What is the Real AC Voltage Phase Shift Across A Capacitor??

What is the Voltage Phase Shift Across A Capacitor in a series RC Circuit with AC Power.

  • 180 Degrees

    Votes: 1 16.7%
  • 90 Degrees

    Votes: 4 66.7%
  • Depends on Measurment Reference Point

    Votes: 0 0.0%
  • Depends on My Specific Formula (Relative to Power Source Ground Ref)

    Votes: 1 16.7%
  • Depends on the exact AC Waveform (sinus, square, sawtooth, etc.)

    Votes: 0 0.0%
  • All of the Above

    Votes: 0 0.0%
  • None of the Above

    Votes: 0 0.0%
  • Don't Know, Don't Care

    Votes: 0 0.0%

  • Total voters
    6

MrAl

Joined Jun 17, 2014
11,389
Assuming I didn't goof this up, here is an expression for (phi) the phase difference between the voltages on the plates of a capacitor in this series RC circuit. Setting Va as the reference for 0 degree phase, Vb phase will be equal to phi.

phi = pi - ArcSin(sin(phi))


\\ View attachment 110071 View attachment 110072
Hello again,

Thanks for taking the time to show what you are thinking about. It appears that we are probably talking about two different things, but i will look over your work to see if i can understand your point of view. That's the least i can do i guess because that's what i was asking of you too :)

In the mean time, perhaps you can provide an *explicit* formula for what you are calculating, even if it means limiting the range or something. The formula you seem to have provided is implicit.

Also, quick question...
My circuit is similar to yours except i have the resistor on top and cap on bottom.
If we call the top of the cap 'a' and bottom 'b' (as you did) in either circuit, would you say, true or false, that the phase shift across the cap is the phase shift measured at 'a' to ground minus the phase shift at 'b' measured from 'b' to ground. True or false.
 

DGElder

Joined Apr 3, 2016
351
Hello again,

In the mean time, perhaps you can provide an *explicit* formula for what you are calculating, even if it means limiting the range or something. The formula you seem to have provided is implicit.
OK, I think I see what you mean. To keep the formula from getting too messy I didn't put in explicit expressions for theta and Vr. If you want to calculate an explicit value for phi as I did in my Excel sheet then you need to first calculate Vr and theta by including variables f,C and R.

Xc = 1/(2*pi*f*C)
Vr = R*V/(R+Xc); theta = Atan(Xc/R).

.....then plug those in my formula.

If I spent some time messing around with it a while I might be able to simplify the formula somewhat, but it works as is.

If we call the top of the cap 'a' and bottom 'b' (as you did) in either circuit, would you say, true or false, that the phase shift across the cap is the phase shift measured at 'a' to ground minus the phase shift at 'b' measured from 'b' to ground. True or false

True. I am calculating the phase difference between voltages on the two cap plates, not the phase of the voltage across the cap with respect to the source, the current or anything else.
 
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MrAl

Joined Jun 17, 2014
11,389
OK, I think I see what you mean. To keep the formula from getting too messy I didn't put in expressions for theta and Vr. If you want to calculate a value for phi as I did in my Excel sheet then you need to first calculate Vr and theta from variables f,C and R.
Vr = R*V/(R+Xc); theta = Atan(Xc/R).

If I spent some time messing around with it a while I might be able to simplify the formula somewhat, but to it works as is.

My circuit is similar to yours except i have the resistor on top and cap on bottom.
If we call the top of the cap 'a' and bottom 'b' (as you did) in either circuit, would you say, true or false, that the phase shift across the cap is the phase shift measured at 'a' to ground minus the phase shift at 'b' measured from 'b' to ground. True or false



True. I am calculating the phase difference between voltages on the two cap plates, not the phase of the voltage across the cap with respect to the source or anything else.

Hello again,

Thanks for that information as i am still trying to narrow down how you are choosing to look at this problem/circuit.
So we agree on one thing now, that the phase shift we are looking for is the phase at 'a' minus the phase at 'b', both measured with ground being the common.

Didnt you state that the phase you were looking for was phi though, not theta? Or did i get mixed up on that :)

Also what might help is if you can tell me where you got the:
-V sin(wt)

part of Vb in your drawing, or what motivated you to use that term in the entire expression. I think it will help if i knew that.

I may have made a mistake though, when i posted that formula as there should be a '2' inside the atan() like so:
Ph=-atan(2*w*R*C)

and unfortunately that sill differs with your 126 degrees because with 10kHz and 150 ohms and 1uf i get close to -84 degrees, so we still dont have the same idea yet but i have a feeling we can clear this up because you've been so patient. Some of these problems really require a lot of that :)
 
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DGElder

Joined Apr 3, 2016
351
Hello again,

Thanks for that information as i am still trying to narrow down how you are choosing to look at this problem/circuit.
So we agree on one thing now, that the phase shift we are looking for is the phase at 'a' minus the phase at 'b', both measured with ground being the common.

Didnt you state that the phase you were looking for was phi though, not theta? Or did i get mixed up on that :)

Also what might help is if you can tell me where you got the:
-V sin(wt)

part of Vb in your drawing, or what motivated you to use that term in the entire expression. I think it will help if i knew that.

I may have made a mistake though, when i posted that formula as there should be a '2' inside the atan() like so:
Ph=-atan(2*w*R*C)

and unfortunately that sill differs with your 126 degrees because with 10kHz and 150 ohms and 1uf i get close to -84 degrees, so we still dont have the same idea yet but i have a feeling we can clear this up because you've been so patient. Some of these problems really require a lot of that :)
Theta is just the phase of the voltage across the resistor (Vr) with respect to the source which is the same as the phase of the current with respect to the source. phi is the phase of Vb with respect to Va.

-V sin(wt) = V sin(wt-180deg) which is the voltage at the bottom of the source.

Vb is the voltage on the bottom of the capacitor. Vb = vector sum of -Vs and Vr. I need that to compare it to Va. In the phasor drawing you can see that phi is the angle between Vb and Vs. Note Vs = Va and they have a magnitude of V. So phi is the angle Vb - Va.

Yes, it is rather confusing.
 
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MrAl

Joined Jun 17, 2014
11,389
Theta is just the phase of the voltage across the resistor with respect to the source which is the same as the phase of the current with respect to the source. phi is the phase of Vb with respect to Va.

-V sin(wt) = V sin(wt-180deg) which is the voltage at the bottom of the source.

Vb is the voltage on the bottom of the capacitor. I need that to compare it to Va.
In the drawing you can see that phi is the angle between Vb and Vs. Remember Vs = Va and they have a magnitude of V. So phi is the angle Vb - Va.

Yes, it is rather confusing.
Hello again,

Thanks again.

I thought we agreed that the phase shift was the phase at 'a' minus the phase at 'b'. If you change that to b-a then we have to change the result of the phase.
So you would rather state that you want to calculate the phase shift by the phase at 'b' relative to ground minus the phase at 'a' relative to ground? I guess that's ok as long as you want to do it that way and dont change it later if possible.

So you can actually state an explicit formula for your calculation of phi? I just wanted to be able to run some numbers so that i can quickly compare to my formula to try to figure out what is going on here.

I have another question but i do have to say that i made another little mistake. In stating the phase shift of -84 degrees i used the old formula, with the new formula i actually get -87 degrees. Sorry about that, although this still differs from your 126 degree phase shift with the same component values, but we should be able to figure out why we have a difference here.

Also i have one more question...
You seem to agree that the phase we are looking for is the phase at 'a' to ground minus the phase at 'b' to ground, or the opposite, which is the phase at 'b' to ground minus the phase at 'a' to ground. I'll take the latter as your result since you stated that last.
But if you know that the phase at 'b' minus the phase at 'a' (both relative to ground) is the phase we are after, then you probably agree that the voltage of that phase is the voltage at 'b' minus the voltage at 'a'.
Is this true?

Also, are you sure you want to use b-a instead of a-b because 'a' is on top (closer to +Vs) so is most positive and 'b' is on the bottom so is most negative. What do you think?
 

DGElder

Joined Apr 3, 2016
351
Something I noticed that could cause some confusion. Vb is a time varying function which I define at the beginning. But I also use Vb as the magnitude of that function in my phasor diagram. I should have notated the magnitude as |Vb| or something else. It all works out, it's just that my notation might cause some cognitive dissonance.

Va = Vs = V sin(wt) which has zero phase shift by definition. So the phase of Vb is the same as the phase difference Vb-Va. If I used Vb as my zero phase shift reference then Va would have a phase shift, which can not be since Va=Vs and the source has no phase shift.
 
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MrAl

Joined Jun 17, 2014
11,389
Something I noticed that could cause some confusion. Vb is a time varying function which I define at the beginning. But I also use Vb as the magnitude of that function in my phasor diagram. I should have notated the magnitude as |Vb| or something else. It all works out, it's just that my notation might cause some cognitive dissonance.

Va = Vs = V sin(wt) which has zero phase shift by definition. So the phase of Vb is the same as the phase difference Vb-Va. If I used Vb as my zero phase shift reference then Va would have a phase shift, which can not be since Va=Vs and the source has no phase shift.

Hi again,

Ok, well, we are trying to calculate the phase which you call 'phi' which is fine, but i need an explicit function for that such as:
phi=This*That+Thisotherthing+f(whaterver)

as so far all i have is an implicit function:
phi=pi-asin(sin(phi))

which doesnt seem to help because when i calculate asin(sin(x)) i get, at least for some range of x:
x=asin(sin(x))

wherefore the function above at least for some phi would come out to:
phi=pi-phi

and solving:
phi+phi=pi
2*phi=pi
phi=pi/2

but the actual range of equality seems to be:
pi/2 to 3*pi/2

which is 90 to 270 degrees which basically tells us that the exact phase angle is not computable to any specific value. So i dont understand what you were trying to show here.

If i had the explicit function i could try it myself :)

Also, a side issue, is the declaration of the voltages at the terminal of the AC source. But we can get to that later, i just dont want to forget so i am writing this note :)
 

DGElder

Joined Apr 3, 2016
351
Mr. Al,

I think some of your confusion could come from approaching this like differences in voltages and not difference in phase. The difference in phase between point b and point a is not the phase you get when you subtract the voltage of point a from the voltage of point b.

Let's say you put your CH1 o-scope probe on point a, you see a sinewave, lets make that the reference signal for comparing phase. You put CH2 probe on point b and you see a sinewave of different amplitude and shifted with respect to CH1 by 150 degrees. So we say that the phase difference is 150 degrees. But, if you subtract CH1 signal from CH2 signal and then look at the phase angle of the resultant sinewave with respect to the reference signal (CH1) it will not be shifted by 150 degrees.

Remember in AC circuits you can't subtract phasors if they have different phase angles.

P.S.
I gave you all the formulas you need to explicity solve for phi, given V,f,C,R of your choice. Go back and look at my posts #99 and #103.
 
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MrAl

Joined Jun 17, 2014
11,389
Hello again,

Ok, well sorry to say that i can not discuss this with you if you are going to keep changing the definitions of things like phase shift, especially when we had agreed upon that already, and now you are refusing to hand me a formula for your idea when that should be easy for you if you understand it as well as you believe you do.
I dont know why but all of a sudden you seem to have gotten very uncooperative. That means we would have a lot of trouble discussing this further.

It's up to you if you want to stop here, i dont mind as this has taken a lot of my time too :)

I have to admit my attention has been somewhat divided as i was working on something else that came up recently too.

It has been interesting talking to you about this, and I still look forward to discussing other topics with you in the future, and i wish you the best of luck with your projects and other endeavors.
 

nsaspook

Joined Aug 27, 2009
13,086
Hello again,

Ok, well sorry to say that i can not discuss this with you if you are going to keep changing the definitions of things like phase shift, especially when we had agreed upon that already, and now you are refusing to hand me a formula for your idea when that should be easy for you if you understand it as well as you believe you do.
I dont know why but all of a sudden you seem to have gotten very uncooperative. That means we would have a lot of trouble discussing this further.

It's up to you if you want to stop here, i dont mind as this has taken a lot of my time too :)

I have to admit my attention has been somewhat divided as i was working on something else that came up recently too.

It has been interesting talking to you about this, and I still look forward to discussing other topics with you in the future, and i wish you the best of luck with your projects and other endeavors.
Translation: I've been proven wrong. We've all been there.
 

nsaspook

Joined Aug 27, 2009
13,086
Mr. Al,

I think some of your confusion could come from approaching this like differences in voltages and not difference in phase. The difference in phase between point b and point a is not the phase you get when you subtract the voltage of point a from the voltage of point b.

Let's say you put your CH1 o-scope probe on point a, you see a sinewave, lets make that the reference signal for comparing phase. You put CH2 probe on point b and you see a sinewave of different amplitude and shifted with respect to CH1 by 150 degrees. So we say that the phase difference is 150 degrees. But, if you subtract CH1 signal from CH2 signal and then look at the phase angle of the resultant sinewave with respect to the reference signal (CH1) it will not be shifted by 150 degrees.

Remember in AC circuits you can't subtract phasors if they have different phase angles.

P.S.
I gave you all the formulas you need to explicity solve for phi, given V,f,C,R of your choice. Go back and look at my posts #99 and #103.
Thanks for your efforts here.
 

DGElder

Joined Apr 3, 2016
351
Well, taking a fresh look at my work this morning I found a few errors. :eek:

First the dumb error in post#103: The formula for Vr is wrong, it should equal RI = R*2V/sqrt(Xc^2+R^2)

Then after studying the phasor diagram (using that new factor of 2 for Vr) it is evident that the possible range for phi is 0>phi<180, where I thought phi would always be an obtuse angle. The problem is that the Law Of Sines will then produce ambiguous results. So my Excel formula works till phi gets down to 90 degrees then after that it gives results moving back toward 180 degrees again instead of zero. The law of cosines can also produce ambiguous results.

Moan......

Perhaps I will just have to break it up into two formula to cover cases for phi less than and more than 90 degrees. Or go get some useful work done.
 
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DGElder

Joined Apr 3, 2016
351
Hello again,

Ok, well sorry to say that i can not discuss this with you if you are going to keep changing the definitions of things like phase shift, especially when we had agreed upon that already, and now you are refusing to hand me a formula for your idea when that should be easy for you if you understand it as well as you believe you do.
I dont know why but all of a sudden you seem to have gotten very uncooperative.
Where the heck did that come from?
Jeez.
 

DGElder

Joined Apr 3, 2016
351
It seems the only additional correction I needed was a conditional change to the last formula:

if theta >=45 deg:
phi = pi - ArcSin(sin(phi))

if theta <45 deg:
phi = ArcSin(sin(phi))

And after plugging in the component values and employing some identities the whole thing reduced to the simple equation:
phi = 2 * theta

A simplifying realization is that Vb = Va = V :rolleyes:
So V, Vr and Vb form an isosceles triangle making it easy to find phi and see it is 2*theta.

I checked the results on my bench with 4 different values of R to get theta =0, 45, 90 and 22.5 degrees. . All the (phi) phase shifts matched the calculations: 0,90,180,45 deg respectively - plus or minus a degree or two.

 
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MrAl

Joined Jun 17, 2014
11,389
Hello,

I need an explicit formula for 'phi' from you that contains only the circuit elements.
Also, you need to state how you would measure the phase in a real circuit, ideally with an oscilloscope.
 
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MrAl

Joined Jun 17, 2014
11,389
Translation: I've been proven wrong. We've all been there.
Hello again,

Funny, you make judgements but i dont see any explicit formula from you either.
Care to render an actual solid opinion here or just keep on judging without any personal proof?

Here is my solution, again:

CapPhase=-atan(2*w*R*C)

and that is my final solution. After 40 years of circuit analysis i cant believe something this simple could be questioned so much.

LATER:
That formula uses two resistors not just one, that is the reason i ended up with -atan(w*R*C) the first time, because that is with one resistor.
 
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MrAl

Joined Jun 17, 2014
11,389
Hi again,

Little note here...

The formula:
CapPhase=-atan(2*w*R*C)

is when placing the cap in between TWO resistors not just one, so to make up for the difference in resistance the "2" appears inside. The resistor R here will be 1/2 the value of this next formula so R=75 in the examples but with two resistors in series which again makes up 150 ohms so the formulas both come out with the same result.

CapPhase=-atan(w*R*C)

and that is with R=150 in the examples.

Since i verified this i have to state the phase of the cap as -84 degrees not -87 degrees, but that error of 3 degrees is nothing compared to what kind of difference we are seeing between other people's formulas, or at least the published numbers.
I have a feeling they will change that soon :)

Values used for that -84 degrees:
R=either 75 or 150 ohms, depending on formula,
C=1uf
w=2*pi*f
f=frequency in Hz

It's been fun :)
 

DGElder

Joined Apr 3, 2016
351
Hi again,

Little note here...

The formula:
CapPhase=-atan(2*w*R*C)

is when placing the cap in between TWO resistors not just one, so to make up for the difference in resistance the "2" appears inside. The resistor R here will be 1/2 the value of this next formula so R=75 in the examples but with two resistors in series which again makes up 150 ohms so the formulas both come out with the same result.

CapPhase=-atan(w*R*C)

and that is with R=150 in the examples.

Since i verified this i have to state the phase of the cap as -84 degrees not -87 degrees, but that error of 3 degrees is nothing compared to what kind of difference we are seeing between other people's formulas, or at least the published numbers.
I have a feeling they will change that soon :)

Values used for that -84 degrees:
R=either 75 or 150 ohms, depending on formula,
C=1uf
w=2*pi*f
f=frequency in Hz

It's been fun :)

You seem to be a man with a solution in search of a problem to fit it. Nowhere do you even mention what this is a phase of nor with respect to what reference phase. Capacitor don't have phase, waveforms have phase and, like voltage, phase is meaningless unless you site the reference waveform. The formula pops out of nowhere with no definitions and no derivations and unspecified frequency mentioned in the numerical result You say it matches some unspecified simulation with an unspecified frequency and ambiguous resistance. You should know your formula is wrong because it can not produce the near 180 deg phase shift in this simulation?

upload_2016-8-9_13-1-29.png


But I recognize the formula, it is textbook for a voltage phase shift of a first order low pass filter. You still have not addressed the question of this thread.
 

DGElder

Joined Apr 3, 2016
351
Here, in one post, is my mathematical proof that the difference in respective phase of the voltages on the two terminals of a capacitor in a balanced AC RC circuit can be between 0 and 180 degrees. And it is twice the phase angle of the current with respect to the source voltage. Also it shows that the peak voltages on each of the cap plates are equal.

The math is able to replicate the posted simulator results and my bench results.

Capture2 small.png


Phasor Diagram.PNG

PHI form1.PNG

PHI form2.PNG


Edit: I just noticed a transcription error on one line in the middle of the first page. I used theta where I should have used phi. The equation should have read:
sin(phi)= (|Vr| / |Vb|) sin(theta)
The error was not carried forward so it did not affect the result.

Also I had trouble with scanning these pages - the border margins were occluded so the one page number and the equation reference numbers are missing.

Changed Vs symbol in schematic to be more clear about zero V reference.
 
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DGElder

Joined Apr 3, 2016
351
Here is an example of calculated and measured results.

PHI Calc Scope.PNG


edit: changed source schematic symbol to be more clear about ground reference.

--
 
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