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CHAPTER 3
Nodal and Loop Analysis
Techniques
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Be able to calculate all currents and voltages in circuits
that contain multiple nodes and loops. Learn to employ Kichhoffs current law (KCL) to perform
a nodal analysis to determine all the node voltages in a
circuit.
Learn to employ Kichhoffs voltage law (KVL) to performa loop analysis to determine all the loop currents in a
network.
Be able to ascertain which of the two analysis techniques
should be utilized to solve a particular problem.
The learning goals for this chapter
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3.1 Nodal analysis
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Reference Node
One node is selected as the reference node and iscommonly called ground.
All other node voltages are defined with respect to the
reference node. Using Ohms Law we can find the current of each
element.V12SV
V2
3bV
V3aV
V
8
3cV
ground
k
VV
k
VI
k
VV
k
VI
k
VV
k
V
I
cb
ba
aS
99
33
99
55
33
1
1
k
VI
k
V
I
b
a
4
06
0
4
2
3.1 Nodal analysis: the variables in the circuitare selected to be the node voltages
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Ohms Law
In general, if we know the node voltages in a circuit,
we can calculate the current through any resistive
element using Ohms law
3.1 Nodal analysis
R
vvi Nm
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The manner of finding node voltages in a circuit
with N nodes.
One of the N nodes is selected as the reference node.
N1 unknown node voltages are remaining.
We can write N of KCL equations for N nodes. But using network topology, only the N1 linearly
independent KCL equationsare required.
Using Ohms law, the KCL equations can be written in
terms of the node voltages.
Find N1 node voltages from a set of N1 equations.
3.1 Nodal analysis
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Specify a reference
is the voltage at node 1 with respect to thereference node 3.
Similarly, is the voltage at node 2 with respect to
the node 3. The voltage at node 1 with respect to node 2 is .
The voltage at node 2 with respect to node 1 is .
The current in is from top to bottom.
The current in is from bottom to top.
3.1 Nodal analysis
V41 V
V22 V
V6
V6
1R
3R 7
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Specify a reference
For example,
If a man were hanging in midair with one hand on one lineand one hand on another, and the dc line voltage of each
line was exactly same.
The voltage across his heart would be zero. He would be safe.
If he let his feet touch the ground,
The dc line voltage would then existfrom his hand to his
foot with his heart in the middle.
He would be dead.
3.1 Nodal analysis
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Circuits Containing only Independent Current
Sources (1)
This network contains 3 nodes and one of them is selected asthe ground.
N1=31=2 linearly independent KCL equations will be required.
Using Ohms law,
3.1 Nodal analysis
Solved by Gaussian elimination, matrix analysis, MATLAB 9
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Example 3.1
3.1 Nodal analysis
Determine all node voltages and branch currents.
We can write 2 KCL equations
Using Ohms law Simplify eqs, From the 1st equation,
Substituted into the 2nd eq.
To find the currents, use Ohms law,
3I
2I
1I
mA121 II
4m6k6k
1m6k12k
221
211
VVV
VVV
mA432 II
V152 V
V61 VmA
2
5
6mA
2
3
6mA
2
1 23
212
1
11
k
VI
k
VVI
R
VI
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Example 3.1 (sol2)
3.1 Nodal analysis
Use the matrix analysis to show
the solution:
Place in the matrix form:
Do the matrix algebra:
||
)( 1
A
AAdjA
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Circuits Containing only Independent Current
Sources (2)
This network contains 4 nodes and 41=3 linearly independentKCL equations are required.
Reordering terms,
3.1 Nodal analysis
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Circuits Containing only Independent Current
Sources (2) The equations can also be written in matrix form as
nodetoconnectedeConductanc 2&1betweeneConductanc
3&1betweeneConductanc
3&2betweeneConductanc
3.1 Nodal analysis
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Symmetrical form of the equations
For circuits with only resistors and independentcurrent sourcesthe matrix is always symmetric.
The diagonal elements are positive and the others arenegative.
3.1 Nodal analysis
3
2
1
3
2
1
333231
232221
131211
i
ii
v
vv
aaa
aaaaaa
NaNN nodetoconnectedeConductanc
2&1betweeneConductanc, 2112 aa
3&1betweeneConductanc, 3113 aa
3&2betweeneConductanc, 3223 aa
NiN nodeenteringsourceCurrent
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Example 3.2
Determine the node voltages:
Using the values, solve the matrix,
3.1 Nodal analysis
mA.2andmA,4,k1,k4,k2 54321 BA iiRRRRR
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Circuits Containing Dependent Current Sources
The dependent source destroy the symmetrical form of nodalequations.
3.1 Nodal analysis
Replacing Write N1=3-1=2 KCL equations
It can be represented by the matrix
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N-1=3 KCL equations
Replace , and simplify
the equations.
Example 3.4 Compute the node voltages.
3.1 Nodal analysis
2mA,4mA,2,k4,k2,k1 4321 BA iiRRRR
Using the values, solve the equations.
V99111 .V V99152 .V V99153 .V18
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Note that the voltage sources are connected
between and nodes and ground.
KCL equation for node
Circuits Containing Independent Voltage Sources
Example 3.5 Determine all node voltages.
3.1 Nodal analysis
V121V V63 V
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Example 3.6 Find the current in two resistors. -An independent V source is
connected between two non-reference nodes.
3.1 Nodal analysis
We cannot write directly KCL equations because we do not know the
current in the voltage source.
So, apply KCL to the surface(dashed line).
Two nodes is constrained by the voltage source.
From those two equations,
Using Ohms law
0A4
126
A621 m
k
V
k
Vm
V621 VV
V4V,10 21 VV
super node
mA
3
1mA,
3
521 II
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Example 3.7 Determine the current
3.1 Nodal analysis
0I
and can be easily found.
We cannot apply KCL at each node and
because we do not know the current in the voltage
source.
Apply KCL at the super node where
V12V6 42 VV
02kk1
12
1k
)6(
2k
12
2k
)6(33311
VVVVV
Solve this equation,
V1231 VV
V7
63 V
To find , use Ohms law0I
A7
3
2k
7
6
oI
super node
1V 3V
2V 4V
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Circuits Containing Dependent Voltage Sources
Example 3.8 Find .
3.1 Nodal analysis
oI
xkIV 21 KCL at node
For the controlling variable
From those equations,
k
VIx
1
2
mA42
21
k
VVIoV8V,16 21 VV
2V
xI
22
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Super node constraint
where
KCL at the super node and node
Combining all the equations,
Example 3.10
Find the voltage .
3.1 Nodal analysis
oV
V44V
xVVV 221 xx VVVV 32 21
Solve these equations,
01
43
1
3
11
2 33
k
V
k
VV
k
VV
k
V
k
xxxx
kk
VV
k
VV xx 2
11
3 33
224
628
3
3
VV
VV
x
x
V13
V5andV2
3
3
VVV
VV
xo
x
super node
3V
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Super node constraint,
KCL at the super node and node ,
For the control parameters,
Combining all the equations,
Example 3.11 Find .
3.1 Nodal analysis
oI
641 VV
k1,12 41
VIVV xx
Solve these equations,
V122V xVV 23
super node
01k1k1k
21k1k
12 54434311
VVVVV
IVVV
x xIVVV
21k1k
545
023
6
3652
54
41
541
VV
VV
VVV
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3.1 Nodal analysis
Problem solving strategy
Nodal analysis for an N-node circuit Determine the number N of nodes in the circuit. Select one
node as the reference node. Assign N1 node voltages. N1
linearly independent equations must be written to solve for the
node voltages.
Write the constraint equations in terms of the assigned node
voltage. Each constraint equation represents one of the
necessary linearly independent equations, and N voltage
sources yield N linearly independent equations.
Use KCL to formulate the remaining N1N linearlyindependent equations. First, apply KCL at each non-reference
node not connected to a voltage source. Second, apply KCL at
each supernode.
v
v
v
26
