AC Theory: Sine Waves, Frequency, Phase, and Power Factor

What Is Alternating Current?

Direct current (DC) flows in one direction only. Alternating current (AC) reverses direction periodically, following a smooth, repeating pattern called a sine wave. Almost all power delivered to industrial and marine facilities is AC because it can be stepped up or down efficiently with transformers, reducing energy loss over long distribution distances.

The Sine Wave

A sine wave is a graph of voltage (or current) plotted against time. Starting at zero, the value rises to a positive peak, returns through zero, falls to a negative peak, and returns to zero again — completing one full cycle. The shape is described mathematically as:

> v(t) = V_peak × sin(2πft)

You do not need to memorize the formula, but you do need to understand the key quantities it contains.

Peak value (V_peak or I_peak) is the maximum positive (or negative) value reached during one cycle. An oscilloscope displays this directly.

Period (T) is the time, in seconds, for one complete cycle.

Frequency (f) is the number of complete cycles per second, measured in hertz (Hz). In North America, utility power runs at 60 Hz. Frequency and period are reciprocals of each other:

> f = 1 / T

RMS Values — What the Meter Reads

Because AC voltage and current swing positive and negative, simply averaging them gives zero — useless for calculating power. Engineers use the root-mean-square (RMS) value, which represents the equivalent DC value that would produce the same heating effect in a resistor.

For a pure sine wave:

> V_RMS = V_peak × 0.7071

> I_RMS = I_peak × 0.7071

The factor 0.7071 is 1/√2. Your DMM reads RMS by default. North American 120 V service has a peak voltage of approximately 170 V; your meter reads 120 V RMS.

Phase Relationships

In a purely resistive load (such as a heating element), voltage and current reach their peaks at exactly the same instant — they are said to be in phase.

In an inductive load (motors, transformers, solenoids), the magnetic field opposes changes in current. Current lags behind voltage — voltage peaks first, then current peaks later. This lag is expressed as a phase angle (θ), measured in degrees.

In a capacitive load, the opposite occurs — current leads voltage. Capacitors are commonly added to industrial systems to correct for the lagging effect of motors.

Phase relationships are visualized with phasor diagrams — arrows whose length represents magnitude and whose angle from horizontal represents phase.

Power in AC Circuits

Three types of power appear in AC systems with reactive loads:

Apparent power (S) is what the utility and wiring must supply. It is measured in volt-amperes (VA) and calculated as:

> S = V_RMS × I_RMS

True (real) power (P) is the power actually converted to useful work (heat, motion). It is measured in watts (W) and calculated as:

> P = V_RMS × I_RMS × cos(θ)

Reactive power (Q) is power stored and returned by inductors and capacitors each cycle — it does no useful work but stresses wiring and transformers. It is measured in volt-amperes reactive (VAR):

> Q = V_RMS × I_RMS × sin(θ)

These three form a right triangle (the power triangle): S² = P² + Q².

Power Factor

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

> PF = P / S = cos(θ)

Power factor ranges from 0 to 1 (or 0% to 100%). A PF of 1.0 means all supplied power does useful work. A low PF (common with large inductive motor loads) means the system draws more current than the load actually needs, wasting capacity and increasing utility costs. Industrial plants are often required by utilities to maintain PF above a specified minimum or face penalty charges.

Improving power factor is done by adding capacitor banks in parallel with inductive loads, reducing reactive power and bringing PF closer to 1.0.

Safety Notes

• Always treat AC circuits as energized until verified de-energized with a calibrated meter. Apply lockout/tagout (LOTO) procedures per OSHA regulations before working on any panel or motor.
• When using an oscilloscope with a step-down isolation transformer in lab, confirm the transformer secondary voltage before connecting probes. Never connect an oscilloscope probe ground clip to a live conductor.
• Wear appropriate PPE — safety glasses and insulated gloves rated for the voltage level present. Verify glove rating before use.
• Even low-voltage AC can cause ventricular fibrillation. Respect all conductors.
• PC-based oscilloscopes and isolated probes reduce shock risk in classroom demonstrations; follow the instructor's setup exactly.