Background
KVL + KCL + Ohm’s law
− sufficient in analyzing any resistive circuits
− difficult to use as the complexity of circuits increased
Nodal Analysis
Nodal Analysis
Mesh Analysis
− developed based on Kirchhoff’s laws
− provide two systematic methods for describing circuits
Nodal Analysis
Steps to Determine Node Voltages
• In nodal analysis, the circuit variables are node
voltages
R1
iA
R2
iB
R3
1. Select a reference node.
2. Assign node voltages to the other nodes.
3. Apply KCL to nonreference nodes
--- express currents in terms of node voltages.
4. Solve the resulting linear equations.
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Nodal Equations by Visual
Inspection
Nodal Analysis
• Node voltages
– The voltage at the reference node is assumed to
be zero
− We select node voltages as being positive w.r.t.
the reference node
− If the node voltage is actually negative w.r.t. the
reference node, the analysis will indicate it.
• Once node voltages are known, we can
calculate any branch current.
(only for circuits containing resistors and
independent current sources)
• Left-hand side:
− Diagonal terms is the sum of conductances of all
resistors connected to the node.
− Off-diagonal terms are the negatives of the
conductances connected between the nodes.
• Right-hand side:
− Algebraic sum of the currents entering the node
Example
Example
1Ω
5A
0.25 Ω
0.5 Ω
4Ω
0.5 Ω
2Ω
6Ω
10 A
1Ω
4A
1A
2
Nodal Analysis with Dependent
Sources
• Dependent sources are handled the same way we
handled independent sources
• The node voltage equations must be
supplemented with an additional equation resulting
from the dependent source
• Observations from visual inspection don’t apply for
circuits containing dependent sources
Nodal Analysis
with Voltage Sources
3 cases:
I. The voltage source connects one of the nodes
and the ground
II. The voltage source lies between two nonreference nodes
III. The voltage source has a series resistor
Examples
10 kΩ
Io
10 kΩ
4 mA
2 Io
10 kΩ
Nodal Analysis with Voltage Sources (I)
The voltage source connects one of the nodes and the ground:
Solution: node voltage = voltage of the voltage source
9 kΩ
12 kΩ
12 V
12 kΩ
6 kΩ
6V
3
Nodal Analysis with Voltage Sources
(II)
Nodal Analysis with Voltage Sources
(II)
The voltage source is connected between two nonreference nodes:
0.5 A
Problem: the current through the voltage source is unknown
Solution: form a “supernode” and apply KCL+KVL to the supernode
2Ω
6V
12 kΩ
6 kΩ
6 mA
4 mA
1Ω
.
.
2V
5Ω
1A
4Ω
2A
Nodal Analysis with Voltage Sources (III)
The voltage source has a series resistor:
The voltage across the resistor can be determined from the
voltage of the voltage source and the node voltage at the other
end (there is no need to define an extra node voltage for the
node that connects the two elements)
20 Ω
V1
20 Ω
24 V
V2
Ix
80 Ω
+
-
4V2
2Ix
+
V2
-
40 Ω
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