1. Problems With Assistance Module 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 Next slide Go straight to the Problem Statement Go straight to the First Step

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 Next slide Go straight to the Problem Statement Go straight to the First Step

3. Textbook Coverage The material for this problem is covered in your textbook in the following chapters: Circuits by Carlson: Chapter 6 Electric Circuits 6 th Ed. by Nilsson and Riedel: Chapter 9 Basic Engineering Circuit Analysis 6 th Ed. by Irwin and Wu: Chapter 8 Fundamentals of Electric Circuits by Alexander and Sadiku: Chapter 9 Introduction to Electric Circuits 2 nd 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, and DPKC_Mod08_Part03. Next slide

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

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

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

8. Your Choice for First Step – Apply superposition 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.try again The circuit given is in steady-state. It is operated at an angular frequency of  = 2,000[rad/s]. a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors.

9. Your Choice for First Step – Find the open-circuit voltage 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.try again The circuit given is in steady-state. It is operated at an angular frequency of  = 2,000[rad/s]. a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors.

10. Your Choice for First Step – Find the short-circuit current 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.try again The circuit given is in steady-state. It is operated at an angular frequency of  = 2,000[rad/s]. a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors.

11. Your Choice for First Step – Convert the circuit to the phasor domain 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.Let’s convert The circuit given is in steady-state. It is operated at an angular frequency of  = 2,000[rad/s]. a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors.

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

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

14. Finding the Thevenin Impedance – V oc and I sc 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.try again The circuit given is in steady-state. It is operated at an angular frequency of  = 2,000[rad/s]. a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors.

15. Finding the Thevenin Impedance -- Equivalent Z’s 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.try again The circuit given is in steady-state. It is operated at an angular frequency of  = 2,000[rad/s]. a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors.

16. Finding the Thevenin Impedance – Apply a Test Source 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.apply the test source The circuit given is in steady-state. It is operated at an angular frequency of  = 2,000[rad/s]. a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors.

17. Applying a Test Source – 1 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 I t, and then take the ratio, The circuit given is in steady-state. It is operated at an angular frequency of  = 2,000[rad/s]. a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. Next slide

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

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

20. Applying a Test Source – 4 With this solution for V c, we can find I t. The circuit given is in steady-state. It is operated at an angular frequency of  = 2,000[rad/s]. a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. Next slide

21. Applying a Test Source – 5 Finally, we plug in this value for I t to get Z Thev, which is The circuit given is in steady-state. It is operated at an angular frequency of  = 2,000[rad/s]. a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. Next slide

22. Finding a Circuit Model – 1 Now, to solve part b, we examine the nature of the equivalent circuit. We found Z Thev, 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, R m, value will be simply the real part of the impedance, and thus R m = 0.412[  ]. 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. The circuit given is in steady-state. It is operated at an angular frequency of  = 2,000[rad/s]. a) Transform the circuit to the phasor domain, and find the Thevenin’s equivalent impedance, as seen at terminals a and b. b) Find an appropriate model for this circuit at this frequency, using a combination of resistors, inductors and capacitors. Next slide

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