West Virginia University
College of Engineering & Mineral Resources
Lane Department of Computer Science and Electrical Engineering
EE224
Electric Circuits Laboratory
S 2006
Experiment No.2
“The Natural and Step Responses of Series RLC Circuit”
Objective: This experiment will study the response of a series RLC circuit to a driven pulse
train with a frequency between 0 Hz and 500 Hz. In particular the under-damped response will
be explored. The dependence of the frequency on the component values will be
verified. Designed circuit has many applications and one of them is as a sound circuit in the
racecar where it is shown that obtained oscillatory response can produce more pleasing “motor
sound” to driver than the pulse train with a frequency between 0 Hz and 500 Hz produced
directly by sensor on the motor shaft.
Theoretical Background: The series RLC circuit is a classic dynamic circuit in the studies
of electrical engineering. One of the primary applications of this circuit is for a band-pass filter
which only allows signals to pass in a particular range of frequency. (This specific case will be
studied later in the following experiments). This is extremely important in radios since people
want to receive stations at certain frequencies but not at the others.
We will first derive the expression for the current response in a source free series RLC circuit.
Differential equation that describes the current in the circuit has the form (Kirchoff’s Voltage
Law has been applied to the closed path in the circuit):
t
di 1
Ri + L + ∫ idτ +V0 = 0
dt C 0
(1)
if we differentiate (1) with respect to t to get:
R
di
d 2i i
+L 2 + =0
dt
C
dt
(2)
which we can rearrange as:
d 2 i R di
i
+
+
=0
2
L dt LC
dt
(3)
the solution for this equation is given in this form:
1
R
s+
=0
L
LC
s2 +
(4)
Equation (4) is called the characteristic equation of the differential equation because the roots of
this quadratic equation will determine the mathematical character of i(t). The roots are:
2
s1, 2
R
1
⎛ R ⎞
=−
± ⎜ ⎟ −
2L
LC
⎝ 2L ⎠
or s1, 2 = −α ± α 2 − ω 0
2
(5)
(6)
Where:
α = is neper frequency or damping factor.
ωo = resonant frequency.
The diagram below represents a basic series RLC circuit:
The neper frequency α for the series RLC circuit is:
α=
R −1
s
2L
(7)
whereas the expression for the resonant radian frequency is:
ωo =
1
LC
rad / s (8)
The current response will be :
ω o2 < α 2 (overdamped),
i (t ) = B1e −αt cos ω d t + B2 e −αt sin ω d t for
ω o2 > α 2 (underdamped),
i (t ) = A1e s1t + A2 e s 2t
for
i (t ) = D1te −αt + D2 e −αt
for
ω o2 = α 2 (critically damped).
Similarly, we can obtain the voltage across the capacitor vc(t) in the form of three different
damped solutions, again by applying Kirchhoff’s voltage law to the same RLC circuit.
The following figure shows plots of these responses for a source free 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 in the neper frequency, or
damping factor α. If α is large compared with the resonant frequency ωo, the voltage or current
approaches its final value without oscillation, and the nonoscillatory response is called
overdamped. 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. The critical value of α occurs when α = ωo; in this
case, the response is on the verge of oscillation and is called the critically damped response.
All three responses start out at some initial condition (e.g. 15V) and all eventually decay to
zero. The temporal decay of the responses is caused by energy loss in the circuit and is called
damping. The case A response does not change sign and is called the overdamped response. The
case C response undershoots and then oscillates about the final value. This response is said to be
underdamped because there is not enough damping to prevent these oscillations. The case B
response is said to be ciritically damped since it is a special case at the boundary between
overdamping and underdamping.
Experimental Procedure:
(1) We have in our lab the following component values available:
R = 10 ohms, C =0.1uf, L = 15000uH.
(2) With these values of R, C, and L, find out ‘damping constant’ and also the ‘frequency of
oscillations’ for the series RLC circuit.
(3) Determine the nature of response this circuit will produce with the given component values.
(4) Determine the expression for voltage across capacitor, Vc(t).
(5) Connect the circuit on a bread board.
(6) Observe the voltage across capacitor on the oscilloscope, when the circuit is driven by pulse
train with a 100 Hz frequency.
(7) Simulate the same circuit in Pspice.
(8) Write a program in Matlab that plots a graph of Vc vs. time.
(9) For your report: Construct a single graph of the oscillations obtained in the lab. Plot Vc vs.
time.
Note: Please follow the format of the report given in the syllabus.