Lecture 8: Linearity and Equivalent CircuitsEvery circuit which is composed of ideal independent voltage and current sources, linear dependent sources, and resistors, has a linear I-V relationship.There is a simpler circuit with the same I-V relationship.IvThevenin Equivalent CircuitThe Thevenin equivalent circuit is composed of a voltage source in series with a resistor:It can model any circuit except a pure independent current source, through choice of VTand RT.IvVTH-VTH/RTHabNorton Equivalent CircuitThe Norton equivalent circuit is composed of a current source in parallel with a resistor:It can model any circuit except a pure independent voltage source, through choice of INand RN.IvINRN-INabTwo Points Define a LineTo find the Thevenin or Norton equivalent for a circuit, all we need to do is: Find two points on the I-V graph for the circuit. Set the voltage V and find the corresponding I Set the current I and find the corresponding V Find the x-intercept and y-intercept of the graph. Find the VTHand RTH, or the INand RNthat replicate this line.Our Favorite Two Points on the I-V Graph We can find the x-intercept directly by finding the V that occurs when I = 0. This means finding the V that occurs when there is air between the circuit terminals. This voltage is called the open-circuit voltage, VOC. VTH= INRN= VOC We can find the y-intercept directly by finding the I that occurs when V = 0. This means finding the I that occurs when there is a wire between the circuit terminals. This current is called the short-circuit current, ISC. IN= VTH/ RTH= -ISCUseful IdentitiesIvINRN-INIvVTH-VTH/RTHVTH= INRNIN= VTH/ RTHRTH= RNRN= VTH/ INRTH= VTH/ INExample (Nilsson & Riedel text)Find the Thevenin and Norton circuits.Example (Nilsson & Riedel text)Find the Theveninand Norton circuits.VTHand INCome From Independent Sources If there are no independent voltage or current sources in a circuit, VTH= 0 V and IN= 0 A. If there is no independent voltage or current present in a circuit (only resistors and linear dependent sources), all currents and voltages in the circuit are zero. In this situation, you know that the I-V graph goes through the origin. However, the slope of the graph, 1/RTH, still must be determined. It cannot be found using RTH= VTH/ IN.No Independent Sources? Test for RTH A simple example of a circuit with no independent sources is a resistor. One cannot determine the resistance by measuring voltage and current—a resistor has no voltage or current on its own. An ohmmeter applies a test voltage and measures the resulting current to find resistance. Do the same to find RTH: Set V using an independent voltage source, and measure I. Or, set I using an independent current source, and measure V. RTH= V / I Here, you are finding an additional point on the I-V graph.ExampleFind the Thevenin and Norton circuits.RTHComes From Resistors and Linear Dependent Sources The value of RTHdoes not depend on the values of independent voltage and current sources in a circuit. I can turn a 12 V source into a -12 V source, or a 0 V source, and the value of RTHremains the same. When looking for RTHin a circuit that has no dependent sources, it is often easier to: Turn off all independent sources (change voltage sources to 0 V wire and current sources to 0 A air) Simplify remaining resistors using series/parallel combinations to find RTHExampleFind RTH.Source TransformationsOne can change back and forth between Thevenin and Norton:==Source TransformationsOne can use source transformations to simplify a circuit just like using series/parallel rules to simplify resistors. Remember that:Example (Nilsson & Riedel text)Find the Thevenin and Norton
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