Lesson 7:
Nodal Analysis I
Learning Objectives

Apply Ohm’s Law using nodal voltages.
Apply the Nodal Analysis method to
determine an unknown node voltage and
branch currents in a simple DC circuit

Nodal analysis
Based on a systematic application of KCL, nodal
analysis allows us to obtain a set of simultaneous
equations to solve for voltages and current in any
linear circuit.
Nodal analysis steps

In nodal analysis we are interested in finding
node voltages.

Given a circuit with n nodes,
1. Select a reference node. Assign voltages
va, vb, vc …. to the remaining nodes.
2. Apply KCL to each node expressing branch
currents in terms of node voltages.
3. Solve the resulting simultaneous equations to
obtain unknown node voltages.
1. Identify the nodes

What are the voltages of the three nodes a,b,c
below?
Va  6V
Vb  unknown
Vc  12V
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!
2. Write equations for branch currents


You can make the problem simpler by always
arbitrarily assuming current leaves each node
This simplifies writing 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
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
You can solve this with some simple algebra…
Vb  6 
4
Vb Vb  12 
0
 
8
3
 1 1 1   6 12 
Vb        
 4 3 8 4 8 
Vb  0.7083  3
Vb  4.235V
Example Problem 1
Solve for node voltage Vb
Example Problem 2
Solve for node voltage Vb
Making it tougher…


The polarity of the voltage supplies changes how you
write the equations
Notice the inverted supply on the right and how you write
the equation
i1 
i2 
i3 
Vb  Va  Vb  6a 
4
Vb  0 


2
Vb  Vc 
8
4
Vb 

2
Vb   8 
8

Vb  8
8
You can solve this with some simple algebra…
Vb  6 
4
Vb Vb  8 
 
0
2
8
1 1 1 6 8
Vb        
 4 2 8 4 8
Vb  0.875   0.5
Vb  0.571V  571mV
Example Problem 3
How many nodes in the circuit below have unknown
voltages?
Determine IUNK
Example Problem 4
Find the voltage at node b and the unknown current.