Problems With Assistance Module 2 – Problem 4

1. Problems With AssistanceModule 2 – Problem 4 Filename: PWA_Mod02_Prob04.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: • Equivalent Circuits • Series and Parallel Combinations of Resistors • Delta-to-Wye Transformations 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 & 4 • Electric Circuits 6th Ed. by Nilsson and Riedel: Chapter 3 • Basic Engineering Circuit Analysis 6th Ed. by Irwin and Wu: Chapters 2 & 10 • Fundamentals of Electric Circuits by Alexander and Sadiku: Chapter 2 • Introduction to Electric Circuits 2nd Ed. by Dorf: Chapter 3 Next slide

4. Coverage in this Module The material for this problem is covered in this module in the following presentations: • DPKC_Mod02_Part01. Next slide

5. Problem Statement Find the resistance seen at terminals A and B of the circuit shown. Next slide

6. Solution – First Step – Where to Start? Find the resistance seen at terminals A and B of the circuit shown. How should we start this problem? What is the first step? Next slide

7. Find the resistance seen at terminals A and B of the circuit shown. Problem Solution – First Step • We are asked to find the ratio of voltage to current at the terminals; this is the same thing as the resistance. What is the best first step? • Attach a source to terminals A and B. • Define currents and voltages for each of the elements in the circuit. • Write a series of KVL and KCL equations. • Simplify the circuit by removing resistors that do not contribute to the solution. • Combine resistors in series and in parallel to simplify the circuit. • Perform a delta-to-wye transformation.

8. Your Choice for First Step –Attach a Source to the Terminals Find the resistance seen at terminals A and B of the circuit shown. This is not the best choice for the first step. We could indeed apply a source, with the intention of then finding the ratio of the voltage at the terminals to the current through the terminals. This ratio would be the resistance. However, this would require solving the circuit, and we can do the problem more easily. Go back and try again.

9. Your choice for First Step was –Define Currents and Voltages for each of the Elements in the Circuit Find the resistance seen at terminals A and B of the circuit shown. 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. At some point we will need to define variables, but it is best to wait until you have a good idea of which ones you need. Go back and try again.

10. Your Choice for First Step –Write a Series of KVL and KCL Equations Find the resistance seen at terminals A and B of the circuit shown. This is not the best choice for the first step. We could indeed write a set of KVL and KCL equations, once we had defined voltages and currents. However, without sources all voltages and currents would be zero. The only way to meaningfully do this would be to apply a source, and then solve for the ratio of voltage to current at the terminals. There is a better approach. Go back and try again.

11. Your Choice for First Step Was –Simplify the circuit by removing resistors that do not contribute to the solution. Find the resistance seen at terminals A and B of the circuit shown. This one way to describe the best choice for the first step. The goal is to simplify the circuit, to make the solution easier and faster. It turns out that there are resistors in this circuit that do not contribute to the solution. In other words, we can take these resistors out, and the answer will be the same. Certainly, this is a good thing to do when we can, since it will simplify the circuit. The key is to do it correctly. Let’s begin that process.

12. Your Choice for First Step Was –Combine Resistors in Series and in Parallel to Simplify the Circuit Find the resistance seen at terminals A and B of the circuit shown. This one way to describe the best choice for the first step. The goal is to simplify the circuit, to make the solution easier and faster. Since all we really need in this problem is the current through the voltage source, we can get this by converting the circuit connected to the source to a single resistor. We can do with using equivalent circuits, specifically by repeatedly combining resistors in series and parallel. Let’s begin that process.

13. Your Choice for First Step –Perform a Delta-to-Wye Transformation Find the resistance seen at terminals A and B of the circuit shown. This is not the best choice for the first step. We could indeed perform a delta-to-wye transformation here. One possible delta configuration is marked in red in the circuit here. However, this transformation is complicated; generally we are best served by avoiding this unless there is no other choice. There are several other choices here. We recommend that you go back and try again.

14. How to Begin Simplifying This Circuit Find the resistance seen at terminals A and B of the circuit shown. • We have reached this point, by either of two different approaches. Each approach involves simplifying the circuit, and in particular, the circuit “seen” by terminals A and B. Which of the following statements are true? • Resistors R2 and R5 are in series. • Resistors R8 and R10 are in series. • Resistor R1 is in series with an open circuit. • Resistors R8 and R10 are in parallel with a short circuit.

15. Which Statements Were True? Find the resistance seen at terminals A and B of the circuit shown. • It turns out that all of the following statements are true. We are going to start by pointing out what statements c) and d) mean. Pick one to start with. • Resistors R2 and R5 are in series. • Resistors R8 and R10 are in series. • Resistor R1 is in series with an open circuit. • Resistors R8 and R10 are in parallel with a short circuit.

16. In Series with an Open Circuit Find the resistance seen at terminals A and B of the circuit shown. We said that resistor R1 was in series with an open circuit. To understand this, we need to remember that we are finding the resistance seen at terminals A and B. In other words, we are thinking about what would happen if we connected a source to those two terminals. However, we are not going to connect anything to terminal C. This is implied by the wording of the problem, and is very important. Thus, no current will flow through R1, even when a source is connected. We could say that resistor R1 can be replaced with the series equivalent resistance of infinite value. We could also say that the resistor has no current through it, and no voltage across it, so it will have no effect. Either way, the resistor R1 can be removed.

17. In Parallel with a Short Circuit Find the resistance seen at terminals A and B of the circuit shown. We said that resistors R8 and R10 were in parallel with a short circuit. Here, there is a wire across one or more resistors. If we are looking from outside this combination, which we are here, then we can apply the parallel equivalent resistor formula. A zero resistance in parallel with any resistance will give a zero resistance. Stated another way, no voltage will be across R8 and R10, even when a source is connected. We could say that this combination can be replaced with the parallel equivalent resistance of zero value. We could also say that the resistors have no voltage across them, and therefore no current through them. Either way, the resistors R8 and R10 can be removed.

18. Simplifying the Circuit Find the resistance seen at terminals A and B of the circuit shown. • Next, we are going to simplify the circuit. This means that we can remove R1, R8 and R10. Our simplified circuit is shown here. Having made this simplification, we can next work with statement a). • Resistors R2 and R5 are in series. We can replace them with an equivalent resistor. Next slide

19. Series Combination Replacement Find the resistance seen at terminals A and B of the circuit shown. Resistors R2 and R5 were in series. We have replaced them with an equivalent resistor, which we call R11. Next, we recognize that R4 and R11 are in parallel, and that that parallel combination is in series with R3. Performing both of these two steps at once, we can move to the next equivalent, on the next slide, where we include the equivalent resistor R12. Next slide

20. Two More Steps Taken Find the resistance seen at terminals A and B of the circuit shown. • We have shown the equivalent, with resistor R12 in place. Then, the next step is to find replacements for this circuit. Which of the following statements about this circuit are true? • Resistor R7 is in series with the parallel combination of resistors R6 and R12. • Resistor R12 is in parallel with resistor R6. • Resistor R6 is in parallel with resistor R7. • The three resistors R6, R7, and R12 are all in parallel.

21. You Said That Resistor R7 Is in Series With the Parallel Combination of Resistors R6 and R12 Find the resistance seen at terminals A and B of the circuit shown. This is not correct. While resistors R6 and R12 are indeed in parallel, that parallel combination is not in series with R7. This is a fairly subtle point made more difficult for some students by the diagonal wire. Still, we need to get this correct. Here is the point: To be in series, the same current must flow through the series combination. That is not the case here, because the current iA is not zero. Remember, that a source can be connected between terminals A and B. This is not a correct step. Go back and try again. As you look back, you may notice that we said that the current through terminal A was not zero, but the current through terminal C (now removed) was zero. What’s the difference? The difference is that we are looking for the resistance between terminals A and B, which means that we are conceptually placing a source across those two terminals.

22. You Said That Resistor R12 Is in Parallel With Resistor R6 Find the resistance seen at terminals A and B of the circuit shown. This is correct as far as it goes. The resistor R12 is indeed in parallel with R6. However, we will save some time if we recognize that R7 is also in parallel with each of these two other resistors. We can combine all three of them in the same step, and save some time. Remember, however, that the product-over-sum rule does not work for more than two resistors; rather we must use the inverse of the sum of the inverses rule, at right. Let’s use this rule to replace these three resistors by a single resistor, which we will call R13.

23. You Said That Resistor R6 Is in Parallel With Resistor R7 Find the resistance seen at terminals A and B of the circuit shown. This is correct as far as it goes. The resistor R7 is indeed in parallel with R6. However, we will save some time if we recognize that R12 is also in parallel with each of these two other resistors. We can combine all three of them in the same step, and save some time. Remember, however, that the product-over-sum rule does not work for more than two resistors; rather we must use the inverse of the sum of the inverses rule, at right. Let’s use this rule to replace these three resistors by a single resistor, which we will call R13.

24. You Said That The Three Resistors R6, R7, and R12 Are All in Parallel Find the resistance seen at terminals A and B of the circuit shown. This is correct. We can combine all three of them in the same step, and save some time. Remember, however, that the product-over-sum rule does not work for more than one resistor; rather we must use the inverse of the sum of the inverses rule, at right. Let’s use this rule to replace these three resistors by a single resistor, which we will call R13.

25. Combining the Three Resistors R6, R7, and R12 in Parallel Find the resistance seen at terminals A and B of the circuit shown. We have combined all three of them in the same step, replacing these three resistors by a single resistor, which we have called R13. Now, it is probably clear the R9 and R13 are in series, and our final answer is given the next slide.

26. Combining Resistors Yet Again Find the resistance seen at terminals A and B of the circuit shown. We have combined the series resistors, and replaced them with an equivalent resistor, which we called R14. At this point it is clear that the resistance between the terminals A and B, which we will call REQ, is R14, or Go to Comments Slide

27. What if I don’t see that resistors are in series with open circuits, or in parallel with short circuits? • This is a difficult question, because this situation is so common for beginning students. However, any other way of combining these resistors will yield the wrong answer. So, the reply, unfortunately, has to be that you need to reconsider the rules of series and parallel until you do see these things. There is only one right answer, and that is the only answer we want. • Much of this insight comes with practice. Therefore, it is important to practice on simple problems first, and then on more difficult problems, until these issues become clear. Go back to Overviewslide.