School of Physics

Experiment 6. Transmission Linesc©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 equip-ment is not operating correctly, turn it off and turn off the power at the mains switch, thenconsult a tutor.

2 Objective

In this experiment you will investigate the following properties of transmission lines

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• 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 toanother. This is a consequence of the conservation of charge; the charge that enters theequipment by one path exits by the other. This is commonly achieved using a cable, inwhich both conducting paths are incorporated into the one structure.

This experiment uses a coaxial cable in which one conductor is a hollow cylinder and theother is a wire along its axis, with the space between filled by a dielectric material, usuallya solid polymer. You will have encountered coaxial cables in the laboratory as the standardway of connecting one piece of equipment to another. Coaxial cables are also commonlyused 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 thepropagation time along a cable, the cable behaves as a transmission line. If we ignore theresistance of the conductors, a transmission line is characterised by its inductance per unitlength, L′ and its capacitance per unit length C ′. Such a lossless transmission line can berepresented 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 aseries inductance of L′∆z and a parallel capacitance C ′∆z per unit length, where L′ and C ′ are theseries inductance and parallel capacitance per unit length

4 The experiment

The equipment used consists of pulse and waveform generators, various cables, adaptorsand components. For measurements, an oscilloscope, current transformer and LCR bridgeare 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 andmany automatic measurement options. How to transfer stored images and data from theoscilloscope 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 forthis size of cable; they are called BNC connectors1. There is a wide variety of adaptorsavailable (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 donot introduce impedance discontinuities that would cause partial reflection or attenuation ofthe signal.

4.4 LCR bridge

The 6401 LCR Databridge can measure the resistance, capacitance and inductance of acomponent at either 100 Hz or 1 kHz. The series option should be used for inductance mea-surements, the parallel option for capacitance measurements. The manufacturer’s specifiedaccuracy is 0.25% ± 1 digit.

4.5 Current transformer

An adaptor (blue box with clear lid) for insertion between the signal generator and the cablehas two outputs: one for voltage, the other from a current transformer with calibration of0.5 AV−1.

4.6 Helical delay cable

The delay cable used in this experiment has an inner helix wound on a flexible magneticcore, which is a solid suspension of a ferrite in a suitable dielectric plastic such as polyethy-lene. This cable exhibits easily observed dispersion.

1BNC 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 Prework

1. A key property of a transmission line is its characteristic impedance. Explain themeaning 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 (L′) and the capaci-tance per unit length (C ′).

3. Find expressions for

(a) L′, and

(b) C ′

of a coaxial cable in terms of the cable dimensions and the relative permittivity (alsocalled 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 theinsulating material.

5. Appendix A contains a discussion of the reflection coefficients for voltage and currentat the end of a coaxial cable of characteristic impedance Z0 when it is terminated byan 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 thecable is terminated by a short circuit (Z = 0)?

(b) What are the values of the reflection coefficients for voltage and current when thecable 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. Findgeneral expressions for each in terms of the angular frequency ω = 2πf and thewavenumber 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 ob-serving the pulse output on the oscilloscope. In particular, familiarise yourself withusing cursors for voltage measurements (e.g. pulse amplitude) and time measure-ments (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 longRG-58 coaxial cable as well as to the oscilloscope. Connect the other end of the cableto the other input of the oscilloscope. You will have to give some thought to suitablechoice of pulse frequency and width in order to see the transmitted and reflectedpulses. It may help to estimate the time for the pulse to propagate along the cable andreturning by assuming a speed comparable to the speed of light.

3. Measure the propagation speed of the pulse in the cable. Hence determine the relativepermittivity of the RG-58 cable’s dielectric insulator.

4. Consult the cable specifications, and compare your value for relative permittivity withtabulated values2. Note: do tabulated values vary with frequency? If so what value ismost appropriate for comparison with your measurements?

Question 1: Explain why the pulse at the end of the cable is close to twice the amplitude ofthe pulse from the pulse generator.

Question 2: With the aid of Appendix B explain why the shape of the reflected pulse differsfrom 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 coax-ial cable.

(a) Compare the amplitudes of the pulse produced by the Hameg pulse generatorwhen the cable is connected to the T-piece and then disconnected. Noting thatthe output impedance of the generator is 50 Ω, obtain a value for the character-istic impedance (and estimate an uncertainty!).

(b) With the aid of the adaptor incorporating a current transformer, directly measurethe voltage and current amplitudes of the pulse from the pulse generator, andhence obtain a value for the cable’s characteristic impedance. Does it matter ifthe 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 vari-able resistor to make another measurement of the characteristic impedance. Inthis case how do you determine an uncertainty?

(d) Use the LCR bridge to determine the inductance per unit length, L′, and thecapacitance per unit length C ′ of the coaxial cable3 and hence obtain a valuefor the characteristic impedance of the cable Z0. Compare with your directlymeasured values.

(e) Confirm that the assumption ωL′ << 1/ωC ′ is justified. Explain why thiscondition allows measurement of cable inductance and capacitance as describedin the footnote.

2Consult, for example, Kaye and Laby or the CRC Handbook.3If we assume that the impedance of L′ is much less than that of C′ (i.e ωL′ << 1/ωC′, the equivalent

circuit in Fig. 6-1 shows that the total parallel capacitance of the cable will be measured when the end is opencircuit, and the total series inductance will be measured when it is short circuit.

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Question 3: Compared with the incident pulse, how is the polarity (inverted, notinverted) 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. Providea simple physical explanation of your observations, i.e., explain in simple physicalterms how the termination affects the reflected pulse.

Question 4: Given the dimensions of the RG-58 coaxial cable (outer diameter 2.95mm, inner diameter 0.81 mm) calculate the characteristic impedance of the coaxialcable, 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 thecharacteristic 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 ofsinusoidal signals in the coaxial cable.

1. Terminate the 40.5 m coaxial cable with the 50 Ω terminator supplied; set the sinewavegenerator to 500 kHz and with the aid of T-pieces at each end observe the signal ateach 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 basedon the propagation speed.

4. Increase the generator frequency until the phase difference is π. What is the wave-length of the signal in this situation?

4The 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 areprevented from going further by the generator’s 20 MHz frequency limit. For eachcase note the wavelength of the sinewave.

6. Plot angular frequency (ω = 2πf ) as a function of wavenumber (k = 2π/λ) anddetermine the propagation speed (i.e phase velocity). How does this compare withthe propagation speed measured earlier using pulses?

7. What can you say about the relationship between phase velocity and group velocityfor 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 forthe 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 squarepulse after propagation along the delay line.

Question 5: What can you conclude about the way phase velocity varies with frequencyfrom the detailed shapes of the input and output pulses? Estimate a “high frequency” delayand a “low frequency” delay.

Question 6: Which of the above delays agrees with the value calculated from the measuredvalues of L′ and C ′? 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 generatorand the load end of the cable is about 1 radian; confirm that this agrees with theobserved phase delay.

2. Increase the frequency until the phase shift is exactly 2π; what can you deduce aboutthe wavelength of the sinewave in the cable.

3. Repeat the measurements for phase shifts of 4π, 6π, 8π, etc until a frequency of about10 MHz.

4. Plot ω as a function of k, and comment on how phase velocity and group velocitychange throughout the frequency range.

5. From your plot find values for the group velocities for “high frequencies” and “lowfrequencies” . Hence deduce “high frequency” and “low frequency” delays to com-pare with those measured previously using the oscilloscope.

C4 .

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A Termination of the cable

When a signal reaches the end of a cable it is reflected in a way that depends upon theterminating impedance Z - the impedance that connects one conductor of the cable to theother5. From transmission line theory the reflection coefficients for voltage ρv and currentρi 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 prop-agates along the cable, with the attenuation increasing with increasing frequency of thesignal. 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 MHzfor a 30 m long RG-58 coaxial cable.

C Agilent DSO 1002A: saving a screen dump to a USB memorystick

1. Insert USB into the front panel slot. The DSO should instantly recognize the USB.

2. Hit the Save/Recall button5This could, for example, be the input impedance of another piece of equipment

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3. Hit Waveform, and select PNG file output. Also worth turning Para(meter) SaveON to save DSO settings. You can also save the traces as comma separated variablefiles (.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 asNewFilex.txt, where x automatically increments by 1)

Sample output (pulse propagation through the 40.5 m coaxial cable)

Analog Ch State Scale Position Coupling BW Limit InvertCH1 On 1.00V/ -3.12V DC Off OffCH2 On 1.00V/ -3.08V DC Off Off

Analog Ch Impedance ProbeCH1 1M Ohm 1XCH2 1M Ohm 1X

Time Time Ref Main Scale DelayMain Center 100.0ns/ 0.000000s

Trigger Source Slope Mode Coupling Level HoldoffEdge CH2 100.0ns/ Auto DC 3.04V 500ns

Acquisition Sampling Memory Depth Sample RateNormal Realtime Normal 1.000GSa

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D LCR Databridge: recommended settings

Component Frequency SER or PARCapacitor < 1µF 1 kHz PARCapacitor ≥ 1µF (non-electrolytic) 100 Hz PARCapacitor ≥ 1µF (electrolytic) 100 Hz SERInductor < 1 H 1 kHz SERInductor ≥ 1 H 100 Hz SERResistor < 10 kΩ 100 Hz SERResistor ≥ 10 kΩ 100 Hz PAR

Uncertainty: ±0.25% of reading ±1 digit.

Notes

1. Although the Databridge provides the option of displaying equivalent series or paral-lel component values, under adverseQ conditions it may not be possible to obtain thebasic 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 buttonmay 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 2000H, can only be measured to the basic accuracy at a measurement frequency of 100Hz. Similarly, capacitors in the range 200 pF to 2 nF, and inductors in the range200 µH to 2 mH, can only be measured to the basic accuracy at a measurementfrequency of 1 kHz. If an attempt is made to measure components in these rangesat a frequency which prevents the best accuracy from being attained, a frequencychange will be indicated by flashing the frequency indicator LED. The Databridgewill provide a measurement at the inappropriate frequency, but the basic accuracymay not be achieved.