2. eecs230_fall2023_lab2 (1)

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EECS 230 ELECTROMAGNETICS I LABORATORY 2 The University of Michigan Ann Arbor, MI, 48109-2122 Fall 2023 Last revised August, 2023 © The University of Michigan, 2023

2 Lab Exercise 2: The Smith Chart Objective The objective of this exercise is to further familiarize students with the network analyzer and its Smith Chart tool. The concepts of time delay, lossy versus lossless lines and impedance matching will also be explored. General concepts to be covered: ● Relative dielectric constant and its relationship to the phase velocity, u p ● Transmission line parameters ● Smith Chart display on the network analyzer and load impedance measurements ● Lossy versus lossless lines on the Smith Chart ● Single-stub matching technique 1. Pre-Lab Assignment 1. Read the supporting material in Chapter 2 of the textbook. 2. Read through the entire lab exercise. 3. Prior to the lab, summarize the experimental procedures in your pre-lab write-up (1 paragraph per experiment): (a) Experiment 2.1 (b) Experiment 2.2 (c) Experiment 2.3 (d) Experiment 2.4 4. Answer Questions 1 and 3 of Experiment 2.1 and Question 1 of Experiment 2.3. 2. Introduction You have seen in class that the wave propagation properties of transmission lines are governed by several parameters. One of these parameters is the relative dielectric constant, ε r . In this lab, you will measure ε r for a lossless transmission line. Using the measured value of ε r , you will determine the phase velocity u p and phase constant β of the line. There are many formats used to display the reflections caused by mismatched loads. Some of these formats include magnitude and phase, SWR, and the Smith Chart. In this lab, we will focus on the Smith Chart. The Smith Chart is a very useful tool in microwave engineering. It allows the microwave engineer to represent complicated equations governing the reflections at load mismatches in a very compact graphical format. The network analyzer has the Smith Chart as one of the display options. In this lab, several different phenomena will be examined using the Smith Chart.

3 In addition, the concept of matching will be introduced. There are several techniques that are widely used to match loads to transmission lines. Among these techniques, three of the most common distributed- element techniques are: quarter-wave transformers, single stub tuners, and double stub tuners. The first two methods are covered in this course. The last method, double stub tuning, is covered in more advanced courses. 3. Useful Equations ࠵? = ࠵? ࠵? ࠵? ( ࠵? ) ࠵? " = ࠵? ࠵? ( ࠵? ࠵? * ࠵? # = + ࠵? ࠵? " - $ ࠵? % = / ࠵?′ ࠵?′ (࠵?) ࠵? = ࠵? & ࠵?࠵? = ࠵? ’ ࠵?′࠵?′ ( ࠵?࠵?࠵? ࠵? ) ࠵? !" = ࠵? # ࠵? $ + ࠵?࠵? # ࠵?࠵?࠵? (࠵?࠵?) ࠵? # + ࠵?࠵? $ ࠵?࠵?࠵?(࠵?࠵?) (࠵?) 4. Equipment Item Part # Cables & Connectors - Calibration Kit HP 85032E Network Analyzer HP 8712C Lossy Line Simulator - Single Stub - Various Loads -

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4 5. Experiment 2.1: Line Parameters of a Lossless Line In this experiment, you will use the network analyzer to compute the line parameters of a lossless line (β (rad/s), C’ (F/m), L’ (H/m), u p (m/s), and ε r ) by making measurements to determine ε r . Although the cable that you will use is a coaxial cable which does have some inherent loss, we will make the assumption for this experiment that the attenuation constant, ࠵? , is approximately 0. We are justified in making this assumption since the loss for this particular cable is 4.1 dB/100 ft. Since the piece of cable you are using is approximately 2 ft. in length, the loss is 0.082 dB or less than 1%. The relative permittivity, ε r , for the transmission line is determined by measuring phase velocity, u p , for the transmission line and comparing it to u p for free space (vacuum). Phase velocity can be determined by measuring the time, Δt, that a wave takes to travel down a transmission line of length ࠵? and return to the network analyzer. The value of ε r can be found from u p , which satisfies the following relation: ࠵? ! = 2࠵? ࠵?࠵? ( ࠵? ࠵? + (2.1) To measure the time delay of the cable, you will use the electrical delay feature of the network analyzer and terminate the patch cord with a load of known reflection coefficient. In this experiment, the transmission line will be terminated in a short. Recall that the reflection coefficient is Γ=-1 for a short- circuit termination. When a transmission line is placed between the network analyzer and a load termination, an electrical delay is added to the load response. The term electrical delay means that the signal has an added phase shift from traversing the transmission line. In particular, if you look at a short termination, you will see that the response is a set of concentric circles wrapping around the outside edge (|Γ|=1 circle) of the Smith Chart. When you add electrical delay, you are actually adding a phase correction to the received signal. The phase correction is given in terms of time. When you have added the correct amount of electrical delay, the response of the short termination will collapse to a point. In reality, the response will collapse to what is termed a “point-like” response (an example of which is demonstrated in Fig. 2-1). This means that the response is not a true point, but is more spread out. This is due to the finite loss in the cable and connectors.

5 Fig. 2-1: Example of a “point-like” response on the Smith Chart Setup This experiment requires the network analyzer, a patch cord (N plug cable), open circuit and short circuit terminations. Refer to lab exercise 1 if you don’t recall how to perform any of the following steps. Set up the network analyzer by configuring the network analyzer: ● Press Preset on the network analyzer followed by OK . ● Set the start frequency to 100 MHz ● Set the stop frequency to 3.4 GHz Procedure 1. Select a patch cord of length 1–2 ft. Measure the exact physical length from the center of the connectors and add 2.6 cm for the N jack to N jack adapter. 2. Attach the patch cord to channel 1 of the network analyzer as shown in Fig. 2-2. Fig. 2-2: Block diagram for Experiment 2.1 3. Connect the short termination to the end of the patch cord using an N jack to N jack adapter.

6 4. Take the transform of the frequency response ● Press Analysis ● Press Transform ● Press Start and change the start time to 0 by pressing 0 and Enter ● Press Transform so that it says Transform ON 5. Place a Maker at the peak of the reflection ● Press Marker ● Press Marker 1 ● Use the knob to adjust the marker to the peak of the time domain reflection. The amount of time delay before the reflection can be read on the network analyzer’s display ● Record the displayed delay time ● Turn Transform to OFF again and record the frequency 6. Perform a quick calculation of u p to ensure the value you collected is reasonable (it should be on the order of magnitude of the speed of light c=3x10 8 m/s). If the value does not make sense, please ask your lab instructor for assistance. 7. Capture the display on the network analyzer. 8. Return the network analyzer to its original state by pressing Preset and OK . Measured Data Use the following to fill in the measured data. If you are missing any data, please repeat the necessary parts of this experiment before proceeding. Patch Cord Length ______ (m) Added Electrical Delay ______ (ns)

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7 Analysis 1. Using Eq. 2.1, compute u p for the patch cord using the experimentally determined electrical delay. 2. Using c = 3*10 8 m/s, and the value of u p computed in the previous question, compute ε r for the patch cord. 3. Using the computed value of ε r , compute β for this transmission line (assume μ = μ 0 and recall what the frequency of operation is). 4. Using Z 0 = 50 Ω, compute parameters L’ and C’ of the transmission line. Questions 1. (Pre-Lab) No measurements were made to compute R’ and G’. Why? 2. What is the difference between a lossless line and a dispersionless (or distortionless) line? Would this technique for measuring the line parameters work for a dispersionless line? Why or why not? 3. (Pre-Lab) When performing the electrical delay measurements, why did we choose the short-circuit to terminate the cables instead of the matched load? 4. When computing u p using the electrical delay technique, why did we have to use twice the length of the cable?

8 6. Experiment 2.2: Lossy and Lossless Lines on the Smith Chart In this experiment, you will use the network analyzer in the Smith Chart display format to show graphically how a wave behaves on a lossy and a lossless line. At low frequencies, it is very difficult to make a lossy line. In this lab, you will be using a lossy line simulator (see Fig. 2-3) that has been made to demonstrate the response of a lossy line on a network analyzer. The lossy line simulator consists of a piece of wire soldered to a BNC connector and is housed in a metal box. Carbon absorber has been placed in the box to make the system more lossy. The wire acts essentially as a small antenna element. The impedance of the antenna element changes with frequency. In addition, the line radiates energy into the carbon absorber where the energy is converted to heat (a tiny amount). These two effects give the lossy line simulator the same characteristic response as a lossy line. Fig. 2-3: The lossy line simulator Setup This experiment uses the network analyzer, a patch cord and the lossy line simulator. Procedure 1. Using the Smith Chart (R + jX) network analyzer settings. To set the network analyzer to measure reflection in smith chart form on channel 1, press FORMAT and then the Smith Chart softkey and then R+jX softkey to display the data on a Smith Chart. Connect the patch cord (lossless line) to channel 1 of the network analyzer leaving its other end open. Capture the resulting display. 2. Connect the lossy line simulator to the end of the patch cord using an N jack to BNC plug adapter (shown below). Capture the resulting display.

9 3. Place one marker at 500 MHz and another one at 1 GHz. Observe the trend of the curve on the network analyzer. Pay attention to the magnitude of reflection coefficients at these two frequencies. Capture the resulting display. Measured Data There were no measurements made for this section, only the two printouts. Analysis 1. Qualitatively comment on the display produced by the lossless patch cord. Why do you see a series of circles as the response? Are the circles relatively stationary in their position on the Smith Chart, that is, do they overlap each other or is there a “spiral” pattern? Why? 2. Qualitatively comment on the display produced by the lossy line simulator. Are the circles relatively stationary in their position on the Smith Chart, that is, do they overlap each other or is there a “spiral” pattern? Why? Questions 1. From the results you observed at 500 MHz and 1 GHz, what can you conclude about the attenuation constant ࠵? in a lossy transmission line in general? How does the attenuation constant change with the frequency? 2. If the lossy transmission line was nearly infinite in length, to where would the response spiral? Why? Explain in terms of reflected power.

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10 7. Experiment 2.3: Impedance Measurements using the Smith Chart In this experiment you will use the network analyzer in the Smith Chart display format to make load impedance measurements. Setup This experiment uses the network analyzer and various loads. ● Connect the patch cord to channel 1 of the network analyzer. ● Press PRESET and OK . ● Set the start frequency to 200 MHz. ● Set the stop frequency to 1.2 GHz. ● Set the display format to Smith Chart. ● Perform a one-port calibration following the procedure outlined in Exercise 1. Procedure 1. Connect the short termination to the end of the patch cord using an N to N adapter. Capture the resulting display. Record the values of the impedances at 400 MHz, 600 MHz, 800 MHz, and 1.0 GHz, as shown in the table in the Measured Data section that follows. ( Hint : To speed up the measurement process, you can place a marker at each of the frequencies by using the MARKER function and using the softkeys to add more markers. The values of the markers will be displayed on the right side of the display when you press MARKER . The first value is the real part of the measured impedance and the second value corresponds to the imaginary part. ) 2. Connect the open termination to the end of the patch cord. Capture the resulting display. Record the impedances at 400 MHz, 600 MHz, 800 MHz, and 1.0 GHz. 3. Connect the resistive termination to the end of the patch cord (You will need to remove the N to N adapter). Record the resistor value listed in Table 2-1 according to your station number. Capture the display. Record the values of the impedances at 400 MHz, 600 MHz, 800 MHz, and 1.0 GHz. Station R (Ω) C (pF) 1 22 39 2 100 10 3 10 39 4 68 15 5 39 1 Table 2-1: Resistor and capacitor values for the resistive, capacitive, series, and parallel loads 4. Connect the capacitive termination to the end of the patch cord. Record the capacitor value listed in Table 2-1. Capture the display. Record the values of the impedances at 400 MHz, 600 MHz, 800 MHz, and 1.0 GHz. 5. Connect the series termination (shown below) to the end of the patch cord. Capture the display. Record the values of the impedances at 400 MHz, 600 MHz, 800 MHz, and 1.0 GHz.

11 6. Connect the parallel termination (shown below) to the end of the patch cord. Capture the display. Record the values of the impedances at 400 MHz, 600 MHz, 800 MHz, and 1.0 GHz. Measured Data Use the following and fill in the measured data. If you are missing any data, please repeat the necessary parts of this experiment before proceeding. Resistor ______ (Ω) Capacitor ______ (pF) Load Impedances Record the impedance values by filling in the following table. Your impedance recordings should be complex numbers expressed in Ω. Load 400 MHz 600 MHz 800 MHz 1 GHz Short Open Resistive Capacitive Series Parallel Analysis

12 1. For each of the loads, compute the theoretical impedance at each frequency (400 MHz, 600 MHz, 800 MHz, and 1.0 GHz). The capacitor and resistor values used in the series and parallel loads are the same as the values used in the resistive and capacitive loads. Assume that the length of the microstrip line (a type of transmission line) between the load and the end of the patch cord is 2 cm and ε r =7 and Z 0 =50 Ω. For the short and open terminations, assume 0 cm between the calibration point and the measurement point. 2. Enter each of the theoretical impedances on a Smith Chart. Use a different Smith Chart for each load, and represent each frequency using a different color. Draw the associated constant |Γ| circles. 3. Compare the theoretical and measured values of the load impedance for each of the loads at each frequency. Comment on your results. If the measured and theoretical values do not agree, determine if the measured values make sense and explain. ( Hint : what does the constant |Γ| circle represent?) 4. For each of the loads, comment on the impedance response as a function of frequency (using the printouts). Is this what you expect to see? Why or why not? Questions 1. (Pre-Lab) Assuming that you did not have a short termination, explain using the Smith Chart how you could make a short circuit from an open circuit and a transmission line of appropriate length. How long does the transmission line need to be? Is your solution frequency dependent? 2. Based on your observations, is the microstrip line used in the resistive termination a wideband or a narrowband circuit? Why? You may assume that all connectors and loads are ideal. Α wideband system has a response which is approximately constant with frequency. A narrowband system, on the other hand, has a response which varies greatly with frequency.

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13 8. Experiment 2.4: Impedance Matching using a Single Stub In this experiment, you will use a single stub to try to match an unknown load to a patch cord. Setup This experiment uses the network analyzer, a patch cord, a single stub, and an unknown load. ● Press PRESET and OK . ● Attach a patch cord to channel 1 of the network analyzer. ● Set the start frequency to 450 MHz and the stop frequency to 550 MHz. ● Set up the network analyzer to measure reflection. ● Perform a one-port calibration and remove the N to N adapter. Procedure 1. Connect the unknown load (shown below) to the end of the patch cord. Capture the display. Record the value of the impedance at 500 MHz using a marker on the Smith Chart. 2. Insert the stub between the patch cord and the unknown load as shown in Fig. 2-4. Record the distance between the load and the stub, measuring from the base of the stub (the “Tee”) to the box of the unknown load.

14 (a) Setup for Experiment 2.4 (b) Block diagram of setup Fig. 2-4 Setup for Experiment 2.4 3. Adjust the position of the sliding short on the single stub tuner until the reflection coefficient is as close to that of a matched load (Γ=0) at 500 MHz as you can make it. This is can be done more easily using the Log Mag display format. Record the resulting stub length. Capture the resulting display and record the value of the resulting impedance at 500 MHz (using the Smith Chart tool on the network analyzer). Note : The length of the stub is adjusted by sliding the short up and down the line. Twist the short to the left to unlock it. Slide the position of the short to the bottom of the stub (near the connector). Slowly move the adjustable short up the stub until a match is achieved. To lock the short in place, twist the short to the right. The resulting stub length is measured from the base of the single stub tuner (the “Tee”) to the center of the adjustable short. Measured Data

15 Use the following and fill in the measured data. If you are missing any data, please repeat the necessary parts of this experiment before proceeding. Impedance before matching ____________ (Ω) Distance between load and stub ____________ (cm) Length of stub ____________ (cm) Impedance after matching ____________ (Ω) Note : All measurement were made at 500 MHz. Analysis 1. In this experiment, you only had control of the length of the stub, namely the distance from the load was fixed. In general, if you were to design a matching circuit using a single stub tuner, you would be able to choose both the length of the stub and the distance from the load. Compute the theoretical parameters of the single stub tuner. ● Compute the normalized impedance z l of the unknown load (Z 0 = 50 Ω). ● Enter this point on a Smith Chart and label it as A . ● Draw the constant SWR circle that passes through A . ● Label the point on the Smith Chart that corresponds to the normalized load admittance y l = 1/z l as point B . ● Compute the required distance, d, between the load and the stub in terms of wavelengths (f = 500 MHz, ε r = 1). ● Label the point on the Smith Chart where the constant SWR circle meets the g=1 circle as point C and record the corresponding admittance. ● Determine the input admittance of the stub, y stub , that is required to cancel out the imaginary part of y C . Label this as point D on the Smith Chart. ● Mark the point on the Smith Chart that corresponds to the admittance of a short circuit as point E . ● Determine the length of the stub in terms of wavelength by finding the distance between points D and E on the Wavelengths Toward Generator (WTG) scale. ● Compute the physical length of the stub. Is this length uniquely determined? Why or why not? 2. Compare the experimentally measured stub length to the theoretical one and comment on the results. 3. Compute the input impedance of the theoretical circuit (matching circuit and load) and compare it to the measured impedance after matching. Questions

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16 1. Were you able to achieve a perfect match with the single stub? If not, why not? 2. Would a quarter wave transformer have achieved a better match for the unknown load? Why or why not? 3. Suggest a way that you could use a stub and a quarter wave transformer to achieve a better match than with either of them alone. 9. Post-Lab Write-Up For each experiment of the lab exercise, include the following items in your post-lab write-up: ● Measured data ● Calculations (show your work and the equations that you used) ● Tables, printouts, and Smith Charts ● Answers to Analysis and Questions (excluding those that were part of the Pre-Lab)

2. eecs230_fall2023_lab2 (1)

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