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1

Basic Laws

Discussion D2.1

Chapter 2Sections 2-1 2-6, 2-10

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2

Basic Laws

Ohm's Law

Kirchhoff's Laws

Series esistors an! "o#ta$e Di%ision

&ara##e# esistors an! Current Di%ision

Source (chan$e

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)

Georg Simon Ohm*1+ 1/

httpwww-histor3.4cs.st-an!rews.ac.u5histor3&ictDisp#a3h4.ht4#

7er4an professor who pu8#ishes a 8oo5

in 12+ that inc#u!es what is now 5nown

as h4's #aw.

h4's Law 9he %o#ta$e across a resistor

is !irect#3 proportiona# to the currect

f#owin$ throu$h it.

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/

esistance

A

l: #en$th

esistance R l A=

= resisti%it3 in h4-4eters

7oo! con!uctors *#ow Copper, 7o#!

7oo! insu#ators *hi$h 7#ass, &aper

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h4's Law

v iR= viR

=

;nits of resistance, R, is h4s *

vRi

=

R: 0 short circuit R= open circuit

1v i R= 1* i i=

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6

;nit of 7 is sie4ens *S,

Con!uctance, 7

1G

R=

iv

G= i Gv=

iG

v=

1 S : 1

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+

&ower

< resistor a#wa3s !issipates ener$3= it transfor4s

e#ectrica# ener$3, an! !issipates it in the for4 of heat.

ate of ener$3 !issipation is the instantaneous power2

2 * * * * * 0v t

p t v t i t Ri tR

= = =

22 * * * * * 0

i tp t v t i t Gv t

G= = =

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Basic Laws

h4's Law

Kirchhoff's Laws

Series esistors an! "o#ta$e Di%ision

&ara##e# esistors an! Current Di%ision

Source (chan$e

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7usta% o8ert Kirchhoff *12/ 1+

httpwww-histor3.4cs.st-an!rews.ac.u5histor3&ictDisp#a3Kirchhoff.ht4#

Born in &russia *now ussia, Kirchhoff

!e%e#ope! his >#aws> whi#e a stu!ent in

1/. 9hese #aws a##owe! hi4 to

ca#cu#ate the %o#ta$es an! currents in4u#tip#e #oop circuits.

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10

CIRCUIT TOPOLOGY

Topology ?ow a circuit is #ai! out.

< branchrepresents a sin$#e circuit *networ5

e#e4ent= that is, an3 two ter4ina# e#e4ent.

< noeis the point of connection 8etween two or4ore 8ranches.

< loopis an3 c#ose! path in a circuit *networ5.

< #oop is sai! to 8e inepenen! if it contains a8ranch which is not in an3 other #oop.

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11

@un!a4enta# 9heore4 of Aetwor5 9opo#o$3

1b l n= +

@or a networ5 with b8ranches, nno!es

an! lin!epen!ent #oops

(a4p#e

b

=n=

l=

"

#

#

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1/

Kirchoff$s C%rren! Law*KCL

9he a#$e8raic su4 of the currents enterin$ a

no!e *or a c#ose! 8oun!ar3 is ero.

1

0N

n

n

i=

=

where N: the nu48er of 8ranches connecte! to

the no!e an! in : the nth current enterin$

*#ea%in$ the no!e.

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1

1

0N

n

n

i

=

=1i

.i

2i

)i

/i

Sign con&en!ion Currents enterin$ the no!e are positi%e,

currents #ea%in$ the no!e are ne$ati%e.

1 2 ) / 0i i i i i+ + =

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16

Kirchoff$s C%rren! Law*KCL

9he a#$e8raic su4 of the currents enterin$

*or #ea%in$ a no!e is ero.

1i

.i

2i

)i

/i1 2 ) / 0i i i i i+ + =

1 2 ) / 0i i i i i + + =

9he su4 of the currents enterin$ a no!e iseua# to the su4 of the currents #ea%in$ a no!e.

1 2 / ) i i i i i+ + = +

nterin$

Lea%in$

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1+

Kirchoff$s ol!age Law*K"L

9he a#$e8raic su4 of the %o#ta$es aroun!

an3 #oop is ero.

1

0M

m

m

v=

=

where M : the nu48er of %o#ta$es in the #oop

an! vm: the mth%o#ta$e in the #oop.

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1

Sign con&en!ion 9he si$n of each %o#ta$e is the po#arit3 of the

ter4ina# first encountere! in tra%e#in$ aroun! the #oop.

9he !irection of tra%e# is ar8itrar3.

C#oc5wise

Counter-c#oc5wise

0 1 2 0V V V + + =

2 1 0 0V V V + =

0 1 2V V V= +

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1

Basic Laws

h4's Law

Kirchhoff's Laws

Series Resis!ors an ol!age (i&ision

&ara##e# esistors an! Current Di%ision

Source (chan$e

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20

0 1 2 1 2V V V IR IR= + = +

( )1 2I R R= +

sIR=

1 2sR R R= +

Series Resis!ors

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21

V0

I

R1

R2

V1

V2

A

ol!age (i&ier

0 0

1 2s

V VI

R R R= =

+

( )02 2 2

1 2

VV IR RR R= = +

( )2

2 0

1 2

R

V VR R= +

( )1

1 0

1 2

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22

Basic Laws

h4's Law

Kirchhoff's Laws

Series esistors an! "o#ta$e Di%ision

Parallel Resis!ors an C%rren! (i&ision

Source (chan$e

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2)

VR

1

I

R2

I1 I2

1 2

1 2

V VI I I

R R= + = +

Parallel Resis!ors

1 2

1 1V

R R

= +

p

V

R=

1 2

1 1 1

pR R R= +

1 2

1 2

p

R RR

R R=

+

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2/

Current Di%ision

1 2

1 2

* * * p

R Rv t R i t i t

R R= =

+

21

1 1 2

* * * Rv ti t i t R R R

= =+

1

2

2 1 2

*

* *

Rv t

i t i t R R R= = +

C%rren! i&ies in in&erse propor!ion !o !he resis!ances

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2

Current Di%ision

Nresistors in para##e#

1 2

1 1 1 1

p nR R R R

= + + + * * p

v t R i t =

* * *

p

j

j j

Rv ti t i t

R R= =Current in jth8ranch is

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26

Basic Laws

h4's Law

Kirchhoff's Laws

Series esistors an! "o#ta$e Di%ision

&ara##e# esistors an! Current Di%ision

So%rce )*change

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2+

Source (chan$e

+e can always replace a &ol!age so%rce in series wi!h a

resis!or by a c%rren! so%rce in parallel wi!h !he same resis!or

an &ice,&ersa-

(oing !his. howe&er. ma/es i! impossible !o irec!ly fin !he

original so%rce c%rren!-

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2

Source (chan$e &roof

ol!age across an c%rren! !hro%gh any loa are !he same

( )L

L s

s L

Rv v

R R=

+

( )s

a

s L

vi

R R=

+

( )' s sa a

s L s

R vi i

R R R= =

+

( )' LL a L s

s L

Rv i R v

R R= =

+