2.1 the node voltage method1

7/31/2019 2.1 the Node Voltage Method1

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CHAPTER 2:

RESISTIVE NETWORKANALYSIS


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LEARNING OBJECTIVESExplain and contrast the meanings of the

node voltage method.

Solve the node voltage method analysis.

SUBCHAPTER 2.1

THE NODE VOLTAGE METHOD


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Node Voltage Method

The node voltage method is based on defining thevoltage at each node as an independent variable.

One of the nodes is selected as a reference node

and each of the other node voltage is referenced to

this node.

Once each node voltage is defined, the current

flowing in each branch is determined by Ohms

Law.Each branch current is expressed in terms of one or

more node voltages.


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NODAL ANALYSIS

Nodal Analysis finds the node voltages by

first performing KCL at the essential nodes

in terms of the node voltages. By solvingthe equations obtained from KCL, we can

find the node voltages.


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NODAL ANALYSIS (cont..)

Current flows from a higher potential to a

lower potential in a resistor.

We can express this principle as:

R

vvi lowerhigher


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Vs

RI

V

R

VVI s

Vs

RI

V

_

+

R

VVI s

R

VV s


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NODAL ANALYSIS (cont..)V

I

R

R

V

R

VI

0

V1

RI

V2

RVVI 21


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V

1 A

I

I = - 1 A

V

1 A

I

I = 1 A

V

VssVV


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V1 V2

2 V+

-

+

-

+-

+

-

R1 R3

R2

R5

3 VR4

Vs

5 VA

B

D

C


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BASIC STEPS

80 5 A

40

15

4 25 3 A

Assume that we are trying to find the voltageacross and the current through all the elements


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STEP 1: MARK ESSENTIAL NODE

80 5 A

40

15

4 25 3 A


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STEP 2: REFERENCE NODE

Mark the reference node with the earth signor

downward arrow .A reference node is the node from where all the other node

voltages the node that is considered to be at 0 V.


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STEP 3: ASSIGN UNKNOWNNODE VOLTAGES

80 5 A

40

15

4 25 3 A

V1 V2

V3


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STEP 4: DECIDE ON NUMBER OFEQUATIONS REQUIRED

Decide on the number of equations

required to solve the circuit.

Referring to the example, there are

3 unknowns (i.e. v1, v2 and v3).


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STEP 5: PERFORM KCL AT THESELECTED NODES

54080

211

vvv

325440

323212

vvvvvv

025415

23233 vvvvv

KCL: Node 2:

KCL: Node 3:

(1)

...(2)

(3)

KCL: Node 1:


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STEP 6: SOLVE THE EQUATIONS

This may be done by solving the

simultaneous equation or applying

Cramer's Rule.


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EXAMPLE 1

Given:

Find: The node voltages in the circuit shown.


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EXAMPLE 1 (cont..)

Solution:


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EXAMPLE 1 (cont..)

At node 1, applying KCL and Ohms Law gives

Multiplying each term in the last equation by 4:

or

321iii

2

0

45 121

vvv

121220 vvv

203 21 vv (1)


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EXAMPLE 1 (cont..)

At node 2, applying KCL and Ohms Law gives

Multiplying each term by 12 results in:

or

5142iiii

6

0510

4

221

vvv

22126012033 vvv

605321 vv (2)


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EXAMPLE 1 (cont..)

METHOD 1: ELIMINATION TECHNIQUEUsing elimination technique, add equation (1)

and (2).

Substituting the value of v2 in equation (1) gives

20321

vv

6053 21 vv +

8042

v V202

v

20203 1 v 13.333V3

40

1v


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EXAMPLE 1 (cont..)

We now obtain v1 and v2 as

giving us the same result as did the elimination

method.

V333.13

12

60100560

120

1

1

v

2

2

3 20

3 60 180 6020 V

12v


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EXAMPLE 2

Given:

Find: The voltages at the nodes in the figure

shown below


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EXAMPLE 2 (cont..)

Solution:


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EXAMPLE 2 (cont..)

At node 1,

Multiplying by 4 and rearranging terms:

xii 1324

3 2131vvvv

1223321

vvv (1)


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EXAMPLE 2 (cont..)

At node 2,

Multiplying by 8 and rearranging terms:

32iiix

4

0

82

23221

vvvvv

074321

vvv (2)


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EXAMPLE 2 (cont..)

At node 3,

Multiplying by 8, rearranging terms, and

dividing by 3:

xiii 221

2

2

84

213231vvvvvv

032321

vvv

(3)


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EXAMPLE 2 (cont..)

METHOD 1: ELIMINATION TECHNIQUEUsing elimination technique, add equation (1)

and (3).

or

1223321

vvv

032321 vvv

+

125521

vv

2.45

12

21 vv (4)


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EXAMPLE 2(cont..)

Add equation (2) and (3) gives

Substituting equation (5) into (4) yields

while

074321 vvv

032321 vvv

042 21 vv

+

21 2vv (5)

V4.22

v4.22

22 vv

V8.4221

vv


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EXAMPLE 2 (cont..)

From equation (3),

Thus,

12323 vvv

22343 vvv

23 vv

V8.41

v V4.22

v

V4.23

v

V4.23

v


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EXAMPLE 2 (cont..)

METHOD 2 : CRAMERS RULETo use Cramers Rule, equation (1) and (3) need

to be put in matrix form as

From this, we obtain

0

0

12

132

174

123

3

2

1

v

v

v

3

3

2

2

1

1,, vvv


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EXAMPLE 2 (cont..)

where are the determinants. To

calculate the determinant of a 3 by 3 matrix, we

repeat the first two rows and cross multiply.

321 ,,


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EXAMPLE 2 (cont..)

NODAL ANALYSIS WITH


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NODAL ANALYSIS WITH

VOLTAGE SOURCES

Now consider how voltage sources affect nodal

analysis.


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NODAL ANALYSIS WITH


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NODAL ANALYSIS WITH

VOLTAGE SOURCES (cont..)

CASE 2:

If the voltage source (dependent or independent)

is connected between two nonreference nodes,the two nonreference nodes from a generalized

node or supernode; we apply both KCL and KVL to

determine the node voltages.

NODAL ANALYSIS WITH


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KCL must be satisfied at a supernode like any

other node. Hence, at the supernode in previous

figure,

or

3241iiii (2a)

6

0

8

0

42

323121 vvvvvv

(2b)

NODAL ANALYSIS WITH


NODAL ANALYSIS WITH


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NODAL ANALYSIS WITH


To apply KVL to the supernode in previous

figure, the circuit need to be redrawn as shown

below.

NODAL ANALYSIS WITH


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NODAL ANALYSIS WITH


Going around the loop in clockwise direction

gives

From equation (1), (2b) and (3), the node

voltages can be obtained.

05 32 vv 532 vv (3)

NODAL ANALYSIS WITH


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A supernode is formed by enclosing a (dependant or

independent) voltage source connected between

two nonreference nodes and any elements

connected in parallel with it.

Properties of a supernode:

1. The voltage source inside the supernode provides aconstraint equation needed to solve for the node voltages.

2. A supernode has no voltage of its own.

3. A supernode requires the application of both KCL and KVL.

NODAL ANALYSIS WITH



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EXAMPLE

Given:



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EXAMPLE (cont..)

Apply KVL to the circuit in figure (b) in order toget the relationship between v1 and v2. Going

around the loop, we obtain

From equation (1) and (2),

0221 vv

1122202 vvv

212 vv (2)


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212

vv

or

and

2231

v V333.71 v

V333.52 v

EXAMPLE (cont..)


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EXERCISE

Given:



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Thank You

2.1 the node voltage method1

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