Mesh current method (steps 1 to 3)
About this video
The mesh current method analyses a circuit by assigning clockwise loop currents to each window, or mesh, of the circuit. Some mesh currents are fixed directly by current sources, and the rest are found by writing Kirchhoff's voltage law around each mesh. Shared elements carry the superposition of neighbouring mesh currents, which is why their subtraction matters.
You'll be able to
- define a mesh current and draw meshes in a circuit
- use superposition to find the current through an element shared by two meshes
- write KVL equations around meshes, including terms with shared resistors
Key words
- Mesh
- An open window or loop in a planar circuit, bounded by branches with no other branch passing through it.
- Mesh current
- A hypothetical current that circulates around a single mesh.
- Element current
- The actual current through a component, found by superimposing the mesh currents that flow through it.
- Kirchhoff's voltage law (KVL)
- The sum of voltage rises and drops around any closed loop equals zero.
- Superposition
- Combining the effects of several sources by adding their individual contributions.
Transcript
- [Voiceover] Now we're gonna discuss the second of two popular ways to analyze circuits, and this one is called the mesh current method. This is actually one of my favorite. There's a fun spot here where we make up currents flowing around in circles inside the circuit, and it's interesting.
In the mesh current method, what we do is we define mesh currents, so we need to define that word. A mesh current. A mesh current is a current that flows in a mesh. And a mesh is the open windows of the circuit. Here's one of the meshes in our circuit. There's another mesh here, it's an open window, and this third one here is another one.
And this will be our mesh current number one. We'll call it i1, and we can define these other mesh currents, like this. And I'm gonna define them all to go around the same direction. They're all going clockwise, around our circuit. Now let me take a second to label the other components in our circuit here.
We'll call this V, we'll call this R1, R2, R3, R4, and this has a current source in it. We'll call that I. Makes it a little bit more interesting, and this is one of the reasons we use the mesh current method. The word mesh actually comes from like screen doors or window screens where there's wires and there's spaces between the wires, so a space in a wire network is called a mesh.
That's where the name comes from. These mesh currents are kind of interesting. In some sense they exist, and then, but it doesn't seem like they really do. How can these currents be flowing around in circles? Let's look a little closely. It's pretty clear that the current flowing through our voltage source, V, is equal to mesh current i1.
And also, when we get up here to R1, mesh current i1 is flowing through R1 in this direction here. What happens when we get to R2? We have one mesh current flowing down through R2, and another mesh current flowing up through R2. How can this make sense? Let me go over here, and we'll draw, redraw R2.
Here's R2, and now I'll redraw the mesh currents. So we have a mesh current, i1 going down like that, and we have a mesh current, i2, like that. And I'll label on here, I'll label the element current, for R2, we'll call that iR2. And what's gonna happen, this is a really wonderful application of the principle of superposition.
i1 and i2 superimposed inside R2 to create iR2. So if we wanted to know iR2, it would be the sum of these two currents. Now i1 is flowing in the same direction as iR2, so this equals i1, and i2 is going in the opposite direction, so it gets a minus sign. And if we somehow knew i1 and i2, this is how we would compute the actually current that's flowing in R2.
Next, since this is a method, I want to list out the steps of the method. The mesh current method is the first thing is to draw the meshes. And we did that over here, when we drew i1, i2, and i3. And the second step is to solve the easy ones. When we say an easy one, here's an example of an easy one.
The current source I, right here, defines what i3 is. So i3 is an example of an easy current to solve for. And then the third step in the method is to write Kirchhoff's voltage law. KVL. Write Kirchoff's voltage law equations, for the two remaining mesh currents.
And the fourth step is solve. So in this video we're gonna do the first three steps, and then in the next video, we'll go the full solve. Okay, so let's run our mesh current method. Number one, draw the meshes. Here's step one, happening right here. So we could check off, we did step one.
Step two is solve for the easy ones. And with think, i3 counts as an easy one. i3 flows down, through the current source, and so, step number two, so step two, is i3 equals negative I, because they're pointing in opposite directions. But that's it, that's the easy one.
We've done that. Now we get to the next step. So now let's start step three. We're gonna write KVL equations on each of these meshes. We'll start here, and we're gonna go around this mesh in this direction and write Kirchhoff's voltage law. So the first thing we come to is a voltage rise.
We go in the minus side and out the plus. That's a voltage rise of V volts. Next we get to R1, and R1 has a current flowing through it, a mesh current, and it will have a voltage on it like that, because the current's coming in from this side. And that will give us a voltage, we go in the plus, out the minus, so that's gonna give us a voltage drop of...
a drop of R1 times i1. Right. Now we're over to here, we go around the corner, and we go through R2. Now, R2 is odd because it has i1 and i2 flowing into it, so what are we gonna do? Well we already figured out this element current is written down right over here.
There it is right there. So if I label the current going down here, I have a plus and a minus, this is gonna be another voltage drop going around the loop. So it's gonna have a value of R2 times this value here. i1 minus i2. And now we've made it all the way back to home, to where we started from, and so that's our KVL equation, which we set equal to zero.
All right. Next step, let's do the other mesh. We're gonna go start here, and we're gonna go around this mesh, in the same way. First, the first term, we're gonna see a voltage, oh look, it's a voltage rise, so I'll have a plus here. A voltage rise, and it's the same term that we had in the previous equation.
It'll be plus R2 times i1 minus i2. That's the voltage rise going from here to here. Now we go through R3. Now R3 just has one mesh current. It just has i2 flowing through it. So we could put that there. So that counts as a voltage drop, gets a minus sign, and it's value is R3 times i2.
That's that one. Now we go over to R4. R4, this is the same thing with two currents that we had over here with R2. So we do the same thing again. This is gonna be a... we'll define this as a voltage drop, there's the element current. And it's the same form as we had here when we calculated the current for R2.
So this is gonna be a voltage drop, give it a minus sign, and it's gonna be R4 times, this time it's i2 going down. i2 minus i3. And do we get home? Now we're all the way home, so our KVL equation is all that equal to zero. So these are our two simultaneous equations that represent our current, and we wrote them using the mesh current method, so we get to check off step three.
And in the next video, we'll solve these two equations.
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