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