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Electric Circuit Analysis
Lec4: Nodal analysis
Ahsan [email protected] 102Department of Electrical Engineering
Steps of Nodal Analysis1. Choose a reference (ground) node.2. Assign node voltages to the other nodes.3. Apply KCL to each node other than the reference
node; express currents in terms of node voltages.4. Solve the resulting system of linear equations for
the nodal voltages.
Common symbols for indicating a reference node, (a) common ground, (b) ground, (c) chassis.
1. Reference Node
The reference node is called the ground node where V = 0
+
–
V 500W
500W
1kW
500W
500WI1 I2
Steps of Nodal Analysis1. Choose a reference (ground) node.2. Assign node voltages to the other nodes.3. Apply KCL to each node other than the reference
node; express currents in terms of node voltages.4. Solve the resulting system of linear equations for
the nodal voltages.
2. Node Voltages
V1, V2, and V3 are unknowns for which we solve using KCL
500W
500W
1kW
500W
500WI1 I2
1 2 3
V1 V2 V3
Steps of Nodal Analysis1. Choose a reference (ground) node.2. Assign node voltages to the other nodes.3. Apply KCL to each node other than the reference
node; express currents in terms of node voltages.4. Solve the resulting system of linear equations for
the nodal voltages.
Currents and Node Voltages
500W
V1
500WV1 V2
50021 VV
5001V
3. KCL at Node 1
500W
500WI1
V1 V2
500500
1211
VVVI
3. KCL at Node 2
500W
1kW
500W V2 V3V1
0500k1500
32212
VVVVV
3. KCL at Node 3
2323
500500I
VVV
500W
500W
I2
V2 V3
Steps of Nodal Analysis1. Choose a reference (ground) node.2. Assign node voltages to the other nodes.3. Apply KCL to each node other than the reference
node; express currents in terms of node voltages.4. Solve the resulting system of linear equations for
the nodal voltages.
Typical circuit for nodal analysis
• Find the node voltages in the circuit shown below.
• At node 1
2
0
45 121
321
vvv
iii
• At node 2
6
0
45 212
5142
vvv
iiii
Practice
• Determine the voltage at the nodes in Fig. below
• At node 1,
243
3
2131
1
vvvv
ii x
• At node 2
4
0
8223221
32
vvvvv
iiix
• At node 3
2
)(2
84
2
213231
21
vvvvvv
iii x
• In matrix form:
0
0
3
8
3
8
9
4
38
1
8
7
2
14
1
2
1
4
3
3
2
1
v
v
v
Spot quiz…
Example:Use nodal analysis to find the voltage at each node of this circuit.
• Step 1:Identify and label, each node in the circuit. Ground has been chosen for you.
• Step 2:Write the Nodal equation for each node identified in Step 1. • Node 1:Because the voltage at node 1 is known with respect to our
reference point, the equation is:V1 = 71 volt
• Node 2:At node 2, assume all currents are leaving the node as shown in Figure 5.
All currents are assumed to be leaving node V2,
so all terms are positive.
Node 2 I (1) + I (2) + I (3) = 0
Ohm's Law I (1) = (V2 - V1)/2
I (2) = (V2 – 0)/11
I (3) = (V2 - V3)/10
Substituting: (V2 - V1)/2 + V2/11 + (V2 - V3)/10 = 0
• I (4) and I (5) are assumed to be leaving node V3, so these terms are positive. But Is (the 2A source) is entering, so it is assigned a negative sign.
Node 3 -Is + I (4) + I (5) = 0
Ohm's Law I (4) = (V3 - V2)/10
I (5) = V3/5
Substituting:-2 + (V3 - V2)/10 + V3/5 = 0
• The three equations are shown below:
Node 1: V1 = 71v
Node 2 : (V2 - V1)/2 + V2/11 + (V2 - V3)/10 = 0
Node 3: -2 + (V3 - V2)/10 + V3/5 = 0
• After solving we find that the node voltages are:
V1 = 71v
V2 = 55v
V3 = 25v
Example:Use nodal analysis to find the voltage at each node of this circuit.
