Wave Propagation in Linear Media | re-examined

3.3 Re ection due to termination mismatch
1.5
1.25
1
0.75
0.5
0.25
Ω
0 2 4 6 8 10 12 14
Figure 3.3: Dispersion relations for the delay line of g. 3.2 for L =1. The solid curve corresponds
to the symmetric low-pass mode, the dashed curve to the antisymmetric band-pass mode. The thin
lines are the linear approximations for low and high frequencies. Note that the diagonal line marks
the velocity of light, =K = 1. Thus the band-pass mode exhibits a phase velocity greater than c
slightly above the lower frequency limit. It is clear to see that the relation vp vg = c 2 mentioned in
the previous chapter is not valid here.
with X(!) taken from (3.23). The proof of the identity ve = vg is then straightforward but
tedious.
3.3 Re ection due to termination mismatch
If we want to examine evanescence, it is not very sensible to consider a steady-state excitation
of the transmission line without re ection. In such a case, the characteristic impedance (3.5)
as well as the wave number (3.4) are imaginary, resulting in total internal re ection, but no
energy transport. Therefore we deem it meaningful to study the simple example of re ection
due to an ohmic load. The transmission line is again described by the equivalent circuit of
g. 3.1, but has now a nite length and a terminating resistor ( g. 3.4).
Before we start with evanescence, let us brie y investigate the behaviour of the pass band
to gain some physical insight. If we solve the di erential equations (3.2) for the boundary
condition U2 = RI2 at the end of the line, we obtain the well-known spatial voltage and
current evolution with respect to the voltage across the load,
U = U2 cos k(l , x)+j Z0
R
I = U2
Z0
Z0
R
sin k(l , x)
cos k(l , x)+jsin k(l , x) :
37
K
(3.27)