Complex numbers, the quick method in electricity and electronics, 'Vidéo 2, 2'

Why are complex or imaginary numbers used in electricity?

When we solve a differential equation, for example, we try to find a solution using several methods, from the simplest to the most complicated, or the most difficult, so to speak. The systematic method is obviously Laplace's method, which consists of replacing products by sums via logarithms. If that's the case, we'll use series expansion to solve a differential equation (it's done!), and of course if series expansion still doesn't find a solution to the differential equation, we'll use the complex numbers, which normally leads to a suitable solution, bearing in mind that solving differential equations is reputedly difficult and frightens off the uninitiated.
In other words, the complex number is just a tool invented by the imagination to solve a mathematical problem where the solution is not obvious if we reason in a ‘conventional’ way using real numbers.
Solving a problem using complex numbers is not a philosophical exercise, but .... no more and no less than a sort of sleight of hand.

The Laplace integral, transform and inverse transform are completely derived from the complex number, contain a complex number and evoke the frequency, since the Laplace integration variable is " s “ with s=W.j , where W is in [Rad/s] and ” j " represents the imaginary. This is why they lend themselves wonderfully well to frequency domain calculations, as we will see in transient analysis and analysis of all kinds.

In electricity and electronics, complex numbers can be used to simplify calculations. A sinusoidal signal is replaced by a complex number in several forms: Cartesian form, polar form (or modulus and phase), exponential form. Instead of performing the long and tedious operations of adding and multiplying trigonometric functions in the time domain, we transform them into moduli and phases in the frequency domain. The same operations performed using moduli and phases are extremely simple. Similarly, using Laplace transforms, we can transform an entire circuit from the time domain to the frequency domain and analyse it in the frequency domain using complex numbers. When a solution is found in the frequency domain, we use the inverse Laplace transform to return to the time domain.

To visualise a complex number, we use the complex plane with an orthonormal reference frame consisting of two axes:

1] A horizontal axis on which real numbers such as -10, 25, 0, 11, etc. are placed. This axis also serves as a 0° reference for angles.
2] A vertical axis, commonly called the j-axis or imaginary axis. Imaginary numbers are placed on this axis, such as (2.j , -100.j , etc ) . The product of a real number and the imaginary j , such as 2.j , for example, is called the imaginary part of a complex number.

A complex number Z can be expressed in several different forms.

A ] Cartesian form: Z = a + j.b where a is a real number, or the real part, located on the real axis and j.b represents the imaginary part located on the imaginary axis.

B ] Polar form: Z = (Modulus) . (Phase°) where (Modulus) = root of 2 (a²+b²) ,
where (Modulus) is always positive or zero and (Phase°) = ARCtan( a/b ) in degrees. (Phase°) or the phase angle can be positive or negative with respect to the reference axis 0°, either (Phase°) or (-Phase°).

e^(j.x)=(Phase°) and e^(-j.x)=(-Phase°).
Hence :
Z = (Module) . (Phase°) = (Module).cos(Phase°) + j.(Module).sin(Phase°) and
Z = (Module) . (-Phase°) = (Module).cos(Phase°) - j.(Module).sin(Phase°).
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