What is the difference between instantaneous power and average power in an AC circuit?

Instantaneous power is the product of instantaneous voltage and current at any given moment in time, $P(t) = V(t)I(t)$. It constantly changes and can even be negative, indicating energy being returned to the source. Average power, on the other hand, is the net power dissipated over a complete cycle. It represents the useful power converted into heat or work and is calculated as $P_{avg} = V_{rms}I_{rms}cosphi$. This average value is what meters typically measure and what we pay for.

Why is the power factor important in AC circuits?

The power factor ($cosphi$) is crucial because it indicates how efficiently electrical power is being utilized. A power factor of 1 (unity) means all the apparent power is useful power, while a power factor close to 0 means most of the power is reactive and does no useful work. A low power factor leads to higher current for the same useful power, causing increased $I^2R$ losses in transmission lines, larger conductor sizes, and reduced efficiency of the power system. Industries often face penalties for low power factors.

What is 'wattless current' and when does it occur?

Wattless current, also known as reactive current, is the component of the total current that does not contribute to the average power dissipation in an AC circuit. It occurs when the current and voltage are $90^circ$ out of phase, as in purely inductive or purely capacitive circuits. In these cases, the power factor $cosphi = cos(90^circ) = 0$, leading to $P_{avg} = 0$. Even though current flows, no net energy is consumed over a cycle; energy is merely exchanged between the source and the reactive component.

How does resonance affect power in an LCR series circuit?

At resonance in an LCR series circuit, the inductive reactance ($X_L$) becomes equal to the capacitive reactance ($X_C$). This causes the net reactance $(X_L - X_C)$ to become zero, making the impedance $Z$ equal to the resistance $R$. Consequently, the phase angle $phi$ becomes $0^circ$, and the power factor $cosphi$ becomes 1. This means that at resonance, the circuit behaves like a purely resistive circuit, dissipating maximum average power for a given RMS voltage, $P_{avg} = V_{rms}^2/R$.

Can the instantaneous power in an AC circuit be negative? What does it mean?

Yes, the instantaneous power in an AC circuit can indeed be negative. This occurs during parts of the cycle when the voltage and current have opposite polarities (e.g., voltage is positive while current is negative, or vice versa). A negative instantaneous power signifies that energy is being returned from the circuit's reactive components (inductors or capacitors) back to the source, rather than being consumed by the load. This energy exchange is characteristic of reactive components and contributes to the 'wattless' component of current.

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Power in AC Circuit

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Version 1Updated 22 Mar 2026

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In an alternating current (AC) circuit, the instantaneous power is the product of the instantaneous voltage and instantaneous current. However, unlike direct current (DC) circuits where power is constant, in AC circuits, both voltage and current vary sinusoidally with time, and often, they are not in phase. This phase difference between voltage and current significantly impacts the average power d…

Quick Summary

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.

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Resonance and Maximum Power

In a series LCR circuit, resonance occurs when the inductive reactance equals the capacitive reactance ($X_L…

Instantaneous Power —

P(t) = V(t)I(t)

Average Power —

P_{avg} = V_{rms}I_{rms}cosphi

RMS Values —

V_{rms} = V_0/sqrt{2}

I_{rms} = I_0/sqrt{2}

Power Factor —

cosphi = R/Z

Impedance —

Z = sqrt{R^2 + (X_L - X_C)^2}

Reactances —

X_L = omega L

X_C = 1/(omega C)

Phase Angle —

anphi = (X_L - X_C)/R

Pure R Circuit —

phi = 0^circ

cosphi = 1

P_{avg} = V_{rms}I_{rms}

Pure L/C Circuit —

phi = pm 90^circ

cosphi = 0

P_{avg} = 0

(Wattless current)

Resonance ($X_L = X_C$) —

phi = 0^circ

cosphi = 1

Z=R

P_{avg,max} = V_{rms}^2/R

P-A-W: Power Always Watts (only in Resistors). Remember the formula: Power = Voltage * In-phase Current. (P = V_rms * I_rms * cos phi). The 'C' in 'Current' reminds you of 'cos phi'.

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