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Problems With Assistance Module 2 – Problem 8

Problems With Assistance Module 2 – Problem 8. Filename: PWA_Mod02_Prob08.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: Meters. Go straight to the First Step.

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Problems With Assistance Module 2 – Problem 8

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  1. Problems With AssistanceModule 2 – Problem 8 Filename: PWA_Mod02_Prob08.ppt Go straight to the First Step Go straight to the Problem Statement Next slide

  2. Overview of this Problem In this problem, we will use the following concepts: • Meters Go straight to the First Step Go straight to the Problem Statement Next slide

  3. Textbook Coverage The material for this problem is covered in your textbook in the following chapters: • Circuits by Carlson: Chapters 2 & 3 • Electric Circuits 6th Ed. by Nilsson and Riedel: Chapter 3 • Basic Engineering Circuit Analysis 6th Ed. by Irwin and Wu: Chapter 2 • Fundamentals of Electric Circuits by Alexander and Sadiku: Chapter 2 • Introduction to Electric Circuits 2nd Ed. by Dorf: Chapters 1 & 3 Next slide

  4. Coverage in this Module The material for this problem is covered in this module in the following presentations: • DPKC_Mod02_Part03. Next slide

  5. A measurement of an unknown resistor, RX, is attempted using a voltmeter and an ammeter, as shown in the circuit here. The intention is to simultaneously measure the current through the resistor with the ammeter, and the voltage across the resistor with the voltmeter. The resistance will then be obtained from the ratio of these two readings. For this problem, assume that the meter resistance for the ammeter is 100[W], and for the voltmeter is 10[kW]. • Find the ammeter reading, if RX is 2.7[kW], for the circuit given. • Find the voltmeter reading, if RX is 2.7[kW], for the circuit given. • Find the ratio of the voltmeter reading to the ammeter reading. • How much error is there in this resistance measurement? • What resistance is this technique really measuring? Problem Statement Next slide

  6. A measurement of an unknown resistor, RX, is attempted using a voltmeter and an ammeter, as shown in the circuit here. The intention is to simultaneously measure the current through the resistor with the ammeter, and the voltage across the resistor with the voltmeter. The resistance will then be obtained from the ratio of these two readings. For this problem, assume that the meter resistance for the ammeter is 100[W], and for the voltmeter is 10[kW]. • Find the ammeter reading, if RX is 2.7[kW], for the circuit given. • Find the voltmeter reading, if RX is 2.7[kW], for the circuit given. • Find the ratio of the voltmeter reading to the ammeter reading. • How much error is there in this resistance measurement? • What resistance is this technique really measuring? Solution – First Step – Where to Start? How should we start this problem? What is the first step? Next slide

  7. Problem Solution – First Step • A measurement of an unknown resistor, RX, is attempted using a voltmeter and an ammeter, as shown in the circuit here. The intention is to simultaneously measure the current through the resistor with the ammeter, and the voltage across the resistor with the voltmeter. The resistance will then be obtained from the ratio of these two readings. For this problem, assume that the meter resistance for the ammeter is 100[W], and for the voltmeter is 10[kW]. • Find the ammeter reading, if RX is 2.7[kW], for the circuit given. • Find the voltmeter reading, if RX is 2.7[kW], for the circuit given. • Find the ratio of the voltmeter reading to the ammeter reading. • How much error is there in this resistance measurement? • What resistance is this technique really measuring? • How should we start this problem? What is the best first step for part a)? • Replace the meters with their resistances. • Write a system of equations for RX. • Define currents and voltages for each of the elements in the circuit. • Define a variable for the current through the ammeter.

  8. Your Choice for First Step –Replace the Meters With Their Resistances. • A measurement of an unknown resistor, RX, is attempted using a voltmeter and an ammeter, as shown in the circuit here. The intention is to simultaneously measure the current through the resistor with the ammeter, and the voltage across the resistor with the voltmeter. The resistance will then be obtained from the ratio of these two readings. For this problem, assume that the meter resistance for the ammeter is 100[W], and for the voltmeter is 10[kW]. • Find the ammeter reading, if RX is 2.7[kW], for the circuit given. • Find the voltmeter reading, if RX is 2.7[kW], for the circuit given. • Find the ratio of the voltmeter reading to the ammeter reading. • How much error is there in this resistance measurement? • What resistance is this technique really measuring? • This is one of the best choices for the first step. • We will not be able to solve for the ammeter reading without replacing the meters with their resistances. So, this is a good choice. Now, if we do this first, then the second step would be which of the following? • Write a system of equations for RX. • Define currents and voltages for each of the elements in the circuit. • Define a variable for the current through the ammeter.

  9. Your Choice for Second Step –Write a System of Equations for RX • A measurement of an unknown resistor, RX, is attempted using a voltmeter and an ammeter, as shown in the circuit here. The intention is to simultaneously measure the current through the resistor with the ammeter, and the voltage across the resistor with the voltmeter. The resistance will then be obtained from the ratio of these two readings. For this problem, assume that the meter resistance for the ammeter is 100[W], and for the voltmeter is 10[kW]. • Find the ammeter reading, if RX is 2.7[kW], for the circuit given. • Find the voltmeter reading, if RX is 2.7[kW], for the circuit given. • Find the ratio of the voltmeter reading to the ammeter reading. • How much error is there in this resistance measurement? • What resistance is this technique really measuring? This is not a good choice for the second step. After we replace the meters with their resistances, we are not in a position to write a system of equations for RX, since the variables are not defined. Even more important, we already know RX, so we do not need to solve for it. This is not a reasonable second step. Go back and try again.

  10. Your Choice for Second Step –Define Currents and Voltages for Each of the Elements • A measurement of an unknown resistor, RX, is attempted using a voltmeter and an ammeter, as shown in the circuit here. The intention is to simultaneously measure the current through the resistor with the ammeter, and the voltage across the resistor with the voltmeter. The resistance will then be obtained from the ratio of these two readings. For this problem, assume that the meter resistance for the ammeter is 100[W], and for the voltmeter is 10[kW]. • Find the ammeter reading, if RX is 2.7[kW], for the circuit given. • Find the voltmeter reading, if RX is 2.7[kW], for the circuit given. • Find the ratio of the voltmeter reading to the ammeter reading. • How much error is there in this resistance measurement? • What resistance is this technique really measuring? This is not a good choice for the second step. After we replace the meters with their resistances, we do not really need to solve for every voltage and current in the circuit. This is not a reasonable second step. Go back and try again.

  11. Your Choice for Second Step –Define a Variable for the Current Through the Ammeter • A measurement of an unknown resistor, RX, is attempted using a voltmeter and an ammeter, as shown in the circuit here. The intention is to simultaneously measure the current through the resistor with the ammeter, and the voltage across the resistor with the voltmeter. The resistance will then be obtained from the ratio of these two readings. For this problem, assume that the meter resistance for the ammeter is 100[W], and for the voltmeter is 10[kW]. • Find the ammeter reading, if RX is 2.7[kW], for the circuit given. • Find the voltmeter reading, if RX is 2.7[kW], for the circuit given. • Find the ratio of the voltmeter reading to the ammeter reading. • How much error is there in this resistance measurement? • What resistance is this technique really measuring? This is the best choice for the second step. After we replace the meters with their resistances, we need to define a variable to solve for. The key variable is the current through the ammeter, since that current will be the current the ammeter reads. Let’s define this variable, and replace the meters with their resistances.

  12. Your Choice for First Step Was –Write a System of Equations for RX • A measurement of an unknown resistor, RX, is attempted using a voltmeter and an ammeter, as shown in the circuit here. The intention is to simultaneously measure the current through the resistor with the ammeter, and the voltage across the resistor with the voltmeter. The resistance will then be obtained from the ratio of these two readings. For this problem, assume that the meter resistance for the ammeter is 100[W], and for the voltmeter is 10[kW]. • Find the ammeter reading, if RX is 2.7[kW], for the circuit given. • Find the voltmeter reading, if RX is 2.7[kW], for the circuit given. • Find the ratio of the voltmeter reading to the ammeter reading. • How much error is there in this resistance measurement? • What resistance is this technique really measuring? This is not the best possible choice for the first step. We are not in a position to write a system of equations for RX, since the variables are not defined. Even more important, we already know RX, so we do not need to solve for it. This is not a reasonable first step. Go back and try again.

  13. Your Choice for First Step Was –Define Currents and Voltages for Each of the Elements • A measurement of an unknown resistor, RX, is attempted using a voltmeter and an ammeter, as shown in the circuit here. The intention is to simultaneously measure the current through the resistor with the ammeter, and the voltage across the resistor with the voltmeter. The resistance will then be obtained from the ratio of these two readings. For this problem, assume that the meter resistance for the ammeter is 100[W], and for the voltmeter is 10[kW]. • Find the ammeter reading, if RX is 2.7[kW], for the circuit given. • Find the voltmeter reading, if RX is 2.7[kW], for the circuit given. • Find the ratio of the voltmeter reading to the ammeter reading. • How much error is there in this resistance measurement? • What resistance is this technique really measuring? This is not the best choice for the first step. In general, we do like to define currents and voltages. However, if it is clear that we are not going to be using the variables we define, then this is not a good use of our time. In this problem, there is a better approach. Go back and try again.

  14. Your Choice for First Step Was –Define a Variable for the Current Through the Ammeter • A measurement of an unknown resistor, RX, is attempted using a voltmeter and an ammeter, as shown in the circuit here. The intention is to simultaneously measure the current through the resistor with the ammeter, and the voltage across the resistor with the voltmeter. The resistance will then be obtained from the ratio of these two readings. For this problem, assume that the meter resistance for the ammeter is 100[W], and for the voltmeter is 10[kW]. • Find the ammeter reading, if RX is 2.7[kW], for the circuit given. • Find the voltmeter reading, if RX is 2.7[kW], for the circuit given. • Find the ratio of the voltmeter reading to the ammeter reading. • How much error is there in this resistance measurement? • What resistance is this technique really measuring? • This is one of the best possible choice for the first step. It is important to define the variables that we need to solve the problem, and there is no question but that we need to define the current through the ammeter. The current through the ammeter will be the current the ammeter reads. Let’s define this variable. After we do, what will be the second step? • Replace the meters with their resistances. • Write a system of equations for RX. • Define currents and voltages for each of the elements in the circuit.

  15. Your Choice for Second Step –Replace the Meters With Their Resistances. • A measurement of an unknown resistor, RX, is attempted using a voltmeter and an ammeter, as shown in the circuit here. The intention is to simultaneously measure the current through the resistor with the ammeter, and the voltage across the resistor with the voltmeter. The resistance will then be obtained from the ratio of these two readings. For this problem, assume that the meter resistance for the ammeter is 100[W], and for the voltmeter is 10[kW]. • Find the ammeter reading, if RX is 2.7[kW], for the circuit given. • Find the voltmeter reading, if RX is 2.7[kW], for the circuit given. • Find the ratio of the voltmeter reading to the ammeter reading. • How much error is there in this resistance measurement? • What resistance is this technique really measuring? This is the best choice for the second step. We will not be able to solve for the ammeter reading without replacing the meters with their resistances. So, this is a good choice. Let’s define the current through the ammeter, and replace the meters with their resistances.

  16. Your Choice for Second Step –Write a System of Equations for RX • A measurement of an unknown resistor, RX, is attempted using a voltmeter and an ammeter, as shown in the circuit here. The intention is to simultaneously measure the current through the resistor with the ammeter, and the voltage across the resistor with the voltmeter. The resistance will then be obtained from the ratio of these two readings. For this problem, assume that the meter resistance for the ammeter is 100[W], and for the voltmeter is 10[kW]. • Find the ammeter reading, if RX is 2.7[kW], for the circuit given. • Find the voltmeter reading, if RX is 2.7[kW], for the circuit given. • Find the ratio of the voltmeter reading to the ammeter reading. • How much error is there in this resistance measurement? • What resistance is this technique really measuring? This is not a good choice for the second step. After we replace the meters with their resistances, we are not in a position to write a system of equations for RX, since the variables are not defined. Even more important, we already know RX, so we do not need to solve for it. This is not a reasonable second step. Go back and try again.

  17. Your Choice for Second Step –Define Currents and Voltages for Each of the Elements • A measurement of an unknown resistor, RX, is attempted using a voltmeter and an ammeter, as shown in the circuit here. The intention is to simultaneously measure the current through the resistor with the ammeter, and the voltage across the resistor with the voltmeter. The resistance will then be obtained from the ratio of these two readings. For this problem, assume that the meter resistance for the ammeter is 100[W], and for the voltmeter is 10[kW]. • Find the ammeter reading, if RX is 2.7[kW], for the circuit given. • Find the voltmeter reading, if RX is 2.7[kW], for the circuit given. • Find the ratio of the voltmeter reading to the ammeter reading. • How much error is there in this resistance measurement? • What resistance is this technique really measuring? This is not a good choice for the second step. After we replace the meters with their resistances, we do not really need to solve for every voltage and current in the circuit. This is not a reasonable second step. Go back and try again.

  18. Taking the First and Second Steps • A measurement of an unknown resistor, RX, is attempted using a voltmeter and an ammeter, as shown in the circuit here. The intention is to simultaneously measure the current through the resistor with the ammeter, and the voltage across the resistor with the voltmeter. The resistance will then be obtained from the ratio of these two readings. For this problem, assume that the meter resistance for the ammeter is 100[W], and for the voltmeter is 10[kW]. • Find the ammeter reading, if RX is 2.7[kW], for the circuit given. • Find the voltmeter reading, if RX is 2.7[kW], for the circuit given. • Find the ratio of the voltmeter reading to the ammeter reading. • How much error is there in this resistance measurement? • What resistance is this technique really measuring? We decided that we needed to define the current through the ammeter, and replace the meters with their resistances. This is done in the circuit here. We have also inserted the value for RX for part a). We have called the current through the ammeter iAM, the resistance of the ammeter RAM, and the resistance of the voltmeter RVM. Solving for iAM at this point is fairly straightforward. Let’s note that RX and RVM are in parallel, and replace them with their equivalent resistance.

  19. Next slide Replacing the Parallel Combination • A measurement of an unknown resistor, RX, is attempted using a voltmeter and an ammeter, as shown in the circuit here. The intention is to simultaneously measure the current through the resistor with the ammeter, and the voltage across the resistor with the voltmeter. The resistance will then be obtained from the ratio of these two readings. For this problem, assume that the meter resistance for the ammeter is 100[W], and for the voltmeter is 10[kW]. • Find the ammeter reading, if RX is 2.7[kW], for the circuit given. • Find the voltmeter reading, if RX is 2.7[kW], for the circuit given. • Find the ratio of the voltmeter reading to the ammeter reading. • How much error is there in this resistance measurement? • What resistance is this technique really measuring? We have replaced the parallel combination of resistors with their equivalent, which we have called RP. Now we can solve directly for iAM, recognizing that these three resistors are in series, and writing This is the reading of the ammeter, the answer to part a).

  20. Next slide Find the Voltmeter Reading • A measurement of an unknown resistor, RX, is attempted using a voltmeter and an ammeter, as shown in the circuit here. The intention is to simultaneously measure the current through the resistor with the ammeter, and the voltage across the resistor with the voltmeter. The resistance will then be obtained from the ratio of these two readings. For this problem, assume that the meter resistance for the ammeter is 100[W], and for the voltmeter is 10[kW]. • Find the ammeter reading, if RX is 2.7[kW], for the circuit given. • Find the voltmeter reading, if RX is 2.7[kW], for the circuit given. • Find the ratio of the voltmeter reading to the ammeter reading. • How much error is there in this resistance measurement? • What resistance is this technique really measuring? Now, let us recognize that voltage between terminals A and B is the voltage across the voltmeter. If necessary, look back at the original circuit and confirm this. We can find this voltage, which we call vVM, as This is the reading of the voltmeter, the answer to part b).

  21. Next slide Find the Ratio of the Readings • A measurement of an unknown resistor, RX, is attempted using a voltmeter and an ammeter, as shown in the circuit here. The intention is to simultaneously measure the current through the resistor with the ammeter, and the voltage across the resistor with the voltmeter. The resistance will then be obtained from the ratio of these two readings. For this problem, assume that the meter resistance for the ammeter is 100[W], and for the voltmeter is 10[kW]. • Find the ammeter reading, if RX is 2.7[kW], for the circuit given. • Find the voltmeter reading, if RX is 2.7[kW], for the circuit given. • Find the ratio of the voltmeter reading to the ammeter reading. • How much error is there in this resistance measurement? • What resistance is this technique really measuring? We are ready now to solve for part c). We have values for the reading of the voltmeter and the reading of the ammeter, and can easily divide them. However, if we look at the equation we wrote to get the voltmeter reading, we can easily see that if we divide the voltmeter reading vVM by the ammeter reading iAM, we will get the parallel resistance RP, This is the answer to part c).

  22. Next slide Finding the Error • A measurement of an unknown resistor, RX, is attempted using a voltmeter and an ammeter, as shown in the circuit here. The intention is to simultaneously measure the current through the resistor with the ammeter, and the voltage across the resistor with the voltmeter. The resistance will then be obtained from the ratio of these two readings. For this problem, assume that the meter resistance for the ammeter is 100[W], and for the voltmeter is 10[kW]. • Find the ammeter reading, if RX is 2.7[kW], for the circuit given. • Find the voltmeter reading, if RX is 2.7[kW], for the circuit given. • Find the ratio of the voltmeter reading to the ammeter reading. • How much error is there in this resistance measurement? • What resistance is this technique really measuring? The error in the measurement could be expressed in different ways. Clearly the difference between the actual resistance of the resistor, 2.7[kW], and the measurement of the resistance, 2.1[kW], is 0.6[kW]. We could also show the error as a percentage. Doing this, we would say that This is the answer to part d).

  23. Go To Comments Slide What is this Technique Really Measuring? • A measurement of an unknown resistor, RX, is attempted using a voltmeter and an ammeter, as shown in the circuit here. The intention is to simultaneously measure the current through the resistor with the ammeter, and the voltage across the resistor with the voltmeter. The resistance will then be obtained from the ratio of these two readings. For this problem, assume that the meter resistance for the ammeter is 100[W], and for the voltmeter is 10[kW]. • Find the ammeter reading, if RX is 2.7[kW], for the circuit given. • Find the voltmeter reading, if RX is 2.7[kW], for the circuit given. • Find the ratio of the voltmeter reading to the ammeter reading. • How much error is there in this resistance measurement? • What resistance is this technique really measuring? As we saw in step c), the technique is actually measuring RP, which is the parallel combination of the resistor being measured, and the resistance of the voltmeter. This is the answer to part e). We saw that with the values given here, this resulted in an error of about -22%. This would be unacceptable in many situations. The size of the error depends on the relative values of the resistor being measured, and the resistance of the voltmeter. If the voltmeter resistance had been much larger than the resistor value, the error might be negligible.

  24. What is the point of this problem? • The point to this problem is that a relatively simple technique for measuring resistance using a voltmeter and an ammeter, has the potential for errors. The problem in this scheme was that the ammeter did not really measure the current through the resistor being measured, but rather the current through the parallel combination of the resistor and the voltmeter. Go back to Overviewslide. Go To Next Comments Slide

  25. Is there another way? • Yes, it is possible to make this kind of measurement in another way. If we place the ammeter directly in series with the resistor being measured, we will measure the current accurately. However, in this case the voltmeter will measure the voltage across the ammeter and the resistor. Thus, the measurement will yield the series resistance of the resistor and that of the ammeter. Depending on the values of the resistor, and the meter resistances, this might be a better measurement. Go back to Overviewslide.

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