Impedance vs frequency

in this video we're going to continue talking about AC analysis and the concept of impedance as the the ratio of voltage to current in an AC situation and just as a reminder of the assumptions we've made for AC analysis we've assumed that all of our signals are of the form of some kind of a sinusoid like that cosine is the typical when we pick cosine Omega T plus some offset angle and we're also going to use Euler's equation which says we can decompose our cosine into two complex Exponential's so one of the complex Exponential's is e to the plus J Omega T plus V and J multiplies both of them plus e to the minus J Omega T plus V and when we use Euler's theorem what we do is we love putting these things through differential equations so we track each of these one at a time through our circuits and typically what I'll focus on is the the plus term but the minus term all the math is the same except there's that one little minus sign and so when we made up the concept of impedance we assumed that V or I was of this form this complex exponential and that may when we put that through our components we came up with the idea of impedance and as a review for a resistor the impedance of a resistor turned out to be just R that's the ratio of V to I in a resistor and for an inductor we decided Z of an inductor was equal to J Omega L and finally for a capacitor a value C C C equals 1 over J Omega C so those are the three forms of those are the three impedances of our three favorite passive components and the units of these the units are in ohms are is measured in ohms that's the normal homes law and we use the same units for impedance of inductors and capacitors so this is in ohms and this is in ohms that may seem a little funny that because there's this frequency term in here but these this is what it is so in the rest of this video I want to look at qualitatively look at the value of these impedances and specifically look at what happens when there's a range of values of the Omega term what happens at zero frequency and low frequency and high frequency and infinite frequency so let's go ahead and do that so we're going to look at the impedance terms at different frequencies and we'll measure frequency as Radian frequency in this so let's let's talk about zero frequency and we'll talk about low frequency these are qualitative boundaries we'll talk about high frequency and let's talk about infinite frequency so we're going to build a little chart here alright so these are the value of Omega here and now we'll do our component so I'm going to do first we'll do our resistor and we're going to fill in the table for Zr and we know that equals R so at any frequency R is just R couldn't be simpler at zero frequency which is just D called DC or a battery R is R at any low frequency R as our RS are at infinite frequency so there's no dependence on frequency in R so now what now let's do the inductor and we decided that Z of an inductor was J Omega L and do something very specific I'm going to do I'm going to get rid of this J I'm going to basically say I want to just look at the magnitude of the reason of the impedance if we just look at the magnitudes the magnitude of Z inductor is Omega L so now let's fill our table in for Omega L so when Omega is zero the magnitude of the impedance is zero for an inductor and when the frequency is low when Omega is low the impedance is going to be relatively low and as the frequency gets high then Omega L becomes a larger number so it becomes high and if the frequency if we let the frequency go to infinity then Omega becomes infinity and this becomes infinity so we see in an inductor from low frequency to high frequency the impedance of the inductor goes from zero up to infinity all right let's fill in the last one here here's our capacitor and the Z of a capacitor equals one over J Omega C and the magnitude of the impedance is just 1 over Omega C all right so let's plot that out what is 1 over Omega C when Omega is 0 well it's infinity and what if Omega is a small number if Omega is a small number then this is a large number this is a high number and if the frequency gets very high the higher Omega gets the smaller the magnitude of Z gets so we get low here you see it's anti-symmetric with the inductor and finally if we let the frequency go to infinity 1 over infinity times c is what is about is zero with this chart I can show you some of the words that we use some of the sort of slang words or jargon words that we use in electrical engineering to describe the behavior of L and C at different frequencies so as we said R is R at any frequency so resistance is constant an inductor it changes over frequency over this big range of frequency and at zero frequency or low frequencies the impedance is very low so this leads to the expression we say an inductor looks like a short an inductor looks like a short at low frequency at zero frequency or low frequency and now let's go to high frequency the impedance gets high and it eventually goes to infinity that's the impedance of an open circuit so we say l equals and open at high F high frequency and it's a short at low frequency let's do the same sort of jargon investigation here for our capacitor a capacitor at zero frequency has an infinite impedance so it looks like what it looks like an open at low frequency so let's go look over it at high frequencies the impedance becomes very low the and at infinite frequency it becomes zero and that's the impedance of a perfect short so the expression the slang expression you'll hear is that a capacitor is a short circuit at high frequency so the reason I'm telling you about this very often experienced engineers will talk to beginners and use these kind of terms that Oh an inductor looks like a short at low frequency and I wanted to show you where these things come from where these terms come from and what they mean and this is this is not a simple idea you saw how many assumptions we made and how much work we did to get to this idea of these sort of comment or familiar expressions for how our components work I hope that helped for you to see how how this comes about