EGR 2201 Unit 4
Mesh Analysis
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Read Alexander & Sadiku, Sections 3.4
to 3.10.
Homework #4 and Lab #4 due next
week.
Quiz next week.
Mesh Analysis

We’ve seen that nodal analysis is a
systematic method for analyzing
circuits.
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It’s based on Kirchhoff’s current law (KCL).
It gives us the node voltages in a circuit.
Once we have these node voltages, we can
find any other voltage or current.
Mesh analysis is another systematic
method for analyzing circuits.


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It’s based on Kirchhoff’s voltage law (KVL).
It gives us the mesh currents in a circuit.
Once we have these mesh currents, we
can find any other current or voltage.
Meshes Versus Loops


Recall that a loop is any closed path in a
circuit.
 Example: This circuit has six loops.
A mesh is a loop that does not contain any
other loop within it.
 Example: The circuit above has three
meshes.
Mesh Currents Versus Branch
Currents

A mesh current is a current that we imagine
to travel around a mesh.

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You can imagine them to travel in either direction,
but most people assume clockwise.
A branch current is a current that passes
through a branch (i.e., an element).
If we know the values of all the mesh currents
in a circuit, we can compute any branch
current.
In many diagrams, our textbook uses:


Lowercase i and a looping arrow for mesh currents
Uppercase I and a straight arrow for branch currents
Example: Mesh Currents Versus
Branch Currents
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In this circuit, the
mesh currents are
labeled i1 and i2.
The branch currents
are labeled I1, I2,
and I3.
Suppose you were
given the values of
the mesh currents.
Then you could easily
compute the branch currents, since:
I1 = i1
and
I2 = i2
and
I3 = i1  i2
Steps in Performing Mesh Analysis on
a Circuit with No Current Sources

Given a circuit with n meshes, without
current sources, follow these steps:
1.
2.
3.
Assign mesh currents i1, i2, …, in to the n
meshes.
Apply KVL to each of the n meshes. Use
Ohm’s law to express the voltages in terms
of mesh currents. Then simplify the
equations.
Solve the resulting n simultaneous
equations to obtain the unknown mesh
currents.
Example: Step 1 (Assign the Mesh
Currents)
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
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Consider this circuit
from the book’s
Example 3.5.
Step 1 has
already been
performed for
us, since the
mesh currents
are labeled i1 and i2.
If an assumed current direction is
wrong, that’s no problem. The math
will still work out.
Example: Step 2 (Apply KVL)
Part 1 of 2
Apply KVL (and
Ohm’s law) to
mesh 1:
15 + 10i2 =
5i1 + 10i1 + 10
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
Apply KVL (and
Ohm’s law) to mesh 2:
10 + 10i1 = 10i2 + 6i2 + 4i2
Example: Step 2 (Apply KVL)
Part 2 of 2
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
Next we use algebra to simplify our
equations.
For mesh 1:
15 + 10i2 = 5i1 + 10i1 + 10
becomes

For mesh 2:
10 + 10i1 = 10i2 + 6i2 + 4i2
becomes

15𝑖1 − 10𝑖2 = 5
10𝑖1 − 20𝑖2 = −10
We now have our two equations in two
variables.
Example: Step 3 (Solve)
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Next we use any of our methods—
substitution, Cramer’s rule, matrix
inversion, MATLAB—to solve our two
equations in two variables.
Using MATLAB, the solution to
15𝑖1 − 10𝑖2 = 5
10𝑖1 − 20𝑖2 = −10
is
i1 = 1 A
and
i2 = 1 A
Example: Extending the Analysis
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Mesh analysis has
given us the values
of the mesh
currents i1 and i2.
We can find all
other currents
and voltages in the
circuit once we
know these mesh currents.
Example: Knowing that i1 = 1 A and
i2 = 1 A, how would we find I3?
Review: Steps in Performing Mesh
Analysis on a Circuit with No Current
Sources

Given a circuit with n meshes, without
current sources, follow these steps:
1.
2.
3.
Assign mesh currents i1, i2, …, in to the n
meshes.
Apply KVL to each of the n meshes. Use
Ohm’s law to express the voltages in terms
of mesh currents. Then simplify the
equations.
Solve the resulting n simultaneous
equations to obtain the unknown mesh
currents.
Circuit Analysis with Simulation
Software
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Section 3.8 discusses computer software
named PSpice, which lets you simulate
circuits.
Sinclair’s computers don’t have PSpice
installed, but we have a similar program
named Multisim.
Both are based on open-source code
named SPICE (Simulation Program with
Integrated Circuit Emphasis), which was
first written at the University of
California, Berkeley in the 1970’s.
Starting Multisim
on Our Computers

Start Menu

National Instruments

Circuit Design Suite 12.0
 Multisim 12.0
Where to Find Some Components
in Multisim

DC Independent Voltage Source
Group=Sources > Family=POWER_SOURCES >
DC_POWER
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Ground
Group=Sources > Family=POWER_SOURCES > GROUND
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DC Independent Current Source
Group=Sources > Family=SIGNAL_CURRENT_SOURCES >
DC_CURRENT
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Resistor
Group=Basic > Family=RESISTOR
Where to Find Some Measuring
Instruments in Multisim
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Voltmeter
Group=Indicators > Family=VOLTMETER
> VOLTMETER_H
Shown is a horizontal voltmeter. Also available are a
vertical version, as well as versions with the leads
reversed.

Ammeter
Group=Indicators > Family=AMMETER
> AMMETER_H
Shown is a horizontal ammeter. Also available are a
vertical version, as well as versions with the leads
reversed.
Multisim Resources
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Tutorial: Inside the program, click Help >
Getting Started.
National Instruments’ Multisim website:
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Main site: http://www.ni.com/multisim/
Student version:
http://www.ni.com/multisim/student-edition/
Free 30-day trial: http://www.ni.com/multisim/try/
The Lessons page of the ANGEL course site
has a brief introductory video and handout
from two other Sinclair instructors.
What About Circuits with Current
Sources?
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As described above, our procedure
applies only to circuits without
current sources.
But it’s not hard to extend the
procedure to circuits with current
sources.
The way you handle a current source
depends on whether the source is
located in only one mesh or is shared
by two meshes….
Case 1: A Current Source Located
in Only One Mesh

A current source
located in only
one mesh
is easy to
handle,
because it
immediately
reveals the mesh current in that mesh.
 Example: In the circuit shown, we can
immediately see that i2 = 5 A.
Case 2: A Current Source Shared by
Two Meshes
A current source
shared by two
meshes is
trickier.
 To handle it,
we create a
supermesh
by excluding the
current source
and any elements
in series with it.

How to Handle a Supermesh
We apply KCL and
KVL to the supermesh to get two
equations.
 Example: In the
circuit shown, KCL gives
i2 = i1 +6
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And KVL around
the supermesh gives
20 = 6i1 + 10i2 + 4i2
We Still Get Enough Equations
If this circuit did
not have a supermesh, we would
get one equation
by applying KVL
to mesh 1 and
another by applying
KVL to mesh 2.
 With the supermesh, we get one equation
by applying KCL and another by applying
KVL to the supermesh.

Nodal Analysis and Mesh Analysis
“By Inspection”
With practice, you’ll become good at
writing down the set of simultaneous
equations that describe a circuit
using either nodal or mesh analysis.
 As discussed in Section 3.6, there is
a shortcut way to write down the
equations quickly by looking at a
circuit without even thinking in
terms of KCL or KVL.

I
won’t expect you to learn this shortcut
method, but you can use it if you wish.
Which Should You Use: Nodal
Analysis or Mesh Analysis?
Many circuits can be analyzed using
either method, and both methods will
give the same results.
 But as discussed in Section 3.7, in
some cases you’ll get the answer with
less work if you’re smart about picking
the better method for your circuit.
 See next slide for examples.
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Examples: Should You Use Nodal
Analysis or Mesh Analysis?
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For example:
 Use nodal analysis for circuits with
fewer nodes than meshes, and use
mesh analysis for circuits with
fewer meshes than nodes.
How many nodes
does this circuit
have?
How many meshes?
More Examples: Should You Use
Nodal Analysis or Mesh Analysis?
As also noted in Section 3.7, mesh
analysis cannot be applied to
nonplanar circuits.
 A circuit is planar if it can be drawn
on a plane with no branches crossing
each other; otherwise it is
nonplanar.
 We will only deal with planar circuits
in this course.

Example of a Planar Circuit

This circuit, in which branches cross,
can be redrawn with no crossing
branches (as on the right below), so
it is a planar circuit.
Crossing branches:
No intersections here.
Example of a Nonplanar Circuit

There is no way to redraw this circuit
without crossing branches, so it is a
nonplanar circuit.
Crossing branches:
No intersections here.