Transcript Chapter 3 - Loop Analysis(PowerPoint Format)

Loop (Mesh) Analysis
• Loop (mesh) analysis is the systematic application of KVL
around various loops in a circuit.
• The KVL equations are written in terms of loop currents,
common to all elements in a loop.
• The result will be a system of equations in which the
unknowns are these loop currents.
• The solution of these equations will, therefore, yield values
for the loop currents.
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Definition of a loop
Definition:
A loop is a path which satisfies both of the following
conditions:
• the path begins and ends at the same point (node)
• no element is traversed more than once.
There are six (6) loops in the circuit below. Three loops are
shown, each marked by a
a
b
c
R1
Vs
+
-
R2
R4
d
R3
R5
R6
#1: a-b-e-a
#2: b-c-e-b
#3: c-d-e-c
e
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More Loops
The remaining three are loops shown below
#4: a-b-c-d-e-a
#5: a-b-c-e-a
#6: b-c-d-e-b
a
b
c
R1
Vs
+
-
R2
R4
d
R3
R5
R6
e
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Clockwise Convention
The diagram is rather “busy” when all six loops are shown.
Although loops can be drawn in either direction, the
convention is to use a clockwise direction, as shown.
a
b
c
R1
Vs
+
-
R2
R4
d
R3
R5
R6
e
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Mesh Versus Loop
A mesh is a special type of loop, one which contains no other
loops inside.
Below, meshes are marked by a
Loops are marked by a
a
b
c
R1
Vs
+
-
R2
R4
d
R3
R5
R6
e
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Mesh Analysis
• To keep the analysis as systematic as possible, we use (if
possible) only meshes to perform loop analysis (mesh analysis).
• There are cases (when a mesh traverses a current source) when
we must use a loop in place of a mesh.
a
b
c
R1
Vs
+
-
R2
R4
d
R3
R5
R6
e
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Mesh Currents
• A mesh current is assigned to each mesh as shown.
• A mesh current is the amount of current common to all elements
in the mesh.
• Mesh current is not (necessarily) the same as branch current.
• Branch current is the actual current in an element and may be
made up of more than one mesh current.
a
b
c
R1
Vs
+
-
I1
R4
d
R2
R3
I2
I3
R5
R6
e
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Mesh vs. Branch Currents
Branch (element) currents can be written in terms of mesh
currents:
Iab = I1
Icb = -I2
Icd = I3
Ibe = I1 - I2
Iec = I3 - I2
Iab
a
b
Icb
R1
Vs
+
-
I1
R4
c
Icd d
R2
Iec R3
I2
I3
R5
R6
Ibe
e
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Loop (Mesh) Analysis and KVL
Loop analysis applies KVL around each loop writing the voltage
drops in terms of the loop currents. The convention is to assume
all resistive voltages are voltage drops as one follows a loop
current around the loop. The resulting equation for loop #1 is:
Vs = R1I1 + R4(I1 -I2) = I1 (R1+R2) - I2  R4
a
+ -
b
c
R1
Vs
+
-
I1
+
R4
-
d
R2
R3
I2
I3
R5
R6
e
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Loop #2
The resulting equation for mesh #2 is:
0 = R2I2 + R5(I2 -I3) + R4 (I2-I1)
0 = - I1 R4 + I2  (R2+R4+R5) - I3 R5
a
+-
b
R1
Vs
+
-
I1
c
R2
R4
+
I2
d
R3
R5
+
-
I3
R6
e
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Loop #3
The resulting equation for mesh #3 is:
0 = R3I3 + R6I3 + R5(I3 -I2)
0 = - I2 R5 + I3  (R3+R5+R6)
a
b
c
R1
Vs
+
-
I1
R2
I2
R4
+-
d
R3
R5
+
I3
R6
+
-
e
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Loop Equations – Matrix Form
The equations represented in matrix form are:
 R4
0
 R1  R 4
  I1  VS 
  R4
  I 2   0 
R 2  R 4  R5
 R5

   
 R5
R3  R5  R6   I 3   0 
 0
a
Vs
+
-
b
R1
I1
c
R2
I2
R4
R3
d
I3
R5
R6
e
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Matrix Structure
Notice the structure: the coefficients of I1, I2, and I3 in their
respective mesh equations are the sum of the resistors in that mesh.
 R4
0
 R1  R 4
  I1  VS 
  R4
  I 2   0 
R 2  R 4  R5
 R5

   
 R5
R3  R5  R6   I 3  0 
 0
a
Vs
+
-
b
R1
I1
c
R2
I2
R4
R3
d
I3
R5
R6
e
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Matrix Structure
The off-diagonal elements are equal to the negative of the resistor
shared by two meshes:
 R4
0
 R1  R 4
  I1  VS 
  R4
  I 2   0 
R 2  R 4  R5
 R5

   
 R5
R3  R5  R6   I 3   0 
 0
a
Vs
+
-
b
R1
I1
c
R2
I2
R4
R3
d
I3
R5
R6
e
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Matrix Structure
The elements on the forcing function matrix (on the right-hand
side) are the voltage source values; positive if the mesh current
travels through a voltage rise and negative if it is a voltage drop.
 R4
0
 R1  R 4
  I1  VS 
  R4
  I 2   0 
R 2  R 4  R5
 R5

   
 R5
R3  R5  R6   I 3   0 
 0
a
Vs
+
-
b
R1
I1
c
R2
I2
R4
R3
d
I3
R5
R6
e
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Using Mesh Currents
Mesh (loop) analysis generates N independent equations, one for
each mesh (loop) current. These equations can be solved for the
mesh currents. Any voltage or branch current can be found from
the values of the mesh currents. For example,
Vce = R5(I2 - I3)
and
Vde = I3  R5
a
Vs
+
-
b
R1
c
R2
I2
I1
R4
R5
R3
+
Vce
-
I3
d
+
R6Vde
-
e
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