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## Wave Propagation in a Coaxial Cable: Instructor's Guide

### Main Ideas

• Transmission and reflection of waves inside a coax cable.
• Damping.
• Speed of an electromagnetic wave inside a medium.

Estimated Time: 90-120 minutes

1. The students will measure the speed of propagation by sending a voltage pulse from the top of the coax cable using a function generator.
2. The students will measure the size of the voltage signal across resistor, RTOP, and resistor, RTERM, (the position of the discontinuity) as a function of varying RTERM, relative to the voltage pulse applied by the function generator.
3. The students will use a single-frequency sinusoidal voltage and measure the voltage between the center and ground at the “top” of the cable for a number of frequencies f.

### Prerequisite Knowledge

Non-dispersive wave equation Reflection and transmission coefficients

### Activity: Introduction

#### Day 1: lab

Students should have read the lab handout prior to coming to class. Explain that the lab is divided into two parts: (i) to develop familiarity with the equipment and to measure the speed of propagation (ii) to measure the height of a voltage pulse at two points on the cable and to model that phenomenon. Stress that the exercise will stretch over two days, plus homework, and there will be frequent mini-wrap up points to keep everyone on track, and to monitor progress. In particular, there are written instructions to bring graphed data to class the following day.

#### Day 2: class follow up

Students will have brought to class 2 copies (one with name to be turned in; one with random number for peer review)

1. Tabulated data (voltage signal size as a function of terminating resistance)
2. Graphed data (voltage signal size as a function of terminating resistance)

Instructor gathers anonymous data (keeping groups intact) as distributes tables and graphs to another group for comment. Comments must address the reasonableness of the results (the physics) and also the presentation (are the data easy to interpret?).

### Activity: Student Conversations

#### Day 1: lab

1. Use of an oscilloscope
• Which part of the pulse should I measure? Does it matter?
• What is a trigger? calibration?
2. How do waves propagate?
• How can waves propagate if the outer braided cable is not electrically connected to the inner solid one?
• Why do we have a resistor on the end, and not another cable?
• Is the impedance of the cable the same as the end-to-end resistance?
3. Why does the reflected wave change sign under some conditions?
4. What experimental parameters can we identify with the symbols in the equations we learned in class?
• What am I supposed to model? How do I know if I'm right?
• Why are there two reflections if the terminating resistance near the source is changed?
• why are we using a pulse rather than a sinusoidal wave?
• what causes damping, the terminating resistor? the resistance of the inner cable? the outer cable?

Instructor observations

• More than 3 people per station encourages “stand-and-watch” mode. At least 3 people allows real discussion and avoids (mostly) one student dominating the discussion. Require students to swap roles.
• Some students work faster and comprehend more quickly. There is a danger of others becoming frustrated. Lack of understanding of an oscilloscope and general inexperience with trouble-shooting techniques leads to slowness on the part of some groups - very frustrating. Demonstrate troubleshooting techniques - start with the simplest thing, then gradually add complexity). Reassure students that the equipment will remain available after hours, and that only about 8-10 measurements are necessary - a 10-minute commitment once they've understood all the components of the experiment. Ask students from groups whose experiment is running smoothly to investigate the problems others are having. If you haven't had problems, you'll never learn to trouble-shoot. Everyone will be on track the next day.
• Keep checking each group for the main problems that can derail the entire experiment - uncalibrated time base, uncalibrated voltages, incorrect grounds, failure to remove the terminating resistance before measuring the termination resistance, and noisy, jumpy signals (managed by keeping connecting cables as short as possible).

#### Day 2: class follow up

1. Evaluation by students of peer presentation
• students should interpret the tables and graphs of peers, pointing out where presentation leads to misinterpretation, or where information has been left out. Significant figures, units, axis labels, clarity of presentation (point size, font size, line type etc. are typically commented upon).
• Students often try to fit a logarithmic function to the voltage vs. terminating resistance data because the data “look logarithmic”. This is a good time to discuss data fitting, and why it is important to fit functions that have some basis in a physical model.
2. Physics of reflection and transmission at abrupt boundaries
• the sign of the voltage reflections is often not understood - this is the time to discuss how the model predicts the sign.
• having conducted and tried to analyze an experiment, the students are brimming with questions. Take a quick poll to find the most common questions and organize the class to answer them.
3. Incorporation of damping
• A model that considers only reflection at a boundary represents qualitatively, but not quantitatively. Discussion of the resistance (not impedance) of the wires in the cables is important
• modify the wave equation to include light damping (recall results from “Oscallations” course. students should incorporate the damping in their model as part of the lab write up.

### Activity: Wrap-up

#### Day 1: lab

There are many conversations that the instructors must have with the students, but these are the ones that warrant interruption of the class at the appropriate time to ensure that all the students understand.

1. Propagation of waves, and their speed
• Students mostly view electric current as the result of electron drift, which is appropriate for dc propagation, but not ac propagation. A kinesthetic activity to help visualize a voltage wave is to have students line up in two parallel rows each opposite another, and students in the rows evenly spaced. They represent electrons in two parallel wires. In one row, students remain evenly spaced. In the other, the students move from their “equilibrium positions”. They then decide if the “voltage” between the two wires is greater or less than it was “in equilibrium”. They must consider the local density of “electrons” to decide. (The comparison to pressure is helpful). Then, the class can discuss how a voltage pulse can propagate down a pair of wires without the electrons moving very far from their equilibrium positions.
2. Terminating resistor as proxy for second cable
• When an electrical disturbance in a cable encounters a second cable, part of the energy is reflected and part transmitted. In this context, the second cable is merely a device to remove some energy from the incident wave. It can as well be represented by a resistor that dissipates exactly the “right” amount of energy - i.e. the amount of energy transmitted into the second cable. Finding a resistor that dissipates all the incident incident energy constitutes “matching”.
3. Identification of parameters, and recoding/graphing data for the next day's discussion
• In any real (uncanned) experiment, discovery is part of the process, and goals may change depending on the way the experiment has gone. Students must practice identifying what they know, what they don't know, what they are trying to find. In the less-structured environment of the lab, it is easy for students to lose sight of goals. Action items for the next day should be identified.

#### Day 2: class wrap up

1. Physics
• The energy in a wave incident on a boundary is partially reflected and the rest is transmitted (in principle into a second cable, but in practice in this experiment, dissipated by a resistor).
• In any real system, some of the wave energy is also dissipated within the medium itself (in this case the resistance (nt impedance) of the wires causes the dissipation.
• A physical medium is characterized by an impedance; mismatched impedances result in energy reflection that can be quantitatively modeled. The resistance of the wires causes some energy dissipation, which in most cases must be incorporated into the model.

### Extensions

1. Change the incident pulse to a sinusoid whose wavelength is of order the cable length. Watch the waveform at the cable ends as the terminating resistor is varied. At an appropriate frequency/wavelength, “standing waves” should in principle be generated. On fact, there is never a true node because the damping in the cable causes attenuation, so there is no superposition of left-and right-moving waves of equal amplitude. However, the amplitude of the reflected wave undergoes oscillations as the terminating impedance is steadily increased. This makes a nice project for advanced students or if the course is also offered at a 500-level, as ours is.
2. Model the experiments in Maple/Mathematica.

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