Electronics/Phasors

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Sinusoids and Phasors

Sinusoidal signals can be represented as $A\sin (\omega t+\varphi )$ where A is the amplitude, $\omega$ is the frequency in radians per second, and $\varphi$ is the phase angle in radians (phase shift). The signal is completely characterized by A, $\omega$, and $\varphi$.

Using Euler's formula,

$A{e}^{j(\omega t+\varphi )}=A\cos (\omega t+\varphi )+jA\sin (\omega t+\varphi )$

so

$A\cos (\omega t+\varphi )=\Re 𝔢(A{e}^{j(\omega t+\varphi )})$

Note: In electrical engineering, the symbol j is used to denote the imaginary unit rather than the symbol i because i is used to denote current, especially small signal current.

A complex exponential can also be expressed as

$A{e}^{j(\omega t+\varphi )}=A{e}^{j\varphi }{e}^{j\omega t}$

The quantity $\tilde{A}=A{e}^{j\varphi }$ is a phasor. It contains information about the magnitude and phase of a sinusoidal signal, but not the frequency. This simplifies use in circuit analysis, since most of the time, all quantities in the circuit will have the same frequency. (For circuits with sources at different frequencies, the principle of superposition must be used.)

Another phasor notation is $\tilde{A}=A\mathrm{\angle }\varphi$. Note that this is simply a polar form, and can be converted to rectangular notation by:

$\tilde{A}=A\mathrm{\angle }\varphi =A\cos \varphi +jA\sin \varphi$

and back again (with care to place the angle in the right quadrant) by:

$\tilde{A}=X+jY=\sqrt{X^2+Y^2}\mathrm{\angle }{\tan }^{-1}\frac{Y}{X}$

For the moment, consider single-frequency circuits. Every steady state current and voltage will have the same basic form:
$\widetilde{A_i}{e}^{j\omega t}$ where $\widetilde{A_i}$ is a phasor. So we can "divide through" by $e^{j\omega t}$ to get phasor circuit equations. We can solve these equations for some phasor circuit quantity $\tilde{Y}$, multiply by $e^{j\omega t}$, and convert back to the sinusoidal form to find the time-domain sinusoidal steady-state solution.

This type of phasor circuit analysis requires knowledge of Electronics/Impedance.