Wave Propagation in Linear Media | re-examined

3.1 Model of a transmission line
the foundation of the well-known correspondence @
@t 7! j! for stationary oscillations and
consequently the basis for the equivalent circuit in g. 3.1.
We then obtain easily the propagation constant
and the characteristic impedance
k = p X 0 B 0 (3.4)
Z0 = U
I =
r X 0
: (3.5)
B0 Wave propagation is possible only if X 0 B 0 > 0, i. e. the signs of the distributed reactances
and susceptances must be equal. From the dispersion relation (3.4) we nd the reciprocal of
the group velocity
1
vg
= dk
d!
= 1
2k
X0 dB0
d!
+ B0 dX0
d!
: (3.6)
Remark (Direction of propagation) As can be seen from (3.6), X 0 > 0 and B 0 > 0
result in sign vp = sign vg, so that the wave crests and wave packets move in the same
direction. If we choose negative impedances, X 0 < 0 and B 0 < 0, respectively, the phase
and group velocities have opposite signs, which ischaracteristic for a backward wave.
In the absence of re ection, which was our initial assumption, the average of the propagated
energy is simply determined by the amplitude of the current I and the characteristic
impedance,
P = I2
2
r X 0
: (3.7)
B0 On the other hand, the mean stored energy per unit length, with U as voltage amplitude, is
given by
W = 1
4
2 dX0 2 dB0
I + U
d! d!
I2 dX0 dB0
= B0 + X0
4B0 d! d!
; (3.8)
where we used the de nition of the characteristic impedance (3.5) to obtain the second expression.
From (3.7) and (3.8) we see immediately that the energy velocity ve = P=W equals
the group velocity (3.6).
Now let us brie y examine the situation when the transmission line is characterised by a
frequency-independent inductance L 0 and a capacitance C 0 per unit length. We then have
X 0 = !L 0 ; B 0 = !C 0 : (3.9)
33