∙In an R-L-C series circuit the magnitude of the phase angle is 54.0^∘, with the source voltage

step 1 We have the magnitude of the phase angle, Phi, equaling 54 .0 degrees. We have the capacitive re

∙In an R-L-C series circuit the magnitude of the phase angle...

$\bullet$ In an $R-L-C$ series circuit the magnitude of the phase angle is $54.0^{\circ},$ with the source voltage lagging the current. The reactance of the capacitor is $350 \Omega,$ and the resistor resistance is 180$\Omega .$ The average power delivered by the source is 140 $\mathrm{W}$ . Find (a) the reactance of the inductor; (b) the rms current; (c) the rms voltage of the source.

$\bullet$ In an $R-L-C$ series circuit the magnitude of the phase
angle is $54.0^{\circ},$ with the source voltage lagging the current. The
reactance of the capacitor is $350 \Omega,$ and the resistor resistance is 180$\Omega .$ The average power delivered by the source is 140 $\mathrm{W}$ . Find (a) the reactance of the inductor; (b) the rms current; (c) the rms voltage of the source.

Key Concepts

Series RLC Circuit

A series RLC circuit is an electrical circuit composed of a resistor (R), an inductor (L), and a capacitor (C) connected in series. In such circuits, the total impedance is the vector sum of the individual resistive and reactive (inductive and capacitive) impedances, which determines both the magnitude and phase relationship between the source voltage and current.

AC Impedance

Impedance in AC circuits is a complex quantity that combines resistance and reactance (both inductive and capacitive) into a single value. It is expressed as a combination of real and imaginary parts and determines how much the circuit resists the flow of alternating current at a particular frequency, influencing both amplitude and phase shift.

Phase Angle

The phase angle in an AC circuit represents the angular difference between the voltage and the current waveforms. This angle is determined by the balance between inductive and capacitive reactances, and it affects the power factor of the circuit. A lagging phase (where the voltage lags the current) indicates a dominance of inductive effects over capacitive ones.

RMS Quantities

Root mean square (rms) values of voltage and current are used in AC circuit analysis to provide equivalent measures of the effective or heating value, similar to a DC circuit. RMS values allow for accurate calculation of power in AC systems by representing the equivalent DC value that would produce the same power dissipation in a resistor.

AC Power Calculation

The average power delivered in an AC circuit is calculated by multiplying the rms values of voltage and current along with the cosine of the phase angle, known as the power factor. This formula specifically accounts for both the magnitude and phase relationship of the voltage and current waveforms, highlighting that only the in-phase component contributes to the average power dissipation.

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