How do you convert frequency to period?
Divide 1 by the frequency in hertz: T = 1 ÷ f. A 1 kHz signal has a period of 1 ms and a 16 MHz clock has a period of 62.5 ns.
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Frequency to period calculator
Convert frequency to period or period to frequency, from Hz to seconds, ms, µs or ns, with the angular frequency in radians per second. Enter a duty cycle to get the on and off time, or start from the on and off time to get the frequency.
Start from
Examples
Hz
%
On time as a percentage of the period.
Period
1 ms
- Angular frequency ω
- 6.28319 krad/s
- On time
- 500 µs 50 % duty
- Off time
- 500 µs
- Half period
- 500 µs
- Cycles per minute
- 60000
Frequency in Hz and angular frequency in rad/s differ by 2π. Mixing them up is a factor of 6.28 error, so check which one a formula asks for.
How do you convert frequency to period?
Period is the reciprocal of frequency: T = 1 ÷ f. A 1 kHz signal repeats every 1 ms, a 16 MHz clock every 62.5 ns and 60 Hz mains every 16.667 ms. Going the other way, f = 1 ÷ T, and angular frequency is ω = 2πf.
Frequency to period formula, angular frequency and duty cycle
Period and frequency are reciprocals, so the tool above also works as a period to frequency calculator: set Start from to Period. Angular frequency measures the same repetition in radians per second, and duty cycle divides the period between its on and off parts.
T = 1 ÷ f f = 1 ÷ T
- f
- frequency in Hz
- T
- period in s
- ω
- angular frequency in rad/s
- D
- duty cycle as a fraction (25 % = 0.25)
ω = 2π × f f = ω ÷ 2π
t_on = D × T t_off = (1 − D) × T D = t_on ÷ (t_on + t_off)
How to convert frequency to period and back
Example 1: 60 Hz mains
T = 1 ÷ 60 = 16.667 ms. The angular frequency is ω = 2π × 60 = 376.99 rad/s, the figure that appears in capacitor and inductor reactance at 60 Hz.
Example 2: 20 kHz PWM at 25 % duty
T = 1 ÷ 20,000 = 50 µs. The on time is 0.25 × 50 µs = 12.5 µs and the off time is 0.75 × 50 µs = 37.5 µs.
Example 3: a servo pulse
A pulse of 1.5 ms followed by 18.5 ms off has a period of 20 ms, so f = 50 Hz and the duty cycle is 7.5 %.
Hz to seconds table
Period, half period and angular frequency for frequencies that come up often in electronics.
| Frequency | Common use | Period | Half period | Angular frequency |
|---|---|---|---|---|
| 1 Hz | Seconds tick | 1 s | 500 ms | 6.2832 rad/s |
| 50 Hz | Mains in much of the world | 20 ms | 10 ms | 314.16 rad/s |
| 60 Hz | Mains in North America | 16.667 ms | 8.3333 ms | 376.99 rad/s |
| 400 Hz | Aircraft power | 2.5 ms | 1.25 ms | 2.5133 krad/s |
| 1 kHz | Audio test tone | 1 ms | 500 µs | 6.2832 krad/s |
| 32.768 kHz | RTC crystal | 30.518 µs | 15.259 µs | 205.89 krad/s |
| 100 kHz | I²C standard mode | 10 µs | 5 µs | 628.32 krad/s |
| 400 kHz | I²C fast mode | 2.5 µs | 1.25 µs | 2.5133 Mrad/s |
| 1 MHz | 1 µs | 500 ns | 6.2832 Mrad/s | |
| 8 MHz | MCU crystal | 125 ns | 62.5 ns | 50.265 Mrad/s |
| 12 MHz | USB and MCU crystal | 83.333 ns | 41.667 ns | 75.398 Mrad/s |
| 16 MHz | MCU clock | 62.5 ns | 31.25 ns | 100.53 Mrad/s |
| 24 MHz | MCU crystal | 41.667 ns | 20.833 ns | 150.8 Mrad/s |
| 25 MHz | Ethernet PHY clock | 40 ns | 20 ns | 157.08 Mrad/s |
| 48 MHz | USB clock | 20.833 ns | 10.417 ns | 301.59 Mrad/s |
| 100 MHz | 10 ns | 5 ns | 628.32 Mrad/s | |
| 1 GHz | 1 ns | 500 ps | 6.2832 Grad/s | |
| 2.4 GHz | Wi-Fi and Bluetooth band | 416.67 ps | 208.33 ps | 15.08 Grad/s |
Period to frequency table
The reverse lookup, handy when reading a period off an oscilloscope.
| Period | Frequency | Angular frequency |
|---|---|---|
| 1 s | 1 Hz | 6.2832 rad/s |
| 100 ms | 10 Hz | 62.832 rad/s |
| 20 ms | 50 Hz | 314.16 rad/s |
| 1 ms | 1 kHz | 6.2832 krad/s |
| 100 µs | 10 kHz | 62.832 krad/s |
| 10 µs | 100 kHz | 628.32 krad/s |
| 1 µs | 1 MHz | 6.2832 Mrad/s |
| 100 ns | 10 MHz | 62.832 Mrad/s |
| 10 ns | 100 MHz | 628.32 Mrad/s |
| 1 ns | 1 GHz | 6.2832 Grad/s |
Using these numbers
- Check whether a formula wants f or ω. Reactance, resonance and filter equations are often written with ω, and the SI Brochure warns that confusing the two is a factor of 2π error.
- A 50 % duty square wave has edges every half period, so the toggle rate of a GPIO is twice the output frequency.
- The period of a real clock differs from nominal by its frequency error: a 20 ppm clock has a period 20 ppm off, which matters for long counts but not for one cycle.
- On an oscilloscope, measure across several cycles and divide, which reduces the error from reading the edges.
Glossary
- Frequency (f)
- The number of cycles of a periodic signal per second, in hertz (Hz).
- Period (T)
- The time for one complete cycle, the reciprocal of the frequency.
- Angular frequency (ω)
- The rate of change of phase, in radians per second: 2π times the frequency.
- Duty cycle
- The on time of a pulse waveform as a percentage of its period.
- Half period
- The time between edges of a 50 % duty square wave, T ÷ 2.
Related guides
- Frequency to wavelength calculator: the same frequency as a distance.
Frequently asked questions
How do you convert Hz to ms or seconds?
T = 1 ÷ f gives the period in seconds, so 1 Hz is 1 s, 50 Hz is 20 ms and 1 kHz is 1 ms. The same division handles MHz to ns: 16 MHz is 62.5 ns.
How do you calculate frequency from period?
Divide 1 by the period in seconds: f = 1 ÷ T. A period of 20 ms is 50 Hz, and a period of 10 ns is 100 MHz.
What is the period of 60 Hz?
16.667 ms, and for 50 Hz it is 20 ms.
What is the angular frequency formula?
Angular frequency ω is the rate of change of phase in radians per second, ω = 2πf. 50 Hz is 314.16 rad/s. The numbers differ from hertz by 2π, so a formula written with ω gives a wrong answer if you enter f.
How do you convert rad/s to Hz?
Divide by 2π: f = ω ÷ 2π. 1,000 rad/s is 159.15 Hz.
How do I find on time and off time from duty cycle?
On time = T × D and off time = T × (1 − D), with D as a fraction. At 20 kHz and 25 % duty, the period is 50 µs, on for 12.5 µs and off for 37.5 µs.
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