Why does AC circuit analysis use complex notation?

AC voltages and currents are sinusoidal.  A typical signal looks like V(t) = V_{0_real}cos(ωt + φ).
Sinusoids are annoying to differentiate and integrate repeatedly.  But complex exponentials behave nicely, dexp(iωt)/dt = iω exp(iωt).  Therefore we represent a sinusoid by a complex exponential,
V(t) = V₀exp(iωt), V₀ = V_{0_real}exp(iφ),
use the math to do the work, and then take the real part at the end.  This turns calculus into algebra.

The complex‑number method only works because AC circuits are linear systems.
Resistors obey V = IR, Inductors obey V = LdI/dt, Capacitors obey I = C dV/dt.  All of these relationships are linear differential equations.  When you replace d/dt with multiplication by iω, they become linear algebraic equations

Because the system is linear superposition works, impedances add like resistances, Kirchhoff's laws apply directly to the complex amplitudes (phasors).  For nonlinear systems (diodes, transistors, saturation) this method does not work.

Complex notation is a "transform", a representation. Be cautious in the following scenarios.

• Non-Linear Elements:  For nonlinear systems (diodes, transistors, saturation) the representation does not work.
• Multi-Frequency Circuits:  A complex amplitude is defined for a single frequency ω.  If your circuit has sources at 60 Hz and 120 Hz, you cannot add their complex representations directly.  You must solve the circuit for each frequency separately and then add the results in the time domain (using superposition).
• Power Calculations: This is the most common pitfall.  You cannot simply multiply the complex voltage by the complex current to get real power. You must use the complex conjugate of the current.  P_{instantaneous} = VI*.
• RMS vs. Peak:  Always check if your notation assumes peak values V₀ or RMS values V_rms.  In power engineering, phasors almost always use RMS, while in physics textbooks, they often use peak.  Mixing them up will lead to a √2 error.