Module 1 – Problem 5

1. Module 1 – Problem 5 Filename: PWA_Mod01_Prob05.ppt This problem is adapted from: Quiz #2 – ECE 2300 – June 17, 1998 Department of Electrical and Computer Engineering University of Houston Houston, TX, 77204-4793 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: • Kirchhoff’s Voltage Law • Kirchhoff’s Current Law • Ohm’s Law 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 1.3 & 1.4 • Electric Circuits 6th Ed. by Nilsson and Riedel: Sections 2.2 & 2.4 • Basic Engineering Circuit Analysis 6th Ed. by Irwin and Wu: Section 2.1 & 2.2 • Fundamentals of Electric Circuits by Alexander and Sadiku: Sections 2.2 & 2.4 • Introduction to Electric Circuits 2nd Ed. by Dorf: Sections 3-2 & 3-3 Next slide

4. Coverage in this Module The material for this problem is covered in this module in the following presentation: • DPKC_Mod01_Part04 A similar problem is worked in: • PWA_Mod01_Prob04 Next slide

5. Problem Statement A device can be modeled using a current source in parallel with a resistor. This device was connected to a 5[V] voltage source, as shown in Figure 1.5a), and a current i5 of 0.93[A] resulted. The same device was then connected to a 4[A] current source as shown in Figure 1.5b), and a voltage v4 of –34[V] resulted. Find the voltage v3 if a 3[W] resistor, as shown in Figure 1.5c), is connected to the same device. Figure 1.5a) Figure 1.5b) Figure 1.5c) Next slide

6. Solution – First Step – Where to Start? A device can be modeled using a current source in parallel with a resistor. This device was connected to a 5[V] voltage source, as shown in Figure 1.5a), and a current i5 of 0.93[A] resulted. The same device was then connected to a 4[A] current source as shown in Figure 1.5b), and a voltage v4 of –34[V] resulted. Find the voltage v3 if a 3[W] resistor, as shown in Figure 1.5c), is connected to the same device. Figure 1.5a) Table 1.4 How should we start this problem? What is the first step? Figure 1.5b) Figure 1.5c) Next slide

7. Problem Solution – First Step A device can be modeled using a current source in parallel with a resistor. This device was connected to a 5[V] voltage source, as shown in Figure 1.5a), and a current i5 of 0.93[A] resulted. The same device was then connected to a 4[A] current source as shown in Figure 1.5b), and a voltage v4 of –34[V] resulted. Find the voltage v3 if a 3[W] resistor, as shown in Figure 1.5c), is connected to the same device. • How should we start this problem? What is the first step? • Find the power delivered by the voltage source and by the current source • Find the power absorbed by the device • Draw the model for the device, with names for the components • Draw a plot of the voltage versus current Figure 1.5a) Figure 1.5c) Figure 1.5b)

8. Your choice for First Step –Find the power delivered by the voltage source and by the current source A device can be modeled using a current source in parallel with a resistor. This device was connected to a 5[V] voltage source, as shown in Figure 1.5a), and a current i5 of 0.93[A] resulted. The same device was then connected to a 4[A] current source as shown in Figure 1.5b), and a voltage v4 of –34[V] resulted. Find the voltage v3 if a 3[W] resistor, as shown in Figure 1.5c), is connected to the same device. This is not a good choice. The power delivered by the voltage source can be found. Since i5 and 5[V] are in the active convention, pdel,5[V] = 5[V]i5 = 4.65[W]. Similarly, for the current source pdel,4[A] = -4[A]v4 = 136[W]. The device is absorbing this power in each case, and the values are not the same. The device is not providing (or absorbing) the same power to everything connected to it, so this information does not help us solve this problem. Go back and try again. Figure 1.5a) Figure 1.5c) Figure 1.5b)

9. Your choice for First Step –Find the power absorbed by the device A device can be modeled using a current source in parallel with a resistor. This device was connected to a 5[V] voltage source, as shown in Figure 1.5a), and a current i5 of 0.93[A] resulted. The same device was then connected to a 4[A] current source as shown in Figure 1.5b), and a voltage v4 of –34[V] resulted. Find the voltage v3 if a 3[W] resistor, as shown in Figure 1.5c), is connected to the same device. This is not a good choice. The power delivered by the voltage source can be found. Since i5 and 5[V] are in the active convention, pdel,5[V] = 5[V]i5 = 4.65[W]. Similarly, for the current source pdel,4[A] = -4[A]v4 = 136[W]. The device is absorbing this power in each case, and the values are not the same. The device is not providing (or absorbing) the same power to everything connected to it, so this information does not help us solve this problem. Go back and try again. Figure 1.5a) Figure 1.5c) Figure 1.5b)

10. Your choice for First Step was –Draw a plot of the voltage versus current This could be helpful, but is not the best choice. The plot of the voltage versus current for this data will be a straight line. However, you may or may not know why this is true. If you don’t know why it is true, then all you really get are two points in an unknown relationship. If you do know why the relationship is a straight line, then this approach will move you towards the answer. For now, let’s assume that you do not know that v vs. i is straight line. Go back and try again. A device can be modeled using a current source in parallel with a resistor. This device was connected to a 5[V] voltage source, as shown in Figure 1.5a), and a current i5 of 0.93[A] resulted. The same device was then connected to a 4[A] current source as shown in Figure 1.5b), and a voltage v4 of –34[V] resulted. Find the voltage v3 if a 3[W] resistor, as shown in Figure 1.5c), is connected to the same device. Figure 1.5a) Figure 1.5c) Figure 1.5b)

11. Your choice for First Step was –Draw the model for the device, with names for the components This is the best choice. The first step is to get this to look like a circuits problem. When it is a circuits problem, we can use our circuits techniques to solve it. This is called modeling. We have been told that we can model the device, and what to use. The key is use this model, and work from there. Let’s try it. A device can be modeled using a current source in parallel with a resistor. This device was connected to a 5[V] voltage source, as shown in Figure 1.5a), and a current i5 of 0.93[A] resulted. The same device was then connected to a 4[A] current source as shown in Figure 1.5b), and a voltage v4 of –34[V] resulted. Find the voltage v3 if a 3[W] resistor, as shown in Figure 1.5c), is connected to the same device. Figure 1.5a) Figure 1.5c) Figure 1.5b)

12. A device can be modeled using a current source in parallel with a resistor. This device was connected to a 5[V] voltage source, as shown in Figure 1.5a), and a current i5 of 0.93[A] resulted. The same device was then connected to a 4[A] current source as shown in Figure 1.5b), and a voltage v4 of –34[V] resulted. Find the voltage v3 if a 3[W] resistor, as shown in Figure 1.5c), is connected to the same device. Drawing the Model for the Device We want a model for the device, and are told that a current source in parallel with a resistance will work. Let’s draw this model, and assign names to the components. The polarity of the current source that we choose does not matter; we just need need to keep that polarity the same with respect to the voltages and currents connected to it. I have called these values iD and RD. I can pick almost anything. The one thing I cannot pick is i5. This name is already in use, and it is a different current. Figure 1.5a) Figure 1.5c) Figure 1.5b) Next step

13. A device can be modeled using a current source in parallel with a resistor. This device was connected to a 5[V] voltage source, as shown in Figure 1.5a), and a current i5 of 0.93[A] resulted. The same device was then connected to a 4[A] current source as shown in Figure 1.5b), and a voltage v4 of –34[V] resulted. Find the voltage v3 if a 3[W] resistor, as shown in Figure 1.5c), is connected to the same device. Drawing the Model for the Device – Note We draw the model, and assign names to the components. We said that the one name we cannot pick for the current source we cannot pick is i5. This name is already in use, and it is a different current. Is this clear? Is iD different from i5? Yes, it is. For example, when we connect the 5[V] voltage source to the device, the current entering the top terminal of the device is i5. However, some of this current goes through the resistor RD. So, iD is different from i5. Figure 1.5a) Figure 1.5c) Figure 1.5b) Next slide

14. Next slide A device can be modeled using a current source in parallel with a resistor. This device was connected to a 5[V] voltage source, as shown in Figure 1.5a), and a current i5 of 0.93[A] resulted. The same device was then connected to a 4[A] current source as shown in Figure 1.5b), and a voltage v4 of –34[V] resulted. Find the voltage v3 if a 3[W] resistor, as shown in Figure 1.5c), is connected to the same device. Connecting the Device to the 5[V] Voltage Source We have attached the 5[V] voltage source. We write KCL at the top node, to get Figure 1.5a) Figure 1.5c) We used Ohm’s Law to write the current in the resistor in terms of the voltage across it, which is 5[V]. Figure 1.5b)

15. Next slide A device can be modeled using a current source in parallel with a resistor. This device was connected to a 5[V] voltage source, as shown in Figure 1.5a), and a current i5 of 0.93[A] resulted. The same device was then connected to a 4[A] current source as shown in Figure 1.5b), and a voltage v4 of –34[V] resulted. Find the voltage v3 if a 3[W] resistor, as shown in Figure 1.5c), is connected to the same device. Connecting the Device to the 4[A] Current Source We have attached the 4[A] current source. We write KCL at the top node, to get Figure 1.5a) Figure 1.5c) We used Ohm’s Law to write the current in the resistor in terms of the voltage across it, which is -34[V]. Figure 1.5b)

16. Next slide 2 Equations and 2 Unknowns A device can be modeled using a current source in parallel with a resistor. This device was connected to a 5[V] voltage source, as shown in Figure 1.5a), and a current i5 of 0.93[A] resulted. The same device was then connected to a 4[A] current source as shown in Figure 1.5b), and a voltage v4 of –34[V] resulted. Find the voltage v3 if a 3[W] resistor, as shown in Figure 1.5c), is connected to the same device. We have obtained two equations with two unknowns. They are: and Thus, we can solve these equations. Figure 1.5a) Figure 1.5c) After some algebra, our solution for the model is: iD = -0.3[A], and RD = 7.9[W]. Figure 1.5b)

17. Next slide Solve for the Voltage Across the 3[W] Resistor A device can be modeled using a current source in parallel with a resistor. This device was connected to a 5[V] voltage source, as shown in Figure 1.5a), and a current i5 of 0.93[A] resulted. The same device was then connected to a 4[A] current source as shown in Figure 1.5b), and a voltage v4 of –34[V] resulted. Find the voltage v3 if a 3[W] resistor, as shown in Figure 1.5c), is connected to the same device. We plug these values in, and we have the device as a circuit model. Now we can connect the device to a 3[W] resistor and solve. Writing KCL, and using Ohm’s Law, we have Figure 1.5a) Figure 1.5c) Figure 1.5b)

18. Next slide Why is the plot of voltage vs. current for the device a straight line? Part 1 A device can be modeled using a current source in parallel with a resistor. This device was connected to a 5[V] voltage source, as shown in Figure 1.5a), and a current i5 of 0.93[A] resulted. The same device was then connected to a 4[A] current source as shown in Figure 1.5b), and a voltage v4 of –34[V] resulted. Find the voltage v3 if a 3[W] resistor, as shown in Figure 1.5c), is connected to the same device. We mentioned that for this problem a plot of the voltage across the device versus the current through the device is a straight line. The key in this problem is understanding why this is true for this device. We can prove this by writing KCL for the device below, having assigned names for the voltage and current. Figure 1.5a) Figure 1.5c) We can write: Figure 1.5b)

19. Next slide Why is the plot of voltage vs. current for the device a straight line? Part 2 A device can be modeled using a current source in parallel with a resistor. This device was connected to a 5[V] voltage source, as shown in Figure 1.5a), and a current i5 of 0.93[A] resulted. The same device was then connected to a 4[A] current source as shown in Figure 1.5b), and a voltage v4 of –34[V] resulted. Find the voltage v3 if a 3[W] resistor, as shown in Figure 1.5c), is connected to the same device. We look at the equation we obtained in this case. As long as the device can be modeled as current source in parallel with a resistor, the equation will have this form. Note, that the equation is the equation for a straight line, as long as iD and RD are known. Figure 1.5a) Figure 1.5c) We can write: Figure 1.5b)

20. Next slide Why is the plot of voltage vs. current for the device a straight line? Part 3 A device can be modeled using a current source in parallel with a resistor. This device was connected to a 5[V] voltage source, as shown in Figure 1.5a), and a current i5 of 0.93[A] resulted. The same device was then connected to a 4[A] current source as shown in Figure 1.5b), and a voltage v4 of –34[V] resulted. Find the voltage v3 if a 3[W] resistor, as shown in Figure 1.5c), is connected to the same device. So, as long as the device can be modeled as current source in parallel with a resistor, the equation will be a straight line. Note that we can write Figure 1.5a) Figure 1.5c) Note that in this case, if we plot vt vs. it, then we have a straight line with a slope of RD and an intercept of iDRD. The signs of the slope and intercept depend on the signs of the values of iD and RD, and on the way we chose polarities for it and vt. Figure 1.5b)

21. Why is the plot of voltage vs. current for the device a straight line? Part 4 We have a straight line with a slope of RD (7.9[W]) and an intercept of iDRD. (-0.3[A].7.9[W] = -2.37[V]) We wrote Go to notes

22. Why do we have to worry about modeling? • Modeling is a very important concept. It really is fundamental to most of the things that we do in circuits. Resistors are models for things where the ratio of voltage to current is constant, but is not exact. We model sources with combinations of resistors and ideal sources. Essentially, everything that we do is a model of reality, which we use because the answers we get for our models are close to the ones for real devices. • In this problem we model a device, and we model the things we connect to the device with resistors. This is a way of thinking we need to be familiar with. Go back to Overviewslide.