Electrical Power Systems

6.6 VOLTAGE WAVES
Characteristics and Performance of Transmission Lines 141
By using eqns. (6.26) and (6.27), we obtain,
V(x) = C1.e ax .e jbx + C2.e –ax .e –jbx
...(6.49)
Transforming eqn. (6.49) to time domain, the instantaneous voltage as a function of t and
x becomes
V(t, x) = 2 Real {C 1e ax e j(wt + bx) } + 2 Real {C 2.e –ax .e j(wt – bx) } ...(6.50)
Note that V(x) in eqn. (6.49) is the rms phasor value of voltage at any point along the line.
As x increases (moving from receiving end to sending end), the first term becomes larger
because of e ax and is called the incident wave. The second term e –ax becomes smaller and is
called the reflected wave. At any point along the line, voltage is the sum of two components.
V(t, x) = V 1 (t, x) + V 2 (t, x) ...(6.51)
where
V 1 (t, x) = 2 C 1 e ax cos(w t + bx) ...(6.52)
V 2 (t, x) = 2 C 2 e –ax cos(wt – bx) ...(6.53)
As we move along the line, eqns. (6.52) and (6.53) behave like travelling waves. Now
consider the reflected wave V 2 (t, x) and imagine that we are riding along with the wave. or
observing instantaneous value, peak amplitude requires that
wt – bx = 2 k p
\ x = w
b
The speed can be given as
\
dx
dt
= w
b
2k
p
t - t
b
Thus, the velocity of propagation is given by v = w
b
= 2 f
p
b
...(6.54)
...(6.55)
...(6.56)
A complete voltage cycle along the line corresponds to a change of 2p radian in the angular
argument bx. The corresponding line length is defined as the wavelength. If b is expressed in
rad/mt.
B l = 2p
\ l = 2p
b
...(6.57)
When line losses are neglected, i.e., when g = 0 and g = 0, then the real part of the
propagation constant a = 0. rom eqn. (6.27)
g = a + jb = zy = ( r + jwL) ( g + jwc) \ b = w LC ...(6.58)