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Lesson 11: Nodal Analysis II
Learning Objectives
Apply Nodal Analysis to circuits with current sources.
Solve for multiple unknown voltages in a complex DC circuit.
Use calculator to solve Equations.
1. Identify the nodes
What are the voltages of the three nodes a,b,c below?
6
12
a
b
c
V V
V unknown
V V
REVIEW
2. Write equations for branch currents
In nodal analysis, we usually write the branch currents directly in terms of node voltages and branch resistances.
Care must be taken in keeping the polarity correct!
REVIEW
You can make the problem simpler by arbitrarily assuming current leaves each node
This simplifies the resulting equations and prevents polarity errors.
1
2
3
6
4 40
3 312
8 8
b a b a
b b
b c b
V V Vi
V Vi
V V Vi
2. Write equations for branch currentsREVIEW
3. Substitute into KCL and solve for the unknowns
1 2 3 0
6 120
4 3 8b bb
i i i
V VV
current out = current in
REVIEW
Current Sources A current source makes the equations simpler, since
now you know what the branch current is. Pay attention to POLARITY!
1
2
3
3
0
4 40
12 12
b b
b b
i A
V Vi
V Vi
Example Problem 1
Determine Vb in the circuit below
123 03 8
1 1 123
3 8 8
4.59.81
0.458
bb
b
b
VV
V
V V
Example Problem 2
Write the equation for the two nodes in the circuit below:
Mathematica Solution
SOLVE function for multiple equations
SOLVE (3 + a/6 + (a-b)/5 = 0 and (b-a)/5 + b/8 + (b+12)/6=0,{a,b})
0 36 5
120
5 8 6
a ba
b a bb
V VV
V V VV
Example Problem 3
Solve for IUNK.
Mathematica Solution
Example Problem 4
Solve for Vab.
Mathematica Solution