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Filename: PWA_Mod04_Prob05.ppt Dave Shattuck University of Houston © Brooks/Cole Publishing Co. A device can be modeled with sources and resistors, and has two terminals. When this device is connected to an ideal voltmeter, as shown in Figure 1, the voltmeter reads vX = 11.4[V]. When this same device is disconnected from the voltmeter, and connected to an ideal ammeter, as shown in Figure 2, the ammeter reads iX = 120[mA]. Find the power absorbed by a 50[W] resistor that is connected to this device. Problems With Assistance Module 4 – Problem 5 A Device + vX A Voltmeter Device iX Figure 1 B Figure 2 Go straight to the First Step Go Ammeter straight to the Problem Statement B Next slide Dave Shattuck University of Houston © Brooks/Cole Publishing Co. Overview of this Problem In this problem, we will use the following concepts: • Equivalent Circuits • Thévenin’s Theorem Go straight to the First Step Go straight to the Problem Statement Next slide Dave Shattuck University of Houston © Brooks/Cole Publishing Co. Textbook Coverage The material for this problem is covered in your textbook in the following sections: • Circuits by Carlson: Sections #.# • Electric Circuits 6th Ed. by Nilsson and Riedel: Sections #.# • Basic Engineering Circuit Analysis 6th Ed. by Irwin and Wu: Section #.# • Fundamentals of Electric Circuits by Alexander and Sadiku: Sections #.# • Introduction to Electric Circuits 2nd Ed. by Dorf: Sections #-# Next slide Dave Shattuck University of Houston © Brooks/Cole Publishing Co. Coverage in this Module The material for this problem is covered in this module in the following presentation: • DPKC_Mod04_Part02 Next slide Next slide Dave Shattuck University of Houston Problem Statement © Brooks/Cole Publishing Co. A device can be modeled with sources and resistors, and has two terminals. When this device is connected to an ideal voltmeter, as shown in Figure 1, the voltmeter reads vX = 11.4[V]. When this same device is disconnected from the voltmeter, and connected to an ideal ammeter, as shown in Figure 2, the ammeter reads iX = 120[mA]. Find the power absorbed by a 50[W] resistor that is connected to this device. A Device + vX A Voltmeter Device iX Figure 1 B Figure 2 B Ammeter Dave Shattuck University of Houston © Brooks/Cole Publishing Co. Solution – First Step – Where to Start? How should we start this problem? What is the first step? A device can be modeled with sources and resistors, and has two terminals. When this device is connected to an ideal voltmeter, as shown in Figure 1, the voltmeter reads vX = 11.4[V]. When this same device is disconnected from the voltmeter, and connected to an ideal ammeter, as shown in Figure 2, the ammeter reads iX = 120[mA]. Find the power absorbed by a 50[W] resistor that is connected to this device. Next slide A Device + vX A Voltmeter Device iX Figure 1 B Figure 2 B Ammeter Dave Shattuck University of Houston How should we start this problem? What is the first step? © Brooks/Cole Publishing Co. Problem Solution – First Step A device can be modeled with sources and resistors, and has two terminals. When this device is connected to an ideal voltmeter, as shown in Figure 1, the voltmeter reads vX = 11.4[V]. When this same device is disconnected from the voltmeter, and connected to an ideal ammeter, as shown in Figure 2, the ammeter reads iX = 120[mA]. Find the power absorbed by a 50[W] resistor that is connected to this device. a) Define the open-circuit voltage. b) Attach a 50[W] resistor to the device. c) Define the short-circuit current. d) Model the device using Thévenin’s Theorem. e) Model the device using Norton’s Theorem. A Device + vX A Voltmeter Device iX Figure 1 B Figure 2 B Ammeter Dave Shattuck University of Houston Your choice for First Step – Define the open-circuit voltage © Brooks/Cole Publishing Co. A device can be modeled with sources and resistors, and has two terminals. When this device is connected to an ideal voltmeter, as shown in Figure 1, the voltmeter reads vX = 11.4[V]. When this same device is disconnected from the voltmeter, and connected to an ideal ammeter, as shown in Figure 2, the ammeter reads iX = 120[mA]. Find the power absorbed by a 50[W] resistor that is connected to this device. This is not a good choice for the first step. Actually, the open-circuit voltage has already been defined. The ideal voltmeter is effectively an open circuit. Thus, vX is the open-circuit voltage. It does not need to be defined again. Please go back and try again. A Device + vX A Voltmeter Device iX Figure 1 B Figure 2 B Ammeter Dave Shattuck University of Houston Your choice for First Step – Attach a 50[W] resistor to the device. © Brooks/Cole Publishing Co. This is not a good choice for the first step. A device can be modeled with sources and resistors, and has two terminals. When this device is connected to an ideal voltmeter, as shown in Figure 1, the voltmeter reads vX = 11.4[V]. When this same device is disconnected from the voltmeter, and connected to an ideal ammeter, as shown in Figure 2, the ammeter reads iX = 120[mA]. Find the power absorbed by a 50[W] resistor that is connected to this device. We can attach the 50[W] resistor to the device, and in fact will need to do this later. However, if we do it now, we will not be able to solve the circuit that results. We will not know the voltage across it, as it will not be 11.4[V], and the current through it will not be 120[mA]. Please go back and try again. A Device + vX A Voltmeter Device iX Figure 1 B Figure 2 B Ammeter Dave Shattuck University of Houston Your choice for First Step – Define the short-circuit current © Brooks/Cole Publishing Co. A device can be modeled with sources and resistors, and has two terminals. When this device is connected to an ideal voltmeter, as shown in Figure 1, the voltmeter reads vX = 11.4[V]. When this same device is disconnected from the voltmeter, and connected to an ideal ammeter, as shown in Figure 2, the ammeter reads iX = 120[mA]. Find the power absorbed by a 50[W] resistor that is connected to this device. This is not a good choice for the first step. Actually, the short-circuit current has already been defined. The ideal ammeter is effectively an short circuit. Thus, iX is the short-circuit current. It does not need to be defined again. Please go back and try again. A Device + vX A Voltmeter Device iX Figure 1 B Figure 2 B Ammeter Dave Shattuck University of Houston Your choice for First Step – Model the device using Thévenin’s Theorem © Brooks/Cole Publishing Co. A device can be modeled with sources and resistors, and has two terminals. When this device is connected to an ideal voltmeter, as shown in Figure 1, the voltmeter reads vX = 11.4[V]. When this same device is disconnected from the voltmeter, and connected to an ideal ammeter, as shown in Figure 2, the ammeter reads iX = 120[mA]. Find the power absorbed by a 50[W] resistor that is connected to this device. A Device + vX Voltmeter This is a good choice for the first step. Notice that we are told that the device has two terminals, and it is made up of sources and resistors. Thus, we can use Thévenin's theorem to model the device. Once we have the Thévenin equivalent, it will be easy to find the voltage across a 50[W] resistor attached to the device. Let’s find the equivalent. A Device iX Figure 1 B Figure 2 B Ammeter Your choice for First Step – Model the device using Norton’s Theorem Dave Shattuck University of Houston © Brooks/Cole Publishing Co. This is a good choice for the first step. Notice that we are told that the device has two terminals, and it A device can be modeled with sources and resistors, is made up of sources and resistors. and has two terminals. When this device is Thus, we can use Norton’s theorem connected to an ideal voltmeter, as shown in Figure to model the device. Once we have the Norton equivalent, it will be 1, the voltmeter reads vX = 11.4[V]. When this easy to find the voltage across a same device is disconnected from the voltmeter, 50[W] resistor attached to the and connected to an ideal ammeter, as shown in device. For this particular problem, Figure 2, the ammeter reads we have chosen to use Thévenin's iX = 120[mA]. Find the power absorbed by a 50[W] Theorem. However, your choice resistor that is connected to this device. was just as good. A A Device + vX Voltmeter Device iX Figure 1 B Figure 2 B Ammeter Dave Shattuck University of Houston Modeling the Device and Attaching the Voltmeter © Brooks/Cole Publishing Co. A device can be modeled with sources and resistors, and has two terminals. When this device is connected to an ideal voltmeter, as shown in Figure 1, the voltmeter reads vX = 11.4[V]. When this same device is disconnected from the voltmeter, and connected to an ideal ammeter, as shown in Figure 2, the ammeter reads iX = 120[mA]. Find the power absorbed by a 50[W] resistor that is connected to this device. Device RTH Now, will this voltage, vTH, be the voltage across the 50[W] resistor? Click on yes or no. A + + - vTH vX - Figure 3 B Let’s model the device using a Thévenin equivalent. We have done this in the circuit in Figure 3. Note that when we attach the voltmeter to the device, as in Figure 3, there is no current through RTH, in this case. Thus, vTH = vX = 11.4[V]. In other words, an ideal voltmeter reads the open-circuit voltage. Voltmeter vX vTH 11.4[V]. Dave Shattuck University of Houston You Said That: Yes, vTH is the voltage across the 50[W] resistor © Brooks/Cole Publishing Co. A device can be modeled with sources and resistors, and has two terminals. When this device is connected to an ideal voltmeter, as shown in Figure 1, the voltmeter reads vX = 11.4[V]. When this same device is disconnected from the voltmeter, and connected to an ideal ammeter, as shown in Figure 2, the ammeter reads iX = 120[mA]. Find the power absorbed by a 50[W] resistor that is connected to this device. Device RTH This answer is not correct. When we remove the voltmeter and connect a 50[W] resistor to the device, the voltage drop across RTH will mean that the voltage across the resistor will not be vTH. The correct answer is no, the voltage across the resistor will depend on RTH. A + + - vTH vX - Figure 3 B Voltmeter vX vTH 11.4[V]. Dave Shattuck University of Houston You Said That: No, vTH is not the voltage across the 50[W] resistor © Brooks/Cole Publishing Co. A device can be modeled with sources and resistors, and has two terminals. When this device is connected to an ideal voltmeter, as shown in Figure 1, the voltmeter reads vX = 11.4[V]. When this same device is disconnected from the voltmeter, and connected to an ideal ammeter, as shown in Figure 2, the ammeter reads iX = 120[mA]. Find the power absorbed by a 50[W] resistor that is connected to this device. Device RTH vTH A vX - Figure 3 B When we remove the voltmeter and connect a 50[W] resistor to the device, the voltage drop across RTH will mean that the voltage across the resistor will not be vTH. The correct answer is no, the voltage across the resistor will depend on RTH. Let’s find RTH. To do this, we will apply the ammeter to the device. Let’s apply the ammeter. + + - This answer is correct. Voltmeter vX vTH 11.4[V]. Dave Shattuck University of Houston Modeling the Device and Attaching the Ammeter © Brooks/Cole Publishing Co. A device can be modeled with sources and resistors, and has two terminals. When this device is connected to an ideal voltmeter, as shown in Figure 1, the voltmeter reads vX = 11.4[V]. When this same device is disconnected from the voltmeter, and connected to an ideal ammeter, as shown in Figure 2, the ammeter reads iX = 120[mA]. Find the power absorbed by a 50[W] resistor that is connected to this device. Device + - RTH A iX vTH= 11.4[V] Figure 4 B We have modeled the device using a Thévenin equivalent. Then, we attach the ammeter to the device, as in Figure 4. Our goal here is to solve for RTH. Clearly, the current through RTH in this circuit is iX. The key then is to find the voltage across RTH. Remember that an ideal ammeter has no voltage across it. Thus, by KVL, the voltage across the resistor must be vTH. See the next slide. Ammeter vTH 11.4[V]. Dave Shattuck University of Houston Voltage Across RTH © Brooks/Cole Publishing Co. A device can be modeled with sources and resistors, and has two terminals. When this device is connected to an ideal voltmeter, as shown in Figure 1, the voltmeter reads vX = 11.4[V]. When this same device is disconnected from the voltmeter, and connected to an ideal ammeter, as shown in Figure 2, the ammeter reads iX = 120[mA]. Find the power absorbed by a 50[W] resistor that is connected to this device. Device + + - RTH vRTH A iX + vAM iX vTH= 11.4[V] Ammeter Figure 4 B To find the voltage across RTH, we write KVL around the loop, and we get vTH vRTH v AM 0, and vAM 0, and so vRTH vTH 11.4[V]. Now, we have the voltage across the resistor, and the current through the resistor, and we have them in the active sign convention. Thus, we can write RTH vTH 11.4[V] iX 120[mA] RTH 95[W]. Next slide Dave Shattuck University of Houston Attaching 50[W] Resistor © Brooks/Cole Publishing Co. A device can be modeled with sources and resistors, and has two terminals. When this device is connected to an ideal voltmeter, as shown in Figure 1, the voltmeter reads vX = 11.4[V]. When this same device is disconnected from the voltmeter, and connected to an ideal ammeter, as shown in Figure 2, the ammeter reads iX = 120[mA]. Find the power absorbed by a 50[W] resistor that is connected to this device. Device A + v50 RTH= -95[W] + - 50[W] vTH= 11.4[V] Since we now have the equivalent circuit for the device, we can connect the 50[W] resistor to the equivalent circuit. We get the circuit shown in Figure 5. Now, we can use the VDR to find the voltage across the 50[W] resistor, v50, as 50[W] v50 11.4[V] , or 95[W] 50[W] v50 12.7[V]. Figure 5 B Next slide Dave Shattuck University of Houston Power Absorbed by the 50[W] Resistor © Brooks/Cole Publishing Co. A device can be modeled with sources and resistors, and has two terminals. When this device is connected to an ideal voltmeter, as shown in Figure 1, the voltmeter reads vX = 11.4[V]. When this same device is disconnected from the voltmeter, and connected to an ideal ammeter, as shown in Figure 2, the ammeter reads iX = 120[mA]. Find the power absorbed by a 50[W] resistor that is connected to this device. Device + v50 50[W] vTH= 11.4[V] v50 12.7[V] , or 50[W] 50[W] 3.21[W]. 2 A RTH= -95[W] + - Finally, with the voltage across the 50[W] resistor, v50, we can find the power absorbed by the resistor. We have pabs ,50[ W ] pabs ,50[ W ] 2 Figure 5 B Next slide Go to Comments Slide Dave Shattuck University of Houston The Solution © Brooks/Cole Publishing Co. A device can be modeled with sources and resistors, and has two terminals. When this device is connected to an ideal voltmeter, as shown in Figure 1, the voltmeter reads vX = 11.4[V]. When this same device is disconnected from the voltmeter, and connected to an ideal ammeter, as shown in Figure 2, the ammeter reads iX = 120[mA]. Find the power absorbed by a 50[W] resistor that is connected to this device. Go back to Problem Statement pabs ,50[ W ] 3.21[W]. A Device + vX A Voltmeter Device iX Ammeter Figure 1 B Figure 2 B Next slide Dave Shattuck University of Houston © Brooks/Cole Publishing Co. What Happened Here? • This seemed like another strange problem, with the Thévenin resistance ending up negative. Not only that, but the voltage from the voltage divider ended up being larger in magnitude than the source, and with an opposite sign. Is this really possible? • The answer is yes, but we should remember that to do this we generally need to have a dependent source. Remember that we know almost nothing about the device in this problem. It could easily have a dependent source (an amplifier) inside of it. In some cases, this means a negative resistance can occur. • It is due to the dependent source that we can have a larger magnitude voltage at the terminals than we had at the source of the equivalent. The dependent source can provide positive power. That’s what the negative resistance means. Go back to Overview slide.