2.2j.2 Phasor

Phasor

A phasor is a complex number representation of a sinusoidal function that helps simplify the analysis of sinusoidal steady-state behavior in electrical circuits. It represents both the amplitude and the phase of a sinusoidal wave. In mathematical terms, a phasor can be expressed as:
$X=|X|e^{j\varphi }⟺X=|X|\angle \varphi ,$
where $|X|$ is the phasor’s peak amplitude, and $\varphi$ is its phase angle.

Using Euler’s theorem, any time-varying sinusoidal signal $x(t)$ can be expressed in the form
$x(t)=\mathcal{R}e\{X{e}^{j\omega t}\},$
where $x(t)=\mathcal{R}e\{|X|e^{j\varphi }{e}^{j\omega t}\}=\cos (\omega t+\varphi ).$ Only the real part of its phasor is taken, as its imaginary part does not have any physical-world implications.

To gain some intuition, provided is a table of some standard time-variant functions and their phasor representations:

| $x(t)$ | $X$ |
| :--------------------------------------: | :----------------------------: |
| $A\cos (\omega t)$ | $A$ |
| $A\cos (\omega t+\varphi )$ | $A{e}^{j\varphi }$ |
| $-A\cos (\omega t+\varphi )$ | $A{e}^{j(\varphi \pm \pi )}$ |
| $A\sin \omega t$ | $A{e}^{-j\pi /2}=-jA$ |
| $A\sin (\omega t+\varphi )$ | $A{e}^{j(\varphi -\pi /2)}$ |
| $-A\sin (\omega t+\varphi )$ | $A{e}^{j(\varphi +\pi /2)}$ |
| $\frac{d}{dt}x(t)$ | $j\omega X$ |
| $\frac{d}{dt}[A\cos (\omega t+\varphi )]$ | $j\omega A{e}^{j\varphi }$ |
| $\int x(t)dt$ | $\frac{1}{j\omega }X$ |
| $\int A\cos (\omega t+\varphi )dt$ | $\frac{1}{j\omega }A{e}^{j\varphi }$ |

Note! $x(t)$ in the time domain is equivalent to multiplication of its phasor counterpart $X$ by $j\omega$ in the phasor domain. Similarly, integration of $x(t)$ in the time domain is equivalent to dividing its phasor $X$ by $j\omega$ in the phasor domain.

Differentiation of a time function

Intuition

The phasor (frequency) domain is generally used for simplification of mathematical derivations. In the phasor domain, as seen above, operations such as differentiation and integration become straightforward.