Steady-State Value for vO(t)

1. Problems With AssistanceModule 8 – Problem 5 Filename: PWA_Mod08_Prob05.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: • Phasor Analysis • 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 Find the steady-state value for vO(t). This problem is significant because it represents the model of a transistor amplifier. Don’t worry; you do not need to know anything about transistors to solve this problem. Next slide

6. Solution – First Step – Where to Start? Find the steady-state value for vO(t). How should we start this problem? What is the first step? Next slide

7. Problem Solution – First Step Find the steady-state value for vO(t). • 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 Find the steady-state value for vO(t). This is not a good choice for the first step. Superposition will not help with this problem. Superposition can be useful when we have more than one independent source. We have only one independent source. Go back and try again.

9. Your Choice for First Step –Find the open-circuit voltage Find the steady-state value for vO(t). This is not a good choice. In fact, we are asked to find the open circuit voltage. However, this is the goal, not the first step. Go back and try again.

10. Your Choice for First Step –Find the short-circuit current Find the steady-state value for vO(t). This is not a good choice. The short-circuit current will not help us solve this problem. This would be useful if we were to find the Thevenin Equivalent. We are not trying to do that. Go back and try again.

11. Your Choice for First Step –Convert the circuit to the phasor domain Find the steady-state value for vO(t). This is a good choice for the first step. Since we have only one independent source, and it is sinusoidal, and since we are only interested in the steady-state solution, the Phasor Transform approach is appropriate. Let’s convert.

12. Converting to the Phasor Domain Find the steady-state value for vO(t). We have converted the circuit to the phasor domain, using the angular frequency of 6,000[rad/s]. Notice that the variable for the dependent source, Ib, has also been converted to a phasor. We want to solve for Vo. One way to approach this kind of problem is with the node-voltage method. There are 4 essential nodes, so we will have three simultaneous equations plus one for the dependent source. Let’s set up to write the equations.

13. Phasor Domain Circuit with Node Voltages Find the steady-state value for vO(t). We have already defined the node voltages, but here the reference node is added, along with naming of the essential nodes. These names are chosen to align with the names of the three terminals of the bipolar junction transistor, which are the base (B), the emitter (E), and the collector (C). We are ready to write the equations.

14. Node-Voltage Equations – 1 Find the steady-state value for vO(t). Writing the node-voltage equations, starting with node B, we have Next slide

15. Node-Voltage Equations – 2 Find the steady-state value for vO(t). Writing the node-voltage equations, next using node E, we have Next slide

16. Node-Voltage Equations – 3 Find the steady-state value for vO(t). Writing the node-voltage equations, next using node C, we have Next slide

17. Node-Voltage Equations – 4 Find the steady-state value for vO(t). Finally, we write the equation for the variable Ib, on which the dependent source depends, Next slide

18. Node-Voltage Equations – Solved Our four equations, with four unknowns, are: Our solutions, from MathCAD in file PWA05_Mod08_soln.mcd, are: Vb = (1.99804x10-3 + j3.59802x10-6)[V], Ve = (1.98239x10-3 - j1.13422x10-4)[V], Vc = (-5.0225x10-3 - j0.03741)[V], and Ib = (1.56462x10-8 + j1.1702x10-7)[A]. Next slide

19. Find the steady-state value for vO(t). Solving for Vo We found that Vc = (-5.0225x10-3 - j0.03741)[V]. We can find Vo from this, using the complex version of the voltage divider rule, Next slide

20. Find the steady-state value for vO(t). Solution for Vo Simplifying our answer, we get Next slide

21. Find the steady-state value for vO(t). Solution for vO(t) The answer on the previous slide is the phasor. The steady-state value for vO(t) is Note that this answer is a voltage which is much larger than the input, vS(t). Indeed the magnitude has increased by a factor of 37.8/2. The magnitude of the output is about 19 times bigger than the input. This is the effect of the transistor amplifier. Go to Comments Slide

22. I’m curious. Where was the transistor in this circuit? • We did not have to know where the transistor was in this circuit. However, if you wish to know, we have marked it in the circuit here. This transistor is called a BJT, and has three terminals, called the base, the emitter, and the collector. Go back to Overviewslide.