superpositiontheorem-2kk00814030159.pptx

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1. Superposition Theorem Prepared By Mhamad Khder Sara Nadr Dlxwaz Aziz Supervidor : Ms. Zainab Pshtiwan Bashdar Anwer

2. Superposition Theorem It is used to find the solution to networks with two or more sources that are not in series or parallel The current through, or voltage across, an element in a linear bilateral network is equal to the algebraic sum of the currents or voltages produced independently by each source. This theorem allows us to find a solution for a current or voltage using only one source at a time . Once we have the solution for each source, we can combine the results to obtain the total solution. If we are to consider the effects of each source, the other sources obviously must be removed.

3. Superposition Theorem Setting a voltage source to zero volts is like placing a short circuit across its terminals. Therefore, when removing a voltage source from a network schematic, replace it with a direct connection (short circuit) of zero ohms. Any internal resistance associated with the source must remain in the network. Fig1. Removing a voltage source to permit the application of the superposition theorem

4. Superposition Theorem Setting a current source to zero amperes is like replacing it with an open circuit. Therefore, when removing a current source from a network schematic, replace it by an open circuit of infinite ohms. Any internal resistance associated with the source must remain in the network. Fig2. Removing a current source to permit the application of the superposition theorem

5. Superposition Theorem Since the effect of each source will be determined independently, the number of networks to be analyzed will equal the number of sources. For a two-source network, if the current produced by one source is in one direction, while that produced by the other is in the opposite direction through the same resistor, the resulting current is the difference of the two and has the direction of the larger. If the individual currents are in the same direction, the resulting current is the sum of two in the direction of either current.

6. Superposition Theorem Similarly, if a particular voltage of a network is to be determined, the contribution to that voltage must be determined for each source. When the effect of each source has been determined, those voltages with the same polarity are added, and those with the opposite polarity are subtracted; the algebraic sum is being determined. The total result has the polarity of the larger sum and the magnitude of the difference.

7. Example 1: Determine the branches current using Superposition Theorem Solution: W e begin by calculating the branch current caused by the voltage source of 120 V. By substituting the ideal current with open circuit, we deactivate the current source, as shown in Figure 4 . 120 V 3 6 12 A 4 2 i 1 i 2 i 3 i 4 Figure 3

8. 120 V 3 6 4 2 i ' 1 i ' 2 i ' 3 i ' 4 v 1 Figure 4 To calculate the branch current, the node voltage across the 3Ω resistor must be known. Therefore v 1 120 v 1 v 1 = 6 3 2 4 where v 1 = 30 V Example1

9. 1 2 30 6 30 = 15 A i' 2 = 3 30 = 10 A i ' 3 = i ' 4 = 6 = 5 A In order to calculate the current cause by the current source, we deactivate the ideal voltage source with a short circuit, as shown in figure 5 3 6 12 A 4 2 1 i " 2 i " i 3 " 4 i " i' 1 = Figure 5 Example1 The equations for the current in each branch,

10. To determine the branch current, solve the node voltages across the 3Ω an d 4Ω resistors as shown in figure 6. The two node voltages are v 3 & v 4 . 12 A 6 2 + + v 3 3 v 4 4 - - 2 2 4 v 3 v 3 v 3 v 4 3 6 v 4 v 3 v 4 12 = = Figure 6 Example1

11. By solving these equations, we obtain v 3 = -12 V v 4 = -24 V Now we can find the branches current as shown in figure 5 . Example1

12. To find the actual current of the circuit, add the currents due to both the current source and voltage source. For reference, figure 4 and figure 5 are shown on right side. Figure 4 Figure 5 Example1

13. Example2. Using the superposition theorem, determine the current through the 12 Ω resistor in Fig 7 . Considering the effects of the 54 V source requires replacing the 48 V source by a short-circuit equivalent as shown in Fig. 8 Figure 7

14. Figure 8. Using the superposition theorem to determine the effect of the 54 V voltage source on current I 2 The result is that the 12 Ω and 4 Ω resistors are in parallel. The total resistance seen by the source is therefore, R T = R 1 + R 2 || R 3 = 24 + 12 || 4 = 24 + 3 = 27 Ω Example2

15. and the source current is Using the current divider rule results in the contribution to I 2 due to the 54 V source: = Now replace the 54 V source by a short-circuit equivalent, the network in Fig. 9 results. The result is a parallel connection for the 12 and 24 resistors.   Example2

16. Therefore, the total resistance seen by the 48 V source is R T = R 3 + R 2 || R 1 = 4 + 12 || 24 = 4 + 8 = 12   Figure9. Using the superposition theorem to determine the effect of the 48 V voltage source on current I 2 Example2

17. And the source current is Apply the current divider rule results in = Current due to each source has a different direction, as shown in Fig 11. The net current therefore is the difference of the two and in the direction of the larger as follows: = - = 2.67 A - 0.5 A = 2.17 A   Figure 10. Using the results of Fig8 and Fig9 to determine current   Example2

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