AC frequency from period:
formula and worked example

Two ways of describing the same repetition

An alternating waveform repeats itself over and over. You can describe that repetition by asking how long one repeat takes, which is the period, or by asking how many repeats fit into one second, which is the frequency. They carry the same information, and converting between them is one of the most common small calculations in electronics and power work.

Which one you meet depends on the instrument. A frequency counter or a mains specification gives hertz directly. An oscilloscope shows a waveform stretched across a time axis, so what you read off the screen is the time between peaks, the period, and you must convert it yourself.

• Period s: the time for one complete cycle, in seconds.
• Result: frequency in hertz, meaning cycles per second.
• Related: angular frequency in radians per second is 2π times the frequency.

The reciprocal relationship

Because the two are reciprocals, the arithmetic works in both directions: the period is 1 divided by the frequency. A fast signal has a short period, and a slow one has a long period. Only the units need care, and the formula wants seconds, not milliseconds.

Prefixes make the work easier. A period in milliseconds gives a frequency in kilohertz directly, since 1 ÷ 1 ms = 1 kHz, and a period in microseconds gives megahertz.

Worked example: a 0.02 s period

An oscilloscope trace shows one full cycle of a mains waveform spanning four divisions on a 5 ms-per-division setting. What frequency is that?

1. 1Convert the screen reading to time: 4 divisions × 5 ms = 20 ms.
2. 2Convert to seconds: 20 ÷ 1,000 = 0.02 s.
3. 3Take the reciprocal: 1 ÷ 0.02 = 50 Hz.

Fifty hertz means fifty complete cycles every second. Each cycle has two half-cycles, so the voltage reverses direction 100 times a second.

Reading the result against familiar signals

Having a few reference points in mind helps you notice when a reading has gone wrong. Mains supply is conventionally 50 Hz in some regions and 60 Hz in others, which correspond to periods of 20 ms and about 16.7 ms.

• 50 Hz: 20 ms period, with each half-cycle lasting 10 ms.
• 60 Hz: about 16.67 ms period.
• 1 kHz: 1 ms period, the common test tone in audio work.
• 1 MHz: 1 µs period, typical of the clock scale of simple digital logic.

If the number you compute is very far from what the system should be doing, suspect the unit before suspecting the equipment. A period read as 20 instead of 0.02 gives 0.05 Hz, which no mains system produces.

Why the change is not symmetric

Lengthening the period by 10%, from 0.02 s to 0.022 s, does not reduce the frequency by 10%. It falls from 50 Hz to 45.455 Hz, a drop of about 9.1%. Going the other way, a 10% shorter period raises the frequency by about 11.1%.

The asymmetry is small for small changes but grows quickly. A period that doubles halves the frequency, and a period cut to a tenth multiplies the frequency by ten. When tracking drift in a clock or a generator, work in whichever quantity you measure and convert at the end.

Slips when converting between them

• Leaving the period in milliseconds, so 20 ms goes into the formula as 20 and gives 0.05 Hz.
• Measuring across several cycles and not dividing: 10 cycles in 0.2 s is a period of 0.02 s, not 0.2 s.
• Measuring peak to trough, which is half a cycle, instead of peak to peak, giving a frequency twice as high.
• Assuming the waveform is steady: on a noisy or jittery signal, a single-cycle reading can differ from the average.

A scenario: checking a generator output

A portable generator is supposed to supply 50 Hz. A scope shows nine complete cycles spanning 186 ms. The period is 186 ÷ 9 = 20.67 ms, so the frequency is 1 ÷ 0.02067 = 48.4 Hz, about 3% low.

Frequency in a generator tracks engine speed, so a low reading usually points to a governor or load issue, not to a faulty meter. Some motors and clocks that rely on the supply frequency will run slow at that figure, which is why the check is worth making.