Electronics
AC voltage and RMS explained
AC (alternating current) is a voltage that swings one way, then the other, instead of holding steady like a battery's DC. The smoothest swing is a sine wave, described by its peak, its frequency and its RMS value. For a sine, RMS is the peak ÷ √2, about 0.707 of it: the steady voltage that would heat a resistor just as much.
Four numbers for a sine
- Peak (Vpeak): the highest voltage, measured from zero
- Peak-to-peak (Vpp): top to bottom, 2 × Vpeak
- Period (T): the time for one cycle, in seconds
- Frequency (f): cycles per second, in hertz (Hz)
Mains in Europe runs at 50 Hz, so one cycle takes 1 ÷ 50 = 0.02 s, or 20 ms. North America's 60 Hz gives about 16.7 ms. Frequency and size are separate: a faster wave isn't a bigger one.
Live experiment
Spin the wheel
Pip rides a point on a spinning wheel, and his height traces a sine wave, shown 50 times slower than life. Change how fast (25, 50 or 100 Hz) and how big (5, 10 or 15 V peak), read the period, peak-to-peak and RMS, and switch on the RMS line.
“Sine waves” is lesson 1 of 6 in Oscillation Coast, Electronics World 4. Free with an account. Lessons unlock in order, so you’ll do the 21 before it first.
RMS: the heating value
A sine spends half its time negative, so its average is zero. Yet it still boils a kettle. So AC is described by its RMS (root mean square) value. For a sine wave:
The √2 comes from power going with voltage squared: averaged over a cycle, sin² is exactly ½.
Mains is quoted in RMS. 230 V RMS peaks at 230 × √2 ≈ 325 V, and 120 V RMS peaks at about 170 V. A multimeter's AC range shows RMS too.
Why RMS makes power easy
Put RMS values into the DC power formulas and they just work:
A 2 kW kettle on 230 V RMS has an element of 230² ÷ 2000 ≈ 26 Ω. Using the peak instead would double the answer for power, which is wrong.
The √2 rule is for sine waves only. A square wave that flips between +5 V and −5 V has an RMS of 5 V, its peak. Cheap meters assume a sine; "true RMS" meters work for any shape.
Free lesson
Now build it yourself
Each lesson comes in three short parts: see the idea, try it on a live board, then build it yourself, with quick checks as you go.
“Sine waves” is lesson 1 of 6 in Oscillation Coast, Electronics World 4. Free with an account. Lessons unlock in order, so you’ll do the 21 before it first.
Or start from the very first lesson, no account neededWork it out
Practice questions
Three quick questions from the lessons. Pick an answer to check it.
Show the answer
Answer: the steady voltage that heats a resistor the same. A sine's average over a cycle is zero, since it spends half its time negative. RMS is the steady DC voltage that would heat a resistor just as much, about 0.707 of the peak.
- Why not “its highest value”? That's the peak. For a sine the RMS is lower: the peak ÷ √2, about 0.707 of it.
- Why not “its average over a cycle”? A sine spends half of each cycle negative, so its plain average over a cycle is zero. RMS squares the voltage first, so both halves count.
Show the answer
Answer: 50 Hz. f = 1 over T = 1 over 0.02 s = 50 Hz. Forgetting to turn 20 ms into seconds gives 0.05 Hz, and 20 Hz is the period's number, not a frequency.
- Why not “0.05 Hz”? That's 1 ÷ 20, with the period left in milliseconds. Turn 20 ms into 0.02 s first, then 1 ÷ 0.02 = 50 Hz.
- Why not “20 Hz”? 20 is the period's number, in milliseconds. Frequency is cycles per second, 1 ÷ T, not the period itself.
Show the answer
Answer: an ideal capacitor. An ideal capacitor stores energy for a quarter of a cycle and hands it all back the next, so it limits the current without getting warm. A resistor or a lamp turns its power into heat and light.
- Why not “a resistor”? A resistor turns electrical energy into heat in both halves of every cycle, so it always takes average power.
- Why not “a lamp”? A lamp's filament is a resistor that gets hot enough to glow, so it takes power from the source in both halves of every cycle.
“Sine waves” is lesson 1 of 6 in Oscillation Coast, Electronics World 4. Free with an account. Lessons unlock in order, so you’ll do the 21 before it first.
Questions people ask
What does RMS mean?
Root mean square. It's the steady DC voltage that would heat a resistor just as much as the AC does. Square the voltage, average it over a cycle, and take the square root. Mains and multimeter AC ranges are quoted in RMS.
How do you convert peak voltage to RMS?
For a sine wave, divide the peak by √2, or multiply by about 0.707. To go back, multiply the RMS by √2. This only holds for sines: a square wave's RMS equals its peak.
What is the peak voltage of 230 V and 120 V mains?
230 V RMS peaks at 230 × √2 ≈ 325 V, twice every cycle, once each way. 120 V RMS peaks at about 170 V.
What's the difference between frequency and period?
The period, T, is the time one cycle takes. The frequency, f, is cycles per second, in hertz. They're reciprocals: f = 1 ÷ T. At 50 Hz each cycle takes 20 ms; at 60 Hz, about 16.7 ms.
Related topics
- Electrical power explained: P = V × IP = V × I: how fast energy turns into heat, and why a resistor has a power rating.
- Low-pass filter explainedA resistor and a capacitor that pass the bass and fade the treble, and where the cutoff sits.
- Decibels explaineddB as a ratio on a log scale: 10 log for power, 20 log for voltage, and why gains just add.