Complex numbers utilities

We have already studied a bit about complex numbers. As we cannot measure them as well as 1 that can be simply applied to 1 meter, complex numbers seem to be imaginary and not very useful. However, many engineering areas employ them. Some topics that involve their use are electrical current, wavelength, liquid flow in relation to obstacles, analysis of stress on beams, the movement of shock absorbers in cars, the study of resonance of structures, the design of dynamos and electric motors, and the manipulation of large matrices used in modeling.

We are only going to focus on the electrical part as it can be more or less easy understood by university students.

When electrical engineers analyzed alternating current circuits, they found that voltage, current and resistance, called impedance, were not the scalar quantities used when measuring DC (direct current) circuits.  These quantities that alternate in direction and amplitude have other dimensions (frequency and phase shift) that must be taken into account.

That is why in order to analyze AC circuits, it became necessary to represent multi-dimensional quantities.  Then scalar numbers were abandoned and complex numbers were used to express the two dimensions of frequency and phase shift at one time. Finally, instead of the typical “i” used by mathematicians to represent these numbers, in electronics it is employed “j” (z=a+bj) so that it is not confused with “I” which stands for intensity of a circuit.