Nodal Analysis 10 V - Computer Action...

14
Review of Basic Concepts: Voltage 10 V - + - + + - + - 2k2k5k7kv 1 v 2 v 3 v 4 The voltage drop from one node to another is the same, no matter what path is chosen Kirchhoff’s voltage law Portland State University ECE 221 Analysis Methods Ver. 1.66 3 Analysis Methods Overview Solving Linear Equations Nodal Analysis Supernodes (Nodal Analysis with Voltage Sources) Mesh Analysis Supermeshes (Mesh Analysis with Current Sources) Introduction to BJT Transistors This is a very important chapter. Portland State University ECE 221 Analysis Methods Ver. 1.66 1 Resistors in Parallel with Voltage Sources Circuit R V s v o - + Circuit V s v o - + What is v o in each case? What effect does the resistor have on the current pumped into the circuit? Portland State University ECE 221 Analysis Methods Ver. 1.66 4 Review of Basic Concepts: Current i 4 i 5 i 3 i 2 i 1 What goes in, has to come out Kirchhoff’s current law Similar to conservation of mass Conservation of electrons Portland State University ECE 221 Analysis Methods Ver. 1.66 2

Transcript of Nodal Analysis 10 V - Computer Action...

Page 1: Nodal Analysis 10 V - Computer Action Teamweb.cecs.pdx.edu/~ece2xx/ECE221/Lectures/AnalysisMethodsx4.pdf · Nodal Analysis: Step 1 – Identify Essential Nodes 10 V 2 mA 1k Ω 2k

Review

ofB

asic

Concep

ts:Volta

ge

10 V- +

- ++

-+

-

2kΩ

2kΩ

5kΩ

7kΩ

v1

v2

v3

v4

•T

he

voltagedrop

fromon

enode

toan

other

isth

esam

e,no

matter

what

path

isch

osen

•K

irchhoff

’svoltage

law

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63

Analysis

Meth

ods

Overview

•Solvin

gLin

earEquation

s

•N

odal

Analysis

•Supern

odes

(Nodal

Analysis

with

Voltage

Sou

rces)

•M

eshA

nalysis

•Superm

eshes

(Mesh

Analysis

with

Curren

tSou

rces)

•In

troduction

toB

JTTran

sistors

This

isa

veryim

portan

tch

apter.

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Resisto

rsin

Para

llelw

ithVolta

ge

Sources

Circuit

RV

sv

o- +

Circuit

Vs

vo- +

•W

hat

isv

oin

eachcase?

•W

hat

effect

does

the

resistorhave

onth

ecu

rrent

pum

ped

into

the

circuit?

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64

Review

ofB

asic

Concep

ts:Curren

t

i4i5

i3i2

i1

•W

hat

goes

in,has

tocom

eou

t

•K

irchhoff

’scu

rrent

law

•Sim

ilarto

conservation

ofm

ass

•Con

servationof

electrons

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Page 2: Nodal Analysis 10 V - Computer Action Teamweb.cecs.pdx.edu/~ece2xx/ECE221/Lectures/AnalysisMethodsx4.pdf · Nodal Analysis: Step 1 – Identify Essential Nodes 10 V 2 mA 1k Ω 2k

Exa

mple

1:

Term

inolo

gy

20 V2 A

R1

R2

R3

R4

R4

R6

R7

R8

35ip

ip

Iden

tifyth

efollow

ing

inform

ationN

odes:

Essen

tialN

odes:

Bran

ches:

Essen

tialB

ranch

es:EB

’sw

ithU

nkn

own

Curren

t:M

eshes:

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67

Resisto

rsin

Series

with

Curren

tSources

Circuit

IsC

ircuitIs

Rio

io

•W

hat

isio

ineach

case?

•W

hat

effect

does

the

resistorhave

onth

evoltage

seenby

the

circuit?

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65

Exa

mple

2:

Circu

itA

nalysis

The

Hard

Way

10 V

i1i3

i2i4

2 mA

1kΩ

2kΩ

5kΩ

10

Can

solvew

ithK

CL

&K

VL.Fou

runkn

owns.

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Netw

ork

Term

inolo

gy

Plan

arC

ircuit

Acircu

itth

atcan

be

draw

non

aplan

ew

ithno

crossing

branch

es

Node

Poin

tor

portion

ofa

circuit

where

2or

more

elemen

tsare

joined

Essen

tialN

ode

Poin

tor

portion

ofa

circuit

where

3or

more

elemen

tsare

joined

Bran

chPath

that

connects

2nodes

Essen

tialB

ranch

Path

that

connects

2essen

tialnodes

w/o

passin

gth

rough

anessen

tialnode

Loop

Path

with

lastnode

same

asstartin

gnode

that

does

not

crossitself

Mesh

Loop

that

does

not

enclose

any

other

loop

s

Note:

this

isn’t

inth

etext.

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Page 3: Nodal Analysis 10 V - Computer Action Teamweb.cecs.pdx.edu/~ece2xx/ECE221/Lectures/AnalysisMethodsx4.pdf · Nodal Analysis: Step 1 – Identify Essential Nodes 10 V 2 mA 1k Ω 2k

Exa

mple

2:

Contin

ued

(1)

i1−

i2−

i3=

0−

i3+

i4=

2m1k

i1+

5ki2

=10

+5k

i2−

2ki3

−10k

i4=

0

InM

atrixform

this

becom

es

1−

1−

10

00

−1

11k

5k0

00

5k−

2k−

10k i1i2i3i4

= 0

2m100

or

Ai=

b

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Solvin

gLin

earEquatio

ns

•M

uch

ofou

rcircu

itan

alysisw

illfo

cus

onfindin

ga

setof

linear

equation

san

dsolvin

gth

eseeq

uation

s

•N

eedas

man

yeq

uation

sas

there

areunkn

owns

•T

hree

possib

leap

proaches

–A

lgebra(elim

ination

,su

bstitu

tion,etc.)

–Cram

er’sru

le

–Lin

earalgebra

•Last

iseasiest

and

leastsu

sceptib

leto

errors

•Req

uires

use

your

scientifi

ccalcu

lators

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Exa

mple

2:

Contin

ued

(2)

Ai=

bw

here

A=

1−

1−

10

00

−1

11k

5k0

00

+5k

−2k

−10k

i=

i1i2i3i4 b

= 0

2m100

•You

rcalcu

latorsh

ould

be

able

tosolve

this

directly

•You

shou

ldon

lyneed

toen

terA

and

b

•You

rcalcu

latorw

illretu

rna

vectori

•Sim

ultan

eously

solvesfor

allth

eunkn

own

variables

•M

uch

fasterth

anCram

er’sru

leor

brute-force

algrebra

•Read

the

man

uals

foryou

rcalcu

lators

•T

his

will

saveyou

time

(hom

ework

&exam

s)an

dred

uce

errors

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612

Exa

mple

2:

Solvin

gLin

earEquatio

ns

i1=

i2+

i3

i4=

i3+

2m10

=1k

i1+

5ki2

5ki2

=2k

i3+

10ki4

Rew

riteso

variables

arein

consisten

tord

eron

leftsid

ean

dcon

stants

areon

the

right

side

i1−

i2−

i3=

0−

i3+

i4=

2m1k

i1+

5ki2

=10

+5k

i2−

2ki3

−10k

i4=

0

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610

Page 4: Nodal Analysis 10 V - Computer Action Teamweb.cecs.pdx.edu/~ece2xx/ECE221/Lectures/AnalysisMethodsx4.pdf · Nodal Analysis: Step 1 – Identify Essential Nodes 10 V 2 mA 1k Ω 2k

NodalAnalysis:

Step

1–

Iden

tifyEssen

tialN

odes

10 V2 m

A

1kΩ

2kΩ

5kΩ

10

•Som

eessen

tialnodes

may

inclu

de

portion

sof

the

circuit

(pieces

ofw

ire)

•Circle

the

entire

node

topreven

terrors

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615

Exa

mple

2:

Contin

ued

(3)

Lin

earEquation

s:

1−

1−

10

00

−1

11k

5k0

00

5k−

2k−

10k i1i2i3i4

= 02m100

Calcu

latorsh

ould

return

:i1i2i3i4

= +

0.909+

1.818−

0.909+

1.091 m

A

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613

NodalAnalysis:

Step

2–

Pick

aReferen

ce

10 V2 m

A

1kΩ

2kΩ

5kΩ

10

•Secon

dstep

isto

pick

areferen

cenode

•Is

ofteneasiest

toch

oose

the

node

that

intercon

nects

the

most

branch

es

•M

ust

be

anessen

tialnode

•U

sually

isat

bottom

ofcircu

it

•Lab

elw

ithth

esam

esym

bol

used

forgrou

nd

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NodalAnalysis:

Intro

ductio

n

•T

here

isan

anoth

erway

tosolve

forcu

rrents

and

voltages

–Easier

–M

orem

ethodical

–Still

based

onO

hm

’slaw

,K

VL,&

KCL

•N

odal

analysis

ison

eof

two

keym

ethods

•M

eshan

alysisis

the

other

•W

ew

illdiscu

ssnodal

analysis

first

•B

asedon

KCL

•M

ust

understan

dterm

inology

intro

duced

earlier

•U

seto

solvefor

voltages

•A

llvoltages

have

acom

mon

reference

poin

t

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614

Page 5: Nodal Analysis 10 V - Computer Action Teamweb.cecs.pdx.edu/~ece2xx/ECE221/Lectures/AnalysisMethodsx4.pdf · Nodal Analysis: Step 1 – Identify Essential Nodes 10 V 2 mA 1k Ω 2k

NodalAnalysis:

Step

5–

Solve

Lin

earEquatio

ns

Lin

earEquation

s:

Solu

tion(from

calculator):

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619

NodalAnalysis:

Step

3–

Label

Oth

erEssen

tialN

odes

10 V2 m

A

1kΩ

2kΩ

5kΩ

10

•A

lsoa

bit

easierif

voltagesare

labeled

•A

llvoltages

arem

easured

relativeto

the

reference

node

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617

NodalAnalysis:

Step

6–

Solve

for

Varia

bles

ofIn

terest

10 V2 m

A

12

- +v2

- +v1

i1i3

i2i4

1kΩ

2kΩ

5kΩ

10

i1=

i2=

i3=

i4=

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NodalAnalysis:

Step

4–

Apply

KCL

All

Labeled

Nodes

10 V2 m

A

12

- +v2

- +v1

1kΩ

2kΩ

5kΩ

10

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618

Page 6: Nodal Analysis 10 V - Computer Action Teamweb.cecs.pdx.edu/~ece2xx/ECE221/Lectures/AnalysisMethodsx4.pdf · Nodal Analysis: Step 1 – Identify Essential Nodes 10 V 2 mA 1k Ω 2k

Exa

mple

3:

NodalA

nalysis

144 V

- +v2

- +v1

3 A

10

Ω

80

Ω

Solve

forv1

and

v2 .

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623

NodalAnalysis:

Review

ofStep

s

•Step

1:Id

entify

essential

nodes

•Step

2:Pick

areferen

ce

–M

ust

be

anessen

tialnode

–A

lways

label

with

the

ground

symbol

–B

estto

pick

essential

node

with

most

branch

es

–O

ftenat

the

bottom

ofth

ecircu

itdiagram

•Step

3:Lab

eloth

eressen

tialnodes

•Step

4:A

pply

KCL

toall

labelled

nodes

except

reference

node

•Step

5:Solve

linear

equation

s

–G

enerates

voltageat

eachnode

(relativeto

reference

node)

•Step

6:Solve

forvariab

lesof

interest

–U

sually

easyafter

Step

5

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621

Exa

mple

3:

Work

space

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NodalAnalysis:

Use

ofLaw

s

•A

llth

reelaw

sare

used

•K

CL

isap

plied

ateach

labelled

node

except

the

reference

node

•O

hm

’slaw

isused

todeterm

ine

the

curren

tin

branch

esth

atcon

tainresistors

•K

VL

isused

todeterm

ine

the

voltagedrop

acrossth

eresistors

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622

Page 7: Nodal Analysis 10 V - Computer Action Teamweb.cecs.pdx.edu/~ece2xx/ECE221/Lectures/AnalysisMethodsx4.pdf · Nodal Analysis: Step 1 – Identify Essential Nodes 10 V 2 mA 1k Ω 2k

Exa

mple

5:

Dep

enden

tVolta

ge

Source

50 V- +

10

Ω

10

Ω

30

Ω39

Ω78

Ω

v/5

v

Solve

forv

.

•W

hat

effect

does

the

10Ω

resistorhave

onth

ecircu

it?

•W

hat

isth

ecu

rrent

flow

ing

throu

ghth

edep

enden

tsou

rce?

•H

owcan

we

apply

KCL

atth

eessen

tialnodes

with

out

this

inform

ation?

•A

ns:

One

extravariab

le

•Im

plies

we

need

anextra

equation

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627

Exa

mple

4:

NodalA

nalysis

20 mA

- +v2

- +v1

- +v3

5 V2

2.7

2.7

3.3

4.7

10

Solve

forv1 ,

v2 ,

and

v3 .

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nd

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625

Exa

mple

5:

Contin

ued

50 V- +

10

Ω

10

Ω

30

Ω39

Ω78

Ω

v/5

v

Solve

forv

.

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nd

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teU

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ersity

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628

Exa

mple

4:

Work

space

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626

Page 8: Nodal Analysis 10 V - Computer Action Teamweb.cecs.pdx.edu/~ece2xx/ECE221/Lectures/AnalysisMethodsx4.pdf · Nodal Analysis: Step 1 – Identify Essential Nodes 10 V 2 mA 1k Ω 2k

Exa

mple

6:

Dep

enden

tSource

Contin

ued

50 V- +

10

Ω

10

Ω

30

Ω39

Ω78

Ω

v/5

v

Solve

forv

.U

sea

supern

ode.

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631

Exa

mple

5:

Work

space

Portla

nd

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629

Exa

mple

6:

Work

space

Portla

nd

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632

NodalAnalysis

and

Supern

odes

•Supern

odes

elimin

ateth

eneed

toin

troduce

anextra

variable

(unkn

own

curren

t)

•N

ecessaryw

hen

avoltage

source

isbetw

eentw

olab

elednodes

(excludin

greferen

cenode)

•Still

need

touse

voltagesou

rceto

generate

one

ofth

eeq

uation

s

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nd

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630

Page 9: Nodal Analysis 10 V - Computer Action Teamweb.cecs.pdx.edu/~ece2xx/ECE221/Lectures/AnalysisMethodsx4.pdf · Nodal Analysis: Step 1 – Identify Essential Nodes 10 V 2 mA 1k Ω 2k

Exa

mple

8:

NodalA

nalysis

11 mA

i1

20 Vi2

10 Vi3

250

Ω

500

Ω1kΩ

25

Solve

fori1 ,

i2 ,an

di3 .

Portla

nd

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ersity

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635

Exa

mple

7:

Dep

enden

tVolta

ge

Source

20 V

+-

20

Ω40

Ω80

Ω3.1

25v

v

35iφ

Fin

dth

epow

erdevelop

edby

the

20V

source.

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633

Exa

mple

8:

Work

space

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636

Exa

mple

7:

Work

space

Portla

nd

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634

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Mesh

Analysis:

Intro

ductio

n

•Recall:

There

isan

easierway

tosolve

forcu

rrents

and

voltagesth

anap

plyin

gK

VL

and

KCL

directly

•N

odal

analysis

ison

eof

two

keym

ethods

•M

eshan

alysisis

the

other

–A

pplies

KV

Lto

solvefor

curren

ts

–M

oreab

stract

–W

orkw

ithim

aginary

curren

ts

–O

nly

applies

toplan

arcircu

its

Portla

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639

Exa

mple

9:

NodalA

nalysis

1 A

3i

i

- +v

Solve

forv.

Portla

nd

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teU

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ersity

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637

Mesh

Analysis:

Step

1–

Label

Mesh

es

40 V64 V

iaic

ib

1.5

Ω2

Ω

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Page 11: Nodal Analysis 10 V - Computer Action Teamweb.cecs.pdx.edu/~ece2xx/ECE221/Lectures/AnalysisMethodsx4.pdf · Nodal Analysis: Step 1 – Identify Essential Nodes 10 V 2 mA 1k Ω 2k

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Mesh

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3–

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ns

[50

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= [4064 ]

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9.8A

i2=

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Page 12: Nodal Analysis 10 V - Computer Action Teamweb.cecs.pdx.edu/~ece2xx/ECE221/Lectures/AnalysisMethodsx4.pdf · Nodal Analysis: Step 1 – Identify Essential Nodes 10 V 2 mA 1k Ω 2k

Exa

mple

11:

Mesh

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3 A

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mple

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Mesh

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12 V

110 V70V

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Exa

mple

11:

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18 V15 V

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Page 13: Nodal Analysis 10 V - Computer Action Teamweb.cecs.pdx.edu/~ece2xx/ECE221/Lectures/AnalysisMethodsx4.pdf · Nodal Analysis: Step 1 – Identify Essential Nodes 10 V 2 mA 1k Ω 2k

Exa

mple

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Exa

mple

13:

Mesh

Analysis

200 V

4.3 id

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50

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100

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Page 14: Nodal Analysis 10 V - Computer Action Teamweb.cecs.pdx.edu/~ece2xx/ECE221/Lectures/AnalysisMethodsx4.pdf · Nodal Analysis: Step 1 – Identify Essential Nodes 10 V 2 mA 1k Ω 2k

Exa

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14:

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Nodalversu

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odal

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Exa

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1.5 mA

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