Node Voltage Method

Node Analysis with Voltage Sources• When voltage sources are added to the circuit the node

voltage analysis will be slightly different as the current through a voltage source Is not directly related to the

voltage across it• There are three methods to deal with a voltage source

• First method: replace the voltage source and series resistance with equivalent current source and parallel

resistance

Node Analysis with Voltage Sources

Second method: • Node B is selected as the reference

node therefore VB=0

• VA=Vs and therefore he voltage of

node A is known. No node-voltage

equation at node A is needed

• The node voltage equations at the

remaining N-2 nodes are written and

solved in the usual way to find the N-

2 unknown node voltages

Node Analysis with Voltage SourcesThird method: • Neither node A nor node B is selected as

reference

• Node A and B are combined into one

supernode

• The node voltage equations in the

remaining N-3 nodes plus the supernode

are written

• The equation related to the voltage

source inside the supernode will provide

the last equation: sBA vvv

Example 1

Find the voltage Vo using node-voltage analysis

Example 1

321

2211

22113212

2

1

1

321

2

2

31

1

111

0

necessary equations 12-N be willthere

sources voltage twoand nodes 3 are thereSince

analysis. voltage

nodein used is method second The

GGG

GvGvv

GvGvGGGvR

v

R

v

RRRv

R

vv

R

v

R

vv

ssO

ssOss

O

sOOsO

Solution:

Example 2

Find the input resistance of the circuit.

Solution: Node A has the voltage of the source therefore two node voltage equations are required

RR

i

vR

R

v

R

vv

R

vvi

i

v

i

vR

v

GG

GG

GvG

GvG

vv

GG

GG

GGv

GGv

v

GvGvGv

GvGvGvvv

GvGvGv

GvGvGv

IN

sIN

sCsBsIN

IN

s

IN

ININ

ss

s

CC

ss

s

BB

B

sCB

sCBsA

CBA

CBA

872.075.11

25.1025.10

75.11

2/2 :KCL

25.10

25.6

5.35.0

5.03

25.0

5.03

25.10

75.2

5.35.0

5.03

5.32

5.05.0

vand vfind tosolved are unknowns twoand equations twoofset The

25.35.0

5.05.03

05.35.02 :C Node

05.035.0 :B Node

C

Example 3

Find Vo when element E is 10kΩ resistance.

Example 3

Solution:

V 2.53 V 97.3

:equations theslovingAfter

0105

0541

5

:equations voltageNode

OA

OAO

OAAA

vv

vvv

vvvv