Node-Voltage Method for Circuit Analysis

Problems With AssistanceModule 3 – Problem 6 Filename: PWA_Mod03_Prob06.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 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, & PWA_Mod03_Prob03 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. • Use your equations to solve for vX. Problem Statement

How should we start this problem? What is the first step? • 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. • Use your equations to solve for vX. Solution – First Step – Where to Start? 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. Problem Solution – First Step • 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. • Use your equations to solve for vX. Next slide

How many essential nodes are there in this circuit? Your answer is: • 7 essential nodes • 8 essential nodes • 9 essential nodes • 10 essential nodes • 11 essential nodes Identify the Essential Nodes

Your choice for the number of essential nodes – 7 This is not correct. Try again.

Your choice for the number of essential nodes – 8 This is not correct. Try again.

Your choice for the number of essential nodes – 9 This is correct. The essential nodes are marked with red in this schematic. There are three non-essential nodes, which are marked with dark blue. 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 – 10 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.

Your choice for the number of essential nodes – 11 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 lower left as the reference node. There is more than one good choice here. There are three different nodes with five connections. Since many people are familiar with reference nodes being drawn at the bottom of schematics, we choose a bottom node. Next, we define the node-voltages. Choosing the Reference Node

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 a total of ten equations, one for each non-reference essential node, and one for each of the two variables that dependent sources depend on. Defining the Node-Voltages

Writing the Node-Voltage Equations – 1 Next equation The equation for Node A is pretty straightforward. We have a current source, but that does not cause any problems in the Node-Voltage Method. Note that the R1 and R3 resistors are in series, and they add.

Writing the Node-Voltage Equations – 2 Next equation The B node is similar to the A node. We can write the equation fairly easily.

Writing the Node-Voltage Equations – 3 – Part 1 The equation for Node C is more complicated. This is why this circuit is included here, to illustrate what can happen with voltage sources. We have a voltage source, vS1, which is a part of two non-reference essential nodes. According to our rules, this should mean that we will use a supernode. However, if we were to assign a supernode to surround vS1, as shown in the dashed red line, we find that we do not have an expression for the current through the vS2 voltage source. Next slide

Writing the Node-Voltage Equations – 3 – Part 2 To get the equation for Node C, we do the same kind of thing that we do with the supernode approach. The concept is the same. Instead of using a supernode around a single voltage source, we simply extend our closed surface to enclose all of the adjacent voltage sources. Some students have called this a super-duper-node. Whatever. With an extra-large supernode to surround the three voltage sources vS1, vS2, and vS3, as shown in the dashed green line, we can write a KCL equation for that closed surface. Next slide

Writing the Node-Voltage Equations – 3 – Part 3 Let’s write a KCL equation for the closed surface given by the dashed green line. It will be long and complicated, but as long as we take a systematic approach, we should be able to do it. Next equation

Writing the Node-Voltage Equations – 4 Next equation For Node D, we note that in the name we used for the last equation we had C+D+E+F. There are four nodes involved in the supernode equation we just wrote. To be able to solve, we need three more equations. We get one from each voltage source. Starting with the vS1 voltage source, we can write the equation here.

Writing the Node-Voltage Equations – 5 Next equation For the vS2 voltage source, we can write the equation here.

Writing the Node-Voltage Equations – 6 Next equation For the vS3 voltage source, we can write the equation here.

Writing the Node-Voltage Equations – 7 – Part 1 The equation for Node G is also somewhat more involved. We have a voltage source, vS5, which is a part of two non-reference essential nodes. According to our rules, this should mean that we will use a supernode. However, if we were to assign a supernode to surround vS5, as shown in the dashed red line, we find that we do not have an expression for the current through the vS4 voltage source. Next slide

Writing the Node-Voltage Equations – 7 – Part 2 The key for writing an equation for Node G is to recognize that the vS4 voltage source is between the reference node and the G Node. Thus, the vS4 voltage source determines the node voltage. The equation can be written as given here. Next equation

Writing the Node-Voltage Equations – 8 Then, to write an equation for Node H, we use the vS5 voltage source in the same way that we write a constraint equation. The equation can be written as given here. Next equation

Writing the Node-Voltage Equations – 9 – Part 1 Now, we need to write an equation for the variables iX and vX. Let’s start with iX. How can we write an expression for iX? Think about your answer before going on. Next slide

Writing the Node-Voltage Equations – 9 Part 2 Next equation We have already written an expression for iX in our first equation. The equation is:

Writing the Node-Voltage Equations – 10 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 – 10 Part 2 Next step How can we write an expression for vX? There are several ways. Here, we will use the node voltages to get an expression for the voltage across the two series resistors. The voltage vX is then found by using the voltage divider rule with that voltage. The equation is:

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 ten simultaneous equations. This is shown in a MathCAD file called PWA_Mod03_Prob06_Soln.mcd which should be available in this module. When we solve, we find that vA = 13.08191[V] vB = 1.08191[V] vC = 12.08191[V] vD = 6.08191[V] vE = 11.08191[V] vF = 1.08191[V] vG = 0 vH = 12[V] iX = 0.25[A] vX = 0.

Wouldn’t it be better to use the Mesh-Current Method on this circuit? • Yes, it would probably be better to use the Mesh-Current Method with this circuit. One of our equations was quite long, with 11 terms. • However, we have worked this problem using the Node-Voltage Method because we wanted to demonstrate that it can be made to work for any circuit. We showed what should be done when essential nodes are connected by a series of voltage sources. • Remember that a voltage source between essential nodes always determines the relationship between the node voltages for those nodes. Go back to Overviewslide.

Node-Voltage Method for Circuit Analysis