PHYSICS 112 LABORATORY
Gary A. Morris
Fall 1997

Purpose: 1) To find experimentally the resonant frequency of an RLC circuit, and to compare it with the value predicted by theory; 2) to study the behavior of the current, the inductive reactance and the capacitive reactance at frequencies near resonance.

Reference: Serway & Faughn, College Physics, 4th. ed., Section 21.4.

Theory: Fig. 1 shows a resistor, a capacitor, and an inductor in series, connected to an AC source.

The peak voltage across each element is given by:

V_R = i_maxR (1)

V_C = i_maxX_C (2)

where X_C is the capacitive reactance, given by

(3)

and

V_L = i_maxX_L (4)

where X_L is the inductive reactance is given by

X_L = 2pfL (5)

For the phase relations between these voltages, see Fig. 21.9 in your text and recall your results from Experiment 16 for the phase relationships between the voltages across resistors, capacitors, and inductors.

The peak voltage of the AC source is given by:

V_max = i_maxZ (6)

where Z is the impedance of the circuit in ohms (W) and is given by

(7)

where R is value of the resistor.

Note that in Equations 1, 2, 4, and 6 above the maximum values of the AC currents and voltages are used rather than the effective values I and V. Since the equations are linear and I = 0.707 i_max and V = 0.707 v_max, we are free to use either set of values. It is easier to obtain i_max and V_max from the screen of the oscilloscope. Therefore, we use the maximum values in this experiment.

(X_L - X_C) is the equivalent reactance between points a and b in Fig. 1. This part of the circuit consists of an inductor and a capacitor in series. The minus sign of X_C indicates that the voltage across the inductor is 180^o out of phase with the voltage across the capacitor. The maximum voltage V_L+C between points a and b is given by:

V_L+C = i_max (X_L-X_C) (8)

There is a specific frequency, called the resonant frequency, f_res, at which X_L = X_C. The theoretical value of f_res is thus obtained by setting

X_L = 2pf_resL equal to

leading to

(9)

At resonance, according to Equation (7) and Fig. 2, Z = R. Therefore, i_max = V_max/R and the current in the circuit has its maximum value. Also, Equation (8) predicts that V_L+C = 0 since X_L - X_C = 0. In practice, V_L+C is a minimum rather than zero at resonance because of the small amount of resistance between points a and b due to the resistance of the inductance coil.

In this experiment, we will find f_res for a particular RLC circuit by noting where v_L+C is a minimum. This experimental value of f_res will then be compared to the theoretical value given by Equation (9).

Apparatus: Fixed-value inductor, capacitance box, resistance box, digital multimeter, oscilloscope, audio oscillator; banana plug wires.

Procedure:

1. Calculate a value for the capacitor which, when combined in series with the inductor you have, will yield a resonant frequency somewhere between 1,000 and 10,000 Hz. Use Eq. (7) to get an approximate value of C corresponding to whatever f_res you choose.
2. Record the values of C, L, and R.
3. Set up the circuit of Fig. 1.
4. Connect the cathode ray oscilloscope (CRO) across points a and b in the circuit of Figure 1. (This will enable you to measure V_L+C.)
5. Sweep through a range of frequencies output from the audio oscillator (AO) below and above f_res. Notice that V_L+C is a minimum at f_res. (It is not zero as Eq. (6) predicts because there is a non-negligible amount of resistance in the inductor.)
6. Now use the CRO to measure V_L+C at ten different frequencies, 5 below and 5 above f_res. (Before you change frequencies, look at steps 4 and 5 below! Perform steps 3, 4 and 5 for each frequency chosen.)
7. Use the AC ammeter setting on the digital multimeter to measure the current in the circuit for the each of the 10 frequencies used in step 6.
8. Measure the capacitive reactance X_C and the inductive reactance X_L for the same frequencies as in step 6. (To do this, measure the current in the series circuit with the ammeter, and the voltage across the inductor with the oscilloscope. Repeat this procedure for X_C. Since the AC ammeter measures effective current, the maximum current must be calculated by multiplying the reading on the ammeter by .
9. On a sheet of ordinary graph paper, plot V_L+C vs. f for the ten frequencies at which you measured V_L+C.
10. Find the experimental value of f_res from this plot.
11. Compare your experimental value with the theoretical value you calculated using Equation (7). Calculate the percentage difference between the experimental and theoretical values of f_res.
12. On a separate sheet of graph paper, plot the values of X_L and X_C obtained above as a function of frequency.
13. On the same sheet of graph paper, plot the current as a function of frequency.
14. Compare the following three results from your data:

a. The frequency at which V_L+C is a minimum.
b. The frequency at which X_L = X_C.
c. The frequency at which the current is a maximum.

      15. From your results suggest alternative definitions of the resonant frequency of an RLC series circuit.