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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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電容 I-V (Capacitor i-v equations)

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    Xiang-hao Lin   發佈於 2019 年 11 月 12 日
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