Prerequisites · read first

AC/DC · frequency · sine

Sine waveFrequency·RMSRectifywiki/embedded-ac-dc-frequency

TL;DR

Electricity comes in two kinds. DC keeps a steady voltage over time (a battery); AC swings + and − over time (a wall outlet), and the base shape of that swing is the sine wave. A sine is fixed in one line $v (t) = V_{p} sin (2 π f t + φ)$ , giving frequency $f$ (Hz, swings per second), period $T = 1/ f$ , and angular frequency $ω = 2 π f$ . The RMS value $V_{r m s} = V_{p} / 2$ is AC's 'real working voltage' (your 220V outlet actually peaks at 311V), and rectification, filtering, and PWM all build on this.

DC and AC, as a picture

DC means the voltage doesn't change over time — on a graph it's just a flat line. Batteries, USB, and logic rails are DC, and the kickboard pack ( $32 \sim 45 V$ ) is too.
AC means the voltage swings between + and − over time — like the curve below. Wall outlets, audio signals, and motor-drive currents are AC. The base shape of that swing is the smooth sine curve.

AC (sine)DC

DC is a flat line, AC is a swinging sine

Reading the sine in one line

One swinging AC curve is fully fixed by just three numbers: how big (amplitude $V_{p}$ ), how fast (frequency $f$ ), and when it starts (phase $φ$ ). One equation ties them together: $v (t) = V_{p} sin (2 π f t + φ)$ .
That phase $φ$ is exactly the 'timing offset' the $j$ on the impedance page pointed to — two signals at the same frequency can still start at different moments, giving a phase difference.

v (t) = V_{p} sin (2 π f t + φ)

$V_{p}$: amplitude — how big (peak)
$f$: frequency — how fast (Hz)
$φ$: phase — when it starts (timing)

the three knobs of a sine

Period, frequency, angular frequency — same thing, three names

The time for one swing is the period $T$ (seconds); how many swings per second is the frequency $f$ (Hz). They're inverses, so $f = 1/ T$ . Korea's grid is $50 Hz$ , so one swing takes $T = 1/50 = 20 ms$ .
There's also angular frequency $ω = 2 π f$ — just $f$ times $2 π$ because the sine runs in radians. This $ω$ shows up verbatim in $X_{C} = 1/ (ω C)$ and $X_{L} = ω L$ — which is why frequency underlies impedance.

Quantity	Formula	at 50 Hz
Period T	T = 1/f (s)	20 ms
Frequency f	f = 1/T (Hz)	50 Hz
Angular ω	ω = 2πf (rad/s)	≈ 314

at 50 Hz — three faces of one fact

RMS — AC's 'real voltage'

AC swings + and − equally, so its plain average is $0$ — yet a heater clearly gets hot. So we define RMS separately: 'the DC voltage that would do the same work measured by heating.'
For a sine, $V_{r m s} = \frac{V _{p}}{2} \approx 0.707 V_{p}$ . The '220V' at your outlet is this RMS value; the actual peak reaches $220 \times 2 \approx 311 V$ . So choose a part's voltage rating against the peak, not the RMS, or it blows.

V_{r m s} = \frac{V _{p}}{2} \approx 0.707 V_{p}

$V_{p}$: peak — true top (rate parts here)
$V_{r m s}$: RMS — heating-equivalent DC

220V (RMS) peaks at ≈311V

Rectification — turning AC into DC

Most circuits need steady DC, but the wall gives swinging AC. So a diode (a one-way part) chops off the − side, leaving only the + humps — the bumpy 'pulsing DC' shown below.
Smooth those humps with a capacitor (filtering) and you finally get DC a circuit can use. Rectification and filtering are all about 'the shape and frequency of AC,' so this page sits underneath them.

After rectifying — only + humps remain

At a glance — from AC to clean DC

Outlet AC → diode rectify (one way only) → pulsing DC → capacitor smoothing → clean DC. This flow happens inside nearly every power adapter.
PWM and switching supplies are likewise 'toggle DC fast to set an average, then filter out the high-frequency parts' — again, a story told on the frequency axis.

AC in±, sine

diode rectifyone way

pulsing DC+ humps

🔋cap smoothingclean DC

what happens inside a power adapter

Pitfalls & gotchas

When you say 'voltage,' always pin down peak ( $V_{p}$ ) vs RMS ( $V_{r m s}$ ) — datasheet ratings are usually peak, outlets and meters usually RMS. They differ by $2 \approx 1.41 \times$ , so confusing them just blows the part. And don't mix frequency $f$ with angular frequency $ω$ — the one that goes into impedance formulas is $ω = 2 π f$ .