12-10.
1 PULSES AND TRANSIENTS
If a pulse of electromagnetic waves is sent along a transmission line, discontinuities
on the line reflect waves back (echoes), giving information about their location
(from delay time) and magnitude (from echo strength). The principle is the same
as with radar.
As an introduction to pulses and transients on transmission lines, let us
consider what happens when a dc impulse is impressed on a two-conductor line
of characteristic resistance R
o
with dc generator (battery) and load as in Fig.
12-34-' The load resistance is R
L
, the generator (battery) internal resistance R
G
,
and the generator (battery) open-circuit voltage VG' If the switch, (sw) is closed at
time I = 0, a voltage
(s) (2)
where [ = line length, m
v = velocity of propagation on line, ms- I
At I, the voltage at the load is the sum of the incident and rellected voltages,
or
v" = V
o
+ PLVO = (I + pJV
o
(3)
where PL = voltage rellection coefficient of load = (R
L
- Ro)/(R
L
+ R
o
).
The reflected voltage PLVO bounces back from the load toward the generator and
is superimposed on the voltage already on the line.
R, FIGURE 12-34
Transmission line or characteristic resistance R
o
and length I
with load resistance RI.' generator (bauery) internal resistance
R
G
and generator (battery) open-circuit voltagc V
G
applied at
---....>-11 l = 0 by closing switch (sw).
R
O
V
o
1
R,
SW
x
----+-
1<
f R. K. Moore, "Traveling Wave Engineering," p. 99, McGraw-HilI, New York. 1960.
528 WAVES AND TRANSMISSION LINES [CHAP. 12
At time 1= 21, the impulse arrives back at the generator, resulting in a total
voltage at the generator of
v", = V
o
+ pLV
O
+ PLPoV
o
= (I + PL + PLPo)Vo (4)
where Po = voltage reflection coefficient of generator = (R
o
- Ro)(R
o
+ R
o
).
An impulse of magnitude PLPo V
o
now travels from the generator to the load, and
the cycle repeats with a period 21,.
If the transmission line parameters are
Length = 300 m
Characteristic resistance R
o
= 100 11
Load resistance R
L
= 200 11
Generator internal resistance R
o
= 2011
Generator open-circuit voltage V
o
= 10 V
we have from (I) that the incident voltage
RoV
o
100 x 10
Vo = = 20 100 = 8.333 V
R
o
+ R
o
+
Assuming that the wave velocity on the line equals the velocity of light,
300
1 = = 10-
6
S = I )Js
, 3 x 10
For the reflection coefficients we have
(5)
(6)
200-100 I
200 + 100 =3
(7)
and
R
o
- R
o
20 - 100 2
Po = R
o
+ R
o
20 + 100 3
Thus, after the first reflection from the load the line voltage is
v = Vo(l + pJ = Vo(l +t) = 8.33(1) = 11.1 V
and after the first reflection from the generator it is
V = Vo(l + PL + PLPo) = 8.33(lf) = 9.26 V
(8)
(9)
(10)
After 6 or 7 )JS (6 or 7 bounces) the voltage settles to a value of 9.091 V, which is
substantially the same as the voltage after an infirtite time and is the value obtained
from de circuit theory as
R
L
200
V = R
L
+ R
o
V
o
= 200 + 20
10
= 9.091 V
(11)
SEC. 12.10.1] PULSES AND TRANSIENTS 529
This voltage is independent of the characteristic resistance R
o
of the transmission
line. However, during the interval of a few microseconds, while transient voltages
are bouncing back and forth, the characteristic resistance is a factor.
Consider next the transmission line in Fig. 12-3Sa which has the same line,
load, and dc generator (battery) values but is shorter (100 m) and has a different
switch. Moving the switch to connect the battery at t =0 for lOOns and then
instantaneously back to the upper contact, a lOO-ns pulse starts upward along the
line to the load. At the SO-m point (halfway between generator and load), the
voltage V. across the line as a function of time is shown by the solid line in Fig.
12-3Sb as produced by the BOUNCING PULSES Computer Program II App.
B. With each reflection, the pulse magnitude decreases and finally becomes zero'
Fig. 12-3Sb also shows (dashed line) the voltage fluctuation at the S0-m point as
200 rl
(a)
- - - -1 V.. if SW closed down 8B
I I II Cpennanenlly I I 8i
Y
I
y
_8.33 Y , I I __ J""
--- l!J!y ,- 29 I
I Initial I I I 1
9
0 I VVo Final
,
I pulse I 1 I I I I I Y,' 9.09
I I'3
V
o I Pulse after IPulse afterl Pulse after I (b)
I reflection reflectIon I 2d reflection 14
from load from geo from gen. ,\lfi
V
o
l.-_..J
/ _y
2 Pulse after 27 0 Final
-gV
o
2d reflection Vz.O
from load
-
5
0C--------;500=-------:,c:OOOs::==----""500'=::--'
lime (ns)
E
8.
E
g 5
;;;
>0'
f
100 rW\/'o
R
L
R
a
-
'00 rl
FIGURE 12-35
(0) DC transmission line 100 m long showing progress of lOO-ns pulse with time. (b) Magnitude of
pulse with time (solid line) as observed at the 5Om point and voltage magnitude when battery is
connected permanently to line at time t = 0 (dashed line). Pulse diagram generated by the BOUNCING
PULSES Program II or App. B.
t The BOUNCING PULSES program introduces a IS-ns rise time and a 15-05 falloff time so that
the pulse base width is 115 ns.
530 WAVES AND TRANSMISSION LINES
(CHAP. 12
produced by the BOUNCING PULSES program if the switch is connected to the
battery at r = 0 and remains connected, illustrating ihe transient response of a line
when it is energized as discussed in the first part of this section. The longer the
line and the smaller the losses, the longer the transients persist. However, if the
line has a matched load, transients disappear after reaching the load.
