1. Fakulti Kejuruteraan Elektrik

2. - with both independent voltage and current sources
Fakulti Kejuruteraan Elektrik
Methods of Analysis
Nodal Analysis
Mesh Analysis
- with both independent voltage and current sources
- With both dependent voltage and current sources

3. NODAL ANALYSIS

4. Nodal Analysis
It provides a general procedure for analyzing circuits using node voltages as the circuit variables.
Example 1

5. Nodal Analysis Steps to determine the node voltages:
Select a node as the reference node.
Assign voltages v1,v2,…,vn-1 to the remaining n-1 nodes. The voltages are referenced with respect to the reference node.
Apply KCL to each of the n-1 non-reference nodes. Use Ohm’s law to express the branch currents in terms of node voltages.
Solve the resulting simultaneous equations to obtain the unknown node voltages.

6. Reference node: zero potential

7. Circuit with current sources only1 2 + v2 - + v1 - i2 v1 v2 i1 i3

8. i1, i2 and i3 : in terms of node voltages
Ohm’s law:
i1, i2 and i3 : in terms of node voltages i2 v1 v2 i1 i3
Applying KCL
At node 1:
At node 2

9. i2 v1 v2 i1 i3

10. In terms of the conductances:
In matrix form:
Solve for the node voltages
cramer’s rule
Substitution method
elimination method
Calculators
Software packages

12. Practice problem 3.1 (pg 85) 3

14. Practice Problem 3.2 (page 88)

15. Nodal analysis with voltage sources
Case 1: If a voltage source is connected between the reference node and the non-reference node, we simply set the voltage at the non-reference equal to the voltage of the voltage source.
Case 2: When a voltage source (dependent or independent) is connected between two nonreference nodes, we can combine those nodes to form a supernode:
KCL and KVL will be applied to determine the node voltages

16. Node 2 and 3 form a supernode
1st – Applying KCL to the supernode;(Supernode eqns)
2nd – Applying KVL to get the relationship between the two nodes. (Supporting eqns)

17. At the supernode:
KCL at supernode
KVL to the supernode

19. Practice Problem 3.3 (page 90)

21. Practice problem 3.4 (page 93)

22. MESH ANALYSIS

23. Mesh Analysis Only applicable to a planar circuit only
Planar Circuit: the circuit that can be drawn in a plane with no branches crossing one another, otherwise it is nonplanar.

24. Planar circuit

25. Planar circuit

26. Nonplanar circuit

27. Nonplanar

28. Mesh Analysis
Mesh analysis provides another general procedure for analyzing circuits using mesh currents as the circuit variables.
Nodal analysis applies KCL to find unknown voltages in a given circuit, while mesh analysis applies KVL to find unknown currents.
A mesh is a loop which does not contain any other loops within it.

29. Mesh Analysis Steps to determine the mesh currents:
Assign mesh currents i1, i2, …, in to the n meshes.
Apply KCL to each of the n meshes. Use Ohm’s law to express the voltages in terms of the mesh currents.
Solve the resulting n simultaneous equations to get the mesh currents.

30. i2 i1
Applying KVL to mesh 1
Applying KVL to mesh 2

31. Example 3.5

32. Practice problem 3.5 (page 96)

33. Example 3.6

34. Practice Problem 3.6 (page 98)

35. Mesh analysis with Current sources
Case 1: When a current source exists only in one mesh, we simply take the value for that loop/mesh equal to the value stated for the current source

36. Mesh analysis with Current sources
Case 2: If the current source (dependent or independent) exists between two meshes, we create a supermesh by excluding the current source and any elements connected in series with it.

37. Mesh 1 and 2 form a supermesh
1st – Applying KVL to the supermesh;(Supermesh eqns)
2nd – Applying KCL to a node in the branch where the two meshes intersect. (Supporting eqns)

39. Practice problem 3.7
a) Determine i1, i2 and i3
b) Determine the current through each element.

40. Problem 3.38
a) Appy mesh analysis to obtain Io

41. Problem 3.27
Use nodal analysis to find voltages V1, V2 and V3

42. Problem 3.31
Find the node voltages for the circuit below

43. Problem 3.64
Find vo and io