Problems With Assistance Module 4 – Problem 5 Filename: PWA_Mod04_Prob05.ppt 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 Go straight to the First Step 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: Equivalent Circuits Thévenin’s Theorem 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 #.# 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 #-# 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 presentation: DPKC_Mod04_Part02 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

Next slide Problem Statement 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 the basic problem. We will take it step by step.

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 Try to decide on the first step before going to the next slide.

Problem Solution – First Step How should we start this problem? What is the first step? Define the open-circuit voltage. Attach a 50[W] resistor to the device. Define the short-circuit current. Model the device using Thévenin’s Theorem. Model the device using Norton’s Theorem. 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. Click on the step that you think should be next.

Your choice for First Step – Define the open-circuit voltage 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 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.

Your choice for First Step – Attach a 50[W] resistor to the device. This is not a good choice for the first step. 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 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.

Your choice for First Step – Define the short-circuit current 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 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.

Your choice for First Step – Model the device using Thévenin’s Theorem 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 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.

Your choice for First Step – Model the device using Norton’s Theorem 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 Norton’s theorem to model the device. Once we have the Norton equivalent, it will be easy to find the voltage across a 50[W] resistor attached to the device. For this particular problem, we have chosen to use Thévenin's Theorem. However, your choice was just as good. 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.

Modeling the Device and Attaching the Voltmeter 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. 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. Now, will this voltage, vTH, be the voltage across the 50[W] resistor? Click on yes or no.

You Said That: Yes, vTH is the voltage across the 50[W] resistor 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 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.

You Said That: No, vTH is not the voltage across the 50[W] resistor 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 answer is 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. Let’s find RTH. To do this, we will apply the ammeter to the device. Let’s apply the ammeter.

Modeling the Device and Attaching the Ammeter 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 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.

Voltage Across RTH To find the voltage across RTH, we write KVL around the loop, and we get 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. 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 Next slide

Attaching 50[W] Resistor 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 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

Power Absorbed by the 50[W] Resistor 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. Finally, with the voltage across the 50[W] resistor, v50, we can find the power absorbed by the resistor. We have Next slide

Go to Comments Slide The Solution Go back to Problem Statement 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

Go back to Overview slide. 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.