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