2. Problems With Assistance Module 3 – Problem 1 Filename: PWA_Mod03_Prob01.ppt 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 presentation: DPKC_Mod03_Part01 and DPKC_Mod03_Part02 Next slide

6. Problem Statement Next slide Use the node- voltage method to solve for the voltage v X.

7. Solution – First Step – Where to Start? How should we start this problem? What is the first step? Next slide Use the node- voltage method to solve for the voltage v X.

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 Use the node- voltage method to solve for the voltage v X.

9. Your choice for First Step – Write KCL for each node This is not a good choice for the first step, although we will write KCL equations for most nodes soon. One purpose of the node- voltage method is to find a systematic way of writing the correct number of equations. It is important, then, to know how many equations we are going to write. Go back and try again.try again Use the node- voltage method to solve for the voltage v X.

10. Your choice for First Step – Write KVL for each loop This is not a good choice. The node-voltage method involves writing KCL equations, not KVL equations. While we may write KVL equations as needed for constraint equations, it is not the systematic step that we take in using the node-voltage method. This is not the way to start this method. Go back and try again.try again Use the node- voltage method to solve for the voltage v X.

11. Your choice for First Step was – Pick the reference node 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 Use the node- voltage method to solve for the voltage v X.

12. Your choice for First Step was – Combine resistors in parallel or series 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, you may have noted that R 4 and R 3 are in parallel, and can be combined into a single resistor. We will not even lose any dependent source variables. However, the node-voltage method does not require that we simplify the circuits, and sometimes we cannot do so. Therefore, we recommend that you go back and try again.try again Use the node- voltage method to solve for the voltage v X.

13. Your choice for First Step was – Identify the essential nodes This is the best choice. The first step is to make sure that we have identified every essential nodes, and only the essential nodes. 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 d)6 essential nodes6 essential nodes e)7 essential nodes7 essential nodes Use the node- voltage method to solve for the voltage v X.

14. Your choice for the number of essential nodes – 3 This is not correct. Try again.Try again Use the node- voltage method to solve for the voltage v X.

15. Your choice for the number of essential nodes – 4 This is correct. The essential nodes are marked with red in this schematic. Use the node- voltage method to solve for the voltage v X. The next step is to pick one of them as the reference node. Which one should we pick?Which one should we pick?

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. Use the node- voltage method to solve for the voltage v X.

17. Your choice for the number of essential nodes – 6 This is not correct. Try again.Try again Remember that two nodes connected by a wire were really only one node. Remember, that just because we have a black circle, does not mean that it is a separate node. There are 6 black circles, but some of them are connected by wires, so are not separate essential nodes. These marks are placed in some places to indicate that crossing wires are connected together. In other places, they are not required, but are included to reinforce that a connection is present. Use the node- voltage method to solve for the voltage v X.

18. Your choice for the number of essential nodes – 7 This is not correct. Try again.Try again Remember that two nodes connected by a wire were really only one node. Remember, that just because we have a black circle, does not mean that it is a separate node. There are 6 black circles, but some of them are connected by wires, so are not separate essential nodes. These marks are placed in some places to indicate that crossing wires are connected together. In other places, they are not required, but are included to reinforce that a connection is present. Use the node- voltage method to solve for the voltage v X.

19. Choosing the Reference Node Use the node- voltage method to solve for the voltage v X. The next step is to pick one of them as the reference node. We have chosen the node at the top left as the reference node. This is considered to be the best choice, since it has 5 connections to it, one of which is a voltage source. The equations will probably be easier to write with this as reference node. Next, we define the node-voltages.define the node-voltages

20. 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 four equations, one for each non-reference essential node (3) and one for the dependent source variable v X.write the Node-Voltage Method Equations Use the node- voltage method to solve for the voltage v X.

21. Writing the Node-Voltage Equations – 1 The equation for Node A is: Use the node- voltage method to solve for the voltage v X. Next equation

22. Writing the Node-Voltage Equations – 2 The equation for Node B is obtained by noting that there is a voltage source between the B node and the reference node. Be careful about the signs. Use the node- voltage method to solve for the voltage v X. There is no negative sign in this equation, because the polarities of the voltage source and v B are aligned. Next equation

23. Writing the Node-Voltage Equations – 3 The equation for Node C is: Use the node- voltage method to solve for the voltage v X. Next equation

24. Writing the Node-Voltage Equations – 4 The equation for the dependent source variable v X is: Use the node- voltage method to solve for the voltage v X. Make sure that you agree with this equation for the dependent source variable v X. It can be obtained by writing KVL around a loop including these voltages. We think of it as follows: v X is the voltage at B with respect to the reference (v B ), minus the voltage at A with respect to the reference (v A ). Next step

25. Writing the Node-Voltage Equations – All Use the node- voltage method to solve for the voltage v X. The next step is to solve the equations. We can do this by substituting back in, pretty easily. Let’s solve. Next step

26. Solving the Node-Voltage Equations Use the node- voltage method to solve for the voltage v X. The next step is to solve the equations. We can do this by substituting back in, pretty easily. Let’s solve. See Note

27. What happened? We didn’t even need the C equation! It is true that in this problem, we did not need to write the KCL equation for the C node. We were able to solve for v X without it. This raises an important point. The Node-Voltage Method is not optimal in all circuits. However, from the solution it gives us, we would be able to find any voltage or current in the circuit. The four equations we obtained were optimal in that sense. There would be other voltages and currents where the C-node equation would be needed. Go back to Overview slide. Overview See next note

28. What if I picked the bottom node as the reference? Is that a big deal? In principle, it does not matter which node we pick as reference. More importantly, sometimes the reference is picked for us. We can always write the equations using any choice of reference nodes. The idea here was to suggest that some choices make the problem easier; the simpler our set of equations, the better off we are. To illustrate the difference, the problem is worked out again on the next slides to show how another choice would work out.the problem is worked out againnext slide Go back to Overview slide. Overview

29. Writing the Node- Voltage Equations – Bottom Node as Reference Here, we have written the four equations that would result if we had used the bottom node as reference. We have a super-node situation for nodes D & E. Next step

30. Solution Using Bottom Node as Reference Now, we solve these equations. Note that the first and last terms in the D&E equation cancelled. We now have: Next step When we solve, we find: v D = -3.88[V], v E = 1.12[V], v F = 1.21[V], and v X = -0.097[V]. This is the same solution we had previously. Here, we have used MathCAD to solve this set of equations.

31. Which Choice of Reference Was Better? This is the set of equations we obtained using the bottom node as reference: This is the set of equations we obtained using the upper left node as reference: Which set of equations would you prefer to write? Which set of equations would you prefer to solve? We would prefer the one obtained using the upper left node. Go back to Overview slide. Overview