1. Methods of Analysis PSUT 1 Basic Nodal and Mesh Analysis Al-Qaralleh

2. Methods of Analysis PSUT Methods of Analysis2 Introduction Nodal analysis Nodal analysis with voltage source Mesh analysis Mesh analysis with current source Nodal and mesh analyses by inspection Nodal versus mesh analysis

3. Lect4 EEE 202 3 Steps of Nodal Analysis 1.Choose a reference (ground) node. 2.Assign node voltages to the other nodes. 3.Apply KCL to each node other than the reference node; express currents in terms of node voltages. 4.Solve the resulting system of linear equations for the nodal voltages.

4. PSUT Methods of Analysis4 Common symbols for indicating a reference node, (a) common ground, (b) ground, (c) chassis.

5. Lect4 EEE 202 5 1. Reference Node The reference node is called the ground node where V = 0 + – V 500  1k  500  I1I1 I2I2

6. Lect4 EEE 202 6 Steps of Nodal Analysis 1.Choose a reference (ground) node. 2.Assign node voltages to the other nodes. 3.Apply KCL to each node other than the reference node; express currents in terms of node voltages. 4.Solve the resulting system of linear equations for the nodal voltages.

7. Lect4 EEE 202 7 2. Node Voltages V 1, V 2, and V 3 are unknowns for which we solve using KCL 500  1k  500  I1I1 I2I2 123 V1V1 V2V2 V3V3

8. Lect4 EEE 202 8 Steps of Nodal Analysis 1.Choose a reference (ground) node. 2.Assign node voltages to the other nodes. 3.Apply KCL to each node other than the reference node; express currents in terms of node voltages. 4.Solve the resulting system of linear equations for the nodal voltages.

9. Lect4 EEE 202 9 Currents and Node Voltages 500  V1V1 V1V1 V2V2

10. Lect4 EEE 202 10 3. KCL at Node 1 500  I1I1 V1V1 V2V2

11. Lect4 EEE 202 11 3. KCL at Node 2 500  1k  500  V2V2 V3V3 V1V1

12. Lect4 EEE 202 12 3. KCL at Node 3 500  I2I2 V2V2 V3V3

13. Lect4 EEE 202 13 Steps of Nodal Analysis 1.Choose a reference (ground) node. 2.Assign node voltages to the other nodes. 3.Apply KCL to each node other than the reference node; express currents in terms of node voltages. 4.Solve the resulting system of linear equations for the nodal voltages.

14. Lect4 EEE 202 14 + – V 500  1k  500  I1I1 I2I2 4. Summing Circuit Solution Solution: V = 167I 1 + 167I 2

15. PSUT Methods of Analysis15 Typical circuit for nodal analysis

16. PSUT Methods of Analysis16

17. PSUT Methods of Analysis17

18.  Calculus the node voltage in the circuit shown in Fig. 3.3(a) PSUT Methods of Analysis18

19.  At node 1 PSUT Methods of Analysis19

20.  At node 2 PSUT Methods of Analysis20

21.  In matrix form: PSUT Methods of Analysis21

22. Practice PSUT Methods of Analysis22

23.  Determine the voltage at the nodes in Fig. below PSUT Methods of Analysis23

24.  At node 1, PSUT Methods of Analysis24

25.  At node 2 PSUT Methods of Analysis25

26.  At node 3 PSUT Methods of Analysis26

27.  In matrix form: PSUT Methods of Analysis27

28. 3.3 Nodal Analysis with Voltage Sources  Case 1: The voltage source is connected between a nonreference node and the reference node: The nonreference node voltage is equal to the magnitude of voltage source and the number of unknown nonreference nodes is reduced by one.  Case 2: The voltage source is connected between two nonreferenced nodes: a generalized node (supernode) is formed. PSUT Methods of Analysis28

29. 3.3 Nodal Analysis with Voltage Sources PSUT Methods of Analysis29 A circuit with a supernode.

30.  A supernode is formed by enclosing a (dependent or independent) voltage source connected between two nonreference nodes and any elements connected in parallel with it.  The required two equations for regulating the two nonreference node voltages are obtained by the KCL of the supernode and the relationship of node voltages due to the voltage source. PSUT Methods of Analysis30

31. Example 3.3  For the circuit shown in Fig. 3.9, find the node voltages. PSUT Methods of Analysis31 i1 i2

32. PSUT Methods of Analysis32 Find the node voltages in the circuit below.

33.  At suopernode 1-2, PSUT Methods of Analysis33

34.  At supernode 3-4, PSUT Methods of Analysis34

35. 3.4 Mesh Analysis  Mesh analysis: another procedure for analyzing circuits, applicable to planar circuit.  A Mesh is a loop which does not contain any other loops within it PSUT Methods of Analysis35

36. PSUT Methods of Analysis36 (a) A Planar circuit with crossing branches, (b) The same circuit redrawn with no crossing branches.

37. PSUT Methods of Analysis37 A nonplanar circuit.

38.  Steps to Determine Mesh Currents: 1.Assign mesh currents i 1, i 2,.., i n to the n meshes. 2.Apply KVL to each of the n meshes. Use Ohm’s law to express the voltages in terms of the mesh currents. 3.Solve the resulting n simultaneous equations to get the mesh currents. PSUT Methods of Analysis38

39. Fig. 3.17 PSUT Methods of Analysis39 A circuit with two meshes.

40.  Apply KVL to each mesh. For mesh 1,  For mesh 2, PSUT Methods of Analysis40

41.  Solve for the mesh currents.  Use i for a mesh current and I for a branch current. It’s evident from Fig. 3.17 that PSUT Methods of Analysis41

42.  Find the branch current I 1, I 2, and I 3 using mesh analysis. PSUT Methods of Analysis42

43.  For mesh 1,  For mesh 2,  We can find i 1 and i 2 by substitution method or Cramer’s rule. Then, PSUT Methods of Analysis43

44.  Use mesh analysis to find the current I 0 in the circuit. PSUT Methods of Analysis44

45.  Apply KVL to each mesh. For mesh 1,  For mesh 2, PSUT Methods of Analysis45

46.  For mesh 3,  In matrix from become we can calculus i 1, i 2 and i 3 by Cramer’s rule, and find I 0. PSUT Methods of Analysis46

47. 3.5 Mesh Analysis with Current Sources PSUT Methods of Analysis47 A circuit with a current source.

48.  Case 1 ●Current source exist only in one mesh ●One mesh variable is reduced  Case 2 ●Current source exists between two meshes, a super- mesh is obtained. PSUT Methods of Analysis48

49.  a supermesh results when two meshes have a (dependent, independent) current source in common. PSUT Methods of Analysis49

50. Properties of a Supermesh 1.The current is not completely ignored ●provides the constraint equation necessary to solve for the mesh current. 2.A supermesh has no current of its own. 3.Several current sources in adjacency form a bigger supermesh. PSUT Methods of Analysis50

51.  For the circuit below, find i 1 to i 4 using mesh analysis. PSUT Methods of Analysis51

52.  If a supermesh consists of two meshes, two equations are needed; one is obtained using KVL and Ohm’s law to the supermesh and the other is obtained by relation regulated due to the current source. PSUT Methods of Analysis52

53.  Similarly, a supermesh formed from three meshes needs three equations: one is from the supermesh and the other two equations are obtained from the two current sources. PSUT Methods of Analysis53

54. PSUT Methods of Analysis54

55. 3.6 Nodal and Mesh Analysis by Inspection PSUT Methods of Analysis55 (a)For circuits with only resistors and independent current sources (b)For planar circuits with only resistors and independent voltage sources The analysis equations can be obtained by direct inspection

56.  the circuit has two nonreference nodes and the node equations PSUT Methods of Analysis56

57.  In general, the node voltage equations in terms of the conductances is PSUT Methods of Analysis57 or simply Gv = i where G : the conductance matrix, v : the output vector, i : the input vector

58.  The circuit has two nonreference nodes and the node equations were derived as PSUT Methods of Analysis58

59.  In general, if the circuit has N meshes, the mesh- current equations as the resistances term is PSUT Methods of Analysis59 or simply Rv = i where R : the resistance matrix, i : the output vector, v : the input vector

60.  Write the node voltage matrix equations PSUT Methods of Analysis60

61.  The circuit has 4 nonreference nodes, so  The off-diagonal terms are PSUT Methods of Analysis61

62.  The input current vector i in amperes  The node-voltage equations are PSUT Methods of Analysis62

63.  Write the mesh current equations PSUT Methods of Analysis63

64.  The input voltage vector v in volts  The mesh-current equations are PSUT Methods of Analysis64

65. 3.7 Nodal Versus Mesh Analysis  Both nodal and mesh analyses provide a systematic way of analyzing a complex network.  The choice of the better method dictated by two factors. ●First factor : nature of the particular network. The key is to select the method that results in the smaller number of equations. ●Second factor : information required. PSUT Methods of Analysis65

66. 3.10 Summery 1.Nodal analysis: the application of KCL at the nonreference nodes ●A circuit has fewer node equations 2.A supernode: two nonreference nodes 3.Mesh analysis: the application of KVL ●A circuit has fewer mesh equations 4.A supermesh: two meshes PSUT Methods of Analysis66