Lesson 8:
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

REVIEW
What are the voltages of the three nodes a,b,c
below?
Va  6V
Vb  unknown
Vc  12V
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
2. Write equations for branch currents


You can make the problem simpler by arbitrarily
assuming current leaves each node
This simplifies the resulting equations and
prevents polarity errors.
i1 
i2 
i3 
Vb  Va  Vb  6a 
4
Vb  0 


3
Vb  Vc 
8
4
Vb 

3
Vb  12 
8
REVIEW
3. Substitute into KCL and solve for the
unknowns
current out = current in
Vb  6 
4
i1  i2  i3  0
Vb Vb  12 
 
0
3
8
Current Sources


A current source makes the equations simpler, since
now you know what the branch current is.
Pay attention to POLARITY!
i1  3 A
Vb  0 
Vb
i2 

4
4
Vb  0  Vb
i3 

12
12
Example Problem 1
Determine Vb in the circuit below
Vb Vb  12 
3 

0
3
8
12
1 1
Vb     3 
8
3 8
4.5
Vb 
 9.81V
0.458
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})
Va Va  Vb 
0  3  
6
5
Vb  Va  Vb Vb  12 
0
 
5
8
6
Example Problem 3
Solve for IUNK.
Mathematica Solution
Example Problem 4
Solve for Vab.
Mathematica Solution