"Veritasium" (YT) - "The Big Misconception About Electricity" ?

[bsfeechannel]

In RF, you will see Z = R + jX all the time in impedance calculations.

That's what the Smith Chart is all about.



Found in whatever cheap ass VNA, whose mention you find scattered all over this forum, for instance.



Or in professional multi hundred thousand dollar equipment.

[adx]

Paradoxical blinking stare.

I will have to have a think about it. But I stand by everything I said because I "feel" I am right - for now. Your post(s) is a perfect illustration of my point.

To Timfox, I didn't say RF engineers don't use j (actually I did a couple of posts back, but in different context), just that I suspect they don't believe it, having risen above it, to the point they realise sqrt(-1) has no physical relevance, with j being the unit vector that I say it is.

bsfeechannel: Seen your post come in. Yes j appears in a couple of places, as an annotation. I was thinking of cheap ass VNAs and Smith charts when I made my claim.

[HuronKing]

Paradoxical blinking stare.

I will have to have a think about it. But I stand by everything I said because I "feel" I am right - for now. Your post(s) is a perfect illustration of my point.

Well you're not in my classroom. I'd do a much better job of explaining it there than in a forum post whipped up over lunch break at work. 

To Timfox, I didn't say RF engineers don't use j (actually I did a couple of posts back, but in different context), just that I suspect they don't believe it, having risen above it, to the point they realise sqrt(-1) has no physical relevance, with j being the unit vector that I say it is.

I know you're talking to TimFox but I'm telling you that you don't need to suspect anything. I got training from RF engineers - they do believe it (you have to enter it into the simulation software!) and it has immense physical significance, just as 'zero' and 'negative' have immense physical significance. 

[PlainName]

Or in professional multi hundred thousand dollar equipment.

Blimey, and they couldn't even number the ports sequentially. The mere thousands of dollars ones probably leave the labels off.

[penfold]

[...]
Now this I absolutely agree with. The difficulties are in the pedagogy. sqrt(-1) is called an 'imaginary' number or a 'complex' variable but these names are strictly historical. We can blame Rene Descartes for coining the term 'imaginary' as a derogatory term to imply they are not useful numbers. Those names have no bearing on what the sqrt(-1) actually represents - and it IS a physical phenomena. It's no less 'real' than negative numbers are 'real...' or how some ancient mathematicians regarded zero as a meaningless number...

Like, what if I asked you to calculate the power supplied by a voltage source? But then you did everything right and discovered the value of the wattage is negative! Is that not a 'real' answer? Of course it is. All it means is that I tricked you in the problem statement - the voltage source is absorbing power instead of delivering power.
[...]

Nice try... maths just isn't that simple. It is kinda unrelated to negative and zero numbers: the algebra and arithmetic of real numbers and vectors as we know them today are defined, metric spaces, isomorphisms, and all that are properly axiomatically defined; the ancient interpretations were more from ill-formed and contentious philosophical bases which would lead to a disagreement. In contrast to that, the 'imaginary' unit, more generally, abstractly-describes translations between the mathematical representations of two 'real' quantities, e.g. phase angle... the imaginary unit is in itself not a physical quantity.

[bsfeechannel]

Blimey, and they couldn't even number the ports sequentially. The mere thousands of dollars ones probably leave the labels off.

They're not oscilloscope channels.

[HuronKing]

[...]
Now this I absolutely agree with. The difficulties are in the pedagogy. sqrt(-1) is called an 'imaginary' number or a 'complex' variable but these names are strictly historical. We can blame Rene Descartes for coining the term 'imaginary' as a derogatory term to imply they are not useful numbers. Those names have no bearing on what the sqrt(-1) actually represents - and it IS a physical phenomena. It's no less 'real' than negative numbers are 'real...' or how some ancient mathematicians regarded zero as a meaningless number...

Like, what if I asked you to calculate the power supplied by a voltage source? But then you did everything right and discovered the value of the wattage is negative! Is that not a 'real' answer? Of course it is. All it means is that I tricked you in the problem statement - the voltage source is absorbing power instead of delivering power.
[...]

Nice try... maths just isn't that simple. It is kinda unrelated to negative and zero numbers: the algebra and arithmetic of real numbers and vectors as we know them today are defined, metric spaces, isomorphisms, and all that are properly axiomatically defined; the ancient interpretations were more from ill-formed and contentious philosophical bases which would lead to a disagreement. In contrast to that, the 'imaginary' unit, more generally, abstractly-describes translations between the mathematical representations of two 'real' quantities, e.g. phase angle... the imaginary unit is in itself not a physical quantity.

This is veering really close to the question of "is mathematics physical?" and that's a big question! 

I'm saying that the terminology associated with 'imaginary' numbers is something we inherited from ancient mathematicians who didn't really know what they were dealing with - we got over it with negative numbers and zero, but sqrt(-1) is still something to be struggled with by students. I don't actually blame the ancient mathematicians - it's just unfortunate their prejudices about how to philosophically interpret these definitions have cursed students of today who hear something like 'imaginary numbers' versus 'real numbers' and assume these labels, by themselves, have something to do with physicality. They don't, at least in my opinion. 

In the case of electric circuits, we know the impedance of an inductor is Z = jwL and the impedance of a capacitor is Z = 1/jwC (thanks Steinmetz!)

Those impedances have physical effects and meaning on our circuits even though they have a weird looking j out in front. And while it is challenging to learn it's not so mysterious. As you said, it just means the incident current and incident voltage undergo a phase shift in time.

[bsfeechannel]

Would it have been doomed to also replicate "until then engineers could only produce boat anchors" had Bardeen not suggested surface states? Was he a 'proper' physicist, or an electrical engineer who went back to do some physics papers - or someone above classification, and the transistor was waiting for him and his particular set of interests?

John Bardeen had a degree in electrical engineering, but he took all the graduate courses in physics and mathematics that had interested him, and he graduated in five years instead of the usual four.

After that he applied and was accepted to the graduate program in mathematics at Princeton University. Then as a graduate student, Bardeen studied mathematics and physics. Under physicist Eugene Wigner, he ended up writing his thesis on a problem in solid-state physics.

At Harvard University, he worked with to-be Nobel laureates in physics John Hasbrouck van Vleck and Percy Williams Bridgman on problems in cohesion and electrical conduction in metals, and also did some work on level density of nuclei. He received his Ph.D. in mathematical physics.

As you can see, Bardeen was a full fledged physicist and went on to win TWO Nobel Prizes in advanced hacking.

And again, did physicists not have that same four decades to come up with the transistor?

Yes. And when the opportunity presented itself, they were prepared for the challenge. Engineers were not.

When analysing that situation, the effect you want to confirm seems lost in the noise and bias, and only one thing shines through (apart from cleverness persistence and teamwork of course): The almighty dollar.

No surprise, here. Science costs money. That's the whole point of the Nobel Prize.

If this thread has shown us anything, it is that most electronics engineering is devoid of any direct use of physics and math,

This thread has shown that electronics engineering devoid of math and physics reduces to a bunch of stupid misconceptions and dogmas bordering pseudo-science.

[adx]

j: It did get me thinking about the 'reality' of the unitary minus operator. It isn't 'real' either: I am quite within my rights as an engineer to answer the power supply question as "it isn't delivering power" or "it delivers no power" or perhaps "it delivers zero power, but...".

Unitary minus is a thing, not a number. It represents the taking away of something, a negative number is composed of this thing and a quantity. Zero has no thing, and no quantity. I can consider negative numbers to be real, but they are as unreal as zero. They work together though very nicely to model the behaviour of quantities, as do rational and irrational numbers. They can be treated as a continuous sequence.

Complex numbers are composed of (up to) 2 things and 2 quantities. They can't be ordered. Phasors add to that an assumption of a sine wave.

I didn't for a moment suggest that reactance has no physical reality - just the square root of minus one as a descriptor.

I forgot to mention, that my position is essentially that i <> j.

I stand my ground firm, on the grounds that it is firm and unyielding.

[bsfeechannel]

bsfeechannel: Seen your post come in. Yes j appears in a couple of places, as an annotation. I was thinking of cheap ass VNAs and Smith charts when I made my claim.

Let me get this straight. Because j doesn't appear explicitly on the VNA display, does it mean it is not there? Isn't the display representing a two-dimensional vector space? Aren't VNAs, VECTOR Network Analyzers?

Your point seems moot.

[penfold]

[...]
This is veering really close to the question of "is mathematics physical?" and that's a big question! 

I'm saying that the terminology associated with 'imaginary' numbers is something we inherited from ancient mathematicians who didn't really know what they were dealing with - we got over it with negative numbers and zero, but sqrt(-1) is still something to be struggled with by students. I don't actually blame the ancient mathematicians - it's just unfortunate their prejudices about how to philosophically interpret these definitions have cursed students of today who hear something like 'imaginary numbers' versus 'real numbers' and assume these labels, by themselves, have something to do with physicality. They don't, at least in my opinion. 

In the case of electric circuits, we know the impedance of an inductor is Z = jwL and the impedance of a capacitor is Z = 1/jwC (thanks Steinmetz!)

Those impedances have physical effects and meaning on our circuits even though they have a weird looking j out in front. And while it is challenging to learn it's not so mysterious. As you said, it just means the incident current and incident voltage undergo a phase shift in time.

Hang on, that's not a very big question, and the answer is relatively simple. Maths itself is not physical, or it is only as physical as any language in which you can express logic, it's conceptual. The links between that language and quantities defined within is also defined and there is an observable consistency between the results of additive processes in 'nature' and in the mathematical system etc... hence why one should always include units against any number with physical significance because that defines the process by which one takes the number on paper and stacks calibrated metre-rules end-on-end to reach a distance. It's all defined, we're safe.

The middle bit was just a segue, I guess it comes across as prejudicial in how dismissive philosophers and mathematicians were, but it wasn't until mid-1800s to early 1900s when the definitions of maths were really put in place to settle the philosophical absurdity of a negative number of chickens (being difficult enough to convey the attribute of "owning no chickens" without also listing everything else you don't own, let alone being able to express such a severe lack of chickens to consider it negative). That is a hugely important split because the same principles of ratiocination that form(ed) mathematics also form the scientific method's basis. So without the formal definitions, the link between reality and squiggles on paper become incredibly tenuous and (logically) absurd.

The concept of reactance in terms of the imaginary unit is defined for sine-waves of fixed frequency, where the derivative preserves the wave shape (sin, cos, -sin, -cos etc) but is translated with respect to the parameter, the imaginary unit facilitates that translation by a simple multiplication... for sine-waves. The decomposition into sine-waves is nice like that for letting us do that, but it isn't unique; we could decompose a signal into squares, wavelet-type things or any other orthogonal basis and for each of those we would have a new complex-number-type-operator, it would just be a far bit more grotesque than an 'i'. In each of those cases, it would be representative of a transformation of the representation of voltage and not in itself a quantity.

[adx]

bsfeechannel: Seen your post come in. Yes j appears in a couple of places, as an annotation. I was thinking of cheap ass VNAs and Smith charts when I made my claim.

Let me get this straight. Because j doesn't appear explicitly on the VNA display, does it mean it is not there? Isn't the display representing a two-dimensional vector space? Aren't VNAs, VECTOR Network Analyzers?

Your point seems moot.

Yep. There's degrees, ratios, other numbers, the concept of a 2D vector space, but no j (a bit like expecting there to be a "y" on a plot of voltage vs current), and certainly no sqrt(-1).

[bsfeechannel]

Yep. There's degrees, ratios, other numbers, the concept of a 2D vector space, but no j (a bit like expecting there to be a "y" on a plot of voltage vs current), and certainly no sqrt(-1).

And of course we should remove the study of Cartesian coordinate system from electronics engineering because y doesn't appear on the screen of any oscilloscope despite the fact that its display IS a Cartesian coordinate system and that any engineer who deserves to be called by that name must be capable of interpreting a Cartesian coordinate system. All of this because y doesn't appear on a plot of voltage vs current.

Gimme a break. Stop giving us that kind of crap only to justify your misconceptions and your precarious understanding of math and physics when applied to engineering.

[adx]

That's not what I'm saying, and the fact you can't work out what I mean has been a surprising insight into the way the human mind works.

Actually no scratch that and I'll remain true to form; it is what I kind of mean (and of course that insight is no surprise to me). Yes, stop studying Cartesian coordinates and silly unit vector formulas. Forget about y. Ignore "functions". Teach the oscilloscope display for what it is.

Prove my misconception. Explain what is the direct relevance of sqrt(-1) to engineering without reference to waffley texts.

[bsfeechannel]

Hang on, that's not a very big question, and the answer is relatively simple. Maths itself is not physical, or it is only as physical as any language in which you can express logic, it's conceptual.

Well, if the universe appears to behave consistently logical in certain circumstances, math can provide a convenient description of what is going on and even help to predict future discoveries.

[bsfeechannel]

That's not what I'm saying, and the fact you can't work out what I mean has been a surprising insight into the way the human mind works.

Yes. That's what you're saying, and that's what we are fighting against vigorously.

Yes, stop studying Cartesian coordinates and silly unit vector formulas. Forget about y. Ignore "functions".

And what you'll have is not engineering anymore. We are totally against this kind of movement, because it only serves to create half-assed "engineers" who like to oppose any knowledgeable person that points out their misconceptions.

Teach the oscilloscope display for what it is.

The oscilloscope display is an application of the Cartesian coordenate system. And whoever invented the oscilloscope had that in mind. That's what it is. If you're taught differently, you were duped.

Prove my misconception. Explain what is the direct relevance of sqrt(-1) to engineering without reference to waffley texts.

HuronKing has already done that on the previous page, showing how Steinmetz revolutionized the solution of AC systems with its approach. Engineering is not a simple subject. You can't simplify it beyond a certain point. If you don't want the "waffley texts", do not get near engineering. It's not for you.

[aetherist]

[...]Steinmetz knew that by doubling the dia or by halving the dia then the magnetic field did not change.
But STR said that the magnetic field did change.
STR said that u could get a magnetic field 1000 times stronger by simply using a very thin wire.

Define "stronger". The field strength at the surface of wires of different diameters carrying the same current will be different, but the field at a common distance from the center's line of each wire will be the same. STR doesn't say anything to the contrary... I think your arithmetic is in error, at least the "reduction" to proportionalities much earlier on was in error, so I guess whatever led to it was too.

I am having a good look at Purcell, re the STR explanation for magnetism near a wire with current.
If Purcell invokes reasonable & internally self consistent aspects of STR (& even praps GTR) then that will be ok.
But i have already seen at least one lie/error. I will explain later today. Really, this should have its own thread. Which would be mainly re the Faraday Disc Paradox.

[penfold]

Hang on, that's not a very big question, and the answer is relatively simple. Maths itself is not physical, or it is only as physical as any language in which you can express logic, it's conceptual.

Well, if the universe appears to behave consistently logical in certain circumstances, math can provide a convenient description of what is going on and even help to predict future discoveries.

Yeah... but it is no more "physical" than a verbal description of a rock, I could describe all the properties of a "rock" in such beautiful detail that suddenly people all over the world suddenly begin to see "rocks". The tangible object to which I attributed the name "rock" and properties of being kinda round, a bit jaggedy, hard... ya know, rock things... will have existed before my description, just more rocks had been identified and unfortunately a few tortoises... until that description gets refined. My verbal description remains purely verbal and not at all physical, it just describes a physical object.

Maths is a descriptive language in which the natural phenomena are described, from those descriptions we can hypothesize, test, and refine new theories... the phenomena, including the big bang, relativity, quantum, etc all existed before humans and maths... yet that curiously happened. The language in which these descriptions are encoded - since it can be communicated verbally... is not exclusively physical.

[SandyCox]

bsfeechannel: Seen your post come in. Yes j appears in a couple of places, as an annotation. I was thinking of cheap ass VNAs and Smith charts when I made my claim.

Let me get this straight. Because j doesn't appear explicitly on the VNA display, does it mean it is not there? Isn't the display representing a two-dimensional vector space? Aren't VNAs, VECTOR Network Analyzers?

Your point seems moot.

The concept of something being "physical" or "real" is in the eye of the beholder.

If you don't believe in complex numbers then just don't use them. You will still be able to do a large part of Electrical and Electronic Engineering by solving the underlying differential equations in the time domain and use a lot of trigonometric identities which will become extremely tedious.

If you chuck out the complex number then you also chuck out Phasor analysis and the whole frequency-domain perspective. You will also loose the Nyquist stability criterion which relies on Cauchy's argument principle. How would you do antenna theory without residues and branch cuts? What about root-locus analysis and design?

I find it strange that you have problems with the Complex numbers but apparently accept the axiom of choice.

And i = j. It is just a difference in notation. Its unclear why we use i for the current shouldn't it be a?

[adx]

This is all a bit silly - it started with a gentile troll about i vs j, then we're now back to arguments over half-arsed engineering.

That's not what I'm saying, and the fact you can't work out what I mean has been a surprising insight into the way the human mind works.

Yes. That's what you're saying, and that's what we are fighting against vigorously.

That's what I said;

Actually no scratch that and I'll remain true to form; it is what I kind of mean ...

Yes, stop studying Cartesian coordinates and silly unit vector formulas. Forget about y. Ignore "functions".

And what you'll have is not engineering anymore. We are totally against this kind of movement, because it only serves to create half-assed "engineers" who like to oppose any knowledgeable person that points out their misconceptions.

Uh uh. We'll have the reality of the industrial engineer, what you're complaining about is not misconceptions, but work. So run with the reality, and stop assuming students need to "study" Cartesian coordinate systems (why?!) and teach the concepts.  Maybe then there won't be so many half-assers about. At least they'll get a head start towards reality. All this mathematics and (dare I say it) physics, does no good.

But as with most silly arguments, there are two sides to them. Rather than maximising the troubles, I can try to minimise. I'll allow the students to learn all the words and notation (claptrap), have them study what they'll never use, make it so confusing they can't see the wood for the trees and therefore have to go search for it themselves. That's a proper education. Problem is, you want us all to be mathematicians and physicists, while denigrating the research abilities of engineers, thereby maintaining a false tension with which to maintain your special ideas.

That's why I'm not totally against that type of movement.

Teach the oscilloscope display for what it is.

The oscilloscope display is an application of the Cartesian coordenate system. And whoever invented the oscilloscope had that in mind. That's what it is. If you're taught differently, you were duped.

I was going to let you have that one, it's plausible that chart recorders etc were invented by a mathematician fanboi/grrl of Descartes. Ancient civilisations drew graphs and plotted grain weights through clairvoyance - through the fog of impracticality the letters "T e k t r o n i x" slowly fade into being, and hello what's this, a "graticule"? Looks handy for the cave wall, illuminated by the light of the fire. Then the quick (albeit foggy) glance over to the timebase knob to check if it goes up to 1 moon / div. "Oh well, maybe those future civilisations will just have to wait for the right advances to come along, I hope for their sakes they don't stop studying Cartesian coordinates along the way."

Prove my misconception. Explain what is the direct relevance of sqrt(-1) to engineering without reference to waffley texts.

HuronKing has already done that on the previous page, showing how Steinmetz revolutionized the solution of AC systems with its approach. Engineering is not a simple subject. You can't simplify it beyond a certain point. If you don't want the "waffley texts", do not get near engineering. It's not for you.

All wrong.

Simple question, met with handwaving. Once again, this is about sqrt(-1), not vectors. That expensive thing you showed is called a vector network analyser, not a really complex mathematical network analyser.

[adx]

bsfeechannel: Seen your post come in. Yes j appears in a couple of places, as an annotation. I was thinking of cheap ass VNAs and Smith charts when I made my claim.

Let me get this straight. Because j doesn't appear explicitly on the VNA display, does it mean it is not there? Isn't the display representing a two-dimensional vector space? Aren't VNAs, VECTOR Network Analyzers?

Your point seems moot.

The concept of something being "physical" or "real" is in the eye of the beholder.

If you don't believe in complex numbers then just don't use them. You will still be able to do a large part of Electrical and Electronic Engineering by solving the underlying differential equations in the time domain and use a lot of trigonometric identities which will become extremely tedious.

If you chuck out the complex number then you also chuck out Phasor analysis and the whole frequency-domain perspective. You will also loose the Nyquist stability criterion which relies on Cauchy's argument principle. How would you do antenna theory without residues and branch cuts? What about root-locus analysis and design?

I find it strange that you have problems with the Complex numbers but apparently accept the axiom of choice.

And i = j. It is just a difference in notation. Its unclear why we use i for the current shouldn't it be a?

Getting late so a quick reply.

Not liking complex numbers in electrical engineering is down to their physicality for me, so if that is in the eye of the beholder then I don't have any choice, because it is my eye not me who doesn't like it. I just don't think the concept of sqrt(-1) can have physical reality, where I do think (or feel) that the concept of negation can.

But ultimately it isn't that I don't like them, but they (I can only assume) catch many students out - given that some people do unquestioningly think the imaginaryness has some (or immense) physical reality.

I am perfectly happy with solving underlying differential equations in the time domain, and trig identities or evaluation, so long as I don't have to do it (a computer can). It's more direct, it's more physical.

I personally think chucking out the complex number is the best thing, or define j to be a unit vector which works with the same operations as i (or lie to students and tell them that, while still using complex numbers). Phasors are inherently 2D, there is no squaring and -1 involved. It, and frequency domain analysis (for real signals, Hermitian transform) still work like they always did - except the sins and cosses are explicit, and not some arcane 'exponentially' complicated notation (kind of like the weirdness of the C language that people get used to).

I'm going to have to pull an aetherist and say I don't know about root-locus analysis. I don't remember it being too horrible, so I might have forgotten it due to disuse, rather than blocking it out. I remember not liking it very much.

i for intensity? I never worked that out either. I was thinking c for current and call capacitors twangductors or something. It's the right thread for that sort of thing.

[TimFox]

Here is a very simple circuit involving resistors and capacitors.
I was looking for a slightly more complicated circuit (RIAA equalization), but couldn't locate the file quickly.
It is very easy to solve for the ratio of (output voltage) / (input voltage) as a function of frequency, using elementary complex algebra (with X_C = -j/wC).
Of course, one could derive the differential equation for the time-dependent output voltage with an input voltage equal to a (real-valued) sine function, and then solve it (perhaps with a LaPlace transformation).
Or, one could do an .AC analysis in SPICE (which is strictly algebraic and undoubtedly uses complex algebra internally).

I'll defer to HuronKing above about pedagogy.  Fifty years ago, I had difficulty understanding or applying complex variables to electrical circuits, but after studying linear algebra and other useful mathematical subjects, I learned to appreciate the usefulness of "complex notation" for real problems.  At the end of a practical calculation, one can express the voltage as a real-valued function of time. 

I have several two-phase lock-in amplifiers, that display the "In-phase" and "Quadrature" components of an input signal with respect to a (coherent) reference signal.  To confuse, these are called "I" and "Q", respectively, but are often referred to as "Real" and "Imaginary" components.

[HuronKing]

[...]
This is veering really close to the question of "is mathematics physical?" and that's a big question! 

Hang on, that's not a very big question, and the answer is relatively simple. Maths itself is not physical, or it is only as physical as any language in which you can express logic, it's conceptual. The links between that language and quantities defined within is also defined and there is an observable consistency between the results of additive processes in 'nature' and in the mathematical system etc... hence why one should always include units against any number with physical significance because that defines the process by which one takes the number on paper and stacks calibrated metre-rules end-on-end to reach a distance. It's all defined, we're safe.

Okay, okay, now you hang on! 
The question I've been answering, from adx, is this one,

Is there any place in engineering, anywhere, where sqrt(-1) has any physical relevance at all? The only place I've ever seen it doing something useful (beyond being an arcane convenience for mathematicians) is in a Feynman lecture where it quasi-continuously described a wave function inside and out of an energy well or something (I can't find it now).

If you're suggesting that sqrt(-1) has no physical relevance because MATHEMATICS has no physical relevance... then yea... okay let's go with that, sqrt(-1) has no physical relevance because it's part of mathematics which inherently has no physical relevance.... it's kind of a tautology and one I don't find that terribly helpful for 1) engineering students or 2) actual engineers trying to devise logical frameworks to relate phenomena to a method of describing and predicting them.

I read adx's question as, if we use sqrt(-1) in our engineering calculations, what does it mean? Does it have a physical meaning? Or is it just something used to torture students with 'claptrap' and is useless for all us manly-men practically practical-minded engineers? That's a valid question but totally independent of the philosophical question 'is mathematics physical' which I don't really care about (I mean I do, but not here). 

Now in answering adx's question, in engineering, does sqrt(-1) have physical meaning? The answer is a resounding YES!!!

Just because people got confused by bad pedagogy in school (I'm included in that) or there are specific engineering lines of work or problem solving techniques that don't use sqrt(-1) is TOTALLY independent of my answers to his question.

Does sqrt(-1) have physical meaning in engineering? Yes. Steinmetz proved it (and I hope references to 'waffley' texts aren't in reference to Steinmetz's treatise). And Edith Clarke literally wrote the book on AC Power Analysis. She was hired as the first woman electrical engineer in the USA in an age of extreme sexism by General Electric to solve power problems stumping their engineers - some of these were problems no one else could figure out:
https://www.google.com/books/edition/Circuit_Analysis_of_A_C_Power_Systems/JB4hAAAAMAAJ?hl=en&gbpv=1&printsec=frontcover

I mean it... she literally solved problems no one else could figure out by using hyperbolic functions and complex impedances. She is a big reason our long-distance energy grid can even exist:
Steady-state stability in transmission systems calculation by means of equivalent circuits or circle diagrams
https://ieeexplore.ieee.org/document/6534694

You can read the paper here:
https://speakingwhilefemale.co/wp-content/uploads/2020/09/Clarke_Transmission.pdf

And I've already shared MY experience in RF engineering and antenna design that sqrt(-1) has tremendous physical meaning and application. If complex phasors and impedances and sqrt(-1) is all worthless claptrap for engineers - then don't use AnSys HFSS simulation software and stay away from RF, I guess? 

If the response to all this is 'nuh uh, I've never needed it..." well then... fine. Good for you. But don't be deluded into thinking that other engineers aren't using it and ascribing physical meaning to it all the time and changing the world. BTW I used complex impedances last night in my class explaining the origins of harmonics in motors and how to interpret a 3-phase phasor diagram like you'd see on a Keysight Power Analyzer:


I really don't think I need to provide more examples of the 'physical relevance' of sqrt(-1). Take it or leave it. The power engineers and RF engineers are quite happy with it.

One last thing about sqrt (-1), I REFUSE to let ourselves be biased against the attribution of physical meaning to sqrt(-1) because freaking Rene Descartes decided to be a smartass and call them 'imaginary numbers' as if they were 'less real' than other numbers in mathematics (which has been argued that NONE of the other numbers in math are real/physical either so the distinction is irrelevant).

So adx, I submit to you that your issue with the 'physicality' of sqrt(-1) is because of an idiotic naming convention.

This is a mess and as a teacher/working engineer/former student who also got confused, I hate it. Screw you Descartes and screw all the math teachers in the intervening centuries who perpetuated this tragedy of a ridiculous name. That bastard Descartes didn't even know how to take a derivative or do a surface integral (he did help us get there though). He shouldn't be allowed to confuse students for centuries because of his antiquated philosophical biases.   

At least when the Big Bang got a derogatory name it was kinda cool sounding... but it also has confused people about what cosmologists actually think about it. 

[SandyCox]

Some people are of the opinion that only the natural numbers (excluding 0) have "physical meaning". (whatever "physical meaning" might be?)
Does 0 have physical meaning?
Do the negative numbers have physical meaning?
What about sqrt(2) which is an irrational number?

Extending the real numbers to an algebraic structure in which the square root of minus 1 exists is brilliant. Furthermore, all polynomials can be factored into monomials. How great is that?

The field of Complex numbers is not the same as the vector space of two-dimensional vectors over the real numbers. The multiplication is different. j isn't a unit since its square isn't equal to j. 1 is the unit of the Complex numbers.

In my opinion, trying to assign "physical meaning" to mathematical concepts only works for very simple problems. Just trust the Mathematics and look at what the theory tells you. Sometimes our intuition fails horribly. Trust the math.

[PlainName]

Blimey, and they couldn't even number the ports sequentially. The mere thousands of dollars ones probably leave the labels off.

They're not oscilloscope channels.

Yeah, I thought the smiley was implied but on reflection I shouldn't have left that kind of thing to chance. Here's a belated one: