EEVblog 1470 - AC Basics Tutorial Part 3 - Complex Numbers

[EEVblog]

[golden_labels]

Nothing wrong with Dave’s video, but I believe three additional comments may be useful to some people. In particular younger and less experienced audience, who may be a bit confused by careless switching between different representations and seemingly unexpected formulas.

Polar form multiplicat and division
While multiplying/dividing amplitude parts is quite obvious, summing angles may be not. To see the logic behind this part, realize that: multiplying something by any complex number with absolute value equal to 1 is the same as rotating that around (0, 0). For example multiplying 1+3i by i means nothing more than rotating it by 90° counterclockwise. That’s the meaning behind i² = -1, the definition of i. You multiply twice by i, you rotate by 180° (2·90°) and rotating 1 by 180° (that is: multiplying by i²) is -1!

Rotating by angle is just adding angles. If you rotate by 30° and then 45°, you rotate by 75°, right? Hence there is a plus sign with angles, for multiplication. The division, being inverse of multiplication, works opposite direction and gives the minus sign.

Jumping between representations:
In the two AC basics videos there is a lot of jumping between sinusoids and complex number in both cartesian and polar form. Even some calculations showing how one comes from another. But being able to come from one equation to another is not enough. The property, which makes those tricks possible, is different. But it is commonly found in maths. It is also used in electronics with Fourier and Laplace transforms.

The critical part is: switching between representation is reversible. You can switch to one, do some operations, switch back to the initial form. The simplest of such operations are mirroring or rotating. You may mirror negative numbers to positive, perform operations,⁽¹⁾ convert back. -4 + -7 is the same as mirroring that, calculating 4 + 7 = 11, mirroring back: -11.⁽²⁾ For any non-negative real number you may square it, multiply, take a square root and it will be the same as if you simply multiplied it. If you take many horizontal lines, after the first step they will all be parabolas: that’s the new space in which you may operate. After doing your job, you square root them, returning to the normal, horizontal lines. Stop here, focus on that image: realize that we have just “reshaped” the entire universe for a moment and then reshaped it back. It was not a single value that was squared, but everything — all possible values, at once.

Of course the above examples have no practical use. However, many such “reshapings” are interesting, because operations that are cumbersome in a straighforward representation are trivial after such a step. Older audience may still recall slide rules, which were doing exactly that: switching to logarithmic scale, performing multiplication by a simple addition, switching back. Phasors are exactly the same idea. We “reshape” our world to something that may not even make much physical sense, so some complicated operations become simple. Then we go back.⁽³⁾

Raising e to imaginary powers
If you are used to thinking of a^b as doing b times multiplication of a, raising something to a power that is a complex number will blow your mind. The thing is… we do not. It’s a notation that actually has a different meaning. e^x is an infinite sum for k from 0 to +∞:

^x0/_0!  +  ^x1/_1!  +  ^x2/_2!  + … +  ^xk/_k!  + …

I believe that at this point you see, that x can be anything. Even a complex number, as there is nothing weird in taking powers of a complex number. As it happens, for x being an integer it does exactly what you would expect from exponentiation you know. It is just more general.

⁽¹⁾ Remembering, that not only values, but also operations must be mapped.
⁽²⁾ At which point I remind about the previous footnote: operations must be transformed too. Coincidentally summation works the same, but multiplication is more complicated.
⁽³⁾ Aforementioned transforms are similar. Fourier transform represents a signal as a sum of sinusoids, yielding representation extremely useful while working with frequencies is of importance. Good luck doing it otherwise. Laplace transform represents signals in a manner that makes both the frequency and amplitude decay/growth obvious, turning ugly differential equations into basic arithmetic.

[bsfeechannel]

Very good. I think Dave nailed it. The only thing that I would change is that what he calls the real and complex planes are actually the real and imaginary axes, respectively. The complex plane is the surface formed by these two axes.

Real numbers are part of the complex plane, as are the imaginary numbers and any other complex number formed by the sum of those two.

I. e., real is not imaginary, but is complex.

[EEVblog]

Very good. I think Dave nailed it. The only thing that I would change is that what he calls the real and complex planes are actually the real and imaginary axes, respectively. The complex plane is the surface formed by these two axes.

Yes, the joys of having no script and just pressing record and talking.

[Nominal Animal]

I too think Dave hit the topic pretty well.

Polar (\$z = r \, \begin{array}{|l}\theta\\\hline\end{array}\$) and Cartesian (\$z = x + j y\$) representations of the same complex number \$z\$ can also be considered "transforms" of the other representation, with the logic that
 - in polar form, multiplication and division are simple, but addition and subtraction complicated;
 - in Cartesian form, addition and subtraction are simple, but multiplication and division can become arduously long and complicated.
You can always transform between the two.  When \$\theta\$ is in radians, the polar form is the same as the exponential form, \$z = r e^{j \, \theta}\$.

Tangentially related (pun not intended):

This ties in to the comment about the importance of (logical) transforms, recently.  Fourier transform (and its inverse) might be the most well known to EEs, transforming for example an AC signal between the time domain (as a function of time) and the frequency domain (as a function of frequency).  However, using complex numbers to represent a time-varying signal with two components – like alternating current with voltage and current – can be considered such a transform, a logical transform of the problem at hand; and picking the representation, polar or Cartesian, and even switching between the two a couple of times while solving a particularly hard problem, a logical transform of the way we describe the signal.

(I just wish the boffins had called them "complex values" instead of "complex numbers", because it is easier to intuitively grasp that a value can consist of multiple components.  That a "complex number" actually refers to something that has two separate components, and to write one out usually takes two numbers, is just unnecessary complexity, unnecessary cognitive load.)

If we use the word "domain" to describe the set of assumptions we have and the way we describe the problem, we can say something truly important: choose your domain wisely.  If there is no single "domain" that works well enough, then split the problem into smaller parts, and apply transforms as needed to solve each sub-problem in a suitable domain.

In computer programming, especially embedded or microcontroller programming, choosing the "domain" (so that you only need a small number of "fast" arithmetic operations to achieve the task at hand) wisely can mean the difference between a sub-$1 MCU/CPU and a $10+ one needed.
In systems and applications programming, in the long term, "maintainability" is a crucial detail in choosing the proper "domain".

Jumping between "domains" (via "transforms", like I described above) is like thinking outside the box; or, like looking at the problem from different angles.

If you only have one domain you feel comfortable in, it is like only having a hammer, thus making all problems nails.  If there is a single MCU one always uses, or a single family, is it because that one is a familiar one, or because it is a good fit to the problem at hand all considered?

I cannot put any of this into words well enough, much less into a video.  But, this is an important idea that spans not just all fields applying math, but basically all engineering and science fields.  I do not see it stated directly or indirectly often, but I do see experienced people apply it all the time; including Dave.  ("If you do this, then Bob's yer uncle, and the solution is plain obvious!")

This is extremely useful in practice: just because a problem looks difficult in one domain, it does not mean the problem is actually difficult; often, it is only difficult in that particular domain.  If you have a sufficient toolbox of transforms in your belt, and aren't afraid of using them, you can often find some other domain you can transform the original problem to and from, where the problem is simple or at least straightforward to solve.