1. LRC Circuit EX-9915 Page 1 of 10
1LRC CIRCUIT
EQUIPMENT
INCLUDED:
1 AC/DC Electronics Laboratory EM-8656
3 Voltage Sensors CI-6503
NOT INCLUDED, BUT REQUIRED:
1 ScienceWorkshop 750 Interface CI-7650
1 DataStudio Software CI-6870
INTRODUCTION
A square wave voltage is applied to an LC circuit and the period of oscillation of the voltage
across the capacitor is measured and compared to the theoretical value. Then three AC circuits
are examined: A sinusoidal voltage is applied individually to a resistor, a capacitor, and an
inductor. The amplitude of the current and the phase difference between the applied voltage and
the current are measured in each of the three circuits to see the effect each component has on the
current. Finally, a sinusoidal voltage is applied to an inductor, resistor, and capacitor in series.
The amplitude of the current and the phase difference between the applied voltage and the
current are measured and compared to theory.
THEORY
LC Oscillations
A low frequency square wave voltage is applied to an inductor and capacitor series circuit. This
charges the capacitor. Then the capacitor discharges through the inductor and the voltage across
the capacitor oscillates at the resonant frequency of the LC circuit,
1
ω= . (1)
LC
Resistive Circuit
A resistive circuit is one in which the dominant component is resistance. In this experiment, a
voltage V = Vmaxsin(ωt) is applied to a resistor alone. The current, I, through the resistor is in
phase with the applied voltage:
I = Imaxsin(ωt) (2)
where the maximum current, Imax, is equal to Vmax/R. (3)
Written by Ann Hanks

2. LRC Circuit EX-9915 Page 2 of 10
Capacitive Circuit
A capacitive circuit is one in which the dominant component is capacitance. In this part of the
experiment, a voltage V = V maxsin(ωt) is applied to a capacitor alone. Since the voltage across
the capacitor is
Q
V = = V max sin(ωt )
C
where Q is the charge on the capacitor and C is the capacitance. Solving for Q and using
dQ
I = , the phase difference between the current through the capacitor and the applied voltage
dt
is π/2:
 π
I = I max cos ωt = I max sinωt +  (4)
 2
Vmax 1
where I = and the capacitive reactance is χC = . (5)
χC ωC
Inductive Circuit
An inductive circuit is one in which the dominant component is inductance. In this part of the
experiment, a voltage V = V maxsin(ωt) is applied to an inductor alone. Note that the resistance of
the inductor is ignored in the theory. The voltage across the inductor is
dI
V =L = Vmax sin(ωt )
dt
where I is the current through the inductor and L is the inductance. Solving for I using
I = Ι( Vmax/L)sin(ωt)dt,
 π
I = −I max cos ωt = I max sin ωt −  (6)
 2
Vmax
where I max = and the inductive reactance is χ L = ωL . (7)
χL
The phase difference between the current through the inductor and the applied voltage is -π/2.
LRC Circuit
A voltage V = V maxsin(ωt) is applied to an inductor, capacitor, and resistor in series. The
resulting current is given by
I = Imaxsin(ωt + φ) (8)
ξmax
where I max = and Z is the impedance, Z = R 2 + ( X C − X L ) 2 . (9)
Z
The phase constant (φ) is the phase difference between the current and applied voltage. Note that
with the sign convention used here, φ is positive when the current leads the emf (V).
XC − XL
tan φ = (10)
R
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LC Oscillations
SETUP
1. Connect the output voltage to the input banana jacks on the AC/DC circuit board using
banana plugs. Connect the inductor and 10µF capacitor in series with the applied voltage
as shown in Figure 1. IMPORTANT: Insert the iron core into the inductor.
L
Applied C
Voltage
Figure
1: LC Circuit
2. Open the DataStudio file called "LC Circuit". Set the 750 signal generator on a 3-Volt
square wave having a frequency of 30 Hz. The 750 interface automatically measures the
applied voltage and the resulting current. No Voltage Sensors are needed for this part of
the experiment.
PROCEDURE
1. Click on START. Adjust the time axis on the oscilloscope so that about one cycle of the
square wave is visible. Click STOP.
2. For more exact measurements, click on the Applied Voltage box on the right of the
oscilloscope and click on the data export button at the top of the oscilloscope. This will
make a data file for the voltage.
3. Click on the Current box on the right of the oscilloscope and click on the data export
button at the top of the oscilloscope. This will make a data file for the current.
4. Click on the data graph and use the Smart Cursor to measure the period of the LC
oscillation. Use f = 1/T to find the frequency.
5. Using the theoretical resonant frequency in Equation (1) and assuming the capacitance is
its stated value, calculate the inductance of the inductor with the core. Note that the
equation has the angular frequency but the linear frequency was found from the graph.
π
Convert the angular frequency to linear frequency using f = .
2
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4. LRC Circuit EX-9915 Page 4 of 10
Resistive Circuit
SETUP
1. Change the circuit to a 10 Ω resistor in series with the applied voltage as shown in
Figure 2.
Applied R
Voltage C
Figure 2: Resistive Circuit
2. Open the DataStudio file called "R Circuit". Set the 750 signal generator on a 3-Volt sine
wave having a frequency of 100 Hz. The 750 interface automatically measures the
applied voltage and the resulting current. No Voltage Sensors are needed for this part of
the experiment.
PROCEDURE
1. Click on START. Adjust the time axis on the oscilloscope so that one or two cycles of
the wave are visible. Click STOP.
2. Use the Smart Cursor to examine the phase difference between the applied voltage and
the current. The phase difference is the angle between the peaks of the two sine waves.
It is calculated using
∆T ∆T
φ= x360 o or φ= x 2π radians. (11)
T T
ΔT is the time between the peak of one wave and the peak of the other wave. T is the
period of the waves which is the inverse of the applied frequency.
3. What is the phase difference between the resistor voltage and the current?
4. Increase the frequency of the applied voltage to 1000 Hz. Does the phase difference
change? Does the magnitude of the current change?
Written by Ann Hanks

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Capacitive Circuit
SETUP
1. Change the circuit to a 10 µF capacitor in series with the applied voltage as shown in
Figure 3.
Applied C
Voltage C
Figure 3: Capacitive Circuit
2. Open the DataStudio file called "C Circuit". Set the 750 signal generator on a 3-Volt sine
wave having a frequency of 100 Hz. The 750 interface automatically measures the
applied voltage and the resulting current. No Voltage Sensors are needed for this part of
the experiment.
PROCEDURE
1. Click on START. Adjust the time axis on the oscilloscope so that one or two cycles of
the wave are visible. Click STOP.
2. Use the Smart Cursor to examine the phase difference between the applied voltage and
the current. Calculate the phase difference using Equation (11).
3. What is the phase difference between the capacitor voltage and the current?
4. Increase the frequency of the applied voltage to 1000 Hz. Does the phase difference
change? Does the magnitude of the current change?
Inductive Circuit
SETUP
1. Change the circuit to the inductor with L the
iron core in the coil in series with the
applied voltage as shown in Figure 4. Applied C
Voltage
Figure 4: Inductive Circuit
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2. Open the DataStudio file called "L Circuit". Set the 750 signal generator on a 3-Volt sine
wave having a frequency of 1000 Hz. The 750 interface automatically measures the
applied voltage and the resulting current. No Voltage Sensors are needed for this part of
the experiment.
PROCEDURE
1. Click on START. Adjust the time axis on the oscilloscope so that one or two cycles of
the wave are visible. Click STOP.
2. Use the Smart Cursor to examine the phase difference between the applied voltage and
the current. Calculate the phase difference using Equation (11).
3. What is the phase difference between the inductor voltage and the current?
4. Decrease the frequency of the applied voltage to 100 Hz. Does the phase difference
change? Does the magnitude of the current change?
LRC Circuit
SETUP
1. Change the circuit to the inductor with the iron core in the coil, the 10 µF capacitor, and
the 10 Ω resistor in series with the applied voltage as shown in Figure 5.
L C
C
Applied R
Voltage C
Figure 5: LRC Circuit without Voltage Sensors
2. Open the DataStudio file called "Resonance Curve". Set the 750 signal generator on a 3-
Volt sine wave having a frequency of 20 Hz. The 750 interface automatically measures
the applied voltage and the resulting current. No Voltage Sensors are needed for this part
of the experiment.
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PROCEDURE
1. Click on START. Adjust the time axis on the oscilloscope so that one or two cycles of
the wave are visible. Allow the oscilloscope to run throughout this part of the
experiment.
2. Click on the Current box on the right of the oscilloscope and then use the Smart Cursor to
measure the maximum current. Enter the signal generator frequency and the maximum
current into the table labeled "10 Ohm".
3. Increase the signal generator frequency by 20 Hz and find the new current and record the
results in the table. Continue to increase the frequency by increments you choose by
watching the resulting resonance graph. If necessary, go back and take data points at
lower frequencies to fill in the resonance curve. If you go back, insert new rows into the
table at the appropriate points so the connecting line on the graph will connect the points
in order of frequency.
4. Continue until you reach 3000 Hz. Be sure you get enough data points around the peak.
5. Determine the frequency corresponding to the peak in the maximum current. Compare
this frequency to the theoretical resonance frequency given by Equation (1).
6. Measure the phase difference between the current and the applied voltage at a frequency
of 100 Hz, at 3000 Hz, and at the resonant frequency.
7. Replace the 10 Ω resistor with a 100 Ω resistor and repeat Steps 2 through 4. Take a
smaller number of data points for this resistor, concentrating on the lower frequencies,
the resonant frequency, and the higher frequencies so the general shape of the curve is
distinguished.
QUESTIONS
1. Does the frequency of the peak maximum current correspond to the frequency of the LC
oscillations? Does changing the resistance change the frequency of the peak?
2. Using Equation (9), calculate the theoretical peak current at resonance. Why is the actual
peak current at resonance less than the theoretical?
3. Why is the peak lower for the greater resistance?
4. Explain why the phase shifts are what they are at low frequency and at high frequency
and at the resonant frequency. In each case, does the current lead or lag the applied
voltage? To what value does the phase shift go as the frequency goes to zero or as it goes
to infinity? Which components dominate the circuit at these different frequencies?
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8. LRC Circuit EX-9915 Page 8 of 10
5. Draw phasor diagrams (approximately to scale) for 100 Hz, 3000 Hz, and the resonant
frequency.
LRC VOLTAGES
SETUP
1. Change the circuit to the inductor with the iron core in the coil, the 10 μF capacitor, and
the 10 Ω resistor in series with the applied voltage as shown in Figure 6. Add 3 Voltage
Sensors to the circuit: One across the inductor plugged into Channel A, one across the
capacitor plugged into Channel B, and one across the resistor plugged into Channel C.
Be careful to plug the sensors into the assigned channels on the interface and pay
attention to the polarity of each sensor.
Voltage Sensor
V
L
L C V Voltage
Sensor
C C
Applied
Voltage Voltage
R V Sensor
C R
Figure 6: LRC Circuit with Voltage Sensors
2. Open the DataStudio file called "LRC Voltages". Set the 750 signal generator on a 3-
Volt sine wave having a frequency of 200 Hz. The 750 interface automatically measures
the applied voltage and the resulting current.
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PROCEDURE
1. Click on START. Adjust the time axis on the oscilloscope so that one cycle of the waves
is visible. Click on STOP. Note that all the voltage scales are set at 1V/division so the
amplitudes of the voltages can be easily compared.
2. To compare the voltages in greater detail, the oscilloscope data must be exported to a
graph. To do this, click on the "Voltage across L" box on the right of the oscilloscope
and click on the data export button at the top of the oscilloscope. This will make a data
file for the voltage. Repeat this procedure for each of the boxes on the right of the
oscilloscope.
3. Drag each of the voltage data in the Data list at the left to a single graph.
4. Click the Lock button at the top of the graph to lock all the time axes together.
5. One by one, click on each of the Legends on the graph and then click on the Smart
Cursor at the top of the graph to turn on a cursor for each wave. Move the cursors so
they seek the data points in each wave and are locked into each other.
6. Place the cursor on the Applied Voltage on a point where the wave crosses the voltage
axis. For this particular time, read the voltages across the inductor, capacitor, and
resistor. Algebraically add these three voltages together. Does the sum equal the
Applied Voltage at that time? Does Kirchhoff's Loop Rule apply to AC circuits?
7. Using the cursors, determine the phase difference between the Applied Voltage and the
Current. Calculate the theoretical phase shift using Equation (10). Calculate the percent
difference between the measured and theoretical values.
8. Calculate the impedance at this frequency using Equation (9) and use it to calculate the
theoretical maximum current. Measure the maximum current on the graph and compare.
NOTE: If an ohmmeter is available, the resistance of the inductor can be measured and
included in the total resistance of the circuit for these calculations.
9. Measure the phase differences between the Applied Voltage and the voltages across each
component in the circuit (L, C, and R). Compare to theory.
QUESTIONS
1. How does the resistance of the inductor affect the phase difference between the current
and the applied voltage? Draw a phasor diagram which includes the fact that the voltage
across the inductor includes the resistance of the inductor.
2. Why is the measured phase difference for the capacitor and applied voltage the same as
theory while the phase for the inductor is not in agreement with theory?
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10. LRC Circuit EX-9915 Page 10 of 10
3. Why is the phase difference between the current and the resistor voltage zero?
4. Is this circuit being driven at resonance, or below resonance, or above resonance? How
do you know? Give at least two reasons.
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3. Why is the phase difference between the current and the resistor voltage zero?
4. Is this circuit being driven at resonance, or below resonance, or above resonance? How
do you know? Give at least two reasons.
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12. LRC Circuit EX-9915 Page 10 of 10
3. Why is the phase difference between the current and the resistor voltage zero?
4. Is this circuit being driven at resonance, or below resonance, or above resonance? How
do you know? Give at least two reasons.
Written by Ann Hanks