University of Surrey, Department of Physics
2nd Year Electromagnetism Laboratory
Transmission Lines ... or ...
There’s more than one way to measure the speed of light! - 2 weeks
1. Introduction
A transmission line is any structure which is used to guide the flow of energy
from one point to another, for example, a co-axial cable or a waveguide.
Electromagnetic waves (i.e., variations in the vectors E and B) propagate down the
wire at a determined speed c, which can be related to the speed of light, c0. Good
discussions of transmission lines can be found in Refs. [1] and [2]. Transmission
lines provide us with a variety of entirely electrical methods for measuring c, the
results of which can be compared with other optical methods, such as the
Michelson-Morely experiment.
Transmission lines are also important for studying the fundamental wave
properties of reflection and superposition. The same equations occur in many
branches of physics where wave propagation is involved (e.g., quantum mechanics,
vibrating strings, ultrasound imaging, lens manufacture) and so a good
understanding of the concepts involved will help you in other areas, too.
2. Aims of the Practical
This practical will introduce you to the basic concepts of transmission line theory.
You will perform a set of three experiments to deduce the value of c and in the
process learn the following ideas and practical skills:
 use of pulse generator and oscilloscope;
 the idea of a terminating impedance;
 use of a vector impedance meter;
 analysis of data and interpretation in terms of a physical model;
 estimation of errors.
3. Theory
3.1 Voltage and Current Waves
The statement that waves can propagate up and down a transmission line seems
strange at first sight. After all, if you connect one terminal of the line to a positive
voltage (+V) and the other to earth and put a load on the other end (see Fig. 1), you
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would expect the voltage all the way along the top line to be constant at +V and to
be 0 all the way along the bottom line. After all, that is the purpose of transmission
lines: to transmit the power without losing any on the way! If the voltage is an
alternating one, the situation should be the same — the top line should have a
voltage alternating everywhere at cos(t), whilst the bottom one remains at 0V.
Where are these waves that the Introduction was talking about?
+V
+V
0
Figure 1: Elementary idea of a transmission line
Transmission lines are not simply the ideal, isolated conductors which the above
argument implies. There are four aspects which complicate life:
(i) The lines are made of real wire with a finite resistance.
(ii) The wires also have a finite (though small) self-inductance.
(iii) The wires lie next to each other in the transmission line. This means that
currents and charges moving in one wire influence the voltage in the other.
There is a finite capacitance between the two wires.
(iv) The material isolating one wire from the other is not a perfect insulator and
there is a leakage current between the two wires.
If we redraw a short section of the line in Fig. 1 as the equivalent circuit for a real
transmission line, we get the diagram of Fig. 2(a). In any unit cell of length x, we
have a resistance R, an inductance L, a capacitance C and a leakage resistance
G. In practice, R and G are very small and can be taken to be zero for the
purposes of this theory. However, C and L are significant — see Fig. 2(b).
Consider the current and voltage changes between two adjacent unit cells. V
changes because of the back e.m.f. caused by the inductor. I changes because charge
moves onto and off the capacitor. Remembering that the normal formula for back
e.m.f is E= L(dI / dt) and Q = CV, so that I = C(dV / dt), we have
V  E  L 
I
t
and
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I = C 
V
t
.
[1]
(a)
R
L
V(0)
G
C
Unit Cell
x
(b)
I(x)
L
I(x+x)
V(0)
C
V(x)
V(x+x)
Unit Cell
Figure 2: (a) Realistic model of transmission line with resistive, inductive, capacitative and
leakage characteristics; (b) simplified model without “losses” (i.e., leakage currents
and resistance)
Now a uniform transmission line will have constant inductance per unit length and
capacitance per unit length; we shall call these L and C. So L = L0x and C =
C0x. Substituting these values into [1] and taking the limit as x0, we obtain the
following set of coupled differential equations, sometimes called the Telegraphist’s
Equations.
V
I
  L0
x
t
and
I
V
  C0
.
x
t
[2]
From these two equations, we can obtain very simply the wave equation.
 2V
 2V
.

L
C
0 0
x 2
t 2
[3]
This describes travelling waves, with a speed c = 1/(L0C0), i.e., a speed which
depends on the electrical properties of the transmission line. Waves can travel both
forwards and backwards along the transmission line and the individual solutions
(which you should recognise from your course on wave motion) may be written as:
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V ( x, t )  V ei (t  kx )
forward travelling
V ( x, t )  V ei (t  kx )
backward travelling
[4]
It is possible to write a similar set of equations for the current, too. We find that
the current and voltage are connected by a quantity Z0 = (L0/C0), which is called
the characteristic impedance of the transmission line.
I ( x, t ) 
V i (t  kx )
e
Z0
forward travelling
V
I ( x, t )    ei (t  kx )
Z0
[5]
backward travelling (N. B.  )
3.2 Reflected Waves and Standing Waves
It is easy to see how the forward travelling wave might be produced by a signal
generator connected to one end of the transmission line. But what is the significance
of the backward travelling component? It turns out that this is caused by a reflection
of the forward (incident) wave at the termination of the line (or some discontinuity
along the line).
We can quantify the reflection by defining a reflection coefficient .
( x ) 
V e i (t  kx )
Reflected voltage at point x
  i (t  kx )
Incident voltage at point x
V e

V 2 ikx
e .
V
[6]
If we choose x=0 at the termination of the wire, then the reflection coefficient there
is simply 0=(V/V). It can be shown that the amount of reflection is directly
related to the difference in impedance between the transmission line and whatever is
at the end of it (the load ZL).
0 
ZL  Z0
.
ZL  Z0
In this practical, we shall concentrate on three important cases:
(i) Short circuit termination:
ZL = 0  0 = 1
The wave is reflected with the same amplitude and a 180 change in phase.
(ii) Open circuit termination:
ZL =   0 = 1
The wave is reflected with the same amplitude and no change in phase.
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[7]
(iii) Termination with characteristic impedance: ZL = Z0  0 = 0
No reflection occurs. All the incident power is absorbed in the terminating
impedance.
If we make a very short pulse of voltage, then it is possible to see a reflected
signal. This is the basis of Experiment 1. However, normally, it is not possible to
observe the incident and reflected waves separately. Any measuring device, such as
an oscilloscope wil measure only the sum of the forward and backward travelling
waves at a given point. I.e., with a ’scope, we measure Re[V(x)]. It can be shown
this sum is given by

Re[V ( x)]  Re V ei (t  kx)  Vei (t  kx)



V 2 0 cos kx eit  (1  0 )ei (t  kx)

=
V - 2  sin kx eit  (1   )ei (t  kx)
0
0
 

0  0

.
[8]
0  0
This means that the sum of two waves travelling in opposite directions is, in
general, a stationary wave and a travelling wave. Notice that when 0 = 1, there is
no travelling wave component and we have only a stationary wave pattern, whose
amplitude varies with x. This is the basis of Experiment 2.
3.3 The difference between impedance and characteristic impedance
In a transmission line where only a forward-going wave exists, we have (as in
Eqs. [4] and [5])
V ( x )  V e i (t  kx )
and
I( x) 
V i (t  kx )
e
.
Z0
[9]
I.e., V(x) = I(x) Z0. This should remind you of Ohm’s Law, V=IR. In general,
however, when we have the reflected wave to consider as well, V(x)  I(x) Z0. Let
us instead define a new quantity called the impedance of the transmission line at a
point x, which does obey Ohm’s Law.
Impedance at point x =
Voltage at point x
Current at point x
[10]
Z ( x )  V ( x ) / I ( x ).
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This means that for a uniform transmission line, Z(x) varies along the length and, at
the end of the line, has the value of the termination impedance. Fig. 3 summarises
the situation. It can be shown that the variation of the impedance in the transmission
line itself is given by
Z( x)  Z0 
Z L  iZ 0 tan kx
.
Z 0  iZ L tan kx
[11]
Consider now the impedance when a line of characteristic impedance Z0 has the
three different termination described earlier.
 Z(x) = iZ0 tan kx
(i) Short circuit termination:
ZL = 0
(ii) Open circuit termination:
ZL =   Z(x) = iZ0 cot kx
(iii) Termination with characteristic impedance: ZL = Z0  Z(x) = Z0
The impedance can be measured using a piece of equipment called a vector
impedance meter. This is the subject of Experiment 3.
I(x)
I(0) = IL
~
V(0) = VL
V(x)
VL = IL ZL
x=0
V(x) = I(x) Z(x)
x
Figure 3: Concept of the impedance at a point. Note that x=0 is defined conventionally at the
termination of the transmission line, so that V(0)=VL and I(0)=IL. This has the
consequence that for a point at a distance d “upstream” of the termination, x=d and
the formulae after Eq. [11] tend to be written as Z(d) = -iZ0 tan (kd) = iZ0 tan kd,
etc.
3.4 Finding the speed of light
In Experiment 1, we find the speed of light by measuring the time between the
incident and reflected pulse of voltage. We have
 
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2l
,
c
[12]
where l is the length of the transmission line and is the length of time taken for the
pulse of voltage to make the return journey. Note that this is the speed of
propagation in the transmission line. It can be shown that for any pair of parallel
conductors arranged to form a transmission lines, we have
LC     0  r   0  r
[13]
and, since the speed of light in vacuo is given by c0 = 1/(00), we have
c0  c  r  r .
[14]
In Experiment 2, you will use a signal generator to provide a voltage oscillating
at angular frequency . This leads to a standing wave pattern in which the voltage
amplitude is V+ cos kx. By measuring the voltage at various different points x, you
will be able to deduce k. It can be shown that

k

1
 c.
L0 C0
[15]
By measuring k for several values of , you can find c and hence c0, as above.
Experiment 3 gives a different method of finding k. Here, the impedance is
measured for a constant length of transmission line, but with a large number of
different frequencies. The impedance depends on kx as in Eq. [11] and hence k and
c can be found.
4. Experimental Procedure
There may be two groups performing this practical at any one time. Because the
apparatus is expensive, it must be shared. Therefore, Group I should perform the
experiments in the order 1, 2, 3. Group II should do them in the order 3, 2, 1.
4.1 Experiment 1: Pulses and Reflections
1.
Connect the pulse generator to the oscilloscope. Set the pulse generator to the
following settings: period 150 ns, delay 600 ns, width 10–100 ns. Set the output
knob to about 1/3 of maximum. The lead from the oscilloscope should be
plugged into the connection labelled A+B on the pulse generator, with the
switch in the up (A) position. Set the timebase of the ’scope to 0.2 s/div. and
the vertical scale to 0.5 V/div. Automatic triggering should be used (not
external), with the TV switch set to off.
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2.
Play around with these settings to ascertain the function of all the different
controls.
3.
Now connect in the long transmission line on the reel, to give the arrangement
of Fig. 3. Do five successive experiments, terminating your transmission line
with (i) the 50 impedance (N.B. the characteristic impedance Z0 of the cable
is 50), (ii) an open circuit, (iii) a short circuit, (iv) the capacitor provided, (v)
the inductor provided.
In each case, sketch the traces you obtain on the oscilloscope, paying attention
to your x- and y-scales
Explain the traces by referring to Eq. [7]. (The case of the capacitor and
inductor are not covered in the theory section — ask a demonstrator.)
4.
Go back to the open circuit termination. Measure on the ’scope screen the time
between the emission of the pulse and the reception of the reflected pulse. Use
Eqs. [12]–[14] to estimate the speed of light, given that r = 2.30.1 and
r=10.0001.
Oscilloscope
Coaxial Transmission Line
Pulse Generator
ZL
Figure 3: Apparatus for Experiment 1
4.2 Experiment 2: Stationary Waves
1.
The wooden board with 20 parallel wires is in effect a single transmission line
of length 20m. It has been built in this way in order to be able to “tap into” the
wire at different points along the path of the signal. The inner tapping points
are connections to the earth part of the transmission line, whilst the outer
terminals correspond to the inner core of the coax. The equivalent piece of
cable with wires exposed is available for you to see.
Set up the signal generator and oscilloscope so that the circuit is equivalent to
that shown in Fig. 4. Initially, terminate the cable by an open circuit.
2.
Set the signal generator to a frequency between 10 and 20 MHz. Measure the
peak-to-peak voltage at all the tapping points. Plot your results.
3.
Perform the same experiment again with the cable terminated by the
characteristic impedance.
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4.
Explain your results by reference to Eq.[8] or use c = f.
5.
From the experimentally measured wavelength of the stationary waves,
calculate the speed of light using Eq. [15].
Oscilloscope
Ch. B
Tapping Points
Ch. A
ZL
Signal Generator
Coaxial Transmission Line
Figure 4: Apparatus for Experiment 2
4.3 Experiment 3: Impedance Measurement
1.
Disconnect everything from the wooden board with the 20 parallel lengths of
wire. You will now measure the impedance of the transmission line on the
board at a number of different frequencies.
You will be using a piece of equipment called a vector impedance meter. This
is a very expensive piece of apparatus. The probe is very delicate and should be
handled with great care — it costs approximately £2000, so do not break it!
The probe should never be connected to any circuit in which an external
voltage (or charged capacitor) is present.
Terminate the transmission line with an open circuit and connect the vector
impedance meter probe, via its built-in BNC connection to the input of the
transmission line.
2.
Dial up the frequency 1 MHz on the front right of the vector impedance meter.
Set the impedance range to 1Kand the phase angle range to 90. Now turn
on the meter.
3.
Adjust the impedance range scale until the needle gives a measureable reading,
i.e., not off the scale. Measure both the impedance and the phase angle. What
does the phase angle signify?
4.
Repeat step 3 for about 30 frequencies linearly spaced between 1 and 15 MHz.
Plot your results for Z, assuming that when the phase angle is negative, Z
should be plotted as a negative number.
5.
You should obtain a graph of roughly the form Z(f) = cot f. If we relate this
to Eq. [11], we see that k = 2f / c and that the x of Eq. [11] is d, the length of
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the transmission line, because of the definition of the zero point. This means
that  = 2d / c.
6.
Now terminate the line with a short circuit and do the same experiment.
Determine a suitable frequency range and number of measurements from the
theory above and justify this in your write-up.
7.
From your graphs you can find the difference in f between the zero-crossing
points. This difference f corresponds to a difference k via Eq. [15]. Use Eq.
[11] and the length of the transmission line to work out what k should be and
hence use your measurements to calculate c. c0 follows as before.
8.
What would you expect to happen if you repeat the experiment with the cable
terminated with its characteristic impedance? Verify your prediction.
[9. Optional extra questions/experiments:
(a) Why is there a phase shift in your results compared with theory?
(b) Connect the vector impedance meter probe to tapping point 19. Terminate
the transmission line with the 50 characteristic impedance. Leave the other
end labelled “Input” open-ciruit. Re-do the impedance measurements from 1–
15 MHz and plot your results. Explain what is happening.
(c) Connect the vector impedance meter to tapping point 9. Leave both ends
open-ciruit. Re-do the impedance measurements, going from 1–20MHz,
sampled quite finely. What is going on? ]
5. References
[1] Electromagnetism, Grant and Phillips (Wiley)
[2] The Feynman Lectures on Physics, Vol. 2, Feynman, Leighton and Sands,
Ch.24
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