AC Basics — Sine Waves, Frequency and RMS Voltage

AC vs DC: the two kinds of current

Direct current (DC) flows one way, at (nearly) constant value — what batteries and USB chargers supply. Alternating current (AC) reverses direction periodically, driven by a voltage that swings positive and negative.

AC won the 19th-century 'war of currents' for one decisive reason: transformers only work with changing current, so AC voltage can be stepped up for efficient long-distance transmission and stepped back down for safety. Your wall outlet, the power grid, and anything with a spinning generator all run on AC.

The sine wave and its numbers

Mains AC follows a sine wave — not by accident, but because generators make it that way: a coil rotating at constant speed in a magnetic field naturally produces a sinusoidal voltage.

$$V(t) = V_0\sin(\omega t)$$

Three numbers describe the wave:

• Peak voltage $V_0$: the maximum swing (US outlets: ~170 V).
• Frequency $f$: cycles per second in hertz (US: 60 Hz; much of the world: 50 Hz).
• Period $T = 1/f$: time per cycle (US: about 16.7 ms).

The angular frequency $\omega = 2\pi f$ is the same information in radians per second — the language of the math, while $f$ in hertz is the language of the label.

Peak vs RMS: the two voltages

Meters read 120 V on a US outlet, but an oscilloscope shows peaks of ±170 V. Both are right — they measure different things.

$$V_{\text{rms}} = \frac{V_0}{\sqrt{2}} \approx 0.707\,V_0$$

RMS (root mean square) is the DC-equivalent value: 120 V AC heats a resistor exactly as much as 120 V DC. The true average over a cycle is zero (halves cancel), so engineers invented RMS to get a number that actually predicts power: $P = V_{\text{rms}} I_{\text{rms}}$ for a resistor.

Same for current: a 10 A RMS current peaks at about 14.1 A. Whenever a rating is given without qualification — 120 V mains, 230 V European mains — it is RMS.

How R, C, and L respond to AC

Ohm's law survives AC intact, in RMS terms: $V_{\text{rms}} = I_{\text{rms}} R$. A resistor does not care which way current flows — it heats up either way, and voltage and current stay perfectly in step.

Capacitors and inductors are a different story:

• Capacitor: passes AC easily but blocks DC. Its opposition, capacitive reactance $X_C = 1/(\omega C)$, falls as frequency rises — fast wiggles slip charge on and off the plates with ease.
• Inductor: resists changes in current. Its inductive reactance $X_L = \omega L$ grows with frequency — the coil fights faster reversals harder.

At high frequency a capacitor looks like a wire and an inductor like an open circuit; at low frequency they swap roles. That frequency dependence is the basis of every audio crossover, radio tuner, and filter ever built.

Worked example: reading a mains label

A label reads '120 V, 60 Hz'. Find the peak voltage and the period.

Peak: $V_0 = V_{\text{rms}}\sqrt{2} = 120 \times 1.414 \approx 170\ \mathrm{V}$. The voltage actually swings between +170 V and −170 V.

Period: $T = 1/f = 1/60 \approx 0.0167\ \mathrm{s} = 16.7\ \mathrm{ms}$.

In one period the voltage goes 0 → +170 V → 0 → −170 V → 0. Sixty of those cycles happen every second — far too fast for a light bulb's filament to cool between peaks, which is why AC-powered bulbs glow steadily instead of flickering.

Variables and units

Common mistakes

Real-world applications

• The entire power grid runs on AC: generators spin coils in magnetic fields to produce the sinusoidal voltage your outlets carry.

• Phone and laptop chargers rectify AC from the wall into the DC their electronics need.

• Radio and audio signals are AC waveforms — their frequency carries the information (stations, sound pitch).

• AC transformers step voltage up for efficient long-distance transmission, then back down for safe home use.