Phasor transform

The phasor transform 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: $v(t)=\mathrm{Re}\{V{e}^{j\omega t}\}$. It’s 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 is what sets up the phasor representation.

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. 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 payoff: 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.

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.