2.1 Power in AC Circuits & Power Factor

Every alternating-current (AC) circuit that supplies motors, transformers, ballasts, or other coil-based loads involves three distinct but related measures of power. A master electrician who thinks only in watts will undersize conductors, overload transformers, and be unable to explain a utility power-factor penalty on a client's electric bill. This section builds the vocabulary, formulas, and calculation skills needed both for the RA 7920 licensure exam and for real troubleshooting on an industrial or commercial job site.

The Three Powers in an AC Circuit

In a purely resistive AC circuit, voltage and current rise and fall together — they are said to be in phase. But motors, transformers, solenoids, and fluorescent/HID ballasts contain coils that store energy in a magnetic field, and capacitors store energy in an electric field. These reactive components shift the current out of phase with the voltage by an angle θ (theta), and that phase shift produces three related quantities that every electrician must be able to name and calculate.

Real (Active) Power — P

Real power, also called active or true power, is the power that actually performs useful work: turning a motor shaft, producing heat or light, running a control circuit. It is what a wattmeter measures and what a utility bills for as energy consumption (kilowatt-hours). Real power is measured in watts (W), or kilowatts (kW, thousands of watts) for larger loads. For a single-phase circuit:

P = V × I × cos(θ)

where V and I are RMS voltage and current, and θ is the phase angle between voltage and current.

Reactive Power — Q

Reactive power is the power that inductors and capacitors exchange back and forth with the source every cycle without performing any net work. An induction motor's windings store energy in a magnetic field during part of each cycle and return that energy to the source during another part. This continuous exchange of energy still requires current to flow through every conductor, transformer winding, and generator in the system — it simply never converts into mechanical work or heat. Reactive power is measured in volt-amperes reactive (VAR), or kilovolt-amperes reactive (kVAR) for larger quantities:

Q = V × I × sin(θ)

Apparent Power — S

Apparent power is the straightforward product of RMS voltage and RMS current, without regard to phase angle. It represents the total current-carrying burden a load places on the conductors, transformer, and generator supplying it — and it is this number, not real power, that actually determines the ampacity of the wire or the kVA rating of the transformer needed. Apparent power is measured in volt-amperes (VA) or kilovolt-amperes (kVA):

S = V × I

The Power Triangle

Because P, Q, and S are related through the phase angle θ, they can be drawn as a right triangle called the power triangle: real power P along the horizontal (adjacent) leg, reactive power Q along the vertical (opposite) leg, and apparent power S as the hypotenuse. The angle between P and S is the same phase angle θ that appears in the P and Q equations. By the Pythagorean theorem:

S² = P² + Q², so S = √(P² + Q²)

Power Factor

Power factor (PF) is the ratio of real power to apparent power:

PF = P / S = cos(θ)

Power factor is a decimal between 0 and 1 (often expressed as a percentage, e.g., 0.85 or 85%). A PF of 1.0 (unity) means voltage and current are perfectly in phase and all delivered apparent power is converted into useful work. Almost every real electrical load — anything with a motor, transformer, or discharge-lamp ballast — has a power factor below unity.

Lagging vs. Leading Power Factor

When current lags behind voltage, the load is said to have a lagging power factor. This is the condition produced by inductive loads — induction motors, transformers, welders, and magnetic ballasts — and it is by far the most common condition in industrial and commercial facilities, since motors make up most of the connected load.

When current leads voltage, the load is said to have a leading power factor. This condition is produced by capacitive loads — banks of power-factor-correction capacitors, some electronic power supplies, and lightly loaded, long underground or submarine cable runs whose capacitance dominates.

Why Power Factor Matters

Power factor is not just an academic concept — it has direct financial and engineering consequences:

• Line losses. For a given real power delivered, a lower power factor means higher current, and conductor heating loss (I²R) rises with the square of current. A facility running at 0.7 PF instead of 0.95 PF draws roughly 36 percent more current for the same useful output, and dissipates proportionally more heat in cables and windings.
• Equipment capacity. Transformers, generators, switchgear, and conductors are all rated in kVA (apparent power), not kW. A low power factor forces the utility — and the building's own electrical system — to be sized and operated for more current than the useful power actually delivered would require.
• Utility penalties. Many electric utilities assess a power-factor surcharge or demand penalty on commercial and industrial accounts once PF falls below a threshold, commonly around 0.85 to 0.90, to recover the cost of the extra system capacity that a poor power factor forces them to carry.

Power Factor Correction

Because most industrial and commercial loads are inductive (lagging), power factor is corrected by connecting capacitors in parallel with the load — at an individual motor, at a distribution panel, or centrally at the service switchboard. A capacitor supplies leading reactive power (capacitive kVAR) that cancels part of the load's lagging reactive power, reducing the net reactive power drawn from the source. This shrinks apparent power (S) and pushes the phase angle θ toward zero — raising PF toward unity — without changing the real power (kW) the load actually consumes. Correction is normally targeted at a PF around 0.95, not a full 1.0, because overcorrecting into a leading power factor can cause overvoltage and resonance problems on lightly loaded circuits.

Worked Example: Sizing a Power Factor Correction Capacitor Bank

A 100 kW induction motor load operates at 0.75 lagging power factor. Find the apparent power, the existing reactive power, and the capacitive kVAR needed to correct the power factor to 0.95 lagging.

Step 1 — Apparent power (S): S = P / PF = 100 kW / 0.75 = 133.3 kVA

Step 2 — Existing reactive power (Q₁): θ₁ = cos⁻¹(0.75) = 41.4° Q₁ = P × tan(θ₁) = 100 × tan(41.4°) = 100 × 0.882 = 88.2 kVAR

Step 3 — Target reactive power (Q₂) at PF = 0.95: θ₂ = cos⁻¹(0.95) = 18.2° Q₂ = P × tan(θ₂) = 100 × tan(18.2°) = 100 × 0.329 = 32.9 kVAR

Step 4 — Required capacitor bank: Qc = Q₁ − Q₂ = 88.2 − 32.9 ≈ 55.3 kVAR

A capacitor bank rated at approximately 55.3 kVAR, connected across the motor terminals or feeder, raises the power factor from 0.75 to 0.95 without changing the 100 kW of real power the motor consumes.

Summary Table: P, Q, S, and PF