An air conditioner connected to a 120 V rms ac line is equivalent to a 12.0Ωresistance and a 1.3

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An air conditioner connected to a 120 V rms ac line is...

An air conditioner connected to a 120 $\mathrm{V} \mathrm{rms}$ ac line is equivalent to a 12.0$\Omega$ resistance and a 1.30$\Omega$ inductive reactance in series. Calculate (a) the impedance of the air conditioner and (b) the average rate at which energy is supplied to the appliance.

An air conditioner connected to a 120 $\mathrm{V} \mathrm{rms}$ ac line is equivalent to a 12.0$\Omega$ resistance and a 1.30$\Omega$ inductive reactance
in series. Calculate (a) the impedance of the air conditioner and
(b) the average rate at which energy is supplied to the appliance.

Key Concepts

Series R-L Circuit Impedance

In AC circuit analysis, a circuit with resistors and inductors in series has an impedance that combines both resistance and inductive reactance. This total impedance is found using the formula Z = ?(R² + X?²), where R is the resistance and X? is the inductive reactance. This impedance determines how much the circuit opposes the alternating current.

Inductive Reactance

Inductive reactance is the opposition that an inductor presents to a change in current in an AC circuit. It is frequency dependent and is given by the formula X? = 2?fL, where f is the frequency and L is the inductance. This reactance causes a phase shift between the voltage and the current in the circuit.

Root Mean Square (RMS) Values

RMS values are used in AC circuits to represent the effective or equivalent DC value that would produce the same power dissipation in a resistive load. The RMS voltage is a standard way to quantify an AC voltage, making it easier to calculate power and perform circuit analyses by using formulas analogous to those in DC circuits.

Average Power in AC Circuits

The average power supplied to an AC circuit is calculated using the RMS values of voltage and current, along with the power factor which accounts for the phase difference between them. In resistive components, the formula P = V_rms² * (R/|Z|²) or alternatively, P = I_rms² * R, is used to determine the power dissipated as heat in the circuit.

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