Anderson Video - LRC Circuits and Impedance

Professor Anderson
10 views
Was this helpful ?
0
<font color="#ffffff">An LRC circuit has all of these elements in it,</font> <font color="#ffffff">so it's got to look like this: voltage source V, we have an L,</font> <font color="#ffffff">we have an R, and we have a C.</font> <font color="#ffffff">This is why it's called an LRC circuit, you have an inductor, you have a resistor,</font> <font color="#ffffff">and you have a capacitor. How do we deal with this?</font> <font color="#ffffff">Well, we know that the voltage drop</font> <font color="#ffffff">has to consist of the voltage drop across the inductor,</font> <font color="#ffffff">the voltage drop across the resistor,</font> <font color="#ffffff">the voltage drop across the capacitor.</font> <font color="#ffffff">That looks good except we know there's this funny business of</font> <font color="#ffffff">the current sloshing back and forth</font> <font color="#ffffff">differently in the inductor, than the driving, and differently in the capacitor,</font> <font color="#ffffff">than the driving. And so, we have to be very careful</font> <font color="#ffffff">when we add these things up, and to add them up appropriately</font> <font color="#ffffff">we actually have to use something called phaser diagrams --</font> <font color="#ffffff">and this is hopefully where your second cup of coffee is going to kick in</font> <font color="#ffffff">because this is a little bit tricky.</font> <font color="#ffffff">Let's say we have this two-dimensional space</font> <font color="#ffffff">where we're going to map out what these different voltages are.</font> <font color="#ffffff">Okay, the voltage across the resistor is going to be this:</font> <font color="#ffffff">directly to the right in this space. We don't know exactly what this space is yet.</font> <font color="#ffffff">The voltage in the inductor</font> <font color="#ffffff">is directly north, the voltage in the capacitor</font> <font color="#ffffff">is directly south.</font> <font color="#ffffff">This is what we mean by 90 degrees.</font> <font color="#ffffff">It's 90 degrees out of phase there, it's 90 degrees out of phase there,</font> <font color="#ffffff">and now it's this entire system that rotates around</font> <font color="#ffffff">at frequency F, or omega.</font> <font color="#ffffff">All three of those things are rotating around in this space.</font> <font color="#ffffff">Now in the senior level physics course when we talk about this stuff,</font> <font color="#ffffff">we talk about the complex space with real and imaginary parts,</font> <font color="#ffffff">just think of it as vectors, right?</font> <font color="#ffffff">The voltage of the resistor pointing this way, the voltage of the inductor is pointing</font> <font color="#ffffff">this way, and the voltage of the capacitor is pointing down.</font> <font color="#ffffff">All three of those things are rotating around</font> <font color="#ffffff">but they always maintain that relationship,</font> <font color="#ffffff">and so now when we add up things to calculate</font> <font color="#ffffff">overall resistance, reactance, impedance, we can't just add them linearly,</font> <font color="#ffffff">we have to add them like vectors.</font> <font color="#ffffff">So in an LRC circuit, we have the following:</font> <font color="#ffffff">V is equal to I times Z.</font> <font color="#ffffff">We've introduced yet a new thing and this thing is called the impedance</font> <font color="#ffffff">of the LRC circuit.</font> <font color="#ffffff">Okay, this is nothing more than ohm's law again</font> <font color="#ffffff">but now we have this complex relationship between these different devices.</font> <font color="#ffffff">All right, what is Z?</font> <font color="#ffffff">Well, if i'm going to add these things up</font> <font color="#ffffff">then it looks like VL and VC are exactly opposite each other,</font> <font color="#ffffff">so I can just subtract those.</font> <font color="#ffffff">But VR is at a right angle to those things</font> <font color="#ffffff">and so i'm going to have to add that in quadrature,</font> <font color="#ffffff">and so what we really get is something that looks like this.</font> <font color="#ffffff">Okay, this is not the impedance yet, this is</font> <font color="#ffffff">how the voltages go, let's see how this applies to the</font> <font color="#ffffff">individual reactances and the individual currents.</font> <font color="#ffffff">We know that Z has to look like ohms,</font> <font color="#ffffff">so we've got to have an R there, that's this thing, okay?</font> <font color="#ffffff">But we also have to have Xl and Xc,</font> <font color="#ffffff">like so.</font> <font color="#ffffff">Okay, this is the overall impedance of the circuit</font> <font color="#ffffff">and to be technically correct it's called the complex impedance.</font> <font color="#ffffff">All right, that looks complicated enough</font> <font color="#ffffff">but come on this is physics, let's make it even more complicated.</font> <font color="#ffffff">We know what Xl is we know what Xc is,</font> <font color="#ffffff">we can rewrite this as the following:</font> <font color="#ffffff">Z is R squared plus Xl -- we said was omega times L,</font> <font color="#ffffff">Xc was 1 over omega times C.</font> <font color="#ffffff">Okay, this is the complex impedance of your LRC circuit.</font>
<font color="#ffffff">An LRC circuit has all of these elements in it,</font> <font color="#ffffff">so it's got to look like this: voltage source V, we have an L,</font> <font color="#ffffff">we have an R, and we have a C.</font> <font color="#ffffff">This is why it's called an LRC circuit, you have an inductor, you have a resistor,</font> <font color="#ffffff">and you have a capacitor. How do we deal with this?</font> <font color="#ffffff">Well, we know that the voltage drop</font> <font color="#ffffff">has to consist of the voltage drop across the inductor,</font> <font color="#ffffff">the voltage drop across the resistor,</font> <font color="#ffffff">the voltage drop across the capacitor.</font> <font color="#ffffff">That looks good except we know there's this funny business of</font> <font color="#ffffff">the current sloshing back and forth</font> <font color="#ffffff">differently in the inductor, than the driving, and differently in the capacitor,</font> <font color="#ffffff">than the driving. And so, we have to be very careful</font> <font color="#ffffff">when we add these things up, and to add them up appropriately</font> <font color="#ffffff">we actually have to use something called phaser diagrams --</font> <font color="#ffffff">and this is hopefully where your second cup of coffee is going to kick in</font> <font color="#ffffff">because this is a little bit tricky.</font> <font color="#ffffff">Let's say we have this two-dimensional space</font> <font color="#ffffff">where we're going to map out what these different voltages are.</font> <font color="#ffffff">Okay, the voltage across the resistor is going to be this:</font> <font color="#ffffff">directly to the right in this space. We don't know exactly what this space is yet.</font> <font color="#ffffff">The voltage in the inductor</font> <font color="#ffffff">is directly north, the voltage in the capacitor</font> <font color="#ffffff">is directly south.</font> <font color="#ffffff">This is what we mean by 90 degrees.</font> <font color="#ffffff">It's 90 degrees out of phase there, it's 90 degrees out of phase there,</font> <font color="#ffffff">and now it's this entire system that rotates around</font> <font color="#ffffff">at frequency F, or omega.</font> <font color="#ffffff">All three of those things are rotating around in this space.</font> <font color="#ffffff">Now in the senior level physics course when we talk about this stuff,</font> <font color="#ffffff">we talk about the complex space with real and imaginary parts,</font> <font color="#ffffff">just think of it as vectors, right?</font> <font color="#ffffff">The voltage of the resistor pointing this way, the voltage of the inductor is pointing</font> <font color="#ffffff">this way, and the voltage of the capacitor is pointing down.</font> <font color="#ffffff">All three of those things are rotating around</font> <font color="#ffffff">but they always maintain that relationship,</font> <font color="#ffffff">and so now when we add up things to calculate</font> <font color="#ffffff">overall resistance, reactance, impedance, we can't just add them linearly,</font> <font color="#ffffff">we have to add them like vectors.</font> <font color="#ffffff">So in an LRC circuit, we have the following:</font> <font color="#ffffff">V is equal to I times Z.</font> <font color="#ffffff">We've introduced yet a new thing and this thing is called the impedance</font> <font color="#ffffff">of the LRC circuit.</font> <font color="#ffffff">Okay, this is nothing more than ohm's law again</font> <font color="#ffffff">but now we have this complex relationship between these different devices.</font> <font color="#ffffff">All right, what is Z?</font> <font color="#ffffff">Well, if i'm going to add these things up</font> <font color="#ffffff">then it looks like VL and VC are exactly opposite each other,</font> <font color="#ffffff">so I can just subtract those.</font> <font color="#ffffff">But VR is at a right angle to those things</font> <font color="#ffffff">and so i'm going to have to add that in quadrature,</font> <font color="#ffffff">and so what we really get is something that looks like this.</font> <font color="#ffffff">Okay, this is not the impedance yet, this is</font> <font color="#ffffff">how the voltages go, let's see how this applies to the</font> <font color="#ffffff">individual reactances and the individual currents.</font> <font color="#ffffff">We know that Z has to look like ohms,</font> <font color="#ffffff">so we've got to have an R there, that's this thing, okay?</font> <font color="#ffffff">But we also have to have Xl and Xc,</font> <font color="#ffffff">like so.</font> <font color="#ffffff">Okay, this is the overall impedance of the circuit</font> <font color="#ffffff">and to be technically correct it's called the complex impedance.</font> <font color="#ffffff">All right, that looks complicated enough</font> <font color="#ffffff">but come on this is physics, let's make it even more complicated.</font> <font color="#ffffff">We know what Xl is we know what Xc is,</font> <font color="#ffffff">we can rewrite this as the following:</font> <font color="#ffffff">Z is R squared plus Xl -- we said was omega times L,</font> <font color="#ffffff">Xc was 1 over omega times C.</font> <font color="#ffffff">Okay, this is the complex impedance of your LRC circuit.</font>