Sinusoidal steady state and phasors

Phasors: sinusoids become arithmetic

• At one frequency a sinusoid is a phasor $V_{m}\angle φ$, and each element is an impedance: $Z_{R} = R$ flat, $Z_{L} = jωL$ up 90°, $Z_{C} = 1/(jωC)$ down 90°. Every DC method — series, parallel, nodal, Thévenin — carries over unchanged.
• Verify every answer in the other representation: $30 + j40 = 50\angle 53.13^{\circ}$ Ω under 100 V draws $2\angle -53.13^{\circ}$ A, and in rectangular $I = 100(30 - j40)/2500 = 1.2 - j1.6$ A with $|I| = 2$ — the two forms agree exactly.

{exercise:uni-ee4x-phasor}