23.1

Capacitors and Capacitive Reactance

Our experience with capacitors so far has been in dc circuits. As we have seen in Section 20.13, charge flows in a dc circuit only for the brief period after the battery voltage is applied across the capacitor. In other words, charge flows only while the capacitor is charging up. After the capacitor becomes fully charged, no more charge leaves the battery. However, suppose the battery connections to the fully charged capacitor were suddenly reversed. Then charge would flow again, but in the reverse direction, until the battery recharges the capacitor according to the new connections. What happens in an ac circuit is similar. The polarity of the voltage applied to the capacitor continually switches back and forth, and, in response, charges flow first one way around the circuit and then the other way. This flow of charge, surging back and forth, constitutes an alternating current. Thus, charge flows continuously in an ac circuit containing a capacitor.
To help set the stage for the present discussion, recall that, for a purely resistive ac circuit, the rms voltage Vrms across the resistor is related to the rms current Irms through it by (Equation 20.14). The resistance R has the same value for any frequency of the ac voltage or current. Figure 23-1 emphasizes this fact by showing that a graph of resistance versus frequency is a horizontal straight line.

Figure 23-1 The resistance in a purely resistive circuit has the same value at all frequencies. The maximum emf of the generator is V0.

For the rms voltage across a capacitor the following expression applies, which is analogous to :

| (23.1)

The term XC appears in place of the resistance R and is called the capacitive reactance. The capacitive reactance, like resistance, is measured in ohms and determines how much rms current exists in a capacitor in response to a given rms voltage across the capacitor. It is found experimentally that the capacitive reactance XC is inversely proportional to both the frequency f and the capacitance C, according to the following equation:

| (23.2)

For a fixed value of the capacitance C, Figure 23-2 gives a plot of XC versus frequency, according to Equation 23.2. A comparison of this drawing with Figure 23-1 reveals that a capacitor and a resistor behave differently. As the frequency becomes very large, Figure 23-2 shows that XC approaches zero, signifying that a capacitor offers only a negligibly small opposition to the alternating current. In contrast, in the limit of zero frequency (i.e., direct current), XC becomes infinitely large, and a capacitor provides so much opposition to the motion of charges that there is no current.

Figure 23-2 The capacitive reactance XC is inversely proportional to the frequency f according to .

Example 1 illustrates the use of Equation 23.2 and an important Problem solving insight: The capacitive reactance XC is inversely proportional to the frequency f of the voltage, so if the frequency increases by a factor of 50, as it does in Example 1, the capacitive reactance decreases by a factor of 50.

| Example 1 | | | A Capacitor in an AC Circuit

For the circuit in Figure 23-2, the capacitance of the capacitor is , and the rms voltage of the generator is 25.0 V. What is the rms current in the circuit when the frequency of the generator is (a) and (b) ? Reasoning The current can be found from , once the capacitive reactance XC is determined. The values for the capacitive reactance will reflect the fact that the capacitor provides more opposition to the current when the frequency is smaller.
Solution

a. | At a frequency of , we find

| (23.2)

| (23.1)

b. | When the frequency is , the calculations are similar:

| (23.2)

| (23.1)

We now consider the behavior of the instantaneous (not rms) voltage and current. For comparison, Figure 23-3 shows graphs of voltage and current versus time in a resistive circuit. These graphs indicate that when only resistance is present, the voltage and current are proportional to each other at every moment. For example, when the voltage increases from A to B on the graph, the current follows along in step, increasing from to during the same time interval. Likewise, when the voltage decreases from B to C, the current decreases from to . For this reason, the current in a resistance R is said to be in phase with the voltage across the resistance.

Figure 23-3 The instantaneous voltage V and current I in a purely resistive circuit are in phase, which means that they increase and decrease in step with one another.

For a capacitor, this in-phase relation between instantaneous voltage and current does not exist. Figure 23-4 shows graphs of the ac voltage and current versus time for a circuit that contains only a capacitor. As the voltage increases from A to B, the charge on the capacitor increases and reaches its full value at B. The current, however, is not the same thing as the charge. The current is the rate of flow of charge and has a maximum positive value at the start of the charging process at . It is a maximum because there is no charge on the capacitor at the start and hence no capacitor voltage to oppose the generator voltage. When the capacitor is fully charged at B, the capacitor voltage has a magnitude equal to that of the generator and completely opposes the generator voltage. The result is that the current decreases to zero at . While the capacitor voltage decreases from B to C, the charges flow out of the capacitor in a direction opposite to that of the charging current, as indicated by the negative current from to . Thus, voltage and current are not in phase but are, in fact, one-quarter wave cycle out of step, or out of phase. More specifically, assuming that the voltage fluctuates as , the current varies as . Since radians correspond to 90° and since the current reaches its maximum value before the voltage does, it is said that the current in a capacitor leads the voltage across the capacitor by a phase angle of 90°.

Figure 23-4 In a circuit containing only a capacitor, the instantaneous voltage and current are not in phase. Instead, the current leads the voltage by one-quarter of a cycle or by a phase angle of 90°.

The fact that the current and voltage for a capacitor are 90° out of phase has an important consequence from the point of view of electric power, since power is the product of current and voltage. For the time interval between points A and B (or and ) in Figure 23-4, both current and voltage are positive. Therefore, the instantaneous power is also positive, meaning that the generator is delivering energy to the capacitor. However, during the period between B and C (or and ), the current is negative while the voltage remains positive, and the power, being the product of the two, is negative. During this period, the capacitor is returning energy to the generator. Thus, the power alternates between positive and negative values for equal periods of time. In other words, the capacitor alternately absorbs and releases energy. Consequently, the average power (and, hence, the average energy) used by a capacitor in an ac circuit is zero.

Problem solving insight

It will prove useful later on to use a model for the voltage and current when analyzing ac circuits. In this model, voltage and current are represented by rotating arrows, often called phasors, whose lengths correspond to the maximum voltage V0 and maximum current I0, as Figure 23-5 indicates. These phasors rotate counterclockwise at a frequency f. For a resistor, the phasors are co-linear as they rotate (see part a of the drawing) because voltage and current are in phase. For a capacitor (see part b), the phasors remain perpendicular while rotating because the phase angle between the current and the voltage is 90°. Since current leads voltage for a capacitor, the current phasor is ahead of the voltage phasor in the direction of rotation.

Figure 23-5 These rotating-arrow models represent the voltage and the current in ac circuits that contain (a) only a resistor and (b) only a capacitor.

Note from the two circuit drawings in Figure 23-5 that the instantaneous voltage across the resistor or the capacitor is . We can find this instantaneous value of the voltage directly from the phasor diagram. Imagine that the voltage phasor V0 in this diagram represents the hypotenuse of a right triangle. Then, the vertical component of the phasor would be . In a similar manner, the instantaneous current can be found as the vertical component of the current phasor.

Check Your Understanding 1

One circuit contains only an ac generator and a resistor, and the rms current in this circuit is IR. Another circuit contains only an ac generator and a capacitor, and the rms current in this circuit is IC. The maximum, or peak, voltage of the generator is the same in both circuits and does not change. If the frequency of each generator is tripled, by what factor does the ratio IR/IC change? (The answer is given at the end of the book.)
Background: An understanding of how the resistance R and the capacitive reactance XC change with frequency is the key to answering this question, because changes in R and XC play the central role in how the current changes.
For similar questions (including calculational counterparts), consult Self-Assessment Test 23.1, which is described at the end of Section 23.2.

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