PX157 - E1 - complex representation

V (t) = V_{0} \cos (ω t)

$ω =$ angular frequency $= 2 π f$

polar representation of a complex number

z = x + i y = r e^{i θ}

$r = \sqrt{x^{2} + y^{2}}$
$θ = \arctan (\frac{y}{x})$

here, $θ = ω t :$

z = r e^{i ω t} = r (\cos (ω t) + i \sin (ω t))

the physical solution is $R e (z)$
notation:
- real voltage: $V = V_{0} \cos (ω t)$
- complex representation: $\tilde{V} = V_{0} e^{i ω t}$
- ' $j$ ' is used instead of ' $i$ ', $j = \sqrt{- 1}$
phasor ( $\tilde{V}$ ):

V (t) = V_{0} \cos (ω t) \to \tilde{V} = V_{0} e^{j ω t}

phase differences:

\begin{aligned} V_{1} & = V_{0} \cos (ω t) \\ V_{2} & = V_{0} \cos (ω t + ϕ) \end{aligned}

- $V_{2}$ leads $V_{1}$ by a phase difference, $ϕ$

in complex notation:

\begin{aligned} {\tilde{V}}_{1} & = V_{0} e^{j ω t} \\ {\tilde{V}}_{2} & = V_{0} e^{j ω t + j ϕ} \\ = {\tilde{V}}_{1} e^{j ϕ} \end{aligned}

eg: ${\tilde{V}}_{2} = {\tilde{V}}_{1} e^{j \frac{π}{2}} = j {\tilde{V}}_{1}$

magnitudes

$\tilde{z} = x + j y$
$| \tilde{z} | = \sqrt{x^{2} + y^{2}}$

$\tilde{z} = \frac{{\tilde{z}}_{1}}{{\tilde{z}}_{2}}$
$| \tilde{z} | = \frac{| {\tilde{z}}_{1} |}{| {\tilde{z}}_{2} |}$