Problems With Assistance Module 8 – Problem 5 Filename: PWA_Mod08_Prob05.ppt Next slide Go straight to the Problem Statement Go straight to the First Step

Overview of this Problem In this problem, we will use the following concepts: Phasor Analysis Equivalent Circuits in the Phasor Domain Next slide Go straight to the Problem Statement Go straight to the First Step

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

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

Problem Statement Next slide Find the steady-state value for v O (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.

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

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 Find the steady-state value for v O (t).

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 useful when we have more than one independent source. We have only one independent source. Go back and try again.try again Find the steady-state value for v O (t).

Your Choice for First Step – Find the open-circuit voltage 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.try again Find the steady-state value for v O (t).

Your Choice for First Step – Find the short-circuit current 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.try again Find the steady-state value for v O (t).

Your Choice for First Step – Convert the circuit to the phasor domain 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. Let’s convert Find the steady-state value for v O (t).

Converting to the Phasor Domain 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, I b, has also been converted to a phasor. We want to solve for V o. 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.Let’s set up to write the equations Find the steady-state value for v O (t).

Phasor Domain Circuit with Node Voltages 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.write the equations Find the steady-state value for v O (t).

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

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

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

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

Node-Voltage Equations – Solved Our four equations, with four unknowns, are: Our solutions, from MathCAD in file PWA05_Mod08_soln.mcd, are: V b = ( x j x10 -6 )[V], V e = ( x j x10 -4 )[V], V c = ( x j )[V], and I b = ( x j1.1702x10 -7 )[A]. Next slide

Solving for V o We found that V c = ( x j )[V]. We can find V o from this, using the complex version of the voltage divider rule, Find the steady-state value for v O (t). Next slide

Solution for V o Simplifying our answer, we get Find the steady-state value for v O (t). Next slide

Solution for v O (t) The answer on the previous slide is the phasor. The steady-state value for v O (t) is Note that this answer is a voltage which is much larger than the input, v S (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. Find the steady-state value for v O (t). Go to Comments Slide

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 Overview slide. Overview