2. Problems With Assistance Module 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 Next slide Go straight to the Problem Statement Go straight to the First Step

3. 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 Next slide Go straight to the Problem Statement Go straight to the First Step

4. 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 6 th Ed. by Nilsson and Riedel: Sections 4.2 through 4.4 Basic Engineering Circuit Analysis 6 th 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 2 nd Ed. by Dorf: Sections 4-2 through 4-4 Next slide

5. 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

6. Problem Statement Next slide Find v 2, v 4, and the power supplied by each source.

7. Solution – First Step – Where to Start? How should we start this problem? What is the first step? Next slide Find v 2, v 4, and the power supplied by each source.

8. Problem Solution – First Step How should we start this problem? What is the first step? a)Write KCL for each nodeWrite KCL for each node b)Identify the essential nodesIdentify the essential nodes c)Write KVL for each loopWrite KVL for each loop d)Pick the reference nodePick the reference node e)Combine resistors in parallel or seriesCombine resistors in parallel or series Find v 2, v 4, and the power supplied by each source.

9. Your choice for First Step – Write KCL for each node Find v 2, v 4, and the power supplied by each source. 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.try again

10. Your choice for First Step – Write KVL for each loop 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.try again Find v 2, v 4, and the power supplied by each source.

11. Your choice for First Step was – Pick the reference node Find v 2, v 4, and the power supplied by each source. 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.try again

12. Your choice for First Step was – Combine resistors in parallel or series Find v 2, v 4, and the power supplied by each source. 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.try again 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.

13. Your choice for First Step was – Identify the essential nodes Find v 2, v 4, and the power supplied by each source. 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: a)3 essential nodes3 essential nodes b)4 essential nodes4 essential nodes c)5 essential nodes5 essential nodes

14. 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. Try again Find v 2, v 4, and the power supplied by each source.

15. 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? Which one should we pick? Find v 2, v 4, and the power supplied by each source.

16. Your choice for the number of essential nodes – 5 This is not correct. Try again.Try again Remember that two nodes connected by a wire were really only one node. Find v 2, v 4, and the power supplied by each source.

17. 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.define the node-voltages Find v 2, v 4, and the power supplied by each source.

18. Defining the Node-Voltages 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. write the Node-Voltage Method Equations Find v 2, v 4, and the power supplied by each source.

19. Writing the Node-Voltage Equations – 1 The equation for Node 2 is given here. Note that we used an expression for the current in R 1 to express the current in the voltage source. The resistor R 1 and the voltage source are in series. Next equation Find v 2, v 4, and the power supplied by each source.

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

21. Writing the Node-Voltage Equations – All The next step is to solve the equations. Let’s solve. Next step Find v 2, v 4, and the power supplied by each source.

22. Solving the Node-Voltage Equations When we solve, we find that v 2 = 36[V], and v 4 = 36[V]. Find v 2, v 4, and the power supplied by each source. Next step 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.

23. Using the Node-Voltages to Solve for Desired Quantities – Part 1 We found that v 2 = 36[V], and v 4 = 36[V]. Find v 2, v 4, and the power supplied by each source. We can use this to find the power supplied by the current source directly. Note that v 4 is the voltage across the current source. Note also that v 4 and i S are in the active convention for this source. Therefore, we can write: p del,iS = v 4 i S = 36[V]3[mA] = 108[mW] Next step

24. Using the Node-Voltages to Solve for Desired Quantities – Part 2 We found that v 2 = 36[V], and v 4 = 36[V]. Find v 2, v 4, and the power supplied by each source. 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 i 1. We write the expression for this just as we had when writing the KCL expression for node 2. Next step

25. Using the Node-Voltages to Solve for Desired Quantities – Part 3 See Note We found that v 2 = 36[V], and v 4 = 36[V]. Find v 2, v 4, and the power supplied by each source. Now, we can find the power supplied by the voltage source. Note that v S and i 1 are in defined in the active convention for the voltage source. Thus, we can write:

26. 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 v S, i S, R 1, R 2 and R 4 to make this happen. You can prove to yourself that R 3 makes no difference in this case by varying its value, and solving again. For all nonzero values of R 3, 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 v 2 =v 4, and solve for values of v S, i S, R 1, R 2 and R 4 to make this happen. The node-voltage technique, once in hand, has many uses. Go back to Overview slide. Overview