Nodal & Mesh Analysis

NODAL ANALYSIS:



Steps to Determine Node Voltages

• Select a node as the reference node. Assign voltages v1, v2, · · · , vn − 1
to determine the remaining n − 1 nodes.
The voltages are referenced with respect to the reference
(datum) node (ground).
• 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.

Nodal Analysis with Voltage Sources

• Voltage source between reference node and a non reference node.
Set the voltage of the non reference node to that of the voltage
source.
• Voltage source between two non reference nodes.
Form a supernode (generalized node) and apply both KVL and
KCL to determine the node voltages.
A supernode is formed by enclosing a (dependent or independent)
voltage source connected between two non reference nodes
and any elements connected in parallel with it.
Properties of a supernode:




BASIC NODAL AND MESH ANALYSIS
– The voltage source inside the supernode provides a constraint
equation needed to solve for the node voltages.
– A supernode has no voltage of its own.

– A supernode requires the application of both KCL and KVL.




MESH ANALYSIS:
-A mesh is a loop that does not contain any other loops within it.

Steps to determine mesh currents:
• Assign mesh currents i1, i2, · · · , in to the meshes.
• Apply KVL 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.



Mesh Analysis with Current Sources
• Current source exists only in one mesh. Here mesh current = ±
current source.
• Current source between two meshes, form ’Supermesh’ by excluding
the current source any element connected in series with it.
• A Supermesh results when two meshes have a (dependent or independent)
current source in common.

Properties of a supermesh
-The current source in the supermesh provides the constraint
equation necessary to solve for the mesh currents.
– A supermesh has no current of its own.

– A supermesh requires the application of both KVL and KCL.





Nodal and Mesh Analysis by Inspection

Gkk = sum of the conductances connected to node k
Gkj = Gjk= negative of the sum of the conductances directly connecting
vk = unknown voltage at node k
ik = sum of all independent current sources directly connected to no
In matrix form, G~v =~i

G is called the conductance matrix.

Rkk = sum of the resistances in mesh k
Rkj = Rjk= negative of the sum of the resistances in common with meshes
ik = unknown mesh current for mesh k in the clockwise direction.
vk = clockwise sum of all independent voltage sources in mesh k, with
In matrix form, R~i = ~v
R is called the resistance matrix.