Why do Electrical Engineers use imaginary numbers in circuit analysis? 

We have nice imagination I'd say

the horror... the horror....

Actually, it’s just so that we can have a squared quantity come out negative.

you misspelled beautiful

​@@cougar2013Not really, it's more profound than that; if we just wanted a root of -1 you can work in ℝ[X]/(X²+1) and there's your root We use complex numbers so that a single complex exponential Ae^(jωt) can represent a sinusoidal wave and encode its frequency, phase and amplitude at once. And all that is to be able to work with ideal sine waves and then apply the superposition principle (corresponding to the Fourier transform) with which you deduce what happens to your circuit with any kind of signal

Why electrical and not something more trendy like Artificial Intelligence?

​@@Anonynomymon-fh8wyYou know electrical engineers can do that too when specified in the field, right? That's the deal of electrical engineering, many majors. You can specify in things like robotics and programming. You pick one or two and specialize. I'm a student yet but I started learning coding in lessons. Then I got curious and learnt some languages. I wasn't interested in AI but anyone who does can do a job about it pretty sure

@@Anonynomymon-fh8wy Artificial intelligence is considered too concentrated and specialized, EE's can also do Ai jobs and do a masters in Ai.

May I ask why? I'm interested in Physics and the statistical component of mathematics, I'd love to hear about your experience.

​​@@Anonynomymon-fh8wyML is an overlap of electrical engineering and computer engineering. But still, EE is the most versatile and broad field of engineering. My master's will be an overlap of math models, machine learning, and digital signal processing

It is when the inductors and capacitors have some stored energy. We usually analyse the circuit when they are in steady state means the transient response has died out long ago.

Quando se adotou números complexos na Engenharia Elétrica, não tinha como se detectar um fenômeno de frações de segundo.

These circuits can be generalized to the s-domain, where the transient response is not ignored. Using purely complex exponentials is a stepping stone to doing that

I got chills the moment my mind made the link between the Laplace transform and the imaginary numbers.

It was in the 1980's when I learned about Laplace transform and complex numbers for electrical engineering during my polytechnical college education. Later on, I also did some eddy-current testing, in which this theory is put into practice. It was mesmerizing to observe how all of these concepts actually work, touching the objects and manipulating the sensors to act as they are supposed to do. It's not just theory for its own sake.

How did he become a professor?

@@nicolaignazio it’s really not that uncommon. He was perfectly competent at his research and still is world renowned in control theory. It was so second nature to him to think of current as complex that he didn’t see the need to keep the justification for it around…. 🤷‍♂️

Have you considered the possibility that the professor was messing with you?

@@ivanmirones9220 He definitely wasn’t. This was 1999, in one of the first lectures. I’d done a year out, so was a bit more confident, everyone else was straight from school. Eventually I gave up. I got to know him more later, he just literally did not give a shit about the physical reality of systems, or considered it a distraction - he was completely consumed by ‘pure’ control theory.

He's metaphorically right: It's called "displacement current" and correspond on the woobling of electrons inside insulators as an effect of an alternate electric field. That result in a 90° shifted alternate current that is the virtual continuation, inside the dielectric, of the alternate current flowing within a capacitor. You can - at that point- represent it as a flow of imaginary virtual particles. If math works, why not?

When you realize free energy exist

​@@lukiepoole9254 when you steal a line from your neighbours or charge your phone at restaurant/hotel.

same here!

@@mandarbamane4268 Haha no. Parametric resonance exist and electrical engineers are NOT taught about it.

Welcome to poverty world of poors ...

Welcome to hell then.

It’s the coolest stuff I’ve seen in the physical world. Linear systems and controls are damn fascinating! Well, so is waveguide and antenna theory, and communication theory! This video shows how you do steady state system analysis, but laplace transforms, which aren’t too different, let you do transient analysis using just algebra too. And what is MIND BLOWINGis that since you’re really just solving ODEs, all this stuff applies to dynamic mechanical (or chemical!!!) systems too! Eg a spring is an inductor, a shock absorber is a capacitor!

This is absolutely true lol. Whenever i see complex number in physics it starts to get confusing af 😂

because I sat there in class and they just spewed it at us

Because real numbers are too complicated for us*

@@nestorv7627 yes, sometimes the longer way is actually the shortcut

​@@nestorv7627If I could code, I'd do an example of RLC analysis using phasor diagrams. Voltage, current, & power, with resonance visuals if ya like seeing humps/dips and cutoffs. Put the values on sliders so you can see what them numbers do. Str8 vectors and waves using real numbers.

I would love a video on this topic

@@rennoc6478 facts

Man nerds

had us in the first half not gonna lie

Actually this Is something different. It's called Steinmetz Transform, look It up. The connections are incredibles.

Thank you for this. As a mathematician switching to EE for a fresh start, it scares me how engineers sometimes ignore nuances like these

Why do we ignore the imaginary part?

@@nestorv7627 No problem. There was actually an error in my logic caused from coping and pasting the text. I wouldn't have noticed it, if you never commented, so thank you.

@@leonhardeuler9839 I think Zach did it because loves his phasor analysis too much. There is also half as much work to show, and also half the work to grade, so professors and students may like it for that reason. The alternative is to differentiate the last expression of v = cos(wt) = RE(Ae^(jwt)) = (A/2)*(e^(jwt) + e^(-jwt)) and multiply by C, then you don't need to worry about "ignoring" the imaginary part; it will just cancel out in the end anyways. I think the real question is why convert it to complex exponentials in the first place, and the answer is all about the basics of frequency domain analysis.

@@nestorv7627 It was my dream to become a mathematician, but my DE prof pulled me aside to talk (he probably didn't get many students that score perfectly on both exams and hw); I told him that I wanted to study math, but he persuaded me to get into engineering instead. I graduate EE next sem, and the only time I felt that I learned any real engineering was the second/third week of EM waves when we were solving problems using the derivative form of maxwell's equations. Every other time, they just lump the necessary parameters, needed to actually solve the DEs directly, into empirical ones, then only expect you to use the "formula" or "procedure" instead of actually understanding the actual physical process. Am I missing something, or would you also say that, for the most part, engineers suck at understanding the actual math used in their field?

interestingly enough, further into a circuits course you end up using the laplace transform to do exactly that. The laplace domain actually takes advantage of imaginary numbers as well, the S variable ends up being an imaginary number.

That class is bread and butter EE!

😂😂😂😂😂😂

This is basically the same way they teach in highschool before complex numbers. Note that by "keeping track of the angles" and using special formulas to add there values with different phases you are basically doing the same calculations as if you were to do it with complex numbers by hand (using trigonometry and pythagoras' theorem) While simples to understand, this method has the disadvantage of having different formulas (for example you cant just calculate the parallel resistance with XY/(X+Y), as you would have no way of accounting for the phases). Also, if there is both series and parallel connections, this can quickly spiral down into lots of trigonometry just so you can keep the phases, while with complex numbers, you are basically hiding all this nasty arithmetic by using a single 2D number to store both the amplitude and phase (and mathematically this allows you to use the EXACT SAME formulas as you would for DC, so you don't need to learn any new formulas except for the generic complex arithmetic)

Ok

I thought we EEs were sharp til I started studying fluids. I don’t EVER want to have to find any PDE BVP solution in closed form….

No. I disagree. The hidden but necessary assumption is that the analysis takes place at a constant frequency. All passive components are modeled as being linear. There is a phasor representation for the Fourier series, if ... at least... non-linearity can be modeled around harmonics of a fundamental frequency.

Pretty sure he has

i think the ideal representation would be as a vector with two components in a geometric algebra, because then we get the benefit of being able to multiply by a constant to get a phase shift as well as the intuition associated with vectors

i think it would cause a bit of confusion as well, bc they don't behave as vectors all the time, for example, there is no dot product or cross product. It makes sense to think of them like vectors, but i think we should do only that, not redefine how we represent them

@@andredetoni897 There is nothing wrong in redefining how we represent something. It all depends on context of the problem and whichever convention you start with you should also finish with, just don't change midstream. It's kind of like whipping up some batter. You're not going to start to stir or whip with a fork and while still whipping it transition the fork into a whisk. That just doesn't happen. Intuition and ingenuity is what allows us to elaborate such models and representations. If people didn't redefine things throughout time then we wouldn't have the tools of calculus that we do today. Here's a prime example; how many times throughout the last 200 years has the model of an atom changed? How many times has various equations, sequences, series, summations, etc... changed due to someone redefining them in different terms? We wouldn't have the complex number system today if people didn't redefine things! Having various representations of the same thing can give us many different perspectives of their properties and behaviors.

@@skilz8098 you're right about all of that man, absolutely. I just think that in this example particularly, it's not necessary. We already treat complex numbers like vectors in a multitude of ways (like calculating power relationships) but they're not fully vectors. I think it's tough to redefine a complex number as a vector when it follows only some of the vectors property, that's all. And I understand completely your point, it would be very convenient. I'm studying to be a professor, so I have kind of strong opinions about how we should teach topics (it doesn't mean they're right tho), and for what I've seen, it's better to define clearly that complex numbers ARE NOT vectors. You make great points thoigh, and I agree with you

@@andredetoni897 As for me, I see numbers and vectors as being the same thing. A number that has a value has a magnitude. Also the sign of that number implies its direction. Take for example +3 and -3. Their magnitude is the same, a weighted or measurable value of ||3||. Yet they differ in sign. One is positive and the other is negative. They are facing or pointing in the opposite direction. If you take the dot product over the product of their magnitudes it is equal to the cos of the angle between them. In other words cos(t) = a dot b over | a | | b |. We can take the points (3,0) and (-3,0) and we end up with cos(t) = (3,0)dot(-3,0) / |3| |-3| which would give us 180 degrees or PI radians. This is the angle of a straight line. So when we look at any number on the number line even the number 1. It is unit vector with its tail at (0,0) and its head at (1,0). Which can be simplified to the unit vector or simply . So in truth all scalars are vectors where their magnitude is emphasized and their directions is their sign. Vectors and numbers are not mutually different. In fact I tend to think that they are the same exact thing because even the number 1 itself is a linear translation of 1 unit in some arbitrary direction. Did you know that the Unit Circle and the Pythagorean Theorem are embedded both in the expression y = x as well as the simplest and first arithmetic expression 1+1 = 2? I could show proof here but his is quite long as it is... If you're interested just le me know and I can show how all mathematics as well as all applied mathematics (science) are related well particularly physics...

watch the video before commenting next time 😂😂😂😂😂😂

Yes he should have told that it's steady state analysis