# Phasors and Impedance

## Phasor Representation

Parameters in AC systems have three variable factors as described in the general form:

$$ v(t) = \hat V \sin (\omega t) $$

Since the frequency in a system is usually constant, the remaining two factors (amplitude and phase) can be represented by a vector, which in relation to electronics, is called a phasor.

Phasors can be mapped onto a Cartesian plane where the the length represents amplitude and angle from the positive \( x \) axis represents phase.

The \( j \) operator is introduced to simplify calculation. It has the property of rotating a phasor by \( \frac{\pi}{2} \) and allows complex number theory to be applied to calculations.

### Impedance

Current and voltage relations in AC system components can be written using the \( j \) operator.

#### Resistor

$$ V_R = R I_R $$

#### Capacitor

$$ V_L = j \omega L I_L $$

#### Inductor

$$ V_C = \frac{1}{j \omega C} I_C $$

The complex reactances \(j \omega L \) and \( \frac{1}{j \omega C} \) are known as impedances.

### Complex and Polar Forms

A phasor, like a complex number, can be expressed in both complex and polar forms. While the complex form is simpler for addition and subtraction, polar form is more suited to multiplication and division.

#### Converting from Complex to Polar Form

$$ V = a + jb = \sqrt{a^2 + b^2} \angle \tan ^{-1} \bigg(\frac{b}{a} \bigg) $$

#### Converting from Polar to Complex Form

$$ V = |V| \angle \theta = |V| \cos( \theta ) + j |V| \sin ( \theta ) $$

### Phasor Operations

Phasor addition and subtraction follow the same rules as complex numbers.

#### Addition and Subtraction

#### Multiplication and Division

$$ V = |I| \angle \theta \times |Z| \angle \phi = |I||Z| \angle (\theta + \phi ) $$

$$ Z = \frac{|V| \angle \theta}{|I| \angle \phi} = \frac{|V|}{|I|} \angle (\theta - \phi ) $$