First Order Transient Circuit Analysis

several months ago I made a video on first-order transient circuits and in this video I'm going to expand upon that and provide a more formal approach to solving these kinds of circuits first thing you want to do is determine if you have a first-order transient circuit to determine this you need to answer first of all do I have a transient in this particular circuit we have a transient given by this flipping of a switch the second question that you need to answer is do I have a single energy storage device energy storage devices in linear for linear components are inductors and capacitors recall that capacitors store a voltage in an electric field and inductors store current in a magnetic field at least that's one way of thinking of them and they are energy storage devices so we appear to have three in our circuit however they can be collapsed into a single effective capacitance so we have effectively a single energy storage device all first-order linear systems will be modeled by this particular equation and we'll take a look at that in a moment maybe the hardest thing to grasp is that for these first second and higher-order transient circuits we need to remember that the current through an inductor is a continuous function which cannot which is time-based and cannot change instantly similarly the voltage across a capacitor cannot change instantly this is to contrast with Ohm's law equation which basically describes that the voltage and the current in a resistor can change instantly and finally we're gonna solve for these four unknowns we'll take a look at this in just a minute the time constant tau for RC and L our circuits are given here and if you were to actually go through and try and solve a system you would get this equation and while you were solving that you would see that tau actually is one of these values I'm not going to demonstrate that right now though so the first thing we'll do is see my circuit I asked the question is it a first-order circuit yes it is I want to draw it as a circuit with a single energy storage device and a single transient there it is okay I just combined the capacitors next thing I'm going to do is remind myself what's the equation that I'm trying to solve here and the reason I do this is because the more you write this equation down the more you'll get it memorized so we have X infinity plus X naught minus X infinity yi minus t minus t0 over tau then what I like to do is write down all of the unknowns X 0 X infinity t 0 and tau and I often try and solve them from easiest to most difficult generally you'll find that t0 is the easiest thing to solve usually it's going to be zero one case I can think of where it won't is if you're dealing with the unit step function you might be asked to find the equation at a particular point in time the next easiest one or the one that I generally solve next anyway is the time constant tau to do the time constant tau you want to look at the resistance the Thevenin resistance seen by the energy storage device after the transient has occurred okay in this particular case after for T greater than zero so when T is greater than zero this opens up we don't have any dependent sources so we can just treat this as a short not that it matters right because this is open so there's no current through that resistor so if essentially I don't really need to write this but I I will anyway I like to draw lots of pictures that kind of help me think my way through it okay this is our seven in through the capacitor and it's gonna simply just be six K so our time constant because we're dealing with a capacitive and resistive network is going to be RC but one thing that I sometimes do is I'll do one over tau don't have to do this if you don't want to it'll do 1 over RC the reason I do that is because then I can move tau or I can move this answer that I get into the numerator and it'll make my final equation look a little bit cleaner okay so this is one over six K times for micro and if you do that all out you'll find that tau equals or one over tau equals minus sorry equals forty one point six six repeating okay so I'm going to just kind of draw this in here one over T equals 41.67 repeat our six seven okay now the next thing I'll do is I'll solve for X infinity that's usually the easier of these two to solve for X infinity you just want to imagine what happens when I flip the switch well what happened before I flip the switch let's let's imagine what's happening over all of time here so some point infinitely in the past this thing this circuit was created and there's a battery attached to it and the battery causes current to flow it causes the capacitor to get charged up eventually the capacitor gets fully charged and then it begins to act like an open and no more current travels through that through that branch of the circuit so all the current flows only through the resistors in this outer loop then we flip the switch now the battery gets disconnected from the circuit so does the 6k resistor and the capacitor which had some charge on it begins to drive a current through these two resistors and eventually all of the energy stored in the capacitor gets dissipated through the resistors so in the end there's no current traveling in this loop there's no current there's nothing happening anywhere in the circuit and therefore recall to by the way that we're looking for V naught of T right so we want to know what happens at the resistor because there's no current flowing from the capacitor there's no current flowing through the resistor and therefore this equals zero okay but we're but notice what I what I just did here I was interested in what happens at V of T but all of my attention really goes to the energy storage device what's happening with that energy storage device will answer the question for both of these so this is exit infinity equals equals zero okay so now what we'll do is we'll try and answer the question what happens to the voltage at the moment T equals zero we're calling that the voltage across the capacitor can change instantly but what can't change is the voltage across the capacitor so that's what we want to determine is what is the voltage across the capacitor at T equals zero minus and T equals zero plus so let's let's draw that if I have a circuit that's been completely charged up like this the capacitor gets treated as an open so let's see what the voltage across the capacitor is going to be and we can just do a simple KVL equation here before I do that KVL equation let me find the current right because I'm gonna need the current to know the voltage drop across this resistor so that's going to be 6 K + 4 K plus 2 K is 12 K 12 divided by 12 K will tell me that I equals 1 milliamp okay so KVL around this loop will show you that VC equals 6 volts right 12 - six volts plus VC or 12 - 6 volts minus VC gives you 6 volts so we know that VC at tea might 0 - equals 6 volts and because we know that the energy can't change instantly we know that the moment after the flip switch is flipped that the voltage on the capacitor is also 6 volts so that brings us to the next circuit diagram in which the battery has been disconnected and our circuit now looks like this and we're trying to answer that question again right V not at at T equals zero 2k and 4k and keep in mind now that this capacitor can kind of just be treated like a battery one thing that you can get messed up on on some of these problems don't forget it how does the capacitor get charged so when you put charge on that capacitor make sure that you know how that charge is distributed because sometimes you'll find that the charge is driving a current in a way that will make your that will give you a negative sign that you may not have anticipated okay so keep your eyes peeled for that in this particular place situation we used the passivity rule to determine that the capacitor was going to accumulate a voltage using this particular sine polarity and we're gonna employ that same polarity over here and now we're gonna think of it as a battery okay and we can now just use a voltage division to solve this so V naught at zero by voltage division is going to be six volts from the capacitor times 4k over the sum of the resistances which is 6k so it's going to be two thirds of six which is four four volts so V naught equals four volts okay up here I was calling them X naught and X infinity because those are generic terms now that we're ready to put together our final equation we want to speak of this in the proper units X infinity is zero so we can ignore those terms and we get for e to the minus 41.67 t and that is in volts and it also is only true for T greater than zero okay so that is our final solution and I hope that was helpful good luck

Channel: Matthew Araujo

Views: 151,953

Rating: 4.6952996 out of 5

Keywords: electrical engineering, first order transient circuit, circuit analysis, differential equations, Electronics

Id: ZThYkQhcPEE

Length: 15min 54sec (954 seconds)

Published: Mon Mar 18 2013