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Topics to be Discussed

Next. Topics to be Discussed. Superposition Theorem. Thevenin’s Theorem. Norton’s Theorem. Maximum Power Transfer Theorem. Maximum Power Transfer Theorem for AC Circuits. Millman’s Theorem. Reciprocity Theorem. Tellegen’s Theorem. Next. Network Theorems.

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Topics to be Discussed

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  1. Next Topics to be Discussed • Superposition Theorem. • Thevenin’s Theorem. • Norton’s Theorem. • Maximum Power Transfer Theorem. • Maximum Power Transfer Theorem for AC Circuits. • Millman’s Theorem. • Reciprocity Theorem. • Tellegen’s Theorem. Ch. 4 Network Theorems

  2. Next Network Theorems • Some special techniques, known as network theorems and network reduction methods, have been developed. • These drastically reduce the labour needed to solve a network. • These also provide simple conclusions and good insight into the problems. Ch. 4 Network Theorems

  3. Next Superposition Principle Ch. 4 Network Theorems

  4. Next Superposition Theorem • The response (current or voltage) in a linear network at any point due to multiple sources (current and/or emf) (including linear dependent sources), • can be calculated by summing the effects of each source considered separately, • all other sources “turned OFF” or “made inoperative”. Ch. 4 Network Theorems

  5. Next “Turning off” the sources Ch. 4 Network Theorems

  6. Next Ch. 4 Network Theorems

  7. Next Linear Dependent Source • It is a source whose output current or voltage is proportional only to the first power of some current or voltage variable in the network or to the sum of such quantities. • Examples : Ch. 4 Network Theorems

  8. Next Application • Problem : Consider two 1-V batteries in series with a 1-Ω resistor. Let us apply the principle of superposition, and find the power delivered by both the batteries. • Solutions : Power delivered by only one source working at a time isP1 = 1 W Ch. 4 Network Theorems

  9. Next • Therefore, the power delivered by both the sources, P = 2P1= 2 W • The above answer is obviously wrong, because it is a wrong application of the superposition theorem. Ch. 4 Network Theorems

  10. Find the current I in the network given, using the superposition theorem. Next Example 1 Ch. 4 Network Theorems

  11. Next Solution : Ch. 4 Network Theorems

  12. Next Ch. 4 Network Theorems

  13. Using superposition theorem, find current ix in the network given. Next Example 2 Ch. 4 Network Theorems

  14. Next Solution : Ch. 4 Network Theorems

  15. Next Ch. 4 Network Theorems

  16. Next Ch. 4 Network Theorems

  17. Next Ch. 4 Network Theorems

  18. Next Benchmark Example 3 Find voltage v across 3-Ω resistor by applying the principle of superposition. Ch. 4 Network Theorems

  19. Next Solution : Using current divider, Ch. 4 Network Theorems

  20. Next Using current-divider,the voltage v5 across 3-Ω Ch. 4 Network Theorems

  21. Next By voltage divider, Ch. 4 Network Theorems

  22. Example 4 Find current i2across R2 resistor by applying the principle of superposition. Where R1=R2=R3=1-Ω and VS=10V, Vb= 5V, α = 2. Ch. 4 Network Theorems

  23. Next Thevenin’s Theorem • It was first proposed by a French telegraph engineer, M.L. Thevenin in 1883. • There also exists an earlier statement of the theorem credited to Helmholtz. • Hence it is also known as Helmholtz-Thevenin Theorem. • It is useful when we wish to find the response only in a single resistance in a big network. Ch. 4 Network Theorems

  24. Next Thevenin’s Theorem • Any two terminals AB of a network composed of linear passive and active elements may by replaced by a simple equivalent circuit consisting of • an equivalent voltage sourceVoc,and • an equivalent resistanceRthin series. Ch. 4 Network Theorems

  25. Next • The voltage Voc is equal to the potential difference between the two terminals AB caused by the active network with no external resistance connected to these terminals. • The series resistance Rthis the equivalent resistance looking back into the network at the terminals AB with all the sources within the network made inactive, or dead. Ch. 4 Network Theorems

  26. Using Thevenin’s theorem, find the current in resistor R2 of 2 Ω. Next Illustrative Example 3 Ch. 4 Network Theorems

  27. Solution : Next 1. Designate the resistor R2 as “load”. Ch. 4 Network Theorems

  28. Next 2. Pull out the load resistor and enclose the remaining network within a dotted box. Ch. 4 Network Theorems

  29. Next 3. Temporarily remove the load resistor R2, leaving the terminals A and B open. Ch. 4 Network Theorems

  30. Next 4. Find the open-circuit voltage across the terminals A-B, 5. This is called Thevenin voltage, VTh = VAB = 11.2 V. Ch. 4 Network Theorems

  31. Next 6. Turn OFF all the sources in the circuit Find the resistance between terminals A and B. This is the Thevenin resistance,RTh. Thus, Ch. 4 Network Theorems

  32. Next 7. The circuit within the dotted box is replaced by the Thevenin’s equivalent, consisting of a voltage source of VTh in series with a resistor RTh, Ch. 4 Network Theorems

  33. Next • 8. The load resistor R2 is again connected to Thevenin’s equivalent forming a single-loop circuit. • The current I2 through this resistor is easily calculated, Important Comment The equivalent circuit replaces the circuit within the box only for the effects external to the box. Ch. 4 Network Theorems

  34. Next Example 4 • Using Thevenin’s Theorem, find the current in the ammeter A of resistance 1.5 Ω connected in an unbalanced Wheatstone bridge shown. Ch. 4 Network Theorems

  35. Next Solution : Ch. 4 Network Theorems

  36. Next Ch. 4 Network Theorems

  37. Next • Ans. -1 A Ch. 4 Network Theorems

  38. Next Benchmark Example 5 Again consider our benchmark example to determine voltage across 3-Ω resistor by applying Thevenin’s theorem. Ch. 4 Network Theorems

  39. Next Solution : • We treat the 3-Ω resistor as load. • Thevenin voltage VTh is the open-circuit voltage • (with RL removed). • We use sourcetransformation. Ch. 4 Network Theorems

  40. Next Ch. 4 Network Theorems

  41. Next To compute RTh, we turn off all the sources in the circuit within box and get the circuit Thus, RTh= 3 Ω. Ch. 4 Network Theorems

  42. Next Ch. 4 Network Theorems

  43. Next Thevenin’s Theorem for dependent sources Case-I : When circuit contain both dependent and independent sources. • The open circuit voltage is determined as usual with the sources activated or alive. • A sort circuited is applied across the terminal ab and the value of sort circuit current isc is found as usual. • Now the thevenin’s resistance Rth = Voc/isc Ch. 4 Network Theorems

  44. Next Thevenin’s Theorem for dependent sources Case-II : When circuit contain only dependent sources. • In this case, Voc = 0. • We connect 1A source to terminal ab and calculate the value of Vab. • Now the thevenin’s resistance Rth = Vab/1 Ch. 4 Network Theorems

  45. WORKED EXAMPLE 3 Find Thevenin’s Equivalent circuit across terminal ab. Ch. 4 Network Theorems

  46. Ch. 4 Network Theorems

  47. Next Norton’s Theorem • It is dual of Thevenin’s Theorem. • A two terminal network containing linear passive and active elements can be replaced by an equivalent circuit of a constant-current source in parallel with a resistance. Ch. 4 Network Theorems

  48. Next • The value of the constant-current source is the short-circuit current developed when the terminals of the original network are short circuited. • The parallel resistance is the resistance looking back into the original network with all the sources within the network made inactive (as in Thevenin’s Theorem). Ch. 4 Network Theorems

  49. Next Example 6 • Obtain the Norton’s equivalent circuit with respect to the terminals AB for the network shown, and hence determine the value of the current that would flow through a load resistor of 5 Ω if it were connected across terminals AB. Ch. 4 Network Theorems

  50. Next Solution : When terminals A-B are shorted Ch. 4 Network Theorems

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