LRC Resonance EX5537 Page of
Equipment
1

Resistor/Capacitor/Inductor Network

UI5210

1

850 Output BNCtoBanana Cable

UI5119

3

Voltage Sensor

UI5100

1

Patch Cords (Set of 5)

SE7123


Required but not included:


1

850 Universal Interface

UI5000


PASCO Capstone Software

UI5400

Introduction
The current through a series LRC circuit is examined as a function of applied frequency and the effects of changing the values of the resistance, inductance, and capacitance are observed.
The phase difference between the applied voltage and the current is measured below resonance, at resonance, and above resonance.
Theory
An inductor, a capacitor, and a resistor are connected in series with a sine wave generator. If the applied voltage is
(1)
then the current in the series LRC circuit is
(2)
where (3)
and (4)
The impedance Z is given by
(5)
where the capacitive reactance is
(6)
and the inductive reactance is
(7)
At resonance, the current is maximum and thus the impedance is at its minimum. The minimum impedance is equal to R. This occurs when X_{L} = X_{C}. This yields the resonant frequency:
(8)
(9)
To measure the phase difference, φ, between the current and the applied voltage, we can measure the phase of the current by measuring the phase of the resistor voltage, since the current is in phase with the voltage across the resistor.
The phase difference between the resistor voltage and the applied voltage is the amount (in radians) that resistor voltage peak is shifted from the applied voltage peak. If the peaks are shifted by one wavelength, that would correspond to one cycle or 2π radians. One cycle corresponds to one period (T). So, to determine the amount of shift, we measure the amount of time the peaks are shifted (Δt) and calculate the ratio Δt/T to find what percentage of a full cycle they are shifted. Thus the phase difference, φ, is equal to this percentage times 2π radians.
(10)
Since f = 1/T,
(11)
Setup

Connect the BNCtoBanana cord to Signal Generator #2 on the 850 Universal Interface and connect the red cord to one end of the 2.5 mH inductor on the circuit board. Connect the other end of the inductor to the 560 pF capacitor in series and the 3.3 kΩ in series. Then connect the black cord to the open end of the resistor.

Connect a Voltage Sensor to Channel A on the 850 interface and attach the leads across the resistor, making sure the black cable from the voltage sensor is connected to the side of the resistor that is attached to the black cable from the 850 output.

Connect a Voltage Sensor to Channel B on the 850 interface and attach the leads across the leads of the Output #2 cable, making sure the black cable from the voltage sensor is connected to the black side of the signal generator.

Open the Signal Generator 850 Output 2 and choose the Sine Wave at a frequency of 10,000 Hz and an amplitude of 7 V. Put the output on AUTO.

In PASCO Capstone, set up a table with columns for frequency, output voltage, resistor voltage, the ratio of resistor voltage divided by output voltage, and the amount of time that the two voltage are shifted from each other.
Procedure
You will vary the frequency of the applied voltage and record the response (current) of the circuit. The response is measured by measuring the voltage across the resistor since the current is in phase with this voltage and it is proportional to it.
One further complication is that you must divide the resistance voltage by the output voltage to account for any changes in the output voltage.
You will also measure the phase difference between the output voltage and the current by measuring the phase between the output voltage and the resistor voltage.

Begin with the signal generator set on 10 kHz. Record this frequency in Table I. Click on Monitor and click the trigger on the oscilloscope. Adjust the vertical and horizontal scales on the scope so you can easily measure the amplitudes of each voltage and the phase shift between the resistor voltage and the output voltage.

Stop monitoring and use the coordinates tool to measure the amplitude of each of the voltages and type the results in Table I.

To find the phase shift between the two voltages, use the delta tool on the coordinates tool to measure the difference in time between adjacent peaks of the two voltages. Record this phase shift in time in Table I.

Increase the frequency of the output by 10 kHz and repeat the measurements.

Continue to increase the frequency in steps of 10 kHz up to 500 kHz. Once you find the peak of the graph of the ratio of voltages versus frequency, adjust the frequency to take some more points near the peak to get more detail.

Exchange the 1 kΩ resistor for the 3.3 kΩ resistor and repeat the entire procedure.
Analysis of the Resonance Curve

Display the runs for both of the resistors on the graph.

Measure the height of the peak for each curve and the frequency of each peak. Record them in the box below.

Measure the width of the curve at half the height for each resistor.
Questions

How do the height and width of the curves change when you increase the resistance?

Calculate the theoretical resonant frequencies and compare them to the measured values with a percent difference. Remember that the frequency of the signal generator, f, which is related to the theoretical frequency, ω, by

Does the resistance change the resonant frequency? How does the resistance of the inductor affect the results?

Why don't the peaks equal one at the resonant frequency?

Why isn't the resonant curve symmetrical about the resonant frequency?
Analysis of the Phase Curve
1. At what frequency is the phase shift zero? What is the impedance at this frequency?
2. As the frequency goes to zero, to what value does the phase shift go? Look at your graph and at the theoretical equation and consider the limit as the frequency goes to zero.
(4)
where
(6)
and
(7)
3. As the frequency goes to infinity, to what value does the phase shift go? Look at your graph and at the theoretical equation and consider the limit as the frequency goes to infinity.
4. Does the current lead or lag behind the applied voltage below the resonant frequency?
5. Which is larger above the resonant frequency: the capacitive reactance or the inductive reactance?
6. How does changing resistance affect the phase vs. frequency graph?
Written by Ann Hanks
