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Mesh Analysis
Introducing Supermeshes!!!
Mesh Analysis
A mesh is a loop with no other loops within
it; an independent loop.
Mesh analysis provides another general
procedure for analyzing circuits, using
mesh currents as the circuit variables.
This is more convenient and reduces the
number of equations that must be solved
simultaneously.
Mesh Analysis (cont.)
Mesh analysis applies KVL to find
unknown currents and is not as general
because it only applies to a planar circuit.
A planar circuit is one that can be drawn in
a plane and no branches that cross.
Steps to Determine Mesh Currents
1. Assign mesh currents i1, i2, … , in to
the n meshes.
2. Apply KVL to each of the n meshes.
Use Ohm’s Law to express the voltages in
terms of the mesh currents.
3. Solve the resulting n simultaneous
equations to get the mesh currents.
Examples
Step 1: Assign mesh currents i1, i2,
… in to the n meshes.
Step 2: Apply KVL to each of the
meshes.
1: -15 + 5i1 + 10(i1 – i2) + 10 = 0
2: -10 + 10(i2 – i1) + 10i2 = 0
Step 3: Solve the equations to get
the mesh currents
1: 15i1 – 10i2 = 5
2: -10i1 + 20i2 = 10
i1 = 1 A
i2 = 1 A
I1 = 1 A
I2 = 1 A
I3 = 0 A
NOTE: Remember, currents flowing
in opposite directions are
subtracted.
Examples
Note: io = i1 – i2
Also, there is a big difference
between branch currents and loop
currents. They are equal only when
the loop current is the only current
involved.
For example in loop 2, the currents
i2 are the same through the 24 ohm
resistor, but they are different at the
4 and 10 ohm resistors in the same
loop because those resistors are
being shared with other loops.
Examples
Solve for i1 and i2.
Mesh Analysis with Current
Sources
Applying Mesh Analysis
to circuits with current
sources is easier
because the current
sources reduce the
number of equations.
Case 1: When a current
source exists only in one
mesh, set that mesh
current to the source
current, being aware of
the direction of that
current.
Case 2: When a current
source exists between
two meshes, we create a
supermesh.
Supermesh
With supermeshes, exclude the current
source and any elements connected in
series with it.
We treat supermeshes differently because
mesh analysis applies KVL and we do not
know the voltage across a current source
in advance.
Supermesh (cont.)
Note the properties of a supermesh:
1. The current source provides the
constraint equation necessary to solve for
mesh currents.
2. A supermesh has no current of its own.
3. A supermesh requires KVL and KCL.
Examples
Case 1: When a current source
exists only in one mesh, set that
mesh current to the source current,
being aware of the direction of that
current.
Therefore, since the 5 A source
appears only in the second mesh, i2
= -5 A because the currents are
going in opposite directions.
The first equation is written as
before,
-10 + 4i1 + 6(i1 – i2) = 0;
Solving yields i1 = -2 A
Examples
Use mesh analysis to solve for Vx.
Examples
Case 2: When a current source
exists between two meshes, we
create a supermesh.
Note: Keep the currents separate
when proceeding through the
supermesh. The currents do not
overlap into the other loops.
KVL:
-20 + 6i1 + 10i2 + 4i2 = 0
KCL (currents at the bottom node):
i2 = i1 + 6
Solve equations as usual:
6i1 + 14i2 = 20
-i1 + i2 = 6
i1 = -3.2 A
i2 = 2.8 A
Example
Use mesh analysis to determine Vo.