1. ECEN 301Discussion #9 – Equivalent Circuits1 Equivalence - Equality Mosiah 29: 38 38 Therefore they relinquished their desires for a king, and became exceedingly anxious that every man should have an equal chance throughout all the land; yea, and every man expressed a willingness to answer for his own sins.

2. ECEN 301Discussion #9 – Equivalent Circuits2 Current Sources All current sources can be modeled as voltage sources (and vice-versa) Many sources are best modeled as voltage sources (batteries, electric outlets etc.) There are some things that are best modeled as current sources: Van de Graaff generator Behaves as a current source because of its very high output voltage coupled with its very high output resistance and so it supplies the same few microamps at any output voltage up to hundreds of thousands of volts

3. ECEN 301Discussion #9 – Equivalent Circuits3 Lecture 9 – Equivalent Circuits Thévenin Equivalent Norton Equivalent

4. ECEN 301Discussion #9 – Equivalent Circuits4 Network Analysis Network Analysis Methods: Node voltage method Mesh current method Superposition Equivalent circuits Source transformation Thévenin equivalent Norton equivalent

5. ECEN 301Discussion #9 – Equivalent Circuits5 Equivalent Circuits It is always possible to view a complicated circuit in terms of a much simpler equivalent source and equivalent load circuit. vava R1R1 R3R3 +–+– R 2 R5R5 R 4 i +v–+v– Load +v–+v– i Source

6. ECEN 301Discussion #9 – Equivalent Circuits6 Equivalent Circuits It is always possible to view a complicated circuit in terms of a much simpler equivalent source and equivalent load circuit. vava R1R1 R3R3 +–+– R 2 R5R5 R 4 i +v–+v– Load +v–+v– i Source

7. ECEN 301Discussion #9 – Equivalent Circuits7 Equivalent Circuits Equivalent circuits fall into one of two classes: Thévenin: voltage source v T and series resistor R T Norton: current source i N and parallel resistor R N vTvT +–+– RTRT Load +v–+v– i iNiN RNRN +v–+v– i Thévenin Equivalent Norton Equivalent NB: R T = R N

8. ECEN 301Discussion #9 – Equivalent Circuits8 Equivalent Circuits Thévenin Theorem: when viewed from the load, any network comprised of independent sources and linear elements (resistors), may be represented by an equivalent circuit.  Equivalent circuit consists of an ideal voltage source v T in series with an equivalent resistance R T vTvT +–+– RTRT Load +v–+v– i “View from load” A fancy way of saying: “The circuit that includes everything except for the load”

9. ECEN 301Discussion #9 – Equivalent Circuits9 Equivalent Circuits Norton Theorem: when viewed from the load, any network comprised of independent sources and linear elements (resistors), may be represented by an equivalent circuit.  Equivalent circuit consists of an ideal current source i N in parallel with an equivalent resistance R N “View from load” A fancy way of saying: “The circuit that includes everything except for the load” iNiN RNRN Load +v–+v– i

10. ECEN 301Discussion #9 – Equivalent Circuits10 Thévenin and Norton Resistances Computation of Thévenin and Norton Resistances: 1.Remove the load (open circuit at load terminal) 2.Zero all independent sources Voltage sources short circuit (v = 0) Current sourcesopen circuit (i = 0) 3.Compute equivalent resistance (with load removed)

11. ECEN 301Discussion #9 – Equivalent Circuits11 Thévenin and Norton Resistances Example1: find the equivalent resistance as seen by the load R L i s =0.5A, v s = 10V, R 1 = 4Ω, R 2 = 6Ω, R 3 = 10Ω, R 4 = 2Ω, R 5 = 2Ω, R 6 = 3Ω, R 7 = 5Ω isis R1R1 R3R3 R 2 R6R6 R 5 vsvs +–+– R 4 R 7 RLRL

12. ECEN 301Discussion #9 – Equivalent Circuits12 Thévenin and Norton Resistances Example1: find the equivalent resistance as seen by the load R L i s =0.5A, v s = 10V, R 1 = 4Ω, R 2 = 6Ω, R 3 = 10Ω, R 4 = 2Ω, R 5 = 2Ω, R 6 = 3Ω, R 7 = 5Ω isis R1R1 R3R3 R 2 R6R6 R 5 vsvs +–+– R 4 R 7 1.Remove the load

13. ECEN 301Discussion #9 – Equivalent Circuits13 Thévenin and Norton Resistances Example1: find the equivalent resistance as seen by the load R L i s =0.5A, v s = 10V, R 1 = 4Ω, R 2 = 6Ω, R 3 = 10Ω, R 4 = 2Ω, R 5 = 2Ω, R 6 = 3Ω, R 7 = 5Ω R1R1 R3R3 R 2 R6R6 R 5 R 4 R 7 2.Zero all independent sources short circuit voltage sources Open circuit current sources

14. ECEN 301Discussion #9 – Equivalent Circuits14 Thévenin and Norton Resistances Example1: find the equivalent resistance as seen by the load R L i s =0.5A, v s = 10V, R 1 = 4Ω, R 2 = 6Ω, R 3 = 10Ω, R 4 = 2Ω, R 5 = 2Ω, R 6 = 3Ω, R 7 = 5Ω R1R1 R3R3 R 2 R6R6 R 5 R 4 R 7 3.Compute equivalent resistance

15. ECEN 301Discussion #9 – Equivalent Circuits15 Thévenin and Norton Resistances Example1: find the equivalent resistance as seen by the load R L i s =0.5A, v s = 10V, R 1 = 4Ω, R 2 = 6Ω, R 3 = 10Ω, R 4 = 2Ω, R 5 = 2Ω, R 6 = 3Ω, R 7 = 5Ω R EQ1 R3R3 R6R6 R EQ2 R 7 3.Compute equivalent resistance

16. ECEN 301Discussion #9 – Equivalent Circuits16 Thévenin and Norton Resistances Example1: find the equivalent resistance as seen by the load R L i s =0.5A, v s = 10V, R 1 = 4Ω, R 2 = 6Ω, R 3 = 10Ω, R 4 = 2Ω, R 5 = 2Ω, R 6 = 3Ω, R 7 = 5Ω R EQ3 R6R6 R EQ2 R 7 3.Compute equivalent resistance

17. ECEN 301Discussion #9 – Equivalent Circuits17 Thévenin and Norton Resistances Example1: find the equivalent resistance as seen by the load R L i s =0.5A, v s = 10V, R 1 = 4Ω, R 2 = 6Ω, R 3 = 10Ω, R 4 = 2Ω, R 5 = 2Ω, R 6 = 3Ω, R 7 = 5Ω R EQ4 R6R6 R 7 3.Compute equivalent resistance

18. ECEN 301Discussion #9 – Equivalent Circuits18 Thévenin and Norton Resistances Example1: find the equivalent resistance as seen by the load R L i s =0.5A, v s = 10V, R 1 = 4Ω, R 2 = 6Ω, R 3 = 10Ω, R 4 = 2Ω, R 5 = 2Ω, R 6 = 3Ω, R 7 = 5Ω R EQ5 R 7 3.Compute equivalent resistance

19. ECEN 301Discussion #9 – Equivalent Circuits19 Thévenin and Norton Resistances Example1: find the equivalent resistance as seen by the load R L i s =0.5A, v s = 10V, R 1 = 4Ω, R 2 = 6Ω, R 3 = 10Ω, R 4 = 2Ω, R 5 = 2Ω, R 6 = 3Ω, R 7 = 5Ω R EQ 3.Compute equivalent resistance

20. ECEN 301Discussion #9 – Equivalent Circuits20 Thévenin Voltage Thévenin equivalent voltage: equal to the open-circuit voltage (v oc ) present at the load terminals (load removed) vTvT +–+– RTRT i = 0 + – v oc = v T vTvT +–+– RTRT Load +v–+v– i Remove load

21. ECEN 301Discussion #9 – Equivalent Circuits21 Thévenin Voltage Computing Thévenin voltage: 1.Remove the load (open circuit at load terminals) 2.Define the open-circuit voltage (v oc ) across the load terminals 3.Chose a network analysis method to find v oc node, mesh, superposition, etc. 4.Thévenin voltage v T = v oc

22. ECEN 301Discussion #9 – Equivalent Circuits22 Thévenin Voltage Example2: find the Thévenin voltage v s = 10V, R 1 = 4Ω, R 2 = 6Ω, R 3 = 10Ω R2R2 R 1 RLRL R 3 vsvs +–+– iLiL

23. ECEN 301Discussion #9 – Equivalent Circuits23 Thévenin Voltage Example2: find the Thévenin voltage v s = 10V, R 1 = 4Ω, R 2 = 6Ω, R 3 = 10Ω +R2–+R2– +R 1 –+ R 3 – vsvs +–+– 1.Remove the load 2.Define v oc + v oc –

24. ECEN 301Discussion #9 – Equivalent Circuits24 Thévenin Voltage Example2: find the Thévenin voltage v s = 10V, R 1 = 4Ω, R 2 = 6Ω, R 3 = 10Ω +R2–+R2– +R 1 –+ R 3 – vsvs +–+– 3.Choose a network analysis method Voltage divider + v oc – i i 3 = 0

25. ECEN 301Discussion #9 – Equivalent Circuits25 Thévenin Voltage Example2: find the Thévenin voltage v s = 10V, R 1 = 4Ω, R 2 = 6Ω, R 3 = 10Ω +R2–+R2– +R 1 –+ R 3 – vsvs +–+– 4.v T = v oc + v oc – i i 3 = 0

26. ECEN 301Discussion #9 – Equivalent Circuits26 Thévenin Equivalent Circuit Computing Thévenin Equivalent Circuit: 1.Compute the Thévenin resistance R T 2.Compute the Thévenin voltage v T vTvT +–+– RTRT Load +v–+v– i

27. ECEN 301Discussion #9 – Equivalent Circuits27 Thévenin Equivalent Circuit Example3: find i L by finding the Thévenin equivalent circuit v s = 10V, R 1 = 4Ω, R 2 = 6Ω, R 3 = 10Ω, R L = 10Ω R2R2 R 1 RLRL R 3 vsvs +–+– iLiL

28. ECEN 301Discussion #9 – Equivalent Circuits28 Thévenin Equivalent Circuit Example3: find i L by finding the Thévenin equivalent circuit v s = 10V, R 1 = 4Ω, R 2 = 6Ω, R 3 = 10Ω, R L = 10Ω 1.Compute R T Remove R L +R2–+R2– +R 1 –+ R 3 – vsvs +–+–

29. ECEN 301Discussion #9 – Equivalent Circuits29 Thévenin Equivalent Circuit Example3: find i L by finding the Thévenin equivalent circuit v s = 10V, R 1 = 4Ω, R 2 = 6Ω, R 3 = 10Ω, R L = 10Ω 1.Compute R T Remove R L Zero sources +R2–+R2– +R 1 –+ R 3 –

30. ECEN 301Discussion #9 – Equivalent Circuits30 Thévenin Equivalent Circuit Example3: find i L by finding the Thévenin equivalent circuit v s = 10V, R 1 = 4Ω, R 2 = 6Ω, R 3 = 10Ω, R L = 10Ω 1.Compute R T Remove R L Zero sources Compute R T = R EQ R EQ

31. ECEN 301Discussion #9 – Equivalent Circuits31 Thévenin Equivalent Circuit Example3: find i L by finding the Thévenin equivalent circuit v s = 10V, R 1 = 4Ω, R 2 = 6Ω, R 3 = 10Ω, R L = 10Ω 1.Compute R T 2.Compute v T (previously computed) R2R2 R 1 RLRL R 3 vsvs +–+– iLiL

32. ECEN 301Discussion #9 – Equivalent Circuits32 Thévenin Equivalent Circuit Example3: find i L by finding the Thévenin equivalent circuit v s = 10V, R 1 = 4Ω, R 2 = 6Ω, R 3 = 10Ω, R L = 10Ω R2R2 R 1 RLRL R 3 vsvs +–+– iLiL vTvT +–+– RTRT RLRL iLiL NB: i L is the same in both circuits

33. ECEN 301Discussion #9 – Equivalent Circuits33 Thévenin Equivalent Circuit Example3: find i L by finding the Thévenin equivalent circuit v s = 10V, R 1 = 4Ω, R 2 = 6Ω, R 3 = 10Ω, R L = 10Ω vTvT +–+– RTRT RLRL iLiL

34. ECEN 301Discussion #9 – Equivalent Circuits34 Norton Current Current equivalent current: equal to the short-circuit current (i sc ) present at the load terminals (load replaced with short circuit) Short circuit load iNiN RNRN Load +v–+v– i iNiN RNRN i N = i sc

35. ECEN 301Discussion #9 – Equivalent Circuits35 Norton Current Computing Norton current: 1.Replace the load with a short circuit 2.Define the short-circuit current (i sc ) across the load terminals 3.Chose a network analysis method to find i sc node, mesh, superposition, etc. 4.Norton current i N = i sc

36. ECEN 301Discussion #9 – Equivalent Circuits36 Norton Current Example4: find the Norton equivalent current i N v s = 6V, i s = 2A, R 1 = 6Ω, R 2 = 3Ω, R 3 = 2Ω, R L = 10Ω isis R1R1 R2R2 RLRL R 3 – + vsvs

37. ECEN 301Discussion #9 – Equivalent Circuits37 Norton Current Example4: find the Norton equivalent current i N v s = 6V, i s = 2A, R 1 = 6Ω, R 2 = 3Ω, R 3 = 2Ω, R L = 10Ω isis R1R1 R2R2 R 3 – + vsvs 1.Short-circuit the load 2.Define i sc i sc

38. ECEN 301Discussion #9 – Equivalent Circuits38 Norton Current Example4: find the Norton equivalent current i N v s = 6V, i s = 2A, R 1 = 6Ω, R 2 = 3Ω, R 3 = 2Ω, R L = 10Ω isis +R1–+R1– +R2–+R2– + R 3 – – + vsvs i sc 3.Choose a network analysis method Node voltage Node a Node b Node d i1i1 i2i2 i3i3 1.v a is independent 2.v b is dependent (actually v a and v b are dependent on each other but choose v b ) v b = v a + v s iviv

39. ECEN 301Discussion #9 – Equivalent Circuits39 Norton Current Example4: find the Norton equivalent current i N v s = 6V, i s = 2A, R 1 = 6Ω, R 2 = 3Ω, R 3 = 2Ω, R L = 10Ω isis +R1–+R1– +R2–+R2– + R 3 – – + vsvs i sc 3.Choose a network analysis method Node voltage Node a Node b Node d i1i1 i2i2 i3i3 iviv

40. ECEN 301Discussion #9 – Equivalent Circuits40 Norton Current Example4: find the Norton equivalent current i N v s = 6V, i s = 2A, R 1 = 6Ω, R 2 = 3Ω, R 3 = 2Ω, R L = 10Ω isis +R1–+R1– +R2–+R2– + R 3 – – + vsvs i sc 3.Choose a network analysis method Node voltage Node a Node b Node d i1i1 i2i2 i3i3 iviv

41. ECEN 301Discussion #9 – Equivalent Circuits41 Norton Current Example4: find the Norton equivalent current i N v s = 6V, i s = 2A, R 1 = 6Ω, R 2 = 3Ω, R 3 = 2Ω, R L = 10Ω isis +R1–+R1– +R2–+R2– + R 3 – – + vsvs i sc 3.Choose a network analysis method Node voltage Node a Node b Node d i1i1 i2i2 i3i3 iviv

42. ECEN 301Discussion #9 – Equivalent Circuits42 Norton Current Example4: find the Norton equivalent current i N v s = 6V, i s = 2A, R 1 = 6Ω, R 2 = 3Ω, R 3 = 2Ω, R L = 10Ω isis +R1–+R1– +R2–+R2– + R 3 – – + vsvs i sc 3.Choose a network analysis method Node voltage Node a Node b Node d i1i1 i2i2 i3i3 iviv

43. ECEN 301Discussion #9 – Equivalent Circuits43 Norton Current Example4: find the Norton equivalent current i N v s = 6V, i s = 2A, R 1 = 6Ω, R 2 = 3Ω, R 3 = 2Ω, R L = 10Ω isis +R1–+R1– +R2–+R2– + R 3 – – + vsvs i sc 4.i N = i sc i1i1 i2i2 i3i3 iviv

44. ECEN 301Discussion #9 – Equivalent Circuits44 Norton Equivalent Circuit Computing Norton Equivalent Circuit: 1.Compute the Norton resistance R N 2.Compute the Norton current i N iNiN RNRN Load +v–+v– i

45. ECEN 301Discussion #9 – Equivalent Circuits45 Norton Equivalent Circuit Example5: find the Norton equivalent circuit v s = 6V, i s = 2A, R 1 = 6Ω, R 2 = 3Ω, R 3 = 2Ω, R L = 10Ω isis R1R1 R2R2 RLRL R 3 – + vsvs

46. ECEN 301Discussion #9 – Equivalent Circuits46 Norton Equivalent Circuit Example5: find the Norton equivalent circuit v s = 6V, i s = 2A, R 1 = 6Ω, R 2 = 3Ω, R 3 = 2Ω, R L = 10Ω isis R1R1 R2R2 R 3 – + vsvs 1.Compute R N Remove R L

47. ECEN 301Discussion #9 – Equivalent Circuits47 Norton Equivalent Circuit Example5: find the Norton equivalent circuit v s = 6V, i s = 2A, R 1 = 6Ω, R 2 = 3Ω, R 3 = 2Ω, R L = 10Ω R1R1 R2R2 R 3 1.Compute R N Remove R L Zero sources

48. ECEN 301Discussion #9 – Equivalent Circuits48 Norton Equivalent Circuit 1.Compute R N Remove R L Zero sources Compute R N = R EQ R EQ Example5: find the Norton equivalent circuit v s = 6V, i s = 2A, R 1 = 6Ω, R 2 = 3Ω, R 3 = 2Ω, R L = 10Ω

49. ECEN 301Discussion #9 – Equivalent Circuits49 Norton Equivalent Circuit Example5: find the Norton equivalent circuit v s = 6V, i s = 2A, R 1 = 6Ω, R 2 = 3Ω, R 3 = 2Ω, R L = 10Ω 1.Compute R N 2.Compute i N (previously computed) isis R1R1 R2R2 RLRL R 3 – + vsvs

50. ECEN 301Discussion #9 – Equivalent Circuits50 Norton Equivalent Circuit Example5: find the Norton equivalent circuit v s = 6V, i s = 2A, R 1 = 6Ω, R 2 = 3Ω, R 3 = 2Ω, R L = 10Ω isis R1R1 R2R2 RLRL R 3 – + vsvs iNiN RNRN RLRL