Lesson 1 - Intro To Node Voltage Method (Engineering Circuits)

hello and welcome to the section of the circuit analysis tutor in this section we're going to begin a journey of talking about new and different kinds of circuit analysis techniques the first one that we're going to talk about is called a node voltage method of circuit analysis and it's something that all students that are studying circuits are going to have to master and it really occupies a good chunk of your course it's one of the core fundamental circuit analysis methods so I want to take a few minutes to put it into context with what we've already learned in the previous lessons we've learned a lot about the basics of circuits voltage current resistance Ohm's law Kirchhoff's laws and we learned that we can analyze pretty much any resistive circuit with Kirchhoff's laws and I'm going to tell you right now kirchoff's laws are always valid they're always true they're always something you can go to and use if you need to and we did lots and lots of problems writing down the equations with Kirchhoff's laws and solving them and finding the currents and the voltages everywhere in the circuit which is what we're trying to do when we solve circuits so kerkhof laws are bulletproof they're always there you're not really trying to replace Kirchhoff's laws here it's just that if you remember back as the circuits get a little more complicated and what I mean by complicated is just more branches and more resistors and things like that then it quickly becomes that you're going to have a lot of kerkhof equations remember we're writing those node equations with the currents coming in and out the kerkhof current law and then sometimes we have to write the kirk off voltage laws and it can quickly even for relatively simple circuits it can quickly lead to four or five equations to solve that you have to then go and get a determinate to solve the system if you're going to use a matrix method or if you're going to do other methods of solution it just be kids it gets to be a little bit cumbersome to work with five simultaneous equations to solve a fairly simple circuit the other problem won't so-to-speak problem with Kirchhoff's laws certain solving circuits with Kirchhoff's laws is the fact that there wasn't really sort of like a rigid methodology to it I taught you what the kerkhof current law was I taught you what the kerkhof voltage law was and I taught you how to write those equations but I told you had a lot latitude like maybe you want to write to Kirk off current laws in one voltage law if you need three equations to solve everything well you could also solve everything by making maybe making one Kirk off current law into voltage laws so the exact equations that you write it really is kind of like flexibility up to you solving the problem there's no rigidly defined way to do it which is kind of nice to have freedom but also kind of it kind of leaves a little bit ambiguous exactly how to proceed with the circuit so here we're going to learn about the node voltage method and later on in the course we're going to learn about the mesh current method both of these methods of circuit analysis are just tools in your tool bag that you can pull out and use them when it's necessary needed sometimes it's going to be simple to solve a circuit and easier to use just Kirchhoff's laws so you do that right sometimes going to be easier depending on how the circuits laid out to use the node voltage method and sometimes we'll learn later that the mesh current method is going to be the easiest way to go so it's just like anytime in math you have different tools different techniques so we're learning all of the methods here now the node voltage method mainly does two things for you that really help you the first one and the most important one is when you use the node voltage method almost always you're going to hit need less equations to write down to describe that circuit then you did if you just use Kirchhoff's laws that's really the main reason we use it if you look at two circuits and solve one of them using just kerkoff's laws and then you solve the same circuit again using the node voltage method but the node voltage method is going to require fewer equations fewer simultaneous equations that means if you're using matrix methods smaller matrices smaller determinants faster to write it down on your test everything's going to be faster so node voltage methods vastly superior in that regard the second thing it does for you is it's a little bit more rigidly defined in other words there's an ABC to exactly what equations you write down and that's just the way you have to write them down so really when you compare contrast just using Kirchhoff's laws versus using node voltage method in the Kirchhoff's laws you get lots of flexibility you can write whatever equations you want whatever node you want whatever you know circle circuits you want as far as the KVL you can write whatever you want long as you have enough equations but you're going to require a lot of equations for node voltage it's much more rigidly defined I'm going to show you how to write the equations and you have to write them this way and once you write in this way those are the equations for the node voltage method however even though it's a little more rigid it's bulletproof and it requires fewer equations and that's really the main advantage for it okay so that's basically what the node voltage is requires fewer equations now before we go any further I need to give you a few definition so instead of just tell you a few definitions let me just draw a very very simple circuit to illustrate what I'm talking about when I say definitions here so here's a very very simple circuit let me put two resistors here two resistors here we'll just connect it here and then over here we'll just stick and we're not going to solve this by the way I'm just using this for definition so what we're going to do is come down here so a very simple equation because our very simple circuit because when I when you read a book they're going to use some terms and you really need to know what they're talking about the first thing is and we've talked about this before some but we need to talk about it again what is a node because this is the node voltage method so you really have to understand what the book is talking about when you're looking at your textbook or what your professor is talking about a node quite simply is any interconnection of two components two or more components that's a node so let me give you an example this guy is a node because we're connecting two resistors together this is the wire here so when you build it yeah it's just a wire in between or maybe you just take these two resistors and hook them together but it doesn't matter exactly how you do it but the fact is you have two components connected together so that's a note this guy right here is also called a node so if you're on the test and your professor says name you know know the nodes you know the nodes here in the drawing then you would point definitely to those two things and call them a note now notice this is also a node right here because here I have a resistor a resistor and a resistor three things that are joined here but because it's three or more we have a special name we call it in central node so anytime you have three or more objects we call it an essential node down here we also have an essential live so this is called an essential node okay it's just a definition folks basically a node is anytime two things are connected but you know what nodes are kind of trivial these two resistors it's just two things connected it's it's not really considered or know that you would really worry yourself with when you're solving a circuit because the current going through both of these guys is really going to be the same because it's going through both of them so yes it's a node technically but for the node voltage method you're not going to concern yourself with these little piddly nodes that have two things connected together so here's the punch line for the node voltage method when we talk about node voltages which we're going to talk about in just a second the nodes we're talking about and that we care about aren't only these essential nodes so basically when you look at a circuit you see three or more interconnections that's an essential node those are the nodes that we care about when we write to the node voltage equations that's all we care about so you don't really look at this and you don't really look at this you only look at these two guys being the essential nodes okay so I could write all these things down but honestly I'm going to just tell you quickly some key things about the node voltage method and you're going to understand it and get practice from actually solving problems so instead of writing a bunch of you know definitions and things down 1 2 3 4 5 I'm just going to tell you some very important things here so make sure you understand the following so the node voltage method basically consists of four steps and you're going to get experience with these steps as we solve problems but here they are first one identify the essential nodes all right makes sense it's a node voltage method I just told you all we care about is essential nodes so you need to look at your circuit and identify which ones are the essential nodes and if it helps you put a big fat dot only on top of the ones that are the essential nodes that's very helpful so do that for yourself if you need to then you need to choose one of these essential nodes as what we call a reference node and so it's a little difficult to explain why we have a reference node until we solve a real problem but basically we have to to choose from this one up here in this one that's all we care about for a node voltage method we have to pick one of them to be a reference node right and so it's it's a little difficult again ahead of time to tell you what what this is all about until we actually solve a problem but for the sake of argument here most of the time you're going to choose the essential node with as many connections as you can find on the bottom of your drawing most of the time and it's just a reference remember when we this is an aside remember when we talked a little bit about voltage in general right we said we always have you know when you have a meter and you measure voltage across an object you have that you have a black lead on your on your volt meter and you have a red lead and that voltage that you're measuring with that meter is in reference to whatever you're touching with the black lead that's the reference when you measure the voltage drop across something it has to be relative to something you're measuring the voltage drop across that object from the black lead of your measurement device like your o meter or your volt meter to the red lead so your reference in that case is the black lead you're seeing you're basically seeing how many volts higher than the black lead is the red lead when you measure something in a circuit so for our reference we just need to pick something in order to do these node voltage equations and we need to pick an essential node so you're going to pick almost always the one on the bottom of the drawing that has the most number of interconnections here we have three interconnections so that's going to be the obvious reference node but as we do more problems you're going to get the hang of which one to pick so don't stress out about that too much right now okay the next thing we're going to do first thing we identify the central node second thing we pick a reference node third thing we write node voltage equations okay these node voltage equations are going to be in reference to the reference node we picked so I haven't told you how to do it yet but we're going to write some equations and they're going to be in reference to the reference node that we pick which you just talked about and we'll show you how to do that here in just a second and finally for n number of essential nodes you need to completely describe the circuit in - one node voltage equations so what that means is if you have two essential nodes like we have here you only need one node voltage equation to completely describe that circuit to completely solve that circuit if you have a more complicated drawing with four essential nodes then you only need three node voltage equations to solve it if you have ten essential nodes you only need nine node voltage equations to solve it and if you go back and compare some of these circuits we're going to work in a second to what we've done in the past you'll find that it's it's vastly reduces the number of equations compared to the two the Kirkhof laws so solving the circuit we're kerkhof laws as we did before so if you me if you have you know five essential nodes in a drawing you need four node voltage equations if you solve the same equations or solve the same circuit with the kirchoff's laws you would have actually had to use more equations to describe the same thing alright node voltage method is much like everything else you can talk about it you can play games with it trying to understand it but until you actually do something it's going to be very difficult to really wrap your brain about around so what we're going to do is solve a very simple circuit using the node voltage method a circuit that you already know how to solve with the Kirkhof laws that we talked about before and the ideas of Ohm's law and things like that so it's going to be a simple circuit but it's just to illustrate the point and as we climb our way out we'll see that we have more complicated circuits that we can tackle so for this particular one what I want to do is draw the circuit like this so what we have this is 10 volts ok then we have a resistor right here a resistor right here a resistor there a resistor over here and then a current source like this let me make sure I label everything this is 1 ohm this is 5 ohms this is to arms this is 10 arms and this is 3 amps so the question is solve this circuit a lot of times you know and the tests you might they might tell you you know find the current in this leg we're going to do some problems in a minute where you're asked to find cific current and this leg we'll get to that later for now if you're just given an open-ended problem solve the circuit then for the node voltage method what that means is you have to find the node voltages the voltages at all of the essential nodes is what you're really trying to find and once you do that you know basically the end game is you write these equations you find these node voltages once you find them anything else in the circuit can be found there like the key that unlocks everything let me show you what I'm talking about what we need to do first according to what we says we need to identify the essential nodes so notice this is a node because this is connected to the resistor but it's not essential because it's only two things but this is an essential node so let's help ourselves out put a dot here because it have three items interconnected that's an essential node this is also an essential mood because I have three items interconnected now look at the bottom of the circuit everything is connected together along the bottom all the branches here so this whole entire if you could just kind of draw a dotted line the entire bottom of the circuit is one giant node that's one thing you really need to make sure you you wrap your brain around a lot of times you look at the bottom here and it looks like different nodes because this is a node and this is a node but really when you think about it if you could if you wanted to build the circuit you would literally be joining all of these things at one common point just because it's drawn spread out here it's just the way it's drawn everything's come in to one common point because this is one perfect wire here so really this is one giant note it's not two different nodes it's one giant essential node so we'll just put a dot here just to even though we have to recognize that it encompasses everything we're just going to put the dot here so when we look at this thing the first thing we're told to do is identify your essential nodes these are the essential nodes there's only three essential nodes so right away you know that with the node voltage method you're only going to require how many equations two equations to solve the circuit because we have three essential nodes that means two equations second thing we're going to need to do is pick a reference node I told you it's almost always going to be at the bottom of the drawing and it's you want to try to pick a reference node that has the most number of interconnections so here we have one item two items three items four items interconnected at the point that is just going to make sense for the essential node reference point so we put a little triangle here that represents commonality or reference so whenever you see something like this drawn anywhere it means this is your reference that you're measuring everything relative to in this case we're going to find these node voltages these two node voltages up here with respect to our reference that's very important that you understand this is our reference and what we have figured out is that we only require two node voltage equations and they're going to represent this voltage this is called a node voltage because it's the essential node here and here's another node voltage what we're going to try to do when we solve this guy is find those reference voltages find those node voltages that's what I'm trying to say these two node voltages at the top we'll call them v1 and v2 whatever doesn't matter once you find those node voltages though everything else in the circuit can be found the current can be found here the current can be found here the current can be found here the current can be found everything is described believe it or not now if you go back and think about it for a second if you were to solve this problem with kerkhof current law and Kirchhoff's voltage law you definitely going to need more than two equations so just by looking at it and seeing okay I only need two equations that's a huge help right there so we've identified our essential nodes we've picked one of them to be a reference what we're trying to solve for is what are these voltages here those are called our node voltages that we care about so just to make it a little bit easier to understand I'm going to label this essential node essential node one and I'm going to label this node essential node two you don't really have to do this on your paper but I'm doing it to make it a little easier to reference what I'm talking about we need to find the voltage at node one and voltage at node 2 and because of this I'm going to label this voltage the voltage at this node I'm going to label it V sub 1 because this is node 1 and I'm going to label this one here V sub 2 now I want you to study this before we get too much farther what we've done is we've said okay this is a node voltage I want to find out what that is so I'm going to label it myself call it V sub one and it's going to be the voltage of this node with respect to the reference node we've chosen that's why the minus signs are down here at the bottom very important we choose a reference node and then we define all the node voltages with respect to that reference node so that's why V sub one is measured with respect to this guy down here so if I were taking my vote my voltmeter like I was talking about you put the black lead here and the red lead here that means you're measuring the voltage with respect to the black lead this is the reference that I was talking about V sub 2 is also measured notice I have another negative sign it's also measured with respect to the reference node we've chosen of course this reference node is all connected along the bottom so these sub two is the same as measuring V sub two here because they're all connected together but just for the sake of clarity I'm measuring V 2 with respect to the reference I'm measuring V 1 with respect to the reference so I know I've spent a lot of time dealing with making the drawing look beautiful but I want to do that because it's important for you to do that if you skip these steps and you say okay I'm gonna do node voltage method you get down there and start writing equations you're gonna you're going to screw it up okay because you have to have everything labeled so that you understand what to do next we have a reference node we have the both node voltages on the drawing the next thing we want to do finally is write the node voltage equations so we need two node voltage equation we need two node voltage equations okay so the first one okay I'm going to write it like this at essential node one to write an ode voltage equation let's say this one here to write an ode voltage equation we're basically going to kind of use Kirchhoff's current law that's basically what we're going to do we're going to try to write an equation that expresses the current coming out of this node and away from node one so yes I'm fully aware that this is a current that this is producing current and it's most likely shooting current down here I'm aware of that over here I'm aware that you have current probably coming into this node but for the purpose this is so important for the purpose of the node voltage method you don't care what you think you know about the circuit when you write your node voltage equations you just pretend that all of the current is coming away from this node and you write all of the equations in the same way so when I write the node voltage equation here I'm going to pretend there's coming out of theirs currents coming out of this node out of this node out of this node I'm going to write a node voltage equation they're summing up all of those currents and setting them equal to zero then I'm going to do the same thing over here and even though it's not totally true that all the currents coming out of the node doesn't matter the sign of the answers is going to take care of everything so when I'm here I know that there has to be some current here I'm going to pretend it's coming away from this node how do I write that current well remember remember do a little side here V is equal to I R so if you're solve for the current anywhere it's going to be V over R I is equal to V over R so when I'm looking for the current coming away from this node this is how I'm going to write it I'm going to say v1 minus this voltage 10 over the 1 ohm I have right here this term right here is expressing the current coming away from this node and you might look at that and say what is he talking about what you need to do is figure out what is the voltage drop across this resistor because if you can find the voltage drop across a resistor you know the current flowing in that resistor and then you know that's what's coming out of that node so the way you figure out what this current is is you say okay there's a voltage v1 at this node this is ten volt here so the the difference between these two voltages must be the voltage drop across this resistor and you always again pretend even if it's not true you pretend that the current is coming out away from your node so that's why we always write it v1 minus 10 over one because you're basically pretending for just a moment that v1 is greater than 10 so you have the higher voltage here - this guy that's going to give you a voltage drop this way for the current to come out it's so important that you understand that you'll get it as we do more and more equations more and more node voltage equations but it's so incredibly important that you understand that let me keep going and then what you'll get the hang of it okay plus what is the current coming down this path here well that's a little bit easier because it's just the voltage across this resistor is given by v1 that we've labeled so v1 over v I is equal to V over R so that's the current in this leg again going away from the node plus what's the current going through this resistor again we have to pretend the currents going this way so it's going to be v1 this voltage at this node minus v2 because we walk around this is the voltage with respect to the reference v2 / - that's all the currents and all the legs that must equal to zero what we've basically done here is use Kirchhoff's current law right kerkoff's Clairol is always valid and what we've done is we've sort of used Kirchhoff's current law in combination with this node voltage concept to write a simple law doesn't look so simple but you'll see in a minute relatively simple equation and what we've basically done is we've said okay here's a node kerkoff's current law says all the current summed up at a node must be 0 that's what it has to be everything sum together has to be equal to 0 that's what it has to be everything here is just a term for each branch here the middle one is the easiest to understand because it's just AI is equal to V over R that's the current flowing away from this node so this one should be easy for you to understand the other two is very important that you understand how I've written in here I've done v 1-10 because I'm pretending the currents flowing away so the voltage here v1 I'm sort of pretending even if because I don't know what v1 is but I'm pretending that it's it's the bigger voltage so v1 minus this 10 volt is going to give me the voltage across this resistor going this way so basically go up to your node and back around divided by the resistance so the difference in the voltages is going to give me this voltage across the resistor divided by 1 I is equal to V over R over here again you start from your reference you go up to your node v1 minus v2 that's going to give me the voltage drop across this resistor in this direction V over R where R is 2 ohms so I hope you understand how I've written the equation if you get lost the only thing you have to remember is if you're working on this node just start from your reference go up to your node that's the v1 minus any other voltage drops 10 volts that gives me this voltage drop over R V over R V over R so you pop up to your node that you're working on and you go around here this is another voltage here the reason we use the other node voltage here is because we get around here this since we've defined it to be the voltage drop from this node down to our reference then this is the only other voltage that we care about so notice that we have an equation in v1 and v2 two variables all right so let's write another equation at the other node we'll again we'll find that we have another equation in v1 and v2 two equations two unknowns all right so let's go and do that right now so let's say at node two let's go and write the equation at node two that's this guy right here so the first thing we need to do is write the current flowing out of this node through this branch here so what we're going to have is we go up to our node v2 minus v1 v2 minus v1 over the resistance here is two ohms so I is equal to V over R we just have to subtract these node voltages to give us the voltage drop across our resistor in this two vitu is first because we're basically pretending v2 is now bigger because we're dealing with this note and so everything is in reference to know you're dealing with v2 comes first because it you pop up to v2 you subtract anyone that's going to imply the currents going this way over to plus what's the current in this leg here what does it look like here here we have v2 but again this is all connected so this v2 is really v2 across this resistor so it's going to be v2 over 10 plus and then finally we look in the other leg here we don't even have to do any node voltage business because we know what the current is here but again when we write our node voltage equations we're always writing it with respect to current flowing away from our essential nodes and this current is going into the node so instead of a plus we actually write it as minus 3 and that kind of stuffs important because you might gloss over a sign but if you get the sign wrong you totally don't get the problem right at all so again before we solve this notice we have two equation V 1 and V 2 are the two unknowns and I haven't shown you how yet but once we solve this in find V 1 and V 2 we have found the node voltages once you have those you can find anything else you want in the circuit the circuit is what we called solve alright so again I want to go over this really quickly for this current here V 2 minus V 1 gives us the drop across the 2 ohm resistor that voltage over the current over the resistance is the current there the current here is simply V 2 over 10 I is equal to V over R and the current here is the current flowing in so we make it negative like this all right so let's draw a line here and let's let's make a little bit of simplification so what we're going to do now is work on you can do this anywhere you want here it gets to be a little bit of technique I'm doing it my way but this is just V 1 minus 10 over 1 so really it's V 1 minus 10 it's all divided by 1 so that just stays the same here we have 1 over v we know that when you divide that that's 0.2 so I'll call it 0.2 V 1 1/2 is 0.5 so what I'm going to say is plus 0.5 v1 minus 0.5 D 2 equals 0 this is just 0.5 here and here and I've sort of split it up with v1 minus v2 okay and then finally just simplify this further we have v1 here we have 0.2 v1 here and we have 0.5 v1 here so we add this together we get 1.7 V sub 1 and then for V sub 2 we have negative 0.5 negative 0.5 v sub 2 and then we have minus 10 don't forget that is equal to 0 we'll just leave it like that for now so this is sort of an important simplification and then the next step let's go ahead and change colors to green here let's take this equation and try to simplify it here so this one we have 1/2 so that's point five zero point five V sub two we'll use parentheses to make it clear and then we have minus 0.5 V sub 1 like that this guy 1 over 10 is point one so it's zero point one V sub 2 keep the parentheses for now minus 3 is equal to zero all right and then finally let's change over to black for V sub one all we have is this so we're going to have negative 0.5 v sub 1 V sub 2 point 5 Plus that is point 6 V sub 2 minus 3 is equal to 0 ok so let's go ahead and move over to the other board and let's write down finally our system of equations so here one point seven V 1 minus 0.5 V 2 we have one point seven the 1 minus 0.5 v 2 and then we have minus 10 is equal to zero so what I'm going to do now is I'm going to write it equals positive 10 I just move the 10 over to the other side all right making it look like more of a real equation down here negative 0.5 V 1 negative 0.5 v 1 plus 0.6 V 2 plus 0.6 V 2 is equal to we have a negative 3 here we move it over make it a positive 3 all right now this is two equations and two unknowns we need to solve these guys are really the sort of like the punch line and at this point you're free to do what you want a lot of calculators you can just dump equations system of equations in hit the solve button you get V 1 and V 2 no problem I'm not sure what calculator you're using or even if you know how to do that in your particular calculator so in this course I'm basically going to solve all of these guys using matrix methods which really is just about as fast so what we're going to have here if you remember is the general the general deal for a matrix equation we've done this in the last set of lessons also matrix a times X is equal to B X is a placeholder it just means what X doesn't necessarily mean X because we don't have any X's here it just means whatever I'm hunting for here I'm hunting for v1 and v2 so the way you write this matrix equation is you need a matrix a which is your coefficients on the left so we have one point seven here we have negative 0.5 here we have negative 0.5 here we have 0.6 that closes matrix a which is the coefficients on the left you multiply matrix a by what you're hunting for v1 v2 you write it as a vertical column matrix like this and that equals the right hand side 10 and 3 if this is confusing to you go back to either the previous lessons in this sequence and the circuit analysis I even do a little more background into what a matrix solution is also I have an entire set of matrix matrix algebra tutor that teaches you what a matrix is what a determinant is all the stuff that you're really going to be talking about here I kind of assume that you already know what that is this is the matrix representation of these two equations if you think about it when you multiply this times this in the matrix fashion that's what you get up here and it's equal to ten if you multiply this times this in the matrix fashion that's what you get it's equal to three so this is totally represents what we have here above the reason we do that is because when we have a matrix equation like this then what would basically simply if we have ax is equal to B then X which is this is what we're calling X we're trying to solve for is equal to the inverse of matrix a times matrix B basically this is a matrix equation on the left hand side you multiply by the inverse of matrix a that totally eliminates this on the right hand side you have to multiply by the inverse of matrix a so on the left your left your end up with what you're solving for so the bottom line is V 1 V 2 is equal to the inverse of matrix a times matrix B the inverse of this matrix you just type this in a calculator or computer hit the inverse button every calculator has that nowadays for engineering anyway and then the universe of this matrix is going to be zero point seven seven nine zero point six four nine zero point six four nine two point two one will close that off this is just the inverse of matrix a and then on the right hand side we have ten and three and so finally at the end of the day Z 1 C 2 when you take this matrix multiplied by this matrix and you do the multiplication in the matrix fashion but what you're going to get is nine point seven four and thirteen point one two so you just read it off just like a book basically so what you have is the one is equal to nine point seven four volts V two is equal to thirteen point one two volts this is the answer these are the node voltages that we cared about everything past this point whenever we have the equations written down everything past that is just algebra it's all just how do you solve a system of equation this is just two equations you could have done it substitution heck you could have done it by graphing if you want to to find the intersection points or whatever the to point the points that are common there you could have done that here we use matrix methods I highly recommend that you get comfortable with matrix methods because circuit analysis is all about solving linear equations like this so if this seems foreign to you please go back to either Volume one of the circuit analysis tutor or even go back to my matrix algebra tutor where I did break it down and show you what a matrix even is but basically this is your matrix equation it represents this you're trying to solve for the voltages so to do that you eliminate this by multiplying by the inverse of this matrix on the left that wipes it out you have to do the same thing to both sides so the inverse of this matrix is also multiplied by the right hand side this is the inverse of the matrix over here this a negative one represents inverse this matrix times the right-hand side gives you what you're solving for on the left so it's just like any equation right 3x is equal to 4 you know you divide by 3 you have to do it to both sides here I'm trying to get rid of this stuff to solve for what I have so I am multiplied by the inverse on the left that wipes it out I have to do the same thing on the right once you get that point you get the voltages the node voltages that we care about so if your professor says find the node voltages that's it that's the answer but I also told you we're done with this problem okay but what I did what I told you is that you once you have the node voltages you can find anything else you want in this circuit let's go take a look at that we now know what v1 is we now know what v2 is fact let's write it down v1 is 9 point 4 volts we know that now we know v2 is 13 point one 2 volts right so let's look at this for a second now that we know what v1 is if we wanted to find the current in this leg all it would be with I is equal to V over R V is 9 point 4 R is 5 you divide them that gives you the current here if you wanted to find the current in this leg remember this is all connected so v2 is really over here it would just be I is equal to V over R 13 point 1 2 volt divided by 10 that gives you the current there so that's that then you might say well what about current over here well that's easy because now we know this is 9 point 4 volts okay so we take 9 point 4 volts minus 10 that's going to give us the voltage drop across here that's kind of that voltage V over R is going to give us a current and that's going to give us the current here same thing over here if we wanted that current we would take the v1 minus v2 that's going to give us a voltage divider it's going to be voltage across this resistor divided by 2 is going to give us the current there so once we have the node voltages everything else can really be found you can find the currents in all the branches and once you know the currents and all the branches you can find the voltages in all the branches now notice here we just stopped and found the node voltages I want to make it clear that if you're trying to find currents and legs if you if you get negative values for the current in a leg it just means that the current is going the opposite way then you you originally drew it so for instance if we were trying to find the current here in this guy let's say that was what the problem was then if you did it by if you took v1 minus v2 then we have a 9 point 4 volt here right minus 10 it's going to give you a negative number and then you divide by r you're going to get a negative current so what that means is see when you take v1 minus v2 that implies that the current is going this way because you're kind of coming up here is the bigger voltage subtracting this one off and then you're going to get a negative voltage that means that the current really doesn't go this way at all it actually goes this way that's just because the way the node voltage is landed so you don't have to concern yourself so much with that when you write your node voltage equations you need to concern yourself with finding the node voltages so but if you get in the end of the get in at the end of the day when you're trying to find current somewhere if you do a subtraction you get a negative number it just means the currents going opposite of the way you kind of assumed when you when you did that calculation so again to recap you identify your essential nodes here we have three of them we know you're going to need two equations as a result you identify your reference node and then you write equations of the other nodes using these voltages that we've labeled these node voltages here you're basically summing up currents going away from your other essential notes and we've shown how to do that here we've shown that how to do that here once you do that the rest of it is algebra beating those equations in the shape putting them in a matrix or if you have a calculator that can do it you can use a calculator finding V 1 and V 2 a lot of problems are going to be over at that point if they say find the node voltage here you're done right but sometimes you'll be asked oh what's the current over here what's the voltage over here once you know the node voltages the entire circuit can be solved and that's basically the node voltage method in a nutshell I've intentionally chosen a very simple problem to start with one that you can solve with other methods but even I think you can see that is sort of a pain as it is to learn a new technique it's still only two equations and that does save you a lot of algebra if you were to do this with Kirchhoff's laws it would take more equations which would take more effort on your part so take your time to watch this lesson as many times as you need to understand what we're talking about to understand the sign conventions to understand how to write the node voltage equations then follow me on to the next section and watch the next solution of the next problem in the next problem we'll do quite a few of these things because it's so very important and the other bit of motivation I will tell you so far all of the circuits we're doing are with DC right with constant voltage sources and they all have just resistors but later on when we get into AC right with alternating current and when we get into capacitors and we get into inductors and you have a lot of things changing because the voltage is changing and going back and forth AC right the node voltage method is still going to be used there's some differences that we'll talk about later but the basic idea you're going to use over and over and over and over so make sure you understand it here because you'll be using it from here on out

Channel: Math and Science

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Keywords: node voltage, node voltage analysis, node analysis, circuit analysis, node voltage method, circuit theory, electrical engineering, nodal analysis, node voltage problems, node, lesson, electronics, node method circuit analysis, nodal voltage method, nodal voltage analysis, node voltage method example, node voltage method dependent source, node voltage example, node voltage calculator, node voltage method steps, node voltage equations, node voltage practice problems

Id: -wCGiSNk5tw

Length: 41min 15sec (2475 seconds)

Published: Thu Feb 04 2016