Methods of Analysis

1
Chapter 3

• Methods of Analysis

2
Introduction

• Techniques for circuit analysis
• Nodal Analysis
• ? Based on KCL
• Mesh Analysis
• ? Based on KVL

3
Nodal Analysis without Voltage Source

• Make use of node voltages as circuit variables.
• Given a circuit with n nodes
• Select a node as the reference node.
• Reference node ? Node with 0 potential.
• 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 Ohms Law to express the branch currents in
  terms of node voltages.
• Solve to determine the unknown node voltages. Use
  substitution method or Cramers Rule.

4
Nodal Analysis without Voltage Source

• Example 1
• Obtain the node voltages.
• Answer
• v1 -2v
• v2 -14v

5
Nodal Analysis without Voltage Source

• Example 2
• Find the voltages at the three non-reference
  nodes.
• Answer
• v1 80v
• v2 -64v
• v3 156v

6
Nodal Analysis with Voltage Source

• Two cases to consider
• Case 1 A voltage source is connected between
  reference node and non-reference node.
• Set the voltage at the non-reference node equal
  to the voltage source.
• Case 2 A voltage source connected between two
  non-reference node
• The two non-reference nodes form a super node.
• Apply both KCL and KVL to determine the node
  voltages.
• Super node is formed by enclosing a (dependent
  or independent) voltage source connected between
  two non-reference nodes and any elements
  connected in parallel with it.

7
Nodal Analysis with Voltage Source

• Properties of super node
• The voltage source inside the super node provides
  constraint equation needed to solve for the node
  voltages.
• A super node has no voltage of its own.
• A super node requires the application of both KCL
  and KVL.

8
Nodal Analysis with Voltage Source

• Example 1 Find v and i
• v-0.2v, i 1.4A

9
Nodal Analysis with Voltage Source

• Example 2 Find v1, v2 and v3 using nodal
  analysis
• v1 -3.043 V, v2 -6.956 V, v3 0.6522 V

10
Mesh Analysis

• Make use of mesh currents as circuit variables
• A mesh is a loop that does not contain any other
  loop within it.
• Only applicable to a planar circuit
• Planar circuit
• A circuit that can be drawn in a plane with no
  branches crossing one another otherwise
  non-planar.

11
Mesh Analysis

• Planar circuit
• ?

12
Mesh Analysis

• Non-planar circuit
• No way to redraw it and avoid the branches
  crossing

13
Mesh Analysis without Current Sources

• Given a circuit with n meshes.
• Assign mesh currents i1, i2, .. in to the n
  meshes. Assume mesh current flows clockwise.
• Apply KVL to each of the n meshes. Use Ohms Law
  to express the voltages in terms of mesh currents.

14
Mesh Analysis without Current Sources

• Example 1 Determine mesh currents i1 and i2.
• i1 2/3 A
• i2 0A

15
Mesh Analysis without Current Sources

• Example 2 Determine I0 .
• I0 -5A

16
Mesh Analysis with Current Sources

• Two cases to consider
• Case 1 A current source exists only in one mesh.
• Set the mesh current current source.
• Case 2 A current source exists between the two
  meshes.
• Create super mesh by excluding the current source
  and any elements connected in series with it.
• A super mesh results when two meshes have a
  (dependent or independent) current source in
  common.

17
Mesh Analysis with Current Sources

• Properties of super mesh
• Current source in the super mesh provides the
  constraint.
• Super mesh has no current of its own.
• Super mesh requires the application of both KVL
  and KCL.

18
Mesh Analysis with Current Sources

• Example 1 Determine i1, i2 and i3.
• i13.474
• i2 0.4737
• i3 1.1052

19
Nodal Analysis by Inspection

• Implemented when all sources in the circuit are
  independent current sources.
• Convert resistors to conductors.

20
Nodal Analysis by Inspection

• Example By inspection, determine the
  node-voltage equations for the circuit.

21
Mesh Analysis by Inspection

• Implemented when all sources in the circuit are
  independent voltage sources.

22
Mesh Analysis by Inspection

• Example By inspection, determine the
  mesh-current equations for the circuit.