EGR 2201 Unit 4 Mesh Analysis Read Alexander & Sadiku, Sections 3.4 to 3.10. Homework #4 and Lab #4 due next week. Quiz next week. Handouts: Quiz 3, Unit 4 practice sheet.

Mesh Analysis We’ve seen that nodal analysis is a systematic method for analyzing circuits. It’s based on Kirchhoff’s current law (KCL). It gives us the node voltages in a circuit. Once we have these node voltages, we can find any other voltage or current. Mesh analysis is another systematic method for analyzing circuits. It’s based on Kirchhoff’s voltage law (KVL). It gives us the mesh currents in a circuit. Once we have these mesh currents, we can find any other current or voltage.

Meshes Versus Loops Recall that a loop is any closed path in a circuit. Example: This circuit has six loops. A mesh is a loop that does not contain any other loop within it. Example: The circuit above has three meshes. Do practice question 1.

Mesh Currents Versus Branch Currents A mesh current is a current that we imagine to travel around a mesh. You can imagine them to travel in either direction, but most people assume clockwise. A branch current is a current that passes through a branch (i.e., an element). If we know the values of all the mesh currents in a circuit, we can compute any branch current. In many diagrams, our textbook uses: Lowercase i and a looping arrow for mesh currents Uppercase I and a straight arrow for branch currents

Example: Mesh Currents Versus Branch Currents In this circuit, the mesh currents are labeled i1 and i2. The branch currents are labeled I1, I2, and I3. Suppose you were given the values of the mesh currents. Then you could easily compute the branch currents, since: I1 = i1 and I2 = i2 and I3 = i1  i2 Do practice question 2.

Steps in Performing Mesh Analysis on a Circuit with No Current Sources Given a circuit with n meshes, without current sources, follow these steps: Assign mesh currents i1, i2, …, in to the n meshes. Apply KVL to each of the n meshes. Use Ohm’s law to express the voltages in terms of mesh currents. Then simplify the equations. Solve the resulting n simultaneous equations to obtain the unknown mesh currents.

Example: Step 1 (Assign the Mesh Currents) Consider this circuit from the book’s Example 3.5. Step 1 has already been performed for us, since the mesh currents are labeled i1 and i2. If an assumed current direction is wrong, that’s no problem. The math will still work out.

Example: Step 2 (Apply KVL) Part 1 of 2 Apply KVL (and Ohm’s law) to mesh 1: 15 + 10i2 = 5i1 + 10i1 + 10 Apply KVL (and Ohm’s law) to mesh 2: 10 + 10i1 = 10i2 + 6i2 + 4i2 Do practice sheet questions 3 a,b.

Example: Step 2 (Apply KVL) Part 2 of 2 Next we use algebra to simplify our equations. For mesh 1: 15 + 10i2 = 5i1 + 10i1 + 10 becomes 15 𝑖 1 − 10𝑖 2 =5 For mesh 2: 10 + 10i1 = 10i2 + 6i2 + 4i2 becomes 10 𝑖 1 −20 𝑖 2 =−10 We now have our two equations in two variables. Do practice sheet question 3c.

Example: Step 3 (Solve) 15 𝑖 1 − 10𝑖 2 =5 10 𝑖 1 −20 𝑖 2 =−10 Next we use any of our methods—substitution, Cramer’s rule, matrix inversion, MATLAB—to solve our two equations in two variables. Using MATLAB, the solution to 15 𝑖 1 − 10𝑖 2 =5 10 𝑖 1 −20 𝑖 2 =−10 is i1 = 1 A and i2 = 1 A Do practice sheet question 3d.

Example: Extending the Analysis Mesh analysis has given us the values of the mesh currents i1 and i2. We can find all other currents and voltages in the circuit once we know these mesh currents. Example: Knowing that i1 = 1 A and i2 = 1 A, how would we find I3? Do practice sheet question 3e.

Review: Steps in Performing Mesh Analysis on a Circuit with No Current Sources Given a circuit with n meshes, without current sources, follow these steps: Assign mesh currents i1, i2, …, in to the n meshes. Apply KVL to each of the n meshes. Use Ohm’s law to express the voltages in terms of mesh currents. Then simplify the equations. Solve the resulting n simultaneous equations to obtain the unknown mesh currents. Have them build circuit from Practice question 3 in Multisim and measure I1.

MATLAB’s format command MATLAB’s format command lets you control the way MATLAB displays answers (for example, whether to use engineering notation in answers). For a list of options, type help format in MATLAB or see this web page. By default, the format is set to short. Usually this works well, but sometimes format shortg or format shorteng works better. See example on next slide.

Example: MATLAB’s format command This is telling you to multiply each number displayed below by 0.001. Ugly! Nicer! Much nicer!

What About Circuits with Current Sources? As described above, our mesh-analysis procedure applies only to circuits without current sources. But it’s not hard to extend the procedure to circuits with current sources. The way you handle a current source depends on whether the source is located in only one mesh or is shared by two meshes….

Case 1: A Current Source Located in Only One Mesh A current source located in only one mesh is easy to handle, because it immediately reveals the mesh current in that mesh. Example: In the circuit shown, we can immediately see that i2 = 5 A. Do practice sheet question 4.

Case 2: A Current Source Shared by Two Meshes A current source shared by two meshes is trickier. To handle it, we create a supermesh by excluding the current source and any elements in series with it.

How to Handle a Supermesh We apply KVL and KCL to the super- mesh to get two equations. Example: Here, KVL around the supermesh gives 20 = 6i1 + 10i2 + 4i2 And KCL gives i2 = i1 +6 Have to be careful with KVL because different resistors in the supermesh have different currents.

We Still Get Enough Equations If this circuit did not have a super- mesh, we would get one equation by applying KVL to mesh 1 and another by applying KVL to mesh 2. With the supermesh, we get one equation by applying KCL and another by applying KVL to the supermesh. Do practice sheet question 5.

Nodal Analysis and Mesh Analysis “By Inspection” With practice, you’ll become good at writing down the set of simultaneous equations that describe a circuit using either nodal or mesh analysis. As discussed in Section 3.6, there is a shortcut way to write down the equations quickly by looking at a circuit without even thinking in terms of KCL or KVL. I won’t expect you to learn this shortcut method, but you can use it if you wish.

Which Should You Use: Nodal Analysis or Mesh Analysis? Most circuits can be analyzed using either method, and the results from the two methods will agree with each other. But as discussed in Section 3.7, in some cases you’ll get the answer with less work if you’re smart about picking the better method for your circuit. See next slide for example.

Examples: Should You Use Nodal Analysis or Mesh Analysis? Recommendation: Use nodal analysis for circuits with fewer nodes than meshes, and use mesh analysis for circuits with fewer meshes than nodes. How many nodes does this circuit have? How many meshes?

Dot Convention to Show Intersections Before we look at the next example, note that our textbook usually does not follow the widespread convention of using a dot to show intersection points between wires. Examples from Multisim: If you saw this in Multisim, you would know that these wires intersect. If you saw this in Multisim, you would know that these two wires cross without intersecting.

More Examples: Should You Use Nodal Analysis or Mesh Analysis? As also noted in Section 3.7, mesh analysis cannot be applied to nonplanar circuits. A circuit is planar if it can be drawn on a plane with no branches crossing each other; otherwise it is nonplanar. We will only deal with planar circuits in this course.

Example of a Planar Circuit This circuit, in which branches cross, can be redrawn with no crossing branches (as on the right below), so it is a planar circuit. I prefer to draw dots on junctions to make it clear whether the branches intersect or merely cross. Crossing branches: No intersections here.

Example of a Nonplanar Circuit There is no way to redraw this circuit without crossing branches, so it is a nonplanar circuit. So you cannot use mesh analysis on this circuit. Crossing branches: No intersections here. Crossing branches: No intersections here.

The Rest of Today’s Class Use the remaining time today to: Finish Lab #4. Review Homework #3 (particularly the super-node problems, Problems 3.13 and 3.18) and make sure you understand how to do them. Work on Homework #4, which is due at our next class. In preparation for the Midterm Exam, review earlier homeworks and practice sheets, and work on your crib sheet.