Problems With Assistance Module 8 – Problem 2 Go straight to the First Step Filename: PWA_Mod08_Prob02.ppt This problem is adapted from Quiz 6 from the summer of 1998 in ECE 2300 Circuit Analysis, in the Department of Electrical and Computer Engineering at the University of Houston 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: Phasor Analysis Complex Ohm’s Law Equivalent Circuits in Phasor Domain 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 chapters: Circuits by Carlson: Chapter 6 Electric Circuits 6th Ed. by Nilsson and Riedel: Chapter 9 Basic Engineering Circuit Analysis 6th Ed. by Irwin and Wu: Chapter 8 Fundamentals of Electric Circuits by Alexander and Sadiku: Chapter 9 Introduction to Electric Circuits 2nd Ed. by Dorf: Chapter 11 This is the material in your circuit texts that you might consult to get more help on this problem. Next slide

Coverage in this Module The material for this problem is covered in this module in the following presentations: DPKC_Mod08_Part01, DPKC_Mod08_Part02, and DPKC_Mod08_Part03. 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 The circuit given is in steady-state. The current through an unknown device, iD(t) has been measured, and its value is given with the figure. Find the steady-state value of the vD(t). Find a circuit model for the unknown device. Assume that it is a passive device. This is the basic problem. We will take it step by step. Next slide

Solution – First Step – Where to Start? The circuit given is in steady-state. The current through an unknown device, iD(t) has been measured, and its value is given with the figure. Find the steady-state value of the vD(t). Find a circuit model for the unknown device. Assume that it is a passive device. Try to decide on the first step before going to the next slide. How should we start this problem? What is the first step? Next slide

Problem Solution – First Step The circuit given is in steady-state. The current through an unknown device, iD(t) has been measured, and its value is given with the figure. Find the steady-state value of the vD(t). Find a circuit model for the unknown device. Assume that it is a passive device. How should we start this problem? What is the first step? Write KCL for each node Convert the circuit to the phasor domain Write KVL for each loop Click on the step that you think should be next.

Your choice for First Step – Write KCL for each node The circuit given is in steady-state. The current through an unknown device, iD(t) has been measured, and its value is given with the figure. Find the steady-state value of the vD(t). Find a circuit model for the unknown device. Assume that it is a passive device. This is not a good choice for the first step. If we were to write the KCL for each node, we would end up with a set of simultaneous intregral-differential equations. There are easier ways to solve this problem. Go back and try again.

Your choice for First Step – Write KVL for each loop The circuit given is in steady-state. The current through an unknown device, iD(t) has been measured, and its value is given with the figure. Find the steady-state value of the vD(t). Find a circuit model for the unknown device. Assume that it is a passive device. This is not a good choice. If we were to write the KVL for each loop, we would end up with a set of simultaneous intregral-differential equations. There are easier ways to solve this problem. Go back and try again.

Your choice for First Step was – Convert the circuit to the phasor domain The circuit given is in steady-state. The current through an unknown device, iD(t) has been measured, and its value is given with the figure. Find the steady-state value of the vD(t). Find a circuit model for the unknown device. Assume that it is a passive device. This is the best choice for the first step. We should recognize that this is a steady-state problem with sinusoidal sources, which is best attacked with the phasor analysis method. The first step in using this method is to convert the circuit to the phasor domain. This means converting the voltages and currents to phasors, and replacing passive elements with their impedances. Let’s convert.

Convert the circuit to the phasor domain The circuit given is in steady-state. The current through an unknown device, iD(t) has been measured, and its value is given with the figure. Find the steady-state value of the vD(t). Find a circuit model for the unknown device. Assume that it is a passive device. We have converted this circuit to the phasor domain. Note that the voltages and currents are converted to phasors, even where they are unknown, as is the case with Vd(w). Always be careful to redraw this kind of circuit, showing appropriate notation, and not having any time-domain expressions in the same figure with phasor-domain expressions. Next slide

Next Step in the Phasor Domain The circuit given is in steady-state. The current through an unknown device, iD(t) has been measured, and its value is given with the figure. Find the steady-state value of the vD(t). Find a circuit model for the unknown device. Assume that it is a passive device. What is the next step? Use the node-voltage method Use the mesh-current method Use Thevenin’s Theorem Use Source Transformations

Next Step in the Phasor Domain – Use the node voltage method The circuit given is in steady-state. The current through an unknown device, iD(t) has been measured, and its value is given with the figure. Find the steady-state value of the vD(t). Find a circuit model for the unknown device. Assume that it is a passive device. Your choice for the next step was to use the node-voltage method. This is a good choice. There are only two essential nodes, and so only one node-voltage equation is needed. We will use this approach. The node voltage is already defined. We will define the reference node and write the equation.

Next Step in the Phasor Domain – Use the mesh-current method The circuit given is in steady-state. The current through an unknown device, iD(t) has been measured, and its value is given with the figure. Find the steady-state value of the vD(t). Find a circuit model for the unknown device. Assume that it is a passive device. Your choice for the next step was to use the mesh-current method. This is not a good choice, since there are three meshes in this circuit. There are better ways to solve. Go back and try to find another approach.

Next Step in the Phasor Domain – Use Thevenin’s Theorem or Source Transformations The circuit given is in steady-state. The current through an unknown device, iD(t) has been measured, and its value is given with the figure. Find the steady-state value of the vD(t). Find a circuit model for the unknown device. Assume that it is a passive device. You may have already noticed that Source Transformations are simply a special case of Thevenin’s and Norton’s Theorems. Perhaps your textbook does not even mention Source Transformations. Do not allow this to bother you. The two methods are about the same thing. While this is a reasonable approach to this problem, it is probably not optimal. We could take the Thevenin equivalent of the circuit connected to the “Unknown Device”, or use source transformations to get to this same point. However, even then, we would have to write another equation using this equivalent to solve for the voltage. We can get this in one step with another method. Go back and try again.

Using the Node-Voltage Method The circuit given is in steady-state. The current through an unknown device, iD(t) has been measured, and its value is given with the figure. Find the steady-state value of the vD(t). Find a circuit model for the unknown device. Assume that it is a passive device. We have defined the reference node. We just need to write the equation, We can solve this equation for Vd.

Solving the Node-Voltage Equation The circuit given is in steady-state. The current through an unknown device, iD(t) has been measured, and its value is given with the figure. Find the steady-state value of the vD(t). Find a circuit model for the unknown device. Assume that it is a passive device. Solving requires only complex arithmetic, Solving for Vd yields,

Solution for Part a) The circuit given is in steady-state. The current through an unknown device, iD(t) has been measured, and its value is given with the figure. Find the steady-state value of the vD(t). Find a circuit model for the unknown device. Assume that it is a passive device. To get the time domain expression, vD(t), we need to inverse transform and get,

Approach for Part b) The circuit given is in steady-state. The current through an unknown device, iD(t) has been measured, and its value is given with the figure. Find the steady-state value of the vD(t). Find a circuit model for the unknown device. Assume that it is a passive device. We have been told for part b) that the Unknown Device can be modeled using passive circuit elements, that is, using resistors, capacitors or inductors. We can find the ratio of the phasor voltage to phasor current, and get the impedance for the Unknown Device. This may give us some insight.

Solution for Part b) The impedance for the Unknown Device is The circuit given is in steady-state. The current through an unknown device, iD(t) has been measured, and its value is given with the figure. Find the steady-state value of the vD(t). Find a circuit model for the unknown device. Assume that it is a passive device. The impedance for the Unknown Device is This impedance has the form of the impedance of a capacitor, in that it is imaginary and negative. Thus, we can model the Unknown Device with a capacitor, where

How did you know what to do in Part b)? In part b) we have already found the voltage across, and current through, the device. Therefore, we know the impedance of the device. Since the complex version of Thevenin’s Theorem holds, and we were told that it was a passive device, this means that the source portion of Thevenin’s Equivalent must be zero. From this, we can figure that from the impedance, we can determine a model. Since the impedance was imaginary and negative, it must be capacitive. Go to next comments slide. Go back to Overview slide.

How did you know what to do in Part b)? I mean really? We need to remember that modeling is what we do all the time in circuit analysis. Usually, we are given the model, and asked to solve something. However, if we really understand what we are doing, we should be able to go backwards, and take the solution, and determine a model. It is a part of circuit analysis. This solution is not unique. Other models are fine as long as the same impedance results. This one is simply the simplest model with this impedance. Go back to Overview slide.