# How and why is arithmetic applied to complex numbers in electrical engineering?

Complex numbers are used in electrical engineering, for example, in invoices that include inductors and capacities.

This is about the phase position of the current compared to the voltage.This changes as soon as an L or C is present in the circuit.

For example:

If there is only one ohmic resistor in the circuit, the current and voltage are phase-equal.In other words, the voltage rises, so the current rises at exactly the same time.

If there is only one inductance in the circuit, the whole thing looks a little different: the current ‘lags’ after the voltage by 90°.So if the voltage rises, the current will only rise after a quarter of a period. (A period of a sinusoidal signal equals 360°)

With only one capacity in the circuit it is exactly the other way around, here the current rushes ahead of the voltage by 90°.

This is where the complex numbers come into play, because this connection has to be expressed in some way mathematically.For this purpose, the tension is placed on the positive real axis. The current – depending on what the circuit looks like – will only show positively on the real axis only in a purely ohmic consumer.

But if inductance or capacity is added, phase inequality must be clarified.

To do this, imagine that the tension always revolves counterclockwise (in mathematically positve direction) around the origin.If there is now a purely inductive load (the current lags behind the voltage by 90°), it has the direction -i, so points downwards when the voltage points to the right. In the case of a capacitive load, the current is therefore in the direction of +i, as it is rushing ahead.

But what is the whole ordeal to ask?

Well, L and C also form a resistance.However, this is frequency-dependent and can only be measured indirectly. But he’s there! This is the result of the so-called reactive power. What is to be noted, however, is that this ‘blind resistance’ cannot simply be added to an ohmic resistance, because it is phase-shifted.

So let’s take a circuit with an ohmic and an inductive consumer connected in series.We look at the ohmic current (on the real axis) and the inductive current (pointing downwards) and now we want to know how big the total current is. For this we add the two vectorially, i.e. we walk at the ohmic current to the end and start there with the inductive current. The length between the origin and the point at which we land is the total current.

So we get a right-angled triangle, where the hypothenuse is the total current.And with this we can calculate it directly with the pythagorean theorem.

What do we have at the end of it you can ask yourself.

Take an electric motor.The performance of this is often expressed in VA (sham power). However, this is not the power that can be used in the end, because this is given in W (active power). The apparent power corresponds to the electrical power we give into the motor, i.e. the voltage multiplied by the total current. There in it are the active power and the reactive power in exactly the same context as above the currents, because the reactive power is offset by 90° to the active power.

So we can always calculate the third party from two of the services.If we know sham and reactive power, we also know the active power that we can use (losses of the engine sometimes left out) at the end.

This is just a (very) rough overview of the topic.But I still hope to have shed some light on the dark.