The power triangle and power factor correction
How real, reactive and apparent power form a right triangle, why a low power factor costs money, and how capacitors correct it.
Reading the power triangle
Every AC load can be drawn as a right triangle. The horizontal leg is real power P in watts, the work the load actually performs. The vertical leg is reactive power Q in VAR, energy that flows into magnetic or electric fields and returns each cycle without doing net work. The hypotenuse is apparent power S in volt-amps, the product of RMS voltage and RMS current that the supply must deliver. Because the triangle is right angled, S^2 = P^2 + Q^2, and the angle at the base is the phase angle between voltage and current.
Where the power factor comes from
The power factor is simply the cosine of that base angle, which equals P divided by S. A resistive heater draws current in step with voltage, so Q is zero, the angle is zero, and the power factor is 1. A lightly loaded motor pulls a large magnetizing current that lags the voltage, pushing Q up and the power factor down. Reading the number back, a power factor of 0.8 means the supply must provide 25 percent more current than the real power alone would suggest, since 1 divided by 0.8 is 1.25.
Why utilities care about it
The wires, transformers and generators of a grid are limited by current, which tracks apparent power rather than real power. A customer with a poor power factor draws extra current for the same useful work, wasting capacity and increasing resistive losses in the network. To discourage this, many tariffs bill for reactive energy or apply a penalty when the power factor drops below a threshold such as 0.9. Correcting the power factor can therefore lower a bill even though the real power consumed does not change.
Correcting a lagging load with capacitors
Most industrial loads lag because motors and transformers are inductive. Adding a capacitor bank in parallel supplies leading reactive power that cancels part of the lagging Q, shrinking the vertical leg of the triangle and pulling the power factor back toward 1. To move from a starting power factor to a target, you compute the reactive power needed as Q = P times (tan of the old angle minus tan of the new angle). Sizing the bank slightly under the target avoids overcorrection, which would swing the load capacitive and raise the current again.