Voltage and current phase shifting/Current without voltage?

[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!..

 

[sophiecentaur]

I envy you!..

Yes. A nice way in for Jim. I wonder how many of his cohort did as well as him. I'll make him blush by saying he is a bit special where E Engineers are concerned.

 

[jim hardy]

Gosh, Sophie - THANK YOU SIR !

I do admire (and envy) those like you for whom math seems so natuaral and are at home with it.
I always struggled with it.


old jim

 

[sophiecentaur]

I do admire (and envy) those like you for whom math seems so natuaral and are at home with it.
I always struggled with it.

You overrate my ability. With me, it's not like that. I am pretty rubbish at Maths because I was never bothered enough to learn enough of 'the rules' that you need when actually trying to work things out. It's more a matter of 'faith' that I could if I was bothered to - or at least that "it can be shown that" by someone smarter than me. I did work with a lot of people who could cope with higher levels of Maths than I could and I really believe it's a valid way through. That's just like using an electronic device with an in and an out and some control knobs. Your employer paid £10k for it so we all believe it and it checks out against other stuff that also cost a lot. Then you look at the system as a whole and wave yer arms about a bit for a result.

 

[jim hardy]

I did work with a lot of people who could cope with higher levels of Maths than I could and I really believe it's a valid way through. That's just like using an electronic device with an in and an out and some control knobs.

Amen.
Laplace transform is beyond my comprehension.
It's quite a handy tool, though.
I suppose one doesn't have to understand equations for impulse and momentum to apply a hammer to a nail .