Why do we say 230 V if the voltage peaks at 325 V?

230 V is the rms value: a 230 V direct voltage would heat a resistor exactly as much as this sine wave. It is the only useful value for power calculations, and the one voltmeters display.

What does 50 Hz mean?

That the voltage completes 50 full cycles per second: one period lasts 20 ms, the voltage crosses zero 100 times a second. In North America the mains is 60 Hz.

Reminder sheet

Alternating current

Bastaya

In AC everything oscillates: the voltage changes sign fifty times a second. The RMS value is the one that heats as much as a DC of the same value.

A sinusoidal quantity is written u(t) = Umax·sin(ωt + φ). Its period T (s) and frequency f = 1/T (Hz) relate to the angular frequency ω = 2πf (rad/s). The European mains is 50 Hz (T = 20 ms).

The RMS value Urms = Umax/√2 is the one producing the same heating in DC: AC voltmeters read it (230 V), while the oscilloscope shows Umax (325 V). Two sine waves of the same frequency are phase-shifted by φ; they are drawn as phasors.

Formulas

Sinusoidal quantityu(t) = Umax · sin(ω · t + φ)

Quantity	Symbol	Unit
Peak value	Umax	V
Angular frequency	ω	rad/s
Initial phase	φ	rad

Angular frequency, frequency, periodω = 2 · π · f
f = 1T

Quantity	Symbol	Unit
Frequency	f	Hz
Period	T	s

RMS valueUeff = Umax√2

Quantity	Symbol	Unit
RMS value	Ueff	V
Peak value	Umax	V

Worked example

The mains supplies 230 V rms at 50 Hz. What are the peak voltage, the period and the angular frequency? Write u(t) and compute its value at t = 5 ms.

Umax = Urms · √2 = 230 × 1.414 = 325 V.

T = 1 / f = 1 / 50 = 20 ms; ω = 2π · f = 314 rad/s.

u(t) = 325 · sin(314 · t). At t = 5 ms (a quarter period): 314 × 0.005 = 1.57 rad = π/2, so u = 325 × 1 = 325 V, the peak.

Common mistakes

Confusing Umax and Urms: voltmeters read the rms value, the oscilloscope shows the peak value, √2 times larger.
Calculator in degrees: ω·t is in radians (314 × 0.005 = 1.57 rad, not 1.57°).
Mixing f (Hz) and ω (rad/s): ω = 2π·f, i.e. 314 rad/s for 50 Hz.

IEC symbols

AC source

Illustration

Sine wave: period T, peak value Umax, RMS value Urms = Umax/√2.

Frequently asked questions

Why do we say 230 V if the voltage peaks at 325 V?: 230 V is the rms value: a 230 V direct voltage would heat a resistor exactly as much as this sine wave. It is the only useful value for power calculations, and the one voltmeters display.
What does 50 Hz mean?: That the voltage completes 50 full cycles per second: one period lasts 20 ms, the voltage crosses zero 100 times a second. In North America the mains is 60 Hz.

15 exercises in this topic

RMS value Know · level 1
Frequency and period Know · level 1
Angular frequency Calculate · level 2
From RMS to peak Calculate · level 2
Instantaneous value Calculate · level 4
Phase shift Know · level 1
From period to frequency Calculate · level 1
Angular frequency of a sinusoidal current Calculate · level 2
Phase at a given instant Calculate · level 3
Instantaneous value of a current Calculate · level 3
RMS value read on the oscilloscope Calculate · level 2
Peak current from the energy Calculate · level 3
RMS EMF of an alternator Calculate · level 2
Kettle: amplitude on the oscilloscope Calculate · level 4
Power from the current amplitude Calculate · level 3

Practise this topic

Simulated time00:00:00

Q = I·t-

W = P·t-

Alternating current

Formulas

Worked example

Common mistakes

IEC symbols

Illustration

Frequently asked questions

15 exercises in this topic

Workbench

Circuit diagram CEI 60617

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