ROSE HULMAN INSTITUTE OF TECHNOLOGY Department of Electrical and Computer Engineering ECE 300 Signals and Systems Fall 2005 RDT System Impulse and Step Response Lab 03 Bruce A Ferguson In this laboratory you will investigate basic system behavior by determining the impulse and step response of a simple RC circuit You will also determine the time constant of the circuit and determine its rise time two common figures of merit FOMs for circuit building blocks Objectives 1 Design an experiment by specifying a test setup choosing circuit component values and specifying test 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 well grounded in the circuit theory you have spent so much of your energy learning An important problem in modern high speed digital and wideband analog systems is the response limitations of basic circuit elements in high speed integrated circuits As simple as it may seem the lowly RC lowpass filter accurately 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 circuit design 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 by showing its circuit schematic or by calculating its impulse response or transfer function But in many cases only FOMs are important to determine the adequacy of the system R1 V1 C1 Vout 0 Figure 1 Simple RC lowpass filter circuit Page 1 of 6 EC 300 Signals and Systems Fall 2005 The first FOM is the system time constant Many simple systems display a characteristic exponential decay or rise in their response Since the form of the response is known the only data important to characterize a specific system is the numerical value of its time constant Figure 2 shows a typical impulse response for the RC filter of Figure 1 The characteristic exponential decay is just as we remember from class The time constant of the circuit is defined as the time it takes for the response to decay to 1 e 37 times its initial value Since the functional form of the response is exp t RC the time constant of the circuit can easily be shown to be simply RC 1 e 1 0 368 time constant Figure 2 Impulse response of the RC lowpass filter circuit of Figure 1 showing the definition of the circuit time constant The time constant FOM allows us to determine the behavior of the circuit in a number of important scenarios such as digital signal response or the time until steady state analysis results are valid The second FOM the risetime characterizes the response of the system to a positive step change in input The response of the circuit of Figure 1 to an applied unit step input is shown 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 from 10 to 90 of its final value This measurement in shown in Figure 3 as the time difference between the times t10 and t90 These values can easily be determined given the step response of the system The 10 90 risetime FOM is especially important in digital systems as excessive slurring of the crisp edges of the digital waveform leads to rapid degradation in system performance The risetime is a slightly deceiving measurement for certain circuits since the rise time depends in part on the magnitude of the step change in output the circuit must produce For this reason a related FOM the slew rate is required to describe the maximum voltage change per unit time which the circuit can produce Page 2 of 6 EC 300 Signals and Systems Fall 2005 90 rise time tr t90 t10 10 t10 t90 Figure 3 Step response of the RC lowpass filter circuit of Figure 1 showing the definition of the 10 90 risetime Measuring these two FOMs in the laboratory is actually a relatively simple effort for low bandwidth systems The requirements are the circuit or the system under test CUT or SUT and appropriate impulse and step waveform generation and measurement equipment However some thought needs to go into the waveform specification in order to facilitate measurement 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 cannot create either of these two ideal waveforms in our part of the universe However we can create reasonable approximations function generator scope trigger trigger Circuit input FG output Figure 4 Experimental test setup for measuring the impulse and step response of the SUT Page 3 of 6 EC 300 Signals and Systems Fall 2005 To simulate the impulse waveform we could use a suitably abrupt and high amplitude pulse waveform But what constitutes suitably abrupt Well if we can produce a pulse whose time width is much less than the time constant of the circuit with a suitably large amplitude then the pulse would provide a reasonable approximation of an impulse for that circuit That same pulse might not be short enough in duration to approximate an impulse for a different circuit having a shorter time constant OK now we see the trick How can we approximate a unit step impulse There are three important aspects of the step waveform to consider First the waveform must have a fast transition from off to on Second the waveform must stay on long enough for the system transients to die out Thus we could imagine a suitably wide pulse as being a reasonable approximation to a step input The third aspect has to do with how we create the step change in the waveform Each of the two waveforms discussed above are single shot events or events that occur only once in time The oscilloscopes we use are optimized for periodic waveforms not single shot events No worries we can create a pulse train with our function generators which allow for both control of pulse widths for impulse step waveform design as well as repeating these pulses periodically to optimize the viewing of the waveforms on the oscilloscope screen However the period of the waveform should be long enough to let all transients die out in the circuit before the next impulse arrives Pre Lab Exercises 1 Calculate the impulse
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