An electric water heater draws 20 A rms at 240 V rms and is purely resistive. An AC motor has

step 1 In this problem, we're told that the RMS current is 20 amps. The RMS voltage is 240 volts. And w

An electric water heater draws 20 A rms at 240 V rms and...

An electric water heater draws $20 \mathrm{~A} \mathrm{rms}$ at $240 \mathrm{~V} \mathrm{rms}$ and is purely resistive. An AC motor has the same current and voltage, but its inductance causes the voltage to lead the current by $20^{\circ}$. Find the power consumption in each device.

Key Concepts

RMS Measurements in AC Circuits

RMS (Root Mean Square) values represent the effective voltage or current of an AC waveform, equivalent to the value of a DC source that would deliver the same power to a resistive load. They are fundamental in AC analysis, as most power calculations in AC circuits use RMS quantities to determine effective energy transfer.

Phase Difference and Power Factor

The phase difference between voltage and current in an AC circuit is a measure of how much one waveform leads or lags the other and is critical to determining the power factor. The power factor, defined as the cosine of the phase angle, indicates the fraction of the apparent power that is actually converted into useful work, making it essential for understanding efficiency in AC systems.

Real (Active) Power in AC Circuits

Real power, also known as active power, represents the actual power consumed by the circuit to perform work. It is calculated using the formula P = Vrms × Irms × cos(?), where cos(?) is the power factor. This concept is crucial for distinguishing between the power that performs useful work and the apparent power that includes energy oscillating due to reactive components.

Resistive and Inductive Loads

Resistive loads, such as in a pure resistor, have voltage and current in phase, resulting in a power factor of 1 and maximum real power consumption. Inductive loads, on the other hand, cause the current to lag (or voltage to lead) the current, resulting in a phase difference that reduces the power factor and, consequently, the real power delivered. Understanding these differences is essential for analyzing various types of AC devices.

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