1. General Methods of Network Analysis Node Analysis Mesh Analysis Loop Analysis Cutset Analysis State variable Analysis Non linear and time varying network Linear and time invariant network

2. Node and Mesh analyses Source transformation Basic facts of node analysis Implication of KCL Implication of KVL Node analysis of linear time invariant networks Duality Basic facts of mesh analysis Implication of KVL Implication of KCL Mesh analysis of linear time invariant networks

3. Node and Mesh analyses Fig. 1

4. Source transformation Ideal voltage source branch can be eliminatedFig. 2

5. Source transformation Ideal current source branch can be eliminatedFig. 3

6. Source transformation summary By source transformation we can modify any given network in such a way that each voltage source is connected in series with an element which is not a source and each current source is connected in parallel with an element which is not a source. If a current source in connected in series with a voltage source or an element the voltage source or that element can be ignored in analyzing the circuit. If a voltage source in connected in parallel with a current source or an element the current source or that element can be omitted in analyzing the circuit.

7. Source transformation summary Fig. 4

8. Source transformation summary Fig. 5

9. Basic facts of node analysis For any network with nodes and branches pick an arbitrary node called the datum node. Assign to all node as Where Implications of KCL Apply KCL to nodesa system of linear algebraic equation of unknowns is obtained “The n linear homogenous algebraic equations in obtained by applying KCL to each node except the datum node constitute a set of linearly independent equation.”

10. Basic facts of node analysis At node k For all node KCL is written in the form A is the reduced incident matrix. ( A a with datum node deleted) Fig. 6 (KCL)

11. Basic facts of node analysis Example 1 Consider the graph in Fig. 8. The graph has 4 nodes and 5 branches. Write The node incident matrix A a and the KCL in matrix form. Fig. 8 Incident matrix A a

12. Basic facts of node analysis KCL or

13. Basic facts of node analysis Implications of KVL Let be the node voltage at nodes respect to the datum node. The k th branch voltage is always the difference Between the nodes connected to it. Therefore all branch voltage can be written in the matrix form (KVL) if branch k leaves node i if branch k enters node i

14. Basic facts of node analysis If branch leaves node and enter node then In matrix form If branch leaves node If branch enters node If branch is not incident with node

15. Basic facts of node analysis Example 2 For the graph of example 1 or KVL

16. Basic facts of node analysis Proof of Tellegen theorem From KVLand from KCL

17. Node analysis of linear time invariant networks In linear time invariant network all element except the independent source are linear and time invariant. The combination of branch equations to KCL and KVL forms a general linear simultaneous equation for Resistive network In a resistive network the branch equation takes the form or In matrix form (1) Fig. 9

18. Node analysis of linear time invariant networks is called the branch conductance matrix and it is a diagonal matrix The and are source vectors Substitute and pre-multiply by A in (1) yields or (2) (3)

19. Node analysis of linear time invariant networks Let and be the node admittance matrix be the node current source vector Then the node equation becomes (4) Once the node voltages are known the branch currents can be found from and (5)

20. Node analysis of linear time invariant networks Example 3 Consider the circuit in example 1 with elements shown in Fig.10, solve the circuit for the node voltages and branch currents by node analysis. Fig.10

21. Node analysis of linear time invariant networks 1) Write KCL 2) Write KVL

22. Node analysis of linear time invariant networks 3) Write branch equation in the form Branch 1 Branch 5 Thus

23. Node analysis of linear time invariant networks 4) Make the form

24. Node analysis of linear time invariant networks 5) and 6) The node equation is

25. Node analysis of linear time invariant networks 7) Solve for e 8) Solve for v

26. Node analysis of linear time invariant networks 9) Solve for j

27. Node analysis of linear time invariant networks Node equation by inspection The node equation can be written in scalar form (6)

28. Node analysis of linear time invariant networks = sum of admittance at node = negative sum of admittance between the node and the node where and = equivalent current source injected at node Redraw the circuit in example 3 By inspection

29. Node analysis of linear time invariant networks Sinusoidal steady state analysis In RLC circuit with sinusoid excitations, branch voltage and branch current are in the form of phasors and branch admittances are the function of frequency The branch equation take the form In the matrix form and the node equation becomes (7) (8) (9)

30. Node analysis of linear time invariant networks Example 4 Fig. 11 Consider the circuit shown in Fig.11. The sinusoid current source of phasor is applied at node 1. The inductors are coupled as shown by its inductance matrix Write the node equation of the circuit.

31. Node analysis of linear time invariant networks Branch equations Inductor branch equations

32. Node analysis of linear time invariant networks

33. Branch equation in matrix form

34. Node analysis of linear time invariant networks The node admittance matrix

35. Node analysis of linear time invariant networks The node equation is Solve for E and substitute in and

36. Node analysis of linear time invariant networks Integrodifferential form equations In general node analysis of a linear network lead to a set of integro- differential equation. The equation involves unknown functions, theirs derivatives and integrals. e.g. Example 5 The linear time-invariant network shown in Fig.12 has the reciprocal Inductance matrix

37. Node analysis of linear time invariant networks Fig. 12 KCL:

38. Node analysis of linear time invariant networks KVL: Branch equation

39. Node analysis of linear time invariant networks It is convenient to write and Therefore the branch equations are

40. Note

41. Node analysis of linear time invariant networks In the matrix form Multiple by A and from KVL or

42. Node analysis of linear time invariant networks

43. The node equation The cut set for branches1, 4, 5 gives initial conditions (a)

44. Node analysis of linear time invariant networks Notes If we define new variables Then The node equation becomes as in (a)

45. Node analysis of linear time invariant networks The short cut method If the circuit involves only few dependent sources the node equation can also be written by inspection. Example 6 Fig.13 Write the node equation for Fig. 13 in sinusoid steady state.

46. Node analysis of linear time invariant networks By inspection consider as independent sources

47. Node analysis of linear time invariant networks Rearrange the equation

48. Node analysis of linear time invariant networks Example 7 Fig.14 Write the integro-differential equation by inspection for the circuit in Fig.14

49. Node analysis of linear time invariant networks Since Then by inspection substitute Then

50. Node analysis of linear time invariant networks Duality A graph of a circuit can be drawn in many ways but it has the same results. Planar graph, Meshes, outer mesh A planar graph can be drawn on the plane without branch intersection. A mesh is the smallest closed path (loop) in a graph and a outer mesh Is a loop formed outside the graph. Fig.15

51. Outer Mesh

52. Node analysis of linear time invariant networks Fig.15 (a),(b),and (c) have the same incident matric and of the same graph. But they have different topology. In Fig.15 the loop is not a mesh but this loop is a mesh in (b) while this loop is an outer mesh in (c). Hinged and unhinged graph Hinged graph can be partitioned into two subgraph by one node. Fig.16 Hinged graphUnhinged graph

53. Node analysis of linear time invariant networks The number of meshes is equal to Fundamental property of an unhinged planar graph Each branch belongs to exactly two meshes including the outer mesh Fig.17

54. Node analysis of linear time invariant networks Assigned reference direction for meshes Each mesh has clockwise direction but the outer mesh has counter clockwise direction. A mesh matrix M a can be written its element is defined by If branch is in mesh and their direction coincide If branch is in mesh and their direction opposite If branch is not in mesh

55. Node analysis of linear time invariant networks For the graph in Fig.17, the mesh matrix M a there are four mesh and 8 branches. The elements of the matrix M a are It can be observed that the Mesh matrix M a has the same properties as The node incident matrix A a. It element is 1 or -1 or 0.

56. Duality Dual graph Dual graph has some properties being demonstrated in example 8. Example 8 Consider the linear time invariant circuit in Fig. 18. In sinusoid steady State, write the node equation of the circuit and find its dual circuit. Fig. 18

57. Duality The node equation written by inspection is By changing The node equation becomes

58. Duality These equation are the loop equation for the circuit in Fig.19 Fig.19 Fig.19 is the dual graph of Fig 18.

59. A graph have node branch and the is the dual graph of if Duality  There is a one-to-one correspondence between the meshes of and the node of  Branch between mesh of correspond to branch between node of G G G1G1 G G1G1 G G1G1 G1G1 G algorithm write node in each mesh of and node for the outer mesh for each branch of common to mesh and mesh there is a branch connected between node and node of if the graph is oriented the direction of the branch of is rotated 90 o clockwise G G1G1 GG1G1 G

60. Duality Example 9 From the planar graph G in Fig.20 construct the dual graph G G Fig.20

61. Duality Step 1 assign node of G G Fig.21

62. Duality Step 2 draw branches of G G Fig.22

63. Duality Notes In general a given topological graph G has many duals. But if the datum node and elements belong to the outer mesh is specified, The Dual graph is always unique The correspondence between the graph G and involves G Branches versus Branches Meshes versus Nodes Datum node versus Outer mesh The incident matrix A a of equal the mesh matrix M a of G G

64. Duality Dual network A network is the dual of the network if the topological graph of is dual of the topological graph of G G And the branch equation of obtained form the corresponding equation of by performing the following substitution Whereare voltage, current, charge and flux linkage

65. Duality Example 10 Fig.23 Consider the nonlinear time varying network shown in Fig 23 draw the dual network.

66. Duality The mesh equations are Dual equation

67. Recall that u v vduvduudvudv v2v2

68. Duality The dual network Fig.24

69. Basic facts of mesh analysis If we apply KVL to meshes 1,2,3,… l (omit the outer mesh), a system of l linear homogeneous equations in b unknowns is obtained. The KVL can be written in the matrix form as or If branch is in mesh and their direction coincide If branch is in mesh and their direction opposite If branch is not in mesh Note that the mesh matrix M is obtained from the matrix M a with the outer Mesh deleted.

70. Basic facts of mesh analysis Example 11 Fig.25 Obtain the KVL for the graph shown in Fig.25

71. Basic facts of mesh analysis KVL or

72. Basic facts of mesh analysis Implication of KCL Let be the mesh currents in clockwise direction. These currents are linearly independent. Thus KCL can not be written in terms of mesh Currents. Since each mesh current runs around a loop if it crosses a cutset in a positive direction it will also cut that cutset in a negative direction too. However, the branch current can be calculated by the equation This is some what similar for the node analysis in which

73. Basic facts of mesh analysis Example 12 Write the KCL for the graph in the example 11 Or

74. Mesh analysis of linear time invariant networks Mesh analysis of a linear time-invariant network is the dual of the node analysis. Sinusoidal steady state analysis A linear time invariant network  with branch and node whose graph is unhinged and planar is in sinusoid steady state at frequency. The phasors of voltage and current vector can be used. If the phasors of mesh current vector is chosen, the Kirchhoff’s laws give (KVL) (KCL) Branch eqn.

75. Mesh analysis of linear time invariant networks The matrixis called the branch impedance matrix. Substitution of equation yields or

76. Mesh analysis of linear time invariant networks Example 1 Consider the circuit of Fig 26. the phasor represent the sinusoid voltage write the mesh equation of the circuit. Fig 26

77. Mesh analysis of linear time invariant networks Let the inductance matrix of the branch 3,4,5 is The mesh matrix The branch 3,4,5 voltages

78. Mesh analysis of linear time invariant networks and Then the branch equation becomes

79. Mesh analysis of linear time invariant networks

80. The mesh equation becomes

81. Mesh analysis of linear time invariant networks The properties of mesh impedance matrix  If the network has no coupling elements is diagonal and is symmetric  if there is no coupling element, the mesh impedance matrix can be written by inspection: is the sum of all impedance in mesh i is the negative sum of all impedance common between mesh i and mesh k  Current source is converted to Thevenin source and is the algebraic sum of all voltage sources whose reference direction push the current flows in the mesh k

82. Mesh analysis of linear time invariant networks Integrodifferential Equations Fig. 27 Consider the linear time-invariant circuit shown in Fig. 27 where the Inductance matrix is

83. Mesh analysis of linear time invariant networks Step 1 Write the KVL Step 2 Write the KCL

84. Mesh analysis of linear time invariant networks Step 3 Write the Branch equations Or the form Combine the equation in the form

85. Mesh analysis of linear time invariant networks Or in scalar form

86. Mesh analysis of linear time invariant networks If we define new variables Then The mesh equation becomes