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EENG223 Mesh Analysıs1
Mesh Analysis
Dr. M. K. Uyguroglu
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EENG223 Mesh Analysıs2
Mesh Analysis
Nodal analysis was developed by applying KCL at each
non-reference node.Mesh analysis is developed by applying KVL
around meshes/loops in the circuit.Mesh analysis results in a
system of linear equations which must be solved for unknown
currents.
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EENG223 Mesh Analysıs3
Mesh Analysis
quantity of interest is currenta mesh is a loop that does not
contain another loop within itwork for planar circuit onlyplanar
circuit -> no branch passes over or under other branchM-meshes
-> assign clockwise current for each meshapply KVL around each
mesh
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EENG223 Mesh Analysıs4
Planar Circuit
Nonplanar Circuit
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EENG223 Mesh Analysıs5
Steps of Mesh Analysis
1. Identify meshes.2. Assign a current to each mesh.3. Apply KVL
around each loop to get an
equation in terms of the loop currents.4. Solve the resulting
system of linear
equations.
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EENG223 Mesh Analysıs6
Identifying the Meshes
Mesh 2
1kΩ
1kΩ
1kΩ
V1 V2Mesh 1+–
+–
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EENG223 Mesh Analysıs7
Steps of Mesh Analysis
1. Identify mesh (loops).2. Assign a current to each mesh.3.
Apply KVL around each loop to get an
equation in terms of the loop currents.4. Solve the resulting
system of linear
equations.
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EENG223 Mesh Analysıs8
Assigning Mesh Currents
1kΩ
1kΩ
1kΩ
V1 V2I1 I2+–
+–
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EENG223 Mesh Analysıs9
Steps of Mesh Analysis
1. Identify mesh (loops).2. Assign a current to each mesh.3.
Apply KVL around each loop to get an
equation in terms of the loop currents.4. Solve the resulting
system of linear
equations.
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EENG223 Mesh Analysıs10
Voltages from Mesh Currents
R
I1
+ –VR
VR = I1 R
R
I1
+ –VRI2
VR = (I1 - I2 ) R
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EENG223 Mesh Analysıs11
KVL Around Mesh 1
1kΩ
1kΩ
1kΩ
V1 V2I1 I2+–
+–
+ - +
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-V1 + I1 1kΩ + (I1 - I2) 1kΩ = 0I1 1kΩ + (I1 - I2) 1kΩ = V1
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EENG223 Mesh Analysıs12
KVL Around Mesh 2
1kΩ
1kΩ
1kΩ
V1 V2I1 I2+–
+–
--
+
+
(I2 - I1) 1kΩ + I2 1kΩ + V2 = 0(I2 - I1) 1kΩ + I2 1kΩ = -V2
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EENG223 Mesh Analysıs13
Steps of Mesh Analysis
1. Identify mesh (loops).2. Assign a current to each mesh.3.
Apply KVL around each loop to get an
equation in terms of the loop currents.4. Solve the resulting
system of linear
equations.
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EENG223 Mesh Analysıs14
Matrix Notation
The two equations can be combined into a single matrix/vector
equation.
−
=
Ω+ΩΩ−
Ω−Ω+Ω
2
1
2
1
k1k1k1k1k1k1
VV
II
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EENG223 Mesh Analysıs15
Solving the Equations
Let: V1 = 7V and V2 = 4VResults:
I1 = 3.33 mAI2 = -0.33 mA
FinallyVout = (I1 - I2) 1kΩ = 3.66V
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EENG223 Mesh Analysıs16
Another Example
1kΩ
2kΩ
2kΩ
12V 4mA
2mA
I0
+–
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EENG223 Mesh Analysıs17
1. Identify Meshes
Mesh 2
Mesh 3
Mesh 1
1kΩ
2kΩ
2kΩ
12V 4mA
2mA
I0
+–
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EENG223 Mesh Analysıs18
2. Assign Mesh Currents
I1 I2
I31kΩ
2kΩ
2kΩ
12V 4mA
2mA
I0
+–
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EENG223 Mesh Analysıs19
Current Sources
The current sources in this circuit will have whatever voltage
is necessary to make the current correct.We can’t use KVL around
the loop because we don’t know the voltage.What to do?
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EEE 223 Mesh Analysıs20
Current Sources
The 4mA current source sets I2:I2 = -4 mA
The 2mA current source sets a constraint on I1 and I3:
I1 - I3 = 2 mAWe have two equations and three unknowns. Where is
the third equation?
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EENG223 Mesh Analysıs21
1kΩ
2kΩ
2kΩ
12V 4mA
2mA
I0
I1 I2
I3The Supermesh surrounds this source!
The Supermesh
does not include this
source!
+–
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EENG223 Mesh Analysıs22
KVL Around the Supermesh
-12V + I3 2kΩ + (I3 - I2)1kΩ + (I1 - I2)2kΩ = 0
I3 2kΩ + (I3 - I2)1kΩ + (I1 - I2)2kΩ = 12V
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EENG223 Mesh Analysı s23
Matrix Notation
The three equations can be combined into a single matrix/vector
equation.
−=
Ω+ΩΩ−Ω−Ω−
V12mA2mA4
1k2k2k1k2k101
010
3
2
1
III
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EENG223 Mesh Analysıs24
Solve Using MATLAB
>> A = [0 1 0; 1 0 -1;2e3 -1e3-2e3 2e3+1e3];
>> v = [-4e-3; 2e-3; 12];>> i = inv(A)*vi =
0.0012
-0.0040-0.0008
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EENG223 Mesh Analysıs25
Solution
I1 = 1.2 mAI2 = -4 mAI3 = -0.8 mA
I0 = I1 - I2 = 5.2 mA
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EENG223 Mesh Analysıs26
Class Example
Mesh AnalysisMesh AnalysisMesh AnalysisPlanar CircuitSteps of
Mesh AnalysisIdentifying the MeshesSteps of Mesh AnalysisAssigning
Mesh CurrentsSteps of Mesh AnalysisVoltages from Mesh CurrentsKVL
Around Mesh 1KVL Around Mesh 2Steps of Mesh AnalysisMatrix
NotationSolving the EquationsAnother Example1. Identify Meshes2.
Assign Mesh CurrentsCurrent SourcesCurrent SourcesKVL Around the
SupermeshMatrix NotationSolve Using MATLABSolution
