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Resonant Frequency of the LRC Circuit (Power Output, Voltage Sensor)

Equipment List Qty Items Part Numbers

1 PASCO 750 Interface 1 Voltage Sensor CI-6503 1 AC/DC Electronics Laboratory EM-8656 2 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

through the resistor by I

VR

R. Also, investigate the phase relationship between the applied

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 XL XC 2 R2

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where XL = inductive reactance = L, XC = capacitive reactance = 1C

, R = resistance, and =

angular frequency = 2f (f = linear frequency). The current will be maximum when the circuit is driven at its resonant frequency:

res 1LC

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 pairof 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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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

Tool’ button ( ) in the Scope display 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.

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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 2res

4. Calculate the theoretical resonant angular frequency using the values of the inductance and capacitance:

res 1LC

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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Lab Report: Resonant Frequency of the LRC Circuit Name: ________________________________________________________________ Data Table

Freq (Hz) VR (V) Current (VR/R) A Freq (Hz) VR (V) Current (VR/R) A 10 90 20 100 30 110 40 120 50 130 60 140 70 150 80 Res Freq:

Item Value Inductance 30 mHResistance Capacitance

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 error meas. - theoretical

theoretical100%

2. 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

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in the equation R = V/I (where V is the amplitude of the applied voltage)

Is this resistance equal to 10 ohms? Why not?