6.1 Power Fundamentals & Three-Phase

Why power fundamentals matter

The Power knowledge area is one of the highest-weight topics on the FE Electrical and Computer exam at roughly 8 to 12 questions. Most power problems reduce to applying the power triangle and the balanced three-phase relations from the NCEES FE Reference Handbook. The exam rewards candidates who can locate the correct formula and keep voltage, current, and angle references consistent, rather than candidates who memorize derivations.

Start every power problem by classifying it: single-phase or three-phase, wye or delta, and whether the question wants real power (W), reactive power (VAR), apparent power (VA), or power factor. A wrong classification is the most common reason a correct equation produces a wrong answer.

The power triangle

For an AC load with voltage and current phasors at angle theta apart, three quantities describe power flow:

• Real power P = V·I·cos(theta), measured in watts (W). This is the average power that does useful work.
• Reactive power Q = V·I·sin(theta), measured in volt-amperes reactive (VAR). It oscillates between source and reactive elements and does no net work.
• Apparent power |S| = V·I, measured in volt-amperes (VA), where complex power S = P + jQ.

These form a right triangle: |S|^2 = P^2 + Q^2. Here V and I are RMS magnitudes. Power factor pf = cos(theta) = P/|S|. An inductive (lagging) load has positive Q; a capacitive (leading) load has negative Q.

Power-factor correction

Utilities and the exam both care about power-factor correction because low pf increases line current and losses for the same real power. To raise a lagging pf, add a shunt capacitor that supplies reactive power, reducing the net Q the source must provide.

The needed capacitor reactive power is Q_C = P·(tan(theta_1) - tan(theta_2)), where theta_1 is the original angle and theta_2 is the target angle. Real power P stays constant during correction; only Q and apparent power change. After correction, line current falls because |S| = V·I drops while P is unchanged.

Single-phase vs three-phase

Three-phase systems use three sources 120 degrees apart and deliver constant instantaneous power, which is why they dominate generation and transmission. The two connection schemes are wye (Y) and delta.

For any balanced load, total real power is P = sqrt(3)·V_L·I_L·cos(theta), where V_L and I_L are line quantities and theta is the per-phase impedance angle. The same form gives Q with sin(theta) and |S| with V_L·I_L only.

Per-unit basics

The per-unit (pu) system expresses each quantity as a fraction of a chosen base: pu value = actual value / base value. Engineers pick a base power (often three-phase MVA) and a base voltage per zone; base impedance follows as Z_base = (V_base)^2 / S_base.

The payoff is that transformer turns ratios disappear when bases are chosen by the turns ratio, so a multi-voltage power network reduces to one connected per-unit circuit. Per-unit values also stay within a familiar range near 1.0, making errors easier to spot. To convert an impedance to a new base, multiply by (V_base_old/V_base_new)^2 and (S_base_new/S_base_old).