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Transcript
EECS 420 – Electromagnetics II Lab Lab 3a – Time Domain Reflectometry 1. Objectives: At the completion of the lab, you should have each done the following: a. Examine the reflections from discontinuities on transmission lines through the use of the Agilent E5071C Network Analyzer. b. Determine the type and values of unknown passive loads by measuring the reflection coefficient in the time domain. 2. Pre-lab Homework: [Need not be turned-in, but should be included in the report as theoretical calculations and plots. I strongly recommend that you do this homework before coming to the lab.] a. Read section 11-3-7 in Engineering Electromagnetics, K. R. Demarest. b. Read the lab handout! You need the lab handout to do the pre-lab homework. c. For each of the following loads that you will be measuring in lab, determine the reflection coefficient at t 0 and t for a step response. A capacitor looks like a short circuit at t 0 (high frequencies pass easily through a capacitor) and an open circuit at t (DC signals don’t pass through a capacitor). An inductor looks like an open circuit at t 0 (high frequencies are stopped by an inductor) and a short circuit at t (DC signals pass through an inductor). After you have found the initial value ( t 0 ) of the reflection coefficient and the final value of the reflection coefficient ( t ), roughly sketch the reflection coefficient versus time (just an exponential function starting at the initial value and asymptotically approaching the final value). Also give the time constant for the load assuming that that source impedance is 50 []. Remember the time constant is RC and L/R depending on the storage element type and R is the Thévenin resistance looking out of the two inputs of the storage element. Assume that the characteristic impedance of the transmission line connecting the source to the load is 50 []. Two of the loads have been done for an illustration. Note that the rough sketch can be done by hand and does not need to be plotted using Matlab or similar programs. i. Load is a 33 [] resistor. 1) RL Z 0 33 50 17 0.2048 (for both t 0 and t RL Z 0 50 33 83 because the value RL is constant. RL will be different at t 0 and at t for loads that include an inductor or capacitor). 2) The sketch of the reflection coefficient is just a straight line of value - 0.2048 . 3) No time constant is necessary for a non-reactive load. 1 0.5 0 -0.5 -1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -7 x 10 Figure 3-1: Reflection coefficient of a 33 [] load ii. Load is a 0.39 [H] inductor. 1) RL Z 0 50 R Z 0 0 50 1 at t 0 and L 1 at RL Z 0 50 RL Z 0 50 0 t . 2) The graph is an exponentially decaying function with an initial value of 1 at t 0 and asymptotically approaching –1 at t . 3) The time constant looking out of the inductor is L 0.39 H 7.8 ns where R is the source impedance of 50 []. R 50 Gamma [V/V] 1 0.5 0 -0.5 -1 0 0.1 0.2 0.3 0.4 0.5 time [s] 0.6 0.7 0.8 0.9 1 -7 x 10 Figure 3-2: Plot of reflection coefficient for a 0.39 [H] load iii. Load is a 130 [] resistor. iv. Load is a 220 [pF] capacitor. v. Load is a 0.39 [H] inductor in series with a 130 [] resistor. vi. Load is a 0.39 [H] inductor in parallel with a 130 [] resistor. vii. Load is a 220 [pF] capacitor in series with a 130 [] resistor. viii. Load is a 220 [pF] capacitor in parallel with a 130 [] resistor. 3. Background: Reflections on transmission lines are the causes of many problems in transmission systems. Every connector, if it is not perfectly matched, can cause reflections (good connectors have reflection coefficients of the order of 0.01 or less). In order to determine the reflections caused by the connectors or the discontinuities in the system, we must be able to measure them. This experiment deals with such measurements. a. The Agilent E5071C Network Analyzer has three frequency-to-time transform modes: time-domain bandpass, time-domain low-pass step, and time-domain low-pass impulse. This experiment uses the time-domain low-pass step mode, which simulates the time domain response to a step input by Fourier transforming the swept frequency-measurements into the time domain. As in a traditional time domain reflectometry (TDR) measurement, the distance to the discontinuity in the DUT, and the type of discontinuity (resistive, capacitive, inductive) can be determined. The low-pass gives the best resolution for a given bandwidth in the frequency domain. If a mismatch exists in the system, part of the incident wave is reflected. The reflected wave is readily identified since it is separated in time from the incident wave. The information carried by the reflected wave is valuable since it reveals the nature of the mismatch. The reflection coefficient in the frequency domain, then, is expressed as Z L Z 0 Z L Z 0 where Z L is the impedance looking into a discontinuity. An exact expression of the reflection coefficient in the time domain can be obtained by taking an inverse Fourier transform of . However, a simpler and more direct analysis is possible. As is shown in the text, it is possible to use a Thévenin equivalent circuit of the transmission line as seen by the load to determine the reflected response. This analysis shows that for single time-constant systems, the reflected waveform is always an exponential transition between an initial value ( t 0 ) and a final value ( t ). These initial and final values, along with the timeconstant, are functions of the load configuration, the characteristic impedance of the transmission line, and the voltage level of the incoming step voltage. For example, in the case of the series RL combination, at t = 0 the reflected voltage is +Ei (assume the incident voltage is +Ei). This is because the inductor will not accept a sudden change in current (zero current implies open circuit); it initially looks like an infinite impedance, and = 1 at t 0 . Then current in L builds up exponentially and its impedance drops toward zero. When t goes to infinity, therefore, is determined only by the value of R since R Z0 R Z0 when t . The exponential transition of (t) has a time constant determined by the effective resistance seen by the inductor. Since the source impedance is Z0, the inductor sees Z0 in series with R, and L R Z0 where here is the time constant. The inductor will initially see the characteristic impedance Z0 of the transmission line and then it will see the source impedance. In this case they are equal and therefore no mismatch in impedance needs to be accounted for. Three other cases (resistor and inductor in parallel, resistor and capacitor in series, and resistor and capacitor in parallel) can be analyzed in a similar way. reflection coefficients of these four cases are found to be, for R-L in series, RR ZZ0 1 RR ZZ0 exp t t 0 exp t 0 0 where L R Z0 for R-L in parallel, 1 RR ZZ0 1 exp t t 0 exp t 0 where L R 1 Z 01 1 for R-C in series, 1 RR ZZ0 1 exp t t 0 exp t where R Z 0 C and for R-C in parallel, 0 The RR ZZ0 1 RR ZZ0 exp t t 0 exp t 0 0 where R 1 Z 01 1C 4. Equipment: You will need the following pieces of equipment: a. Agilent E5071C Network Analyzer b. One 2 [ft] transmission line cable c. Calibration Standards d. Three Circuit Boards. The values of components used for the circuit boards are R1 = 33 [, R2 = 130 [, C = 220 [pF] or 180 [pF], and L = 0.39 [H]. The resistor used in combination with C or L is R2. If the capacitor is 180 [pF] it will be marked. If the capacitor is unmarked, it is 220 [pF]. The R39 on the inductor stands for Radix point (generic term for decimal point that is not base-ten specific) followed by 39 with units of [H]. Network Analyzer PORT 1 PORT 2 Cable L R2 R1 C C L R2 R2 R2 Circuit Boards Figure 3-3: Test setup for this lab C R2 L 5. Procedure: 1) Press [ON] to turn on the power of the Network Analyzer. 2) Set the stop frequency to 1.5 [GHz] (use the [STOP] button under stimulus control to do this). 3) Set number of points to measure to 401 by pressing [SWEEP SETUP] [ POINTS] [401] [x1] 4) Press [SYSTEM] [RETURN] [ANALYSIS] [TRANSFORM] [SET FREQ LOWPASS]. (This sets the low-frequency limit so that the start frequency is an exact sub-harmonic of the stop frequency.) 5) Perform an S11 1-port calibration. 6) Press [SYSTEM] [RETURN] [ANALYSIS] [TRANSFORM] [TRANSFORM ON/OFF]. 7) Press [SYSTEM] [RETURN] [ANALYSIS] [TRANSFORM] [TYPE] [LOWPASS STEP]. 8) Press [SYSTEM] [RETURN] [ANALYSIS] [TRANSFORM] [START] [-5] [ G/n] to select a start time of -5 nanoseconds . 9) Press [SYSTEM] [RETURN] [ANALYSIS] [TRANSFORM] [STOP] [60] [ G/n] to select a stop time of 60 nanoseconds . 10) Press [FORMAT] [REAL] to view the step response of the reflectance . The step response should be zero for negative time because the unit step function is zero for negative time. 11) Save your current settings to register [STATE01]. 12) Connect an open termination to the cable and press [SCALE] [AUTO SCALE] to center the display. An open has a reflectance of one, so the output should be one for positive time. 13) Connect a short termination to the cable and press [SCALE] [AUTO SCALE]. Notice the polarity of the step response. A short has a reflectance of negative one, so the output should be negative one for positive time. 14) Connect a matched load termination to the cable and press [SCALE] [AUTO SCALE]. A matched load has a reflectance of zero, so the output should be zero for all time. 15) Connect one of the circuits on the circuit boards to the cable (it is preferred to test the resistors first). 16) Press [SCALE] [AUTO SCALE] to center the display if the trace of the signal is not properly displayed. 17) If the transient time is too short, set stop time at a lower value. If the transient time is too long, set stop time to a higher value. 18) Set marker one at the beginning of the initial response (will be slightly greater than 0 [ns] because of the small delay introduced by the micro-strip line connecting the SMA connector and the load). Set marker two at separate stable point (after the initial response, before the transient response is finished, and away from major ripples). Set marker three at steady state (latest time possible since transient effects decrease as time increases). Read step (21) to make sure you understand what values you will need. 19) Save your measurement screens. 20) Repeat steps 14) to 18) for all eight circuits. 21) From the data you recorded in step 18 and 19, find the values of the resistors, capacitors, and inductors in each circuit. The only assumption that you may make is that the components are in one of the eight different configurations discussed in the lab handout. Compare your results with the given values. Your results should be with in 50% of the nominal value. The following equation will be useful: t 0 exp t For each circuit you should have measured the initial response, (0), and the steady-state response (). You know the form that will take based on the type of step response that you measured (in the lab homework you might have sketched each of the possible responses and you can match the measured response to the corresponding sketch to find the component configuration). And you have another independent point from markers two that can be used to solve for . Finally, you need to determine the two unknowns (RL or RC) using the value and either (0) or () depending on the unknown load. Note that all of your time values have to be shifted so that the initial response lines up with t 0 . The time at which you obtain your initial response (0) is the time offset you should subtract from all of your measurements. This compensates for the additional time delay caused by the micro-strip line between the SMA connector and the load. NOTE: YOU NEED TO SUBMIT ONLY ONE REPORT FOR LABS 3a AND 3b TOGETHER AND IT IS DUE ONE WEEK AFTER YOU FINISH LAB 3b. THIS REPORT CAN BE UPTO 8 PAGES LONG.