Phase (waveform)

Phase is the position of a waveform within its cycle — it specifies when, relative to some reference time, the waveform reaches its peak. It is the third of the three numbers (with amplitude and frequency) that fully describe a sinusoid.

In $v(t)=A\sin (\omega t+\phi )$ the phase is $\phi$, the constant added to the angle inside the sine. Here $A$ is the amplitude, $\omega$ the angular frequency, and $t$ time. At $t=0$ the waveform is not necessarily at zero — it is at $v(0)=A\sin \phi$. The phase tells you how far into the cycle the signal already is at the moment you start your clock.

Phase is a time shift

Pull the $\omega$ out of the argument:

$v(t)=A\sin (\omega t+\phi )=A\sin \left(\omega \right(t+\frac{\phi }{\omega }\left)\right)$

This is the same sinusoid as $A\sin (\omega t)$, just evaluated at a shifted time $t+\phi /\omega$. A positive $\phi$ shifts the waveform earlier (it leads — the peak arrives sooner by $\phi /\omega$ seconds); a negative $\phi$ shifts it later (it lags). So phase is nothing more than a time offset, expressed as an angle so it is independent of the particular frequency. Two sinusoids with identical $A$ and $\omega$ but different $\phi$ are time-shifted copies of one another — same shape, same size, same pitch, just slid along the time axis.

Phase $\phi$ in $v(t)=A\sin (\omega t+\phi )$ is the time position within a cycle; a cosine is a sine shifted by $\pi /2$.

Cosine is a phase-shifted sine

The cleanest concrete example: a cosine is a sine advanced by a quarter cycle.

$A\cos (\omega t)=A\sin (\omega t+\frac{\pi }{2})$

This follows from the trig identity $\cos \theta =\sin (\theta +\pi /2)$ — the cosine curve is exactly the sine curve slid left by $90°$. So sine and cosine are not different functions in any deep sense; they are the same waveform at two different phases. This is why circuit analysis can pick either as the reference: the choice only shifts every phase in the problem by a constant.

Degrees vs radians, and a slide typo

One full period corresponds to one complete trip around the circle: $2\pi$ radians, which is $360°$. Half a period is $\pi$ rad $=180°$, and a quarter period is $\pi /2$ rad $=90°$ — that quarter-cycle is precisely the sine-to-cosine shift above.

The handwritten note on the course slide reads “one T = 2π = 180°“. The "$T=2\pi$" part is fine (a period is $2\pi$ radians of the argument), but $2\pi$ rad is $360°$, not $180°$ — this is a slide typo, acknowledged in the notes. The correct correspondences are $2\pi \text{rad}=360°$ for a full period and $\pi /2\text{rad}=90°$ for the sine→cosine shift.

Why phase matters

Phase carries no information about a signal’s strength or pitch, but it governs how sinusoids combine. Two equal sinusoids exactly in phase ($\phi$ difference $=0$) add to double the amplitude; the same two exactly out of phase ($180°$ apart) cancel to zero. Filters and amplifiers generally shift phase as well as amplitude — an RC highpass filter, for instance, produces a phase shift $\phi =\arctan (1/(2\pi fRC))$ that approaches $90°$ at low frequency. The phase part of a circuit’s Transfer function is half of its Bode plot and matters whenever timing or feedback stability is at stake.