Impedance

Impedance

Impedance is a generalization of resistance…
$Z\left(s\right)=\frac{V\left(s\right)}{I\left(s\right)}$

More notes here $Z=\frac{V}{I}=\frac{V}{I}\left({\theta }_v-{\theta }_i\right)$
Inductors lag voltage, where capacitors lead voltage.

LaPlace Domain

Each one of these impedance representations can also be represented in the LaPlace domain which is useful for controls and systems modelling.

Resistor Impedance

$Z_R=R$
No angle here.

Capacitor Impedance

$X_C=-j\frac{1}{wc}$
With an angle of +90 degrees.
Capacitors have the following relationships:
$i=C\frac{dv}{dt}$
$v=\frac{1}{c}{\int }_{}^{}idt$ $E=C\int vdv=\frac{C}{2}{v}^2$

Inductor Impedance

$X_L=jwL$

With an angle of -90 degrees.
Inductors have the following relationships:
$v=L\frac{di}{dt}$
$i=\frac{1}{L}{\int }_{}^{}vdt$
$E={\int }_{}^{}pdt={\int }_{}^{}vidt=L\frac{i^2}{2}$

Getting Current in a load

I thought you had to use CDR and get Z_eq to get the current in a load but you can just use Ohms law with a given voltage LOL

Series Impedances

You sum impedances in series.

Parallel Impedances

$G=\frac{1}{R}\left(Siemens\right)$
$R_{total}=\frac{1}{\frac{1}{R_1}+\cdots +\frac{1}{R_n}}$
A similar concept exists for complex impedances.
$Y=\frac{1}{Z}$
Where Y is admittance (the opposite of impedance)
Therefore, for impedances in parallel, you can sum the admittances and invert the total.