Understanding Power in a Pure Inductive Circuit

Let's analyze the power consumption in a circuit containing only an inductor (a pure inductive circuit) when connected to an AC voltage source.

What is a Pure Inductive Circuit?

A pure inductive circuit is an electrical circuit that contains only inductance and no resistance or capacitance. In such a circuit, the inductor resists changes in electric current.

Power in AC Circuits

In AC circuits, power can be categorized into:

• Real Power (P): The actual power consumed or dissipated by the circuit, measured in Watts (W). This is the power that does useful work.
• Reactive Power (Q): The power exchanged between the source and the reactive components (inductors or capacitors), measured in Volt-Amperes reactive (VAR). It is associated with energy storage in magnetic or electric fields.
• Apparent Power (S): The vector sum of real and reactive power, measured in Volt-Amperes (VA).

The question asks about "the power," which typically refers to the real power (average power consumed).

Voltage and Current Relationship

In a pure inductive circuit, the current lags behind the voltage by a phase angle of 90 degrees (or $\frac{\pi}{2}$ radians). This phase difference is crucial for determining power consumption.

Calculating Real Power

The formula for real power (P) in an AC circuit is given by:

$$ P = V_{rms} \times I_{rms} \times \cos(\phi) $$

Where:

• $V_{rms}$ is the root-mean-square (RMS) voltage across the inductor.
• $I_{rms}$ is the root-mean-square (RMS) current flowing through the inductor.
• $\phi$ is the phase angle between the voltage and the current.

For a pure inductive circuit, the phase angle $\phi$ is $90^\circ$. The cosine of this angle is:

$$ \cos(90^\circ) = 0 $$

Substituting this value into the power formula:

$$ P = V_{rms} \times I_{rms} \times 0 $$

$$ P = 0 $$

Explanation of Zero Power

The inductor stores energy from the source during one half of the AC cycle (building up its magnetic field) and returns this stored energy back to the source during the other half of the cycle (as the magnetic field collapses). Because energy is only stored and returned, and not dissipated as heat (like in a resistor), the net energy consumed by the inductor over a full cycle is zero. Therefore, the average power, or real power, in a pure inductive circuit is zero.

Conclusion

Based on the phase relationship between voltage and current and the power formula, the power consumed in a pure inductive circuit is consistently zero.