Problems With Assistance Module 8 – Problem 4

1. Problems With AssistanceModule 8 – Problem 4 Filename: PWA_Mod08_Prob04.ppt This problem is adapted from Quiz #6 from the fall of 1998 in ECE 2300 Circuit Analysis, in the Department of Electrical and Computer Engineering at the University of Houston 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: • Phasor Analysis • Thevenin’s Theorem • Equivalent Circuits in the Phasor Domain 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: Chapter 6 • Electric Circuits 6th Ed. by Nilsson and Riedel: Chapter 9 • Basic Engineering Circuit Analysis 6th Ed. by Irwin and Wu: Chapter 8 • Fundamentals of Electric Circuits by Alexander and Sadiku: Chapter 9 • Introduction to Electric Circuits 2nd Ed. by Dorf: Chapter 11 Next slide

4. Coverage in this Module The material for this problem is covered in this module in the following presentations: • DPKC_Mod08_Part01, DPKC_Mod08_Part02, andDPKC_Mod08_Part03. Next slide

5. Problem Statement • The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s]. • Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. • Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. Next slide

6. Solution – First Step – Where to Start? • The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s]. • Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. • Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. How should we start this problem? What is the first step? Next slide

7. Problem Solution – First Step • The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s]. • Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. • Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. • How should we start this problem? What is the first step? • Apply superposition • Convert the circuit to the phasor domain • Find the open-circuit voltage • Find the short-circuit current

8. Your Choice for First Step –Apply superposition • The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s]. • Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. • Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. This is not a good choice for the first step. Superposition will not help with this problem. Superposition can be applied when we have more than one independent source. We do not even have a single independent source. Go back and try again.

9. Your Choice for First Step –Find the open-circuit voltage • The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s]. • Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. • Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. This is not a good choice. If we were to find the open-circuit voltage in this circuit, we would get zero. There are no independent sources. This would not be helpful, even if we were going to solve in the time domain, which we are not. Go back and try again.

10. Your Choice for First Step –Find the short-circuit current • The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s]. • Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. • Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. This is not a good choice. If we were to find the short-circuit current in this circuit, we would get zero. There are no independent sources. This would not be helpful, even if we were going to solve in the time domain, which we are not. Go back and try again.

11. Your Choice for First Step –Convert the circuit to the phasor domain • The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s]. • Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. • Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. This is a good choice for the first step. Indeed, this should be an easy choice, since we are told to do this explicitly in the problem. Even if we had not been told this, we should know it, since we are asked for an equivalent impedance, which means the phasor domain. Let’s convert.

12. Converting to the Phasor Domain • The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s]. • Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. • Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. We have converted the circuit to the phasor domain, assuming that the angular frequency is 2,000[rad/s]. Notice that the variable for the dependent source, VC, has also been converted to a phasor. Next, we need to find the Thevenin impedance at these two terminals, a and b.

13. Finding the Thevenin Impedance • The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s]. • Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. • Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. • We need to find the Thevenin impedance at the two terminals, a and b. How should we find this value? • Find the open-circuit voltage and short-circuit current, and then take the ratio. • Apply a test source, and find the ratio of the voltage to current. • Replace all the components with their equivalent impedances, and combine using series and parallel impedance rules.

14. Finding the Thevenin Impedance – Voc and Isc • The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s]. • Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. • Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. You suggested that to find the Thevenin impedance at the two terminals, a and b, we should find the open-circuit voltage and short-circuit current, and then take the ratio. Unfortunately, this will not work for this circuit. There are no independent sources, and therefore both of these values will be zero, and the ratio will be undefined. This method will not work for this circuit. Go back and try again.

15. Finding the Thevenin Impedance -- Equivalent Z’s • The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s]. • Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. • Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. You suggested that to find the Thevenin impedance at the two terminals, a and b, we should replace all the components with their equivalent impedances, and combine using series and parallel impedance rules. Unfortunately, this is not a straightforward process for this circuit. There is a dependent source present in the circuit, and it is fairly complicated to come up with an accurate equivalent impedance for this source. Generally, this is not a recommended approach. There is an easier, and more straightforward method. Go back and try again.

16. Finding the Thevenin Impedance – Apply a Test Source • The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s]. • Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. • Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. You suggested that to find the Thevenin impedance at the two terminals, a and b, we should apply a test source, and find the ratio of the voltage to current. This is the approach that we recommend for this kind of circuit. While more complicated than the other methods of finding a Thevenin impedance, this approach will always work. Don’t forget that the first step in this approach is to set all independent sources equal to zero. There are no independent sources in this circuit, so we can apply the test source.

17. Applying a Test Source – 1 • The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s]. • Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. • Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. We can apply the test source. Here, we have chosen to apply a 1[V] voltage source. Any source will do; this is simply a convenient choice for this problem. Our goal is to find the current It, and then take the ratio, Next slide

18. Applying a Test Source – 2 • The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s]. • Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. • Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. To solve, we first find Vc. We can write one equation for Vc using KCL, to write Next slide

19. Applying a Test Source – 3 • The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s]. • Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. • Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. Solve for Vc. Next slide

20. Applying a Test Source – 4 • The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s]. • Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. • Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. With this solution for Vc, we can find It. Next slide

21. Applying a Test Source – 5 • The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s]. • Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. • Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. Finally, we plug in this value for It to get ZThev, which is Next slide

22. Finding a Circuit Model – 1 • The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s]. • Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. • Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. Now, to solve part b, we examine the nature of the equivalent circuit. We found ZThev, and it turned out that the imaginary part was positive. Thus, we can model this impedance with a resistor and an inductor in series. The resistor, Rm, value will be simply the real part of the impedance, and thus Rm = 0.412[W]. This is true because the impedance of a resistor is just equal to the resistance. Now, we need to find the inductance of the inductor. Next slide

23. Finding a Circuit Model – 2 • The circuit given is in steady-state. It is operated at an angular frequency of w = 2,000[rad/s]. • Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. • Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. To find the inductance of the inductor, we recognize that the impedance of the inductor is ZLm = jwLm = 20j[W]. Since we know that w = 2,000[rad/s], we can solve this for Lm and we get Go to Comments Slide

24. Does it matter about the polarity of the test source? • Yes, just as it mattered how we applied the test source in the time domain, it matters how we apply it in the phasor domain. • The voltage and current must be assigned in the passive convention for the circuit, because that is how Ohm’s Law is defined. This means that they must be in the active convention for the source you attach. Go back to Overviewslide.