THE EXPERIMENT

A. Preliminary Calculations (do before coming to lab)

Select L, C, R_r components; they could be the same values that you used in Exp. 2. Calculate ₀, Q and (₂ - ₁). Values of ₀ should be between 10⁴ and 5 x 10⁵ sec^-1 (ie, 1.6 and 80 kHz), and you want a Q between 4 and 10. Use the estimated values of the other two resistances in Fig. 1, or use your Exp. 2 values. Compute new values of ₀ and Q after you have measured resistances in the lab.

Figure 5

B. Measure ₀ and Q

Set up the circuit in Fig. 5 and connect the oscilloscope across R (ie, measure V_r). This is proportional to the current flowing in the circuit and should show a large amplitude when you change the frequency of the signal generator (sine wave) so as to pass through resonance, ₀ (= 2f₀). This value of ₀ should be within 1% of your calculated value. Now find ₁ and ₂ by changing frequency until the sine wave amplitude V_R decreases to 0.707 of the resonance value. Verify that Eq. 4 gives you a value of Q within a few percent of what you calculated.

Hints:

• Current is proportional to the voltage across a resistor (no phase shift) so that we can measure the current by measuring the voltage across a known resistor and divide by the value of the resistance.
• The internal resistance of the signal generator will be in series with the LRC circuit so that it must be much smaller than the resistance you add to the circuit (R) if it is not to effect the circuit behavior.

C. Graph of Frequency Response

Now measure several values of V_r at different frequencies, recording them in a table. Make sure that at least 3 points are below ₁, 10 points between ₁ and ₂ and 3 points above ₂ ( include the ₀, ₁, and ₂ points as well). Plot V_R as a function of for your data (with error bars). Fit your data to the functional form of eqn (3) i.e. y = and from the coefficients of the fit, determine values (with error) for ₀ and Q. Compare these values with the expected values determined earlier.

Hints:

• Leave space in the table to add in the phase data from part D.
• The quality factor, Q, of a resonant system, in the time domain, determines the rate at which oscillations will decay after being excited. The higher the Q, the slower the decay. In the frequency domain, Q determines the width, in frequency, of the resonant response to a periodic driving term. The higher the Q, the narrower the resonance.

D. Measure Phase Shifts

Set up the oscilloscope to display V_SG on Channel 1 and V_r on Channel 2; make sure that the trigger is set to "Ch. 1 only" or to "Ext. Trigger" (and then connect a coaxial cable from the signal generator to the oscilloscope "Ext. Trigger" input.). Verify that the two sine waves "Change Phase" with respect to each other as you vary the frequency through the resonance point. Use Fig. 3 for the prescription for measuring phase ( = t). Note that a relative motion of 1/4 cycle is a 90 phase shift, and the two waves should be "in phase" ( = 0) at resonance. Check that a phase shift of /4 ( 45) corresponds to the half-power points (₁ and ₂) as indicated in Fig. 4 and Eq. 2.

Hints:

• In order to make a more accurate phase measurement, you will need to use the square wave rather than the sine wave in Ch. 1 for the "reference". This is because the internal resistance (R_S) of the generator is part of the circuit and causes some of the phase shift of the "entire circuit" with respect to the circuit drive voltage, V_S. The negative going edge of the square wave corresponds to t = 0 of the V_S sine wave; use it to measure for the frequency where the V_r amplitude has decreased to 1/2 of the resonance value. Eq. 2 predicts what this phase shift should be. If you measure both above and below ₀ (plus and minus phase shifts) and take the average or (₁ - ₂)/2, you will obtain a more accurate value of the phase shift.

E. The "Q Multiplier"

Common leads on electronic equipment (scope and signal generator) are usually connected to the power line common and are therefore shorted together. Thus you can not connect the scope across the capacitor in Figure 5. Doing so would short circuit the resistor if the scope common is connected to the node between R_r and C. Rearrange the circuit as shown in Figure 6. This allows you to measure the voltage across the capacitor without "shorting" the resistor.

Now measure V_CO, the amplitude of the voltage across the capacitor at resonance. (Make sure that you stay at the resonant frequency for this section of the experiment). Next, disconnect the entire circuit from the generator and measure a proper value for V_S (the internal voltage of the generator). Determine the Q of the circuit by taking the ratio of the capacitor voltage amplitude at resonance to the drive voltage (refer to eqn (5)). Compare this value of Q to the expected value found earlier. Comment on why the circuit was removed from the signal generator before measuring V_S.

Hints:

• If V_C0 is more than 160 volts peak-to-peak, you'll have to decrease V_S so that you can do the measurement.

Figure 6

F. Phase of the Capacitor Voltage

While the circuit is still connected as in part E, look at the phase of V_C, compared to the square wave as you did in part D. At resonance, the capacitor sine wave should have a phase of = - /2. What do you observe? Now interchange L and C in the circuit, and look at the phase of V_L. At resonance, the inductor sine wave should have a phase of +/2. Again, what do you observe? Since the L and C voltages are 180⁰ (or ) out of phase with each other, the observed voltage across the combination of L and C should be quite small. Measure the voltage across the combination of L and C, what do you observe?

G. How Q Depends on R

Restore your circuit to the original form (Fig. 5), but increase R_r so that R_(new) = 2R_(old), for the total resistance of the circuit. Since you expect that Q = , the new value of Q should be 1/2 of your original Q. Verify by finding the new ₁ and ₂, and check that ₀ does not change.