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