Phasor transform

The phasor transform is the operation that maps a sinusoidal time-domain signal $v(t)=A\cos (\omega t+\varphi )$ to its complex amplitude $V=A{e}^{j\varphi }$, dropping the time dependence. Inverse transform: $v(t)=\mathrm{Re}\{V{e}^{j\omega t}\}$. The transform is one-to-one at a fixed frequency $\omega$, so all sinusoids of frequency $\omega$ correspond bijectively to complex numbers, and operations on sinusoids carry over to operations on those complex numbers — usually much simpler.

This sets up the phasor representation; this note focuses on the transform itself and why it works.

Three observations behind the trick

1. Every sinusoid is the real part of a rotating complex exponential. By Euler’s formula,

$A\cos (\omega t+\varphi )=\mathrm{Re}\left[A{e}^{j(\omega t+\varphi )}\right]=\mathrm{Re}[\underset{V}{\underbrace{A{e}^{j\varphi }}}\cdot e^{j\omega t}].$

The factor $A{e}^{j\varphi }$ depends only on the specific signal’s amplitude and phase; the factor $e^{j\omega t}$ depends only on time and the (common) angular frequency.

2. Linear operations preserve frequency. Adding sinusoids of frequency $\omega$ gives a sinusoid of frequency $\omega$. Differentiating gives one (multiplied by $\omega$, phase-shifted $90°$). Integrating gives one. In a linear circuit driven at one frequency, everything in sight oscillates at that frequency, and the $e^{j\omega t}$ factor is along for the ride.

3. A one-to-one correspondence between sinusoids and complex numbers. At a fixed $\omega$, the sinusoid $A\cos (\omega t+\varphi )$ is completely specified by the pair $(A,\varphi )$, which packages neatly as the single complex number $V=A{e}^{j\varphi }$.

The transform pair

$\boxed{v(t)=A\cos (\omega t+\varphi )⟷V=A{e}^{j\varphi }.}$

To go from time domain to phasor: read off amplitude and phase. To go back: multiply the phasor by $e^{j\omega t}$ and take the real part.

The frequency is not stored in the phasor. The phasor only encodes amplitude and phase — frequency is context.

Differentiation becomes multiplication

The deepest single fact: differentiation with respect to time becomes multiplication by $j\omega$. From $\frac{d}{dt}A\cos (\omega t+\varphi )=-A\omega \sin (\omega t+\varphi )=A\omega \cos (\omega t+\varphi +90°)$ — another sinusoid at $\omega$, amplitude $\omega$ times, phase advanced $90°$. In phasor language, $A{e}^{j(\varphi +90°)}=A{e}^{j\varphi }\cdot e^{j90°}=V\cdot j\omega$ (using $e^{j90°}=j$ and the $\omega$ scaling).

So $dv/dt\leftrightarrow j\omega V$. Integration corresponds to division by $j\omega$. Differential equations in time become algebraic equations in $\mathbb{C}$. Resistors, inductors, capacitors all get impedances that are complex numbers, and Kirchhoff’s laws apply directly. See Phasor relationships for circuit elements.

Worked example

Combine $v_1(t)=12\cos (1000t+30°)$ with $v_2(t)=12\cos (1000t-30°)$. Phasors $V_1=12e^{j30°}$ and $V_2=12e^{-j30°}$. Convert to rectangular: $V_1=6\sqrt{3}+6j$, $V_2=6\sqrt{3}-6j$. Sum: $12\sqrt{3}$. The imaginary parts cancelled (because $V_1$ and $V_2$ are complex conjugates). Inverse transform: $v(t)=12\sqrt{3}\cos (1000t)\approx 20.78\cos (1000t)$.

Two sinusoids symmetric about $\varphi =0$ summed to a sinusoid centered at $\varphi =0$. The phasor picture makes this visually obvious in a way the trigonometric identity doesn’t.

Limitations

The phasor transform only handles sinusoidal steady state — a single sinusoid at one fixed frequency. Transients (the time after a switch closes, before steady state), multi-frequency signals, and non-sinusoidal inputs need other tools: the Laplace transform (transients and exponentials), the Fourier transform (spectral decomposition).

The Laplace transform generalizes the phasor transform: a phasor is what you get when you evaluate the Laplace transform at $s=j\omega$ on a steady-state sinusoid. Adding a real part to $s$ tracks decay rates too. See Complex frequency.