1. Nodal Analysis
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2. Steps of Nodal Analysis
1. 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.
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3. Example: A Summing Circuit
• The output voltage V of this circuit is
proportional to the sum of the two input
currents I1 and I2
• This circuit could be useful in audio
applications or in instrumentation
• The output of this circuit would probably
be connected to an amplifier
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4. 1. Reference Node
500Ω 500Ω
+
I1
500Ω V 1kΩ 500Ω I2
–
The reference node is called the ground node
where V = 0
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5. Steps of Nodal Analysis
1. 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.
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6. 2. Node Voltages
V1 500Ω V 500Ω V3
2
1 2 3
I1 1kΩ 500Ω I2
500Ω
V1, V2, and V3 are unknowns for which we
solve using KCL
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7. Steps of Nodal Analysis
1. 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.
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8. Currents and Node Voltages
V1 500Ω V2 V1
V1
500Ω
500Ω
V1 − V2
500Ω
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9. 3. KCL at Node 1
V1 500Ω V2
I1 V1 − V2 V1
500Ω I1 = +
500Ω 500Ω
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10. 3. KCL at Node 2
V1 500Ω V2 500Ω V3
1kΩ
V2 − V1 V2 V2 − V3
+ + =0
500Ω 1kΩ 500Ω
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11. 3. KCL at Node 3
V2 500Ω V3
V3 − V2 V3
500Ω I2 + = I2
500Ω 500Ω
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12. Steps of Nodal Analysis
1. 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.
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13. 4. Summing Circuit Solution
500Ω 500Ω
+
I1
500Ω V 1kΩ 500Ω I2
–
Solution: V = 167I1 + 167I2
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14. A Linear Large Signal
Equivalent to a Transistor
0.7V
Ib
+ –
+
5V 1kΩ 50Ω 2kΩ
+
100Ib Vo
–
–
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15. Steps of Nodal Analysis
1. 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.
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16. Linear Large Signal Equivalent
0.7V
V1 Ib V 2 V3 V4
1 + –
2 3 50Ω 4 +
1kΩ
5V +
100Ib Vo
–
2kΩ –
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17. Steps of Nodal Analysis
1. 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.
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18. KCL @ Node 4
0.7V
V1 Ib V2 V3 V4
1 + –
2 3 50Ω 4 +
1kΩ
5V +
100Ib Vo
–
2kΩ –
V3 − V4 V4
+ 100 I b =
50Ω 2 kΩ
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19. The Dependent Source
• We must express Ib in terms of the node
voltages:
V1 − V2
Ib =
1 kΩ
• Equation from Node 4 becomes
V3 − V4 V1 − V2 V4
+ 100 − =0
50Ω 1 kΩ 2kΩ
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20. How to Proceed?
• The 0.7-V voltage supply makes it
impossible to apply KCL to nodes 2 and 3,
since we don’t know what current is
passing through the supply
• We do know that
V2 – V3 = 0.7 V
• The above is a needed constraint
equation
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21. KCL at
Supernode
V2 − V1 V3 − V4
+ =0
1kΩ 50Ω
0.7V
V1 V2 V3 V4
1 + –
Ib 4 +
1kΩ 50Ω
+
100Ib Vo
–
2kΩ –
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