Problems With Assistance Module 3 – Problem 3

Problems With AssistanceModule 3 – Problem 3 Filename: PWA_Mod03_Prob03.ppt This problem is adapted from: Exam #2 – Problem #1 – ECE 2300 – July 25, 1998 Department of Electrical and Computer Engineering University of Houston Houston, TX, 77204-4793 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 Node-Voltage Method Go straight to the First Step Go straight to the Problem Statement Next slide

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

Coverage in this Module The material for this problem is covered in this module in the following presentations: • DPKC_Mod03_Part01 & DPKC_Mod03_Part02 Similar problems are worked in: • PWA_Mod03_Prob01 & PWA_Mod03_Prob02 Next slide

Next slide • Use the node-voltage method to write a system of equations that could be used to solve this circuit. Do not attempt to simplify the circuit first. (Note: to be able to solve the system of equations, you want to have a set of independent equations, where the number of variables is equal to the number of unknowns.) • Use your equations to solve for vX. Problem Statement

Use the node-voltage method to write a system of equations that could be used to solve this circuit. Do not attempt to simplify the circuit first. (Note: to be able to solve the system of equations, you want to have a set of independent equations, where the number of variables is equal to the number of unknowns.) • Use your equations to solve for vX. Solution – First Step – Where to Start? How should we start this problem? What is the first step? Next slide

The first step in a problem like this is to identify the essential nodes. This allows us to make a wise choice about the reference node. • Use the node-voltage method to write a system of equations that could be used to solve this circuit. Do not attempt to simplify the circuit first. (Note: to be able to solve the system of equations, you want to have a set of independent equations, where the number of variables is equal to the number of unknowns.) • Use your equations to solve for vX. Problem Solution – First Step Next slide

How many essential nodes are there in this circuit? Your answer is: • 4 essential nodes • 5 essential nodes • 6 essential nodes • 7 essential nodes • 8 essential nodes Identify the Essential Nodes

Your choice for the number of essential nodes – 4 This is not correct. Try again.

Your choice for the number of essential nodes – 5 This is correct. The essential nodes are marked with red in this schematic. There are seven non-essential nodes, which are marked with green. This step may be the most difficult one in this problem. We believe that the best thing to do is to go ahead and label all the essential nodes on the schematic, and count them. As you do this, make sure that you do not have any nodes connected by a wire. The next step is to pick one of them as the reference node. Which one should we pick?

Your choice for the number of essential nodes – 6 This is not correct. Try again. 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.

The next step is to pick one of the essential nodes as the reference node. We have chosen the node at the upper right as the reference node. This is considered to be the best choice, since it has 11 (!?!) connections to it. We would rather not write an equation with 11 terms in it, if we can avoid it. Next, we define the node-voltages. Choosing the Reference Node The perceptive student might recognize that since three of the components have both ends connected to this same node, that the currents will cancel for these components. That is, the current that goes out of the node through the 6[W] resistor at the top of the resistor, enters the same node again at the bottom of the resistor. Thus, six of the terms will cancel out. Still, this equation is complicated, and has 5 terms. We would rather not write this equation.

The next step is to define the node-voltages. We have done so here. The node voltages are indicated in red. Now, we are ready to write the Node-Voltage Method Equations. Even before we do, we can predict that we will need to write eight equations, one for each non-reference essential node, and one for each of the four variables that dependent sources depend on. Defining the Node-Voltages

Writing the Node-Voltage Equations – 1 Next equation The equation for Node A involves two voltage sources. One is in series with two resistors, and one is between two essential nodes. Because of the 5vX dependent source, we will need to write a supernode equation. The supernode is shown. Note that vA-4[V]-vC is the voltage across the series combination of the 20[W] and 12[W] resistors. This led to the last term.

Writing the Node-Voltage Equations – 2 Next equation Because of the supernode, we now need another equation for nodes A and B. We get this using the constraint equation given us by the value of the voltage source.

Writing the Node-Voltage Equations – 3 Next equation The equation for Node C is given here. Here we use the elements in series with both the 4[V] and the 18[V] voltage sources to get currents for these branches.

Writing the Node-Voltage Equations – 4 Next equation For Node D, we note that there is a voltage source between this node and the reference node, which sets this node-voltage. Our equation follows from this.

Writing the Node-Voltage Equations – 5 Part 1 Next step Now, we need to write an equation for each of the variables, iQ, iX, vQ, and vX. We’ll take them one at a time. Let’s start with iQ. How can we write an expression for iQ? Think about your answer before going on.

Writing the Node-Voltage Equations – 5 Part 2 Next equation How can we write an expression for iQ? Here, we can choose the reference node, or the D node for a KCL equation. (Note that we did not write the KCL for the D node yet, because of the voltage source.) We will choose the D node to write the equation, because it looks as though it will be simpler. The equation is:

Writing the Node-Voltage Equations – 6 Part 1 Next step Now, let’s write an expression for iX. How can we write an expression for iX? Remember that iX is the current through a wire within the reference node. There is no voltage across this wire. Can we say that iX is zero? Think about your answer before going on.

Writing the Node-Voltage Equations – 6 Part 2 Next step Now, iX is the current through a wire within the reference node. There is no voltage across this wire. Can we say that iX is zero? The answer is NO! The voltage across all wires is zero, but the current through a wire is not generally zero. We can write KCL for the closed surface below, and write the following equation:

Writing the Node-Voltage Equations – 6 Part 3 Next equation Note that because the 6[W] and 8[W] resistors have both ends connected to the same node, the voltage across them must be zero. We have included these terms here for clarity, but obviously they are not needed in the equation.

Writing the Node-Voltage Equations – 7 Part 1 Next step Now, let’s write an expression using vQ. How can we write an expression for vQ? Think about your answer before going on.

Writing the Node-Voltage Equations – 7 Part 2 Next step How can we write an expression for vQ? Here, we write a KVL equation. We need to write KVL around a loop, and we would like to use as simple a loop as we can. We will use the loop shown below. The equation is:

Writing the Node-Voltage Equations – 7 Part 3 Next equation How did we write this expression for vQ? As we go around the loop, the term (vB-vC) is the voltage across the 13[W] resistor. The last term (vB-vC) is the voltage across the 12[W] resistor. We wrote an expression for the current going down through it, as we did in the first equation, and then multiplied it by 12[W] to get the voltage. Note that because this is a current going down through, the voltage polarity that results from Ohm’s Law requires a “-” sign in this equation.

Writing the Node-Voltage Equations – 8 Part 1 Next step Finally, let’s write an expression for vX. How can we write an expression for vX? Think about your answer before going on.

Writing the Node-Voltage Equations – 8 Part 2 Next step How can we write an expression for vX? Here, we can write a KVL equation for the loop shown below. The equation is: This is the voltage across the 11[W] resistor. We get it in terms of the current through that resistor, which is set by the dependent current source.

Writing the Node-Voltage Equations – All The next step is to solve the equations. Let’s solve. Next step

Go to Notes Solving the Node-Voltage Equations We have used MathCAD to solve the eight simultaneous equations. This is shown in a MathCAD file called PWA_Mod03_Prob03_Soln.mcd which should be available in this module. When we solve, we find that vX = 11.7[V].

Weren’t some of those equations really hard to write? • Some of the equations, particularly those for the variables that the dependent sources used, were difficult. They were difficult in the sense that you need to think carefully, taking one step at a time, and paying attention to the signs of each term. • In some cases, it is worth the time it takes to define a few more variables to make the equations easier to write. Note, though, that this means that we will have more equations to solve. • There are two really important things: • Write the same number of independent equations as unknowns. • Get the sign correct in every term. • The node-voltage method just gives us a systematic way of doing these two things. Go back to Overviewslide.

Problems With Assistance Module 3 – Problem 3