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Current time:0:00Total duration:11:43

Video transcript

let's talk about the idea of the impedance of some simple networks now what I've shown here there's a very simple network it has two impedances in it z1 and z2 and inside these boxes are one of our favorite passive components either an R and L or a C that's what's in both of these things we're going to look at combinations of this and figure out what the impedance of simple combinations are when we talk about impedance what we mean is we take sinusoidal signals we take a sinusoidal voltage and divide it by a sinusoidal current and that is that ratio is impedance and so the voltage is impressed across the two terminals here and then the current the current I flows this way so this will be I and this will be plus or minus V between here and here and the relationship between those two things is called impedance so now we have a circuit here with two impedances in it in series they're connected in series because they're head-to-tail now if both of these impedances were resistors like this if we just make them resistors when we write down the impedance of a resistor we just write down our the vet the impedance of a resistor is R and the impedance of this one is R so we'll call that one r2 and r1 and the overall the effective impedance of the whole thing is resistors in series we know this is r1 plus r2 so nothing new here yet now let's make a little change let's do this let's make a network that looks like a resistor and a capacitor and I want to know the voltage to current ratio or I want to know the impedance the effective impedance of this so we transform this circuit we write down this value is R and the impedance of every of a capacitor is one over J Omega C and if I want to I can write it exactly the same way I could say that is equal to minus J times 1 over Omega C remember 1 over J is the same as minus J so if I want to know the impedance of this network here Z is now this is the great trick of doing this transformation we can use the same laws that we know for resistors on this transformed circuit we transformed it into the frequency domain so the series combination of two impedances is the sum of the impedances R plus 1 over J Omega C this is the impedance of this network here let's do another let's do an inductor combination so we'll do a resistor and an inductor like that so the impedance of a resistor is r the impedance of an inductor is j omega l and i can write the combined impedance of this the same thing it's a series of peden s-- so I can do R plus J Omega L now what I want to do next is is introduce some new terminology that we talked about impedances with so let's look at these two examples down here for the capacitor and the inductor r is a real number 1 over J Omega C that's a comp that say imaginary number and together they make a complex number and over here with the inductor we see the same thing a real part and an imaginary part so the way we write an impedance in general as a rectangular complex number is we say Z equals R plus and the letter we use is X now R is the resistance and X the name for X in general is reactance X is the imaginary part of an impedance and that's referred to as the reactance we also talked about the inverse of resistance 1 over resistance is called conductance that's that's 1 over R and 1 over X is referred to as susceptance now these are all just words that they all sound like they sort of mean the same thing but engineers wanted to have sort of different words for different parts of the of the impedance and these are the words that we use and finally we have another word for the inverse of impedance the general idea of 1 over Z and that's referred to as admittance at MIT admittance this is a little vocabulary we have admittance is the opposite of impedance or the inverse of impedance susceptance is the inverse of reactance and conductance is the inverse of resistance these are all just sort of every word we can think of that that meant that meant resist and let through so now I'm going to roll up here a little bit and we'll do some plots we'll look more carefully at these at these impedance expressions so if these are all complex numbers that means I can plot them on a complex plane so let's do that and see what it tells us see if we can learn anything ok if this is the real and this is the imaginary now for resistance to resistors we just have two real parts so there's some Zi that's the sum of two R's and that's a real number so I would just get a some sort of value like here like that all right add those two vectors together and that would be Z so let's do that for the the other circuit now if I do it for capacitance RC circuit the one resistor is out here like this that's a value of our and see how do we plot C well C remember is 1 over J Omega C is the same as negative J times 1 over Omega C so that's going to give us a negative J it's going to give us a line on the negative J axis here this real imaginary and Z is going to be this value here right there has a magnitude of 1 over J Omega this point right here has a magnitude the length of that vector is 1 over Omega C and that point right there when we do the vector e D that will be Z right there depth at the time like that now let's do the L let's plot the the circuit that has an L in it ok again we have an R so we go out some distance R right there now J Omega L we have a positive J here so it goes up by Omega L so let's say L was kind of small let's say it goes right there so that has a magnitude of Omega L on the imaginary axis and that will give us a point that will be Z for that network now you notice as let's say we change oh may go let's say Omega is varying Omega goes up in this case here Omega it goes up this point will travel up it'll move in the complex plane if Omega changes this one if Omega changes if Omega gets bigger for a capacitor if a mega gets bigger for a capacitor that means that this number gets smaller and this will move in this direction and on the resistor side if Omega changes well there's there's no Omega term here so nothing changes when Omega changes in this resistor picture so this shows us what happens in a graphical way when we change the frequency of a complex impedance now I want to do one more let's do a circuit that looks like this let's do three things let's do an inductor a resistor and a capacitor all in series and what is C so we label it as we did before it J Omega L R 1 over J Omega C so let's now work out the impedance of this it's a series circuit so the impedance is ad so we can say that Z equals J Omega L plus R plus 1 over J Omega C so we can try a plot of these things here's the real here's the imaginary let's do our first R is always on the real axis some value like that J Omega L goes up by some value here's the inductor represented as a vector and here's 1 over J Omega C that goes down on the axis like that so it's going to come down here somehow like this let's make the capacitor small which makes its impedance large the magnitude of that is 1 over Omega C and when we put all these together we do basically we do a vector add of these guys here so I can let's move the L over here let's L goes up like that there's a vector ad of L and then I have a vector ad of this vector here adds onto that back down like that and the final answer is this guy right here and this is Z for this network right here is that complex number right there and you see as we change Omega different things happen and we can basically move this point around moving this point around with frequency is basically the essence of AC analysis we'll see you next time