Experiment 10

The RLC Series Circuit, I

The resonant frequency of an RLC series circuit is found by determining the frequency at

which the current delivered to the circuit is a maximum.

Theory

Suppose that a signal generator produces a sinusoidal voltage V= Vmsin27ft. When this

voltage is delivered to an RLC series circuit, the voltage amplitude can be writtenas

V m = I m Z, (1)

where Im is the current amplitude and Z is the impedance of the circuit. The value of the impedance is

(2)

where R is the resistance in the circuit, and the inductive reactance XL and capacitive reactance Xcare given by

(3)

and

(4)

L is the inductance in the circuit and C is the capacitance.

When (2) is substituted into (1), and the result solved for the current amplitude, the ex-

pression becomes

(5)

Because the inductive and capacitive reactances are frequency dependent, the impedance Z varies with frequency. The maximum current flow then occurs when the impedance is a minimum, and the impedance is a minimum when XL = Xc . This frequency at which occurs is called the resonant frequency and has a value of

. (6)

As the frequency is varied above and below the resonant frequency, the impedance increases, thus reducing the value of the current flow

Instead of directly determining the maximum value of the current amplitude, the ratio of the voltage across the resistor to the voltage across the RLC combination is found. This ratio is directly proportional to the cuffent amplitude and reaches a maximum when the current amplitude is a maximum as is shown below:

(7)

or

(8)

Note that this equation indicates that the ratio VRIV. = I at the resonant frequency. This is only true is the total resistance of the circuit was oigy due to the resistor. However, the signal generator and the inductor both have resistance, and a more proper expression for the ratio is

(9)

where RT = R + (resistance of the inductor + resistance of the signal generator).

Apparatus

oinductor, approximately 20 mH

ocapacitance box, set at approximately 0. 15 F

odecade resistance box, set at approximately 500 92

osignal generator

ooscilloscope with 2 leads

o3 cycle semi-log graph paper.

Procedure

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1)Connect the circuit as shown in Figure 1.

2)Turn on the equipment and set the signal generator to 100 Hz. Be sure the switches on the oscilloscope are in the calibrate positions.

3)Set the vertical mode switch on the oscilloscope so that both channel I and channel 2 traces are

Figure 1. The RLC series circuit.

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displayed on the screen. Adjust the amplitude knob on the signal generator until an 800 mv peak-to-peak signal is displayed on the screen for channel I ' This voltage is Vm.Read and record the peak-to-peak signal for both channel I and channel 2. The voltage across channel 2 is VR. (Remember you can increase the accuracy with which you read the channel 2 voltage by changing the vertical sensitivity for channel 2.)

4)Change the frequency of the signal generator to 200 Hz. Repeat step (3). Continue to repeat step (3) for frequencies of 400 and 700 Hz and multiple of Io above these frequencies up to I.

5)When you are close in frequency to the resonant frequency of the circuit, make more measurements so that the exact resonant frequency can be found.

Analysis

Graph the ratios V R / V m against the corresponding. frequencies on three cycle semi-log graph paper with the frequency graphed on the log scale. Draw a smooth curve through the points and determine the resonant frequency from the graph. Report the experimental value, the theoretical value, and the percentage error between the two in a table of results.

Questions
  1. Draw a phasor diagram for the RLC series circuit showing the phasors that represent the voltage amplitude across the entire circuit, the individual circuit elements, and the current amplitude for your circuit.

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