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Impedance and phasors cheat sheet

Element	Resistor	Inductor	Capacitor
v-i relationship	$v = iR$	$v=L\frac{\text di}{\text dt}$	$i=C\frac{\text dv}{\text dt}$
impedance	$Z_R = R$	$Z_L = j \omega L$	$Z_C = \frac{1}{j \omega C} = j\frac{-1}{\omega C}$
reactance	$X_R = 0$	$X_L = \omega L$	$X_C = \frac{-1}{\omega C}$

where

variable	quantity	number type	SI unit	unit abbreviation
$v$	instantaneous voltage	real	volts	[V]
$i$	instantaneous current	real	amps	[A]
$R$	resistance	real	ohms	[Ω]
$L$	inductance	real	henrys	[H]
$C$	capacitance	real	farads	[F]
$Z$	impedance	complex	ohms	[Ω]
$X$	reactance	real	ohms	[Ω]

Phasors

A phasor is a complex value that represents a sinusoidal voltage or current.

The magnitude of the phasor is the the r.m.s. amplitude of the sinusoid.
The angle of the phasor is the phase angle of the sinusoid.

Phasors are used to analyse circuits in which:

All elements are linear, i.e. resistors, inductors, capacitors.
All voltage and current sources are sinusoidal, with the same frequency f.

To analyse an AC circuit using phasors,

Represent each voltage or current source as a phasor
Represent each R, L, C element as an impedance
Calculate the phasor values of all branch currents and node voltages using Ohm's law, etc.
Convert voltage and current phasors to time-domain form if desired.

Ohm's law for phasors:

$$ V = IZ $$

where

$V$ is a complex-values voltage phasor [volts]
$I$ is a complex-valued current phasor [amps]
$Z$ is a complex-valued impedance [ohms]

Equivalent impedance of two impedances in series:

$$ Z_{eq} = Z_1 + Z_2 $$

Equivalent impedance of two impedances in parallel:

$$ \frac{1}{Z_{eq}} = \frac{1}{Z_1} + \frac{1}{Z_2} $$

ZIPPOS: Z's In Parallel? Product Over Sum!

$$ Z_{eq} = \frac{Z_1Z_2}{Z_1 + Z_2} $$

MATLAB / Octave example phasor calculation

% frequency [Hz] and angular frequency [rad/s]
f = 50;
w = 2*pi*f;

% Elements
R = 1e3;
C = 220e-9;

% Input voltage (reference phasor)
Vs = 230;

% Impedances
Zr = R;
Zc = 1/(j*w*C);

Zeq = Zr + Zc;

% Calculate voltage and current phasors
I = Vs / Zeq;
Vr = I*Zr;
Vc = I*Zc;

% Display rms amplitude, peak amplitude and phase angle for each phasor
I_rms = abs(I)
I_pk = sqrt(2) * I_rms
I_phase = angle(I)

Vr_rms = abs(Vr)
Vr_pk = sqrt(2) * Vr_rms
Vr_phase = angle(Vr)

Vc_rms = abs(Vc)
Vc_pk = sqrt(2) * Vc_rms
Vc_phase = angle(Vc)