Presentation on Thévenin’s and Norton’s Equivalent Circuits and Superposition Theorem, Exercises of Mathematics

Download Presentation on Thévenin’s and Norton’s Equivalent Circuits and Superposition Theorem and more Exercises Mathematics in PDF only on Docsity! 06/02/2016 1 Thévenin’s and Norton’s Equivalent Circuits and Superposition Theorem Thévenin’s and Norton’s Theorems • Thévenin’s Theorem As far as its appearance from outside is concerned, any two terminal network of resistors and energy sources can be replaced by a series combination of an ideal voltage source VOC and a resistor R, where VOC is the open-circuit voltage of the network and R is the resistance that would be measured between the output terminals if the independent energy sources were removed and replaced by their internal resistance. Thévenin’s Voltage VTh is the open-circuit voltage measured at the network output, i.e., VTh = VOC Finding Thévenin’s Voltage (VTh) Thévenin’s Resistance RTh is the resistance that would be measured between the output terminals if the independent energy sources were removed and replaced by their internal resistance (i.e., independent sources are killed). The resistance can be calculated by replacing the load with a test voltage and then measuring the test current after the independent sources are killed. Finding Thévenin’s Resistance (RTh) • Norton’s Theorem As far as its appearance from outside is concerned, any two terminal network of resistors and energy sources can be replaced by a parallel combination of an ideal current source ISC and a resistor R, where ISC is the short-circuit current of the network and R is the resistance that would be measured between the output terminals if the independent energy sources were removed and replaced by their internal resistance Norton’s Current INo is the short-circuit current measured at the network output, i.e., INo = ISC Finding Norton’s Current (INo) 06/02/2016 2 Norton’s Resistance RTh is the resistance that would be measured between the output terminals if the independent energy sources were removed and replaced by their internal resistance (i.e., independent sources are killed). Norton’s Resistance is exactly the same as the Thevenin’s Resistance. The resistance can be calculated by replacing the load with a test voltage and then measuring the test current after the independent sources are killed. Finding Norton’s Resistance (RTh) Relationship between Thévenin’s and Norton’s Theorems From the two equivalent circuits we can deduce the following: Thévenin’s Resistance (RTh) can be also calculated by dividing the open circuit voltage (VOC) by the short circuit current (ISC) measured. You can always replace a Thévenin’s equivalent circuit (i.e., any voltage source) with a Norton’s equivalent circuit (i.e., its equivalent current source). This operation is sometimes called source transformation. Sometimes, one can perform source transformation (i.e., replacing voltage sources with current sources or vice versa) in an electrical circuit in order to simplify the circuit analysis. NOTE: Any resistance in series will contribute the source resistance of a voltage source before transformation. Similarly any resistance in parallel will contribute to the source resistance of the current source before transformation. Determine Thévenin and Norton equivalent circuits of the following circuit. Example: – if nothing is connected across the output no current will flow in R2 so there will be no voltage drop across it. Hence Vo is determined by the voltage source and the potential divider formed by R1 and R3. Hence – if the output is shorted to ground, R2 is in parallel with R3 and the current taken from the source is 30V/15 k = 2 mA. This will divide equally between R2 and R3 so the output current, and so – the resistance in the equivalent circuit is therefore given by – or Solution: – hence equivalent circuits are: Solution: (continued) 06/02/2016 5 Superposition Theorem • Principle of Superposition In any linear network of resistors, voltage sources and current sources, each voltage and current in the circuit is equal to the algebraic sum of the voltages or currents that would be present if each source were to be considered separately. When determining the effects of a single independent source the remaining independent sources are replaced by their internal resistance. IMPORTANT: Dependent sources stay as they are. They are never killed while applying the superposition theorem. In other words, independent voltage and current sources are turned on and off as we apply superposition while dependent sources remain always on. Determine the output voltage V3 in the following circuit using superposition theorem. Example: – First, let us consider the effect of the 15V source alone Solution: – Next consider the effect of the 20V source alone – Finally, the output of the complete circuit is the sum of these two voltages Example : (with dependent sources) For the circuit shown below, determine I1 using the superposition theorem. 06/02/2016 Solution: 1. Let us first Kill the 21 independent voltage souree (nate that depenclent sources always stay on the circ eS | De" th tI. af) | - 2) We noed to fin in the syst nd vol vasa) fest in onder to fire the other ew:ren 4) Firstly, let us use souroe sransformation to replaces 8l-Nortow current sontee with its equivalent The in voltage saree, ie ii) Thea, let us obtain the Thévenia equivalent voltage source and resis tance of the circuit containing 16V-source, 20, 19 and 69 resistors, Le Rn oé ae) Now, we can tine ‘aqsay by solving the follow 1WO7 — Bangs) ~ Laas) ~ 8Tansay ~ 2eagay =O Ligysary = ats) to find one Feaeay © ragaay = A Tange) = LOY 2. Lot us now kill the 8 independent eurrent source (aote that dependent 1b) Now, let us find Vizgiea) a8 Venger) = 4 aagsry +8 Lange) + 2ainay = 9.70 ove, we can fil Aye) as Fallows sources always stay on the circtit) 1a) We need vo find o ) fist in order to find the other currents aad volta 1) First, ot us obtain the Thévenin equivalent voltage source and resis: 2, 19 and 69 resistors, ie. I the AO-resistor Bowing throu 1) Bowing 28 (similar to. Fagg ate (44) rainy = 4 Taney = 145. by Now, let us find Vier Vicueav) = Aaron) + 8 Ferien) + 2taceevy ) Non, we can find Zhy94) a8 follows Vewiawy Viena Mrik ~W-12 Th = Tyesay + Nypavy = 2.4 — 242 = 0.02