When a resistor is connected by itself to an ac generator,...

When a resistor is connected by itself to an ac generator, the average power delivered to the resistor is $1.000 \mathrm{~W}$. When a capacitor is added in series with the resistor, the power delivered is $0.500 \mathrm{~W}$. When an inductor is added in series with the resistor (without the capacitor), the power delivered is $0.250 \mathrm{~W}$. Determine the power delivered when both the capacitor and the inductor are added in series with the resistor. Section 23.4 Resonance in Electric Circuits

Key Concepts

Resonance in Electric Circuits

Resonance in electric circuits occurs when the inductive and capacitive reactances cancel each other out, leaving only the resistive impedance. At resonance, the circuit experiences a maximized or minimized response (depending on configuration) to the input signal, and certain parameters such as power delivery or voltage can peak in value.

Power Factor

The power factor is the cosine of the phase angle between the voltage and the current in an AC circuit. It quantifies the fraction of the total power that is used to do useful work. A power factor of one implies that all the power is effectively used, while lower power factors indicate inefficiencies due to energy stored and later returned to the circuit.

Impedance

Impedance generalizes resistance for AC circuits, combining the effects of both resistance and reactance (from capacitors and inductors) into a complex quantity. It determines how the circuit opposes the flow of AC and affects both the amplitude and phase of the current relative to the voltage.

Average Power in AC Circuits

In AC circuits, the average power represents the time-averaged rate at which energy is delivered to the circuit. Unlike DC circuits, the instantaneous power in AC circuits varies over time due to the sinusoidal nature of current and voltage. Calculating average power involves accounting for the phase difference between the voltage and current, often embodied in the power factor.

Reactance

Reactance is the component of impedance that arises from the presence of capacitors and inductors in AC circuits. Capacitive reactance opposes changes in voltage, while inductive reactance opposes changes in current. Both depend on the frequency of the AC source and contribute to the phase shift between voltage and current.

AC Circuits

AC circuits are electrical circuits powered by alternating current, where voltage and current vary sinusoidally over time. These circuits involve components like resistors, capacitors, and inductors whose behavior in response to AC differs from DC, particularly due to the role of frequency and phase differences between voltage and current.

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