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Electrical Circuits · AC Circuits

AC waveforms

Reading32mLesson 1 of 3 in this module

Describe a sine wave with peak, peak-to-peak, RMS, frequency and period, and convert between them.

Why it matters

The power from a wall outlet, a generator or a utility transformer is alternating current (AC): the voltage swings positive and negative many times a second. Batteries, solar panels and the inside of most electronics use direct current (DC), which stays in one direction. To read a nameplate, choose a meter setting or compare a 120 V supply with a 170 V peak on an oscilloscope, you need the vocabulary of AC waveforms: peak, peak-to-peak, RMS, frequency and period.

DC and AC

  • DC has a steady polarity. A 9 V battery's + terminal is always about 9 V above its − terminal.
  • AC reverses polarity in a repeating pattern. The usual shape is a sine wave, because that is what a rotating generator naturally produces.

AC won for power distribution mainly because transformers can step AC voltage up for efficient long-distance transmission and back down for safe use. You will see why in the transformers lesson. DC is making a comeback in solar, batteries, data centres and long high-voltage links, but buildings are still wired for AC.

Describing a sine wave

A sine-wave voltage can be written as:

v(t) = Vpeak × sin(2π × f × t)
  • Peak (Vpeak): the highest value above zero.
  • Peak-to-peak (Vpp): from the top of the wave to the bottom, Vpp = 2 × Vpeak.
  • Frequency (f, hertz): complete cycles per second. North American power is 60 Hz; most of the rest of the world uses 50 Hz.
  • Period (T, seconds): time for one cycle, T = 1 / f. At 60 Hz, T = 1/60 s ≈ 16.7 ms. At 50 Hz, T = 20 ms.

RMS: the "effective" value

If you connect a heater to AC, the voltage spends most of its time below the peak, so it does not heat as much as a DC supply at the peak voltage would. The RMS (root mean square) value is the DC voltage that would produce the same heating in a resistor.

For a sine wave:

Vrms = Vpeak / √2 ≈ 0.707 × Vpeak
Vpeak = Vrms × √2 ≈ 1.414 × Vrms

The same ratio applies to current. When someone says "120 V AC" or "230 V AC", they mean the RMS value. Power formulas like P = Vrms × Irms (for a resistor) work with RMS values exactly as they do for DC.

The √2 ratio is only for sine waves. A square wave's RMS equals its peak; a triangle wave's RMS is Vpeak / √3. Many cheap meters assume a sine wave; true-RMS meters measure correctly on distorted waveforms, which are common around variable-speed drives and electronic loads.

Worked examples

Example 1. A nameplate says 120 V, 60 Hz. What are the peak, peak-to-peak and period?

Vpeak = 120 × 1.414 = 169.7 V
Vpp   = 2 × 169.7  = 339.4 V
T     = 1 / 60     = 0.0167 s = 16.7 ms

Example 2. An oscilloscope (in a training lab or a simulator) shows a sine wave with a peak of 10 V and one cycle every 2 ms.

f    = 1 / 0.002 = 500 Hz
Vrms = 10 / 1.414 = 7.07 V

Example 3. A 230 V RMS heater has a resistance of 26.45 Ω. What power does it use?

P = Vrms² / R = 230² / 26.45 = 2,000 W

Reading the waveform

  • Phase describes where a wave is in its cycle, in degrees (360° per cycle). Two waves at the same frequency can be shifted; for example, current lags voltage in a motor. Phase shift becomes important in the next two lessons.
  • Angular frequency ω = 2πf (radians per second) appears in reactance formulas. At 60 Hz, ω ≈ 377 rad/s.

Common mistakes

  • Treating 120 V as the peak. It is the RMS value; the peak is about 170 V. Insulation and component voltage ratings must cover the peak.
  • Using the √2 ratio on non-sine waves. Use a true-RMS meter or the right formula for the waveform.
  • Mixing up frequency and period. Hertz is cycles per second; period is seconds per cycle.
  • Confusing peak with peak-to-peak. An oscilloscope often shows Vpp; halve it to get the peak.

Safety note

Mains AC is dangerous at any of these values and can cause a fatal shock or an arc flash. In this course you look at AC waveforms only in simulators or videos. Measuring live mains circuits is done only after employer or school training, with rated meters and protective equipment, under supervision. See NFPA 70E (https://www.nfpa.org/codes-and-standards/nfpa-70e-standard-development/70e) and OSHA's electrical safety pages (https://www.osha.gov/electrical). Checked on 2026-10-23.

Recap

  • DC holds one polarity; AC reverses, usually as a sine wave.
  • T = 1 / f; North America uses 60 Hz, most other places 50 Hz.
  • Vrms = Vpeak / √2 for a sine wave; quoted mains voltages are RMS.

Try it

Open the Do tab and write the RMS formula in a small AC waveform calculator.

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