4 BASIC NODAL AND MESH ANALYSIS
4
35
Basic Nodal and Mesh Analysis
4.1
Introduction
4.2
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.
Current flows from a higher potential to a lower potential in a resistor
i=
4.2.1
vhigher − vlower
R
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:
4 BASIC NODAL AND MESH ANALYSIS
36
– 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.
4.3
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.
4.3.1
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.
4 BASIC NODAL AND MESH ANALYSIS
4.4
37
Nodal and Mesh analysis by Inspection
• Nodal Analysis

 G11


 G21

 .
 ..


Gn1
Gkk
Gkj
vk
ik
G12
G22
...
Gn2
···
···
...
···
G1n
G2n
...
Gnn



 i1
  v1 






 i
  v2 
 =  2

 .

 .
 ..

  ..






in
vn










=
=
=
=
sum of the conductances connected to node k
Gjk = negative of the sum of the conductances directly connect
unknown voltage at node k
sum of all independent current sources directly connected to n
In matrix form, G~v = ~i
G is called the conductance matrix.
• Mesh Analysis

 R11


 R21

 .
 ..


Rn1
Rkk
Rkj
ik
vk
R12
R22
...
Rn2
···
···
...
···
R1n
R2n
...
Rnn


 v1
  i1 






 v
  i2 
 =  2

 .
 . 
 ..
  .. 






vn
in










sum of the resistances in mesh k
Rjk = negative of the sum of the resistances in common with m
unknown mesh current for mesh k in the clockwise direction.
clockwise sum of all independent voltage sources in mesh k, wi
In matrix form, R~i = ~v
R is called the resistance matrix.
4.5
=
=
=
=

Computer-Aided Circuit Analysis
Pick a common node and label it ’0’.
4 BASIC NODAL AND MESH ANALYSIS
4.6
38
Summary and Review
4.6.1
Applications: DC Transistor Circuits
4.6.2
Using Maxima to solve system of equations-hkd7ec4p3
Solve the following three equations in three unknowns:
4 = v1/100 + (v1 − v2)/20 + (v1 − vx)/50
10 − 4 − (−2) = (vx − v1)/50 + (vx − v2)/40
−2 = v2/25 + (v2 − vx)/40 + (v2 − v1)/20
/* hkd7ec4p3.mac */
e1 : 4 = v1/100 + (v1 - v2)/20 + (v1 - vx)/50;
e2 : 10 - 4 - (-2) = (vx - v1)/50 + (vx - v2)/40;
e3 : -2 = v2/25 + (v2 - vx)/40 + (v2 - v1)/20;
sol: linsolve([e1,e2,e3],[v1,v2,vx]);
/* float(sol); [v1 = 264.308, v2 = 183.922, vx = 397.427] */ rearranging the three equations
0.08v1 − 0.05v2 − 0.02v3 = 4
−0.02v1 − 0.025v2 + 0.045v3 = 8
−0.05v
1 + 0.115v2 − 0.025v

 3 = −2 

−0.05 −0.02   v1 
 4 
 0.08











 −0.02 −0.025
0.045   v2  =  8  ⇒ A~x = ~b using MatLab






−2
v3
−0.05 0.115 −0.025
~x = inv(A) ∗ ~b ⇒ ~x′ = [264.3, 183.9, 397.4]