ESE Undergraduate Laboratory
Step response of RLC Circuits


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.


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:

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, in which  is the damping ratio and  is the undamped resonant frequency. The roots of the quadratic equation are equal to, For the example of the series RLC circuit one has the following characteristic equation for the current iL(t) or vC(t), Depending on the value of the damping ratio one has three possible cases:

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

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

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

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,

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:

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 
RT (Ohm) 
. . .
RL (Ohm) 
. . .
Rtot (Ohm) 
. . .
R (Ohm) 
. . .
4. Measure the response of each case.

Case 1: critically damped response.

Case 2: overdamped response.
Figure 6: method to measure the time constant.

Case 3: underdamped response

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.


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.