Voltage and current phase shifting/Current without voltage?

[Purple_Dan]

Hello, I've recently been learning about capacitance and inductance causing the current to lead or lag. But how can the current possibly maintain the same waveform? Shouldn't the zeros of voltage and current always align? How can there be current without voltage?

 

[anorlunda]

Shouldn't the zeros of voltage and current always align? How can there be current without voltage?

Shouldn't the zeros of voltage and current always align?

No they don't. They align only for purely resistive elements. Inductors and capacitors are different.

We have a PF Insights article on just that subject here. Please read that first, then post again if you have questions.

https://www.physicsforums.com/insights/ac-power-analysis-part-1-basics/

 

[Purple_Dan]

No they don't. They align only for purely resistive elements. Inductors and capacitors are different.

We have a PF Insights article on just that subject here. Please read that first, then post again if you have questions.

https://www.physicsforums.com/insights/ac-power-analysis-part-1-basics/

Thanks for the quick reply.
Although interesting, I didn't really get the answer to my question from that article.
I understand how the power in the circuit is 0 at any time the voltage or current is 0. But I'm not too interested in the power for this specific question.

Perhaps it would be easier (for me at least) to think in terms of the movement of electrons.

I'm glad you sent me to that article actually, as it shows exactly the graph I'm having trouble with where the current is shifted 90 degrees.
Have I been taught an untruth at some point? Can there be current without voltage?

 

[jim hardy]

Hello, I've recently been learning about capacitance and inductance causing the current to lead or lag. But how can the current possibly maintain the same waveform? Shouldn't the zeros of voltage and current always align? How can there be current without voltage?

Anorlunda hit it. Check his article..
Remember you're being taught about sinewaves.

Go back to capacitor basics
i = c dv/dt

only for sinewaves do the function and its derivative have the same shape.
d(sine) = cosine , and a cosine is just a sine shifted by 90 degrees.
When one is at zero the other is at peak.

Try it for a triangle wave.

It's good that you question. Bouncing the basics against one another to resolve such apparent conflicts molds your thinking processes and you arrive at robust mental models. Keep up the good work.

 

[anorlunda]

Perhaps it would be easier (for me at least) to think in terms of the movement of electrons.

No, that is seldom a helpful way to do it.

Have you studied derivatives, sines and cosines? If yes, and I say $I=C\frac{dV}{dt}$ for a capacitor, and V=cos(wt), what is I? What is V when I is zero?

 

[cnh1995]

Can there be current without voltage?

No.
When input voltage is zero, the current in the circuit is maintained by the energy stored in the inductor or capacitor. The phase difference is a result of the energy storing ability of the reactive elements and their V-I relationship.

But how can the current possibly maintain the same waveform?

As jim said, derivatives and integrals of a sine-wave signal are also sine waves with some phase difference. Write the KVL equation (integro-differential equation) of an RLC circuit and solve it for sine wave as well as triangular wave.

 

[Purple_Dan]

No.
When input voltage is zero, the current in the circuit is maintained by the energy stored in the inductor or capacitor. The phase difference is a result of the energy storing ability of the reactive elements and their V-I relationship.As jim said, derivatives and integrals of a sine-wave signal are also sine waves with some phase difference. Write the KVL equation (integro-differential equation) of an RLC circuit and solve it for sine wave as well as triangular wave.

Unfortunately, I'm not at liberty to be doing mathematical proofs as I'm at work, however, you may have shed some light on the situation.

So we're basically fudging the graphs by ignoring the voltage across the capacitor when the source voltage is 0?

 

[jim hardy]

So we're basically fudging the graphs by ignoring the voltage across the capacitor when the source voltage is 0?

Not at all.
We're aligning our thinking so our imagination comes into agreement with how Mother Nature built the real world.
We can imagine all sorts of things that aren't right.

I use this quote from Lavoisier a lot.
https://web.lemoyne.edu/giunta/EA/LAVPREFann.HTML

"Instead of applying observation to the things we wished to know, we have chosen rather to imagine them. Advancing from one ill founded supposition to another, we have at last bewildered ourselves amidst a multitude of errors. These errors becoming prejudices, are, of course, adopted as principles, and we thus bewilder ourselves more and more. The method, too, by which we conduct our reasonings is as absurd; we abuse words which we do not understand, and call this the art of reasoning. When matters have been brought this length, when errors have been thus accumulated, there is but one remedy by which order can be restored to the faculty of thinking; this is, to forget all that we have learned, to trace back our ideas to their source, to follow the train in which they rise, and, as my Lord Bacon says, to frame the human understanding anew.

Build your mental picture models slowly and frequently cross checking them with math.
Especially in electrical phenomena for analog electronics is really well described by the equations.
When your mental model leads you to the right formula you're getting someplace.

You have to form the mental habit of that differential-integral relation and it's not easy.
Sine and cosine at any instant are equal* to one another's slope'. That picture won't let you down.
*(well make that proportional related by ω to one another's slope)That Laviosier article is quite interesting. Print a copy and take it home? I kept that snippet framed above my desk for years.

old jim

 

[cnh1995]

So we're basically fudging the graphs by ignoring the voltage across the capacitor when the source voltage is 0?

Which graphs are you talking about? Is it a purely capacitive circuit or an RC circuit?

 

[GrahamN-UK]

Shouldn't the zeros of voltage and current always align?

No. That only happens in the special case of a purely resistive circuit (as anorlunda noted). But as you are only now learning about inductors and capacitors all your previous experience has been with resistive circuits and that special case. The general case is that you have circuits with R, L and C and the voltage and current zero-crossing points don't coincide.

How can there be current without voltage?

In a steady state, sustained case there can't be (well, unless you're playing with superconductors). But here we are not considering a sustained zero voltage, we are looking at sinusoidal waveforms where the instantaneous values are always changing. We get a non-zero current only because the voltage is changing as it passes through the zero crossing point. If we froze the voltage as it reached zero and kept it at zero the current wouldn't keep flowing. This only works because the voltage is always changing in a sine wave.

The differential equations for capacitors posted by Jim Hardy and anorlunda describe how the current flow in or out of a capacitor is proportional to how fast the voltage across the capacitor is changing (with time). A feature of a sine wave is that its value is changing at its fastest as the value crosses zero. So the current flow in or out of a capacitor reaches a maximum as the sinusoidal voltage across it crosses zero. (A positive or negative maximum, depending on which direction the voltage is crossing zero.) Another feature of a sine wave is that its rate of change falls to zero at the peak positive and peak negative values. So the current in a capacitor will pass through zero as the sine voltage across it reaches peak positive and negative values.

The corresponding equation for an inductor describes how the voltage across an inductor is proportional to how fast the current through the inductor is changing. Hence the voltage across an inductor will be at a (pos or neg) maximum as the current through it crosses zero. Similarly the voltage will cross through a zero as the current reaches peak +ve and -ve values.

The voltage across a resistor is just the resistance times the current through it, from Ohm's Law as you are presumably already aware. It doesn't care how fast the current or voltage are changing. When the current is zero the voltage is zero. Hence for a resistive circuit the voltage and current zero-crossing points occur at the same time.

 

[sophiecentaur]

Perhaps it would be easier (for me at least) to think in terms of the movement of electrons.

It would be a good thing to dump that approach. No one uses Electrons when doing this sort of analysis. They don't help at all.
If you associate movement of electrons (I guess you mean current) with Power then you have the problem of deciding which way the Power is actually flowing. It can be either way, depending on the Potential Difference.

Hello, I've recently been learning about capacitance and inductance causing the current to lead or lag. But how can the current possibly maintain the same waveform? Shouldn't the zeros of voltage and current always align? How can there be current without voltage?

It's not clear, from what you write, where you level of knowledge sits for this topic. The theory is very well established and, if you can bring yourself to accept that the Mathematical models of circuits are valid then your questions are answered if you just read a textbook. Trying to apply intuition against Maths usually leads nowhere. If you haven't done the Maths yet then just wait and it will all be clear when you have. The elementary rules for DC circuits do not apply in a straightforward way for time varying circuits with reactive components which Store Energy for a finite time.

 

[jim hardy]

How can there be current without voltage?

Because current through a capacitor is not proportion to voltage. It's in proportion to rate of change of voltage.

But how can the current possibly maintain the same waveform?

It only does that for sine waves. They are a special mathematical case. We get so wrapped up in our studies we forget about that.
https://www.mathsisfun.com/algebra/trig-sin-cos-tan-graphs.html

Differentiating any other waveform will change its shape.

Shouldn't the zeros of voltage and current always align?

Take it to DC. Put 100 VDC across a capacitor.
It charges quickly and current drops to zero.
Right there's proof that Zero Current and Zero Voltage needn't coincide.

 

[sophiecentaur]

I'm afraid that I have to blame all you Power Engineers for this guy's problems. All the rules about sinusoidal waveforms work so well for you and you are so familiar with it that you forget about
1. DC and
2. The exponential decay function that's always involved with Capacitors and Inductors.
You get on with your Phases, Reactive power and Power Factors etc etc and solve very hard probs. KVL and KCL appear to go out of the window for the beginner.
No need to get defensive guys () but the cognitive dissonance no longer gets to you because you are so familiar with it all.
This problem, in its own way, is a bit like the Particle Wave Duality that gave people such problems a hundred years ago. Wear the right hat and the problem just goes away.

 

[Windadct]

I think a very good way to think of this is a pendulum. Where gravity is the Voltage and the movement of the weight is the current.

At max height, the force of gravity (in the direction of movement) is at it's maximum, but you have zero movement ( current) at the point of maximum force. At the bottom there is no force in the direction of movement and gravity is now contributing nothing to the movement. ( so maximum movement with no force) .

This is a general condition of all undampened oscillating systems.

A weight hanging or "bouncing" from a vertical spring may be even clearer, since it is a 1 dimensional model.

 

[Purple_Dan]

I'm afraid that I have to blame all you Power Engineers for this guy's problems. All the rules about sinusoidal waveforms work so well for you and you are so familiar with it that you forget about
1. DC and
2. The exponential decay function that's always involved with Capacitors and Inductors.
You get on with your Phases, Reactive power and Power Factors etc etc and solve very hard probs. KVL and KCL appear to go out of the window for the beginner.
No need to get defensive guys () but the cognitive dissonance no longer gets to you because you are so familiar with it all.
This problem, in its own way, is a bit like the Particle Wave Duality that gave people such problems a hundred years ago. Wear the right hat and the problem just goes away.

Wow thanks! It's nice to have backup. Also, I was thinking the same thing regarding the particle/wave duality. I know it's the best explanation we have at the moment, but it's a bit lazy to say it's sometimes this and sometimes that.

So I don't have a lot of time to post, but I'm trying to keep up with all your replies. I just have the following to say.

In the case of DC being applied to a capacitor, yes, there's no current once the voltage in the capacitor is equal to the supply. But that's voltage without current, not current without voltage.

Furthermore, when I said that the zeroes must align, I meant just the zeroes.
I understand that current might not be always proportional to voltage, but surely there must always be voltage and the current and voltage must always be the same polarity.

I also don't subscribe to the notion that thinking of electricity in terms of the electrons in the wire is not useful. How can ignoring what is actually happening in the system and replacing it with what mathematicians reckon be any more useful? Especially when the maths gives us answers where the voltage is zero and the current is not, and we know that can't happen.

That being said, I haven't looked to see if anyone's observed this on an oscilloscope, so I'll look into that when I have some spare time.

 

[nsaspook]

I also don't subscribe to the notion that thinking of electricity in terms of the electrons in the wire is not useful. How can ignoring what is actually happening in the system and replacing it with what mathematicians reckon be any more useful? Especially when the maths gives us answers where the voltage is zero and the current is not, and we know that can't happen.

You're not the first or last one with this notion. When people say "It would be a good thing to dump that approach" it comes from decades of practical experience of teaching others about electricity because most of the time the person with that notion does not understand what's actually happening.

 

[sophiecentaur]

Wow thanks! It's nice to have backup.

Read what I wrote. it is not backup.

How can ignoring what is actually happening in the system

It is not 'what is actually happening'. The electrons are not 'carrying the power' there is no way that such low mass particles, traveling at 1mm/s could be carrying any appreciable power. It's about Potential Difference and Current, acting together. (Which is why it's called Electromagnetic theory)

replacing it with what mathematicians reckon

The mathematical description is a way of describing what is happening and it gives good, quantitative predictions. Waving arms about and saying that the particles describe it all is demonstrating a very limited knowledge of the evidence. If you don't like the Maths then you just have to stay out of the discussion because there is no better model than the mathematical one.

I was thinking the same thing regarding the particle/wave duality.

If you re-read my words, you will see that I was referring to the situation 100 years ago and that 'duality' is not a current view.

when I have some spare time.

A good idea. When you have some time, do some serious reading and improve the basis for any arguments you use.

 

[jim hardy]

In the case of DC being applied to a capacitor, yes, there's no current once the voltage in the capacitor is equal to the supply. But that's voltage without current, not current without voltage.

Picky, picky. Point was zeroes don't have to align.

I understand that current might not be always proportional to voltage, but surely there must always be voltage and the current and voltage must always be the same polarity.

You have much to unlearn.

 

[Averagesupernova]

.
I understand that current might not be always proportional to voltage, but surely there must always be voltage and the current and voltage must always be the same polarity.

I also don't subscribe to the notion that thinking of electricity in terms of the electrons in the wire is not useful. How can ignoring what is actually happening in the system and replacing it with what mathematicians reckon be any more useful? Especially when the maths gives us answers where the voltage is zero and the current is not, and we know that can't happen.

That being said, I haven't looked to see if anyone's observed this on an oscilloscope, so I'll look into that when I have some spare time.

Viewing it on an oscilloscope is one of the first things I did with a scope in school. Trust me, it's there. Why is it you just know that can't happen concerning the voltage being zero and the current not being zero? How do you "just know"? There is no other way to shift phase between current and voltage without having times throughout the cycle where one is positive and the other negative.

 

[jim hardy]

That being said, I haven't looked to see if anyone's observed this on an oscilloscope, so I'll look into that when I have some spare time.

 

[Tom.G]

Hi @Purple_Dan,

Let's see if this gedanken (mental experiment) helps.

Setup:
As is commonly known, with zero voltage difference there is zero current
As a voltage difference increases, current increases

Now consider:
A voltage sine wave

1. At the instant of zero crossing a discharged capacitor is connected to sine wave
2. Consider an infinitesimal time slice at zero crossing, the voltage has increased some tiny amount
3. With a voltage difference now applied to the capacitor, some current will flow to charge the capacitor
4. Keep in mind that for a sine wave the voltage changes most rapidly around/at the time of zero crossing
5. Now consider the infinitesimal time slice following the first one
6. The voltage difference has increased again but not quite as much as the increase at the zero crossing
7. This slightly lower voltage difference between the sine wave and the capacitor voltage causes a slightly smaller current to flow into the capacitor
8. As the sine wave approaches its peak, the voltage difference decreases between succeding infiniteseimal time slices
9. At the peak of the sine wave there is no voltage difference between time slices
10. Now with no voltage difference between the sine wave and the capacitor, there is no current flowing

That is why the 90 degree phase shift occurs. When the voltage increases rapidly (at zero crossing), the current is highest. Conversly, where the voltage doesn't change (at the peak), the current is zero due to the zero voltage difference.

With an inductor a somewhat similar but more complicated effect takes place due to interaction with the changing magnetic field. I'll let someone else go into details of that if you can't dig out the answer somewhere.

 

[Purple_Dan]

Okay, so regarding my level of knowledge on the subject, I have a BEng in Robotics, which was mainly software and electronics, with a bit of mechanics thrown in. Also an A level in Physics.
So I'm not completely out of my element, but most of my electrical knowledge is DC and nowhere near the experience or knowledge that most of you have. Also, I haven't used Fourier, Laplace or Z transforms in over 3 years, so my maths on the subject is rusty.

I suppose, as I have come to accept in many subjects, a case of unlearning what I've been taught. I had to do a similar thing with the 2-8-8 rule in Chemistry when learning about sub-orbitals.
It's clear that if a bunch of experts are saying it's one way, and some guy is questioning it, it's probably the guy's lack of understanding that is the problem. However, I suppose it's always good to question what you're being taught, else you'll believe anything. But I'm going to flip it on it's head and instead of questioning what I'm being taught now, I'll question what I've been taught before.
A lot of my wife's work (she's a teacher) is unlearning untruths as much as it is teaching truths, and the unlearning bit is difficult because humans generally don't like change.
But I can't be a flat earther forever.

What really swung it for me was:

Thinking about it in terms of the change in time. There's only going to be one slice of time where the voltage is zero, so the change in voltage will always be non-zero.

The pendulum model.

And most importantly, seeing it on an oscilloscope!

Thanks for your time and patience with an arrogant greenhorn.

 

[cnh1995]

With an inductor a somewhat similar but more complicated effect takes place due to interaction with the changing magnetic field. I'll let someone else go into details of that if you can't dig out the answer somewhere.

For a purely inductive circuit, the same thought experiment will do, but the instant of switching should be voltage peak (not voltage zero crossing like in case of a capacitor).

 

[Windadct]

The circuit theory behind an inductor and a capacitor are, to me, exactly the same. They are both act by storing energy, the Capacitor in an E field, and inductor in an M field.

So nether one is more complicated than the other - except that conventional thinking is from a voltage perspective and current is the result, so the capacitor may make more sense initially. But if you accept that they are the same - it may be helpful to realize that they are the same, when your perspective of V and I are swapped. Meaning if you hit a road block with one, ask if it helps to consider the other.

Gotta love MIT's open courseware initiative, reference Formula 1.5 and 1.27 in THIS. -- this whole document is a very good summary of this thread.

E1-> Haha -- also in review I am chuckling a little that they cheated on the plots by just renaming the two traces, and used the same image!

 

[tech99]

Hello, I've recently been learning about capacitance and inductance causing the current to lead or lag. But how can the current possibly maintain the same waveform? Shouldn't the zeros of voltage and current always align? How can there be current without voltage?

Inductance behaves a bit like a heavy trolley on the table, and obeys Newton's Laws. If you push and pull it back and forth, you will notice that when it is a maximum speed (maximum current) you are not pushing at all (zero voltage).

 

[anorlunda]

Would it make explanations easier if we used square wave AC? DC voltage in one direction for a while, then the other direction for a while.

The math is a bit messier because you don't have sin/cos to work with. But might the conceptual level explanations be easier?
Even with square waves we can have both real and imaginary power and all the other essential properties of AC.

Hmm, I'm skeptical but I'll work up examples for an RL circuit and a RC circuit tomorrow and we'll see if it's easier or not. I'll avoid RLC because I don't want to complicate with resonant frequency.

 

[Averagesupernova]

Voltage and current relationships with inductive and capacitive circuits is like bratty kids. You tell them to do something and it always takes a while before they do it.

 

[tech99]

Would it make explanations easier if we used square wave AC? DC voltage in one direction for a while, then the other direction for a while.

The math is a bit messier because you don't have sin/cos to work with. But might the conceptual level explanations be easier?
Even with square waves we can have both real and imaginary power and all the other essential properties of AC.

Hmm, I'm skeptical but I'll work up examples for an RL circuit and a RC circuit tomorrow and we'll see if it's easier or not. I'll avoid RLC because I don't want to complicate with resonant frequency.

If you apply a square voltage waveform to an inductor, the current will grow exponentially and not as a quadrature square wave.

 

[sophiecentaur]

Even with square waves we can have both real and imaginary power

I don't see how that works because the phase changes of harmonics are not the same. I have a problem with so called Imaginary Power in any case. The term seems to be just a convenient one and it only applies in Power AC theory. (I long ago ceased to argue about the term but it really can only apply for a single frequency signal, I feel sure)

 

[jim hardy]

And most importantly, seeing it on an oscilloscope!

HYou from Missouri ?

Would it make explanations easier if we used square wave AC? DC voltage in one direction for a while, then the other direction for a while.

Triangle wave voltage applied to a capacitor gives square wave current
that's a easy to grasp visual effect .

If you apply a square voltage waveform to an inductor, the current will grow exponentially and not as a quadrature square wave.

Well, if by "square" we mean not symmetric about zero.

 

[anorlunda]

I promised to work an example to explore whether using a square wave rather than a sin wave might make AC circuits easier to explain to beginners. As an old time analog guy, I always learn best from study of time plots.

Before showing the results, my conclusion is that the answer to the square wave experiment is "No.. Not useful." A square wave allows one to dodge the fuzzy idea of phase angle, but the spiky plots are just too hard to follow.

I made a numerical example using an RL circuit. Below is the circuit. Note that I labeled the nodes A B and C. On the right is the color legend for the time plots below. Do not get confused by the sign conventions. VAB+VBC+VCA=0 using these sign conventions.

First, below is a familiar DC step response. Time is the horizontal axis. Initial conditions were all zero. At time t=0+, the current is zero and all the voltage drop is across the inductor. In the steady state the voltage across the inductor, VCA, is zero. The ratio R/L is 10.

Now below see the same circuit's response to 2.5 cycles of a square wave in VAB. It approximates AC but not exactly because there are not enough cycles shown to wipe out the arbitrary initial conditions. The power PAB can be seen to be both positive and negative during a single cycle. We also see that the times of current zeroes and voltage zeroes are not the same. That was the misunderstanding of the OP in this thread.

If the time integral of power flow over a whole cycle is exactly zero, we say that the AC power is purely imaginary. (@sophiecentaur, the word imaginary comes from the complex number descriptions of this circuit. Blame the history of mathematics, not the history of EE for the word choice.)

I like to stress that P=VI instantaneously always works for both DC and AC. The only time when AC is different from DC is when we define AC values as averages over a whole cycle (and many cycles after the startup transients have died away.) IMO, if we stressed that more emphatically, then students would have less difficulty in going from DC to AC learning.

Finally, see below the same 2.5 cycles, but changing the values of R and L so that R/L is 0.1 instead of 10.0. With those numbers, the voltage across the inductor approximates the same square wave as VAB but shifted one half cycle. Current maximums are roughly 90 degrees out of phase with the square wave. Current wave shape looks like a triangle wave and power looks like a sawtooth.

p.s. I also did the case with a capacitor instead of an inductor, but I didn't upload them because I think the whole exersize (while fun) is of no value. If someone disagrees and asks, I'll post those too.

 

[jim hardy]

Current wave shape looks like a triangle wave

Triangle wave current through an inductor gives square wave voltage because e = L di/dt

from my old reactor days ( i use this image a lot)

top trace 20 ma p-p current through primary of my rod position indicator coil stack
bottom trace is voltage induced in its secondary
rounded edges are from eddy currents in the nonlaminated steel core
and my function generator current wasn't very linear either
but you get the idea.

to me that's a lot clearer demonstration of the differential relationships than just looking at phase shift of two sine shaped waves.
Students can graphically determine L from di/dt.

Triangle wave voltage across a capacitor gives square wave current because i = C dv/dt

Then one can differentiate a square wave to show the spikes.
It'd be natural enough to go from there to sinewaves and show that dsin = cosine and that's why the phase shift.
That should drive home the fact Sines are a mathematical oddity in that differentiating them doesn't change their shape .

That would help solve Sophie's dilemma that we don't sufficiently impress on students of AC circuits that this is a niche field where the coin of the realm is sinusoidal voltage and current. A big niche, but still a niche .
My old high school electronics teacher very much impressed that on us boys. After we were proficient in phasors and rectangular-polar and watts and reactive VA and power factors he took us into tubes and radio circuits .

I think your triangles would be a good teaching tool, @anorlunda

old jim

 

[anorlunda]

OK if @jim hardy thinks it has value, that's good enough for me. Here is the capacitive version to compare to the inductive version in post #31.

I made a numerical example using an RC circuit. Below is the circuit. Note that I labeled the nodes A B and C. On the right is the color legend for the time plots below. Do not get confused by the sign conventions. VAB+VBC+VCA=0 using these sign conventions.

First, below is a familiar DC step response. Time is the horizontal axis. Initial conditions were all zero. At time t=0+, the voltage on C is zero and all the voltage drop is across the resistor. In the steady state the voltage across the capacitor, VCA, is -VAB, and current is zero The ratio R/C is 10.

Now below see the same circuit's response to 2.5 cycles of a square wave in VAB. It approximates AC but not exactly because there are not enough cycles shown to wipe out the arbitrary initial conditions.

Finally, we rerun the 2.5 cycles case but changing R and C such that R/C=0.1 instead of 10.0. Note that the capacitor voltage VCA (the purple line) is a triangle wave as @jim hardy predicted.

In both the inductive and capacitive cases, if I pushed the frequency high enough, the time responses to a square wave would begin to resemble responses to a sin wave. That is why I say that we can do AC analysis no matter what the wave shape. They differ only in magnitudes, not in principles or in forms of the equations.

Once again, my motivation and my point:

On an instantaneous basis, there is no difference between DC and AC circuit analysis. DC methods apply to both. AC analysis differs only because we use it to describe averages over whole cycles rather than instantaneous values. Teachers should emphasize this vital point when introducing the subject of AC.

I urge the emphasis because I see so many student questions here on PF which make it obvious that the questioner does not understand that vital difference.

 

[anorlunda]

I see now the wisdom of @jim hardy 's square/triangle wave point. Thank you Jim. So I simplified my example and redid it considering one component at a time.

For a capacitor, $I=C\frac{dV}{dt}$. So a square wave current produces a triangle wave voltage. Voltage max/min occur at the times of current zeroes. Instantaneous power is P=VI. Power is + half of each cycle and - the other half. The average power over an entire cycle is zero. Average V and average I are also zero.

For an inductor, $V=L\frac{dI}{dt}$. Everything we said about a capacitor applies here if we just reverse V and I. P remains unchanged.

For a resistor, $V=IR$. A square wave voltage results in a square wave current. Voltage and current zeroes happen at the same times. Power is positive the whole cycle, so the average power = instantaneous power = VI. When V is +, I is + so P is +. When V is - I is - and P remains +. So average V and average I are zero, but average P is plus.

So there we have the essential components and definitions for AC circuit analysis, without using sin/cos and without mentioning phase angle. Does anyone think it would be easier to introduce AC if this example were intermediate?

 

[jim hardy]

Thank you Jim. So I simplified my example and redid it considering one component at a time.

No, I thank YOU !

Your new sketches are SOO much easier to see.

Kids who've not yet had any calculus at all can work it graphically with just algebra .

old jim

 

[sophiecentaur]

I promised to work an example to explore whether using a square wave rather than a sin wave might make AC circuits easier to explain to beginners. As an old time analog guy, I always learn best from study of time plots.

I think the sinewave model will not be superseded. For a start, it deals with just one frequency and requires only a very few axioms to get started. I know it can't be used for explaining 'what really happens' to people who want to feel they know the topic really well.
But I say that they have to go along with the standard AC theory with all its limits or they have to get over the Maths of the Full Monty approach. They can't have it both ways because a difficult subject is . . . . Just DIFFICULT.
Annoying when they have been telling you in school that anything is possible but there it is.

[/B]

 

[anorlunda]

@sophiecentaur , did you read #34 and #35?

I am trying to find a way to introduce AC without phase angles, and without even mentioning frequency. Step 2 would be to move from square waves to sinusoidal waves. So the goal is not to replace the sinusoidal model, but to introduce it in 2 steps.

Remember the confusion of the OP of this thread. He thought that when V is zero, I must be zero simultaneously. That is the kind of misconception that students of traditional teaching come away with. I'm not after better science, but better teaching.

 

[jim hardy]

Shouldn't the zeros of voltage and current always align? How can there be current without voltage?

There's somebody not yet thinking in terms of differentials. Or even the precursor to derivatives, deltas.

But how can the current possibly maintain the same waveform? Shouldn't the zeros of voltage and current always align?

There's somebody not realizing sine and cosine have the same shape just are offset by 90 degrees, and have the strange property that one of them is always proportional to the other one's slope.

He needs to see some graphs. Then do some graphical exercise homework problems that demonstrate above points. The straight lines of triangle and square waves lead the mind directly to thinking about slope and rate of change. It's a small step from there to sinewaves..

Then be reminded "AC Circuit Analysis , having grown out of power field(Steinmetz and Tesla) naturally is based on steady state sine waves.
One must remember that sinewaves are a mathematical special case.
While they're the coin of the realm in power, in other fields of electronics they're not so ubiquitous .
All EE students first learn steady state sinewave circuit analysis . It's a century old tradition. It allows solution by simple algebra.
Non steady state behavior and non sinewave functions require solution by differential equations . Other mathematical tools like Laplace Transform are useful for solving them. .
So buckle up and learn your basic steady state sinewave AC circuit analysis. "

That's how i was taught AC before i took Calculus.

 

[cnh1995]

@jim hardy, @anorlunda, I have a question regarding the capacitor waveforms in #34.
I understood the square-triangular graphs for capacitor and I understand I=CdV/dt for a capacitor.
Now, suppose you take a capacitor and connect it to a triangular-wave voltage source through a switch and keep the switch open. This switch is closed at the instant when the triangular voltage wave reaches its peak. So the voltage is applied across the capacitor when it is at the peak. Call this time as t=0. So from t=0, the capacitor will see a negative rate of change of voltage. What should the current waveform look like in this case?

 

[anorlunda]

@jim hardy, @anorlunda, I have a question regarding the capacitor waveforms in #34.
I understood the square-triangular graphs for capacitor and I understand I=CdV/dt for a capacitor.
Now, suppose you take a capacitor and connect it to a triangular-wave voltage source through a switch and keep the switch open. This switch is closed at the instant when the triangular voltage wave reaches its peak. So the voltage is applied across the capacitor when it is at the peak. Call this time as t=0. So from t=0, the capacitor will see a negative rate of change of voltage. What should the current waveform look like in this case?

The voltage across a capacitor can't jump instantaneously, that would take infinite current. So if you connect an uncharged capacitor to an ideal voltage source, you created a contradiction. The voltage at t=0+ must be V but it also must be zero. Something has to give. In practice, it would likely be that the voltage source is not ideal. It has an internal resistance that limits the current. So the circuit and the initial transient would resemble those in #31 of this thread.

We get similar questions all the time. "What happens to I=V/R when R=0?" The answer is that Ohm's Law and circuit analysis apply only over reasonable ranges of V and I. Infinity is never reasonable.

 

[cnh1995]

So if you connect an uncharged capacitor to an ideal voltage source, you created a contradiction.

Exactly!
This is why the result of the simulation I just ran is showing a very large spike of current, which is an indication of something invalid.

I knew that 'an ideal inductor fed from an ideal current source' is an invalid situation. But it never occurs in ac circuits.

But if a capacitor is energized when the input voltage is at its peak (or any non-zero value), it should create an invalid situation (in ideal case of course).
(It's strange that it didn't occur to me even after knowing about inductors).

Thanks a lot!

 

[jim hardy]

Indeed you'd get a current spike . How big ?
This is where 'ideal' components in thought experiments can get you in trouble
Let's take your capacitor thought experiment
As anorlunda said much earlier, at any instant AC is DC because current flows only one direction at a time
i = c X dv/dt or if you prefer i = c X Δv / Δt
at the instant of switch closure,
Δv is a real number, the triangle wave peak
and Δt is zero.
Sophiecentaur tells us to 'use the maths'
and division by zero does not give infinity, it is undefined. Its limit approaches infinity as denominator approaches zero,
but division by zero is not allowed
and any rigorous math model should blow up (or if a computer program, complain..)

Were there any resistance in the circuit to drop the source voltage you could solve for real and finite current
but the ideal capacitor has none.
Any real capacitor has some resistance in its wires ,
so in your thought experiment as that resistance approaches zero, ,,,,, think about it - resistance will be in the denominator too
as resistance approaches zero current can only approach the mathematical limit which is infinity . But you'll never get all the way there .

The math works out ! (again)

The end result of all this is just what anorlunda said - you can't have finite Δv in zero Δt , you have to take the average over some finite Δt. This is where graphic solutions excel in teaching basics . I talk myself through any approach to math problems before setting pencil to paper to test my logic. Given my Latex Illiteracy Syndrome i especially have to talk through them here.

https://en.wikipedia.org/wiki/Division_by_zero
Historical accidents

• On September 21, 1997, a division by zero error in the "Remote Data Base Manager" aboard USS Yorktown (CG-48) brought down all the machines on the network, causing the ship's propulsion system to fail.[12][13]

 

[cnh1995]

These are some simulation results for inductive ac circuits.

1)Purely inductive circuit:
Switching instant: Voltage(green) zero crossing.

Remarks: Maximum dc offset in current (blue), no damping because of absence of resistance.
Hence, no negative half in the current waveform.

2)Purely inductive circuit:
Switching instant: Voltage peak (almost).

Remarks: Zero dc offset in current, symmetrical current waveform.

3)Purely inductive circuit:
Switching instant: Between voltage zero and peak.
Remarks:Non-zero dc offset (intermediate), no damping.

You can see how V_L=Ldi/dt is satisfied in every waveform.

 

[cnh1995]

Post #43 Continued...

4)R-L circuit:
Switching instant: Voltage(green) zero crossing.
Remarks: Initial dc offset current dies out exponentially (transient) and the phase difference between voltage and current(red) becomes equal to the power factor angle at the end of this transient.

Here's what the entire transient looks like:

If the switching angle is equal to the power factor angle, there is zero dc offset i.e. no transient.

 

[jim hardy]

Thank you @cnh1995

those DC offsets in inductors are real and the math will show them.

They cause saturation in power transformers which gives huge inrush current for first few half cycles should one happen to switch it on near the zero crossing.
That's why they make peak switching solid state relays for inductive loads and zero switching ones for capacitive loads.

Nice demonstration , practical application of theory - thanks again !

old jim

 

[sophiecentaur]

@sophiecentaur , did you read #34 and #35?

I am trying to find a way to introduce AC without phase angles, and without even mentioning frequency. Step 2 would be to move from square waves to sinusoidal waves. So the goal is not to replace the sinusoidal model, but to introduce it in 2 steps.

Remember the confusion of the OP of this thread. He thought that when V is zero, I must be zero simultaneously. That is the kind of misconception that students of traditional teaching come away with. I'm not after better science, but better teaching.

My point is that sometimes there are limits to how easy you can present a subject without losing the meaning or further confusion. What I call AC theory is a way to present a limited set of EE which is more or less self consistent. It is no surprise that it emerged as a field in itself because it does so well as a tool which copes with such a lot of EE problems. If someone wants to 'argue' with the validity of just dealing with 50/60Hz signals then they don't have a leg to stand on. It's proved it's utility.
it is very risky to try half way house approaches without a very thorough treatment.
A really useful radical to teaching method would have to be in the form of a proper textbook, I think. Else it would generate as many new questions as answers. A private exercise is, of course fine but you'd need a very complete picture to justify not doing the whole thing.
Frequency and phase are essentials so what are you proposing to get across whilst avoiding them? And why?
Bending the subject to fit the student is very risky. The way to deal with difficult bits is to study harder or to admit it's too hard and do something else. Mainstream always gets criticized when individuals have problems. What about the vast majority who actually get along with it?

 

[jim hardy]

Frequency and phase are essentials so what are you proposing to get across whilst avoiding them? And why?

I'm not proposing to avoid them
just to postpone leaping into them until we've got students familiarized with slope and graphical methods applied to simple square and triangle waves , so as to lead their minds into the differential relation that's so necessary to understand reactance.

I guess I'm biased because it's how i was taught long before i'd taken any math class that even mentioned Euler and his identity..

 

[sophiecentaur]

I'm not proposing to avoid them
just to postpone leaping into them until we've got students familiarized with slope and graphical methods applied to simple square and triangle waves , so as to lead their minds into the differential relation that's so necessary to understand reactance.

I guess I'm biased because it's how i was taught long before i'd taken any math class that even mentioned Euler and his identity..

This confuses me a bit. Why were you being taught about "waveforms" etc. before circuit theory? What was the context? Perhaps in the services as a trainee operator? That, I could understand and it could be the reason for my 'dissonance' because my education always involved Deferred Gratification - until we had the Maths to deal with things.

 

[jim hardy]

What was the context? Perhaps in the services as a trainee operator?

Probably close. High school electronics class, taught by a retired Merchant Marine radioman turned engineer turned teacher.

After DC circuits analysis we of course moved into AC. Learned first about exponential charging , decay, time constants , effect of differentiator and low pass on step and triangle functions. Became skilled with slide rule and 1/e to work them.
Then he introduced us to rotating phasors to represent sines, real and imaginary components and operator j , and rectangular-polar conversion by slide rule.
He drilled us nearly to death doing sinewave AC circuit analysis with slide rules. He didn't take us into three phase power, instead into tubes and radio, transmission lines and antennas, finally transistors.
Setting was lab environment . We were two boys to a bench each bench with an oscilloscope, meters, power supplies, a "trainer" rack with tube sockets and patch panels to build circuits. By end of 11th grade we boys knew every resistor in an AM or FM radio reciever & transmitter and were handy with Smith charts.
Teacher was a very hands-on type guy , Monday was lecture day, Tuesday and Wednesday a lab covering the previous days' lectures, Thursday we wrote and presented our reports..
Friday was project day, everybody had to build something for his personal use from surplus electronic parts. He had access to leftovers from Cape Canaveral so there was no shortage of those . I built several tube hi-fi amplifiers and a Wheatstone bridge for measuring precision resistors (this was early 60's when digital meters were exotic rarities).

So we learned basic electronics and test equipment;
how to do an experiment and write up an organized report with purpose, method, presentation of data, observations and conclusions, ;
and how to build something starting with a blank aluminum chassis and Greenlee tube socket punch. .

Was it a disservice to launch us boys without the advanced math ?
I think not, for when in college i saw how calculus described what i had been doing with just arithmetic and operator j it was quite a thrill. Made me appreciate the genius of my high school teacher . I found myself explaining things to other students.
I'd not have made it through college antennas class had not we high school boys built that parallel wire transmission line (from #10 copper house wire on a 2X4) and done "slotted line" SWR & reflection coefficient measurements .

Besides, what're you going to do with a bunch of tenth graders who are just learning 2nd year algebra and trigonometry ?
Teach them to work with the tools they have.

As a teacher you know how a good one can affect a kid's life. He got me into EE and over the years i met several other of his students who were similarly influenced.
He made things intuitive for us boys.
Sorry for the ramble - it's not about me it's about " how does one teach ?"
For me something real makes the math intuitive, not the other way round. That's why i say so often here: "When your intuition leads you to the correct math you're beginning to understand".
Math closes the feedback loop and tells me i have achieved a valid mental model for something.

Perhaps my upbringing was weird but it's the only one I've got. I reckon it's why I'm a bit weird.

old jim

 

[cnh1995]

After DC circuits analysis we of course moved into AC. Learned first about exponential charging , decay, time constants , effect of differentiator and low pass on step and triangle functions. Became skilled with slide rule and i/e to work them.
Then he introduced us to rotating phasors to represent sines, real and imaginary components and operator j , and rectangular-polar conversion by slide rule.
He drilled us nearly to death doing sinewave AC circuit analysis with slide rules. He didn't take us into three phase power, instead into tubes and radio, transmission lines and antennas, finally transistors.
Setting was lab environment . We were two boys to a bench each bench with an oscilloscope, meters, power supplies, a "trainer" rack with tube sockets and patch panels to build circuits. By end of 11th grade we boys knew every resistor in an AM or FM radio reciever & transmitter and were handy with Smith charts.
Teacher was a very hands-on type guy , Monday was lecture day, Tuesday and Wednesday a lab covering the previous days' lectures, Thursday we wrote and presented our reports..
Friday was project day, everybody had to build something for his personal use from surplus electronic parts. He had access to leftovers from Cape Canaveral so there was no shortage of those . I built several tube hi-fi amplifiers and a Wheatstone bridge for measuring precision resistors (this was early 60's when digital meters were exotic rarities).

So we learned basic electronics and test equipment;
how to do an experiment and write up an organized report with purpose, method, presentation of data, observations and conclusions, ;
and how to build something starting with a blank aluminum chassis and Greenlee tube socket punch. .

I envy you!..