Phasors

With phasors, you can convert an equation of strictly sinusoidal input at one frequency to the phasor domain, use traditional circuit analysis methods, then convert back to the time domain to solve a circuit.

You can understand phasors geometrically or algebraically. Geometrically, phasors are vectors with both a real component (that represent something like resistance or voltage in phase) and an imaginary component (the reactance/phase shift of the signal). The real component is the x-axis and the imaginary component is the y-axis. Algebraically, phasor conversions are as follows:

Resistor

\[Z_R = R\]

Capacitor

\[Z_C = \frac{-j}{\omega C}\]

Inductor

\[Z_L = j\omega L\]

For a voltage source:

\[v(t) = A\cos(\omega t + \phi) \Rightarrow V = A\angle \phi = A\cos(\phi) + jA\sin(\phi)\]

Note that $j$ is used for the imaginary number $\sqrt{-1}$ since $i$ is conventionally already used for current.

What happened to the frequency?

You will notice that the frequency $\omega$ is missing for the voltage source in the phasor domain. This is because phasors are meant to only work with one frequency and only sinusoidal signals. Additionally, phasors will only give the steady-state response of a circuit.

It is possible to use phasors with different frequencies of sinusoidal signals, but solving will involve superposition. For other types of signals, a Fourier transform may be more appropriate. For the transient response, a Laplace transform is best. The Laplace transform is the most versatile transform.