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Complex numbers in electrical engineering

IntroductionComplex numbersWhere are they used?

Scalar Quantity(Voltage, Current, Resistance)Vector Quantity (Impedance)

Complex numbersLike vectors

Representation of two dimensions with one numberArgand Diagram

In Electronics, i = j because it could be confused with current Main questionwhy is it used? And where?

In DC, V = IRquantities are vectors

In ACthere are wavestwo dimensionsOne wave for Voltage and one for currentBoth are not

in the same time due to resistance. Current is generally later.

Because of waves the resistance in DC is different from resistance in AC - Impedance

Electrical impedance is the measure of the resistance to a current when an AC voltage is applied.

It is the complex ratio of the voltage to the current in an alternating current (AC) circuit. Impedance

extends the concept of resistance to AC circuits

Possesses both magnitude and phase, unlike resistance, which has only magnitude.

When a circuit is driven with direct current (DC) can be thought of as impedance with zero phaseangle.

In AC, V = IZ , where Z = impedance

In general terms, impedance is the ratio V/I (voltage over current). V and I are not necessarily constant

in time, so they are represented by their frequency components (include equation).

So you see, in its basic form, you need two numbers to represent the ratio of two sine waves:

frequency and phase. A complex number is a mathematical convenience to carry over those two

values indirectly in their vector form.

Ac wavesphase differenceAsin (wt + theta), Theta is phase differenceHow late the second wave

is.

Adds the dimension of time. To simplify these dimensions, a complex number is used in order to convey two dimensions in one

number

A + jBrepresents two dimensionsfrequency (real part), Phase shift (complex part)

This doesnt mean that the phase shift is not quantifiable.

In my opinion, "imaginary" numbers could also be called "phase" numbers, or "time-displacement"

numbers because we represent the time difference between the sinusoids as a distance in the

imaginary direction.

Imaginary numbers allow us to treat time as a distance and make the equations much easier to deal

with.

Ease of calculation : Rather than the circuit element's state having to be described by two differentreal numbers V and I, it can be described by a single complex number z =V + i I. Similarly, inductance

and capacitance can be thought of as the real and imaginary parts of another single complex

number w = C + i L. The laws of electricity can be expressed using complex addition and multiplication.

In simpler terms, imagine an electronic piano. Each key produces a different tone. A volume control

changes the amplitude (volume) of all the keys by the same amount. That's how real numbers affect

signals. Now, imagine a filter. It makes some keys sound louder and some keys sound softer,

depending on their frequencies. That's complex numbers -- they allow an "extra dimension" of

calculation.

Another benefitease in solving differential equations. What is differential equationsnow mover

exampleis at certain velocity, gathers more snow, more snow is more weight, more weight is lessvelocity, less velocity, less snow and so on. This is solved quicker using the complex field in

engineering.