UKH

I'm a maths teacher, I saw that a collegue had been teaching some electronics.  

I noted that there was the equation, Z=sqrt(R^2+(xL)^2) written on the board, I asked him "what's Pythagoras doing there, where is the right angle?" "that's just the formula" he said.  I pushed slightly but he didn't have any clue why the calculation of the impedence, would be related to the resistance and the inductive reactance with a pythagorean formula.

Am I entirely barking up the wrong tree, or is there some underlying orthoganality at play with the resistance and inductance which helps us calculate the 'size' of the impedence?

The right angle is between the real and imaginary plane, Z = R + jwL so |Z| = sqrt(R^2+(wL)^2). where w = 2*pi*frequency. Googling phaser diagrams will get you onto the right track.

"The phase angle of reactance, either inductive or capacitive, is always 90o out-of-phase with the resistive component, so the circuit's resistive and reactive values cannot be simply added together arithmetically to give the circuits total impedance value. That is R + X does not equal Z."

https://www.electronics-tutorials.ws/accircuits/impedance.html

What someone will have to answer is if resistive impedance is '90deg out of phase' because it's defined/constructed as such (ie impedance over and above DC resistance) or if there's a more fundamental reason (ETA which lukevf may just have given)

And you'd think the person teaching electronics would know this... I'd be pretty unimpressed with any teacher who says "it's just the formula" and doesn't know how to explain it.

Yes, there is, kinda sorta.  The voltage across a pure resistor is proportional to the current flowing through it, whereas the voltage across a pure inductor is proportional to the rate of change of the current.

If the current is sinusoidal, and at any instant is proportional to sin(t), then the voltage caused by resistance in that instant will also be proportional to sin(t), whereas the voltage caused by inductance in that instant will be proportional to d/dt sin (t) = cos (t).

There's your right angle.  If you represent those two voltages using rotating vectors, there's a 90° phase shift between them, and you could use Pythagoras to work out the magnitude of the vector you get by adding them together.

A resistor is a passive component - the voltage drop across it is simply proportional to the current through it, AC or DC.  [V = IR]

Reactive components that can store and release energy - inductors and capacitors - have a voltage drop across them that  is out of phase to an alternating current passing through them.  This means that their transfer function describing the voltage drop across them is an imaginary quantity, with 1j representing the phase shift they produce between the current flowing through them and the voltage across them. [V = jIX] where 'j' is sqrt(-1) and 'X' is the reactance calculated from "L" if it's an inductor and the frequency of the waveform.

If you combine a passive and a reactive component you get "V = IR + jIX" where V is a complex quantity.   If we want to assign a single complex reactance we can say "V = iZ" where "Z=R + jX".

In this V and i are complex quantities (an AC waveform with a frequency and a phase - with the angle (argument) of the complex Z introducing a phase shift between i and V), 

Now, if you just want the magnitude of the transfer function |Z|, you want to find the magnitude of "R + jX", which I expect you can see is where your right angle comes from.  It's the magnitude of a complex number.

To clarify something that confuses some folks, instantaneous voltage and current are real numbers, but when studying how a pure AC current produces a pure AC voltage drop, using complex numbers allows the relative phase of those waveforms to be studied as well as their amplitudes. Indeed consideration of the phase is required to get the correct results.  This provides a basis for modelling the effect on arbitrary waveforms.

It's fundamental.  The post from deepsoup above it answers it clearly for inductors.  It's similar for capacitors, the rate of change of voltage across a capacitor is proportional to the current through it.  If the driving voltage in either case is a sine wave with respect to time, that means the current is a cosine wave - that is a phase shift of 1/4 of a cycle from the voltage wave.  So, the phase shift is fixed in terms of radians (or whatever) for any frequency.  Edit: This is because the rate of change is a derivative, and when you derive a sine wave you get a cosine wave.

Which is handy.

In terms of the OP, I see no real world or pedagogic value in teaching a formula for the magnitude of reactance.  As the OP’s question shows, it illuminates nothing about anything really.

This has been a brilliant thread so far, thanks to all who have replied.  In response to WT about the value in teaching the formula.  This is on the new T-level course, which is being taught to kids in Y12 who got a GCSE grade 4 (equivalant to a low grade C in old money) so they have no idea about what sin or cos is other than a button on a calculator, they have no idea what a complex number is, and no idea what a vector is other than how far you move along and up.  Given this starting point "it's just the formula" is probably about as far as they can get. 

I agree it's sad that the teacher didn't have the knowledge or intellectual curiosity to know, or find out, however when I asked about the Pythagoras part of the whiteboard he started talking about the cosine function which was there - showing he either didn't know the difference between trigonometry and pythagoras or didn't recognise the formula I have posted about as pythagorean and assumed I was mistakenly talking about the trig and calling it pytagoras - either way, he's not the brightest spark and it's bloody worrying that he's the one teaching this course.

Thank you all again for the insight. 

Complex numbers are a bit abstract, obviously. 

Not so much vectors though, I mean the concept of using coordinates to describe a location on a map, and the idea that the relationship between one point on the map and another involves a direction and a distance is pretty relatable.

"It's just the formula" might be all there's time for, but in the context of AC electricity I think rotating vectors are much more directly relatable to the real world than a Pythagorean formula if you come back to the reason that the waveforms are sinusoidal in the first place - we generate our electricity using rotating machines!

If you draw the rotor of a stylised generator, seen end-on, it just looks like a wheel.  Put a dot on the rim, draw an arrow from the axle to the dot on the rim, plot the height of the dot above/below the axle as the rotor turns - there's your rotating vector and the resultant sinusoidal waveform as a tangible thing in the real world.

If you're dealing with 3-phase electricity, you can draw three dots equidistant around the rim in red/yellow/blue (brown/black/grey if you wanted to relate it to modern wiring) with three vectors pointing towards them, and there's the phase relationship between the three phases.

If you can use Pythagoras to demonstrate why sin(60) = sqrt(3)/2, then it's a short step from there to drawing an arrow from the red dot on the rim to the yellow dot on the rim, bisecting the angle from the axle and showing why the voltage between red-yellow phase is sqrt(3) x single phase voltage of either with respect to ground.

Miiight still be a bit abstract to talk about the 90° phase relationship between resistance and inductance, but at least looking at the vectors it's clear why you use Pythagoras to work out the magnitude of their sum instead of just adding them together.

Well, it allows them to calculate the impedance of an RC or RL network at a given frequency.

Granted, there's a lot more understanding that could be imparted by explaining the origin of the formula, such as how these networks affect phase and amplitude at different frequencies.

I love this thread.

Aside from sharing practical info about climbing, help with just living life,  I'm increasingly finding a trove of 'interesting' stuff in the various forums.

Though I can read the language used in the thread, and can even understand some of the ideas, most of it is way over my head. I remain in awe of experts - whatever their field.

On an average day, I look at three websites for my daily dose of news/views: UKC is the third. I save it until last, cos I know I can be entertained, informed or just impressed by the words of other UKCers; and it's a good antidote to the doom/gloom on the other two sites.

Thanks UKC !

Don't know their background, but as someone who trained as an engineer (not mathematician or physicist) I am definitely interested... and I can see the right angle in this one, but at the end of the day I personally... am happy to work with formulae/algorithms that are plenty good enough and work nicely even if they haven't/can't be proven, I just want to know if it's reliable.

If it gives an answer within the accuracy I need with the kinds of range/size of inputs I have, then I've plenty got other things to worry about and be getting on with, without quibbling over where it came from.

Good enough = Good enough 😉

Phrased a bit tongue in cheek here, but also serious.

I think that's a little harsh. There's always a deeper why. For engineering, I think it's ok to sometimes just say here's an equation that explains the relations of the various things, here's how to use them, now shut up and eat your dinner (to paraphrase https://en.wikipedia.org/wiki/Oliver_Heaviside).

My experience is that the best teachers are ones that are skilful at teaching (understanding what you don't understand, and finding a way to take you to a place where you do) and really only need to be a page to a page and a half ahead in the textbook to achieve that.

I absolutely agree that understanding something is not required to use or teach it, and being a good teacher is a completely different skill. A friend of mine was head of maths at his primary school and by all accounts a great teacher, but he was absolutely hopeless at maths. I am a hopeless teacher and despite understanding a whole bunch of stuff I can't for the life of me explain it to anyone else in a way they understand.

But, I can't see anyone really learning about the subject if they're just teaching equations with no background. It must be dry as hell and no one is going to remember it. Maybe I'm a particularly visual person, but whenever I calculate impedance or anything involving vectors I see it as a 2D picture. I don't think I could do just plugging numbers into a mystery equation.

Yep, no agreed. Blind calculator mashing isn't the aim of the game and you'd always hope to get a deeper understanding, but sometimes things need to be ring fenced a bit.

So then getting good an hypothetical to illustrate how you can mix just plugging numbers into a mystery equation with really giving students a good intuitive feel for something simultaneously: OP said electronics so maybe a lesson on filters (it's inherently mixed frequency so the phasor diagram reaches it's limit). It took us a good few weeks to get to the voltage divider equation V_out/V_in = Z_a / (Z_a + Z_b) with various see-saw analogies and the like. Then ontop, following that when Z is a function of frequency, you therefore get different gains for different frequency signals. That's the crux of a filter. Directly giving Z as a formula doesn't harm that takeaway in my opinion.

I have no clue if my A-level electronics teacher knew why pythagoras was involved or not. Very non-obvious if you haven't also studied electrical engineering, they were a geologist by training so i suspect not. One of the my all time stand out teachers.

(He was good an curious tho, so probably would have engaged with OPs question in a "let's find out together" kind of way.)

An excellent thread!

A few thoughts from another maths teacher (retired). I know nothing of electronics, and so sadly, as I now see, have never used phase diagrams and phase vectors. When teaching, I always loved teaching the stuff about the sum of two sinusoidal waves with the same frequency being another wave, which does seem rather miraculous (and, of course, very important). I had not realised that thinking of a wave as the horizontal component of a rotating vector made it geometrically evident, by the addition of two such vectors, that this is the case - some pupils would have really appreciated this. The school maths "textbook" way of doing this with Asint + Bcost by "doing pythagoras" on A and B to get R and then taking it out as a factor to get R(A/R sint +B/R cost) and then noticing that A/R and B/R are the cos and sin of some angle, say p, so then, using a compound angle formula we get the sum being the wave Rsin(t +p). So the same pythagoras gives the amplitude just as in the phase diagram appoach. Of course if the two waves do not have a phase difference of 90, then, with a phase diagram, the cosine rule is needed to get the amplitude, and, in the "textbook" approach, some jiggery pokery with compound angle formulae is needed. 

Using the real part of a rotating complex numbers in the exp(it) form does, by factoring out exp(it) from Aexp(it) + B exp(i[t + q]) make the calculation of the amplitude and phase routine, sort of on autopilot, and it's satisfying to see pythagoras drop out in the case of q=90 degrees (and the cosine rule otherwise - in fact this could be seen as a roundabout proof of the cosine rule, which is fun)

While on the subject of complex numbers, it seems to me that their use in physics and engineering is really just ever as a clever trick to facilitate calculation - all physically meaningful inputs and outputs of calculations are real and complex numbers are just convenient intermediaries.  Is there anywhere they are actually needed? I suspect that everything could be done rather more laboriously without them. Is this even true of quantum mechanics where the wave funcion is complex? Is the wave function part of reality? After all, the arithmetic of complex numbers is precisely mirrored by a set of real 2 by 2 matrices. 

And if complex numbers are just a sort of technical short cut in physics and engineering, do they really only come in to their full glory in pure maths when it becomes apparent that they are really just as "real" as the real numbers and that without them we are held back by only seeing a sliver of a cross section of the world of numbers. But maybe it is their completeness which makes them a powerful calculating tool in physics.

It can get quite complicated! You may think that a transformer is a simple electrical device, but, check out its phasor diagram ...

https://images.app.goo.gl/JJ8tKy2up1ptv29a7

🙂

You could say exactly the same about fractions, ordered pairs of integers with some special rules to manipulate them, but really just a clever trick to facilitate calculation! All physically meaningful inputs and outputs of calculations could be represented as integers, so fractions are just convenient intermediaries, right? Ditto the reals (as Cauchy sequences or Dedekind cuts), everything could be done, though rather more laboriously, without them...

Of course the difference is we're all introduced to fractions and decimals at a young age and consequently they get baked very deeply into people's intuition of what a "number" is, while complex numbers get left until A-levels so tend to feel like an artificial invention unless you end up in a career that makes daily use of them. Such as say, pure mathematician, physicist, or signal processing engineer.

As a lapsed physicist and current signal processing engineer I'd say their application in both fields has such power and elegance, such a natural fit to the phenomena in question, that asking whether they're "real" or not is a trick question (because the answer makes no difference).

Going one step further than complex algebra, quaternion algebra is used in computer graphics to compose multiple rotations in 3D virtual space. This operation cannot be performed using a matrix of real numbers without risking running into the "gimbal lock" problem, which occurs when two of the rotational axes coincide and a single rotational dimension is "lost".

I believe the theory of electromagnetism was original formulated using quaternion algebra, but it didn't really catch on.

It was originally done in scalar component notation with quaternions being an abortive attempt to simplify it on the path to fit-for-purpose vector notation.

Some light reading:

https://arxiv.org/pdf/2010.09679.pdf

In reply to duchessofmalfi:

Don't be gross.