Problems With Assistance Module 3 – Problem 5 Go straight to the First Step Filename: PWA_Mod03_Prob05.ppt This problem is adapted from: Problem 4.38, page 185 in Circuits by A. Bruce Carlson Brooks/Cole Thomson Learning 2000 ISBN: 0-534-37097-7 You can see a brief introduction starting on the next slide, or go right to the problem. Go straight to the Problem Statement Next slide

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 Mesh-Current Method Go straight to the First Step Go straight to the Problem Statement Next slide

Textbook Coverage This material is covered in your textbook in the following sections: Circuits by Carlson: Sections 4.2 & 4.3 Electric Circuits 6th Ed. by Nilsson and Riedel: Sections 4.1, & 4.5 through 4.7 Basic Engineering Circuit Analysis 6th Ed. by Irwin and Wu: Section 3.2 Fundamentals of Electric Circuits by Alexander and Sadiku: Sections 3.4 & 3.5 Introduction to Electric Circuits 2nd Ed. by Dorf: Sections 4-5 & 4-6 You should also read these sections in your text. This material is intended to complement your textbook coverage, not replace it. Next slide

Coverage in this Module The material for this problem is covered in this module in the following presentations: DPKC_Mod03_Part03 and DPKC_Mod03_Part04 A similar problem is worked in: PWA_Mod03_Prob04 This is the material in this computer module that you might consult for more explanation. These are presentations of key concepts that you should find in this problem. Next slide

Problem Statement Use the mesh-current method to find i1, v1, v2, and v3. This is the basic problem. We will take it step by step. Next slide

Solution – First Step – Where to Start? How should we start this problem? What is the first step? Solution – First Step – Where to Start? Use the mesh-current method to find i1, v1, v2, and v3. Try to decide on the first step before going to the next slide. Next slide

Problem Solution – First Step How should we start this problem? What is the first step? Write KVL for each mesh Identify the meshes and define the mesh currents Write KCL for each node Combine resistors in parallel or series Use the mesh-current method to find i1, v1, v2, and v3. Click on the step that you think should be next.

Your choice for First Step – Write KCL for each node This is not the best choice for the first step. We have been instructed to use the mesh current method, which involves writing KVL equations. We may need to write KCL equations at some point, but the first steps in the mesh-current method do not involve KCL. Go back and try again. Your choice for First Step – Write KCL for each node Use the mesh-current method to find i1, v1, v2, and v3.

Your choice for First Step – Write KVL for each mesh This is not the best choice for the first step. In the mesh-current method we write KVL equations, but we want to do it in terms of mesh currents. Go back and try again. Your choice for First Step – Write KVL for each mesh Use the mesh-current method to find i1, v1, v2, and v3.

Your choice for First Step was – Combine resistors in parallel or series This might be helpful, but is not a good choice for the first step in this problem. Generally, it is a good thing to simplify a circuit, where we can do so. In the mesh-current method we do not need to do this kind of simplification. More importantly, we cannot simplify here since there are no resistors in series or parallel. Therefore, we recommend that you go back and try again. Use the mesh-current method to find i1, v1, v2, and v3.

Your choice for First Step was – Identify the meshes and define the mesh currents This is the best choice. By making sure that we have identified the mesh currents, we can determine how many equations will be needed in the mesh-current method. How many meshes are there in this circuit? Your answer is: 3 meshes 4 meshes 7 meshes 365 days Use the mesh-current method to find i1, v1, v2, and v3.

Your choice for the number of meshes – 4 This is not correct. Try again. Use the mesh-current method to find i1, v1, v2, and v3.

Your choice for the number of meshes – 3 With only 3 meshes, the mesh current method is a good choice, since we will have only 3 simultaneous equations. Compare this with the node-voltage method. How many equations would that method require? (Answer: 3) Use the mesh-current method to find i1, v1, v2, and v3. The next step is to define the mesh currents.

Your choice for the number of meshes – 7 This is not correct. Try again. Remember that meshes cannot enclose other meshes. Seven would be the total number of closed paths, but not the number of meshes. Use the mesh-current method to find i1, v1, v2, and v3.

Defining the Mesh Currents The next step is to define the mesh currents. We have done so here. Now, we are ready to write the Mesh-Current Method Equations. Even before we do, we can predict that we will need to write three equations, one for each mesh. Defining the Mesh Currents Use the mesh-current method to find i1, v1, v2, and v3.

Writing the Mesh-Current Method Equations – 1 Mesh A includes a current source, and that current source is a part of two different meshes. We will need to write a supermesh equation, and a constraint equation. The constraint equation is given here. Use the mesh-current method to find i1, v1, v2, and v3. Next equation

Writing the Mesh-Current Method Equations – 2 The supermesh equation for meshes A & C is given here. The supermesh is drawn in blue in the circuit below. Next equation

Writing the Mesh-Current Method Equations – 3 Mesh B also includes a current source. That current source is a part of only one mesh. We will need to write an equation using the current source. Use the mesh-current method to find i1, v1, v2, and v3. Next step

Writing the Mesh-Current Method Equations – All Use the mesh-current method to find i1, v1, v2, and v3. The next step is to solve the equations. Let’s solve. Next step

Solving the Mesh-Current Method Equations We have used MathCAD to solve the three simultaneous equations. This is shown in a MathCAD file called PWA_Mod03_Prob05_Soln.mcd which should be available in this module. Solving the Mesh-Current Method Equations Use the mesh-current method to find i1, v1, v2, and v3. When we solve, we find that iA = 2[A], iB = 6[A], and iC = 10[A]. Next step

Using the Mesh Currents to Solve for Desired Quantities – Part 1 When we solve, we find that iA = 2[A], iB = 6[A], and iC = 10[A]. Use the mesh-current method to find i1, v1, v2, and v3. We can use this solution to find the quantities requested. Note that iA is the same as i1. Therefore, we can write: i1 = iA = 2[A] Next step

Using the Mesh Currents to Solve for Desired Quantities – Part 2 When we solve, we find that iA = 2[A], iB = 6[A], and iC = 10[A]. Use the mesh-current method to find i1, v1, v2, and v3. We can use this solution to find the quantities requested. Note that v1 is the voltage across a current source, so we have to use the things the current source is connected to in order to find v1. We can write: v1 = (iB-iA)10[W]+(iB-iC)4[W] v1 = 24[V] Next step

Using the Mesh Currents to Solve for Desired Quantities – Part 3 When we solve, we find that iA = 2[A], iB = 6[A], and iC = 10[A]. Use the mesh-current method to find i1, v1, v2, and v3. We can use this solution to find the quantities requested. We can get v2 directly in terms of the mesh currents. We can write: v2 = (iB-iC)4[W] v2 = -16[V] The answer for this problem given in the back of the Carlson text on page 831 is +16[V]. We believe that this is an error. The answer is –16[V]. Next step

Using the Mesh Currents to Solve for Desired Quantities – Part 4 When we solve, we find that iA = 2[A], iB = 6[A], and iC = 10[A]. Use the mesh-current method to find i1, v1, v2, and v3. We can use this solution to find the quantities requested. We can get v3 directly in terms of the mesh currents. We can write: v3 = (iC)2[W] v3 = 20[V] Go to notes slide.

Go back to Overview slide. So, which is better? The Node-Voltage Method or the Mesh-Current Method? – 1 In slide 13 we asked you to compare the Node-Voltage Method to the Mesh-Current Method. Which one is better? First, we will ask an analogous question: Which is better, an automobile or an airplane? You might have a preference between cars and planes, but generally the answer depends on where you are trying to go. If you are trying to go to a store 2 miles (or 3 kilometers) away, you would probably prefer a car. If you are trying to go to a city 5,000 miles (or 8,000 kilometers) away, you would probably prefer a plane. The key is: the answer depends on what you are trying to do. Go back to Overview slide. Go to next notes slide.

Go back to Overview slide. So, which is better? The Node-Voltage Method or the Mesh-Current Method? – 2 In slide 13 we asked you to compare the Node-Voltage Method to the Mesh-Current Method. Which one is better? When we ask which method is better, the answer depends on what circuit we are solving, and what kinds of solutions we are looking for. In this circuit the number of equations are the same, so there is no difference there. One of the equations was very simple in the mesh-current method, so there is an advantage there. Three of the quantities desired were the same as the node-voltages, so there is a different advantage there. In the end, the only general answer to this question is, It depends. Go back to Overview slide.