Power in AC Circuit — Core Principles

Core Principles

Power in an AC circuit is more complex than in a DC circuit due to the sinusoidal variation of voltage and current, and the potential phase difference between them. Instantaneous power, $P(t) = V(t)I(t)$, fluctuates with time and can even be negative.

The crucial quantity for practical applications is the average power, $P_{avg}$, dissipated over a full cycle. This is given by the formula $P_{avg} = V_{rms}I_{rms}cosphi$, where $V_{rms}$ and $I_{rms}$ are the root mean square values of voltage and current, respectively, and $phi$ is the phase angle between them.

The term $cosphi$ is known as the power factor, which indicates the efficiency of power utilization. For purely resistive circuits, $phi = 0^circ$ and $cosphi = 1$, leading to maximum power dissipation.

For purely inductive or capacitive circuits, $phi = pm 90^circ$ and $cosphi = 0$, resulting in zero average power dissipation (wattless current). In LCR series circuits, the power factor is $R/Z$, and at resonance, it becomes 1, maximizing power transfer to the resistance.

Important Differences

vs Power in DC Circuit

Aspect	This Topic	Power in DC Circuit
Nature of Voltage/Current	Constant (steady)	Sinusoidally varying with time
Formula for Power	$P = VI$	$P_{avg} = V_{rms}I_{rms}cosphi$
Phase Difference	Not applicable (voltage and current are always in phase)	Exists between voltage and current, denoted by $phi$
Power Factor	Always 1 (implicitly)	Varies between 0 and 1, $cosphi$
Reactive Components (L, C)	No effect on steady power dissipation	Significantly affect power factor and average power (can cause wattless current)
Energy Dissipation	Always dissipated in resistance	Only dissipated in resistance; reactive components store/release energy

The fundamental distinction between power in DC and AC circuits lies in the time-varying nature of AC quantities and the introduction of phase difference. In DC, power is a constant product of voltage and current, $P=VI$. In AC, we consider average power, $P_{avg} = V_{rms}I_{rms}cosphi$, where the power factor $cosphi$ accounts for the phase angle between voltage and current. This phase difference, absent in DC, means that reactive components (inductors and capacitors) can cause current to flow without dissipating useful power, a phenomenon known as wattless current, which is unique to AC circuits.