Capacitor theory question (why 90 deg out of phase)

[Hiwatt25]

So, I'm having trouble understanding why there is a 90 degree phase difference between current and voltage when an AC signal is applied to a capacitor.

I'm so confuse by this concept that I don't even know if I'm asking the question right. 

[Sir H C]

The phase shift is frequency dependant actually.  It all has to do with:

I = C * dV/dT

So the current is the derrivative of the change in voltage with respect to time, with sine waves that comes out to turning a sine to a cosine (or is it -cosine?), which are 90 degrees out of phase.

[R.G.]

dV/dT doesn't have frequency in it. For a capacitor in isolation, the equation means that all frequencies have a 90 degree phase shift between voltage and current as measured right at that capacitor.

Integrals aren't needed either. The derivative of a sine is cosine, and vice versa as well.

But for Hiwatt25, here's another way to think of it.

A capacitor is like a sealed tank that you can stuff charge into. Using the water analogy, imagine that you have a sealed tank full of air that you're going to try to pressurize using a garden hose. The garden hose gets connected to the tank, but you actually have to pump the tank full of water to bring the pressure inside the tank up. The pressure in the tank lags the change in the rate of water being pumped in or out.

Same with a cap. You have to pump a bunch of charge - electrons - into a cap to get a voltage to appear across it. Then to change the voltage, you have to pump a lot of electrons in or out to change the voltage. So each change in voltage requires a lot of moving charge - current that is - to move in an out of the cap.

Capacitance is defined by charge storage. Capacitance C = Q/V  where Q is the total charge and  V is the voltage. So we can rearrange that a bit to Q = C*V . Literally, the number of electrons stored (OK, the charge on that number of electrons) is proportional to the capacitance in farads times the volts.

If we change the volts, then the charge changes too - delta-Q = C * delta-V.

If we KEEP changing the voltage at x volts/second, then charge changes at delta-Q per second = C times x volts/second.

And that gets us to change in charge per second - moving charge is current. So current equals C times the rate at which voltage changes.

For a sine wave, the instantaneous rate at which the sine wave is changing happens to be a cosine. So if we apply a sine wave voltage to a cap, the current must be ... the cosine.

And sines and cosines are 90 degrees phase shifted.

I know, I know; I beat around the bush about six ways. Did any of them help?

[Hiwatt25]

R.G.-

Yep that helps.  I had to read and re-read your explanation a few times sloooowly  and I don't think I'll be writing any papers on the finer points of capacitance in the near future.  But, I'm getting the gist.

Thanks the insight. 

[db]

dV/dT doesn't have frequency in it.

Well not quite.
say we have a voltage

v=V*sin(wt) - where w = 2*PI*f

i=C*dV/dt as Sir H C correctly pointed out.

therefore,

i=w*C*V*cost(wt)

= w*C*V*sin(wt+90)

i.e. always a 90 degree phase shift as RG pointed out, but the magnitude is directly proportional to frequency.

Now for the interesting bit...(well I think so anyway)
if you take the ratio v/i, you should get the impedance:

v       V*sin(wt)                           tan(wt)
--  = --------------               =      ---------
i      w*C*V*cos(wt)                     w*C

but hang on, this means that the instantaneous impedance varies with time and frequency?  Well, this is where the term "reactance" comes in.  Capacitors and inductors are both reactive components in the sense that they respond according to the applied voltage (or current for inductors).  If you think of resistors, they don't care what the voltage is, the ratio of volts to current is always the same and constant i.e. R = V/I no matter what the shape or frequency of the applied voltage.

Getting back to capacitors, a phase shift of 90 degrees is represented by j where j = sqrt(-1).  So we write the impedance of a capacitor as j/wC i.e. impedance inversely proportional to frequency with a constant 90 degree phase shift.

I hope this helps someone, somewhere.

[Hiwatt25]

My eye just popped out of my head.

[gaussmarkov]

fwiw, i have been working up a description of capacitors on my webpage.  it's not finished yet.  but what's there may help with answering the question about why there is a lag in the voltage across the cap.  it won't help with the 90 degrees business, which is sine/cosine specific.

the description begins at http://gaussmarkov.net/parts/capacitors.php and the page that i am thinking might be helpful here is http://gaussmarkov.net/parts/capacitors.php?page=low pass filters.  tthe discussion is all in terms of voltage changes.  but current is the same through a series of components so you can think of current as proportional to the source voltage in my examples.

hope this helps, gm

[Rob Strand]

If you want to start with fundamentals then there's a couple of steps before Sir H C's equation.    (If you start up stream in the derivation, you just end-up with V = (1/(j w C)) * I  which has the 90deg phase shift from the start!).

From physics,

        i  is defined as i = dq /dt

and  C is defined as q = C v

[ed: by the way q is charge]

So you take the second of these an plug it into the first,

   i = d(Cv)/dt   = C dv/dt; which is  Sir H C's well know equation.

Now you assume v has the form v(t) = V sin(wt)  where V is the magnitude and w is the angular frequency (w = 2 * pi * frequency).

Plug this into i = C dv/dt,

i  = C d(V sin(wt)) /dt
   = C V d (sin(wt)) /dt
   =  w C V  cos(wt)

which is of the form I cos(wt)  where I is the magnitude of the current.

From this you conclude the current magnitude is I = w C V, which you futher deduce the impedance (ratio of magnitudes) V/I = (1/wC).  You also deduce the phase shift between v and i,  v is a sine function and i is a cos function, which are 90 deg out of phase.

(You can go through the calculations with Laplace transforms and complex number j's etc.  Mathematically it is all the same thing but you need to know that stuff to know why it's the same thing.)