Problems With Assistance Module 3 – Problem 7

Problems With AssistanceModule 3 – Problem 7 Filename: PWA_Mod03_Prob07.ppt Go straight to the First Step 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 Next slide

Coverage in this Module The material for this problem is covered in this module in the following presentations: • DPKC_Mod03_Part03 & DPKC_Mod03_Part04 A similar problem is worked in: • PWA_Mod03_Prob04 & PWA_Mod03_Prob05 Next slide

Problem Statement Use the mesh-current method to find vX. Next slide

How should we start this problem? What is the first step? Solution – First Step – Where to Start? Use the mesh-current method to find vX. Next slide

We need to identify the meshes and define the mesh currents. This circuit has already been drawn in planar form, and it is fairly easy to recognize the six meshes. Thus, we will need to write six equations for meshes and two more for dependent source variables. Just to make the comparison, how many equations would be required for the Node-Voltage Method? The First Step is to Identify the Meshes and Define the Mesh Currents Use the mesh-current method to find vX. Next slide

We already decided that we need eight equations for the Mesh-Current Method. How many equations would be required for the Node-Voltage Method? We have eight essential nodes in this circuit. In this circuit, each black dot is an essential node, although this is not always true. Thus, we would need seven equations (8-1) plus two more for the dependent sources. So, the Node-Voltage Method would require nine equations. Thus, for this circuit, the Mesh-Current Method requires fewer equations. How Many Node-Voltage Method Equations Would be Needed? Use the mesh-current method to find vX. Next slide

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. Remember that we will need a total of eight equations. Defining the Mesh Currents Use the mesh-current method to find vX.

Writing the Mesh-Current Method Equations – 1 Mesh A includes a voltage source, but no current sources, so the equation is a straightforward application of KVL. Next equation

Writing the Mesh-Current Method Equations – 2 – Part 1 Mesh B has a voltage source and a current source. The voltage source causes no problems, but the current source requires that we look more carefully. The iS1 current source is a part of two meshes, and we would expect to have to write a supermesh equation and a constraint equation. The supermesh that we expect to use is shown here, as a green dashed line. Next slide

Writing the Mesh-Current Method Equations – 2 – Part 2 Let’s look at the supermesh that we expect to use, shown here as a green dashed line. We do not have an expression for the voltage across the iS2 current source. Thus, we will not be able to write the KVL for this supermesh without adding another variable. What can we do? Next slide

Writing the Mesh-Current Method Equations – 2 – Part 3 We will not be able to write the KVL for this supermesh without adding another variable. What can we do? The answer is to recognize that the iS2 current source is a part of only one mesh, and thus we will have an equation for the C mesh. Then, we can use the iS1 current source to write a constraint equation for meshes B & C. Let’s write the equations. Next slide

Writing the Mesh-Current Method Equations – 2 – Part 4 We can write a constraint equation for meshes B & C. Let’s write the equations. Next equation

Writing the Mesh-Current Method Equations – 3 Next, we can write a constraint equation for mesh C. Here we use the current source, iS2, that is a part of only the C mesh. Next equation

Writing the Mesh-Current Method Equations – 4 – Part 1 Next, we take mesh D. This mesh has a current source, iS3, as a part of this mesh. This current source is a part of two meshes, and we expect to write a supermesh equation. However, the supermesh here, drawn with a green dashed line, again has a current source, iS4, and we do not have an expression for the voltage across that current source. Next slide

Writing the Mesh-Current Method Equations – 4 – Part 2 We do not have an expression for the voltage across that current source, iS4. However, we can extend our supermesh even further, and then we will have a closed path that we can apply KVL for. This closed path, or super-duper-mesh, is shown with a green dashed line here. Next equation

Writing the Mesh-Current Method Equations – 5 We now need two more equations. We can get one each for constraint equations using the current sources, iS3 and iS4. Using iS3 we get this equation. Next equation

Writing the Mesh-Current Method Equations – 6 We can get the second equation from the constraint equations using the current source iS4. Next equation

Writing the Mesh-Current Method Equations – 7 Now, we need equations for the variables that dependent sources depend on. Let’s start with iX. Next equation

Writing the Mesh-Current Method Equations – 8 Finally, we need an equation for the variable vX. We need a closed loop that we can write a KVL equation for, that includes vX. One of many possibilities is shown with a green dashed line in this diagram. Next slide

Writing the Mesh-Current Method Equations – All We can substitute in the values, and get the following set of eight equations in eight unknowns. The next step is to solve the equations. Let’s solve. Next step

We have used MathCAD to solve the three simultaneous equations. This is shown in a MathCAD file called PWA_Mod03_Prob07_Soln.mcd which should be available in this module. Solving the Mesh-Current Method Equations When we solve, we find that iA = 0.98465[A], iB = 15.80922[A], iC = -2[A], iD = 3.1693[A], iE = 7.1693[A], iF = -2.6772[A], iX = -0.98465[A], and vX = -3.56184[V]. Go to notes

So, what do we really do with current sources? • Here we had two different situations, each with two current sources in adjacent meshes. When this happened, the usual approaches did not work. • However, it is important to recognize that what we did was consistent with what we have done all along. We can’t write an expression for the voltage across a current source without introducing additional variables. So, • we write equations using the current sources to determine mesh currents, or the relationship between two mesh currents; • then we write KVL equations for additional closed paths that enclose the current sources (supermeshes), as needed. Go back to Overviewslide.

Problems With Assistance Module 3 – Problem 7