Lectures on Network Theorems (Circuits )

Network Theorems

1. ELL 100 -Introduction to Electrical Engineering
LECTURE7: NETWORKTHEOREMS
THEVENIN& NORTONEQUIVALENTS

2. ❑Introduction
❑Thevenin’stheorem
❑Norton’s theorem
❑Numerical Examples
2
Outline

3. INTRODUCTION
3
Thevenin’sTheoremApplication
•Itoftenoccursinpracticethataparticularelementinacircuitis
variable(usuallycalledtheload)whileotherelementsarefixed.
•Asatypicalexample,ahouseholdoutletterminalmaybeconnected
todifferentappliancesconstitutingavariableload.
•Eachtimethevariableelementischanged,theentirecircuithastobe
analyzedalloveragain.
•Toavoidthisproblem,Thevenin’stheoremprovidesatechniqueby
whichtheentirefixedpartofthecircuitisreplacedbyaverysimple
equivalentofavoltagesourceinserieswithanimpedence.

4. INTRODUCTION
4
Thevenin’sTheoremApplicationExample

5. INTRODUCTION
5
Thevenin’sTheorem
V
th= Thevenin’sequivalent voltage
voltage R
th= Thevenin’sequivalent resistance + -
+
-
V1
V2
I1
R2
R1
R3 R4
R5
R6 Load Terminals
A
B +
-
A
B
Vth
Rth
Load Terminals
Theveninequivalent circuit

6. INTRODUCTION
6
Thevenin’sTheoremapplicationareas

7. INTRODUCTION
7
Thevenin’sTheoremapplicationareas
kΩ
10 kΩ
μApeak

8. THEVENIN’STHEOREM
8
Statement:Alineartwo-terminalcircuitcanbereplacedbyan
equivalentcircuitconsistingofavoltagesourceV
Thinserieswitha
resistorR
Th,whereV
Thistheopen-circuitvoltageattheterminalsand
R
Thistheinputorequivalentresistanceattheterminalswhenthe
independentsourcesareturnedoff.
LinearCircuit:Alinearcircuitisonewhoseoutputislinearlyrelated
(ordirectlyproportional)toitsinputi.e.containingonlylinear
elementseg.R,L,C,transformer

9. THEVENIN’STHEOREM
9
HowtofindTheveninequivalentvoltageV
ThandresistanceR
Th?
•Thetwocircuitsgivenbelowaresaidtobeequivalentiftheyhave
thesamevoltage-currentrelationattheirterminals.
(a)OriginalCircuit (b)Theveninequivalentcircuit

10. THEVENIN’STHEOREM
10
•Iftheterminalsa-baremadeopen-circuited(byremovingtheload),no
currentflows,thentheopencircuitvoltageacrosstheterminalsa-bis
equaltothevoltagesourceV
Th.
•Thus,V
Thistheopen-circuitvoltageacrosstheterminali.e.V
Th=v
oc
•Theinputresistance(orequivalentresistance)ofthedeadcircuit(all
independentsourcesturnedoff)attheterminalsa-bintheFig(a)must
beequaltoR
ThinFig.(b)(inputresistancewithV
Thturnedoff).
•Thus,R
Thistheinputresistanceattheterminalswhenthe
independentsourcesareturnedoff,i.e.R
Th=R
in

11. THEVENIN’STHEOREM
11
FindingtheTheveninequivalentresistanceR
Th:
Case1:WhenthenetworkshasnoDependentSources
•Turnoffalltheindependentsources,thenR
Thistheinputresistance
ofthenetworklookingbetweenterminalsaandb.
Case2:WhenthenetworkshasDependentSources
•Turn off all independent sources, apply voltage source v
oacross
terminals a-b and determine the resulting current i
o. Then R
Th= v
o∕ i
o.
•Alternatively,insertacurrentsourcei
oacrossterminalsa-bandfind
theterminalvoltagev
o.AgainR
Th=v
o∕i
o.

12. THEVENIN’STHEOREM
12
StepstodetermineThevenin’sEquivalentResistance(R
TH)
andVoltage(V
TH):
•RemoveloadresistorR
Loranycomponentconnectedacrossthe
terminalsa-bthroughwhichTheveninequivalentisrequired.
•DetermineR
THbyshortingallvoltagesourcesandopen-
circuitingallcurrentsources,andthencalculatingthecircuit’stotal
resistanceasseenfromtheopenterminalsa-b.
•DetermineV
THbycalculatingthevoltagebetweenopen
terminalsa-b(byusualcircuitanalysismethods).

13. THEVENIN’STHEOREM
13
Example:TofindTheveninequivalentbetweenterminals‘a’and‘b’
Case1:Nodependentsource

14. b
20Ω
10Ω10Ω
20Ω10Ω
10Ω
a THEVENIN’STHEOREM
14
Step1:Removealltheindependentsources.
a)Replacevoltagesourcebyshortcircuit
b)Replacecurrentsourcebyopencircuit

15. THEVENIN’STHEOREM
15
Step2:FindingR
Thb
20Ω
10Ω10Ω
20Ω10Ω
10Ω
a b
10Ω10Ω
10Ω10Ω
10Ω
a

16. THEVENIN’STHEOREM
16
Step2:FindingR
Thb
10Ω10Ω
10Ω10Ω
10Ω
a b
10Ω
30Ω
30Ω
10Ω
a
30Ω
StartoDeltatransformation

17. THEVENIN’STHEOREM
17
Step2:FindingR
Thb
10Ω
30Ω
30Ω
10Ω
a
30Ω 30Ω
7.5Ω
a
7.5Ω
b 30Ω
15Ω
a b
R
Th=30||15=10Ω

18. THEVENIN’STHEOREM
18
Step3:FindingV
Th
SourceTransformationsb
10Ω10Ω
10Ω10Ω
10Ω
a
i1 i230V
50V
10V

19. THEVENIN’STHEOREM
19
Step3:FindingV
Th
Forloop1:
Forloop2:
Onsolving(1)and(2),i
1=0,i
2=2A12
12
305030100
23(1)
ii
ii

 21
12
501030100
63(2)
ii
ii

 b
10Ω10Ω
10Ω10Ω
10Ω
a
i1 i230V
50V
10V

20. THEVENIN’STHEOREM
20
Step3:FindingV
Th
ApplyingKVLtotheoutputloop,
Therefore,12
1030100
10 V
ab
ab
vii
v

 10 V
Thab
Vv
TheveninEquivalent Circuit

21. THEVENIN’STHEOREM
21
Example:TofindTheveninequivalentbetweenterminal‘a’and‘b’
Case2:Withdependentsource
Turn off all
independent sources,
apply voltage source
v
0at terminals aand b
and determine the
resulting current i
0.
Then R
Th= v
0 ∕i
0.

22. THEVENIN’STHEOREM
22
Step1.Removedependentsources
5Acurrentsourceisreplacedbyopencircuitandsetv
0=1V.

23. THEVENIN’STHEOREM
23
Step2.FindR
Th
KVLforloop1:
But,
Therefore,

24. THEVENIN’STHEOREM
24
Meshanalysisforloop2and3,
Onsolvingweget,i
0=−i
3=1/6A.
Therefore,

25. THEVENIN’STHEOREM
25
Step3.FindV
Th
Wehavetofindv
ocforthiscircuit.
Applyingmeshanalysisweget,
and

26. THEVENIN’STHEOREM
26
TheTheveninequivalentcircuitisshownbelow.
Onsolving(1),(2),(3)and(4),weget,i
2=10/3=>

27. NORTON’STHEOREM
27
Statement:Alineartwo-terminalcircuitcanbereplacedbyan
equivalentcircuitconsistingofacurrentsourceI
Ninparallelwitha
resistorR
N,whereI
Nistheshort-circuitcurrentthroughthe
terminalsandR
Nistheinputorequivalentresistanceatthe
terminalswhentheindependentsourcesareturnedoff.
WhyareweusingNorton’sTheorem?
•Simplifiesthenetworkintermsofcurrentsinsteadofvoltages.
•Itreducesanetworktoasimpleparallelcircuitwithacurrent
sourceandaresistor.

28. NORTON’STHEOREM
28
Original circuit Norton equivalent circuit

29. NORTON’STHEOREM
29
StepstodetermineNorton’sequivalentResistance(R
N)
andCurrent(I
N):
•CalculateR
NinthesamewayasR
Th.
•Usingsourcetransformation,theTheveninandNorton
resistancesareequali.e.R
N=R
Th.
•TofindtheNortoncurrentI
N,wedeterminetheshort-circuit
currentflowingfromterminalatob.
•Thisshort-circuitcurrentistheNortonequivalentcurrentI
N.

30. NORTON’STHEOREM
30
CloserelationshipbetweenNorton’sandThevenin’stheorems:

31. NORTON’STHEOREM
31
SinceV
Th,I
N,andR
Th/Narerelated,todeterminetheTheveninor
Nortonequivalentcircuitwefind:
•Theopen-circuitvoltagev
ocacrossterminalsaandb(=V
Th)
•Theshort-circuitcurrenti
scatterminalsaandb(=I
N)
•TheequivalentinputresistanceR
inatterminalsaandb
whenallindependentsourcesareturnedoff(=R
Th/N)

32. NORTON’STHEOREM
32
ExampleofNortonTheorem
Case1:Withoutdependentsource
I
Norton= (28/4) + (7/1) = 14 A
R
Norton= (4||1) = 0.8 Ω

33. NORTON’STHEOREM
33
ExampleofNortonTheorem
Case2:Withdependentsource

34. NORTON’STHEOREM
34
Step1:ComputeR
N.Setthe
independentsourcesequaltozero
andconnectavoltagesourcev
0=1V
totheterminals.
Weignorethe4-Ωresistorbecauseit
isshort-circuited.Hence,i
x=0.
Alsoduetotheshortcircuit,the5-Ω
resistor,thevoltagesource,andthe
dependentcurrentsourceareallin
parallel.

35. NORTON’STHEOREM
35
Step2:ComputeI
N.Short-circuit
terminalsaandbandfindthecurrenti
sc,
asindicatedinthefigure.
Notefromthisfigurethatthe4Ω
resistor,the10Vvoltagesource,the5Ω
resistor,andthedependentcurrent
sourceareallinparallel.
Hence,
KCL at node a:

36. NORTON’STHEOREM
36
7A = 5Ω
Original circuit Norton equivalent

37. NUMERICAL
37
Q1.FindtheTheveninequivalentofthecircuitshownbelowacross
terminalsa-b.ThenfindthecurrentthroughR
L=6Ωand36Ωrespectively.

38. NUMERICAL
38
Soln:
Step1.FindR
Thbyturningoff
the32Vvoltagesource
(replacingitwithashortcircuit)
andthe2Acurrentsource
(replacingitwithopencircuit).

39. NUMERICAL
39
Step2.Makea-bopencircuit.FindV
Thbyapplyingmesh/nodeanalysis.
=>i
1=0.5A

40. NUMERICAL
40
Step 3. Finding current through R
L
a)WhenR
L=6Ω
a)WhenR
L=36Ω
Theveninequivalentcircuit

41. NUMERICAL
41
Q2.FindtheTheveninequivalentofthegivencircuitatterminalsa-b.

42. NUMERICAL
42
Soln:
Step1.FindR
Th.Weexcitethecircuitwith1Acurrentsource.
Thereducedcircuitisshown
= 1 A

43. NUMERICAL
43
KCLatnodea:
Alsowehave,
Substituting(2)in(1),wehave
= 1A
Negativesigninresistanceindicatesthat
thecircuitissupplyingpower.Resistors
cannotsupplypower,it’sthedependent
currentsourcethatsuppliespower.

44. NUMERICAL
44
Step2:FindingV
Th
=>i
1=i
x
=>ν
ab=-2i
x=-4i
x
=>ν
ab=i
x=0
=>V
Th=ν
ab=0
i
1
KCL at node a:
i
1+ i
x= 2i
x
Sincewedon’thaveanyindependent
sourcesinthegivencircuit,thevalueof
TheveninequivalentvoltageV
Thiszero.
Thus,theTheveninequivalentcircuitisjustanegative(-4Ω)
resistanceacrossa-bi.e.a“dependentvoltagesource”withν
ab=4i
ba

45. NUMERICAL
45
Q3.FindtheTheveninequivalentofthegivencircuitatterminalsa-b.

46. NUMERICAL
46
Soln:
Step 1. Finding V
Th.
Applying KVL to the loop,
= 0
But V
o = 10kI ,

47. NUMERICAL
47
Step2:TofindR
Th,removeindependentvoltagesource70Vandapplya
1-Vindependentsourcetoexcitethecircuitattheterminalsa-b
Wehave
And

48. NUMERICAL
48
Q4.DeterminetheTheveninequivalentcircuitbetweenterminalsa-b.

49. NUMERICAL
49
Soln:
Step1:FindingR
Th.
Removeindependentsources
i.ereplacevoltagesourceby
shortcircuitandcurrent
sourcebyopencircuit.
Thenwehave,

50. NUMERICAL
50
Step2:FindingV
Th.
KCLatnode1:
KCLatnode2:
Solving(1)and(2),
=>

51. NUMERICAL
51
Q5.FindtheTheveninandNortonequivalentsatterminalsa-b
ofthecircuitshownbelow.

52. NUMERICAL
52
Soln:
Step1:FindingR
Th/N.Replacecurrentsourcesbyopen-circuitsand
voltagesourcebyshort-circuit.
Step2:FindingV
Th.Applysourcetransformationto1Acurrentsource
andapplynodalanalysis.20 5
5 (14 6)
20 5
4
Th
Th N
R
RR

  

   146
3
1465
8V
Th Th
Th
VV
V




||

53. NUMERICAL
53
And,
Therefore,IN = 2 A
RN = 4 Ω
a
b
Thevenin
Equivalent Circuit
Norton Equivalent
CircuitVTh = 8 V
+
_
RTh = 4 Ω
a
b

54. NUMERICAL
54
Q6.FindtheNortonequivalentatterminalsa-bofthecircuitshown.

55. NUMERICAL
55
Soln:
Step1:FindingR
N.
Settheindependentsources
tozero.Thisleadstothe
reducedcircuitshown.
Thus,

56. NUMERICAL
56
Step2:FindingI
N.Short-circuitterminalsaandb.
Ignorethe5-Ωresistorbecauseithasbeenshort-circuited.
Applyingmeshanalysis,
Onsolving,wegetIN = 1 A
RN = 4 Ω
a
b
Norton’s Equivalent

57. UNSOLVEDNUMERICAL
57
Q1.UsingThevenin’stheorem,findtheequivalentcircuittotheleftofthe
terminalsinthecircuitbelow.ThenfindI.
Ans:V
Th=90V,R
Th=45Ω,I=1.5A.

58. UNSOLVEDNUMERICAL
58
Q2.FindtheTheveninequivalentcircuitofthecircuitshownbelowtothe
leftoftheterminals.
Ans:V
Th=5.333V,R
Th=444.4mΩ.

59. UNSOLVEDNUMERICAL
59
Q3.DeterminetheTheveninequivalentcircuitshownbelow,asseenbythe
7Ωresistor.Thencalculatethecurrentflowingthroughthe7Ωresistor.
Ans:V
Th=80V,R
Th=9.069Ω.

60. UNSOLVEDNUMERICAL
60
Q4.ObtaintheTheveninequivalentatterminalsa-bofthecircuitshown
below.
Ans:V
Th=38V,R
Th=20Ω.

61. UNSOLVEDNUMERICAL
61
Q5.FindNortonequivalentresistanceR
NandcurrentI
Natterminalsa-bof
thecircuitshownbelow.
Ans:I
N=4.5A,R
N=90Ω.

62. UNSOLVEDNUMERICAL
62
Q6.FindtheNortonequivalentwithrespecttoterminalsa-binthecircuit
shownbelow.
Ans:I
N=666.67mA,R
N=10Ω.

63. UNSOLVEDNUMERICAL
63
Q7.FindNortonequivalentresistanceR
NandcurrentI
Natterminalsa-bof
thecircuitshownbelow.
Ans:I
N=10A,R
N=1Ω.

64. UNSOLVEDNUMERICAL
64
Q8.DeterminetheNortonequivalentatterminalsa-bforthecircuitbelow.
Ans:I
N=3A,R
N=-4Ω.

65. UNSOLVEDNUMERICAL
65
Q9.ObtaintheNortonequivalentofthecircuitinfigshownbelow,tothe
leftofterminalsa-b.Usetheresulttofindcurrenti.
Ans:I
N=-0.4A,R
N=10Ω,i=2.4A.

66. UNSOLVEDNUMERICAL
66
Q10.GiventhecircuitinFigbelow,obtaintheNortonequivalentasviewed
fromterminals:(a)a-b(b)c-d.
Ans:(a)I
N=7A,R
N=R
Th=2Ω,V
Th=14V.(b)I
N=12.667A,R
N=R
Th=1.5Ω,V
Th=19V.

67. UNSOLVEDNUMERICAL
67
Q11.ObtaintheNortonequivalentatterminalsa-bofthecircuitinFig.
Ans:I
N=-20mA,R
N=100kΩ.

68. UNSOLVEDNUMERICAL
68
Q12.UseNorton’stheoremtofindVointhecircuitofFig.
Ans:I
N=-1mA,R
N=20kΩ.

69. UNSOLVEDNUMERICAL
69
Q13.ObtaintheTheveninandNortonequivalentcircuitsatterminalsa-bfor
thecircuitinFig.
Ans:(a)I
N=16.667A,R
N=R
Th=10Ω,V
Th=166.67V.

70. UNSOLVEDNUMERICAL
70
Q14.DeterminetheTheveninandNortonequivalentsatterminalsa-bofthe
circuitinFig.
Ans:(a)I
N=1.7778A,R
N=R
Th=22.5Ω,V
Th=40V.

71. REFERENCES
71
1.EdwardHughes;JohnHiley,KeithBrown,IanMcKenzieSmith,
“ElectricalandElectronicTechnology”,10thedition,PearsonEducation
Limited,Year:2008.
2.Alexander,CharlesK.,andSadiku,MatthewN.O.,“Fundamentalsof
ElectricCircuits”,5thEd,McGrawHill,IndianEdition,2013.