23.6

Concepts & Calculations

A capacitor is one of the important elements found in ac circuits. As we have seen in this chapter, the capacitance of a capacitor influences the amount of current in a circuit. The capacitance, in turn, is determined by the geometry of the capacitor and the material that fills the space between its plates, as Section 19.5 discusses. When capacitors are connected together, the equivalent capacitance depends on the nature of the connection—for example, whether it is a series or a parallel connection, as Section 20.12 discusses. The next example provides a review of these issues concerning capacitors.

| Concepts & Calculations Example 7 | | | Capacitors in Ac Circuits

Two parallel plate capacitors are filled with the same dielectric material and have the same plate area. However, capacitor 1 has a plate separation that is twice that of capacitor 2. When capacitor 1 is connected across the terminals of an ac generator, the generator delivers an rms current of 0.60A. What is the current delivered by the generator when both capacitors are connected in parallel across its terminals? In both cases the generator produces the same frequency and voltage.
Concept Questions and Answers Which of the two capacitors has the greater capacitance?

> Answer The capacitance of a capacitor is given by Equation 19.10 as , where is the dielectric constant of the material between the plates, is the permittivity of free space, A is the area of each plate, and d is the separation between the plates. Since the dielectric constant and the plate area are the same for each capacitor, the capacitance is inversely proportional to the plate separation. Therefore, capacitor 2, with the smaller plate separation, has the greater capacitance.

Is the equivalent capacitance of the parallel combination greater or smaller than the capacitance of capacitor 1?

> Answer According to Equation 20.18, the equivalent capacitance of the two capacitors in parallel is , where C1 and C2 are the individual capacitances. Therefore, the value for CP is greater than the value for C1.

Is the capacitive reactance for CP greater or smaller than for C1?

> Answer The capacitive reactance is given by , according to Equation 23.2, where f is the frequency. For a given frequency, the reactance is inversely proportional to the capacitance. Since the capacitance CP is greater than C1, the corresponding reactance is smaller.

When both capacitors are connected in parallel across the terminals of the generator, is the current from the generator greater or smaller than when capacitor 1 is connected alone?

> Answer According to Equation 23.1, the current is given by , where Vrms is the rms voltage of the generator and XC is the capacitive reactance. For a given voltage, the current is inversely proportional to the reactance. Since the reactance in the parallel case is smaller than for C1 alone, the current in the parallel case is greater.

Solution Using Equation 23.1 to express the current as and Equation 23.2 to express the reactance as , we find that the current is

Applying this result to the case where C1 is connected alone to the generator and to the case where the two capacitors are connected in parallel, we obtain

Dividing the two expressions gives

According to Equation 20.18, the effective capacitance of the two capacitors in parallel is , so that the result for the current ratio becomes

Since the capacitance of a capacitor is given by Equation 19.10 as , we find that

We know that , so that the current in the parallel case is

As expected, the current in the parallel case is greater.

In ac circuits that contain capacitance, inductance, and resistance, it is only the resistance that, on average, consumes power. The average power delivered to a capacitor or an inductor is zero. However, the presence of a capacitor and/or an inductor does influence the rms current in the circuit. When the current changes for any reason, the power consumed by a resistor also changes, as Example 8 illustrates.

| Concepts & Calculations Example 8 | | | Only Resistance Consumes Power

An ac generator has a frequency of 1200 Hz and a constant rms voltage. When a resistor is connected between the terminals of the generator, an average power of 0.25 W is consumed by the resistor. Then, a 0.080-H inductor is connected in series with the resistor, and the combination is connected between the generator terminals. What is the average power consumed in the series combination?
Concept Questions and Answers In which case does the generator deliver a greater rms current?

> Answer When only the resistor is present, the current is given by Equation 20.14 as , where Vrms is the rms voltage of the generator. When the resistor and the inductor are connected in series, the current is given by , according to Equation 23.6. In this expression Z is the impedance and is given by Equation 23.7 as , where XL is the inductive reactance and is given by Equation 23.4 as . Since Z is greater than R, the current is greater when only the resistance is present.

In which case is a greater average power consumed in the circuit?

> Answer Only the resistor in an ac circuit consumes power on average, the amount of the average power being , according to Equation 20.15b. Since the resistance R is the same in both cases, a greater average power is consumed when the current Irms is greater. Since the current is greater when only the resistor is present, more power is consumed in that case.

Solution When only the resistor is present, the average power is , according to Equation 20.15b, and the current is given by , according to Equation 20.14. Therefore, we find in this case that

When the inductor is also present, the average power is still , but the current is now given by , according to Equation 23.6, in which Z is the impedance. In this case, then, we have

Dividing this result by the analogous result for the resistor-only case, we obtain

In this expression Z is the impedance and is given by Equation 23.7 as , where XL is the inductive reactance and is given by Equation 23.4 as . With these substitutions, the ratio of the powers becomes

The power consumed in the series combination, then, is

As expected, more power is consumed in the resistor-only case.

At the end of the problem set for this chapter, you will find homework problems that contain both conceptual and quantitative parts. These problems are grouped under the heading Concepts & Calculations.

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