To study the electrical resonance, to find the quality factor of the circuit. 

AC generator, resistor, capacitor, inductor, oscilloscope, connecting wires. 

Please, read carefully Tipler's "Physics", Ch. 28, section 28-5, pp. 912-916. Pay special attention to the picture 28-18 and the equation 28-58.

Consider a series circuit containing a resistor, inductor and capacitor as in Figure 1.

Figure 1.
 

The voltage across the inductor is

the voltage across the capacitor is

the voltage across the resistor is

Phase relations among these voltages are shown in Figure 2. The voltage across the resistor is in phase with the current. The voltage across the inductor leads the current by 90 degrees.

 
Figure 2.

The voltage across the capacitor lags the current by 90 degrees. The total voltage across the resistor, inductor and capacitor should be equal to the emf supplied by the generator.
 

.

From Figure 2 we can see that

.

If we divide both sides of this equation by current, we will get

where (X_L - X_C) is called the total reactance, and Z is called the impedance of the circuit.

We know that the capacitive reactance X_C = 1/wC, and the inductive reactance X_L = wL depend on frequency. The value of frequency when
 

w = 1/ = w₀ = 2pf₀

The frequency f₀ is called the resonance frequency of the circuit. At this frequency the impedance is smallest and the maximum value of the current (and the voltage across the resistor V_R) can be obtained. At this frequency the circuit is said to be at resonance. At resonance the current is in phase with the generator voltage.

If we measure voltage across the resistor, depending on frequency, we will obtain a resonance curve of the circuit as shown in Figure 3.
 
       Figure 3.

 A resonance curve can be characterized by the resonance width Df, the frequency difference between the two points on the curve where the power is half its maximum value or voltage is

V_max/= 0.707 V_max

When the width is small compared with the resonance frequency, the resonance is sharp; that is, the resonance curve is narrow. The circuit can be characterized by the quality factor

If resistance is small and resonance is sharp, the quality factor is large. When the resistor is large, the quality is small. Note, that the Q can not be less than one; for good industrial circuits it can reach thousands.

 
Q is a measure of the rate at which energy is dissipated in the circuit if the AC voltage source across the series circuit was removed. 

1. Connect a series LCR circuit as shown in Figure 4:

Figure 4.

2. Calculate the resonance frequency of your circuit
 

where L is in Hn; C is in F; f is in Hz.

 
3. Apply AC voltage to your circuit having the frequency being close to f₀. Measure the peak-to-peak voltage V_R across your resistor. Also, measure the frequency using the oscilloscope (f = 1/T). You should always use the oscilloscope to measure the frequencies instead of reading the frequency from the AC generator.

4. Decrease the frequency slowly and observe that V_R decreases. Increase the frequency above f₀ and observe that V_R decreases. Estimate the frequencies where V_R stops decreasing both above and below f₀. Your data will be collected between these two frequencies.

5. Take at least 10 readings of V_R for frequencies below f₀. Make sure that you take at least 5 readings in the vicinity of f₀, since the data that is most meaningful is near f₀, in the upper region of the curve shown in figure 3. Record data in the table shown below. Do not forget to record the errors!

6. Take 10 readings of V_R for frequencies above f₀. Make sure to take at least 5 readings in the vicinity of the f₀. Record your data in the table. Do not forget to record the errors.
 

f, Hz	VR, V

1. Plot V_R as a function of the frequency. The graph you get should be similar to one shown in Figure 3.

2. Find the resonance frequency for your circuit, which corresponds to the maximum of your voltage. Compare it with the calculated resonance frequency.

3. On your graph, find the frequencies f1* and f2* at which the voltage V_R is equal to 70.7% of its resonance (maximum) value. At these frequencies the power consumed by the resistance of the circuit is 50% of its maximum at resonance.

4. Calculate the quality factor (Q) of your circuit:

Pay attention to the units of the Q. Find its experimental error. (See Error Propagation techniques in Fundamentals of Data Analysis). What is the physical meaning of 1/Q?

Include all of the original data taken in lab and your graph (remember units, axis labels, etc.). In the conclusions, restate your values for f₀ and Q and their experimental errors. Compare your values of the resonance frequency! Make sure to include your answer to data analysis question 4. Speculate on possible sources of error in this experiment.