West Virginia University
College of Engineering & Mineral Resources
Lane Department of Computer Science and Electrical Engineering
EE224
Electric Circuits Laboratory
S 2006
Experiment No.3
“Series and Parallel RLC Resonant Circuits”
Objective: The purpose of this experiment is to analyze parallel and series RLC resonant
circuits. You will measure the frequency response in the laboratory and from these data will
formulate a model of the circuits, which are frequency-selective. This model of RLC circuit then
will be used to predict the frequency response of the device to other types of inputs.
Theoretical Background:
In addition to coupling circuits and wave shaping circuits, one of the primary applications
of series and parallel circuits containing capacitors, inductors and resistors is filter applications.
Filters are generally described as passing or blocking high or low frequencies or a band of
frequencies. In filter applications, a sine wave source is connected to a circuit and an output is
taken across one (or possibly two) of components in the circuit.
Series and parallel resonant circuits generally operate as band pass and band stop filters.
Band pass filters are used to pass on a band of frequencies between amplifiers in a tuned circuit
application. Band stop filters reject a group of unwanted frequencies. Also, series and parallel
resonant circuits are used as the frequency determining components in oscillator circuit
applications.
The resistance (actually reactance) of capacitors and inductors change with respect to
frequency. The reactance of a capacitor is indirectly related to frequency and the reactance of a
coil is directly related to frequency. This indicates that the reactance of the capacitor will be a
high value at low frequencies and the reactance of the inductor will be high at high frequencies.
Resonance, either series resonance or parallel resonance, is defined as a situation where
the reactance of the capacitor is equal to the reactance of the inductor, or where capacitive
reactance is equal to inductive reactance.
When capacitors and coils are connected in series, the reactances of the components are
180 degrees out of phase, and the reactances cancel, which indicates that a series resonant circuit
is, in effect, a very low resistance, or a short. The output across a series resonant circuit will be
relatively small amplitude at resonance. Therefore, the output across the capacitor and coil in
series will be a band stop filter while the output across a resistor will be a band-pass filter.
When capacitors and coils are connected in parallel, the currents through the components are 180
degrees out of phase, and the reactive currents cancel which indicates that a parallel resonant
circuit has zero amperes of current through it which is indicative of a large resistance, or an open
circuit. The output across a parallel resonant circuit will be relatively large amplitude at
resonance. Therefore, the output across the capacitor and coil in parallel will be a band pass
filter.
In order to quantify the frequency-selectivity properties of series and parallel RLC resonant
circuits, we introduced the concepts of resonance, bandwidth, and quality factor. Some
formulations of these are given below.
Impedance (Z) for a serial RLC circuit is a function of the resistance (R), the inductive reactance
(XL), and the capacitive reactance (XC):
Z = R 2 + (X L − X C )
2
(1)
Inductive reactance is a function of the inductance (L) and frequency (f) of the AC voltage:
X L = 2πfL
(2)
Capacitive reactance is a function of the capacitance (C) and frequency (f) of the AC voltage:
XC =
1
2πfC
(3)
If the sum of XL and XC is zero, then the equation for the resonant frequency in a series RLC
circuit is:
ωo =
1
(4)
LC
The resonance frequency (ωo) is the frequency at which the output is in phase with the input. In
other words, at resonance, circuit is operating at unity power factor (purely resistive circuit).
The bandwidth (β) is defined as the range of frequencies for which the peak amplitude of the
response is at least 1√2 times the maximum peak amplitude. Circuits that are intended to be very
frequency-selective have a narrow bandwidth. How frequency-selective a circuit is depends on
both the bandwidth and the resonant frequency. Thus a circuit with a bandwidth of 10 Hz relative
to a resonant frequency of 1000 Hz is much more selective than a circuit with a bandwidth of 10
Hz relative to a resonant frequency of 100 Hz. The quality factor (Q) of the resonant circuit
recognizes this attribute of frequency selectivity since it is defined as the ratio of the resonant
frequency to the bandwidth (see equations below).
Bandwidth of the series resonant circuit is defined as:
β = ω 2 − ω1 =
R
L
(5)
Quality factor is defined as:
Q=
ωO ωO L
1
1 L
=
=
=
β
R
ω O CR R C
(6)
By keeping the requirement that the sum of XL and XC should be zero, it can be shown that
equation (4) holds for parallel circuits as well as series circuits.
For a parallel circuit, impedance is:
1
1 ⎛
1 ⎞
⎟
= 2 + ⎜⎜ 2πfC −
2
2πfL ⎟⎠
Z
R
⎝
2
(7)
Bandwidth of the parallel resonant circuit is defined as:
β = ω 2 − ω1 =
1
RC
(8)
The quality of the frequency response in parallel resonant circuit is described as:
Q = ω O RC =
R
ωO L
=R
C
L
(9)
Experimental Procedure:
Parallel RLC Resonant Circuits.
1.
2.
3.
4.
5.
Make the connections as shown in the figure above.
Calculate the resonant frequency of the circuit.
Take the necessary readings to fill out the data table below.
Find the frequency at which the output voltage Vo is maximum (Vmax).
Calculate 0.707 of the maximum voltage obtained in 4. Vary the frequency of the
generator above and below resonant frequency until you obtain 0.707 of Vmax.
Record the frequencies as well as the magnitude and phase angle of Vo at these
frequencies (Note: these are the cut off frequencies).
6. Plot Vo Vs f and Adb Vs f on semi-log paper. The frequency must be on the log
scale.
7. Use PSCPICE to solve the circuit. The program should plot Vo Vs f. Be sure to
indicate the resonant frequency and the bandwidth on the program output
8. Use the formulae in the textbook to calculate fo, bandwidth (β) and quality factor (Q)
and compare them with the experimental data.
Series RLC Resonant Circuits.
1.Connect the circuit as shown in the figure.
2. Repeat steps 2 and 8 as for the parallel circuit.
For your Report answer these questions:
1. What is the meaning of 3db points on Adb Vs f curve?
2. Show the duality relationships between the parallel and series resonant circuits. This involves
making comments about the following
[a] The total circuit impedance and admittance at fo.
[b] The currents Is at fo.
[c] The output voltage Vo at fo.
[d] Phase difference between Vo and Is at fo.
[e] Magnitude and phase angle of Vo at half power points.
Note1: Apart from giving the formulae for these parameters indicate if they are maximum or
minimum where applicable.
Note2: Please follow the format of the report given in the syllabus.
Data Table for the RLC Circuit.
f (KHz)
1.0
2.0
3.0
3.2
3.5
4.0
4.2
4.5
5.0
6.0
7.0
8.0
9.0
10.0
Vo (volts)
Adb=20 log Vo/Vin