Circuit Analysis: Node Voltage Method

1. Problems With AssistanceModule 3 – Problem 2 Filename: PWA_Mod03_Prob02.ppt This problem is adapted from: Problem 4.6, page 183 in Circuits by A. Bruce Carlson Brooks/Cole Thomson Learning 2000 ISBN: 0-534-37097-7 Go straight to the First Step Go straight to the Problem Statement Next slide

2. Overview of this Problem In this problem, we will use the following concepts: • Kirchhoff’s Voltage Law • Kirchhoff’s Current Law • Ohm’s Law • The Node-Voltage Method Go straight to the First Step Go straight to the Problem Statement Next slide

3. Textbook Coverage The material for this problem is covered in your textbook in the following sections: • Circuits by Carlson: Sections 4.1 & 4.3 • Electric Circuits 6th Ed. by Nilsson and Riedel: Sections 4.2 through 4.4 • Basic Engineering Circuit Analysis 6th Ed. by Irwin and Wu: Section 3.1 • Fundamentals of Electric Circuits by Alexander and Sadiku: Sections 3.2 & 3.3 • Introduction to Electric Circuits 2nd Ed. by Dorf: Sections 4-2 through 4-4 Next slide

4. Coverage in this Module The material for this problem is covered in this module in the following presentations: • DPKC_Mod03_Part01 and DPKC_Mod03_Part02 A similar problem is worked in: • PWA_Mod03_Prob01 Next slide

5. Problem Statement Find v2, v4, and the power supplied by each source. Next slide

6. Solution – First Step – Where to Start? How should we start this problem? What is the first step? Find v2, v4, and the power supplied by each source. Next slide

7. How should we start this problem? What is the first step? • Write KCL for each node • Identify the essential nodes • Write KVL for each loop • Pick the reference node • Combine resistors in parallel or series Find v2, v4, and the power supplied by each source. Problem Solution – First Step

8. This is not the best choice for the first step, although we will write KCL equations for most nodes soon. It is generally worth while to spend some time looking at the problem and choosing an approach before beginning to write equations. Note that we have six variables defined already, but will not need that many. Go back and try again. Find v2, v4, and the power supplied by each source. Your choice for First Step –Write KCL for each node

9. This is not the best choice for the first step. It is generally worth while to spend some time looking at the problem and choosing an approach before beginning to write equations. Note that we have six variables defined already, but will not need that many. Go back and try again. Find v2, v4, and the power supplied by each source. Your choice for First Step –Write KVL for each loop

10. This will be helpful, but is not the best choice for the first step. The node-voltage method indeed requires that we pick, and label, the reference node. However, it is usually wise to be sure that we know where all the essential nodes are, and how many connections they have, before making this choice. Thus, while it may not be necessary in simple problems like this, we recommend that you go back and try again. Find v2, v4, and the power supplied by each source. Your choice for First Step was –Pick the reference node

11. This might be helpful, but is not the best choice for the first step. Generally, it is a good thing to simplify a circuit, where we can do so. Here, we cannot do so since, there are no resistors in series or parallel. Therefore, we recommend that you go back and try again. Find v2, v4, and the power supplied by each source. Your choice for First Step was –Combine resistors in parallel or series Note to advanced students: We could use delta-to-wye or wye-to-delta transformations, but we are going to take a different approach here.

12. This is the best choice. By making sure that we have identified the essential nodes, we can determine how many equations will be needed in the node-voltage method. • How many essential nodes are there in this circuit? Your answer is: • 3 essential nodes • 4 essential nodes • 5 essential nodes Find v2, v4, and the power supplied by each source. Your choice for First Step was –Identify the essential nodes

13. Find v2, v4, and the power supplied by each source. Your choice for the number of essential nodes – 4 This is not correct. Remember that essential nodes must have at least 3 connections. In addition, remember that two nodes connected by a wire were really only one node. Try again.

14. Your choice for the number of essential nodes – 3 This is correct. The essential nodes are marked with red in this schematic. There is a non-essential node, which is marked with green. With only 3 essential nodes, the node-voltage method is a good choice, since we will have only 2 simultaneous equations. The next step is to pick one of them as the reference node. Which one should we pick? Find v2, v4, and the power supplied by each source.

15. Find v2, v4, and the power supplied by each source. Your choice for the number of essential nodes – 5 This is not correct. Try again. Remember that two nodes connected by a wire were really only one node.

16. Choosing the Reference Node The next step is to pick one of them as the reference node. We have chosen the node at the bottom as the reference node. This is considered to be the best choice, since it has 4 connections to it. The equations will probably be easier to write with this as reference node. In addition, the two node voltages that result are the voltages we were asked to find. Next, we define the node-voltages. Find v2, v4, and the power supplied by each source.

17. The next step is to define the node-voltages. We have done so here. Now, we are ready to write the Node-Voltage Method Equations. Even before we do, we can predict that we will need to write two equations, one for each non-reference essential node. Find v2, v4, and the power supplied by each source. Defining the Node-Voltages

18. Find v2, v4, and the power supplied by each source. Writing the Node-Voltage Equations – 1 The equation for Node 2 is given here. Note that we used an expression for the current in R1 to express the current in the voltage source. The resistor R1 and the voltage source are in series. Next equation

19. Find v2, v4, and the power supplied by each source. Writing the Node-Voltage Equations – 2 The equation for Node 4 is given here. Next step

20. Find v2, v4, and the power supplied by each source. Writing the Node-Voltage Equations – All The next step is to solve the equations. Let’s solve. Next step

21. Find v2, v4, and the power supplied by each source. We have used MathCAD to solve the two simultaneous equations. This is shown in a MathCAD file called PWA_Mod03_Prob02_Soln.mcd which should be available in this module. Solving the Node-Voltage Equations When we solve, we find that v2 = 36[V], and v4 = 36[V]. Next step

22. Find v2, v4, and the power supplied by each source. Using the Node-Voltages to Solve for Desired Quantities – Part 1 We found that v2 = 36[V], and v4 = 36[V]. We can use this to find the power supplied by the current source directly. Note that v4 is the voltage across the current source. Note also that v4 and iS are in the active convention for this source. Therefore, we can write: pdel,iS = v4 iS = 36[V]3[mA] = 108[mW] Next step

23. Find v2, v4, and the power supplied by each source. Using the Node-Voltages to Solve for Desired Quantities – Part 2 We found that v2 = 36[V], and v4 = 36[V]. Next, we want to find the power supplied by the voltage source. For this, we need to find the current through the voltage source, which has already been labeled as i1. We write the expression for this just as we had when writing the KCL expression for node 2. Next step

24. Find v2, v4, and the power supplied by each source. Using the Node-Voltages to Solve for Desired Quantities – Part 3 We found that v2 = 36[V], and v4 = 36[V]. Now, we can find the power supplied by the voltage source. Note that vS and i1 are in defined in the active convention for the voltage source. Thus, we can write: See Note

25. What happened? The two node-voltages were the same! • It is true that in this problem, the two node-voltages were the same. This occurred because Carlson chose the values of vS, iS, R1, R2 and R4 to make this happen. You can prove to yourself that R3 makes no difference in this case by varying its value, and solving again. For all nonzero values of R3, the solution will be the same. • This raises yet another important point. The Node-Voltage Method gives us general equations which apply for the way the circuit is laid out, called the topology. Once you have the equations, you could set v2=v4, and solve for values of vS, iS, R1, R2 and R4 to make this happen. The node-voltage technique, once in hand, has many uses. Go back to Overviewslide.