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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)