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7/29/2019 Unit-2 Network Analysis Part II
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, 102013 Ch. 3 Network Analysis- Part II 1
Topics to be Discussed
Loop-current Analysis. Counting Independent
Loops.
Mesh Analysis.
Supermesh Method.
Limitations of MeshAnalysis.
Planar Network. Procedure for Mesh
Analysis.
Node Voltages Analysis. Supernode.
Counting Independent
Nodes.
Nodal Analysis. Choice Between the
TWO.
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2.1 KIRCHHOFFS CURRENT LAW (KCL):-
It states that the algebraic sum of all currents
entering a node is zero. Mathematically:
Currents are positive if entering a nodeCurrents are negative if leaving a node.
Example:
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Applying Kirchhoff's current law:
I1 + I2 + I3 + I4 = 0
(the negative sign inI2indicates that I2 has a
magnitude of 3A and is flowing in the direction
opposite to that indicated by the arrow)
Substituting:
5 - 3 + I3 + 2 = 0 Therefore, I3 = - 4A (ie 4A
leaving node)
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2.2 KIRCHHOFFS VOLTAGE LAW (KVL):-
It states that the algebraic sum of the voltage drops
around any loop, open or closed, is zero.Mathematically
Example:
Going round the loop in the direction of the current, I,
Kirchhoff's Voltage Law gives:
10- 2I - 3I = 0
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-2Iand -3Iare negative, since they are voltage drops
i.e. represent a decrease in potential on proceeding
round the loop in the direction of I. For the same
reason + 10V is positive as it is a voltage rise or increasein potential.
Concluding:
5 I = 10 Therefore, I = 2A
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Loop-current Analysis Loop analysis is systematic method of network
analysis.
It is a general method and can be applied to anyelectrical 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 networkwhere three or more elements have a commonconnection.
Next
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Before the loop analysis can be applied to a
network, we must first check that it has onlyvoltage 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.
Next
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Example1
Find the voltage across the 2- resistance.
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Recognize the independent loops (which does not pass
through a current source), and mark the loop currents.
This choice reduces labour, as only one currentI1 is
to be calculated.
Next
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Write KVL equations and solve forI1.
A0.435
1
1222
121
79)1)(()3()4(67)2()1)((
I
IIII
III
Next
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Counting Independent Loops
It appears to have two loops. But, these two loops are not independent.
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Suppose that we had marked the two loop currentsI1
andI2 in the standard way,
2 1 2AI I
The values of these two currents are constrainedby
the above relation.
Then,
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We identify independent loops by turning off allsources. We are, then, left with one loop containing
two resistances.
Hence, we have only one independent loop
requiring only one KVL equation.
Next
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For determining the current through 5- resistance,
we should choose
Thus, the single KVL equation is
A0.462
1
11 0)2(8510
I
II
Next
Click
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In case, we are to determine the current through 8-
resistance, we should choose
A1.538
1
1108)2(510
I
II
The single KVL equation then becomes
Next
Click
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BenchmarkExample1
Consider the benchmark example, and solve itby using loop-current analysis.
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Solution :
We note that the givencircuit has one independent
loop and two constrained loops.
Our aim is to determine the voltage across 3-
resistance.
So, we should select the unknown loop currentIpassing through 3- resistance (but not through any
current source).
The two known loop currents of 4 A and 5 A aremarked to flow in the two loops as shown.
Next
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Writing KVL equation around the loop ofI, we get
A650)45(1632 IIII
Therefore, the unknown voltage, v = 3I= 2.5 V.
Next
Click
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Example2
Find the currents i1and i2 in the circuit givenbelow.
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Solution :Applying KVL to the two loops,
2
2
3 2
or 2 3
i
i
1 A
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MESH ANALYSIS
In circuit terminology, a loopis any closedpath.
A meshis a special loop, namely, thesmallest loop one can have.
In other words, a mesh is a loop thatcontains no other loops.
Mesh analysis is applicable only to a planar
network. However, most of the networks we shall need
to analyze are planar.
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Once a circuit has been drawn in planar form,
it often looks like a multi-paned window.
Each pane is a mesh.
Meshes provide a set of independent
equations.
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By definition, a mesh-current is that currentwhich flows around the perimeter of a mesh. It is
indicated by a curved arrow that almost closes onitself.
Branch-currents have a physical identity. They
can be measured. Mesh-currents are fictitious.
The mesh analysis not only tells us the minimum
number of unknown currents, but it also ensuresthat the KVL equations obtained are independent.
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Loop (Mesh) Analysis
Next
http://localhost/var/www/apps/conversion/tmp/scratch_10/Loop%20Analysis%20for%20Ch.%203-Part-II.swfhttp://localhost/var/www/apps/conversion/tmp/scratch_10/Loop%20Analysis%20for%20Ch.%203-Part-II.swfhttp://localhost/var/www/apps/conversion/tmp/scratch_10/Loop%20Analysis%20for%20Ch.%203-Part-II.swf7/29/2019 Unit-2 Network Analysis Part II
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Example 2
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 isautomatically satisfied.
Next
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Resistance Matrix
Mesh current matrix
Source matrix
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ApplyingCrammers rule :
The current in 3-ohm resistor is I1I2 = 64 = 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, asR12 = R21, R13 = R31, etc.
All the elements on the major diagonal have
positive values. The off-diagonal elementshavenegative 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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Planar Network If a network can be drawn on sheet of paper
without crossing lines, it is said to beplanar.
Is it a planar network ?
Yes, it is. Because it can be drawn in a plane,
as shown in the next figure.
Next
Click
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This is definitely non-planar.
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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 themajor diagonal is negative (or zero), and
represents the mutual resistance.
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5.Check the symmetry of resistance matrix aboutthe 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 Calculator.8. Determine the branch currents and voltages.
Next
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Example 3
Determine the currents in various resistances of
the network shown.
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Writing the mesh equationsby inspection,
Solving,
Next
Solution :
Click
we get I1 = 2.55 A, I2= 3.167 A
Click
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Example 4 Find the current drawn from the source in the
network, using mesh analysis.
Next
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Using Calculator, we get Click
1I 6 A
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How to Handle Current Sources
If a circuit has current sources, a modestextension of the standard procedure is
needed.
There are three possible methods.
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First Method
If possible, transform the current sources into
voltage sources.
This reduces the number of meshes by 1 for
each current source.
Apply the standard procedure of meshanalysis to determine the assumed mesh
currents.
Go back to the original circuit, and getadditional equations, one for each current
source,
Next
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Example 5
Solve the following circuit for the three meshcurrents.
Next
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Solution :
We convert the 13-A current source in parallel with 5-
resistor into an equivalent 65-V voltage source inseries with 5- resistor.
This reduces the number of meshes to two.
Next
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We can write the mesh equations in the matrix form just
by inspection,
1
2
9 5 10
5 11 52
I
I
1 2andI I 5 A 7 A
We now go back to the original circuit. Obviously, the
current through the current source is
2 3 3 213A 13 7 13I I I I 6 A
Next
Click
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Second Method
We can assign unknown voltages to each
current source.
Apply KVL around each mesh, and
Relate the source currents to the assumed
mesh currents.
This is generally a difficult approach.
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Third Method
(Supermesh Method) Create a supermeshfrom two meshes that
have a current source as a common element.
The current source is in the interior of thesupermesh.
Thus, the number of meshes is reduced by 1for each current source present.
If the current source lies on the perimeter of
the circuit, then ignore the single mesh inwhich it is found.
Apply KVL to the meshes and supermeshes.
Next
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Example 6
Solve the circuit of Example 5, using
supermesh method.
Solution :
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Going along the dotted arrow, the KVL equation for
this supermesh is
3 1 2
1 2 3
5( ) 6 13 0
or 5 6 5 13
I I I
I I I
The KVL equation for mesh 1 is
1 2 39 0 5 75I I I
We have only two equations for three unknowns.
The third equation is obtained by applying KCL to
either node of the current source
Next
Click
Click
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Thus, we have
1 2 30 13I I I
These three equations can be put in the matrix form,
1
2
3
5 6 5 13
9 0 5 75
0 1 1 13
I
I
I
Using Casio fx-991ES, we directly get
1 2 3, and .I I I 5 A 7 A -6 A
Which is same result as obtained in Example 5.
Next
Click
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Example 7
Apply mesh analysis to determine current
through 7- resistance in the given network.
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Solution :
The given network is a planar networkhaving
independent voltage sources.
It has three meshes for which the mesh currentsI1,I2,
andI3 are marked all with clockwise directions.
By inspection, the matrix equation is written as
1
2
3
3 4 4 0 42 25
4 4 5 6 6 25 57 70
0 6 6 7 70 4
I
I
I
Next
Click
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1
2
3
7 4 0 67
or 4 15 6 1520 6 13 74
I
I
I
7 3
I I 2 A
Solving the above equation forI3,
Next
Click
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Node-Voltage Analysis
Next
N d V l A l i
http://localhost/var/www/apps/conversion/tmp/scratch_10/Nodal%20Analysis%20for%20Ch.%203-Part-II.swfhttp://www.eas.asu.edu/~holbert/ece201/nodalanalysis.htmlhttp://www.eas.asu.edu/~holbert/ece201/nodalanalysis.htmlhttp://localhost/var/www/apps/conversion/tmp/scratch_10/Nodal%20Analysis%20for%20Ch.%203-Part-II.swfhttp://localhost/var/www/apps/conversion/tmp/scratch_10/Nodal%20Analysis%20for%20Ch.%203-Part-II.swfhttp://localhost/var/www/apps/conversion/tmp/scratch_10/Nodal%20Analysis%20for%20Ch.%203-Part-II.swf7/29/2019 Unit-2 Network Analysis Part II
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Node Voltages Analysis
It is dual of the Mesh Analysis.
It involves the application of KCL
equations, instead of KVL.
One of the nodes is taken as reference ordatum or groundnode.It is better to select
the one that has maximum number of
branches connected. The reference node is assumed to be at
ground or zero potential.
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The potentials of all other nodes are defined
w.r.t. the reference node.
KCL equations are written, one for each
node, except the reference node.
The equations are solved to give node
voltages.
Current through any branch and voltage atany point of the network can be calculated.
Next
E l 8
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Example 8
Solve the circuit given, using the node voltagemethod.
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Solution : It has only two nodes. Node 2 has been
taken as reference node. The currents in various
branches have been assumed. Writing the KCLequations,
A
V
67
60,Now
18
047
6012
0
12
1
111
321
VI
V
VVV
III
Next
H H dl V l S
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How to Handle Voltage Sources
If one terminal of a voltage source with a
series resistance is grounded (as in theExample 8), the KCL equation can bewritten in terms of this voltage.
Difficulty arises, if a circuit containsfloating voltage sources.
A voltage source is floating if its neitherterminal is connected to ground.
If possible, first transform the voltagesources into current sources.
Next
C i d N d SUPERNODE
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Constrained Node or SUPERNODE
There is another way which uses the concept of
constrained nodeorsupernode. This method is especially suitable for the circuits having
a floating voltage source with no series resistance.
The two ends of a voltage source cannot make two
independent nodes. Hence, we treat these end nodes together as a
supernode.
The supernode is usually indicated by the region
enclosed by a dotted line.
The KCL is then applied to both nodes at the same
time.
Next
C i I d d N d
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Counting Independent Nodes
Itis a node whose voltage cannot be derived
from the voltage of another node.
First turn off all sources, and then counting all
the nodes separated by resistors.
The number of independent nodes is oneless than this number.
Next
E l 9
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Example 9 Determine the current through 4- resistor in the
circuit given below.
Next
S l i
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Solution :
Here, the voltages at nodes a and b are not independent.
The two node voltages are related as
a b a b c6 or 0 6v v v v v
We can treat the two constrained nodes a and b, as asupernode.
Now, writing KCL for this supernode, we get
a b c
a b c
23 4
or 0.33 0.25 0.25 2
v v v
v v v
Next
Click
A l i KCL d
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Applying KCL to node c
c c b
a b c
7
5 4or 0 0.25 0.45 7
v v v
v v v
Above equations can be written in the matrix form,
a
b
c
1 1 0 6
0.33 0.25 0.25 2
0 0.25 0.45 7
v
v
v
Solve the above equation
Next
Click
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We solve the above equations using calculator to get
b c8.77V and 20.43Vv v
Finally, the current through 4- resistor is
b c 8.77 ( 20.42)4 4
v v 2.9125 A
Next
Click
B h k E l 10
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BenchmarkExample 10
Consider the benchmark exampleand solve it
by using node-voltage analysis.
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Solution :
Nodes c and dare constrained to one another.
To find the number of independent nodes, we turn off
the sources to get the circuit,
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There are three nodes, two of which are independent.
However, if we add the two series resistors to make a 5-
resistor we will have only one independent node (node
a).
Hence we will have to solve only one equation.
The unknown voltage across 3- resistor can then be
determined by applying voltage divider rule.
Next
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Writing KCL equation for node a,
a a
a
(6) (0)4 5
1 5
6 14.17V
1.2
v v
v
Using the voltage divider, the voltage across 3-
resistor is
34.17
2 3v
2.5 V
Next
Click
Click
Example 11
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Example 11
Apply KCL to determine currentISin the circuit
shown. Take Vo = 16 V.
Next
Solution : Applying KCL at nodes 1 and 2
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Solution : Applying KCL at nodes 1 and 2,
Next
Click
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Therefore, the current,
Next
Example 12
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Example 12
Using nodal analysis, determine the current
through the 2- resistor in the network given.
Next
Solution : It is much simpler to write the KCL
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Solution : It is much simpler to write the KCL
equations, if the conductance (and not the
resistances) of the branches are given.
Next
It has 3 nodes. So, we have to write KCL
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It has 3 nodes. So, we have to write KCL
equations for only 2 nodes.
We just equate the total current leaving the nodethrough several conductances to the total
source-current entering the node.
At node 1,
At node 2,
Next
Writing the above equations in matrix form
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Writing the above equations in matrix form,
Next
V51 V
2
3
2.12.0
2.07.0
2
1
V
V
Solving for V1, using Calculator, we getClick
Finally, the current in the 2- resistor,
2.52
5
2
1VI
Nodal Analysis
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Nodal Analysis
The above examples suggests that it ispossible to write the nodal analysis
equationsjust byinspection of the network.
Such technique is possible if the networkhas only independent current sources.
All passive elements are shown as
conductances, in siemens (S).
Next
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In case a network contains a practicalvoltage source, first convert it into an
equivalent practical current source.
Write the Conductance Matrix, Node-Voltage Matrix and the Node-Current
Source Matrix, in the same way as in the
Mesh Analysis.
Next
Example 13
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Example 13
Let us again tackle Example 12, by writing the
matrix equations just by inspection.
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Conductance matrix.
G11 =Self-conductance
of node 1.G12=Mutual conductance
between node 1 and 2.
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Node-voltage Matrix.
Node current-source Matrix.
Note that all the elements on the major diagonalof matrix G are positive.
All off-diagonal elements are negative or zero.
Next
Example 14
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Example 14
Solve the following network using the nodal
analysis, and determine the current through the 2-Sresistor.
Next
Solution :
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Solution :
Next
We can write the nodal voltage equation in matrix
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We can write the nodal voltage equation in matrix
form, directly by inspection :
25
3
11
1124
263
437
or
)25(
)3(
)8()3(
)524(24
2)123(3
43)34(
3
2
1
3
2
1
V
V
V
V
V
V
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Finally, the current through 2-S resistor is
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Using Calculator, we get
V3andV2 32 VV
Example 15
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Example 15
Find the node voltages in the circuit shown.
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Solution :
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Solution :First Method
Transform the 13-V source and series 5-S resistor to
an equivalent current source of 65 A and a parallel
resistor of 5 S
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Now, we can write the nodal equations in
matrix form for the two nodes just by
inspection,
1
2
9 5 10
5 11 52
V
V
1 2
andV V 5 V 7 V
Now, from the original circuit shown, we get
3 2 13 7 13V V 6 V
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Second Method
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We use the concept of supernode. The voltage source is
enclosed in a region by a dotted line, as shown in figure.
The KCL is then applied to this closed surface:
2 3 16 5( ) 13V V V
The KCL equation for node 1 is
1 39 5 75V V
For three unknowns, we need another independent
equation. This is obtained from the voltage drop across
the voltage source,
2 3 13V V
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Writing the above equations in matrix form,
Click
13565 V
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Solving, we get
1 2 3, , andV V V 5 V 7 V 6 V
Which are the same as obtained by first method.
In general, for the supernode approach, the KCL
equations must be augmented with KVL equations the
number of which is equal to the number of the floating
voltage sources.
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13
75
13
110
509
565
3
2
1
V
V
V
Click
Choice Between the TWO
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Choice Between the TWO
We select a method in which the number of
equations to be solved is less.
The number of equations to be solved in
mesh analysis isb(n1)
The number of equations to be solved in
nodal analysis is
(n1)
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Review
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Review
Loop-current Analysis.
Counting Independent
Loops.
Mesh Analysis.
Supermesh Method. Limitations of Mesh
Analysis.
Planar Network.
Procedure for MeshAnalysis.
Node Voltages Analysis. Supernode.
Counting Independent
Nodes.
Nodal Analysis.
Choice Between theTWO.