Complex Numbers

In electrical engineering, complex numbers are used to simplify difficult computations. In Cartesian form, it is expressed in the following way:

a+jb

Where:

a = real component

b = imaginary component

j = \sqrt{-1}

In polar form, it is expressed in the following way:

r e^{j\theta} = r \angle \theta

Where:

\begin{aligned} r &= \sqrt{a^2 + b^2} \\ \theta &= \arctan(\frac{b}{a}) \end{aligned}

Operations

Complex addition and subtraction is easily done in Cartesian form:

(a + jb) + (c + jd) = a + c + j(b + d) \\ (a + jb) – (c + jd) = a – c + j(b – d)

When dealing with polar form, it’s easiest to first convert to Cartesian form and add or subtract from there.

Multiplication and Division

Conversely, complex multiplication and division is easily done in polar form:

r_{1} \angle \theta_{1} \times r_{2} \angle \theta_{2} = r_{1}r_{2} \angle(\theta_{1} + \theta_{2}) \\ r_{1} \angle \theta_{1} \div r_{2} \angle \theta_{2} = \frac{r_{1}}{r_{2}} \angle(\theta_{1} – \theta_{2})

Therefore, when dealing with Cartesian form, it’s easiest to first convert to polar form and multiply or divide from there.