Impedance

now we're going to talk about the idea of impedance this is a really important idea in electronics and it's something that comes from the study of AC analysis and AC analysis is where we limit ourselves to inputs to our circuits that look like sinusoidal of all the signals that we could possibly have in the entire universe we're going to limit ourselves just for the moment to sine waves and there's some great simplifications that emerge from this so in this video we're going to look at we're going to develop basically the IV equations for our three favorite passive components resistor inductor and capacitor and we're going to look at those when the input is a sinusoidal I the voltage of the current is in the shape of a sinusoidal so as we look at our IV equations with sinusoidal e going to breakdown sinusoids into complex Exponential's and when we studied sinusoids we found out that we could disassemble sinusoidal x' using Euler's equation so for example if we have a cosine wave if we have a cosine as a function of time Omega T we could express that in terms of complex Exponential's like this one half e to the plus J Omega T plus e to the minus J Omega T like that and what I'm going to do now is I'm going to say let's look at let's look at what happens when we use this as an input signal this is not a real input signal it's an imaginary rotating vector but if I have two of these I can reassemble them into a cosine wave and we like to use these Exponential's because they go through the differential equations of a circuit really easily these are the these are the inputs that we know how to solve when we do differential equations so what I'm going to do is develop the IV equations for the resistor inductor and capacitor in terms of this kind of an input when the voltage or the current looks like this one of those equations look like so we're going to start with I'll start with the resistor here's a resistor and we know from that that just Ohm's law is V equals I times R and just from the moment I'm going to assume that the current let's assume that I equals e to the plus J Omega T so if this is I what is what is V for a resistor well we just plug I in here and we get V equals R times e to the plus J Omega T all right now I'm going to do something that looks like it's a little too simple but it's going to get interesting soon I want to look at the ratio of voltage to current in this situation where we're driving with this complex exponential so a voltage turned out to be our e to the plus J Omega T and that's the voltage I get if I put a current through the resistor of e to the plus J Omega T and what is that equal to well these two are the same so they cancel and I get the ratio of voltage to current is R for a resistor so for so for a resistor we just proved that V over I equals R so this is in news we did make a discovery here this is just owns law and for a resistor the voltage over the current is always equal to the resistance all right this is going to get more interesting now as we go do inductors and capacitors so let's do an inductor it has a value of L Henry's and for an inductor we know that voltage equals L di DT all right and let's do the same thing again let's let I equal e to the plus J Omega T so it's a complex exponential current that we're forcing through our inductor and let's go ahead and work out what V is so V equals L times D DT of this value here e to the plus J Omega T or V equals now we take the derivative and the J Omega term comes down to multiply L so we get J Omega L times what times the same thing e to the plus J Omega T this is the beautiful thing about Exponential's they give us back themselves all right just so now let's do this let's take once more let's what's the ratio of voltage to current and that equals here's the voltage J Omega L times e to the plus J Omega T and let's divide that by I which is I is right here I is e to the plus J Omega T so those cancel and we get V over I equals J Omega L so now we have an equation for V over I for an inductor and this is interesting this time we have the inductance value which we expected and there's also this this Omega J Omega term that comes in so this tells us this is frequency Omega is frequency so this tells us that the ratio of V to I for an inductor is dependent on frequency now we'll do the same thing for a capacitor so here's a capacitor and there's it's capacitance value in farad's and for the capacitor we know that I equals C times DV DT and this time let's let V of T let's let V equal e to the plus J Omega T so this time we're going to force a voltage across our capacitor that is this imaginary this complex exponential and that gives us let's plug that in here I equals C times D DT of e to the plus J Omega T and let's take that derivative I equals same thing happens J Omega comes down to multiply out in front with C and we get the same thing over here so we get J Omega C times e to the plus J Omega T and now we'll ask the same question again that we did before which is what is what is V over I for a capacitor and we can fill this in v is sitting right here V is e to the J Omega T e to the plus J Omega T and the current is we work that out down here that's J Omega C times e to the plus J Omega T and once again we get this nice cancellation here this cancels with this and for a capacitor we get V over I equals 1 over J Omega C let me put a box around that one too so this says that the ratio of voltage to current in a capacitor it depends on the value of the capacitor of course and it also depends on frequency so just like over it with the inductor we have a frequency term in here so now we're going to give this this ratio of voltage to current in all three cases we're going to give a special name and that name is impedance and the the symbol we often use for impedance is a Z so this word impedance is the general notion of the ratio of voltage to current and we can do that for all three of our passive components for a resistor the impedance is its resistance R so the word impedance is like the word resistance except it's a more general concept it's the general concept of voltage divided by current for a resistor the impedance is the resistance for an inductor the impedance V over I is J Omega L and down here for a capacitor the impedance is 1 over J Omega C for a capacitance so this is where the idea of impedance this word impedance this is where it comes from and the idea includes both the values of the components and the effect that frequency has on on the voltage to current ratio so both of those things are are combined into one idea so just a quick summary of impedance if we'd say the impedance of a resistor that equals our impedance of an inductor equals J Omega L and the impedance of a capacitor equals one over J Omega C and as a reminder of the assumptions we made we said that we're only going to consider sinusoidal inputs and what we did is we broke up our sinusoidal wave into these we looked at how these rotating vectors pass through our our components in the in the form of voltages and currents so there's no new physics here what happened was we took these J Omega s that came out of the Exponential's when we took the derivatives we associated those with the the component itself we did that here and we did that here and you can see it here so we've sort of just done a it's it's it's somewhat of a notational trick we associate this frequency dependence not with the inputs and the voltages and currents but with the components themselves and this is something we call this is referred to as transforming we've transformed our components that's that's the phrase that's used there but from this comes the idea of impedance and in the general sense of the voltage to current ratio