K. A. Saaifan, Jacobs University, Bremen
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4. Basic Nodal and Mesh Analysis
This chapter introduces two basic circuit analysis techniques named nodal analysis
and mesh analysis
4.1 Nodal Analysis
For a simple circuit with two nodes, we often have one unknown “voltage
between two nodes”
To solve the unknown, applying KCL at this node gives
Adding a node should provide an additional unknown, three-node circuit has
2 unknown
N-node circuit has (N-1) voltages with (N-1) equations.
K. A. Saaifan, Jacobs University, Bremen
Nodal technique applies the following step
1- Count the number of nodes (N)
2- Designate a reference node
3- Label the nodal voltages (we have N-1 voltages)
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K. A. Saaifan, Jacobs University, Bremen
4- Write KCL equations for the non-reference nodes (currents in = currents out)
5- Organize the equations
(1)
(2)
6- Solve the system of equations for the nodal voltages
K. A. Saaifan, Jacobs University, Bremen
Using a Cramer's rule and determinants, we have
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K. A. Saaifan, Jacobs University, Bremen
Compute the voltages at each node
Ans:
Write KCL equations for the three nodes
Organize the equations
(1)
(2)
(3)
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K. A. Saaifan, Jacobs University, Bremen
Compute the voltage at each node
Ans:
Solve the system of equations for the nodal voltages
Use a Cramer's rule and determinants to
solve the system
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K. A. Saaifan, Jacobs University, Bremen
4.2 Nodal Analysis with Supernode
A supernode is formed when a voltage source is the only element
connected between two essential nodes
1- Define a current through the source and
write KCL equations for the two nodes
2- We note that there is no need to determine ivs to solve the circuit
(1)
3- Apply KVL between the two nodes
(2)
Thus, the KCL at the supernode is directly given
by
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K. A. Saaifan, Jacobs University, Bremen
Determines the node-to reference voltages
.
Node 1 to reference is supernode
Node 2
Node 3 & node 4
Express vx=v2-v1 and vy=v4-v1 in terms of
nodal voltages and organize the equations
(1)
(2)
(3)
Solve to get
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K. A. Saaifan, Jacobs University, Bremen
4.3 Mesh Analysis
In nodal analysis, circuit variables are node voltages
Nodal analysis applies KCL to find unknown voltages
In mesh analysis, circuit variables are mesh currents
Mesh analysis applies KVL to find unknown currents
Both methods result in a system of linear equations
Mesh analysis is only applicable to a circuit that is planar
Planar vs. Non-planar Circuits
Planar circuit: it can be drawn on a plane surface where no branch cross
any other branch (element)
Non-planar circuit there is no way to redraw it and avoid the branches
crossing
Planar circuit
Non planar circuit
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K. A. Saaifan, Jacobs University, Bremen
Mesh & mesh current
A mesh is a property of a planar circuit and it is defined a loop that does
not contain any other loops within it
The current through a mesh is known as a mesh current
mesh
mesh
K. A. Saaifan, Jacobs University, Bremen
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4.3 Mesh Analysis
1. Determine if the circuit is a planar circuit. If not, perform nodal analysis instead.
2. Count the number of meshes (M)
3. Label each of the M mesh currents (defining all mesh currents to flow clockwise
results in a simpler analysis)
4. Write a KVL equation around each mesh
For mesh 1, we have
or
(1)
For mesh 2, we have
or
The solution is easily obtained
(2)
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K. A. Saaifan, Jacobs University, Bremen
Determine the power supplied by the 2 V source
.
i1
We first define two clockwise mesh currents
For mesh 1, we write the following KVL equation
The same for mesh 2, we write
i2
K. A. Saaifan, Jacobs University, Bremen
Rearranging and grouping terms, we have
and
Solve the both equation yields
i1=1.132 A and i2=-0.1053 A
The 2 V source supplies (2)(i1-i2)=2.4 W
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K. A. Saaifan, Jacobs University, Bremen
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4.4 The Supermesh
Similar to the supernode in a node voltage analysis
A supermesh is formed when a current source is the only element connected
between two meshes
1- Define a voltage across the source and
write KVL equations for the two meshes
and
2- We do not need to evaluate vcs to solve the circuit
3- This leads us to create a supermesh whose interior is that of mesh 1 and mesh 2
4- Finally, the source current is related to the mesh currents,
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K. A. Saaifan, Jacobs University, Bremen
Determine the three mesh currents
.
i1
i2
i1
i2
i1-i2
i1-i2
i3-i2
i3
i3-i2
i3
i1-i3
The 7 A independent current source forms a supermesh between mesh 1 and
mesh 3
Applying KVL over the supermesh gives
or
KVL for mesh 2
or
K. A. Saaifan, Jacobs University, Bremen
Homework Assignment 3
P4.8, P4.10, P4.14, P4.22, P4.26, P4.31, P4.36, P4.44
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