Frequency and period

Period is the time a repeating signal takes to complete one full cycle; frequency is how many cycles it completes per second. They are reciprocals of each other and together describe how fast a sinusoid oscillates — one of the three numbers (with amplitude and phase) that fully specify it.

The three related quantities

For a sinusoid $v(t)=A\sin (\omega t+\phi )$:

• Period $T$ — the time in seconds for the waveform to repeat. After $T$ seconds the waveform looks exactly as it did before: $v(t+T)=v(t)$.
• Ordinary frequency $f$ — cycles per second, measured in hertz (Hz). One hertz is one cycle per second. Frequency is the reciprocal of the period:

$f=\frac{1}{T}$

A signal that completes a cycle every $1\text{ms}$ has $f=1/(10^{-3}\text{s})=1000\text{Hz}=1\text{kHz}$.

• Angular frequency $\omega$ — radians per second. One full cycle of a sine is $2\pi$ radians of its argument, so to advance the argument by $2\pi$ in one period $T$ the rate must be

$\omega =\frac{2\pi }{T}=2\pi f$

with units of rad/s. The factor of $2\pi$ is just the conversion between “cycles” and “radians”: $\omega$ counts how fast the angle inside the sine sweeps, while $f$ counts how many complete loops that makes per second.

These three are different views of the same thing. Given any one of $T$, $f$, $\omega$ you can get the other two: $T=1/f=2\pi /\omega$.

Why both $f$ and $\omega$ exist

$f$ in hertz is the intuitive, measurable quantity — it is what an instrument reads and what a spec sheet quotes. $\omega$ in rad/s is the natural variable for the math: it is what appears directly inside $\sin (\omega t)$ and in circuit expressions like the impedance of a capacitor, [[Capacitive reactance|$Z_C=1/(j\omega C)$]], or the cutoff of an RC filter, ${\omega }_c=1/(RC)$. You convert with $\omega =2\pi f$ whenever you cross between the two. Forgetting the $2\pi$ is one of the most common circuit-analysis mistakes — e.g. a cutoff "${\omega }_c=1/(RC)$" corresponds to "$f_c=1/(2\pi RC)$", not $1/(RC)$.

Physical meaning

Frequency is what the period means in the real world. For an audio signal, frequency is pitch — a $440\text{Hz}$ tone is the musical note A above middle C; higher frequency, higher pitch. For a radio transmission, the frequency is the carrier band — a station “at $99.9\text{FM}$” radiates a carrier at $99.9\text{MHz}$. Two sinusoids of the same amplitude but different frequency are genuinely different signals (different pitch, different channel); two with the same frequency but different phase are merely time-shifted copies of each other. Because any signal decomposes into sinusoids of various frequencies (its Fourier series), frequency is the axis along which we describe what a filter or amplifier does to a signal.