How this instrument works
Cycles per second, abbreviated cps, was the name used for frequency for most of the 20th century — an AC mains supply, a radio dial, or a vibrating string was described in cps well into the 1950s. In 1960, the 11th General Conference on Weights and Measures (CGPM), the international body that governs the SI system of units, adopted 'hertz' as the official name for the SI unit of frequency, honoring physicist Heinrich Hertz. This was a rename, not a redefinition: hertz was set equal to cycles per second exactly, so converting cps to Hz needs no multiplier at all — 1 cps is 1 Hz, always, by definition.
Revolutions per minute takes that same rate and reframes it for anything that spins rather than merely oscillates. Since a minute holds 60 seconds, multiplying a cps figure by 60 gives the number of full turns in a minute — the figure a motor nameplate, a turntable, or a fan speed is usually quoted in. It's also the number behind a genuinely useful physical fact: a two-pole synchronous motor running directly off a 60 Hz AC supply spins at exactly 120 × 60 ÷ 2 = 3,600 rpm, which is why 3,600 rpm shows up so often as the top speed on US-market synchronous and induction motors built for 60 Hz mains.
Radians per second, sometimes called angular frequency, restates the identical rate a third way — in terms of the angle swept around a circle rather than the count of full turns. One complete cycle sweeps exactly 2π radians, so multiplying cps by 2π converts a cycle count into the angular rate that appears directly inside equations like a sine wave's argument, ωt, or an AC circuit's reactance formulas. All three outputs on this page — Hz, rpm, and rad/s — describe one and the same oscillation; only the units of what's being counted change.
- Enter a rate into the Cycles per second (cps) field — it opens at 60, the historic name for US AC mains frequency.
- Read Hertz (Hz) directly below it — this will always equal the cps figure exactly, since the two are the same unit under different names.
- Read Revolutions per minute (rpm) for the same rate expressed as full turns per minute, useful for motors, fans, and turntables.
- Read Radians per second (rad/s) for the angular-frequency form used in oscillation and AC circuit equations.
- Negative cps values are rejected — a cycle rate is a magnitude, so the field only accepts zero or positive numbers.
Worked example — 60 cps, historic name for US mains frequency
US AC mains power was described as '60 cps' for most of the 20th century, before the 1960 rename to hertz took hold in common usage. Enter 60 into Cycles per second (cps): Hertz (Hz) reads exactly 60, since the rename introduced no conversion factor at all; Revolutions per minute (rpm) reads 3,600, matching the synchronous speed of a standard two-pole motor built to run on that supply (120 × 60 ÷ 2 = 3,600); and Radians per second (rad/s) reads 376.991118, the same rate expressed as 60 × 2π.
A 1 cps rate — one full cycle every second, the pace of a clock's ticking second hand — converts to 1 Hz, 60 rpm, and about 6.283185 rad/s (2π). Note that a phonograph nicknamed a '45' spins at 45 revolutions per minute, which is only 0.75 cycles per second, not 45 cps — entering 45 into this tool (as shown in the reference vectors) illustrates the three-way conversion at a round number, but it is not describing an actual 45 rpm record's true rotational rate.
Questions
Is cps the same thing as Hz, or does it need a conversion factor?
They are the same unit — no conversion factor beyond 1 is needed. The 11th General Conference on Weights and Measures (CGPM) renamed the SI frequency unit 'hertz' in 1960, replacing the older name 'cycles per second' for the identical quantity. A device or textbook rated in cps and one rated in Hz are describing the same rate; only the label changed.
Why did the name change from cps to hertz in 1960?
The CGPM, the international body responsible for the SI system, adopted 'hertz' to honor physicist Heinrich Hertz, whose 1880s experiments confirmed the existence of electromagnetic waves. It was a naming decision for an existing unit, not a redefinition — one cps had always equaled what one hertz equals today.
How do you convert cycles per second to RPM?
Multiply by 60, since a minute contains 60 seconds — a rate of 60 cps becomes 3,600 rpm, and a rate of 1 cps becomes 60 rpm. This conversion matters for anything that spins: a two-pole synchronous motor on 60 Hz mains, for instance, runs at exactly 3,600 rpm because of this same relationship.
Why does a 60 Hz motor's speed come out to 3,600 rpm?
Because a two-pole synchronous motor completes one full mechanical revolution per electrical cycle, and 60 cycles per second times 60 seconds per minute gives 3,600 revolutions per minute. Motors with more pole pairs turn proportionally slower for the same supply frequency, but the 120 × frequency ÷ poles relationship always reduces to this same cps-to-rpm conversion at its core.
Is a '45 rpm' record actually 45 cycles per second?
No — a 45 rpm record's name refers to its rotational speed, 45 revolutions per minute, which works out to 0.75 cycles per second (45 ÷ 60), not 45 cps. Entering 45 into the Cycles per second field on this page shows how 45 cps itself would convert (2,700 rpm, 45 Hz), which is a useful illustration of the conversion but not a description of how fast a physical '45' record actually spins.
What is radians per second, and why does this converter show it?
Radians per second, or angular frequency (ω), restates a cycle rate in terms of the angle swept per second rather than the number of full turns — one cycle equals 2π radians, so ω = cps × 2π. It's the form that appears directly in oscillation equations like x(t) = A sin(ωt) and in AC circuit reactance formulas, so having it alongside Hz and rpm saves converting by hand.