İzmir University of Economics
EEE201 Electrical Circuits I Lab
EXPERIMENT 10:
Step Response of RLC Circuits
Objective:
The students will be able to learn:
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How to build RLC circuits
How to measure the inductance of a inductive circuit element
How to analyze the transient response of an RLC circuit to a step input
How to study and measure the overdamped, critically damped and underdamped circuit
response of an RLC circuit
How to design an RLC circuit for obtaining a desired step 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
the quadratic equation are equal to,
is the undamped resonant frequency. The roots of
(1b)
1-1
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.
1-2
Figure 3: Underdamped response to a unit input step function.
In can be shown that the expression for the damping ratio and the undamped resonant
frequency for the circuit of Figure 1 is equal to,
A. In-Lab Experimental Work
Task 1.1: Construct the circuit given in Figure 4. Choose R, C and L yielding damping ratio equal
to 1, 2 and 0.2, respectively. Use a square wave with 1 Vpp (i.e. amplitude of 0.5V with offset
of 0.5V - use the function generator) and frequency of 200 Hz as input voltage.
Figure 4
Task 1.2: On the oscilloscope, measure and save as file both input voltage and the capacitor
voltage for each different set of R, C and L given in Task 1.1.
Task 1.3: Measure the natural frequency and damping ratio for each different set of R, C and L
given in Task 1.1.
Task 2.1: Change the square wave voltage source in the experiment of Tasks 1.1-1.3 with a
sinusoidal wave with 5 Vpp. Observe what happens when the input voltage frequency is close
and also equal to the natural frequency.
B. Post-Lab Tasks: Lab Report
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Explain what you have done in this lab and how do you measure the quantities related
to the experiment.
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