UNIVERSITY of PENNSYLVANIA

Step response of RLC Circuits

Goals

To build RLC circuits and to observe the transient response to a step input. You will study and measure the overdamped, critically damped and underdamped circuit response.

Background

RLC circuits are widely used in a variety of applications such as filters in communications systems, ignition systems in automobiles, defibrillator circuits in biomedical applications, etc. The analysis of RLC circuits is more complex than of the RC circuits we have seen in the previous lab. RLC circuits have a much richer and interesting response than the previously studied RC or RL circuits. A summary of the response is given below.

Lets assume a series RLC circuit as is shown in Figure 1. The discussion is also applicable to other RLC circuits such as the parallel circuit.

Figure 1: Series RLC circuit

By writing KVL one gets a second order differential equation. The solution consists of two parts:

x(t) = xn(t) + xp(t),
in which xn(t) is the complementary solution (=solution of the homogeneous differential equation also called the natural response) and a xp(t) is the particular solution (also called forced response). Lets focus on the complementary solution. The form of this solution depends on the roots of the characteristic equation,
(1)
in which  is the damping ratio and  is the undamped resonant frequency. The roots of the quadratic equation are equal to,

(1b)

For the example of the series RLC circuit one has the following characteristic equation for the current iL(t) or vC(t),
s2 + R/L.s + 1/LC =0. (2)
Depending on the value of the damping ratio one has three possible cases:

Case 1: Critically damped response: two equal roots s= s1= s2

(3)

The total response consists of the sum of the complementary and the particular solution. The case of a critically damped response to a unit input step function is shown in Figure 2.

Case 2: Overdamped response: two real and unequal roots s1 and s2

(4)
Figure 2 shows an overdamped response to a unit input step function.

Figure 2: Critically and overdamped response to a unit input step function.

Case 3: Underdamped response: two complex roots

(5)
Figure 3 shows an underdamped response to a unit input step function.
Figure 3: Underdamped response to a unit input step function.

Pre-lab assignments

1. Review the sections on RLC circuit in textbook (6.3 in Basic Engineering Circuit Analysis, by D. Irwin).

2. Prove that the expression for the damping ratio and the undamped resonant frequency for the circuit of Figure 1 is equal to,

(6)
3. Assume that C=100nF. Find the values of R and L such that  = 10 krad/s for the three cases of damping ratio equal to 1, 2 and 0.2.

4. For the three cases of damping ratio equal to 1, 2 and 0.2 find the expression of the voltage vC(t) over the capacitor using the values of the capacitor, inductor and resistors calculated above. Assume a unit step function vS as the input signal, and initial conditions vC(0)=0 and iL(0)=0. Plot the response for the three cases (preferably using a plotting program such as MATLAB, Maple or a spreadsheet).

In-lab assignments

A. Equipment:

• 1. Agilent Signal Generator
• 2. Agilent Scope
• 3. Protoboard
• 4. Resistor: 5Kohm potentiometer
• 5. Capacitors: 100nF
• 6. Inductor 100mH
• 7. box with cables and connectors
• 8. Scope Probe
• 9. DMM
• 10. RLC Meter
• 10. Multisim software
B. Procedure

1. Simulate the three RLC circuits using Multisim software for the cases of damping ratio equal to 1, 2 and 0.2 (use the values of R, L and C found from the pre-laboratory). Use a square wave with 1Vpp (i.e. amplitude of 0.5V with offset of 0.5V - use the function generator in EWB) and frequency of 200 Hz as input voltage. Compare the waveforms with the one you calculated in the pre-lab. Make a print out.

2. Get the components L and C you will need to build the RLC circuit. A real inductor consists of a parasitic resistor (due to the windings) in series with an ideal inductor as shown in Figure 4. Measure the value of the inductor and the parasitic resistance RL using an RLC meter and record these in your notebook. Measure also the value of the capacitor. For the resistors use a 5 kOhm potentiometer.

Figure 4: Model of an inductor

3. Build the series RLC circuit of Figure 5, using the values for L and C found in the pre-lab corresponding to the damping ratio of 1, 2 and 0.2.

Figure 5: RLC circuit: (a) RTOT includes all resistors in the circuit; (b) showing the different resistors in the circuit.

The total resistor RTOT of the circuit consists of three components: RT which is the output resistance of the function generator (50 Ohm), the parasitic resistor RL and the actual resistor R. First calculate the required resistor R such that the total resistor corresponds to the one found in the pre-lab for each case. Fill out a table similar to the one shown below.

 Damping ratio 1 2 0.2 RT (Ohm) . . . RL (Ohm) . . . Rtot (Ohm) . . . R (Ohm) . . .
4. Measure the response of each case.

Case 1: critically damped response.

a. Set the potentiometer to the value R calculated above corresponding to a damping ratio of 1.

b. Set the function generator to 1Vpp with an offset voltage of 0.5V and a frequency of 200 Hz. Display this waveform on the oscilloscope. Measure the voltage over the capacitor and display the waveform vC(t)on the scope. Measure its characteristics: risetime, Vmin, Vmax, and Vpp. Make also a print out of the display. Compare the measured results with the one from the pre-lab and the simulations.

Case 2: overdamped response.
a. Set the potentiometer to the value R calculated above corresponding to a damping ratio of 2. Measure and display the response over the capacitor and make a print out. Determine the rise time, min and max value of the voltage vC.

b. Calculate one of the time constants of the expression (4). Usually one of the time constants is considerably larger than the other one which implies that the exponential with the smallest time constant dies out quickly. You can make use of this to find the largest time constant. Measure two points on the graph (v1,t1) and (v2,t2) as shown in Figure 6. Choose t1 sufficiently away from the origin so that one of the exponentials has decayed to zero. You can than make use of the following relationship to find the time constant:

(7)
in which Vf is the final value of the exponential (value at the time t=infinite). The expression you derived in the last lab: t=trise/2.2 is a special case of the above expressions (i.e. v1=0.1Vmax; v2=0.9Vmax).

Figure 6: method to measure the time constant.

Case 3: underdamped response

a. Set the potentiometer corresponding to the value R calculated above corresponding to a damping ratio of 0.2. Measure and display the response over the capacitor and make a print out. Determine its characteristics: voltage and time of the first peak, voltage and time of the second peak. Make a print out.

b. Determine the value of t and  wd   from the measured waveform (See Figure 3). Use the expression (7) to determine the value of the time constant (t=1/s).

5. Vary the potentiometer and observe the behavior of the response (display the voltage over the capacitor). Notice when the output goes from underdamped to critically damped and overdamped. In general, a critically damped response is preferred because it does not give overshoot or "ringing" and has a fast rise time. An overdamped response has a slower rise time than the other responses, while the underdamped response rises the fastest, but also give a lot of overshoot which is not desired. Record your observations in you lab notebook.

References:

J. D. Irwin, "Basic Engineering Circuit Analysis," 5th edition, Prentice Hall, Upper Saddle River, NJ, 1996.

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Created by Jan Van der Spiegel: April 15, 1997.
Updated by Sid Deliwala on Jan 11, 2013.