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Experiment 13: AC Series Circuit: Power and Resonance

Objective: To investigate the frequency response and power consumption in an RLC series circuit. Prior to lab: Using equation (2) below, develop a table of 20 frequencies relative to the resonant frequency fo that will map out the resonant frequency curve shown in Figure 1. Plot your data to determine the best use of you information. Use the data for L, C, and r + 10 Ω (remember r is the loss term for the inductor) which you determined for the LRC circuit board last week. Assume the source voltage is 5.0 V. Hint: You should use frequency values which are grouped near resonance as well as some which are more widely spaced when away from the resonance frequency. Apparatus: The same RLC circuit board used last week, audio frequency sine wave generator, 2 DMMs, wires Discussion: The impedance of an RLC series circuit is given by

where XL = ωL and XC = 1/ωC. Both the inductive and capacitive reactances are frequency dependent (ω = 2πf). Since I = V/Z, the current in the circuit will maximize (for a constant voltage source) where the impedance is a minimum. The occurs at resonance when XL = XC. Or

(1)

Furthermore the power dissipated by the circuit (Pav = I2Z cos φ = I2R) will be a maximum at resonance, and because Z = R, the phase (φ) between the source voltage and the current will be zero. The following algebra leads to an expression of the average power as a function of frequency. Start with the expression for average power

Substitute for V and Z

Substitute for Z in terms of R, XL and XC.

Write the reactances in terms of L, C and ω.

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Multiply the capacitive reactance by L/L

Substitute the ωo2 for 1/LC.

Factor out an ωL.

Factor out a 1/ω2.

Simplify and then multiply R by ω2/ω2.

Simplify.

(2)

A graph of this equation is shown in the following figure.

Figure 1. A typical graph of the frequency response of a high Q circuit (graph is not for data from our experimental apparatus).

The quality factor (Q) of the circuit is determined from the graph by finding the ratio of the frequency spread at one half maximum power and dividing it into the resonant frequency as expressed in equation (3) below.

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(3)

The theoretical equations on the previous pages yield Q in terms of R,L and ωo.

(4)

Procedure: Wire the circuit according to the circuit diagram given below (and shown in the picture of the apparatus.

Figure 2. The experimental apparatus wired to measure the source voltage of the Function Generator as in step 2 below.

DMMDCV (for f)

ac source

DMMACV10 Ω

CLr

Figure 3. The wiring diagram of the apparatus pictured in Fig. 2.

Notes: 1. ω = 2π f 2. Use the same circuit board as you used last week. 3. All voltage measurements are rms.

1. The DMM used to measure frequency should be set at 20 DCV.. 2. Connect the other DMM across all the elements of the RLC circuit and set it on the

ACV 20V range. (You will be measuring the source voltage.) Be sure the sine wave function is chosen on the function generator and set the source output voltage to about 5 V. Record the source voltage.

3. Make a table to record frequency, the potential difference (V10) across the 10 Ω resistor.

4. Reconnect the leads of the AC DMM (just used in step 2) to measure the voltage across the 10 Ω resistor.

5. Vary the frequency. For each value of frequency measure the voltage across the 10 Ω resistor. The voltage across the resistor should vary from about 1.0 V to a maximum and then back down to 1.0 V as the frequency is increased. Be sure to collect enough data (a minimum of 20 data points between 20 Hz and 600 Hz) to be able to plot an appropriate resonance curve.

6. Determine the resonant frequency of the circuit by adjusting the frequency output until the potential difference (V10) across the 10 Ω resistor is a maximum. Record these measurements. Include this measurement in your data table.

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Meter measurement uncertainties are 1. AC voltage: 1.5% of reading + 3 digits. 2. Resistance: 0.5% of reading + 1 digit. Analysis:

You will need to use the calculated or measured values for the L r and C from the previous AC series circuit experiment. If you do not have the same circuit board you will have to repeat the previous week’s measurements and calculations.

1. From the previous experiment on AC series circuit (using the same circuit board) calculate the resonance frequency from L and C (Eq. 1), and compare to the measured value. 2. Plot the graph of power (I2Rtotal) vs. frequency where I=V10/R and R is the resistance of the 10 Ω resistor and Rtotal is the total resistance of the circuit. 3. Use the theoretical equation for power (Eq. 2) as function of frequency, and plot its curve on the same graph with the measured power vs. frequency graph. Caution: You must use the (measured) source voltage and the total resistance of the circuit for this calculation. 4. Determine the experimental value of Q. (Eq. 3). 5. Calculate the theoretical value of Q (Eq. 4) for the circuit. You must use the total resistance of the circuit for this calculation. Report: Determine appropriate error, and discuss your results and analysis. There is a lot you have done here, report on all of it.