Thévenin Equivalent Circuit at Terminals A and B

1. Find the Thévenin Equivalent of the circuit at terminals A and B. Problems With AssistanceModule 4 – Problem 2 Filename: PWA_Mod04_Prob02.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 • Thévenin’s Theorem 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 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

4. Coverage in this Module The material for this problem is covered in this module in the following presentation: • DPKC_Mod04_Part02 Next slide

5. Problem Statement Find the Thévenin Equivalent of the circuit at terminals A and B. Next slide

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

7. Problem Solution – First Step • How should we start this problem? What is the first step? • Define the open-circuit voltage. • Replace vS1 and R2 with a current source in parallel with a resistance. • Define the short-circuit current. • Replace iS1 and R1 with a voltage source in parallel with a resistance. • Define the open-circuit voltage and the short-circuit current. Find the Thévenin Equivalent of the circuit at terminals A and B.

8. Your choice for First Step –Define the open-circuit voltage This is a good choice for the first step. To find a Thévenin Equivalent, we need to find two of three possible items. The open-circuit voltage is one of these three. Looking at the circuit, though, it appears that finding the short-circuit current may be even easier than finding the open circuit voltage, since R4 and R5 will be shorted out when the short is applied. So, even though you made a good choice, we suggest that you try again. Find the Thévenin Equivalent of the circuit at terminals A and B.

9. Your choice for First Step –Replace vS1 and R2 with a current source in parallel with a resistance This is not a good choice. Generally, it is reasonable to apply source transformations to find Thévenin’s or Norton’s equivalents. However, in this case, having a current source in paralle with a resistance in place of vS1 and R2 will not simplify this circuit. It just will not help. Please go back and try again. Find the Thévenin Equivalent of the circuit at terminals A and B.

10. Your choice for First Step –Define the short-circuit current This is a good choice for the first step, and the one that we will choose here. To find a Thévenin Equivalent, we need to find two of three possible items. The short-circuit current is one of these three. Looking at the circuit, it appears that finding the short-circuit current may be even easier than finding the open circuit voltage, since R4 and R5 will be shorted out when the short is applied. So, let’s try this. Find the Thévenin Equivalent of the circuit at terminals A and B.

11. Your choice for First Step was –Replace iS1 and R1 with a voltage source in parallel with a resistance This is not a good choice for the first step. The most important point to make here is that iS1 and R4 are not in parallel, and therefore, we cannot replace them with a voltage source in series with a resistance. This is a common error. Some also attempt to replace these with a voltage source in parallel with a resistor, but this is also not valid. Therefore, we recommend that you go back and try again. Find the Thévenin Equivalent of the circuit at terminals A and B.

12. Your choice for First Step was –Define the open-circuit voltage and the short-circuit current This is not a good choice. The implication of this step is that we should do both things at once, that is, define the open-circuit voltage and the short-circuit current in the same diagram. This is not possible. When we open circuit terminals A and B, there is no current. When we short circuit terminals A and B, there is no voltage. We cannot do both at once. Please go back and try again. Find the Thévenin Equivalent of the circuit at terminals A and B.

13. We have defined in this diagram the short circuit current. It is always important to do this, to show which polarity we are solving for. Next, we are going to simplify the circuit. This step is not necessary, but it makes the solution a little bit easier, so we will do so. Let’s go to the next slide and consider the simplifications possible. Defining the Short-Circuit Current Find the Thévenin Equivalent of the circuit at terminals A and B. Next slide

14. Which of the simplifications listed here are valid and useful? • Remove R4 and R5 since they are in parallel with a short circuit. • Remove R1 since it is in series with a current source. • Remove vS1 and R2 since they are in parallel with the vS2 voltage source. What Simplifications are Possible Here? Find the Thévenin Equivalent of the circuit at terminals A and B. • Choices that you may select: • Just 1. • Just 2. • Just 3. • All of the above. • None of the above.

15. You Chose: Just #1 You said that you could remove R4 and R5 since they are in parallel with a short circuit. This is true. However, there are more simplifications that could be chosen here. Please go back and try again. Find the Thévenin Equivalent of the circuit at terminals A and B.

16. You Chose: Just #2 You said that you could remove R1 since it is in series with a current source. This is true. However, there are more simplifications that could be chosen here. Please go back and try again. Find the Thévenin Equivalent of the circuit at terminals A and B.

17. You Chose: Just #3 You said that you could remove vS1 and R2 since they are in parallel with the vS2 voltage source. This is true. However, there are more simplifications that could be chosen here. Please go back and try again. Find the Thévenin Equivalent of the circuit at terminals A and B.

18. You said that you could • Remove R4 and R5 since they are in parallel with a short circuit. • Remove R1 since it is in series with a current source. • Remove vS1 and R2 since they are in parallel with the vS2 voltage source. • These can all be done. Let’s make these changes. You Chose: All of the Above Find the Thévenin Equivalent of the circuit at terminals A and B.

19. You said that you could not • Remove R4 and R5 since they are in parallel with a short circuit. • Remove R1 since it is in series with a current source. • Remove vS1 and R2 since they are in parallel with the vS2 voltage source. • These can all be done. Let’s make these changes. You Chose: None of the Above Find the Thévenin Equivalent of the circuit at terminals A and B.

20. Making the Simplifications • In the diagram shown, we have • Removed R4 and R5 since they were in parallel with a short circuit. • Removed R1 since it was in series with a current source. • Removed vS1 and R2 since they were in parallel with the vS2 voltage source. • The much simpler circuit results. We can solve for iSC, in the next slide. Find the Thévenin Equivalent of the circuit at terminals A and B. Note that none of these simplifications are needed. You can solve for the short-circuit current in the circuit before the simplifications. This just makes it easier.

21. We can write the KCL for the closed surface shown as a dashed red line. We can solve for iSC, in the equation, as shown below. Finding the Short-Circuit Current Find the Thévenin Equivalent of the circuit at terminals A and B. Note that we have expressed the current through R3 as vS2/R3. Why is this true? Choose your answer below. Because R3 is in series with vS2. Because R3 is next to vS2. Because the voltage vS2 is across R3, due to the short circuit.

22. We have expressed the current through R3 as vS2/R3. Why is this true? You said that it was because R3 is in series with vS2. This is not correct. You Chose: Because R3 is in series with vS2 Find the Thévenin Equivalent of the circuit at terminals A and B. This is not correct because R3 is not in series with vS2. The current source iS1 means that they do not have the same current through them. Go back and try again.

23. We have expressed the current through R3 as vS2/R3. Why is this true? You said that it was because R3 is next to vS2. This is not correct. You Chose: Because R3 is next to vS2 Find the Thévenin Equivalent of the circuit at terminals A and B. This is not correct because simply having R3next to the source vS2 does not yield this current. We could write this expression because of Ohm’s Law. Look at the circuit again, and go back and try again.

24. We have expressed the current through R3 as vS2/R3. Why is this true? You said that it was because the voltage vS2 is across R3. This is the correct answer. You Chose: Because the voltage vS2 is across R3 Find the Thévenin Equivalent of the circuit at terminals A and B. This is correct because the source vS2 being across the resistor R3 means that Ohm’s Law applies here. By shorting terminals A and B, the bottom terminal of the voltage source is the same as the right hand terminal of the resistor. This is why we can write this equation. Let’s take the next step.

25. We have found the short circuit current for the circuit below. Now, what is the next step? What is the Next Step? Find the Thévenin Equivalent of the circuit at terminals A and B. • Choose the next step for this problem. • Find the equivalent resistance, REQ. • Find the open-circuit voltage, vOC. • Find the Norton current, iN. • Convert all the current sources to voltage sources, and solve.

26. Your choice for the Next Step –Find the equivalent resistance, REQ This is a good choice for the next step. In general, we could do either this or find the open-circuit voltage. Here, where there are no dependent sources, it is probably easier to find the equivalent resistance. This is particularly true when we have several independent sources, because we will set all of them equal to zero in the first step of finding REQ. So, we will choose this step. Let’s do it. Find the Thévenin Equivalent of the circuit at terminals A and B.

27. Your choice for the Next Step –Find the open-circuit voltage, vOC This is a good choice for the next step. If we found the open-circuit voltage, we would have everything we need to solve this problem. However, while it is a good choice, it is not the best choice. In this situation, it appears that there is a better choice. Find the better choice, and we will explain why it is better in this case. Go back and try again. Find the Thévenin Equivalent of the circuit at terminals A and B.

28. Your choice for the Next Step –Find the Norton current, iN This is not a good choice for the next step. We are not trying to find the Norton equivalent. If we were, we would note that the short-circuit current, which we already found, is the Norton current. Go back and try again. Find the Thévenin Equivalent of the circuit at terminals A and B.

29. Your choice for the Next Step –Convert all the current sources to voltage sources, and solve This is not a good choice for the next step. We could use source transformations to convert the current sources to voltage sources. However, that would be to go down a different path, when we are very near the answer on the path we have already chosen. Go back and try again. Find the Thévenin Equivalent of the circuit at terminals A and B.

30. Finding the Equivalent Resistance, REQ To find the equivalent resistance, the first step is to set all of the independent sources equal to zero. Let’s do it. Remember that the current sources when set equal to zero become open circuits. The voltage sources when set equal to zero become short circuits. This has been done in the next slide. Find the Thévenin Equivalent of the circuit at terminals A and B.

31. Setting Independent Sources Equal to Zero We have set the independent sources equal to zero. The circuit is much simpler, but it can be simplified further. First, note that resistor R1 is open-circuited. That is, the resistor is in series with an open circuit. It can be removed. Then, note that resistor R2 is short-circuited. That is, the resistor is in parallel with a short circuit. It can be removed. Let’s do both simplifications. Find the Thévenin Equivalent of the circuit at terminals A and B.

32. Simplifying the Circuit After these simplifications, we are left with three resistors in parallel. The solution now is straight-forward. We have an equivalent resistance, which is Find the Thévenin Equivalent of the circuit at terminals A and B. Next slide

33. Finding the Thévenin Voltage With these values, we can find the Thévenin voltage, which is Find the Thévenin Equivalent of the circuit at terminals A and B. Because of the polarity we chose when we found iSC, this will be the voltage at A with respect to B, when these terminals are open-circuited. This leads to the answer on the next slide.

34. Go to Comments Slide The Solution The solution is given in either of the two circuits, shown below. Note that it is important to know which terminal is A, and which is B, so that the polarity of the voltage source can be interpreted correctly. Find the Thévenin Equivalent of the circuit at terminals A and B.

35. Was This Worth It? • This is a good question. However, the best answer is, “It depends.” • We have gone through a fair amount of work, but by doing so we have a simpler circuit. Whether it was worth the work depends on what we were going to use the circuit for. • For example, if we were to connect the circuit to 12 different resistors, or to 12 different current sources, it would be much easier to solve the simpler circuit each time, and in the end it would be worth it. For one resistor, it was probably not a good use of our time. • Note, though, that Thévenin’s Theorem has benefits as a way of thinking about a circuit. This will pay off in many areas, among them when we are designing circuits. Go back to Overviewslide.