Thévenin’s and Norton’s Theorems Notes Ch1

ECE 2202 Circuit Analysis II Lecture Set #1 Thévenin’s and Norton’s Equivalents and Dependent Sources Overview Thévenin’s and Norton’s Equivalents and Dependent Sources In this part, we will cover the following topics: • Dependent Sources and Equivalent Resistance • The Test-Source Method • Example of finding an equivalent resistance with a dependent source present Textbook Coverage This material is introduced in different ways in different textbooks. Approximately this same material is covered in the Nilsson and Riedel textbook in the following sections: • Electric Circuits 10th Ed., by Nilsson and Riedel: Section 4.11 Thévenin’s and Norton’s Theorems Reviewed Thévenin’s Theorem and Norton’s Theorem can be stated as follows: Any circuit made up of resistors and sources, viewed from two terminals of that circuit, is equivalent to a voltage source in series with a resistance, or to a current source in parallel with a resistance. The voltage source is equal to the open-circuit voltage for the twoterminal circuit, the current source is equal to the short-circuit current for that circuit, and the resistance is equal to the equivalent resistance of that circuit. RTH A Any circuit made up of resistors and sources ~v TH B + - A ~i B A N RN B Extra note We have shown that for the Thevenin equivalent and for the Norton equivalent, the open-circuit voltage is equal to the short-circuit current times the equivalent resistance. This is fundamental and important. vOC  iSC REQ . Equivalent Resistance Reviewed When we find the equivalent resistance for a Thévenin’s equivalent or a Norton’s equivalent, we set the independent sources equal to zero, and find the equivalent resistance of what remains. When a dependent source is present, trying to find the equivalent resistance results in a situation we have not dealt with yet. What do we mean by the equivalent resistance of a dependent source? The answer must be stated carefully. If the ratio of voltage to current for something is a constant, then that something can be said to have an equivalent resistance, since it is behaving as a resistance. RTH A Any circuit made up of resistors and sources ~v TH B + - A ~i B A N RN B Equivalent Resistance of a Source So, what we mean by the equivalent resistance of a dependent source is that in this case the ratio of voltage to current is a constant. Then the source can be said to have an equivalent resistance, since it is behaving as a resistance. The equivalent resistance of a dependent source depends on what voltage or current it depends on, and where that voltage or current is in the circuit. It is not easy to predict the answer. RTH A Any circuit made up of resistors and sources ~v TH B + - A ~i B A N RN B No Equivalent Resistance for an Independent Source The equivalent resistance of a dependent source, in this case, is the ratio of voltage to current, which is a constant. Then the source can be said to have an equivalent resistance, since it is behaving as a resistance. This will only be meaningful for a dependent source. It is not meaningful to talk about the equivalent resistance of an independent source. The ratio of voltage to current will not be constant for an independent source. RTH A Any circuit made up of resistors and sources ~v TH B + - A ~i B A N RN B Simple Example with a Dependent Source We will try to explain this by starting with a simple example. We wish to find the equivalent resistance of the circuit below, as seen at terminals A and B. This will mean that the ratio of the voltage across the circuit, labeled vQ, to the ratio of the current through the circuit, labeled iQ, must be a constant. Let’s find that constant by finding the ratio. Any circuit made up of resistors and dependent sources iQ A iQ A + vQ + vQ ~ iS = 3 iX iX - RX= 100[W] - B B Simple Example with a Dependent Source – Step 1 We wish to find the equivalent resistance of the circuit below, as seen at terminals A and B. Let’s find the ratio of the voltage across the circuit, labeled vQ, to the ratio of the current through the circuit, labeled iQ. This must be a constant. Let’s look first at the circuit equivalent on the right. We note that from Ohm’s Law applied to RX, we can say vQ  iX RX . Next, we apply KCL at the A node to write that iQ  iX  3iX . Any circuit made up of resistors and dependent sources iQ A iQ A + vQ + vQ ~ iS = 3 iX iX - RX= 100[W] - B B