2 Capacitive Reactance
Friday, May 25, 2018
5:37 AM
If a resistor is connected across a sine wave generator, the current is in phase with the applied voltage. If, instead of a resistor, we connect a capacitor across the generator, the current is not in phase with the voltage. This is illustrated in Figure 6-4.
Figure 6-4
We can understand why the capacitor current leads the capacitor voltage by using the defining v-i relationship for a capacitor. If the capacitor voltage is a sinusoid:
then the capacitor current is:
Note that the current and voltage have exactly the same frequency, but the current is leading the voltage by 1/4 cycle, or 90º. "Leading" means the current peak occurs before the voltage peak - on a graph, it means the current peak is to the left of the voltage peak. Alternatively, we could say that the voltage is lagging the current.
Current in the capacitor is directly proportional to the capacitance and the rate of change of voltage. The largest current is when the voltage change is a maximum. If the capacitance is increased or the frequency is increased, there is more current. This is why a capacitor is sometimes thought of as a "high-frequency short".
Reactance is the opposition to alternating current (AC) and is measured in ohms, like resistance. Capacitive reactance is written with the symbol XC . Looking at the above equation, we can see that there is a relationship between the amplitudes of the sinusoids given by:
If we arrange this equation to get the ratio of voltage to current, we get the definition of capacitive reactance:
where 𝑓 is the generator frequency in hertz and 𝐶 is the capacitance in farads.
Ohm's Law can be generalized to AC circuits. For a capacitor, we can find the voltage across the capacitor by rearranging the above equation:
𝑉𝐶=𝑋𝐶𝐼
Capacitive Reactance
From your kit, retrieve the capacitors whose value is given in Table 6-2 below. Measure and record their capacitance values in Table 6-2. Measure and record the value of resistor R1.
Table 6-2
Component

Listed Value
Measured Value
C1
10 nF
C2
10 nF
R1
1.0 kΩ
Construct the circuit shown in Figure 6-5. Set the FG for a 10 kHz sine wave with a 1.0 V RMS output. Connect the DSO as shown. Check the frequency and voltage with the DSO.
Note: 1.0 VRMS = 2.828 Vpp
Figure 6-5
The circuit is a series circuit, so the current in the resistor is the identical current seen by the capacitor. You can find this current easily by applying Ohm's Law to the resistor. Measure the RMS voltage across the resistor, VR1 , using the DMM on AC Voltage mode. Record the measured voltage in Table 6-3 in the row labelled Capacitor C1. Compute the current in the circuit by dividing the measured voltage by the resistance of  R1 and enter the value in Table 6-3.
Table 6-3
VR1
(measured)
Step 3
I
(computed)
Step 3
VC1
(measured)
Step 4
XC
(computed)
Step 4
C
(computed)
Step 5
Capacitor
C1
Capacitor
C2
Measure the RMS voltage across the capacitor, VC1. Record this voltage in Table 6-3. Then use this voltage to compute the capacitive reactance using Ohm's Law:
𝑋𝐶=𝑉𝐶𝐼
Enter this computed value of the capacitive reactance in Table 6-3.
Using the capacitive reactance found in step 4, compute the capacitance using the equation
Enter the computed capacitance in Table 6-3. This value should agree with the value marked on the capacitor and measured in step 1 within experimental tolerances.
Repeat steps 3, 4, and 5 using capacitor C2. Enter the data in Table 6-3 in the row labelled Capacitor C2.
Now connect C1 in series with C2. The equivalent capacitive reactance and capacitance can be found for the series connection by measuring across both capacitors as if they were one capacitor. Enter the data in Table 6-4 in the row
labelled Series Capacitors. The following steps will guide you:
(a) Find the current in the circuit by measuring the voltage across the resistor (as before) and dividing by the resistance. Enter the measured voltage and the current you compute in Table 6-4.
(b) Measure the voltage across both capacitors. Enter this voltage in Table 6-4. Then use Ohm's Law to find the capacitive reactance of both capacitors.
(c) Compute the total capacitance by using the equation:
Table 6-4
VR1
(measured)
Step (a)
I
(computed)
Step (a)
VCT
(measured)
Step (b)
XCT
(computed)
Step (b)
CT
(computed)
Step (c)
Series
Capacitors
Parallel
Capacitors
Connect the capacitors in parallel and repeat step 7. Assume the parallel capacitors are one equivalent capacitor for the measurements. Enter the data in Table 6-4 in the row labelled Parallel Capacitors.