Part IV: Investigation of an RC Circuit (or an RL Circuit)

In this part of the experiment you are to study voltages and phases in a series RC or a series RL circuit. You will have an opportunity to use both channels of the oscilloscope. Your instructor will assign you either the RL or the RC series circuit.

A. The series RC circuit with its connections to the function generator, F.G., and the oscilloscope, V-222 is shown schematically in Figure 6.1a for the case in which the voltage across the resistor V_R(t) is to be measured simultaneously with the voltage e(t) produced across the entire circuit by the function generator.

The actual circuit board connections required to reproduce the circuit drawn in Figure 6.1a are shown in Figure 6.1b. (Note: If you have the RL circuit, refer to Figures 6.2a and 6.2b instead).

For either the RC or RL circuit use the 2.2 kW resistor and connect the circuit as shown in 6.1b (or 6.2b). The total voltage e(t) as produced across the circuit by the function generator is connected to channel 2 (10) of the oscilloscope. The voltage across the resistor is the channel 1 (9) input. Trigger the scope internally (31) on channel 2 (32). Both channels (oscillating voltages) can be displayed simultaneously as vertical deflections of the electron beam by setting the display mode to alternate (21). Select a function generator sinusoidal output of six volts peak-to-trough, a frequency of 1 kHz and adjust the time base (26) and vertical gains (15) and (16) such that somewhat greater than one cycle fills the screen similar to Figure 6.3.

What you are to investigate first experimentally is the relationship between the time dependent voltage e(t) across the circuit (channel 2) and the time dependent voltage V_R(t) across the resistor (channel 1) including the phase angle F between them. Since both e(t) and V_R(t) are sinusoidal with the same frequency you will actually measure the maximum values of these voltages, e_m and V_Rm.

Measure amplitudes e_m and V_Rm and phase angle F for a large range of frequencies (suggested settings are 1 kHz, 3 kHz, 10 kHz, 30 kHz, 70 kHz and 140 kHz for the RC circuit, and 0.5 kHz, 2.5 kHz, 10 kHz, 30 kHz, 50 kHz and 70 kHz for the RL circuit). Maintain the voltage from the function generator (channel 2) at six volts peak-to-trough for all measurements during the entire laboratory. You may have to adjust the function generator output slightly when you change frequencies to keep e_m at six volts. By recording peak-to-trough voltages you will avoid vertical centering errors (19) and (20). As you change frequency it is important to adjust the scope so as to maintain a pattern similar to Figure 6.3. Changes in the vertical gains (15) and (16) or in the time base (26) do not affect the measured voltages. The purpose of such changes is to present the pattern in its most easily measurable form. When many cycles are displayed you cannot measure F well, for example. Also, remember that voltages cannot be measured when the uncal lamps (17) (18) are on.

To determine the phase angle F, count the number of divisions on the scope corresponding to one wavelength or 360° of phase (this is t in Figure 6.3). Then count the number of divisions by which the waves are shifted. (This is Dt in Figure 6.3). Dt/t is the fraction of one wavelength by which the waves are shifted so that multiplication of Dt/t by 360° gives the phase angle F in degrees. Be sure to record whether e_m leads or lags V_Rm for your circuit. Note that V_a lags V_b in Figure 6.3 because V_a reaches its maximum later than V_b.

Do not panic if you do not see any phase shift at one end of your frequency range. This is correct. If you do not see any phase shift at either frequency extreme, then your circuit or your scope is connected incorrectly.

B. Interchange the positions of the capacitor and resistor (or inductor and resistor) in your circuit to correspond to Figures 6.1c and 6.1d (or 6.2c and 6.2d if you have the RL circuit). This interchange is necessary because, for these oscilloscopes, all voltage measurements must be made relative to the circuit.

Measure V_Cm (or V_Lm) for the same e_m and frequencies as in part A. (Note: Since you will measure only maximum values in this work we will drop the subscript m from now on for convenience.) Be sure to record whether e leads or lags V_C (or V_L) but do not measure the angles.

C. Interpretation of the Data

From class you learned that capacitive and inductive reactance X_C and X_L are given by X_C = 1/2pfC and X_L = 2pfL, respectively. Here f is the sinusoidal frequency of the function generator, C is the capacitance in farads and L is the inductance in henrys. X_C and X_L then have the units of Ohms. Your objective is to show that your data support the statement above that X_C is inversely proportional to frequency (or X_L is proportional to frequency). Since you cannot measured X_C, (or X_L) directly you will have to determine experimental values for them by combining your data in the following way.

X_C = V_C/I and X_L = V_L/I from Ohms law for sinusoidal voltages.

I is the sinusoidal current (unmeasured) at the frequency at which V_C (or V_L) has been measured. If both equations are multiplied top and bottom R, the resistance in the circuit, then

X_C = V_CR/IR and X_L = V_LR/IR.

But V_R = IR so X_C = V_CR/V_R and X_L = V_LR/V_R.

The value of the resistance R does not depend on frequency so you can take its value to be fixed at 2.2 kW. Use this value of R and the measured values of V_C and V_R (or V_L and V_R) that you obtained at the same frequency to determine experimental values of X_C (or X_L) in kW for each of your frequencies.

To see whether your experimental results for X_C (or X_L) have the expected theoretical frequency dependence given by X_C = 1/2pfC (or X_L = 2pfL) note that log X_C = -log f - log 2pC (or log X_L = log f + log 2pL). Thus a log-log plot of X_C versus f should be a straight line of slope -1 (and a log-log plot of X_L versus f should be a straight line of slope +1).

You may plot your data on 3´3 cycle log-log paper and/or by means of the computer program Quarttro Pro. Since it is difficult to measure the voltages to better than several percent on the oscilloscope try as an initial value for the uncertainty in Y (uncertainty in X_C or X_L) a value that is several percent of your measured X_C (or X_L) at 10 kHz. Discuss the results for your slope.

D. Interpretation of the measured phase angles of Part A.

The phase angle F that you measured between e and V_R, as a function of frequency is best understood by reference to the following phasor diagrams.

For the RC circuit e lags V_R by F. For the R_L circuit e leads V_R by F. Check to see if this is what you found.
At 1 kHz frequency:

1. What phase angle F did you measure?
2. How do V_R and e compare in magnitude?
3. How do V_C (or V_L) and e compare in magnitude?
4. How does X_C (or X_L) compare with R?
5. Draw a phasor diagram representing the situation.

At 140 kHz frequency: (70 kHz if you had the RL) circuit.) Answer 1-5 above.

NOTE: If you compare your answers above with someone who had the other circuit, you should see just the "opposite" effects.