AC Reactance of Capacitors and Inductors

EXPERIMENT 8 AC. REACTANCE OF CAPACITORS AND INDUCTORS

OBJECT:

To study the electrical characteristics of an AC series circuit containing a resistor, and inductor, and a capacitor.

APPARATUS:

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

THEORY: ALTERNATING CURRENT IN A RESISTOR

Consider an AC circuit consisting of a generator and a resistor (Figure 1).
Figure 1.

If E is the emf supplied by the generator, the Kirchhoff's loop rule gives

E - VR = 0, where VR = IR. If the generator produces harmonic waves E = E_maxcos wt, we have E_maxcos wt - IR = 0 The current in the resistor is I = (E_max/R) cos wt, and I_max = E_max/R, so I = I_maxcos wt. It means that for such a current, the current and voltage are in phase: both of them are zero at the same instant, both of them pass through their maximum values at the same instant.

Most AC voltmeters and ampermeters are designed to measure root-mean-square (rms) values of voltage and current, rather than their maximum values. The rms value of a current and emf are equal to

I_rms=I_max/

, E_rms = E_max/

and I_rms =E_rms /R.

ALTERNATING CURRENT IN CAPACITORS

If we connect a capacitor across the terminals of a generator (Figure 2),
Figure 2.

Kirhhoffs's loop rule gives us for this circuit:

E-V_C =0 or E-(q/C)=0. Solving for charge we get q=E_maxC cos wt. The current is related to the charge by I = dq/dt = -E_maxC sin wt = I_maxsin wt = I_maxcos(wt + p/2) (1) where I_max= E_maxCw = E_max/(1/wC)= E_max/X_C. Where X_C = 1/wC (1a) is called the capacitive reactance. Correspondingly I_rms =E_rms/X_C

From equation (1) we can see that there is a 90^o phase difference between voltage and current: the maximum value of voltage occurs one-forth period after the maximum value of the current. The voltage drop across the capacitor lags the current by 90^o.

ALTERNATING CURRENT IN INDUCTORS

Figure 3 shows an inductor connected across the terminals of an AC generator.

Figure 3.

Again, Kirchhoff's loop rule gives us

E - V_L = 0 where V_L = L dI/dt. So E = E_maxcos wt = L dI/dt. Solving this equation for the current, we will obtain I = (E_max/wL) sin wt = I_maxsin wt =I_max cos (wt - p/2) (2)

We see that there is a phase difference between the current and the voltage. The maximum value of the voltage occurs at 90o, or one fourth period before the corresponding maximum value of the current. The voltage drop across an inductor is said to lead the current by 90^o.

From equation (2)

I_max = E_max/wL = E_max/X_L, where X_L= wL (2a) is called the inductive reactance. Similarly I_rms =E_rms/wL = E_rms/X_L .

The circuits in Figures 2 and 3 contain only a generator and an inductor or capacitor. In the figures, the voltage drop across the inductor or capacitor equals the voltage of the generator. So we can write for the current in the inductor:

I_rms = V_L,rms/wL = V_L,rms/X_L,

and for the current in the capacitor:

I_rms = V_C,rms/(1/wC) = V_C,rms/X_C.

PROCEDURE

1. Assemble the circuit as shown in Figure 4:
Figure 4.

2. Turn on the power of the generator and adjust the oscilloscope to get an indication of a voltage sine wave.

3. Measure and record the frequency of the AC voltage using the oscilloscope. Remember to record an error associated with this measurement.

4. Change the voltage amplitude on the generator and take 5 readings of the current and the peak to peak voltage across the resistor. Record your data in Table 1. Remember to record the associated errors as well.

5. Change the frequency of the generator, measure and record it again and take another 5 readings of the current and voltage.

Table 1.

f,Hz I,A V_Rpp,V

. . .

6. Assemble the circuit with a capacitor as shown in Figure 5.

Figure 5.

7. Record the voltage across the resistor (V_Rpp) and the capacitor (V_Cpp). Remember the associated errors.

8. Adjust the voltage amplitude on the frequency generator to change the voltage levels. Measure and record these voltages (and the errors) for several values (5 readings) of applied voltages. Do not change frequency, only change output voltage. Measure the frequency using the oscilloscope. Record your data in table 2.

NOTE: THE OSCILLOSCOPE GROUNDS MUST BE PLACED AS INDICATED AND THE GROUND PRONG MUST BE BY-PASSED WITH AN ADAPTER.

Table 2.

f,Hz V_Rpp,V V_Cpp,V

. . .

9. Choose the voltage amplitude in the middle of its respective scale and do not change it. Take readings of the V_Rpp and V_Cpp at different frequencies (5 to 7 readings). Measure frequencies using the oscilloscope. Record your data in table 2.

10. Replace the capacitor with an inductor. Repeat steps 6-8 for this circuit. Measure f, V_Rpp, V_Lpp. Record your data in Table 3.

Table 3.

f,Hz V_Rpp,V V_Lpp, V

. . .

11. Repeat the steps in exercise 9 for this circuit. Change your frequency levels and take 5 readings for f, V_Rpp and V_Lpp. Record your data in Table 3.

DATA ANALYSIS

1. Calculate V_rms= (V_Rpp/2)(0.707) using Table 1. Graph V_rms as a function of I_rms = I (at digital ammeter). Determine the slope of the line and find the value of the resistor (in Ohms), and its experimental error. Repeat this for the second frequency. Does the resistance depend on frequency?

2. Calculate current amplitude (I_a = V_Ra/R) using Table 2. Graph the amplitude of the voltage across the capacitor V_Ca as a function of the current amplitude. Find the slope (which is equal to the reactance of the capacitor) and its experimental error. Compare your result with X_C calculated using formula (1a).

3. Calculate reactance of the capacitor X_C= V_Ca/I_a for different frequencies. Plot X_Cas a function of inverse frequency. Find the slope of the graph a = 1/2 pC, and its experimental error. Find C and its experimental error. Compare your result with the capacitance indicated on your capacitor.

4. Calculate current amplitude (I_a = V_Ra/R) using Table 3. Graph the amplitude of the voltage across the inductor V_La as a function of the current amplitude. Find the slope (which is equal to the reactance of the inductor) and its experimental error. Compare your result with X_L calculated using formula (2a).

5. Calculate reactance of the capacitor X_L = V_La/I_afor different frequencies. Plot X_Las a function of frequency. Find the slope of the graph b = 2pL, and its experimental error. Find L and its experimental error. Compare your result with the inductance indicated on your inductor.

In this lab report, you should include all of the original data and calculations. You will find it helpful to do the data analysis using a data plotting program. Include all graphs from the data analysis. In your results and conclusions, restate all of your final results. Remember to compare the results as asked for in the Data Analysis. Also, be sure to answer the question in Data Analysis 1.