Wave Propagation in Linear Media | re-examined

3Wave propagation in electromagnetic transmission lines
Losses are accounted for by a series resistance R 0 andashunt conductance G 0 . The propagation
constant aswell as the characteristic impedance then become complex,
k = p (!L 0 , jR 0 )(!C 0 , jG 0 )= +j (3.10)
Z 0 =
s !L 0 , jR 0
!C 0 , jG 0 = Z0 + jX0 : (3.11)
With these de nitions, we nd the propagated and stored energies
P =Re UI
2
W = 1
4
and subsequently the energy velocity
= 1
2 I2 Z0
, I 2 L 0 + U 2 C 0 = 1
4 I2 , L 0 + jZ 0j 2 C 0
ve = P
W =
(3.12)
(3.13)
2Z0
L0 +(Z0 2 +X0 2 : (3.14)
)C0 We can now use the relation k = Z 0(!C 0 , jG 0 ) to express Z0 in terms of and . Furthermore,
we take the square of (3.10) to obtain an expression for . The absolute value of Z 0
is readily found from (3.11). Taking all these relations together and inserting them into the
expression for the energy velocity, we get after some algebra the astonishing result
ve = ! ; (3.15)
which is the phase velocity de ned by the real part of the complex dispersion relation. This
observation has also been reported by Mainardi [32] (see section 1.4). Needless to emphasise
that in the absence of losses (R 0 = G 0 = 0), the phase, group, and energy velocities are
identical, and no dispersion occurs. In this connection and for the sake of completeness, we
also recall the well-known condition R 0 C 0 = L 0 G 0 for a dispersion-free transmission line even
in the presence of losses.
Remark (Frequency dependence of losses) At rst sight, the propagation constant
(3.10) reveals an interesting property for ! !1. Within the scope of this model, the
real part is roughly proportional to !, whereas the imaginary part remains constant.
This indicates the existence of a distinct wave front velocity (see section 1.3). Thus a
step in a signal may be attenuated, but the wave front itself would never be distorted.
Unfortunately, the assumption of frequency-independent losses are no longer justi ed
at high frequencies. In fact, the series resistance is increased by the skin e ect (thus
R 0 / p !), whereas the shunt conductance is deteriorated by dielectric losses (G 0 / !).
Consequently, the imaginary part will increase with growing frequency, and the wave
front will be distorted.
An analysis based on an equivalent circuit is incomplete without discussion of its limits of
validity. The derivation of the equivalent circuit relies on the fact that adjacent in nitesimal
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