Page 326 - Engineering Electromagnetics, 8th Edition

308 ENGINEERING ELECTROMAGNETICS

where f is the second derivative of f 1 with respect to its argument. The results in

1
(17) can now be substituted into (13), obtaining
1 f = LC f (18)

ν 2 1 1
We now identify the wave velocity for lossless propagation, which is the condition
for equality in (18):

1
ν = √ (19)
LC

Performing the same procedure using f 2 (and its argument) leads to the same expres-
sion for ν.
The form of ν as expressed in Eq. (19) confirms our original expectation that the
wave velocity would be in some inverse proportion to L and C. The same result will
be true for current, as Eq. (12) under lossless conditions would lead to a solution of
the form identical to that of (14), with velocity given by (19). What is not known yet,
however, is the relation between voltage and current.
We have already found that voltage and current are related through the tele-
graphist’s equations, (5) and (8). These, under lossless conditions (R = G = 0),
become

∂V =−L ∂ I (20)
∂z ∂t

∂ I =−C ∂V (21)
∂z ∂t

Using the voltage function, we can substitute (14) into (20) and use the methods
demonstrated in (15) to write
∂ I 1 ∂V 1 ( f − f ) (22)

∂t =− L ∂z = Lν 1 2
We next integrate (22) over time, obtaining the current in terms of its forward and
backward propagating components:
1 z z
+
I(z, t) = f 1 t − − f 2 t + = I + I − (23)
Lν ν ν
In performing this integration, all integration constants are set to zero. The reason
for this, as demonstrated by (20) and (21), is that a time-varying voltage must lead
to a time-varying current, with the reverse also true. The factor 1/Lν appearing
in (23) multiplies voltage to obtain current, and so we identify the product Lν as
the characteristic impedance, Z 0 ,of the lossless line. Z 0 is defined as the ratio of