Electric Circuit Analysis/Nodal Analysis
Lesson Review 5 & 6:What you need to remember from Kirchhoff's Voltage & Current Law . 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.
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 7: PreviewThis Lesson is about Kirchhoff's Current Law. 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: The following external link has an excellent summary on using Cramer's rule to solve linear equations: After you have satisfied yourself with the above resources, you can go to Part 2. 
Part 2: Nodal analysisLet's start off with some useful definitions:
Basic rule: The sum of the currents entering any point (Node) must equal the sum of the currents leaving.( From KCL in Lecture 6). 
Part 3The following is a general procedure for using Nodal Analysis method to solve electric circuit problems. The aim of this algorithm is to develop a matrix system from equations found by applying KCL at the major nodes in an electric circuit. Cramer's rule is then used to solve the unkown major node voltages.
Remember to consult previous lessons if you are not confident using the circuit analysis techniques that will be employed in this lesson.
1.) Choose a reference node. ( Rule of thumb: take the node with most branches connecting to it ) 2.) Identify and number major nodes. ( Usually 2 or 3 major Nodes ) 3.) Apply KCL to identified major nodes and formulate circuit equations. 4.) Create matrix system from KCL equations obtained. 5.) Solve matrix for unknown node voltages by using Cramer's Rule (Cramer's Rule is simpler, although you can still use the Gaussian method) 6.) Use solved node voltages to solve for the desired circuit entity.

Part 4 : ExampleConsider Figure 7.1 with the following Parameters: Find current through using Nodal Analysis method.
This is the same example we solved in Exercise 6, except that in this case we have added some resistors to increase the complexity of the circuit. Figure 7.2 shows voltages at nodes a, b, c and d. We use node a as the common node (ground if you like). Thus as we did previously.

Part 5 : Example (Continued)Now that we have labelled the currents flowing in this circuit using the Passive Sign Convention, and have identified nodes b; c and d as major nodes, we proceed as follows: KCL at node b:
Thus, by applying Ohm's Law to the above equation we get:
Therefore ............... (1)
Then, by applying Ohm's Law to the above equation we get: 
Part 6 : Example (Continued)
Therefore ............... (2)
Applying Ohm's Law...
Therefore ............... (3) 
Part 7:The next step in this algorithm is to construct a matrix. To do that easily, we substitute all resistances in the above equations 1, 2 and 3 with their equivalent admittances  as follows: etc thus equations 1, 2 and 3 will be rewritten as follows:
Now we can create a matrix with the above equations as follows:
The following matrix is the above with values substituted: →

Part 8:Remember to read *Solutions/Cramer Also, use the provided link for details on working out the determinant of a 3 x 3 Matrix. Solving determinants of:
As follows:

Part 9:
2. 3. Now we can apply Ohm's law to solve for the current through as follows:
As we previously saw, the positive sign in the above current tells us that the effective current flowing through is in fact in the direction we chose when drawing up the circuit in figure 7.2.
As usual the following part is an exercise to test yourself on the content discussed in this lesson. Please refer to Part 11 for further reading material and interesting related external links. 
Part 10:Exercise 7Consider Figure 7.3 with the following parameters: Find the current through using the Nodal Analysis method. 
Part 11:Further Reading & Other Interesting Links: References:
Completion List Once you finish your Exercises you can post your score here! To post your score just email your course coordinator your name and score *Click Here.

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