Assignment: Signals and Systems

Class Room Activity: RLC Impulse and Step Response

Objective of the lab experiment:

The objective of this experiment is to experimentally measure the step response of an RLC circuit and compare it to the response predicted using MATLAB.

Equipment list:

One LM741 op amp
One 1 ? resistor
One 100 ? resistor
One 1 k? resistor
One 0.1 µF capacitor
One 10 mH inductor
One1 NI Elvis II board

Power supply with +15 V and -15 V outputs

Function generator capable of producing unipolar and bipolar square waves and sinusoids up to 5 KHz

Oscilloscope

Three BNC to alligator

One PC running LabVIEW 2010or higher and CodeWarrior 5.9

MATLAB

One 2.0 mm flathead screwdriver

Theory

The previous labs have focused on having a single reactive component in a circuit to create a filter. This lab will focus on what happens if there are two different reactive components in the same circuit. An RLC circuit with two reactive components (an L and a C) is shown below. Using the knowledge gained previously, the transfer function can easily be found.

Note that all of the components are connected in series and the voltage across the resistor is our output.

Using the voltage divider rule here gives the following relationship:

V_out/V_in =Z_R/(Z_R+Z_C+Z_L )=R/(R+1/Cs+Ls)

Now substitute numerical values of R, L, and C. Then simplify the expression to get the transfer function as a ratio of two rational polynomials in s domain. Make the coefficient of the highest power of s unity in both numerator and denominator. If necessary, multiply the whole transfer function by a constant number.

Now write a MATLAB program that takes in the numerical values of R, L, and C; generates the transfer function of the system; and computes its Bode plot, its pole-zero map, and the step response of the system. Use the MATLAB keywords tf, bode, pzmap, and step to get these results.

Note that on the pole-zero map created by the pzmap command, poles are identified with an X and zeroes are identified with an O.

You should see a Bode plot as shown below for R = 100, L = 10 mH, and C = 0.1 µF.

Graph produced by MATLAB Bode plot command

Check the pole-zero map to see if it agrees with the poles and zeroes of the transfer function that was created by calculating the poles and zeroes manually.This kind of step response you will see is known as a decaying sinusoid, which can only appear in a second-order (or higher order) transfer function.

Compare the theoretical step response generated by MATLAB to the actual step response measured with the constructed circuit.

Procedure:

Step 1

Set up the frequency generator with the specifications below.

VPP 5 V
Frequency 200 Hz
Signal Type Square wave

Step 2

Build the following RLC circuit and observe the step response on the oscilloscope.

The observed response should look like the following:

Response of the 1? circuit as seen on an oscilloscope.

Step 3

Input these values into MATLAB using the supplied program and observe the theoretical step response.

Step 4

Change resistor to 1k? and observe the step response. Then use the MATLAB code to observe the theoretical step response.

The observed response should look like the following:

Response of the 1 k? circuit as seen on an oscilloscope

Step 5

Change resistor to 100 ? and observe the step response. Then use the MATLAB code to observe the theoretical step response.

The observed response should look like the following:

Response of the 100 ? circuit as seen on an oscilloscope

Now answer the following questions.

Does the step response from the actual measurement match the theoretical curve from MATLAB for R = 1 ?? Note the magnitude of the peak value of the step response produced by MATLAB and compare it to the peak value produced by the experimentally measured curve. Why is the MATLAB value much higher than the experimental value? (Hint: The inductor is not an ideal inductor.)

What is the mathematical relationship between the impulse and step response of a system?

How will the shape of the step response change as the value of resistance is increased?

Why is it desirable to include an op-amp buffer to drive the RLC circuit rather than driving the circuit directly from the function generator?

Why is it that we never see an oscillatory response in an RC circuit, whose response is always exponential regardless of resistance?

Attachment:- Lab.rar