Network Theorems: Superposition, Thevenin, Norton

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