ELI the ICE man

okay it's time to introduce you to a new friend Eli the Iceman Eli the Iceman is a friend of every electrical engineer and what we've been talking about is AC analysis and in AC analysis we limit ourselves to one type of signal and that's a sinusoidal story we like is called cosine we say cosine of Omega T plus V Omega represents the the Radian frequency of the cosine here it's shown in blue that Radian frequency is Omega and fee is the phase delay or the phase shift and if we look here we see this isn't really a cosine wave because the peak is a little before zero time equals zero so this distance right here is the lead the phase lead and that's fee so when fee is a positive number this whole cosine wave is shifted a little bit to the left that's what we mean by phase shift so when these kind of signals are input into our favorite components we're going to get a relationship between the voltage and the current in those components and this related by the impedance we've defined the idea of impedance as the ratio of voltage to current we gave that the symbol Z now in this video instead of using V as my variable for voltage I'm going to use a different letter I'm going to use e e is short for EMF or electro-motive force and it's really commonly used almost as often as as V for representing voltage and I'll show you why I want to use E in a little bit and another way I can write this just as easily e equals Z times I and this looks a lot like Ohm's law and what we're going to find out here is we can apply this in addition to applying it to resistors we can apply it to capacitors and inductors so first off we're going to look at our friend the inductor and we're going to look at the equation e equals Z I for an inductor I'm going to assign I to be a sinusoidal to some magnitude we'll call it I naught cosine Omega T plus V so I'm going to say my current is a cosine wave of this magnitude with this phase delay and that's shown in blue here so this here is I and now let's write e in terms of this I here so I can write e equals e now what is Z for an inductor the impedance of an inductor is J Omega L and what is I I is sitting right here now I'm going to represent I like this I'm going to represent I as a phasor or a phasor representation and we said that that can be represented as I the magnitude of the current indicated at the angle of fee so these are equivalent representations of I this is the time domain representation and this is the phasor representation now what we have out here in front of I is a scaling factor there's this complex J that will take care of in a second and there's Omega L so Omega is the frequency and L is the size of the inductor now for the purposes of this video when I plot out the voltage over here in orange I'm going to we're going to assume that the scaling factor Omega L is one just so that we can focus on the timing relations ships between the current and the voltage when we talked about complex numbers multiplying by J multiplying something by J represents a rotation of plus 90 degrees and so I can write this as e equals let's put the let's put the scaling factor out there and we'll have I naught which is the original magnitude of the current and feet it's changed your fee changes fee becomes fee and this multiplication by J here corresponds to adding 90 degrees to feed so multiplying by J corresponds to a 90 degree phase shift and if I draw here this is now e and the phase shift we decided this distance right here this distance right here is fie and this distance right here is a phase lead of 90 degrees and you'll notice I off the peaks of these these waveforms because that's the easiest place to see the lead so when I move to the left that corresponds to a lead of plus 90 degrees so in an inductor in an inductor in an inductor we say that e leads I by 90 degrees all right now let's do it for our capacitor and we'll do the same kind of thing here for a capacitor we'll assign the same current will say I equals some current I knot times cosine of Omega T plus fee and now let's work out the voltage across the capacitor so the voltage across the capacitor e is the same thing we have here e equals e I or I can write e in the capacitor equals Z now what is impedance of a capacitor it's 1 over J Omega C that's Z and I we represent the same way as we did before I knot at an angle of fee so now let's carefully do this this multiplication e equals 1 over J times 1 over Omega C times I naught and an angle of V so here's this 1 over J term now I can rewrite 1 over J as minus J now we're multiplying something by minus J and multiplying by minus J corresponds to a rotation of minus 90 degrees so I can write e one more time like this e equals 1 over Omega C there's the scale factor here's the original current magnitude and I get the angle of fee this time minus 90 degrees so this minus sign here corresponds to a lag a phase lag so here's our original current here let me label that here's I and now we have our voltage e looks like this here's E and what we see let me go out here and measure it here here we have a phase lag we're pointing to the right of 90 degrees and that we call a lag we can summarize that we can say in a capacitor we say e lags I and an equivalent way to say this is we could say that I leads e I leads voltage so I can actually put boxes around these two results here and here now there's a lot of sign flipping going on here and there's actually an easy way to remember this and I want to introduce you to someone who can help you remember this and his name is Eli the Iceman so what can Li tell us Eli tells us that in an inductor and L voltage leads current and over here in a capacitor C current leads voltage that's the message from Eli the Iceman he helps us remember the order that voltage and current change in inductors and capacitors he's going to be your friend for a long time