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UNIT-1
DC Circuit
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Node-voltage analysis of
resistive circuit in the
context of dc voltages
and currents
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Objectives
To provide a powerful but simple circuit analysis tool based on Kirchhoff’s
current law (KCL) only.
1 Node voltage analysis
In the previous lesson-4, it has been discussed in detail the analysis of a dc network by
writing a set of simultaneous algebraic equations (based on KVL only) in which the
variables are currents, known as mesh analysis or loop analysis. On the other hand, the
node voltage analysis (Nodal analysis) is another form of circuit or network analysis
technique, which will solve almost any linear circuit. In a way, this method completely
analogous to mesh analysis method, writes KCL equations instead of KVL equations, and
solves them simultaneously.
2.Solution of Electric Circuit Based on Node Voltage Method
In the node voltage method, we identify all the nodes on the circuit. Choosing one of them
as the reference voltage (i.e., zero potential) and subsequently assign other node voltages
(unknown) with respect to a reference voltage (usually ground voltage taken as
zero (0) potential and denoted by ( ). If the circuit has “n” nodes there are “n-1”
node voltages are unknown (since we are always free to assign one node to zero or ground
potential). At each of these “n-1” nodes, we can apply KCL equation. The unknown node
voltages become the independent variables of the problem and the solution of node
voltages can be obtained by solving a set of simultaneous equations.
Let us consider a simple dc network as shown in Figure 5.1 to find the currents
through different branches using “Node voltage” method.
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KCL equation at “Node-1”:
I s1 − I s3 − V1 −V2 − V1 −V3 0 ; → I s1
− I s3 − 1 1 V−
1
1 V−
2
1V 0
3
R R
R4 R2 R2 4 4 R2
I −I G V −G V −G V
s1 s3 11 1 12 2 13 3 (5.1)
where Gii = sum of total conductance (self conductance) connected to Node-1.
KCL equation at “Node-2”:
−V2 −V3 1 1 1V 1
−V2 −Is2 0 ; → − I s 2 − V
− V
V
R R
1
1 2 3
4 3
R R R R
4 3 4 3
− I s 2 − G21 V1 G22 V2 − G23 V3 (5.2)
KCL equation at “Node-3”:
−V V −V V 1 1V 1 1 1
I
s3 R 3V2
1R 3 − 3
R
0 ; → I s3 −
R
V
1 −
R
2 R R R V3
3 2 1 2 3 1 2 3
Is3 − G31 V1 − G32 V2 G33 V3 (5.3)
In general, for the ith Node the KCL equation can be written as
I G V G V
∑ ii − i1 1 − i 2 2 − Gii Vi −
− GiN VN
where,
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UNIT-1
DC Circuit
www.android.previousquestionpapers.com | www.previousquestionpapers.com | www.ios.previousquestionpapers.com
, www.android.universityupdates.in | www.universityupdates.in | www.ios.universityupdates.in
Node-voltage analysis of
resistive circuit in the
context of dc voltages
and currents
www.android.previousquestionpapers.com | www.previousquestionpapers.com | www.ios.previousquestionpapers.com
, www.android.universityupdates.in | www.universityupdates.in | www.ios.universityupdates.in
Objectives
To provide a powerful but simple circuit analysis tool based on Kirchhoff’s
current law (KCL) only.
1 Node voltage analysis
In the previous lesson-4, it has been discussed in detail the analysis of a dc network by
writing a set of simultaneous algebraic equations (based on KVL only) in which the
variables are currents, known as mesh analysis or loop analysis. On the other hand, the
node voltage analysis (Nodal analysis) is another form of circuit or network analysis
technique, which will solve almost any linear circuit. In a way, this method completely
analogous to mesh analysis method, writes KCL equations instead of KVL equations, and
solves them simultaneously.
2.Solution of Electric Circuit Based on Node Voltage Method
In the node voltage method, we identify all the nodes on the circuit. Choosing one of them
as the reference voltage (i.e., zero potential) and subsequently assign other node voltages
(unknown) with respect to a reference voltage (usually ground voltage taken as
zero (0) potential and denoted by ( ). If the circuit has “n” nodes there are “n-1”
node voltages are unknown (since we are always free to assign one node to zero or ground
potential). At each of these “n-1” nodes, we can apply KCL equation. The unknown node
voltages become the independent variables of the problem and the solution of node
voltages can be obtained by solving a set of simultaneous equations.
Let us consider a simple dc network as shown in Figure 5.1 to find the currents
through different branches using “Node voltage” method.
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KCL equation at “Node-1”:
I s1 − I s3 − V1 −V2 − V1 −V3 0 ; → I s1
− I s3 − 1 1 V−
1
1 V−
2
1V 0
3
R R
R4 R2 R2 4 4 R2
I −I G V −G V −G V
s1 s3 11 1 12 2 13 3 (5.1)
where Gii = sum of total conductance (self conductance) connected to Node-1.
KCL equation at “Node-2”:
−V2 −V3 1 1 1V 1
−V2 −Is2 0 ; → − I s 2 − V
− V
V
R R
1
1 2 3
4 3
R R R R
4 3 4 3
− I s 2 − G21 V1 G22 V2 − G23 V3 (5.2)
KCL equation at “Node-3”:
−V V −V V 1 1V 1 1 1
I
s3 R 3V2
1R 3 − 3
R
0 ; → I s3 −
R
V
1 −
R
2 R R R V3
3 2 1 2 3 1 2 3
Is3 − G31 V1 − G32 V2 G33 V3 (5.3)
In general, for the ith Node the KCL equation can be written as
I G V G V
∑ ii − i1 1 − i 2 2 − Gii Vi −
− GiN VN
where,
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