Electrical Engineering: Ch 8: RC & RL Circuits (14 of 43) Current=? in RL Circuit: Ex. 2

welcome to electron line here's their second example of how to find the current in an RL circuit not only do we need to find the current through the inductor we're also being asked to find the voltage across the inductor and the current through the six ohm resistor notice here that we have a 10 volt voltage supply and we have a switch that closes at T equals zero so what's happening before the switch closes what happens when T is less than zero well when T is less than zero the switch is open all the current goes to here 3 3 ohm resistor to the branch point since the inductor will be in a steady state situation that means then that the inductor no longer opposes the current because it only opposes a change in the current the current is now changing all the current will be going to be inductor and none of it will be going to the 6 ohm resistor which means that the inductor will app act like a short and the 6 ohm resistor will act like an open redrawing the circuit with this open right here it'll look as follows we have the 10 volt voltage supply we have a 2 ohm resistor a 3 ohm resistor a short and back to the voltage source 2 ohms 3 ohms and that will be the equivalent circuit before time equals 0 well then to find the current at that moment you could say that the current I which is equal to the voltage divided by the total resistance the total resistance will be 5 ohms 10 volts divided by 5 ohms equals 2 amps which means that before time equals zero there's a 2 amp current flowing from the voltage supply through the two resistors through the inductor and back so the initial current through the inductor is equal to 2 amps let's go ahead and write that so the initial current through the inductor I could write I sub L is equal to 2 amps now that we know that let's now go to situation when the time is after T equals zero so now we can say that T is greater than 0 that means the switch will have closed and the circle will look as follows 10 volt voltage supply the tomb resistor short circuit back to the voltage supply now this side a 3 ohm resistor a 6 ohm resistor and we have an inductor and this is a two conductor alright notice once we close a switch here any current coming out of the 10 volt voltage supply will go through the 2 ohm resistor but then we'll follow the short circuit path back to the voltage supply and none of the current coming from here will go to this part of the circuit which means that this part of the circuit can now be drawn as an equivalent circuit so what let's go ahead and do that the equivalent circuit will not look like this relative to the inductor so we have a 3 ohm 6 ohm and a 2 Henry inductor so that's the equivalent circuit notice that if we want to go one step farther and combine those two resistors notice that these two resistors are in parallel relative to this inductor so this can now be written as or at least drawn as a single resistor with a single inductor the equivalent resistance in this case would be the sum of those stupid remember those are in parallel so be the product over the sum 18 divided by 9 which means 2 ohms is the equivalent resistance here to Henry's is the inductance which means that the time constant being L over R equals to Henry's divided by 2 ohms equals 1 second we know the initial current we know the time constant from that we should be able to draw the equation that determines the current through the inductor after T equals zero that means that I is a function of time which is equal to the initial current as T equals zero so that would be T equal to zero times e to the minus T over tau and so this is equal to two amps of initial current times e to the minus T over one or we simply could write this as two amps times e to the minus T that's the current through the inductor after time equals zero okay couple more things that we need to find we need to find the voltage across the inductor and the current through the six ohm resistor all right so let's find the voltage across the inductor we know that the voltage across an inductor is equal to L times the IDT from that we can say therefore that the voltage across the inductor is equal to L well L is equal to one sec oh I'm sorry L is equal to two Henry's to Henry's multiplied times the D DT of the current and the current is equal to right here two amps times e to the minus T when we take the derivative of that so this would be equal to there will be two times two that would be four and that would be volts times e to the minus T times the derivative of the exponent which is a negative one which means that this is equal to a minus four volts times e to the minus T that would be the voltage across the inductor as a function of time finally we're now ready to find the current through the six ohm resistor notice using Ohm's law that the current through the sixth ohm resistor as a function of time is equal to the voltage across the six ohm resistor as a function of time divided by the resistance of course that would be the 6 ohm resistance resistor but the voltage across the six ohm resistor must be equal to the voltage across the inductor because those two branches are in parallel and the voltage across the inductor is this voltage right here which means that this is equal to minus 4 volts times e to the minus T divided by the 6 ohms and therefore the current I to the 6 ohm resistor would be equal to minus 2/3 amps multiply them to e to the minus T and that's how we also find the current through that resistor the approach is the same we look at the circuit before the switch is closed we realize that this will become an a short circuit this will become an open circuit we redraw the equivalent circuit to find the initial current flowing through the inductor after the switch is closed this becomes a short circuit all the current from the voltage supply will be short circuited through here will not go to the inductor this can then be drawn as an equivalent circuit and then even reduced even further to find the equivalent resistance from that we find the time constant once we know the time constant and the initial current through the inductor we can find the current as a function of time through the inductor then we can find the voltage across inductor our real I think that it's the inductance times have changed we respect the time of the current and then finally we can find the current to the six ohm resistor realizing that the voltage across these two branches has to be exactly the same and that's how it's done

Channel: Michel van Biezen

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Keywords: ilectureonline, ilectureonline.com, Mike, Mike van Biezen, van Biezen, ilecture, ilecture online, Electrical Engineering, RC & RL Circuits, Ch 8, Chapter 6, Introduction, First Order, Voltage, Current, Differential Equation, Capacitor, Resistor, Inductor, General Equation, Initial Charge, Voltage Divider, Parallel, Equilvalent, Circuit, Kirchhoff, Voltage Law, Ohm's Law, L/R Time Constant, RL Circuit, Finding the Current, Current=?

Id: pd8cEL8hhi8

Length: 8min 7sec (487 seconds)

Published: Mon Sep 26 2016