An L-R-C series circuit consists of a 2.50-capacitor, a 5.00-mH inductor, and a 75.0-Ωresistor c

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An L-R-C series circuit consists of a 2.50-capacitor, a...

An $L-R-C$ series circuit consists of a 2.50-$\mu$F capacitor, a 5.00-mH inductor, and a 75.0-$\Omega$ resistor connected across an ac source of voltage amplitude 15.0 V having variable frequency. (a) Under what circumstances is the average power delivered to the circuit equal to $1\over2$ V$_{rms}$ I$_{rms}$? (b) Under the conditions of part (a), what is the average power delivered to each circuit element and what is the maximum current through the capacitor?

Key Concepts

Condition for Specific Power Delivery

The average power delivered in an AC circuit is given by P_avg = V_rms I_rms cos(?), which can also be expressed as P_avg = V_rms I_rms (R/Z) in a series circuit since cos(?) = R/Z. For the average power to equal a specified fraction (for example, one-half) of the product V_rms I_rms, the ratio R/Z must equal that fraction. This condition sets a relationship between the circuit’s resistance and its net impedance, thereby constraining the effective reactance difference (X_L - X_C) relative to R.

Reactance of Inductors and Capacitors

Inductors and capacitors introduce frequency-dependent opposition to current changes in AC circuits, known as reactance. Inductive reactance (X_L = ?L) increases with frequency, while capacitive reactance (X_C = 1/(?C)) decreases with frequency. These reactances determine how the circuit responds at different frequencies and influence the overall impedance and phase relationships.

Phase Angle and Power Factor

The phase angle in an AC circuit indicates the phase difference between the voltage and current waveforms. The power factor, given by cos(?), where ? is the phase angle, represents the fraction of the apparent power (V_rms × I_rms) that is actually dissipated as real power. This concept is critical when relating the theoretical maximum power transfer to the actual average power used in the circuit.

RMS Values

Root-Mean-Square (RMS) values are measures of the effective voltage and current in AC circuits. They represent the DC equivalent values that would deliver the same power as the corresponding AC signals, making them indispensable when calculating average power in circuits with sinusoidally varying quantities.

Impedance in Series RLC Circuits

In AC circuits, impedance is the combined effect of resistance and reactance that opposes current flow. For a series RLC circuit, impedance is calculated as Z = ?(R² + (X_L - X_C)²), where R is resistance, X_L is inductive reactance, and X_C is capacitive reactance. This formula is key to linking voltage, current, and power in AC circuits.

AC Circuit Analysis

AC circuit analysis deals with circuits supplied by alternating current, where both the magnitude and phase of signals vary sinusoidally. This analysis involves understanding how circuit elements such as resistors, inductors, and capacitors interact with AC sources, and it heavily relies on phasor diagrams and frequency-domain techniques to simplify calculations.