School of Physics
Experiment 6.
Transmission Lines
c
School
of Physics, University of Sydney
Updated BWJ March 14, 2016
Online resources:
David K.Cheng, Field and wave electromagnetics, Addison- Wesley, (Reading, Mass, 1992),
Chapter 9: pp370-88. Click here.
1
Safety
There are no particular safety issues with this experiment. If you suspect an item of equipment is not operating correctly, turn it off and turn off the power at the mains switch, then
consult a tutor.
2
Objective
In this experiment you will investigate the following properties of transmission lines
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S ENIOR P HYSICS L ABORATORY
• propagation of pulsed and sinusoidal signals,
• characteristic impedance,
• termination impedance, and
• dispersion
3
Introduction
At least two conducting paths are needed to connect one piece of electrical equipment to
another. This is a consequence of the conservation of charge; the charge that enters the
equipment by one path exits by the other. This is commonly achieved using a cable, in
which both conducting paths are incorporated into the one structure.
This experiment uses a coaxial cable in which one conductor is a hollow cylinder and the
other is a wire along its axis, with the space between filled by a dielectric material, usually
a solid polymer. You will have encountered coaxial cables in the laboratory as the standard
way of connecting one piece of equipment to another. Coaxial cables are also commonly
used to bring the signal from a TV antenna to the radiofrequency input of a TV receiver.
If the duration of a signal pulse (or the period of a sinusoidal signal) is much less than the
propagation time along a cable, the cable behaves as a transmission line. If we ignore the
resistance of the conductors, a transmission line is characterised by its inductance per unit
length, L0 and its capacitance per unit length C 0 . Such a lossless transmission line can be
represented by an equivalent circuit, as shown in Fig. 6-1.
L!Δz
L!Δz
C!Δz
C!Δz
Δz
Fig. 6-1 : Equivalent circuit of a lossless transmission line where each segment of length ∆z has a
series inductance of L0 ∆z and a parallel capacitance C 0 ∆z per unit length, where L0 and C 0 are the
series inductance and parallel capacitance per unit length
4
The experiment
The equipment used consists of pulse and waveform generators, various cables, adaptors
and components. For measurements, an oscilloscope, current transformer and LCR bridge
are available. A short section of delay line is used in the last part of the experiment.
4.1
Signal generators
There are two signal generators:
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• a HAMEG 8035 20 MHz Pulse Wave Generator,and
• a HAMEG 8032 20 MHz Sine Wave generator.
Note: Both signal generators have output impedances of 50 Ω.
4.2
Digital oscilloscope
The Agilent digital oscilloscope has two 1 MΩ inputs and allows signal averaging and
many automatic measurement options. How to transfer stored images and data from the
oscilloscope to a computer is explained in Appendix C.
4.3
Connectors
The connectors at the ends of the coaxial cables used in this experiment are standard for
this size of cable; they are called BNC connectors1 . There is a wide variety of adaptors
available (male-to-male, female-to-female and T-pieces) which allow cables to be joined.
Such connector and adaptor systems are the result of careful design to ensure that they do
not introduce impedance discontinuities that would cause partial reflection or attenuation of
the signal.
4.4 LCR bridge
The 6401 LCR Databridge can measure the resistance, capacitance and inductance of a
component at either 100 Hz or 1 kHz. The series option should be used for inductance measurements, the parallel option for capacitance measurements. The manufacturer’s specified
accuracy is 0.25% ± 1 digit.
4.5
Current transformer
An adaptor (blue box with clear lid) for insertion between the signal generator and the cable
has two outputs: one for voltage, the other from a current transformer with calibration of
0.5 AV−1 .
4.6
Helical delay cable
The delay cable used in this experiment has an inner helix wound on a flexible magnetic
core, which is a solid suspension of a ferrite in a suitable dielectric plastic such as polyethylene. This cable exhibits easily observed dispersion.
1
BNC stands for Baby Neill-Concelman, where these two names refer to the designers of the ”N” and ”C”
coaxial connectors respectively. For more details see http://en.wikipedia.org/wiki/BNC connector.
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5
S ENIOR P HYSICS L ABORATORY
Prework
1. A key property of a transmission line is its characteristic impedance. Explain the
meaning of this term.
2. Find expressions for
(a) the propagation speed (v), and
(b) characteristic impedance (Z0 )
of a transmission line in terms of the inductance per unit length (L0 ) and the capacitance per unit length (C 0 ).
3. Find expressions for
(a) L0 , and
(b) C 0
of a coaxial cable in terms of the cable dimensions and the relative permittivity (also
called dielectric constant) of the insulating material.
4. With the aid of the answers to Question 3, find expressions for
(a) the propagation speed (v), and
(b) the characteristic impedance (Z0 )
of a coaxial cable in terms of the cable dimensions and the relative permittivity of the
insulating material.
5. Appendix A contains a discussion of the reflection coefficients for voltage and current
at the end of a coaxial cable of characteristic impedance Z0 when it is terminated by
an arbitrary impedance Z. Consult the Appendix and answer the following questions:
(a) What are the values of the reflection coefficients for voltage and current when the
cable is terminated by a short circuit (Z = 0)?
(b) What are the values of the reflection coefficients for voltage and current when the
cable is terminated by an open circuit (Z = ∞)?
(c) What value of Z leads to no reflection of a signal from the end of the cable?
6. Explain the difference between phase velocity and group velocity of a wave. Find
general expressions for each in terms of the angular frequency ω = 2πf and the
wavenumber k = 2π/λ of the wave.
6
Procedure
6.1
Propagation of pulses in a coaxial cable
1. Become familiar with the operation of the pulse generator and the oscilloscope by observing the pulse output on the oscilloscope. In particular, familiarise yourself with
using cursors for voltage measurements (e.g. pulse amplitude) and time measurements (e.g. pulse width, time interval between pulses).
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2. Use a BNC T-piece to connect the pulse generator output to the 40.50 ± 0.05 m long
RG-58 coaxial cable as well as to the oscilloscope. Connect the other end of the cable
to the other input of the oscilloscope. You will have to give some thought to suitable
choice of pulse frequency and width in order to see the transmitted and reflected
pulses. It may help to estimate the time for the pulse to propagate along the cable and
returning by assuming a speed comparable to the speed of light.
3. Measure the propagation speed of the pulse in the cable. Hence determine the relative
permittivity of the RG-58 cable’s dielectric insulator.
4. Consult the cable specifications, and compare your value for relative permittivity with
tabulated values2 . Note: do tabulated values vary with frequency? If so what value is
most appropriate for comparison with your measurements?
Question 1: Explain why the pulse at the end of the cable is close to twice the amplitude of
the pulse from the pulse generator.
Question 2: With the aid of Appendix B explain why the shape of the reflected pulse differs
from the shape of the pulse from the generator?
C1 .
6.2
Characteristic impedance
1. There are several different ways to measure the characteristic impedance of the coaxial cable.
(a) Compare the amplitudes of the pulse produced by the Hameg pulse generator
when the cable is connected to the T-piece and then disconnected. Noting that
the output impedance of the generator is 50 Ω, obtain a value for the characteristic impedance (and estimate an uncertainty!).
(b) With the aid of the adaptor incorporating a current transformer, directly measure
the voltage and current amplitudes of the pulse from the pulse generator, and
hence obtain a value for the cable’s characteristic impedance. Does it matter if
the end of the cable is open circuit or short circuit? Explain your observations.
(c) With reference to Question 5(c) of the Prework, devise a way of using the variable resistor to make another measurement of the characteristic impedance. In
this case how do you determine an uncertainty?
(d) Use the LCR bridge to determine the inductance per unit length, L0 , and the
capacitance per unit length C 0 of the coaxial cable3 and hence obtain a value
for the characteristic impedance of the cable Z0 . Compare with your directly
measured values.
(e) Confirm that the assumption ωL0 << 1/ωC 0 is justified. Explain why this
condition allows measurement of cable inductance and capacitance as described
in the footnote.
2
Consult, for example, Kaye and Laby or the CRC Handbook.
If we assume that the impedance of L0 is much less than that of C 0 (i.e ωL0 << 1/ωC 0 , the equivalent
circuit in Fig. 6-1 shows that the total parallel capacitance of the cable will be measured when the end is open
circuit, and the total series inductance will be measured when it is short circuit.
3
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S ENIOR P HYSICS L ABORATORY
Question 3: Compared with the incident pulse, how is the polarity (inverted, not
inverted) of the reflected voltage and current pulse affected by
(i) an open circuit, or
(ii) a short circuit
at the end of the cable.
Do your observations agree with your answers to Question 5 of the Prework. Provide
a simple physical explanation of your observations, i.e., explain in simple physical
terms how the termination affects the reflected pulse.
Question 4: Given the dimensions of the RG-58 coaxial cable (outer diameter 2.95
mm, inner diameter 0.81 mm) calculate the characteristic impedance of the coaxial
cable, using your value for the relative permittivity of the polyethylene dielectric.
2. Local area networks commonly use twisted pair cables (see Figure 6-2). Measure the
characteristic impedance of the twisted pair cable provided.
Fig. 6-2 : Example of a cable consisting of four twisted pairs
C2 .
6.3
Propagation of sinusoidal signals in a coaxial cable
In this section you will use the Hameg sinewave generator to investigate the propagation of
sinusoidal signals in the coaxial cable.
1. Terminate the 40.5 m coaxial cable with the 50 Ω terminator supplied; set the sinewave
generator to 500 kHz and with the aid of T-pieces at each end observe the signal at
each end of the cable on the DSO.
2. Measure the phase difference between the signals at each end of the cable.4 .
3. Show that the measured phase differences is consistent with the value expected based
on the propagation speed.
4. Increase the generator frequency until the phase difference is π. What is the wavelength of the signal in this situation?
4
The oscilloscope has a phase difference measurement mode
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5. Find the frequency values that lead to phase differences of 2π, 3π, . . . until you are
prevented from going further by the generator’s 20 MHz frequency limit. For each
case note the wavelength of the sinewave.
6. Plot angular frequency (ω = 2πf ) as a function of wavenumber (k = 2π/λ) and
determine the propagation speed (i.e phase velocity). How does this compare with
the propagation speed measured earlier using pulses?
7. What can you say about the relationship between phase velocity and group velocity
for this cable in the frequency range examined?
C3 .
6.4
Delay line
1. Use the LCR bridge to determine the inductance and capacitance per unit length for
the delay line and show that Z0 ≈ 1.6 kΩ.
2. Terminate the delay line with a 1.6 kΩ resistor and observe the effect on a square
pulse after propagation along the delay line.
Question 5: What can you conclude about the way phase velocity varies with frequency
from the detailed shapes of the input and output pulses? Estimate a “high frequency” delay
and a “low frequency” delay.
Question 6: Which of the above delays agrees with the value calculated from the measured
values of L0 and C 0 ? Explain why.
6.5
Phase and group velocities for a delay line
1. For the delay line set the sinewave generator to a frequency of 100 kHz. Using the
“low frequency delay” show that the expected phase difference between the generator
and the load end of the cable is about 1 radian; confirm that this agrees with the
observed phase delay.
2. Increase the frequency until the phase shift is exactly 2π; what can you deduce about
the wavelength of the sinewave in the cable.
3. Repeat the measurements for phase shifts of 4π, 6π, 8π, etc until a frequency of about
10 MHz.
4. Plot ω as a function of k, and comment on how phase velocity and group velocity
change throughout the frequency range.
5. From your plot find values for the group velocities for “high frequencies” and “low
frequencies” . Hence deduce “high frequency” and “low frequency” delays to compare with those measured previously using the oscilloscope.
C4 .
6–8
A
S ENIOR P HYSICS L ABORATORY
Termination of the cable
When a signal reaches the end of a cable it is reflected in a way that depends upon the
terminating impedance Z - the impedance that connects one conductor of the cable to the
other5 . From transmission line theory the reflection coefficients for voltage ρv and current
ρi are given by
ρv = −ρi =
Z − Z0
Z + Z0
(1)
where Z0 is the characteristic impedance of the cable.
B
Attenuation of RG-58 cable as a function of frequency
The amplitude of a sinusoidal signal propagating along a coaxial cable decreases as it propagates along the cable, with the attenuation increasing with increasing frequency of the
signal. Figure 6-3 shows attenuation over a 30 m for a RG-58 coaxial cable.
Fig. 6-3 : Attenuation in decibels (dB) of a sinusoidal signal as a function signal frequency in MHz
for a 30 m long RG-58 coaxial cable.
C
Agilent DSO 1002A: saving a screen dump to a USB memory
stick
1. Insert USB into the front panel slot. The DSO should instantly recognize the USB.
2. Hit the Save/Recall button
5
This could, for example, be the input impedance of another piece of equipment
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6–9
3. Hit Waveform, and select PNG file output. Also worth turning Para(meter) Save
ON to save DSO settings. You can also save the traces as comma separated variable
files (.csv), which can be imported into QtiPlot for plotting.
4. Hit External
5. Hit New File (simply accept the default filename)
6. Hit Save (this saves the screen dump as NewFilex.png, and the DSO settings as
NewFilex.txt, where x automatically increments by 1)
Sample output (pulse propagation through the 40.5 m coaxial cable)
Analog Ch
CH1
CH2
State
On
On
Analog Ch
CH1
CH2
Impedance
1M Ohm
1M Ohm
Time
Main
Trigger
Edge
Time Ref
Center
Source
CH2
Acquisition
Normal
Scale
1.00V/
1.00V/
Position
-3.12V
-3.08V
BW Limit
Off
Off
Invert
Off
Off
Probe
1X
1X
Main Scale
100.0ns/
Slope
100.0ns/
Sampling
Realtime
Coupling
DC
DC
Delay
0.000000s
Mode
Auto
Coupling
DC
Memory Depth
Normal
Level
3.04V
Sample Rate
1.000GSa
Holdoff
500ns
6–10
D
S ENIOR P HYSICS L ABORATORY
LCR Databridge: recommended settings
Component
Capacitor < 1µF
Capacitor ≥ 1µF (non-electrolytic)
Capacitor ≥ 1µF (electrolytic)
Inductor < 1 H
Inductor ≥ 1 H
Resistor < 10 kΩ
Resistor ≥ 10 kΩ
Frequency
1 kHz
100 Hz
100 Hz
1 kHz
100 Hz
100 Hz
100 Hz
SER or PAR
PAR
PAR
SER
SER
SER
SER
PAR
Uncertainty: ±0.25% of reading ±1 digit.
Notes
1. Although the Databridge provides the option of displaying equivalent series or parallel component values, under adverse Q conditions it may not be possible to obtain the
basic accuracy in both modes. When a mode change is required to improve accuracy,
this is indicated by the SER or PAR LED flashing. If this occurs, the SER-PAR button
may be operated to change mode and thus improve accuracy.
2. Capacitors in the range 200 µF to 2 mF, and inductors in the range 200 H to 2000
H, can only be measured to the basic accuracy at a measurement frequency of 100
Hz. Similarly, capacitors in the range 200 pF to 2 nF, and inductors in the range
200 µH to 2 mH, can only be measured to the basic accuracy at a measurement
frequency of 1 kHz. If an attempt is made to measure components in these ranges
at a frequency which prevents the best accuracy from being attained, a frequency
change will be indicated by flashing the frequency indicator LED. The Databridge
will provide a measurement at the inappropriate frequency, but the basic accuracy
may not be achieved.