72
Resonant Frequency of the LRC Circuit
(Power Output, Voltage Sensor)
Equipment List
Qty
1
1
1
2
Items
Part Numbers
PASCO 750 Interface
Voltage Sensor
CI-6503
AC/DC Electronics Laboratory
EM-8656
Banana Plug Patch Cord
SE-9750 or SE-9751
1 Iron Core; 1 – 10 Ohm Resistor; 1 - 100-microfarad (µF) capacitor; 1 – Inductor coil
Introduction
The purpose of this activity is to study resonance in an inductor-resistor-capacitor circuit (LRC
circuit) by examining the current through the circuit as a function of the frequency of the applied
voltage. Determine what happens to the amplitude of the current in the LRC circuit when the
frequency of the applied voltage is at or near the resonant frequency of the circuit. Use the
‘OUTPUT’ feature of the PASCO 750 Interface to apply voltage to the circuit. Use the Voltage
Sensor and DataStudio to measure the voltage across the resistor in the circuit as the frequency
of the voltage is changed. The voltage measured across the resistor is related to the current
V
through the resistor by I  R . Also, investigate the phase relationship between the applied
R
voltage and the resistor voltage as you vary the frequency.
Background
When a vibrating mechanical system is set in motion, it vibrates at its natural
frequency. However, a mechanical system can be forced to vibrate at a
different frequency. The amplitude of vibration, and hence the energy
transferred to the system, depends on the difference between the natural
frequency and the frequency of forced vibration. The amplitude becomes very
large when the difference between the natural and forced frequency becomes
very small. This is known as resonance and the natural frequency of the system is sometimes
called the resonant frequency. At resonance, relatively little energy is required to get a large
amplitude. One example of resonance is when a singer’s amplified voice is used to shatter a
glass.
Electrical resonance is analogous to mechanical resonance. The energy transferred to a system is
a maximum at resonance.
The amplitude of the AC current (Io) in a series LRC circuit is dependent on the amplitude of the
applied voltage (Vo) and the impedance (Z).
Io 
Vo
Z
Since the impedance depends on frequency, the current varies with frequency:
Z
PASCO
X
L
 XC
© 2004
 R
2
2
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72 LRC Circuit
Physics Experiment Manual
012-09290
1
, R = resistance, and  =
C
angular frequency = 2f (f = linear frequency). The current will be maximum when the circuit is
driven at its resonant frequency:
where XL = inductive reactance = L, XC = capacitive reactance =
 res 
1
LC
One can show that, at resonance, XL = XC and thus the impedance (Z) is reduced to R. At
resonance, the impedance is the lowest value possible and the current will be the largest value
possible.
SAFETY REMINDER

Follow directions for using the equipment..
Setup
1.
Set up the PASCO 750 Interface and the computer and start DataStudio.
2.
Connect the Voltage Sensor into the interface. Connect banana plug patch cords into the
‘OUTPUT’ ports on the PASCO 750 Interface.
3.
Open the DataStudio with a Scope display of
‘Output Voltage’ and ‘Voltage, Ch A’. It has a
Signal Generator window for controlling the
output from the interface. Set the Voltage Sensor
to 5000 Hz.

The Signal Generator is set to output a sine wave
at 2.97-volt with an initial frequency at 10 Hz. The
Signal Generator is set to ‘Auto’ so it will start and
stop automatically when you start and stop
measuring data.
4.
Set up the circuit shown in the illustration. Put the
iron core inside the inductor coil. Connect a wire
lead from the spring clip on the right-hand side of
the inductor to the spring clip next to the top
banana jack.
5.
Place a 100-microfarad(µF) capacitor in the pair
of spring clips nearest the bottom banana jack on
the board. Connect the 10-ohm resistor (brown,
black, black) vertically above it. Make sure the
capacitor is pointing towards the negative terminal.
6.
Connect the wire lead between the resistor and
the left -hand side of the inductor.
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© 2004
PASCO
012-09290
Physics Experiment Manual
72 LRC Circuit
7.
Put alligator clips on the Voltage Sensor leads and connect the alligator clips at each end of
the resistor.
8.
Plug the banana plug patch cords from the ‘OUTPUT’ port of the PASCO 750 Interface
into the banana jacks.
Procedure
1.
In DataStudio, click ‘Start’ to begin recording data.

Watch the Graph display of voltage versus time.
2.
In the Scope display, determine the voltage,
VR, across the resistor (i.e., voltage from
the sensor). In DataStudio, click the ‘Smart
) in the Scope display
Tool’ button (
toolbar. Move the cursor/cross-hair to a
peak of the signal that shows the voltage
across the resistor, VR.
3.
Record the voltage, VR, in the Data Table in
the Lab Report section next to 10 Hz.
4.
Adjust the function generator to 20 Hz.
Repeat the process to find the new value of
voltage and record it in the Data Table next
to 20 Hz.
5.
Increase the frequency in 10 Hz increments until 150 Hz. Repeat the process of using the
Smart Cursor to find each new value for the resistor voltage, VR. Record each voltage
value in the Data Table.

NOTE: Adjust the Sweep Speed in the Scope display using the Sweep Speed button as needed.
7.
Look at the Data Table and estimate the approximate resonant frequency (where voltage
across the resistor reaches a maximum).
8.
Adjust the function generator to the value of your estimate of the resonant frequency.
Make fine adjustments to the frequency until the trace of voltage from the resistor is in
phase with the trace of the OUTPUT voltage.
9.
Record the new resonant frequency in the Data Table.
10.
Stop measuring data.
•
If you have a meter that can measure inductance, resistance, and capacitance, use it to measure
the inductance of the coil with the core inside, the resistance of the 10 ohm resistor, and the
capacitance of the 100 microfarad capacitor. Record your values in the Data Table.
Analysis
1.
Calculate the current through the resistor for each increment of frequency and record the
values in the Data Table.
PASCO
© 2004
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Physics Experiment Manual
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2.
Graph the current versus the linear frequency. You can use the software or graph paper.
(NOTE: The function generator frequency is the linear frequency.)
3.
Using the resonant frequency found from the Scope display, calculate the resonant angular
frequency and record the value in the Data Table:
 res  2 res
4.
Calculate the theoretical resonant angular frequency using the values of the inductance and
capacitance:
 res 
1
LC
Optional
Determine whether the resonant frequency depends on the resistance.
(To see if the resistance makes a difference, set the Scope to the resonant frequency and
then replace the 10-ohm resistor by a 100-ohm resistor. Does the resonant frequency
increase, decrease, or stay the same?)
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PASCO
012-09290
Physics Experiment Manual
72 LRC Circuit
Lab Report: Resonant Frequency of the LRC Circuit
Name: ________________________________________________________________
Data Table
Freq (Hz)
10
20
30
40
50
60
70
80
Item
Inductance
Resistance
Capacitance
VR (V)
Current (VR/R) A
Freq (Hz)
90
100
110
120
130
140
150
Res Freq:
VR (V)
Current (VR/R) A
Value
30 mH
Calculation
Resonant frequency (linear)
Resonant angular frequency
Theoretical resonant angular frequency
Questions
1.
How does your measured value for resonant angular frequency compare to the theoretical
value for resonant angular frequency?
Remember, Percent HUURU 

2.
meas. - theoretical
theoretical
 100%
Is the plot of current versus frequency symmetrical about the resonant frequency (Plot by hand)?
Explain.
3.
At resonance, the reactances of the inductor and the capacitor cancel each other so that the
impedance (Z) is equal to just the resistance (R). Calculate the impedance (Z) and compare
this value to the resistance of the circuit by using the amplitude of the current at resonance
PASCO
© 2004
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Physics Experiment Manual
012-09290
in the equation R = V/I (where V is the amplitude of the applied voltage)
Is this resistance equal to 10 ohms? Why not?
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© 2004
PASCO