Revised 12 October 2004
Node Voltage Analysis
Nodal Analysis
ENGI 241
Node Voltage Procedure
• Use Node Voltage Analysis when:
– We are solving for node voltages
– The number of independent nodes is less than the number of independent loops
• We begin by
– Assume all branch currents enter the node
– Write a KCL equation for each node
• If the node is driven by a source current, IS is negative if it enters the node
and positive if it leaves the node
– It is usually more convenient to use conductance's instead of resistances
– Collect the terms for each node
– Solve for the unknown node voltage using Ohm’s Law, KVL, and KCL as
appropriate.
• Use Simultaneous Linear Equations for multiple unknown node voltages
Andrew H. Andersen
ENGI 241 - Node Voltage Analysis
ENGI 241 - Node Voltage Analysis
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Revised 12 October 2004
Node Voltage Analysis
Node Voltage Example
A
Find VA
Andrew H. Andersen
ENGI 241 - Node Voltage Analysis
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Node Voltage Example 1A
Write KCL for all currents entering the node
Andrew H. Andersen
ENGI 241 - Node Voltage Analysis
ENGI 241 - Node Voltage Analysis
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Revised 12 October 2004
Node Voltage Analysis
Node Voltage Example 1B
Same Circuit as 1A, but with VS2 reversed
Andrew H. Andersen
ENGI 241 - Node Voltage Analysis
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Node Voltage Example 2
A
B
Write KCL for all currents entering nodes A and B
Andrew H. Andersen
ENGI 241 - Node Voltage Analysis
ENGI 241 - Node Voltage Analysis
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Revised 12 October 2004
Node Voltage Analysis
Node Voltage Example 2
Andrew H. Andersen
ENGI 241 - Node Voltage Analysis
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Node Voltage Example 3
B
A
Andrew H. Andersen
ENGI 241 - Node Voltage Analysis
ENGI 241 - Node Voltage Analysis
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Revised 12 October 2004
Node Voltage Analysis
Node Voltage Example 3
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ENGI 241 - Node Voltage Analysis
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Node Voltage Example 4
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ENGI 241 - Node Voltage Analysis
ENGI 241 - Node Voltage Analysis
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Revised 12 October 2004
Node Voltage Analysis
Nodal Analysis with a Supernode
• A Supernode exists when a voltage source (may be either dependent or
independent) is present between two essential nodes
– These two nodes connected only by the source can be combined to form a
supernode (.2VD between nodes B and C in previous graphic)
• Account for the currents that flow through the supernode (from node C to
node B in the previous graphic)
– In the KCL equation for Node B, the branch currents entering node C flow
though the supernode to node B
• To solve a circuit with a supernode
– Write the KCL equations for the two nodes of the supernode not adjacent to the
reference or adjacent to a voltage source
– Write the KVL equation for the supernode
– Solve for one node of the supernode voltages in terms of the other node
• Solve for VC in terms of VB in the previous graphic
– Collect all terms in the equation
– Solve the Simultaneous Liner Equation
Andrew H. Andersen
ENGI 241 - Node Voltage Analysis
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Node Voltage Example with Supernode
VD = VS1 - VB
0.2VD = VB - VC
VC = VB - 0.2VD
VC = VB - 0.2(VS1 – VB)
VC = 1.2VB - 0.2VS1
Andrew H. Andersen
ENGI 241 - Node Voltage Analysis
ENGI 241 - Node Voltage Analysis
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Revised 12 October 2004
Node Voltage Analysis
Node Voltage Example 5
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ENGI 241 - Node Voltage Analysis
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Node Voltage Example 6 with Supernode
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ENGI 241 - Node Voltage Analysis
ENGI 241 - Node Voltage Analysis
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Node Voltage Analysis
Node Voltage Example 6 with Supernode
VC = VB + 10ID
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ENGI 241 - Node Voltage Analysis
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Example 7
R1
14
rs1
R3
A
rs2
R2
B
C
4.5
7.7
.1
.1
VS1
RL
120Vdc
VS2
.1
120Vdc
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ENGI 241 - Node Voltage Analysis
ENGI 241 - Node Voltage Analysis
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Revised 12 October 2004
Node Voltage Analysis
Bridge Using Node Voltage Analysis
Calculate VA and VB as if RM is not present
VAB = VA – VB
Andrew H. Andersen
ENGI 241 - Node Voltage Analysis
ENGI 241 - Node Voltage Analysis
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