
interested in the TM1 mode. First, find the cutoff frequency of this mode. Then, by
numerically searching for the improper surface-wave solutions, find the splitting point
frequency fs. Then, for a frequency that is 10% lower than the splitting point frequency
(i.e., f = 0.9 fs) numerically search to find the complex TM1 leaky-mode solution (the one
that has a positive attenuation constant). You can use whatever numerical search routine
you want. Note that the secant method works in the complex plane, and is usually a good
choice. This method is allows you to find the complex roots (zeros) of a complex
function f (z). The method is represented by the following iterative formula (which
requires two initial guesses z0 and z1)
()
(
)
() ( )
11
1
n
nnnn
nn
fz
zzzz fz fz
+−
−
⎛⎞
=− − ⎜⎟
⎜⎟
−
⎝⎠
.
3) A TEx leaky mode has a field on the interface (x = 0) due to a line source at z = 0 that is
represented as
(
)
0, LW
z
jk z
y
EzAe
−
=
where LW
zzz
kj
β
α
=− and A is an amplitude constant. The exact electric field above the
interface is
.
where
()
1/2
22
0xz
kkk=− . The branch of the square root is chosen so that that Im 0.
x
k
<
Show that
() ()
1
,0, ,
2xz
jk x jk z
yyzz
Exz E ke e dk
π
∞
−−
−∞
=∫
()
()
lw
2
lw 2
0, 2 z
yz
zz
k
Ek jA
kk
⎛⎞
⎜⎟
=− ⎜⎟
−
⎝⎠
