A series ac circuit contains a 290 Ωresistor, a 14 mH inductor, a 2.5 μF capacitor, and an ac po

step 1 Hi, here in this given problem in the series AC circuit, the value of resistance R, that is 290

A series ac circuit contains a 290 Ωresistor, a 14 mH...

A series ac circuit contains a $290 \Omega$ resistor, a $14 \mathrm{mH}$ inductor, a $2.5 \mu \mathrm{F}$ capacitor, and an ac power source of voltage amplitude $45 \mathrm{~V}$ operating at an angular frequency of $360 \mathrm{rad} / \mathrm{s}$. (a) What is the power factor of this circuit? (b) Find the average power delivered to the entire circuit. (c) What is the average power delivered to the resistor, to the capacitor, and to the inductor?

A series ac circuit contains a $290 \Omega$ resistor, a $14 \mathrm{mH}$ inductor, a $2.5 \mu \mathrm{F}$ capacitor, and an ac power source of voltage amplitude $45 \mathrm{~V}$ operating at an angular frequency of $360 \mathrm{rad} / \mathrm{s}$.
(a) What is the power factor of this circuit? (b) Find the average power delivered to the entire circuit. (c) What is the average power delivered to the resistor, to the capacitor, and to the inductor?

Key Concepts

AC Circuit Analysis

AC circuit analysis involves understanding how circuits powered by alternating current behave. This includes studying sinusoidal voltage and current signals and using techniques like phasor representation to analyze the frequency?dependent behavior of circuit components.

Series RLC Circuit

A series RLC circuit consists of a resistor, an inductor, and a capacitor connected in a single loop. The response of this circuit depends on the interplay between resistance and the frequency?dependent reactances of the inductor and capacitor, which together determine the overall impedance and phase relationships.

Impedance

Impedance is the measure of the total opposition a circuit presents to the flow of alternating current. It combines both the real part (resistance) and the imaginary part (reactance) into a complex number, which helps in analyzing the amplitude and phase shift between voltage and current in AC circuits.

Reactance

Reactance refers to the opposition that capacitors and inductors offer to AC, which varies with frequency. Inductive reactance increases with frequency while capacitive reactance decreases, and the difference between these two determines the net reactive behavior of the circuit.

Power Factor

The power factor is the cosine of the phase angle between the voltage and current in an AC circuit. It indicates the proportion of the electrical power that is used to do useful work. A high power factor means that most of the power is being effectively converted into work, while a low power factor indicates the presence of significant reactive power.

Average Power

Average power in AC circuits—often referred to as real power—is the actual power consumed by resistive elements over a complete cycle. In AC circuits, even though all components contribute to the overall impedance, only resistors dissipate energy as heat, while capacitors and inductors merely exchange stored energy with the circuit.

Power Distribution in AC Components

In AC circuits, power distribution is such that the resistor dissipates energy as heat, representing real power consumption. In contrast, the inductor and capacitor ideally do not dissipate power over a complete cycle because they only store energy temporarily, leading to an average power of zero for these reactive components.

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