Node-Voltage and Mesh-Current Analysis - Notes | ECE 3710, Study notes of Electrical and Electronics Engineering

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Node-­‐Voltage and Mesh-­‐Current

Analysis

ECE 3710, Fall 2011

Text Key

SecCons 2.4 and 2.5 in the book

Node-­‐Voltage Analysis

• Start by defining a reference node
• Next, we label the voltages at each node. The

negaCve reference polarity is the reference

node

• Next we strategically solve for each of the

node voltages

Node-­‐Voltage Example

R

R

R

R4 R

Node 1

Node 2

Node 3

v

z v

x

v

y

v

1 v 2

v

3

Node-­‐Voltage Example

R

R

R

R4 R

Node 1

Node 2

Node 3

v

z v

x

v

y

v

1 v 2

v

3 Assume we know all node voltages, we can draw KVL across the loop

Node-­‐Voltage Example

Looking at KVL, we can write the soluCon around the

loop:

! V

2

+ V

X

+ V

3

What do we know about V

3

What do we know about V

2

It’s the node voltage at Node 3 It’s the node voltage at Node 2

A Few Notes on our Choices

R

R2 R

R4 R

v

v

v

Reference

V1 is the source voltage Make the reference node ground

KCL for Node Voltages

R

R2 R

R4 R

v

v

v

Reference

V

3

R 5

V

2

! V

3

R 3

V

1

! V

3

R 1

Node-­‐Voltages in Standard Form

Node 1:

v 1! v 3 R 1

v 1! v 2 R 2 = ia

Node 2:

v 2! v 1 R 2

v 2 R 3

v 2! v 3 R 4

Node 3:

v 3 R 3

v 3! v 2 R 4

v 3! v 1 R 1

• ib = 0

Standard Form

g

11

v

1

+ g

12

v

2

+ g

13

v

3

= i 1

g

21

v

1

+ g

22

v

2

+ g

23

v

3

= i 2

g

31

v

1

+ g

32

v

2

+ g

33

v

3

= i 3

g 11 g 12 g 13 g 21 g 22 g 23 g 31 g 32 g 33 ! "

$ % & & & &

GV = I V^ =^ G

! 1

I

Standard Form Example

v

v

v

10A 20Ω

ix

Standard Form Example

Node 1:

v 1! v 3 20

v 1 5

v 1! v 2 1

Node 2:

v 2! v 1 1

v 2! v 3 20

Node 3:

v 3! v 1 20

v 3 10

v 3! v 2 20

• ib = 0

1. 25 v 1 ! 1 v 2 !. 05 v 3 = 0 ! 1 v 1

• 1. 05 v 2 !. 05 v 3 =! 10 !. 05 v 1 !. 05 v 2 +. 2 v 3 = 0

1. 25! 1 !. 05 ! 1 1. 05 !. 05 !. 05 !. 05. 2 "

$ $ $ % & ' ' '

How to Draw Mesh Currents

R1 R

R

V1 V

How would we normally solve this circuit using currents? Instead of drawing “branch currents”, we draw “loop currents”! i1 i

How to Draw Mesh Currents

R1 R

R

V1 V

i1 i

v 1 = R 1 ( i 1 ) + R 3 ( i 1 ! i 2 ) ! v 2 =! R 3 ( i 1 ! i 2 ) + R 2 ( i 2 )