Intro to Frequency-Dependent Impedance | Capacitors in Alternating Currents | Doc Physics

so in the corner I've got the 2 phasers for resistors and capacitors I'm going to draw you a little AC circuit here we've got an AC generator and we'll connect immediately to a resistor and then go to a capacitor and this is an RC circuit because we've got some resistance and we've got some capacitance now the voltage across this resistor is well let me talk about V Max V Max C it's going to be changing continuously as a function of time but I want to disregard the time and just set that sine function always to be 1 so I can talk about the maximum voltage that's just the maximum current times the resistance and the voltage maximum across the capacitor well that voltage maximum is going to be well it's the maximum current again times the capacitive reactance which we wrote as X sub C okay so you would think then that if somebody came up to you and said hey I want to know the voltage not between here and here and not between there and there but I want to know the voltage between here and wait for it here that you could add them up but that's not the case because they don't reach their peaks at the same time so what we do is we say this resistor is changing the whole time I guess I should get you some axes under which things can rotate above what you can rotate I mean and we're saying that the Y component is the business end of the phasor and so there is presently no voltage across the resistor and no current going through it and then suddenly huge voltage and huge current and then no voltage and no current and then the voltage is the other direction so the current goes the other direction and it's just going around like this at at a rate Omega and the capacitor is doing a similar thing but it's doing it out of phase where the current through the capacitor is not occurring at the same time as the voltage through the capacitor current though by diarhea current has to be the same through the circuit so I'm going to set this down let's set the capacitor one down on top of the resistor one boom like that and then I'm going to move these guys and of course the length of each of these voltages depends on oh man the length of each of these voltages depends on the value of the resistance and the value of the capacitance so oh and I guess the value of Omega also so we need to draw these suckers I'm going to draw them at some instant in time let's say I draw the capacitive voltage here and remember V max for the capacitor is simply I max times the reactance of the capacitor I should define reactance of the capacitor here I think it was like 1 over Omega times C that says that the capacitor well if the capacitor is big it doesn't have much reactance but the capacitor is small it has a lot of reactants and if the frequency is big the capacitor doesn't give you a lot of trouble in the circuit doesn't have a lot of reactants but if the frequency is small then the capacitor will give you all kinds of trouble so small capacitors give you trouble and small frequencies give you trouble but as far as reactants big capacitance and big frequency high frequency Channel won't give you much trouble so then we've got the maximum voltage across the capacitor and then at a right angle to it remember we saw that boom at a right angle to it we've got the maximum voltage across the resistor being max R and we said that this is just IMAX times R okay and well if we want to know the total voltage wait this is supposed to be the right angle this is the sloppiest right angle it ever drawn there now it's perfect and I'm going to say that we want to add these two vectors so we want to go out here like this and we want to say that this vector is the overall voltage notice that if those guys are rotating together like that then this overall voltage the total is going to be the vector sum of the maximum voltage on the capacitor and the maximum voltage on the resistor but lo and behold that's right angle also and so we can use the Pythagorean theorem we can say that the maximum voltage total is simply the scroot of the vaccin maximum to the vacuum role how square that right next molted on the resistor and the maximum voltage on the capacitor squirm and addemup this means that we can well see that inside here maximum voltage on the resistor is simply V oh no what do we want to say we want to say it's IMAX times ARA so it's I MX square R square and this guy over here is IMAX square times XC score and I'm supposed to screw the whole thing you notice I can factor out an IMAX score and it comes out of the screw as a regular IMAX so this is just I max times the scroot of our square plus X C score and I'll remind you well we can go dot dot dot dot that over here this V Max total then is also IMAX times the screwed up I'm just going to plug in what X sub C is that's R square plus 1 over Omega square C square okay so now I'm going to identify this as looking a whole lot like Ohm's law look at this V Max total is I max total times this thing which is sort of giving giving a difficulty towards letting current flow and so I identify this thing here's my plan identify the scroot of R square plus 1 over Omega C quantity score to be something that we call impedance it's a word that sounds a lot like resistance it's something that's keeping something from happening it's impeding the flow of current but it's a little more subtle because it depends on frequency now the in pedan s' of this circuit depends on frequency and the resistance and the capacitance and you will find that you get all kinds of really beautiful rules for instance v in a circuit max is I max times the impedance just like Ohm's law except if you calculate the impedance you can use it as the resistance yay also if you don't have a capacitor you remember that impedance well just discussed it impedance is our score under the screw plus 1 over Omega C square if you don't have a capacitor then you don't have any capacitance to worry about and then this oh that would look like the capacitance is zero so this is going to be infinite oh gosh no what if what if there is no capacitor then this term just disappears and the impedance becomes the resistance sure but what if we don't have any resistance in that case then the impedance simply becomes these are all special cases if there's no resistance then the impedance just becomes 1 over Omega C because we're going to screw that thing right there and 1 over Omega C is just the capacitive reactance so it's a combination of R and XC but because they're at right angles to each other we need to have them inside of scroot doing some P so the next thing we'll study is awesome the next thing we're going to study is called inductance I mean you've studied inductance already but we're going to bring inductance we're going to bring inductance into this impedance idea and the final thing that I want to point out is that in this circuit oh here we go is a circuit right here in this circuit we are alternately putting energy into the capacitor and taking energy out of the capacitor so the total energy dissipated in this capacitor is 0 capacitors don't dissipate energy they'll store it and they'll give it back the resistor that sucker is dissipating energy but because the voltage and the current are out of phase by 90 degrees what was I able to do I think I drew current here this is current versus time and I also want to draw the voltage as a function of time well the voltage is zero when the current is maximum and it's going to start negative and go away positive and negative C because they are 90 degrees out of phase where sometimes getting positive power we're going to say that power is I times V where sometimes getting positive power that's where they're both positive or where they're both negative and then at other times we're getting negative power where once positive and ones negative so you should work through that because this was not a fair treatment of it but I think you can also intuitively agree that a capacitor isn't going to waste any of your energy it will be this this power supply will be delivering energy to the capacitor and then taking energy back out of the capacitor during the cycle it does that two times each by

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Keywords: problem, understand, help, tutor, solve, Capacitor (Invention), Electrical Impedance, Alternating Current (Invention), Frequency (Dimension), Electric Current (Dimension)

Id: csjiObHwVJQ

Length: 9min 44sec (584 seconds)

Published: Wed Feb 06 2013