ECE 300 Lab 7Impulse and Step ResponseECE 300 Lab 7In this laboratory, you will investigate basic system behavior by determining theimpulse and step response of a simple RC circuit. You will also determine thetime constant of the circuit and determine its rise time, two common figures ofmerit (FOMs) for circuit building blocks.Objectives:1. Design an experiment by choosing circuit component values and waveform details to investigate the impulse and step response.2. Measure the impulse response of the circuit and determine the time constant, validating your theoretical calculations.3. Measure the step response of the circuit and determine the rise time, validating your theoretical calculations.Background:As we introduce the study of systems, it will be good to keep the discussion wellgrounded in the circuit theory you have spent so much of your energy learning.An important problem in modern high speed digital and wideband analogsystems is the response limitations of basic circuit elements, for example in highspeed integrated circuits. As simple as it may seem, the lowly RC lowpass filteraccurately models many of the systems for which speed problems are so severe.There is a basic need to be able to characterize a circuit independent of its circuitdesign and layout in order to predict its behavior. Consider the now-overly-familiar RC lowpass filter shown in Figure 1. We could describe this system byshowing its circuit schematic or by calculating its impulse response or transferfunction. But in many cases only FOMs are important to determine the adequacyof the system.Figure 1. Simple RC lowpass filter circuit.The first FOM is the system time constant. Many simple systems display acharacteristic exponential decay in their response. Since the form of theresponse is known, the only data important to characterize a specific system isthe numerical value of its time constant. Figure 2 shows a typical impulseresponse for the RC filter of Figure 1. The characteristic exponential decay isBAF 1 1/15/19 +C1Vout-+0+-V1+R1ECE 300 Lab 7just as we remember from class. The time constant of the circuit is defined asthe time it takes for the response to decay to 1/e times its initial value. Since thefunctional form of the response is exp(-t/RC), the time constant of the circuit caneasily be shown to be simply RC.Figure 2. Impulse response of the RC lowpass filter circuit of Figure 1, showingthe definition of the circuit time constant.This FOM allows us to determine the behavior of the circuit in a number ofimportant scenarios, such as digital signal response or the time until steady-stateanalysis results are valid. The second FOM is characterizes the response of the system to a step change ininput. The response of the circuit of Figure 1 to an applied unit step input isshown in Figure 3. The rise time most typically used as a FOM is the “10-90%risetime”, which is simply the amount of time necessary for the output to rise from10% to 90% of its final value. This measurement in shown in Figure 3 as thetime difference between the times t10 and t90. These values can easily bedetermined given the step response of the system.BAF 2 1/15/191/e*1=0.368 = time constant1/e*1=0.368 = time constantECE 300 Lab 7Figure 3. Step response of the RC lowpass filter circuit of Figure 1, showing thedefinition of the 10-90% risetime.The 10-90% risetime FOM is especially important in digital systems, asexcessive “slurring” of the crisp edges of the digital waveform leads to rapiddegradation in system performance.Measuring these two FOMs in the laboratory is actually a relatively simple effortfor low bandwidth systems. The requirements are a the circuit, or the systemunder test (SUT), and appropriate impulse and step waveforms. However, somethought need to go into the waveform specification in order to facilitatemeasurement of the two FOMS.The basic test setup is shown in Figure 4. The trick is to be able to create an“impulse” and a “unit step” waveform to test the system. Of course, we cannotcreate either of these two ideal waveforms in our part of the universe. However,we can create reasonable approximations. BAF 3 1/15/19t90t10rise timetr= t90-t1090%10%t90t10rise timetr= t90-t1090%10%ECE 300 Lab 7Figure 4. Experimental test setup for measuring the impulse and step responseof the SUT.To simulate the impulse waveform, we could use a suitably abrupt pulsewaveform. But what constitutes “suitably abrupt”? Well, if we can produce apulse whose time width is much less than the time constant of the circuit, thenthe pulse would provide a reasonable approximation of an impulse for that circuit.That same pulse might not be short enough in duration to approximate animpulse for a different circuit having a shorter time constant.OK, now we see the trick. How can we approximate a unit step impulse. Thereare three important aspects of the step waveform to consider. First, thewaveform must have a fast transition from off to on. Second, the waveform muststay “on” long enough for the system transients to die out. Thus, we couldimagine a suitably wide pulse as being a reasonable approximation to a stepinput. The third aspect has to do with how we create the impulse waveform.Each of the two waveforms discussed above are single-shot events, or eventsthat occur only once in time. The oscilloscopes we use are optimized for periodicwaveforms, not single-shot events. No problem, we can create a pulse train withour function generators which allow for both control of pulse widths forimpulse/step waveform design, as well as repeating these pulses periodically tooptimize the viewing of the waveforms on the oscilloscope screen. However, theperiod of the waveform should be long enough to let all transients die out in thecircuit before the next impulse arrives.Prelab Exercises:1. Calculate the impulse response of the RC lowpass filter shown in Figure 1.Determine the time constant for the circuit.2. Find the step response of the circuit, and determine the 10-90% rise time.3. Compare the time constant and rise time to the bandwidth of the lowpass filter.BAF 4
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