Describe a sinusoid by amplitude, angular frequency, and phase.
Convert between time-domain sinusoids and phasors, both directions.
Explain why differentiation becomes multiplication by jω.
Add sinusoids of the same frequency using phasor arithmetic.
From here on the sources are sinusoidal — the shape of Ontario's grid, of radio, of audio, of every carrier and clock harmonic. Sinusoids hold a privileged position in linear circuits: feed a linear circuit a sinusoid at frequency ω and EVERY voltage and current in it is a sinusoid at that same ω — only amplitudes and phases differ. So at a fixed frequency, a signal is fully described by just two numbers, and the phasor is the bookkeeping that exploits it: a complex number carrying the amplitude and phase, with the ωt spin factored out and understood.
v(t)=Vmcos(ωt+ϕ)⟺V=Vm∠ϕtime domain ↔ phasor
Two sinusoids at one frequency: the second lags the first by 90° — the relationship a phasor captures as a single angle.
The payoff is what happens to calculus. Differentiating V_m cos(ωt + φ) gives −ωV_m sin(ωt + φ) = ωV_m cos(ωt + φ + 90°): the SAME sinusoid, scaled by ω and advanced 90°. In phasor language, scaling by ω and rotating 90° is one operation — multiplication by jω. The differential equations of CA-5 and Lesson 1 collapse into complex-number algebra:
dtd⟺jωthe phasor's entire reason to exist
WORKED EXAMPLE
Both directions
Time → phasor: v(t) = 10 cos(377t + 30°) V becomes V = 10∠30° V. (The ω = 377 rad/s is carried alongside, not inside, the phasor.)
Phasor → time: I = 2∠−45° mA at ω = 1000 rad/s becomes i(t) = 2 cos(1000t − 45°) mA.
377 rad/s is worth recognising on sight: 2π × 60 Hz — the angular frequency of every wall outlet in Ontario.
A phasor is the sinusoid with the time spin undone.WORKED EXAMPLE
Adding sinusoids without trigonometry
Add v_1 = 3 cos(ωt) and v_2 = 4 cos(ωt + 90°) — by trig identities, an ugly afternoon; by phasors, three lines.