WEST VIRGINIA UNIVERSITY
West Virginia University
COLLEGE OF ENGINEERING
College of Engineering & Mineral Resources
Lane Department of Computer Science and Electrical Engineering
EE224
Electric Circuits Laboratory
S 2006
Experiment No.5
“Study of Damped Responses”
Objective: The purpose of this experiment is to design a simple series RLC circuit and
generate the three different kinds of voltage responses namely, the Underdamped, Overdamped
and Critically damped response.
Theoretical Background:
The voltage responses of RLC circuits depend on the values of the resonant frequency
R⎞
1
⎛
(ω0 =
) and alpha or the damper frequency ⎜α =
⎟ . As you will see in this experiment,
2L ⎠
LC
⎝
by varying the values of R, L and C, one should be able to produce the three kinds of voltage
responses. The solution of the differential equation that describes the current or the voltage in the
series RLC circuit is given in the following form:
s2 +
R
1
s+
=0
L
LC
(1)
Equation (1) is called the characteristic equation of the differential equation because the
roots of this quadratic equation will determine the mathematical character of f(t) (Note: function
f(t) represents either voltage or current response). The roots are:
2
s1, 2
R
1
⎛ R ⎞
=−
± ⎜ ⎟ −
2L
LC
⎝ 2L ⎠
or s1, 2 = −α ± α 2 − ω 0
2
(2)
(3)
where:
α = is neper frequency or damping factor.
ωo = resonant frequency.
The nature of the characteristic roots tells us whether the solution for any voltage or
current in the circuit is overdamped, underdamped, or critically damped. There are three
different cases:
1.
2.
3.
Roots are real and distinct, response is overdamped.
Roots are complex, the response is underdamped.
Roots are real and equal, the response is critically damped.
The diagram below represents a basic series RLC circuit:
Figure1. Series RLC circuit
The terms overdamped, underdamped, and critically damped are used to describe the
impact of the dissipative element R on the response. The effect of R is reflected as mentioned
before in the neper frequency or damping factorα.
1.
If α is large compared with the resonant frequency ωo, the voltage or current
approaches its final value without oscillation, and this nonoscillatory response is
called overdamped.
2.
If α is small compared to ωo, the response oscillates about its final value, and this
response is called underdamped. The smaller the value of α is, the longer the
oscillation persists. If the dissipative element is removed from the circuit, α equals
zero and the voltage or current response becomes a sustained oscillation.
3.
If α is equal to ωo, the response is on the verge of the oscillation and is called the
critically damped response.
Experimental Procedure:
1. Work out the conditions required to produce the three voltage responses and calculate the
critical value of the resistance ‘Rcr’.
Assume c=0.1uf and L = 15000uH.
2. To get accurate results, it is advisable to measure the exact value of the component you
are using. Use a multimeter to measure resistance.
3. Set up the circuit as shown in the figure. Try using different resistors. Make sure you start
off with a low resistance value. Set the frequency at some low value (say around 50 Hz).
The output is measured across the capacitor.
4. Vary the resistance (start from a low value) of the circuit and observe the output. Make
observations on the output voltage response as you proceed from the Underdamped to the
Critically damped to the Overdamped case.
For your Report answer these questions:
1.
2.
What are the conditions for producing the Underdamped, Critically damped and the
Overdamped responses?
Does altering the source frequency affect the voltage response of the circuit?
Note2: Please follow the format of the report given in the syllabus.
Data Table for the RLC Circuit.
R
α = R / 2L
ω 0 = 1 /( LC )1 / 2
Type of Response
10 ohms
100 ohms
1K
2K
Critical value of the resistance was calculated to be, Rcr = _____________.