Source Transformations 1

ECE 2201 Circuit Analysis Lecture Set #7 Source Transformations Version 18 Overview Source Transformations In this lecture set, we will cover the following topics: • Equivalent circuits review • Source transformations • Some additional transformations • Example of source transformations Textbook Coverage This material is covered in your textbook in the following sections: • Electric Circuits 8th Ed. by Nilsson and Riedel: Section 4.9 Equivalent Circuits Review Source Transformations are another example of equivalent circuits. To use them well, it is important that we follow the rules for equivalent circuits. This will help us avoid problems. Equivalent Circuits: A Definition Imagine that we have a circuit, and a portion of the circuit can be identified, made up of one or more parts. That portion can be replaced with another set of components, if we do it properly. We call these portions equivalent circuits. Two circuits are considered to be equivalent if they behave the same with respect to the things to which they are connected. One can replace one circuit with another circuit, and everything else cannot tell the difference. We will use a metaphor for equivalent circuits here. This metaphor is that of jigsaw puzzle pieces. The idea is that two different jigsaw puzzle pieces with the same shape can be thought of as equivalent, even though they are different. The rest of the puzzle does not “notice” a difference. This is analogous to the case with equivalent circuits. Equivalent Circuits: A Caution Two circuits are considered to be equivalent if they behave the same with respect to the things to which they are connected. The properties (voltage, current, power) within the circuit may be different. It is important to keep this concept in mind. A common error for beginners is to assume that voltages, currents, or powers within a pair of equivalent circuits are equal. They may not be. These voltages, currents, or powers are only required to be equal if they can be identified outside the equivalent circuit. Go back to Overview slide. Source Transformations Defined The equivalent circuits called the source transformation can be defined as follows: A portion of a circuit where we have a voltage source in series with a resistance is equivalent to current source in parallel with a resistance. The resistances for these two equivalents are equal. These two cases are equivalent as long as the resistances are equal and if the voltage source and current source are related by vS  iS REQ . REQ vS A + A ~ B iS REQ B Notation A portion of a circuit where we have a voltage source in series with a resistance is equivalent to current source in parallel with a resistance. The resistances for these two equivalents are equal. These two cases are equivalent as long as the resistances are equal and if the voltage source and current source are related by vS  iS REQ . We have used the symbol “~” to indicate equivalence here. Some textbooks use a doublesided arrow ( or ), or even a single-sided arrow ( or ), to indicate this same thing. REQ vS A + A ~ B iS REQ B Note 1 A portion of a circuit where we have a voltage source in series with a resistance is equivalent to current source in parallel with a resistance. The resistances for these two equivalents are equal. These two cases are equivalent as long as the resistances are equal and if the voltage source and current source are related by vS  iS REQ . This equivalence can go in either direction. That is, we can replace the circuit on the right with the one on the left, or the other way around. Neither one is simpler; we just prefer one or the other in some situations. REQ vS A + A ~ B iS REQ B Note 2 A portion of a circuit where we have a voltage source in series with a resistance is equivalent to current source in parallel with a resistance. The resistances for these two equivalents are equal. These two cases are equivalent as long as the resistances are equal and if the voltage source and current source are related by This equation is not really Ohm’s Law. It looks like Ohm’s Law, and has the same form. However, it should be noted that Ohm’s Law relates voltage and current for a resistor. This relates the values of sources and resistances in two different equivalent circuits. However, if you wish to remember this by relating it to Ohm’s Law, that is fine. vS  iS REQ . REQ vS A + A ~ B iS REQ B Note 3 A portion of a circuit where we have a voltage source in series with a resistance is equivalent to current source in parallel with a resistance. The resistances for these two equivalents are equal. These two cases are equivalent as long as the resistances are equal and if the voltage source and current source are related by The polarities of the sources with respect to the terminals is important. If the reference polarity for the voltage source is as given here (voltage drop from A to B), then the reference polarity for the current source must be as given here (current flow from B to A). This is one good reason for naming the terminals of these equivalents. vS  iS REQ . REQ vS A + A ~ B iS REQ B Note 4 A portion of a circuit where we have a voltage source in series with a resistance is equivalent to current source in parallel with a resistance. The resistances for these two equivalents are equal. These two cases are equivalent as long as the resistances are equal and if the voltage source and current source are related by As with all equivalent circuits, these two are equivalent only with respect to the things connected to the equivalent circuits. For example, when these two equivalent circuits are connected to an open circuit, in one the resistor dissipates power, and in the other it does not. vS  iS REQ . REQ vS A + A ~ B iS REQ B Note 5 Go back to Overview slide. A portion of a circuit where we have a voltage source in series with a resistance is equivalent to current source in parallel with a resistance. The resistances for these two equivalents are equal. These two cases are equivalent as long as the resistances are equal and if the voltage source and current source are related by These equivalent circuits hold for dependent sources as well as independent sources. The key is that the variable, which the dependent sources depend on, must remain intact. That is, the voltage or current that the dependent sources use must be outside of the circuit being replaced. vS  iS REQ . REQ vS= m vX A A ~ + - iS= (m/REQ)vX REQ B + vX - B + vX - Other Useful Transformations There are some other useful transformations, relating to sources, that can be defined at this point. These transformations do not have a common name, and in a sense they derive from the definitions of ideal voltage sources and ideal current sources. Still, since they are equivalent circuits relating sources, that have much the same form as source transformations, they are listed in the slides that follow.