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• - [Voiceover] We're gonna talk about the equations

• that describe how a capacitor works, and then I'll give you

• an example of how these equations work.

• The basic equation of a capacitor, says that the charge, Q,

• on a capacitor, is equal to the capacitance value,

• times the voltage across the capacitor.

• Here's our capacitor over here.

• Let's say we have a voltage on it, of plus or minus V.

• We say it has a capacitance value of C.

• That's a property of this device here.

• C is equal to, just looking at the equation over there,

• C is equal to the ratio of the charge,

• stored in the capacitor, divided

• by the voltage of the capacitor.

• What we mean by stored charge is,

• if a current flows into this capacitor,

• it can leave some excess charge on the top.

• I'll just mark that with plus signs.

• There will be a corresponding set of minus charges,

• on the other plate of the capacitor.

• This collection of excess charge will be Q ,

• and this down here will be Q-,

• and they're gonna be the same value.

• What we say here, is when the capacitor's in this state,

• we say it's storing this much charge.

• We'll just name one of these numbers here.

• They're gonna be the same, with opposite signs.

• That's what it means for a capacitor to store charge.

• What I want to do now, is develop some sort of expression

• that relates the current through a capacitor,

• to the voltage.

• We want to develop an IV characteristic,

• so this will correspond, sort of like,

• Ohm's Law for a capacitor.

• What relates the current to the voltage.

• The way I'm gonna do that, is to exercise this equation,

• by causing some changes.

• In particular, we'll change the voltage on this capacitor,

• and we'll see what happens over here.

• When we say we're gonna change a voltage, that means

• we're gonna create something, a condition of DV, DT.

• A change in voltage per change in time.

• I can do that by taking the derivative

• of both sides of this equation here.

• I've already done it for this side.

• Over here, what I'll have is DQ, DT.

• I took the derivative of both sides, just to be sure

• I treated both sides of the equation, the same.

• Let's look at this little expression right here.

• This is kind of interesting.

• This is change of charge, with change of time.

• That's equal to, that's what we mean by current.

• That is current.

• The symbol for current is I.

• DQ, DT is current, essentially, by definition, we give

• it the symbol I, and that's gonna be equal to C DV, DT.

• This is an important equation.

• That's, basically, the IV relationship,

• between current and voltage, in a capacitor.

• What it tells us, that the current

• is actually proportional to,

• and the proportionality constant is C,

• the current's proportional to the rate of change of voltage.

• Not the voltage itself,

• but to the rate of change of voltage.

• Now, what I want to do is find a expression

• that expresses V, in terms of I.

• Here we have I, in terms of DV, DT.

• Let's figure out if we can express V,

• in terms of some expression containing I.

• The way I do that is, I need

• to eliminate this derivative here.

• I'm gonna do that by taking the integral of this side

• of the equation, and at the same time, I'll take

• the integral of the other side

• of the equation, to keep everything equal.

• What that looks like is, the integral of I...

• With respect to time, is equal to the integral

• of C DV, DT, with respect to time, DT.

• On this side, I have basically, I do something like this.

• I have the integral of DV.

• This looks like an anti-derivative.

• This is an integral, acting like an anti-derivative.

• What function has a derivative of DV?

• That would be just plain V.

• I can rewrite this side of the equation,

• constant C comes out of the expression,

• and we end up with V, on this side.

• Just plain V.

• That equals the integral of I DT.

• We're part way through, we're developing what's gonna

• be called an integral form of the capacitor IV equation.

• What I need to look at next is,

• what are the bounds, on this integral?

• The bounds on this integral are basically minus time,

• equals minus infinity, to time equals sub-time T,

• which is sort of like the time now.

• That equals capacitance times voltage.

• Let me take this C, over on the other side,

• and actually, I'm gonna move V over here, onto the left.

• Then, I can write this, one over C.

• This is the normal looking version of this equation.

• I DT, minus infinity to time, T.

• Time, big T, is time right now.

• What this says, it says that the voltage on a capacitor

• has something to do with the summation,

• or the integral, of the current, over its entire life,

• all the way back to T equals minus infinity.

• This is not so convenient.

• What we're gonna do instead, is we're gonna pick a time.

• We'll pick a time called T equals zero,

• and we'll say that the voltage on the capacitor

• was equal to, let's say, V not, with some value.

• Then, what we'll do, is we're gonna change

• the limit on our integral here,

• from minus infinity, to time, T, equals zero.

• Then, we'll use the integral from, instead,

• zero to the time, we're interested in.

• That equation looks like this.

• We're just gonna change the limits on the integral.

• We have the integral now, but we have to actually account

• for all the time, before T equals zero.

• What we do there, is we just basically add V not.

• Whatever V not is, that's the starting point,

• at time equals zero, and then the integral takes us,

• from time zero until time now.

• This is the integral form of the capacitor equation.

• I want to actually make one more little change.

• This is the current at V, as a function of T.

• What we really want to write here,

• is we wanna write V of a little T.

• This is just stylistically, this is what we like

• this equation to look like.

• I want the limits on my integral, to be zero to t.

• Now, I need to sort of make a new replacement

• for this T that's inside here.

• I can call it something else.

• I can call it I of, I'll call it tau.

• This is basically just a little fake variable.

• D tau plus V not.

• This is, now, we finally have it,

• this is the integral form of the capacitor equation.

• We have the other form of the equation that goes with this,

• which was I equals C DV, DT.

• There's the two forms of the capacitor equation.

• Now, I want to do an example with this one here,

• just to see how it works, when we have a capacitor circuit.

- [Voiceover] We're gonna talk about the equations

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