How this instrument works
Frequency counts how many complete cycles a repeating signal finishes in one second, measured in hertz. Period is the mirror image of that count: the time a single cycle takes. Because the two describe the same event from opposite directions, one is always the reciprocal of the other — T = 1/f — and neither can be zero without breaking the relationship: zero would demand an infinitely long period, a signal that never finishes even one cycle.
Angular frequency, ω, restates the same rate in radians per second instead of cycles per second. One full cycle sweeps out 2π radians around a circle, so multiplying the cycle rate by that constant converts cycles per second into radians per second: ω = 2πf. This is the number that actually appears inside the sine and cosine of an oscillation, x(t) = A sin(ωt), and inside reactance formulas like X_L = ωL for an inductor — using f in place of ω there is a common slip that throws the answer off by a factor of roughly 6.28.
An audio engineer tuning a synthesizer against concert pitch, an electrician sizing a capacitor bank against 60 Hz mains, and a control engineer writing a transfer function in the Laplace domain each reach for a different one of these three numbers depending on which equation they are feeding. The instrument keeps all three in view at once, so a value read off an oscilloscope in milliseconds and a coefficient needed in radians per second never have to be converted by hand.
- Enter the signal's rate in Hertz in the Frequency field — switch its unit to kHz for radio-range values.
- Read the Period field for the time one full cycle takes, shown in milliseconds by default; switch to seconds for slower signals.
- Read the Angular frequency ω, rad ⁄ s field for the same rate in radians per second, ready to drop into ωt, X_L = ωL, or X_C = 1 ⁄ (ωC).
- Double the Frequency value to check the octave relationship: the Period halves while ω doubles.
Worked example — concert pitch A at 440 Hz
Concert pitch A, the tuning note an orchestra sounds before a performance, is fixed at 440 Hz by international standard. Enter 440 in the Frequency field. The period comes out to T = 1 ⁄ 440 = 0.00227272727273 s, which the instrument displays as roughly 2.2727 ms in its default unit — the time the string or air column takes to complete a single back-and-forth swing.
The angular frequency comes from the second formula: ω = 2π × 440 = 2764.60153516 rad ⁄ s. That is the exact figure a physics or acoustics equation wants when it is written in terms of radians rather than cycles — for instance the string's displacement over time, x(t) = A sin(2764.60t). All three numbers, 440 Hz, 2.2727 ms, and 2764.60 rad ⁄ s, describe one and the same vibration, just measured along different axes.
Questions
What is the difference between frequency and angular frequency?
The first counts cycles per second, in hertz; the second counts radians per second, for the exact same repeating motion. Hertz suits a tuner, a spec sheet, or a mains rating; radians per second suits anything written as a sine, cosine, or complex exponential, since one full cycle equals 2π radians.
Why does the period formula break down at zero?
T = 1 ⁄ f divides by f, and dividing by zero is undefined. Physically that matches reality: a signal that never completes a single cycle has no finite time to call its period. The Frequency field is restricted to positive values for exactly this reason.
Why is there a factor of 2π between f and ω?
Because one cycle is a trip once around a circle, and a full trip around a circle measures 2π radians. Hertz counts trips per second; multiplying by 2π converts that count into radians swept per second, which is what ω means. Skip the 2π and the two quantities are treated as interchangeable, which they are not.
Can I recover a plain Hz value from ω instead of the other way around?
Yes: divide ω by 2π, f = ω ⁄ (2π). This instrument takes f as the input, so to go the other direction, divide a known ω by 2π yourself before entering the Hz value, or read the relationship straight off the formula box.
Does this work for AC mains power, like 50 or 60 Hz?
Yes — mains power is a sinusoidal oscillation like any other, so the same identities hold. At 60 Hz the period is about 16.7 ms per cycle and ω is about 377 rad ⁄ s, the exact coefficient that appears in reactance formulas such as X_L = ωL and X_C = 1 ⁄ (ωC) for circuit components on that supply.
What is the most common mistake people make with ω?
Plugging a plain Hz value into a formula that actually wants ω in radians per second — for example computing an inductor's reactance as fL instead of 2πfL. Because 2π is close to 6.28, the mistaken answer is off by roughly that factor, large enough to be caught immediately by a sanity check against a known circuit.