I was just wondering how complex numbers can be applied in electrical engineering and why we use complex numbers over regular, real numbers for this application (e.g what capabilities does the complex number have that real numbers do not in electrical engineering)?
I have done some research concerning impedances and understand how they are written in complex form, however I am still confused why complex numbers are necessary in this field over regular numbers.
If you consider Real Power and Imaginary Power we are talking about resistive power and reactive power with energy stored in inductors, and capacitors. The vector sum of both is called "apparent power"
Even in mechanical systems there are complex reciprocal devices with stored energy in flywheels or springs. Inductors and Capacitors are similar in that they can store energy , in math called imaginary value.
But when an inductor opens current and arcs, it turns in to real energy similar to shorting out a capacitor into some resistance. Although this is a crude example like putting a crowbar brake across a flywheel.
If you don’t own a copy of the volumes of Feynman’s Lectures on Physics, I would highly recommend one.
He brilliantly introduces complex numbers in Vol. 1, “22-5 Complex Numbers”. But in the next section, “22-6 Imaginary Exponents”, he makes the following famous assertion:
We summarize with this, the most remarkable formula in mathematics:
\begin{equation} \label{Eq:I:22:9} e^{i\theta}=\cos\theta+i\sin\theta. \end{equation} This is our jewel.
There’s too much to cover here, but I refer you to this lecture where he applies the above formula, with regard to AC circuits: Vol 2. 22 - AC Circuits
An excerpt:
We have already discussed some of the properties of electrical circuits in Chapters 23 and 25 of Vol. I. Now we will cover some of the same material again, but in greater detail. Again we are going to deal only with linear systems and with voltages and currents which all vary sinusoidally; we can then represent all voltages and currents by complex numbers, using the exponential notation described in Chapter 23 of Vol. I. Thus a time-varying voltage V(t) will be written
\begin{equation} \label{Eq:II:22:1} V(t)=\hat{V}e^{i\omega t}, \end{equation} where $$\hat{V}$$ represents a complex number that is independent of t. It is, of course, understood that the actual time-varying voltage V(t) is given by the real part of the complex function on the right-hand side of the equation.
Similarly, all of our other time-varying quantities will be taken to vary sinusoidally at the same frequency ω. So we write \begin{equation} \begin{aligned} I&=\hat{I}\,e^{i\omega t}\quad(\text{current}),\\[3pt] \xi&=\hat{\xi}\,e^{i\omega t}\quad(\text{emf}),\\[3pt] E&=\hat{E}\,e^{i\omega t}\quad(\text{electric field}), \end{aligned} \label{Eq:II:22:2} \end{equation} and so on.
Most of the time we will write our equations in terms of V, I, ξ, ... (instead of in terms of V̂, Î, ξ̂, ...) remembering, though, that the time variations are as given in (22.2).
In our earlier discussion of circuits we assumed that such things as inductances, capacitances, and resistances were familiar to you. We want now to look in a little more detail at what is meant by these idealized circuit elements. We begin with the inductance.
For one thing it makes the math a lot easier. For example, think about solving differential equations. It's much simpler to use the Laplace Transform and solve the differential equation, rather than use classical techniques. On the same subject, it gives another perspective to the same problem from a frequency domain point of view.
There are also tools like Bode plots, which easily give quick approximations to how a system behaves in the frequency domain.
If you are doing time domain analysis, everything is expressed in real numbers - voltages, currents, resistances, because these are always simple instantaneous values. When you do frequency domain analysis, that's when complex numbers come in, because quantities like voltages, currents, and impedances have both a magnitude and a phase; expressing such quantities as complex numbers helps when performing calculations.
