Measuring the self-inductance and self-capacitance of simple transmission lines

Prepared by Dr. Aaron Scher
[email protected]
Oregon Institute of Technology

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1. Objectives

To measure the self-inductance and self-capacitance of a simple two-wire transmission line. To illustrate how wire-spacing and the environment effects these parameters. To measure the self-inductance and self-capacitance of a coaxial cable transmission line

2. Equipment

1. Two three feet lengths of #22 gauge hookup wire (It doesn't have to be exactly three feet, just somewhere in this ball park). For this experiment I use Elenco solid hookup wire.
2. Inductance/Capacitance (LC) meter. In the pictures below I use the inexpensive LC 100 A meter (for demonstration purposes). You will perform your measurements with the Keysight U1733C LCR Meter. Set the meter to measure L and C at the lowest frequency possible for the device.
3. Wood board to secure wires to. You can just use a non-metallic table top instead
4. Tape to hold wires down to the board.
5. Wire strippers/cutters.
6. Ruler or measuring tape.
7. A few small alligator clips and extra wire to convert open circuit to short circuit without the need to solder.
8. One 50 Ohm coaxial cable (roughly three feet long) with BNC male connectors on both ends.
9. One BNC female to double banana plug (see picture here)
10. One BNC female terminator with short circuit. For this experiment I use a bulkhead jack chassis mount with a small wire soldered to short the center conductor to ground (see picture here)
11. One BNC male to male adapter.

3. Report guidelines

For the lab report, you will create a PowerPoint presentation (or use a similar presentation program), save it as a PDF, and submit it on-line according to the instructions given in class. The presentation should be tutorial in nature; your target audiences are other engineers and scientists who are interested in learning more about circuits and electromagnetism.

Your presentation will have 13 slides. Please include a slide number in the footer of each slide. To earn full credit your presentation must contain the slides in the order asked for in this lab. If you miss a slide, please leave a blank slide in its place so that you have still have exactly 13 slides total. Your first two slides should be:

• Slide 1: Title slide with our name, student ID number, date, lab name, class number/title.
• Slide 2: A team picture or insignia with the names of your teammates.

4. Procedure

Measuring Self-Capacitance

1. Place two three-feet lengths of #22 gauge hookup wire flat on a non-metallic table (or wooden board) such that the wires are parallel and spaced 2.5 cm apart (about one inch).
2. Connect a capacitance meter at one end of your two-wire transmission line, as illustrated in the diagram and photograph below.
3. Measure the capacitance of the two wire line. The capacitance should be around 5 to 25 pF, depending on the exact length of wires, how they are positioned, and the dielectric properties of the wooden board and/or table top. Fill in the appropriate box in Table 1 with this value (quick shout out to Table generator for making the act of generating tables in HTML so easy). Vary the spacing between wires and continue to fill out the capacitance column (labelled "C [pF]") in Table 1. Table 1 should be presented in Slide 3. With the measured capacitance values and measured length of your transmission line, fill out the capacitance per unit length column (labelled "C/length [F/m]") in Table 1.

Table 1: Measured results for two-wire transmission line

Spacing [cm]	C [pF]	L [nH]	C/length [F/m]	l/length [H/m]	Z0 [Ohms]	VF (velocity factor)
.5	___________	____________
2.5
5
7.5
10

4. Again place the wires parallel and spaced 2.5 cm apart (about one inch). Without disturbing the wires, place your hand in different positions near the transmission line (such as between the two wires, or hovering your hand closely above the two wires). Does your hand effect the value of the measured capacitance? How so? Place a wooden board on top of the transmission line as shown in the figure below. How does the presence of the board effect the value of the measured capacitance? Present our observations for this step in Slide 4.

Measuring Self-Inductance and Resistance

1. With the wires again placed parallel and spaced 2.5 cm apart, connect an inductance meter at one end of your two-wire transmission line and short the other end, as illustrated in the diagram and photograph below.
2. Measure the inductance of the two wire line. The inductance should be around 1 to 3 uH, depending on the exact length of wires and how they are positioned. Fill in the appropriate box in Table 1 with this value. Vary the spacing between wires and continue to fill out the inductance column (labelled "L [nH]") in Table 1. With the measured inductance values and measured length of your transmission line, fill out the inductance per unit length column (labelled "L/length [H/m]") in Table 1.
3. Without disturbing the wires, place your hand in different positions near the transmission line (such as between the two wires, or hovering your hand closely above the two wires). Does your hand effect the value of the measured inductance? How so? Place a wooden board on top of the transmission line. Does the presence of the board effect the value of the measured inductance? Present our observations for this step in Slide 5.
4. Now that you you have both capacitance and inductance measurements, fill out the rest of Table 1. The characteristic impedance is Z0 = sqrt(L/C) and the phase velocity is vp = 1/sqrt(L*C). where for both equations L is the inductance in [H] and C is the capacitance in [F]. The velocity factor VF = vp/c Since the phase velocity should be less than the speed of light (vp < c), we expect VF < 1.
5. In Slide 6 present and discuss how your measured results compare with the line parameters (inductance, capacitance, characteristic impedance, and phase velocity) of an ideal twin wire transmission line suspended in vacuum. For calculating the inductance of the ideal line, you can use Clemson University's Twin Wire Inductance Calculator. Once you have inductance, you can calculate the capacitance of an ideal transmission line by the equation: 1/sqrt(LC)=c₀, where c₀ is the speed of light.
6. With the line still shorted at the end (the spacing of the lines aren't critical), measure the resistance of the two wire line with an ohmmeter or multimeter. The resistance should be very small, so make sure you are careful when taking this measurement.Present our measurements in Slide 7

Measuring Self-Inductance and Self-Capacitance of a Coaxial Cable

1. In this section you will measure the self-capacitance and self-inductance of a coaxial cable using similar methods to those you used to measure the two-wire transmission line (i.e. by opening and shorting the load end, respectively). However, for the coaxial cable we will be more careful in "calibrating out" the effects of the connecting adaptors.
2. Connect the BNC-female-to-double-banana plug to the LC meter as shown in Figure 6. Set the meter to measure capacitance. If your meter has a ZERO button, press it to zero the unit. When the unit is zeroed by pressing the ZERO button, the unit stores the value of the capacitance and subtracts it from subsequent measured values. In our case, this removes excess capacitance contributed by the adapter and leads of the LC meter, which will yield the "true" or "calibrated" capacitance of the coaxial line. If your meter does not have a ZERO button, then record the value of capacitance when the meter is connected to just the adapter alone. You will need to later manually subtract this value from the measured value of the capacitance of the line.

3. Connect the coaxial cable to the LC meter via the BNC-female-to-double-banana plug adaptor, as shown in Figure 7. Make sure the cable is positioned straight (not bent), measure the "true" value of the capacitance, and fill out the appropriate columns in Table 2 with your results. Present this table in Slide 8. Does placing your hand on the cable or bending the cable change the capacitance? Present your observations in Slide 8.

   Table 2: Measured results for coaxial cable

   C [pF]	L [nH]	C/length [F/m]	l/length [H/m]	Z0 [Ohms]	VF (velocity factor)
   ___________	___________

4. Set the LC meter to measure the inductance of the coaxial cable. When making the inductance measurement, you will short out the end of the cable using the BNC female terminator with short circuit (see the Equipment List.) Before you jump into the inductance measurement though, note that the adaptors and the short circuit contribute additional inductance to the measurements. Therefore, you will need to "calibrate out" or remove this excess inductance (using a similar method to the one you used for the capacitance measurements) to yield the "true" inductance of the coaxial line. Present your calibration steps (with photographs) in Slide 10. Measure the "true" inductance of the coaxial cable and fill out Table 2 with your results. Does placing your hand on the cable or bending the cable change the inductance? Present your observations in Slide 11.
5. Now that you you have both capacitance and inductance measurements, fill out the rest of Table 2. The characteristic impedance is Z0 = sqrt(L/C) and the phase velocity is vp = 1/sqrt(L*C), where for both equations L is the inductance in [H] and C is the capacitance in [F]. The phase velocity should be less than the speed of light (but on the same order of magnitude as the speed of light). Hence the velocity factor VF = vp/c < 1.
6. On the side of the coaxial cable should be written the characteristic impedance (for this experiment the characteristic impedance should be 50 Ohms). In Slide 12 discuss how well the characteristic impedance you measured (indirectly from L and C) compares with this ideal value.
7. Present your conclusions and a summary of what you learned in this lab and how it relates to practical circuit design in Slide 13.

5. Theory

A good model (equivalent circuit) of a small segment of two-wire transmission line is shown in the figure below. Here, we define "small" as "electrically small", meaning the length of the line is significantly smaller than a guide wavelength. In this manner, we can safely ignore standing waves on the line, and model the line with discrete lumped elements. In the model, the transmission line is described by the lines self-capacitance (C), self-inductance (L), and resistance (R). Note that this model ignore any dielectric leakage between the two wires, which is usually relatively small anyway. The parameters R,L, and C depend on the physical dimensions and materials of the transmission line. For example, when you double the length of the line, the values for R,L, and C double. When you increase the spacing between the conductors, the inductance L increases while the capacitance C decreases.

To measure the self-capacitance of the electrically-short transmission line, we open one end of the line and measure the capacitance from the other end. As illustrated in the figure below, with one end open, the voltage drop accross the inductor is much smaller than that across the capacitors, and the circuit's behaviour is dominated by its self-capacitance.

To measure the self-inductance of the electrically-short transmission line, we short one end of the line and measure the inductance from the other end. As illustrated in the figure below, with one end short, the currents through the capacitors are very small, and the circuit's behaviour is dominated by its self-inductance.

5. References

1. 1. EMC Education Manual, Education Committee, IEEE Electromagnetic Compatibility Society, July 1992.