Lesson 7Basic Laws of Electric CircuitsMesh Analysis
Basic CircuitsMesh Analysis: Basic Concepts:In formulating mesh
analysis we assign a mesh current to each mesh.Mesh currents are
sort of fictitious in that a particularmesh current does not define
the current in each branchof the mesh to which it is assigned.
Basic CircuitsMesh Analysis: Basic Concepts:Figure 7.2: A
circuit for illustrating mesh analysis.Eq 7.1Around mesh 1:
Basic CircuitsMesh Analysis: Basic Concepts:We are left with 2
equations: From (7.1) and (7.4) we have,Eq 7.5Eq 7.6We can easily
solve these equations for I1 and I2.
Basic CircuitsMesh Analysis: Basic Concepts:The previous
equations can be written in matrix form as:Eq (7.7)Eq (7.8)
Basic CircuitsMesh Analysis: Example 7.1.Write the mesh
equations and solve for the currents I1, and I2.Figure 7.2: Circuit
for Example 7.1.Mesh 14I1 + 6(I1 I2) = 10 - 2Mesh 26(I2 I1) + 2I2 +
7I2 = 2 + 20Eq (7.9)Eq (7.10)
Basic CircuitsMesh Analysis: Example 7.1, continued.Simplifying
Eq (7.9) and (7.10) gives,10I1 6I2 = 8
-6I1 + 15I2 = 22Eq (7.11)Eq (7.12) % A MATLAB Solution R = [10
-6;-6 15]; V = [8;22]; I = inv(R)*VI = 2.2105 2.3509
I1 = 2.2105
I2 = 2.3509
Basic CircuitsMesh Analysis: Example 7.2Solve 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.
Basic CircuitsMesh Analysis: Example 7.2Clearing Equations
(7.13), (7.14) and (7.15) gives,20I1 4I2 10I3 = 30-4I1 + 18I2 11I3
= -18-10I1 11I2 + 30I3 = 20In matrix form:WE NOW MAKE AN IMPORTANT
OBSERVATION!!Standard Equation form
Basic CircuitsMesh Analysis: Standard form for mesh
equationsConsider 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.
Basic CircuitsMesh Analysis: Standard form for mesh equationsR12
= 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.
Basic CircuitsMesh 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)
Basic CircuitsMesh Analysis: With current sources in the
circuitExample 7.4: Consider the following:Figure 7.5: Circuit
diagram for Example 7.4.Use the direct method to write the mesh
equations.
Basic CircuitsMesh Analysis: With current sources in the
circuitThis 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.
Basic CircuitsMesh Analysis: With current sources in the
circuitExample 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 the5 and 20 resistors.
Basic CircuitsMesh Analysis: With current sources in the
circuitExample 7.4: Continued. Equation for mesh 1:10I1 + (I1-I2)5
= 10or15I1 5I2 = 10Equations for mesh 2:2I3 + (I3-I2)20 = 20or-
20I2 + 22I3 = 20Constraint EquationI2 = - 4A
Basic CircuitsMesh Analysis: With current sources in the
circuitExample 7.4: Continued. Express the previous equations in
Matrix form: I1 = -0.667 AI2 = - 4 AI3 = - 2.73 A
