UNIVERSITY of PENNSYLVANIA

DEPARTMENT OF ELECTRICAL AND SYSTEMS ENGINEERING

ESE Undergraduate Laboratory

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 v C.

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: =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 and d from the measured waveform (See
Figure 3). Use the expression (7) to determine the value of the time constant
(=1/).

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.