Hertz (Hz)
Definition, derivation and practical use in the International System of Units
Frequency states how many times a periodic process repeats per unit of time. The quantity runs through acoustics, radio engineering, power systems and computing, and its SI unit, the hertz (Hz), is one of the derived SI units. The unit is named after the German physicist Heinrich Hertz (1857-1894), who demonstrated electromagnetic waves experimentally.
The hertz is also the unit that can be realized with the lowest measurement uncertainty of any in the SI, because it rests directly on the definition of the second. To see why, it is necessary, as for the meter and the kilogram, to distinguish between definition, realization and practical use.
The definition of the hertz in the SI system
Within the International System of Units (SI), the hertz is defined as:
the frequency of a periodic phenomenon that completes one full cycle per second.
In formula form: 1 Hz = 1 s⁻¹. The hertz is thus the only one of the common derived units expressed in a single base unit. The definition rests on:
- the second, defined by giving the hyperfine transition frequency of the caesium-133 atom the exact value 9 192 631 770 Hz,
- the concept of the period T, where the frequency f = 1/T counts the number of periods per second.
The hertz is used only for periodic phenomena. Angular frequency ω = 2πf is stated in radians per second, and radioactive decay, which also has the dimension s⁻¹, is measured in becquerels (Bq). Giving these quantities different units is a deliberate choice in the SI to keep them apart.
Fact box: the history of the hertz
| Year | Event |
| 1886-1888 | Heinrich Hertz experimentally demonstrates the electromagnetic waves predicted by Maxwell. |
| 1930 | The International Electrotechnical Commission (IEC) adopts the name hertz for the unit of frequency. |
| 1960 | The 11th General Conference on Weights and Measures (CGPM) brings the hertz into the SI, replacing “cycles per second”. |
| 1967 | The second is redefined through caesium-133, giving the hertz an atomic foundation. |
| 2019 | The SI revision keeps the caesium frequency as one of the seven defining constants. |
How much is 1 hertz?
One hertz is one cycle per second, a slow pace in most technical contexts. A resting heart beats at about 1 Hz. The mains supply in Sweden and the rest of Europe alternates at 50 Hz, in North America at 60 Hz. Human hearing spans roughly 20 Hz to 20 kHz, and concert pitch A4 has a frequency of 440 Hz.
Higher up the scale, FM radio sits around 100 MHz, Wi-Fi at 2.4 and 5 GHz and processor clocks at several GHz. Visible light oscillates at about 4 to 8 × 10¹⁴ Hz. The relation f = 1/T means that a period of 20 ms corresponds to 50 Hz, which is easy to work out with the time converter.
Definition, realization and use
- The definition states what a hertz is in principle: one cycle per second.
- The realization refers to the atomic clocks and oscillators with which national metrology institutes produce a known frequency.
- The use refers to frequency measurement in telecommunications, power systems, music and electronics development.
For the hertz, definition and realization lie unusually close together, because the second itself is defined as a frequency.
How the hertz is realized
Frequency is the quantity that can be realized with the lowest relative uncertainty in the whole SI. The primary standard is the caesium fountain, an atomic clock that locks a microwave oscillator to the transition at 9 192 631 770 Hz with a relative uncertainty of about 10⁻¹⁶. Hydrogen masers serve as stable secondary references, and International Atomic Time (TAI) is built from a weighted combination of hundreds of such clocks.
Optical atomic clocks, based on transitions in strontium or ytterbium for example, reach relative uncertainties around 10⁻¹⁸. With frequency combs, for which the Nobel Prize in Physics was awarded in 2005, optical frequencies can be counted down to the microwave range and compared directly with the caesium definition. In this way even the meter, defined through the speed of light, is traced to a frequency.
Practical measurement and sources of error
In practice, frequency is measured with frequency counters, which count the number of cycles during a gate time set by an internal reference oscillator. The accuracy of the measurement is limited by several factors:
- The error of the reference oscillator: a quartz oscillator drifts with temperature and age, so demanding measurements use GPS-disciplined or rubidium-based references.
- Stability over time: short-term and long-term variations are described with the Allan deviation, not with a simple standard deviation.
- Doppler shift and aliasing: motion between source and receiver shifts the frequency, and too low a sampling rate makes high frequencies appear falsely as low ones.
The hertz and other units
The hertz links time to several other quantities in the SI:
| Unit | Quantity | Relation to the hertz |
| second (s) | Time | f = 1 / T |
| becquerel (Bq) | Activity | Also s⁻¹, but reserved for radioactive decay |
| radian per second (rad/s) | Angular frequency | ω = 2π · f |
| joule (J) | Energy | Photon energy E = h · f |
| watt (W) | Power | Power = energy per cycle · f |
Multiples and common conversions
The hertz takes the usual SI prefixes: 1 kHz = 10³ Hz, 1 MHz = 10⁶ Hz, 1 GHz = 10⁹ Hz and 1 THz = 10¹² Hz. Each step is a factor of 1,000, so 2.4 GHz = 2,400 MHz = 2,400,000 kHz.
Conversion between frequency and period is done by inversion: 1 kHz corresponds to a period of 1 ms and 1 MHz to a period of 1 µs. The period can in turn be converted between time units with the time converter, and the power a periodic signal carries with the power converter. For rotating machinery the relationship between rpm and hertz is a division by 60, and the frequency converter handles the prefix steps.
Summary
The hertz is the SI unit of frequency and is defined as one cycle per second. The unit is derived directly from the second and is realized with atomic clocks whose relative uncertainty is lower than for any other quantity in the SI.
From the 50 Hz of the mains to the hundreds of terahertz of light, the unit spans more than twelve orders of magnitude. It is the combination of a simple definition, an exceptionally accurate realization and well-understood sources of error that makes the hertz one of the most reliable units in modern measurement.