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## Electrical engineering

### Course: Electrical engineering > Unit 2

Lesson 3: DC circuit analysis- Circuit analysis overview
- Kirchhoff's current law
- Kirchhoff's voltage law
- Kirchhoff's laws
- Labeling voltages
- Application of the fundamental laws (setup)
- Application of the fundamental laws (solve)
- Application of the fundamental laws
- Node voltage method (steps 1 to 4)
- Node voltage method (step 5)
- Node voltage method
- Mesh current method (steps 1 to 3)
- Mesh current method (step 4)
- Mesh current method
- Loop current method
- Number of required equations
- Linearity
- Superposition

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# Node voltage method (steps 1 to 4)

The Node Voltage Method solves circuits with the minimum number of KCL equations. Steps 1 to 4 out of 5. Created by Willy McAllister.

## Want to join the conversation?

- Question about Step 4, when writing KCL, why does i1 = (V1 - V2)/R1 and not just V1/R1? Why is it the difference between that and V2/R2?(26 votes)
- Hello Arig,

The voltage V2 is likely a non zero voltage. And so we must account for the actual voltage drop across R1:

i1 = change_in_voltage / resistance

Recall that KCL is concerned with the currents at a particular node. "The sum of the currents must equal zero!"

i1 + I2 + I3 = 0

where:

i1 = (V2 - V1) / R1

i2 = (V2 - 0) / R2

i3 = iS

Regards,

APD(13 votes)

- I'm a little unsure as to how you could say just from visual inspection that the node voltage V1 = 15volts. If the voltage has to add up to zero would that mean V2 = 0 volts?(5 votes)
- Good question. At6:32, I know V1 = 15V because there is an ideal 15V voltage source connected directly between the reference node at the bottom, and node 1. You asked about summing the loop voltages: "the voltage has to add up to zero." . Be sure to phrase KVL as, "the
**element voltages**around a loop add up to zero." Remember, the element voltages (Vr1 and Vr2) are the voltages across each resistor element. These are the voltages you put into a KVL equation. Element voltages are not always the same as node voltages (v1 and v2). KVL around the loop would be V1 + Vr1 + Vr2 = 0. If you want, you could replace element voltage Vr2 with node voltage V2, since they mean the same thing.(10 votes)

- At8:55, I don't understand clearly why i1 equals (V1-V2)/R1.(5 votes)
- V1 and V2 are "node voltages". That means they are measured with respect to the reference node (ground). This is suggested by the two orange arrows that start at the ground node and curve up to each node.

Resistor R1 is connected between node 1 and node 2. The "element voltage" that appears across R1 is the difference of the two node voltages, V1-V2. Using Ohm's Law you compute the current through R1 as

i = Voltage across resistor / R

i = (V1 - V2) / R(9 votes)

- What is ground meant to be in our circuits, i tries looking up on the internet and it tells me ground is the point in circuit that is meant to be at 0v. However why do we need a point on the circuit that is at 0v, and isn't the negative terminal of the battery at 0v with respect to the positive, so do i take it that the negative terminal of our battery is ground??(3 votes)
- Ground is used to establish a reference voltage somewhere in the circuit. A story: suppose you are on the second floor of a building and want to answer the question, "How high is the top of your head?" One end of the ruler will be at the top of your head. Where does the other end of the ruler go? The floor you are standing on? The ground floor? The center of the Earth? You have to decide on a reference level. Your decision about where to put the other end of the ruler establishes the reference height from which you will measure all heights. The reference height = 0, by definition. That's what ground does in a circuit.(13 votes)

- Of course in Germany (or I guess in Middle Europe in general) the voltage arrows point in the other direction. Usually without a change in sign for the voltage, the current flow remains unchanged. Compare to: https://de.wikipedia.org/wiki/Netzwerkanalyse_(Elektrotechnik) The voltage arrow points from positive potential to negative potential, as the technical current flow. This way, the current flow and the voltage drop in DC circuits match (besides the voltage source kind of). Not exactly a question, but still good to know.(6 votes)
- Interesting! So the custom in your part of the world is for the voltage arrow to point in the direction of the "drop", as opposed to the custom I was taught: pointing in the direction of the voltage "rise".(4 votes)

- Isn`t the node an intersection between 3 branches? Why is 1 a node?(4 votes)
- According to the James W. Nilsson and Susan A. Riedel, a node is: a point where two or more Circuits elements join. Therefore node 1 it is a node since it is the joining of Vs and R1.

While an essential node is the one where 3 or more circuit elements join.(3 votes)

- I think understand why i1 = (V1-V2)/R1, since the i1 is connected to nodes a and b, but how did you know it was V1-V2 and not V2-V1?(2 votes)
- Good question. The reason V1 goes first in (V1-V2) is because the current is entering the resistor from the V1 side. (The current exits the resistor on the V2 side.)

This goes back to the Sign Convention for Passive Components. https://www.khanacademy.org/science/electrical-engineering/ee-circuit-analysis-topic/modal/a/ee-sign-convention, or read an improved version here: https://spinningnumbers.org/a/sign-convention.html

The sign convention says the more positive voltage on the resistor will be on the end where the current goes*in*.

If I had happened to point the i1 current arrow the other way (to the left), the expression for i1 would change to i1 = (V2-V1)/R1.(4 votes)

- What exactly is a current source? I understand a voltage source such as a battery or power supply, but am not clear on what a current source might be.(2 votes)
- Hey, I still do not get why I1 is equal to (V1-V2)/R1, by KCL we know that the current going in the first node is equal to Is, and it is possible to obtain Is by dividing Vs/R1 therefore I1 the value going into the node should be equal to Is. Hope I made my question clear, hope I hear from you soon.(2 votes)
- By KCL we know that the currents flowing
**into**node 2 = currents flowing**out**of node 2. That means i1 = IS + i2. The current flowing in R1 is labeled i1. If you want, you can say the current flowing in R1 is (IS + i2), not just IS.

Try to avoid the temptation of solving the current flows in your head before doing the analysis. Relax and use the steps of the Node Voltage Method and let the math solve the problem for you. THEN, when you know the answer, go back and look at the circuit with new eyes to "see" where the currents flow.(3 votes)

- at6:23you seid "The easy nodes are the ones that are connected directly to a source that goes to the reference node" so why v2 isn't an easy node?(1 vote)
- Oops. Good catch. I left out one word... The easy nodes are the ones connected directly to a
*voltage*source that goes to the reference node.(2 votes)

## Video transcript

- [Voiceover] We're going to talk about a really powerful way to analyze circuits called the Node Voltage Method. Before we start talking
about what this method is, we're going to talk about a
new term called a node voltage. So far, we already have the idea of an element has a voltage
across it, and we refer to that as an element voltage, or
if it's part of a circuit and it's a branch of a circuit, it'd be called a branch voltage. That's a voltage that's associated with a particular element. So now we have the idea of
something called a node voltage. This is still a voltage,
it's not anything strange, but if we go over to our circuit here, and we label the nodes. Let's start labeling the nodes,
we'll call this node here where this junction between
this resistor and this source, we'll call this node one. This is the junction between
these two components here. There's another node that's
these two resistors connected to this current source,
and that's a single distributed node, so
we'll call that node two. And down here, these three
components are connected together in a junction, and that's node three. To define a node voltage,
the first idea we need is to define a reference node,
the idea of a reference node. A good choice for the
reference node is usually one that's connected to the
terminals of the power sources, or it's the node that's
connected to a lot of branches, a lot of elements, and node three here is a good choice for a reference node. The way we mark that is with
a symbol that looks like this, a ground symbol, that's
called ground in this circuit. There's other kinds of ways
to indicate a reference node. That's a common way. You can draw one that
looks like the ground, connected to the ground. Sometimes you'll see it
with just an upside-down T, like that, that's another
way to draw a ground. This symbol on a schematic
indicates the reference node. We've picked a reference node
to be node three, down here. So, a node voltage is
measured between a node and the reference node. In our case we have this voltage here, is the node voltage on
node one, we'll call it V1. This voltage here is
going to be called V2. And in particular, these
voltages are measured with respect to node three,
so there's the minus and plus and minus and plus. We're going to use these node voltages in the Node Voltage Method. First, what I want to do, I want to label my complements here. We're going to call this
Vs, and make it 15 volts. This resistor's going to
be R1, and we'll give it a value of 4kohms. We'll call this R2, and we'll
give it a value of 2kohms. This is the same circuit
that we analyzed when we did application of the fundamental
laws in another video. Oh, and the last guy here,
Is, current source Is, and we'll make that one 3 milliamps. We've analyzed this circuit before. We used Kirchoff's Laws,
KVL and KCL, to figure out what the voltages and
currents were in this circuit. We're going to do the same analysis, but this time we're
going to use what we call the Node Voltage Method. It's basically the same application of the fundamental laws, we
use Ohm's Law, Kirchoff's Laws, but it's in a really
clever, organized way, that is really efficient. Whoever thought this up was pretty bright and I'm really glad
that they wrote it down and shared it with us. What I want to do first is just write down what are the steps of this method? It's not a theory, it's
a method, so it basically a sequence of steps that you go through to analyze the circuit. I'll write the list right here. First step is pick a reference node. We already did that. The second step is to
name the node voltages. We already did that,
we named our nodes V1, that node there is V1 and
that node there is V2, with respect to the reference node, which is down there at node three. Whenever you talk about node voltages, there's always an assumption
that one of the nodes is a reference node. The third step is to solve the easy nodes. I'll show you what that means in a second. The fourth step is to write KCL, Kirchoff's Current Law equations. The fifth step is to solve the equations. That's the Node Voltage Method, and we're going to go
through the rest of this, we've done the first two steps. What does it mean to solve the easy nodes? The easy nodes are the
ones that are connected directly to a source that
goes to the reference node. That's an example of an easy node. So V1 is an easy node. So let's solve for V1. By inspection, I can say V1 is 15 volts. That's Step Three. The other node's not easy, the other node has lots of components and something interesting's
going on over here. So this was step three. Let's label the steps. Here's the Step One. Here's Step Two. And here's Step Three. Now we're ready to go to Step Four, let me move up a little bit. Step Four is write the
Kirchoff's Current Law equations directly from the circuit. We're going to do this in a special way, We're going to perform at
this node here, at node two. We're going to write the
current law for this. That means we got to
identify the currents. There's a current, we'll
call that a current, and that's a current. Let me give some names to these
currents just to be clear. We'll call this one I1
because it goes through R1. We'll call this one here, I2
because it goes through R2. This one is already Is. Now let's write Kirchoff's
Current Law just in terms of I, and we'll say all the
currents flowing into the node add up to zero, so these
two have arrows going out, so they're going to get negative signs when we write Kirchoff's Current Law. Let's do that right here. And we write I1 minus I2 minus Is equals to zero. So right now we're working on Step Four. This is the essence of
the Node Voltage Method. This is where we do something new that we haven't done before. We're going to write these currents in terms of the node voltages. So we can write I1, I1 is
current flowing this way through this current. I1 equals V1 minus V2 over R1. That's the current flowing in resistor R1, in terms of node voltages. The current flowing down through I2, now we have to subtract I2,
so we just apply Ohm's Law directly, which means
that the current in I2 is equal to V2 divided by R2. The last current is Is, minus Is. We'll write that in
terms of Is, like that, and that equals zero. This means we have now
completed Step Four. That is KCL written using the
terminology of node voltages. We could check off that
we've done Step Four.