Mesh current steps 1 to 3

Now we're going to 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 favorites. There's a fun spot in here where we make up currents flowing around in circles inside the circuit, and it's interesting.

So, 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 a 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. This will be our mesh current number one; we'll call it i1. We can define these other mesh currents like this, and I'm going to define them all to go around in 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. That 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, but it doesn't seem like they really do. How could 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. 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 could this make sense? Let me go over here and we'll redraw r2. Here's r2, and now I'll redraw the mesh current. So here we have a mesh current i1 going down like that, and we have a mesh current i2 like that. I'll label the element current for r2; we'll call that ir2.

Now, what's going to happen? This is a really wonderful application of the principle of superposition. i1 and i2 superimpose 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. If we somehow knew i1 and i2, this is how we would compute the actual current that's flowing in r2.

So 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 do i1, i2, and i3. 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.

Then the third step in the method is to write Kirchhoff's Voltage Law (KVL) equations for the two remaining mesh currents. The fourth step is to solve. So in this video, we're going to do the first three steps, and then in the next video, we'll do 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 can check off we did step one. Step two is to solve for the easy ones, and we think i3 counts as an easy one. i3 flows down through the current source, so step number two: 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 going to write KVL equations on each of these meshes. We'll start here, and we're going to 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 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. That will give us a voltage. We go in the plus, out the minus, so that's going to give us a voltage drop of r1 times i1.

Right now, we are 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 going to do? Well, we already figured out that 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 going to be another voltage drop going around the loop, so it's going to have a value of r2 times this value here (i1 minus i2). And now, we've made it all the way back home, to where we started from. So that's our KVL equation, which we set equal to 0.

All right, next step. Let's do the other mesh. We're going to start here, and we're going to go around this mesh in the same way. First, the first term, we're going to see a voltage—oh look, it's a voltage rise. So we'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; it gets a minus sign and its 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 going to be a— we'll define this as a voltage drop; there's the element current. Well, and it's the same form as we had here when we calculated the current for r2.

So this is going to be a voltage drop that gets a minus sign, and it's going to be r4 times this time it's i2 going down (i2 minus i3). Did 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 circuit, 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.