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