An ac generator with a frequency of 1.55 kHz and an rms voltage of 20.8 V is connected in series

step 1 In the first item we have to evaluate the power factor of the circuit. So we have to evaluate th

An ac generator with a frequency of 1.55 kHz and an rms...

An ac generator with a frequency of 1.55 kHz and an rms voltage of 20.8 $\mathrm{V}$ is connected in series with a 2.00 -k\Omega resistor and a
$292-\mathrm{mH}$ inductor. (a) What is the power factor for this circuit?
(b) What is the average power consumed by this circuit?

Key Concepts

Power Factor

The power factor in an AC circuit is the ratio of the real (active) power that performs work to the apparent power that flows in the circuit. It is defined as the cosine of the phase angle (?) between the voltage and current waveforms, with cos ? = R/Z in a series RL circuit. The power factor indicates how efficiently electrical power is being used and is a critical parameter in AC power analysis.

Impedance in AC Circuits

Impedance is the total opposition that a circuit presents to the flow of alternating current and is a complex quantity combining resistance and reactance. In a series resistor-inductor (RL) circuit, the impedance is computed as Z = ?(R² + X_L²), where R is the resistance and X_L is the inductive reactance. This concept is fundamental in understanding how AC circuits behave and how voltage and current are phase-shifted relative to each other.

Average Power Consumption in AC Circuits

Average power consumption in AC circuits refers to the real power that is converted into useful work or heat over time. In circuits with resistive and reactive components, the average power is given by P_average = V_RMS × I_RMS × cos ?, where cos ? is the power factor. It can also be calculated as P_average = V_RMS² × (R/|Z|²) for series circuits, highlighting the role of both resistance and the phase relationship between voltage and current.

RMS Voltage

RMS voltage stands for Root Mean Square voltage, which is a measure of the effective value of an alternating voltage. It is used in AC circuits to represent the DC equivalent voltage in terms of the power delivered; that is, an RMS voltage will produce the same amount of heat in a resistor as a DC voltage of the same magnitude.

Angular Frequency and Inductive Reactance

Angular frequency (?) is calculated as 2? times the frequency and is crucial for determining how components like inductors behave in AC circuits. Inductive reactance (X_L) is the opposition that an inductor offers to a change in current and is given by X_L = ?L, where L is the inductance. This reactance increases with frequency, impacting the overall impedance of the circuit.

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