Complex numbers

Prof. Dr. Thorsten Jungmann (last updated 2026-03-02)

Complex numbers: the mathematical tool that lets you combine amplitude and phase in a single expression.

Complex numbers (German: komplexe Zahlen) extend the real numbers by an imaginary component. In electrical engineering, they are the central computational tool of AC circuit analysis: they allow you to represent sinusoidal voltages and currents as phasors in the complex plane and to reduce circuit calculations to algebraic operations.

This page summarizes the mathematical foundations needed for AC circuit analysis in lesson EE2-03 (Complex AC circuit analysis).

1 Definition

A complex number $\underline{z}$ consists of a real part and an imaginary part:

$$\underline{z}=a+jb$$

where:

• $a=\mathrm{Re}(\underline{z})$ the real part
• $b=\mathrm{Im}(\underline{z})$ the imaginary part
• $j$ the imaginary unit with $j^2=-1$

Difference between mathematics and electrical engineering

In electrical engineering, $j$ (not $i$) is used for the imaginary unit, because $i$ is already reserved for the current.

2 Forms of representation

2.1 Cartesian form (standard form)

$$\underline{z}=a+jb$$

Suitable for addition and subtraction.

2.2 Polar form (exponential form)

$$\underline{z}=r\cdot e^{j\phi }$$

where:

• $r=|\underline{z}|=\sqrt{a^2+b^2}$ the magnitude
• $\phi =\arg (\underline{z})=\arctan \left(\frac{b}{a}\right)$ the argument (angle)

Suitable for multiplication and division.

2.3 Conversion

Polar form → Cartesian form:

$$a=r\cdot \cos \phi ,b=r\cdot \sin \phi$$

Cartesian form → polar form:

$$r=\sqrt{a^2+b^2},\phi =\arctan \left(\frac{b}{a}\right)$$

Take the quadrant into account

When you calculate $\phi$ with $\arctan$, you must take the quadrant into account. $\arctan$ only returns angles in the range $-90°$ to $+90°$. If $\underline{z}$ lies in the second or third quadrant ($a<0$), you must add $180°$.

3 Euler’s formula

Euler’s formula establishes the relationship between the exponential function and the trigonometric functions:

$$e^{j\phi }=\cos \phi +j\sin \phi$$

It is named after Euler and forms the mathematical basis for representing sinusoidal AC quantities as rotating phasors.

Euler’s identity (German: Euler’sche Identität), the special case $\phi =\pi$:

$$e^{j\pi }+1=0$$

4 Rules of arithmetic

4.1 Addition and subtraction

In Cartesian form – add the real parts and the imaginary parts separately:

$${\underline{z}}_1+{\underline{z}}_2=(a_1+a_2)+j(b_1+b_2)$$

4.2 Multiplication

In polar form – multiply the magnitudes, add the angles:

$${\underline{z}}_1\cdot {\underline{z}}_2=r_1\cdot r_2\cdot e^{j({\phi }_1+{\phi }_2)}$$

4.3 Division

In polar form – divide the magnitudes, subtract the angles:

$$\frac{{\underline{z}}_1}{{\underline{z}}_2}=\frac{r_1}{r_2}\cdot e^{j({\phi }_1-{\phi }_2)}$$

4.4 Conjugate

The conjugate ${\underline{z}}^{\ast }$ is obtained by reversing the sign of the imaginary part:

$${\underline{z}}^{\ast }=a-jb=r\cdot e^{-j\phi }$$

Useful, for example, for the complex apparent power: $\underline{S}=\underline{U}\cdot {\underline{I}}^{\ast }$.

5 Application in AC circuit analysis

Physical quantity	Complex representation	Real part =	Imaginary part =
Impedance	$\underline{Z}=R+jX$	Active resistance $R$	Reactance $X$
Voltage	$\underline{U}=U\cdot e^{j{\phi }_U}$	$U\cos {\phi }_U$	$U\sin {\phi }_U$
Current	$\underline{I}=I\cdot e^{j{\phi }_I}$	$I\cos {\phi }_I$	$I\sin {\phi }_I$
Apparent power	$\underline{S}=P+jQ$	Active power $P$	Reactive power $Q$

The complex representation allows you to generalize Ohm’s law to AC circuits:

$$\underline{U}=\underline{Z}\cdot \underline{I}$$

By introducing the complex impedance, you can transfer all methods known from DC circuit analysis (voltage divider, Kirchhoff’s laws, superposition theorem, equivalent sources) directly to AC circuits.

6 Practical/worked example

Given: ${\underline{z}}_1=3+j4$ and ${\underline{z}}_2=1-j2$.

Magnitude of ${\underline{z}}_1$:

$$|{\underline{z}}_1|=\sqrt{3^2+4^2}=\sqrt{25}=\underline{5}$$

Argument of ${\underline{z}}_1$:

$${\phi }_1=\arctan \left(\frac{4}{3}\right)=\underline{53.1°}$$

Polar form of ${\underline{z}}_1$:

$${\underline{z}}_1=5\cdot e^{j\cdot 53.1°}$$

Addition:

$${\underline{z}}_1+{\underline{z}}_2=(3+1)+j(4-2)=\underline{4+j2}$$

Multiplication (in Cartesian form):

$${\underline{z}}_1\cdot {\underline{z}}_2=(3+j4)(1-j2)=3-j6+j4-j^2\cdot 8=3-j6+j4+8=\underline{11-j2}$$