electrical circuits problems and analysis

Electrical circuits

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Circuit Analysis-1Circuit Analysis-1
Subodh Kumar SinghalSubodh Kumar Singhal

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Thought of The Day
If you cannot do
great things,
do small things in a
great way.

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Topics to be DiscussedTopics to be Discussed
•Loop Analysis.
•Mesh Analysis.
•Mesh-currents.
•Resistance Matrix.
•Self-resistance.
•Mutual resistance.
•Limitations of Mesh Analysis.
•Planar Network.
•Procedure for Mesh Analysis.
•Examples.

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Loop Analysis
•Loop analysis is systematic method of
network analysis.
•It is a general method and can be applied to
any electrical network, howsoever
complicated it may be.
•It is based on writing KVL equations for
independent loops.
•A loop is a closed path in a network.
•A node or a junction is a point in the network
where three or more elements have a
common connection.

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Contd..
•Before the loop analysis can be applied to a
network, we must first check that it has only
voltage sources (independent or dependent).
•Any current source must be transformed into
its equivalent voltage source.
•Sometimes, it is a difficult task to identify
independent loops in a network.
•The method of loop analysis can be best
understood by considering some examples.

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Example
•Find the currents i
1
and i
2
in the circuit given
below.

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Solution : Applying KVL to the two loops,
And

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Example
•Find a single voltage source equivalent of the
following circuit.

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Solution : We first replace the current source by
its equivalent voltage source. We then apply
Thevenin’s Theorem.
36 V

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To find Thevenin’s equivalent resistance, we
reduce the network as shown.
Applying KVL, we get
V42



636636
A1
01212
0366432216
IV
I
I
III
PQ

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Equivalent voltage source is
42 V

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Planar Network
•If a network can be drawn on sheet of paper
without crossing lines, it is said to be planar.
•Is it a planar
network ?
•Yes, it is. Because it can be drawn in a
plane, as shown in the next figure.

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•This is definitely non-planar.

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Mesh Analysis
•Mesh is a loop which contains no other loop
within it.
•Mesh analysis is applicable only to a planar
network.
•But most of the networks we shall need to
analyze are planar.
•Once a circuit has been drawn in planar form, it
often looks like a multi-paned window.
•Each plane is a mesh.
•Meshes provide a set of independent equations.

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Contd..
•By definition, a mesh-current is that current
which flows around the perimeter of a mesh. It
is indicated by a curved arrow that almost
closes on itself.
•Branch-currents have a physical identity. They
can be measured.
•The mesh analysis not only tell us the minimum
number of unknown currents, but it also
ensures that the KVL equations obtained are
independent.

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Example
•Let us consider a simple network having only
two meshes.
•Although the directions of the mesh currents
are arbitrary, we shall always choose
clockwise mesh currents.
•This results in a certain error-minimizing
symmetry.
•Note that by taking mesh currents, the KCL is
automatically satisfied.

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Resistance Matrix
Mesh current matrix
Source matrix

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Applying Crammer’s rule :
The current in 3-ohm resistor is I
1 – I
2 = 6 – 4 = 2A

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Three-mesh Network
•Write the three equations for the three
meshes and put them in a matrix form.

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Self-resistance of mesh 1
Mutual resistance
between mesh 1 and 2.

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The Resistance Matrix
•It is symmetrical about the major diagonal,
as R
12 = R
21, R
13 = R
31, etc.
•All the elements on the major diagonal have
positive values.
•The off-diagonal elements have negative
values.
•The mutual resistance between two meshes
will be zero, if there is no resistance common
to them.

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Mesh Analysis Limitations
•It is applicable only to those planar
networks which contain only
independent voltage sources.
•If there is a practical current source,
it can be converted to an equivalent
practical voltage source.

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Procedure for Mesh Analysis
1.Make sure that the network is planar.
2.Make sure that it contains only independent
voltage sources.
3.Assign clockwise mesh currents.
4.Write mesh equations in matrix form by
inspection. An element on the principal
diagonal is the self-resistance of the mesh.
These elements are all positive. An element
off the major diagonal is negative (or zero),
and represents the mutual resistance.

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5. Check the symmetry of resistance matrix
about the major diagonal.
6.An element of the voltage source column
matrix on the right side represents the
algebraic sum of the voltage sources that
produce current in the same direction as
the assumed mesh current.
7.Solve the equations to determine the
unknown mesh currents, using Cramer’s
rule.
8.Determine the branch currents and
voltages.

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Example
• Determine the currents in various resistances of
the network shown.

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Writing the mesh equations by inspection,
Solving, we get I
1
= 2.55 A, I
2
= 3.167 A

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Example
•Find the current drawn from the source in the
network, using mesh analysis.

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ReviewReview
•Loop Analysis.
•Mesh Analysis.
•Mesh-currents.
•Resistance Matrix.
•Self-resistance.
•Mutual resistance.
•Limitations of Mesh Analysis.
•Planar Network.
•Procedure for Mesh Analysis.
•Examples.

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In the Next ClassIn the Next Class
•Node-Pair Voltages Analysis.
•Generalized Matrix Form of
Nodal Equations.
•Examples.
•Choice Between the TWO.