Basic Laws of Electric Circuits

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Basic Laws of Electric Circuits
Mesh Analysis
Lesson 7
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Basic Circuits
Mesh Analysis Basic Concepts
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In formulating mesh analysis we assign a mesh
current to each mesh.
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Mesh currents are sort of fictitious in that a
particular mesh current does not define the
current in each branch of the mesh to which it is
assigned.
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Basic Circuits
Mesh Analysis Basic Concepts
Figure 7.2 A circuit for illustrating mesh
analysis.
Around mesh 1
Eq 7.1
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Basic Circuits
Mesh Analysis Basic Concepts
Eq 7.2
Eq 7.3
Eq 7.4
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Basic Circuits
Mesh Analysis Basic Concepts
We are left with 2 equations From (7.1) and
(7.4) we have,
Eq 7.5
Eq 7.6
We can easily solve these equations for I1 and I2.
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Basic Circuits
Mesh Analysis Basic Concepts
The previous equations can be written in matrix
form as
Eq (7.7)
Eq (7.8)
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Basic Circuits
Mesh Analysis Example 7.1.
Write the mesh equations and solve for the
currents I1, and I2.
Figure 7.2 Circuit for Example 7.1.
Eq (7.9)
4I1 6(I1 I2) 10 - 2
Mesh 1
Eq (7.10)
Mesh 2
6(I2 I1) 2I2 7I2 2 20
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Basic Circuits
Mesh Analysis Example 7.1, continued.
Simplifying Eq (7.9) and (7.10) gives,
10I1 6I2 8 -6I1 15I2 22
Eq (7.11)
Eq (7.12)
    A MATLAB Solution R 10 -6-6
15 V 822 I inv(R)V   I  
2.2105 2.3509    
I1 2.2105 I2 2.3509
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Basic Circuits
Mesh Analysis Example 7.2
Solve for the mesh currents in the circuit below.
Figure 7.3 Circuit for Example 7.2.
The plan
Write KVL, clockwise, for each mesh. Look for a
pattern in the final equations.
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Basic Circuits
Mesh Analysis Example 7.2
Mesh 1 6I1 10(I1 I3) 4(I1 I2)
20 10
Eq (7.13)
Mesh 2 4(I2 I1) 11(I2 I3) 3I2
- 10 - 8
Eq (7.14)
Mesh 3 9I3 11(I3 I2) 10(I3 I1)
12 8
Eq (7.15)
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Basic Circuits
Mesh Analysis Example 7.2
Clearing Equations (7.13), (7.14) and (7.15)
gives,
Standard Equation form
In matrix form
20I1 4I2 10I3 30
-4I1 18I2 11I3 -18
-10I1 11I2 30I3 20
WE NOW MAKE AN IMPORTANT
OBSERVATION!!
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Basic Circuits
Mesh Analysis Standard form for mesh equations
Consider the following
R11
of resistance around mesh 1, common to mesh 1
current I1.
R22
of resistance around mesh 2, common to mesh 2
current I2.
R33
of resistance around mesh 3, common to mesh 3
current I3.
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Basic Circuits
Mesh Analysis Standard form for mesh equations
R12 R21 - resistance common between mesh 1
and 2 when I1 and I2 are
opposite through R1,R2.
R13 R31 - resistance common between mesh 1
and 3 when I1 and I3 are
opposite through R1,R3.
R23 R32 - resistance common between mesh 2
and 3 when I2 and I3 are
opposite through R2,R3.

sum of emf around mesh 1 in the direction of I1.
sum of emf around mesh 2 in the direction of I2.
sum of emf around mesh 3 in the direction of I3.
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Basic Circuits
Mesh Analysis Example 7.3 - Direct method.
Use the direct method to write the mesh equations
for the following.
Figure 7.4 Circuit diagram for Example 7.3.
Eq (7.13)
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Basic Circuits
Mesh Analysis With current sources in the
circuit
Example 7.4 Consider the following
Figure 7.5 Circuit diagram for Example 7.4.
Use the direct method to write the mesh equations.
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Basic Circuits
Mesh Analysis With current sources in the
circuit
This case is explained by using an
example. Example 7.4 Find the three mesh
currents in the circuit below.
Figure 7.5 Circuit for Example 7.4.
When a current source is present, it will be
directly related to one or more of the mesh
current. In this case I2 -4A.
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Basic Circuits
Mesh Analysis With current sources in the
circuit
Example 7.4 Continued. An easy way to handle
this case is to remove the current source as
shown below. Next, write the mesh equations for
the remaining meshes.
Note that I 2 is retained for writing the
equations through the 5 ? and 20 ? resistors.
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Basic Circuits
Mesh Analysis With current sources in the
circuit
Example 7.4 Continued.
Equation for mesh 1
10I1 (I1-I2)5 10
or
15I1 5I2 10
Equations for mesh 2
Constraint Equation
2I3 (I3-I2)20 20
I2 - 4A
or
- 20I2 22I3 20
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Basic Circuits
Mesh Analysis With current sources in the
circuit
Example 7.4 Continued. Express the previous
equations in Matrix form
I1 -0.667 A
I2 - 4 A
I3 - 2.73 A
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circuits
End of Lesson 7
Mesh Analysis