nodal analysis for kcl and kvl

NODAL AND LOOP ANALYSIS TECHNIQUES

LEARNING GOALS

Develop systematic techniques to determine all the voltagesand currents in a circuit

NODAL ANALYSISLOOP ANALYSIS

NODE ANALYSIS

• One of the systematic ways to determine every voltage and current in a circuit

The variables used to describe the circuit will be “Node Voltages” -- The voltages of each node with respect to a pre-selected reference node

1 1 21@ : 4 0

6 12

V V VV mA

k k

USING KCL

2 2 12@ : 2 0

6 12

V V VV mA

k k

BY “INSPECTION”

1 2

1 1 14

6 12 12V V mA

k k k

1 2

1 1 12

12 6 12V V mAk k k

LEARNING EXAMPLE

VV

VV

4

6

4

1

SOURCES CONNECTED TO THE

REFERENCE

SUPERNODE

CONSTRAINT EQUATION VVV 1223

KCL @ SUPERNODE

02

)4(

212

63322

k

V

k

V

k

V

k

Vk2/*

VVV 22332VVV 12

32 add and 3/*

VV 385 3

mAk

VIO

8.323 LAW SOHM'

OIV FOR NEEDED NOT IS

2

LEARNING EXAMPLE

CIRCUITS WITH DEPENDENT SOURCESPRESENT NO SIGNIFICANT ADDITIONAL COMPLEXITY. THE DEPENDENT SOURCESARE TREATED AS REGULAR SOURCES

WE MUST ADD ONE EQUATION FOR EACHCONTROLLING VARIABLE

VOLTAGE SOURCE CONNECTED TO REFERENCE

VV 31

0263

: 212

xI

k

V

k

VV VKCL@ 2

CONTROLLING VARIABLE IN TERMS OF NODE VOLTAGES k

VIx 6

2

REPLACE

062

632212

k

V

k

V

k

VVk6/*

VVVV 602212

mAk

VVIO

13

21

OI FIND

LEARNING EXAMPLE

SUPER NODE WITH DEPENDENT SOURCE

VOLTAGE SOURCE CONNECTED TO REFERENCE

VV 63

SUPERNODE CONSTRAINT xVVV 2

21

KCL AT SUPERNODE

k12/*

062)6(22211

VVVV

CONTROLLING VARIABLE IN TERMS OF NODES

2VV

x

213VV

183321 VV 184

1 V

LEARNING EXAMPLE FIND THE VOLTAGE Vo

AT SUPER NODE

1 22

XV V V

2 3 1 32 14

2 01 1 1 1

V V V VV V VmA

k k k k

3 2 3 1

3@ : 2 0

1 1

V V V VV mA

k k

4 4@ : 4V V V

CONTROLLING VARIABLE2X

V V

1k

1k

SOLVE EQUATIONS NOW

1

1 3

1 3

3

2 2 2 6

2 2

X

X

X

V V

V V V V

V V V V

VARIABLE OF INTEREST 1 3OV V V

LEARNING EXAMPLE Find the current Io

FIND NODES – AND SUPER NODES

2 2@ : 12V V V

3 3@ : 2

XV V V

4 1

1 3 4 3 4 51 2 4

@super node:

6 (constraint eq.)

21 1 1 1 10

X

V V V

V V V V V VV V VI

k k k k k

5 4 5

5@ : 2 0

1 1X

V V VV I

k k

1 2

4

CONTROLLING VARIABLES

1

X

X

V V V

VI

k

7 eqs in 7 variables

VARIABLE OF INTEREST 5

1O

VI

k