Complex Number Representation for AC Circuits

For AC circuits, sinusoidal values such as voltage are generally represented by Phasor Intro, which are based on complex numbers, allowing us to avoid trigonometric functions.

A complex number $A$ can be represented as:

• Rectangular:

$$A=a+jb$$

• Polar:

$$A=|A|\angle \alpha$$

The magnitude can be found with:

$$|A|=\sqrt{a^2+b^2}$$

Angle:

$$\alpha ={\tan }^{-1}\left(\frac{b}{a}\right)$$

Also note that:

$$A=|A|e^{j\alpha }=|A|(\cos \alpha +j\sin \alpha )$$