Node and Mesh Analysis
1
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Circuit Terminology
Name
Definition
Node
A point where two ore more branches meet
Essential node
A node where three or more branches combine
Path
A trace of the adjacent circuit elements, where no element is included more than
once.
Branch
A path that connects two nodes, and contains a single element such as voltage
source or resistor
Essential Branch
Path that connects two nodes without passing through an essential node.
Loop
A closed path in a circuit
Mesh
A loop that does not contain any other loops.
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Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Circuit analysis methods introduced so
far
• Voltage-current relations:
• Ohm’s Law
• Kirchoff’s Current Law (KCL)
• Kirchoff’s Voltage Law (KVL)
• Circuit Reduction
• But circuit reduction is just a way of applying
Ohm’s Law, KCL, and KVL to simplify the analysis
by reducing the number of unknowns!
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Example Circuit
•Circuit reduction
techniques don’t apply
•Large number of
unknowns, if we use
exhaustive application
of KVL, KCL, and
Ohm’s Law
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Two new analysis techniques
• Next:
• Nodal Analysis
• Mesh Analysis
• Nodal analysis and mesh analysis provide
rigorous ways to define a (relatively small)
set of unknowns and write the circuit
governing equations in terms of these
unknowns
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Nodal analysis – overview
• Identify independent nodes
• The voltages at these nodes are the node voltages
• Use Ohm’s Law to write KCL at each
independent node in terms of the node
voltages
• Solve these equations to determine the node
voltages
• Any desired circuit parameter can be
determined from the node voltages
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Mesh analysis – overview
• Identify mesh loops
• The currents around these loops are the mesh
currents
• Use Ohm’s Law to write KVL around each loop
in terms of the mesh currents
• Solve these equations to determine the mesh
currents
• Any desired circuit parameter can be
determined from the mesh currents
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Important observation
• Nodal analysis and mesh analysis are not
fundamentally “new” analysis techniques
• We are still applying KVL, KCL, and Ohm’s Law!
• Nodal and mesh analysis simply allow us to
identify a reduced set of unknowns which
completely characterize the circuit  we can write
and solve fewer equations to simplify our
analysis!
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Nodal Analysis
v1  v2
v2  v1
i1 
and i2 
R
R
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Nodal Analysis
• We will illustrate the nodal analysis
technique in the context of an example
circuit:
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Nodal Analysis
•Step 1: Identify a
reference node
•Label the reference
node voltage as VR =
0V
•The reference node is
arbitrary! You are
merely identifying the
node to which all
subsequent voltages
will be referenced
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Nodal Analysis
Step 2: “Kill” sources
and identify independent
nodes
•Short-circuit voltage
sources
•Open-circuit current
sources
•The remaining nodes
are “independent”
•Label voltages at these
nodes
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Nodal Analysis
•Step 3: Replace
sources and label
“constrained” voltages
•The constrained
voltages are at
dependent nodes
•Voltage sources
“constrain” the
difference in voltage
between nodes they
interconnect
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Nodal Analysis
Step 4: Apply KCL at
each independent
node
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Nodal Analysis
Step 5: Use Ohm’s
Law to write the KCL
equations in terms of
node voltages
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Nodal Analysis
Step 5: continued
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Nodal Analysis
Step 6: Solve the
system of equations
to determine the
node voltages
•The node voltages
can be used to
determine any other
desired parameter in
the circuit
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Nodal Analysis – checking results
• Checking results in step 5:
• In general, in the equation for node “X”, the
multiplicative factor on the node voltage VX will be
the sum of the conductances at node “X”
• The multiplicative factors on all other node
voltages in the equation will be the negative of the
conductances between node “X” and the
respective node voltage
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Nodal Analysis – checking results
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Nodal Analysis – shortcuts
• It is common to combine steps 4 and 5
• Apply KCL and Ohm’s Law simultaneously
• You can, if you wish, choose your current
directions independently each time you
apply KCL
• For example, you can assume that all currents are
leaving the node, each time you apply KCL
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Shortcuts applied to our example
Previous Results:
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Obtain values for the unknown voltages across the
elements in the circuit below.
At node 1
At node 2
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
v1
v1  v2

 3.1
2
5
v2
v v
 2 1  - (-1.4)
1
5
(a) The circuit of Example 4.2 with a
22-V source in place of the 7-W
resistor. (b) Expanded view of the
region defined as a supernode; KCL
requires that all currents flowing
into the region must sum to zero, or
we would pile up or run out of
electrons.
At node 1:
v1  v2 v1  v3
83 

3
4
At the “supernode:”
v2  v1 v3  v1 v3 v2
3  25 

 
3
4
5 1
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Determine the node-to-reference voltages in the circuit below.
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Examples of planar and nonplanar networks; crossed wires without
a solid dot are not in physical contact with each other.
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
(a) The set of branches identified by the heavy lines is neither a path nor a
loop. (b) The set of branches here is not a path, since it can be traversed
only by passing through the central node twice. (c) This path is a loop but
not a mesh, since it encloses other loops. (d) This path is also a loop but
not a mesh. (e, f) Each of these paths is both a loop and a mesh.
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
The Node Analysis
• The node analysis uses the voltages at the nodes as circuit
variables.
• Example 3.1
Determine the voltage v1 and v2 using node analysis.
1
+
1W
10 V
5W
v1
-
2
2W
+
v2
10 W
2A
-
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Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Solution
• The circuit shown in Figure 3.3 contains 3 essential nodes (ne =
3); so we need (ne­-1) node-voltage equations to describe the
circuit.
• The steps in node analysis are as follows:
• A reference node is chosen. Normally, the node with the largest
number of branches will be chosen as the reference node. In
Figure 3.3, the node at the bottom of the circuit (indicated by ▼)
contained the largest number of branches, so it was chosen as the
reference node. The rest of the nodes in the circuit are called
non-reference nodes.
• Using Ohm’s law, we formulate the node voltage equation for
each node. For this circuit, we define the node voltages as v1 and
v2. The node voltage is defined as the voltage increase from the
reference node to the nonreference node.
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Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
• According to Kirchoff’s Current Law, the total current leaving
each branch is equal to zero. Therefore the node voltage
equation at node 1 is:
And at node 2:
v 1  10 v 1 v 1  v 2


0
1
5
2
v 2  v1 v 2

20
2
10
Solve the simultaneous equations to obtain the unknown node voltages,
in this case v1 and v2. The simultaneous equations which describe the
circuit above in terms of v1 and v2. By solving them, we obtain:
v1 = 9.09 V
v2 = 10.91 V
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Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Mesh Current Analysis




V1  I1  I 2 R and V2  I 2  I1 R
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Determine the two mesh currents, i1 and i2, in the circuit below.
For the left-hand mesh,
-42 + 6 i1 + 3 ( i1 - i2 ) = 0
For the right-hand mesh,
3 ( i2 - i1 ) + 4 i2 - 10 = 0
Solving, we find that i1 = 6 A and i2 = 4 A.
(The current flowing downward through
the 3-W resistor is therefore i1 - i2 = 2 A. )
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Find the three mesh currents in the circuit below.
Creating a “supermesh” from meshes 1 and 3:
-7 + 1 ( i1 - i2 ) + 3 ( i3 - i2 ) + 1 i3 = 0
[1]
Around mesh 2:
1 ( i2 - i1 ) + 2 i2 + 3 ( i2 - i3 ) = 0
[2]
Finally, we relate the currents in meshes 1 and 3:
i1 - i3 = 7
[3]
Rearranging,
i1 - 4 i2 + 4 i3 = 7
[1]
-i1 + 6 i2 - 3 i3 = 0
[2]
i1
[3]
- i3 = 7
Solving,
i1 = 9 A, i2 = 2.5 A, and i3 = 2 A.
Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Mesh Analysis
• Figure 3.11 illustrates the example of using mesh current
with the exist of dependent sources. This circuit contains six
branches where the current is unknown and four nodes.
Therefore three mesh current is required to describe the
circuit (6 – (4 – 1) = 3). These currents are shown in Figure
3.12.
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Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
The current equations are:
50 = 5(i1 – i2) + 20(i1 – i3)
0 = 5(i2 – i1) + 1i2 + 4(i2 – i3)
0 = 20(i3 - i1) + 4(i3 – i2) - 15iΦ
(3.18)
(3.19)
(3.20)
If the current that controls the independent
source is expressed in terms of mesh current,
iΦ = i1 – i3
(3.21)
By substituting equation (3.21) into equation
(3.20) and rearranging the variables,
50 = 25i1 - 5i2 - 20i3
(3.22)
0 = -5i1 + 10i2 - 4i3
(3.23)
0 = -35i1 - 4i2 + 9i3
(3.24)
Solving the simultaneous equations,
i1 = -0.57 A, i2 = -1.43A and i3 = -2.86 A.
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Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Summary
• The number of equations is equal to the number of
unknown.
• The Node voltage analysis is based on Kirchhoff’s
Current Law.
• The Mesh analysis is based on Kirchhoff’s Voltage
Law.
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Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia
Thank You
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Subject Matter Expert/Author: Wei Wen Shyang (OUM)
Copyright © ODL Jan 2005 Open University Malaysia