AC Fundamentals

AC Circuits · 10 min read

A battery wakes a light bulb. A wall socket wakes the same bulb a hundred times a second. Both deliver enough energy to glow a filament, but the two sources feel different to every component downstream — a capacitor ignores steady DC and welcomes AC; an inductor is the reverse. Everything in this subject follows from that one change: the voltage doesn't sit still.

DC vs AC

Direct current is a steady push — a battery holds its terminals at a fixed voltage, so current flows in one direction at a constant level. Plot voltage against time and it's a flat horizontal line.

Alternating current swings. Mains electricity in the UK is a 230 V source that oscillates 50 times a second between two peaks roughly 325 V apart — positive peak, through zero, negative peak, back through zero, and so on. Plot it and you see a textbook sine wave.

DC holds a steady level; AC swings between a positive and negative peak many times per second.

The sinusoidal waveform

Mathematically, an AC voltage is:

v(t) = V_p \sin(\omega t + \phi)

Four numbers describe it completely:

Peak amplitude $V_p$ — the maximum value the voltage reaches. Units: volts.
Angular frequency $ω$ — how fast the sinusoid oscillates, in radians per second.
Phase $φ$ — where the waveform sits on the time axis at $t = 0$ . If φ = 0 the wave starts at zero and rises; if φ = 90° it starts at the peak.
Time $t$ — the independent variable. The output $v$ changes as time flows.

One full cycle of a sine wave. Peak amplitude ±Vp, peak-to-peak Vpp = 2Vp, period T = one full oscillation.

Frequency and period

The period $T$ is how long one full cycle takes — from zero, up to $+V_p$ , back through zero, down to $-V_p$ , and back to zero. Its reciprocal is the frequency $f$ , measured in hertz (cycles per second):

f = \frac{1}{T}, \qquad \omega = 2\pi f

UK / EU mains runs at $f = 50\,\text{Hz}$ — one cycle every 20 ms. US mains is $f = 60\,\text{Hz}$ — one cycle every 16.67 ms. Audio signals span roughly 20 Hz to 20 kHz. Wi-Fi radio is around 2.4 GHz — a sinusoid that swings 2.4 billion times a second.

Peak, peak-to-peak, and RMS

You'll see voltages quoted three different ways:

Peak $V_p$ — highest instantaneous value. Mostly useful when designing for component breakdown voltage.
Peak-to-peak $V_{pp}$ = $2 V_p$ — the distance between the positive and negative peaks. What you measure on an oscilloscope by eye.
RMS (root-mean-square) $V_{rms}$ — the equivalent DC voltage that would dissipate the same power in a resistor. For a pure sine wave:
$V_{rms} = \frac{V_p}{\sqrt{2}} \approx 0.707 \, V_p$

When someone says “230 V mains” they mean 230 V RMS. The peak is $230 \cdot \sqrt{2} \approx 325\,\text{V}$ , which is why mains-rated capacitors are labelled for at least 400 V: they have to survive the peak, not just the RMS.

The orange dashed lines at ±0.707 Vp mark V_rms — the DC voltage that would dissipate the same power in a resistor as this sine wave.

An AC voltage is a sinusoid with three knobs — amplitude

V_p

, frequency

f

, phase

φ

. Report voltages in RMS for power arithmetic, peak for insulation design, and peak-to-peak for oscilloscope reading.

Why RMS, not average?

The arithmetic mean of a sine wave over one period is zero — the positive and negative halves cancel exactly. That average is useless for calculating power: a bulb still glows on AC because power depends on $v^2$ , which is always positive. RMS captures this by taking the root of the mean of the squared voltage:

V_{rms} = \sqrt{\frac{1}{T} \int_0^T v(t)^2 \, dt}

For a pure sine this integrates to $V_p / \sqrt{2}$ . The DC-equivalent power dissipated in a resistor $R$ is:

P = \frac{V_{rms}^2}{R} = V_{rms} \cdot I_{rms}

Every “power” calculation you make in AC uses RMS values — unless explicitly told otherwise. Getting this reflex right now is the single biggest win of this topic.