Generating the Sine Wave

The sine wave isn't a mathematical accident — it's the signature of rotation. Spin a wire loop in a magnetic field, and the EMF it produces rides out a pure sinusoid whose amplitude is set by the geometry and whose frequency is set by how fast you spin. Every generator on the power grid is a fancy elaboration of this one idea.

Spinning a coil in a magnetic field

A rectangular coil of $N$ turns, area $A$, rotates at angular velocity $ω$ inside a uniform magnetic field $B$. As it rotates, the magnetic flux through the coil changes as $\Phi(t) = N B A \cos(\omega t)$, and Faraday's law gives the induced EMF:

That's a pure sine of peak amplitude $V_p = N B A \omega$ at frequency $ω$. The output is AC because the rotation keeps flipping the coil relative to the field — half the turn pulls flux one way, the other half pushes it back.

The rotating phasor

Picture a vector of length $V_p$ rotating counter-clockwise at $ω$ radians per second. Its angle at time $t$ is $\theta = \omega t$. Project the tip onto the vertical axis: the signed length of that projection is $V_p \sin(\omega t)$. (Project it onto the horizontal axis instead and you'd get $V_p \cos(\omega t)$ — the same waveform, 90° ahead.)

That is the sine wave. The phasor — the rotating vector — is a book-keeping device that keeps the amplitude and the phase in one object. You'll meet phasors formally in a few topics; today's job is to see the identity: the sine is the vertical component of a uniformly rotating vector.

Phase angle

The angle at $t = 0$ doesn't have to be zero. If the phasor starts at $φ$ instead of 0, the output is:

A positive $φ$ shifts the waveform left on the time axis — the wave reaches its peaks earlier. A negative $φ$ shifts it right — it reaches peaks later. When two signals share the same frequency but differ in phase, the earlier-peaking one is said to lead, and the later-peaking one lags.

Measuring the phase difference is just geometry. Two signals of frequency $f = 50\,\text{Hz}$ (period $T = 20\,\text{ms}$) with a 60° lag are separated in time by:

So v₂ hits every peak ~3.33 ms after v₁ does. Phase is time wearing angular clothing.

Sine and cosine are siblings

A cosine is just a sine with a 90° head start:

So when a book quotes a signal as $10\cos(\omega t)$ V, that's the same amplitude and frequency as $10\sin(\omega t + 90°)$ V — just a different phase reference. For circuit analysis, we'll usually pick sine or cosine as the house convention and stick with it per chapter.