Electric Circuit Analysis/Mesh Analysis
Lesson Review 5 & 6:What you need to remember from Kirchhoff's Voltage & Current Law .
This part of the course onwards will collaborate with the Mathematics Department extensively. Mathematical Theory will be kept minimal as mathematical tools are only used here as a means to an end. Links to relevant Mathematical theories will be supplied to assist the student. Lesson Review 7:What you need to remember from Nodal analysis. If you ever feel lost, do not be shy to go back to the previous lesson & go through it again. You can learn by repetition.


Lesson 8: PreviewThis Lesson is about Mesh Analysis. The student/User is expected to understand the following at the end of the lesson.
Part 1: Prereading MaterialThe student is advised to read the following resources from the Mathematics department: 
Part 2: Mesh AnalysisLet's start off with some useful definitions:

Part 3The following is a general procedure for using Mesh or Loop Analysis method to solve electric circuit problems. The aim of this algorithm is to develop a matrix system from equations found by applying KVL arround Loops or Meshes in an electric circuit. Cramer's rule is then used to solve the unknown Mesh Currents. Once the Mesh Currents are solved, normal circuit analysis methods ( Ohm's law; Voltage and Current Divider principles etc... ) can then be used to find whatever circuit entity. Remember to consult previous lessons if you are not confident in using normal circuit analysis techniques that will be used in this lesson.
1.) Choose a conventional current flow. 2.) Identify and Number loops or meshes. ( Usually 2 or 3 meshes/ loops ) 3.) Apply KVL to identified meshes/loops and formulate ciruit equations. 4.) Create Matrix system from KVL equations obtained. 5.) Solve Matrix for unknown Mesh Currents by using Cramer's rule ( it is simpler although you can still use gaussian method as well ) 6.) Used solved Mesh Currents to solve for the desired circuit entity.
Generally speaking, Mesh Analysis is shorter than Nodal Analysis although not always preffered. Let's try an example to illustrate the above Mesh/Loop analysis algorithm. 
Part 4 : ExampleConsider Figure 8.1 with the following Parameters: Find current through using Mesh Analysis method.
This is the same example we solved in Exercise 7. We can see that there are three closed paths (loops) where we can apply KVL in, Loop 1, 2 and 3 as shown in figure 8.2 We can now apply KVL around the loops remembering Passive Convention when defining Currents and voltages. KVL around abca loop:
Therefore ............(1) 
Part 5 : Example (Continued)KVL around acda loop:
Therefore ............... (2)
Therefore ............... (3)
Now we can create a matrix with the above equations as follows: 
Part 6 : Example (Continued)
The following matrix is the above with values substituted: ?
Solving determinants of:
As follows: 
Part 7: Example (Continued)

Part 8: Example (Continued)
Now we can use the solved determinants to arrive at solutions for Mesh Currents as follows: 1. 2. 3.
The answer is as we expected it to be, flows in the direction of . As expected, mesh analysis and nodal analysis gave equivalent answers. 
Part 9: Exercise 8Consider Figure 8.3 with the following Parameters: Find current through using Mesh Analysis method. 
Part 10:Further Reading Links: References:
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