Lesson 10: 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 analysisBased 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?

6

12

a

b

c

V V

V unknown

V V

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!

You can make the problem simpler by always arbitrarily assuming current leaves each node

This simplifies writing 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 currents

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

You can solve this with some simple algebra…

6 120

4 3 81 1 1 6 12

4 3 8 4 8

0.7083 3

4.235

b bb

b

b

b

V VV

V

V

V V

Example Problem 1

Solve for node voltage Vb

Example Problem 2

Solve for node voltage Vb

1

2

3

6

4 40

2 2

8 8

8 8 8

b a b a

b b

bb c b

V V Vi

V Vi

VV V Vi

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

You can solve this with some simple algebra…

6 80

4 2 81 1 1 6 8

4 2 8 4 8

0.875 0.5

0.571 571

b bb

b

b

b

V VV

V

V

V V mV

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.