Q2Power System - I
Question
Q.2. Explain clearly how the magnetic field energy and the reactive power in an inductive circuit are related.
Answer
Reactive power in an inductive circuit represents the rate at which energy oscillates into and out of the circuit's magnetic field twice per cycle without net energy transfer; it equals ωL·I² (or equivalently I²X), directly proportional to the peak magnetic field energy stored (½LI²) times twice the angular frequency.
In a purely inductive circuit carrying sinusoidal current i(t) = Im·sin(ωt), the instantaneous magnetic field energy stored in the inductor is w(t) = ½L·i(t)² = ½L·Im²·sin²(ωt), which fluctuates between zero (when current is zero) and its maximum value ½L·Im² (when current is at its peak), completing two full oscillations per electrical cycle (since sin² has period π, half that of sin).
The instantaneous power delivered to the inductor is p(t) = v(t)·i(t); with v(t) = L(di/dt) = ωL·Im·cos(ωt) and i(t)=Im sin(ωt), this gives p(t) = ωL·Im²·sin(ωt)cos(ωt) = (ωL·Im²/2)·sin(2ωt), which is a pure sinusoid at twice the supply frequency, oscillating symmetrically between positive and negative values with zero average (net) value over a complete cycle — physically, this means energy flows from the source into the magnetic field during one quarter cycle and flows back out to the source during the next quarter cycle, with no net energy consumed by an ideal inductor.
The reactive power Q is defined as the amplitude of this oscillating power term, Q = ωL·Irms², since Irms = Im/√2, this gives Q = ωL·Im²/2 = I²X (where X=ωL is the inductive reactance and I is the rms current) — showing directly that reactive power Q is proportional to (and in fact numerically equal to 2ω times) the peak magnetic field energy stored: Q = 2ω × (½LIm²/2) after accounting for rms vs peak values, or more directly Q = I²X represents the rate of energy exchange associated with building and collapsing the magnetic field twice every cycle. This is why reactive power, while it does no net useful work, is nonetheless essential — the magnetic fields it sustains are what enable transformers, motors and inductive loads to function, and adequate reactive power supply/compensation is necessary to maintain system voltage levels, since insufficient reactive power support leads to voltage collapse in a power system.