Reflections on a mismatched transmission line
Reflections.doc (4/1/00)
Introduction
The transmission line equations are given by
V  z, t 
I  z, t 
 l
z
t
I  z, t 
V  z, t 
 c
z
t
(1)
(2)
Where, c is the per-unit-length capacitance obtained by taking the capacitance C of some section of
length z and dividing the capacitance C by that length, that is, c  C / z . All other sections of the line will
be characterized by this per-unit-length capacitance since the line is uniform. Similarly, for l, the per-unitlength inductance, where l  L / z . It is of importance to note that both c and l are static field parameters
that operate at dc (zero Hz). These parameters can be calculated using the theory of electrostatic and
magnetostatic field theory. In addition, these parameters (l and c) represent the transverse electromagnetic
(TEM) field structure for non-static excitation, that is, ac.
V(z,t) is the voltage between the two conductors comprising the transmission line, and I(z,t) is the current
that flows through these two conductors.
Time domain analysis of lossless transmission lines
This section will consider signal sources that are general in nature, that is, they have general waveshapes.
The transmission line equations given by equations 1 and 2 can be combined to yield two additional
equations where each new equation involves only the line voltage or the line current only. Differentiating
equation 1 with respect to distance z, and equation 2 with respect to time t, the following is obtained,
 2V ( z, t )
 2 I ( z, t )
 l
z 2
zt
2
2
 I ( z, t )
 V ( z, t )
 c
zt
t 2
(3)
(4)
Substituting equation 4 into equation 3 gives,
 2V ( z, t )
 2V ( z, t )
 lc
z 2
t 2
(5)
 2 I ( z, t )
 2 I ( z, t )

lc
z 2
t 2
(6)
similarly,
Phase velocity
The first thing to note about equations 5 and 6 is the product lc. This has the dimensions of 1 velocity2 .
Thus, the velocity u, of the signal travelling down a transmission line, is taken as,
u
1
lc
m/s
(7)
For transmission lines contained within a homogeneous medium characterized by the permiability  , and
the permittivity  ,
(8)
lc  
Solution to equations 5 and 6 are,
 z
 z
(9)
V ( z, t )  V   t    V   t  
 u
 u
 z
 z
(10)
I ( z, t )  I   t    I   t  
 u
 u
 z
 z
where, V   t   and I   t   represent forward travelling waves of voltage and current as sitting on a
 u
 u
point on each waveform, the point must move in the positive z direction for increasing time in order to keep
the arguments of the functions constant, that is, in order to stay sitting at the same position on the waveform.


Similarly, V   t 
z
z

 and I  t   represent backward travelling waves of voltage and current, moving
u
 u
in the negative z direction.
Characteristic impedance of the transmission line
The forward travelling current and voltage waves are related, as are the backward travelling waves of current
and voltage. This relationship is given by
  z 
 z
V   t    Z0  I   t  
 u
  u 
  z 
 z
V   t    Z0  I   t  
 u
  u 
(11)
(12)
where,
Z0 
l
c

(13)
and Z 0 is the characteristic impedance of the line. Impedance is generally used, but more strictly speaking
the term should be characteristic resistance of the line.
By substituting equations 11 and 12 back into the transmission line equations (equations 1 and 2), the
general form of the solutions to equations 5 and 6 become,
 z
 z
V ( z, t )  V   t    V   t  
 u
 u


V  z V  z
I ( z, t ) 
t   
t  
Z0  u  Z0  u 
(14)
(15)
Load reflection coefficient
L
Consider Figure 1 that shows a parallel wire transmission line of length L, that is terminated in a load
resistance RL , and driven by a voltage source having an open-circuit waveform of VS (t ) and an internal
resistance RS . It is required to determine the voltage at the beginning of the transmission line V  0, t  , the
voltage at the load termination V  L, t  , the current at the beginning of the transmission line I  0, t  and the
current into the load termination I  L, t  as functions of time.
Figure 1
The terminated transmission line
The voltage reflection coefficient at the load
L
Referring to Figure 2, at z = L, the total voltage in terms of the total current is
V  L, t   RL I ( L, t )
(16)
Expressing this condition in terms of forward and reflected waves,
 L
 L R
 L R
 L
V  t   V  t    L V  t    L V  t  
 u
 u  Z0
 u  Z0
 u
(17)
The left and right hand sides of equation 17 are not satisfied unless RL  Z 0 , in which case the line is
matched, and also, there is no backward wave (the V- component is zero). In order to resolve this
discrepancy brought about by equation 17, we must modify this equation. In order to do so, the following
statements are made;
 We have forward and backward travelling waves existing at the load
 Some resolving factor in equation 17 that absorbs the V- component is required.
For the transmission line a voltage reflection coefficient at the load can be defined. This is the ratio of the
backward and forward travelling waves at z = L, given by
L 
V  t  L / u 
V  t  L / u 
(18)
Figure 2
Voltage reflection coefficient at the load  L
Substituting equation 18 into 17, equation 17 becomes
R
 L
 L
V   t   1   L   L V   t   1   L 
Z0
 u
 u
(19)
Where,
RL  Z 0
1   L 
1   L 
(20)
or the reflection coefficient at the load expressed in terms of the characteristic impedance and the load
resistance is
L 
RL  Z 0
RL  Z 0
(21)
The introduction of the reflection coefficient into equation 17 resolves the apparent discrepancy.
It is of interest to note that a current coefficient at the load can also be defined from equation 15 and
equation 18. That is,
I  t  L / u 
 L  
I t  L / u 
(22)
Equation 22 shows that the current reflection coefficient is the negative of the voltage reflection coefficient
at the load.
Forward and backward voltage waves compared with forward and
backward current waves
Forward and backward voltage waves given by equation 14 have the same positive sign, that is
 z
 z
V f ( z, t )  V   t   and Vb  z, t   V   t   . This is because both are in the direction assumed for
 u
 u
 z
 z
the total voltage V ( z, t )  V   t    V   t   .
 u
 u
However, forward and backward current waves have opposite signs. This is shown by equation 15, where
I ( z, t ) 
V  z  V  z 
V  z 
.
The
forward
current
wave
I
(
z
,
t
)

t


t

f


 t   is directed in the


Z0  u  Z0  u 
Z0  u 
positive z direction as is the total line current, but the backward wave I b ( z, t )  
V  z 
 t   is directed in
Z0  u 
the opposite direction (negative z).
The applet shows the reflected wave at the load for both a pulse and more instructively, for a ramp. The
reflection of waves by the load discontinuity can be viewed as a mirror that produces, as a reflected wave V -,
a replica of V+, that is “flipped around” and all points on the V- waveform are the corresponding points on
the V+ waveform multiplied by the load reflection coefficient  L . The total voltage at the load is the sum of
all of the individual voltages existing at the load at each instant in time. The applet shows this total voltage
for all instants over the time that the pulse or ramp traverses back and forth down the line for three
reflections.
The voltage reflection coefficient at the source
S
Consider Figure 3, at z = 0, the portion of the transmission line at the source.
Figure 3
Voltage reflection coefficient at the source  S
When the source is connected to the line a forward travelling wave is propagated down the line. There will
be no backward travelling wave appearing at the source until the initial forward travelling wave has reached
the load and produced a backward travelling wave that has returned back to the source. The time for the
forward wave to reach the load is T = L/u. The portion of the reflected wave requires an additional time T =
L/u to move back from the load to the source. Therefore, for 0  t  2L u , no backward travelling waves
will appear at the source z = 0 and for any time less than 2L/u the total voltage and current at z = 0 will
consist only of the forward travelling wave or if the wave has fully emerged, nothing at all. The applet
allows for the pulse or ramp width and line length to be varied so that an emerging pulse can still be
emerging by the time the reflected voltage wave has returned (back and forth several times if so desired).
Thus, the total voltage and current at the source can be written as
 0
V  0, t   V   t  
 u
0  t  2L u
 0
V  t  
 0
 u
I  0, t   I   t   
Z0
 u
(23)
0  t  2L u
(24)
Since the ratio of the total voltage and current on the line is Z0 for 0  t  2L u , the line appears to have an
input resistance Z0 over this time-interval giving rise to a reduction in the value of the source voltage. That
is, the forward travelling wave is related to the source voltage VS(t) by
V  0, t   VS  t   RS I  0, t   VS  t  
RS
V  0, t 
Zo
so that,
V  0, t  
Z0
VS  t 
Z 0  RS
0  t<
2L
u
(25)
The applet has V  0, t   20 pixels and is fixed at this value regardless of the ratio
Z0
.
Z 0  RS
At the source, similar to the case for the load, a reflection coefficient can be obtained which is given by
S 
RS  Z 0
RS  Z 0
(26)
This source reflection coefficient is the ratio of the incoming incident wave (which is the reflected wave
from the load) and the reflected portion of this wave (that portion that is sent back towards the load). A
forward travelling wave is therefore initiated as the source in the same manner as at the load. As only
resistive dividers are used at the source and at the load, the waveshapes are retained and only reduced in
amplitude by a factor  S . The applet shows the total voltage at the source as the sum of the values of the
individual voltage waves existing at the source at any time during the course of three reflections.
The process of repeated reflections continues as re-reflections at the source and load. At any time the total
voltage (or current) at any point on the line is the sum of the values of the individual voltage waves (current
waves) existing on the line at that point and time. The applet shows this as the “distributed” voltage
history waveform where one is sitting at some point on the front of the voltage waveform and “riding” the
wave.
AT 04/01/2000
References:
“Introduction to Electromagnetic Fields”, Clayton R.Paul, Keith W. Whites, Syed A. Nasar, WCB McGrawHill, third edition, 1998, ISBN: 0-07-046083-3. International Edition ISBN: 0-07-115478-7