UFRGS – ENG04030 ANÁLISE DE CIRCUITOS I – SHaffner Versão: 17/11/2010 Página 1 de 2

Circuitos de duas portas – quadripolos Imitância Híbrido Transmissão

Impedância (z) Normal (h) Normal (a)

1 11 12 1

2 21 22 2

v z z i

v z z i

= ⋅

1 11 12 1

2 21 22 2

v h h i

i h h v

= ⋅

1 11 12 2

1 21 22 2

v a a v

i a a i

= ⋅ −

Admitância (y) Invertido (g) Invertido (b) 1v

+

−2v

+

−

1i

1i

2i

2i

1 11 12 1

2 21 22 2

i y y v

i y y v

= ⋅

1 11 12 1

2 21 22 2

i g g v

v g g i

= ⋅

2 11 12 1

2 21 22 1

v b b v

i b b i

= ⋅ −

Conversão de parâmetros

22 11 2211

21 21 22 11

12 12 1212

21 21 22 11

21 21 2121

21 21 22 11

11 22 1122

21 21 22 11

1

1

1

1

y a b hz

y a b h g

y h gaz

y a b h g

y h gbz

y a b h g

y a b gz

y a b h g

∆= = = = =∆

− −∆= = = = =∆

− −∆= = = = =∆

∆= = = = =∆

22 22 1111

12 12 11 22

12 12 1212

12 12 11 22

21 21 2121

12 12 11 22

11 11 2222

12 12 11 22

1

1

1

1

z a b gy

z a b h g

z h gay

z a b h g

z h gby

z a b h g

z a b hy

z a b h g

∆= = = = =∆

− −−∆ −= = = = =∆

− −− −∆= = = = =∆

∆= = = = =∆

11 22 2211

21 21 21 21

12 11 2212

21 21 21 21

21 22 1121

21 21 21 21

22 11 1122

21 21 21 21

1

1

1

1

z y b ha

z y b h g

b h gza

z y b h g

b h gya

z y b h g

z y b ga

z y b h g

− −∆= = = = =∆

−∆ −= = = = =∆

−−∆= = = = =∆

− − ∆= = = = =∆

22 11 2211

12 12 12 12

12 11 2212

12 12 12 12

21 22 1121

12 12 12 12

11 22 1122

12 12 12 12

1

1

1

1

z y a gb

z y a h g

a h gzb

z y a h g

a h gyb

z y a h g

z y a hb

z y a h g

− −∆= = = = =∆

−∆ −= = = = =∆

−−∆= = = = =∆

∆ −= = = = =∆

12 12 2211

22 11 22 11

12 12 1212

22 11 22 11

21 21 2121

22 11 22 11

21 21 1122

22 11 22 11

1

1

1

1

a b gzh

z y a b g

z y gah

z y a b g

z y gbh

z y a b g

a b gyh

z y a b g

∆= = = = =∆

− −∆= = = = =∆

− −− −∆= = = = =∆

∆= = = = =∆

21 21 2211

11 22 11 22

12 12 1212

11 22 11 22

21 21 2121

11 22 11 22

12 12 1122

11 22 11 22

1

1

1

1

a b hyg

z y a b h

z y hag

z y a b h

z y hbg

z y a b h

a b hzg

z y a b h

∆= = = = =∆

− −−∆ −= = = = =∆

− −∆= = = = =∆

∆= = = = =∆

Propriedades Recíproco se

12 21 12 21

12 21 12 21

1

1

z z h h a

y y g g b

= = − ∆ == = − ∆ =

Simétrico se recíproco e

11 22 11 22

11 22 11 22

1

1

z z a a h

y y b b g

= = ∆ == = ∆ =

11 22 12 21 11 22 12 21

11 22 12 21 11 22 12 21

11 22 12 21 11 22 12 21

z z z z z y y y y y

a a a a a b b b b b

h h h h h g g g g g

∆ = − ∆ = −

∆ = − ∆ = −

∆ = − ∆ = −

Quadripolos A e B Associação de quadripolos

Série

A

B

1 1 1a bv v v

+= +

−2 2 2a bv v v

+= +

−

1 1 1a bi i i= = 2 2 2a bi i i= =

[ ] [ ] [ ]a bz z z= +1 1 1a bi i i= = 2 2 2a bi i i= =

Paralela

A

B

1 1 1a bv v v

+= =

−2 2 2a bv v v

+= =

−

1 1 1a bi i i= + 2 2 2a bi i i= +

[ ] [ ] [ ]a by y y= +1 1 1a bi i i= + 2 2 2a bi i i= +

1av

+

−2av

+

−

1ai

1ai

2ai

2ai

[ ]

[ ]

[ ]

11 12

21 22

11 12

21 22

11 12

21 22

a aa

a a

a aa

a a

a aa

a a

z zz

z z

y yy

y y

a aa

a a

=

=

=

1bv

+

−2bv

+

−

1bi

1bi

2bi

2bi

[ ]

[ ]

[ ]

11 12

21 22

11 12

21 22

11 12

21 22

b bb

b b

b bb

b b

b bb

b b

z zz

z z

y yy

y y

a aa

a a

=

=

=

Cascata

A B

1 1av v

+=−

2 2bv v

+=−

1 1ai i= 2 2bi i=

[ ] [ ][ ]a ba a a=1 1ai i= 2 2bi i=

UFRGS – ENG04030 ANÁLISE DE CIRCUITOS I – SHaffner Versão: 17/11/2010 Página 2 de 2

Circuitos de primeira ordem: análise no domínio tem po Associação de capacitores e indutores

Série

( )( ) ( ) ( ) ( )

11 1 1eq 1 2

eq 1 0 2 0 00N

N

C C C C

v v t v t v t

−− − −= + + += + + +

…

…

( ) ( ) ( ) ( )eq 1 2

eq 1 0 2 0 00N

N

L L L L

i i t i t i t

= + + += = = =

…

…

Paralelo

( ) ( ) ( ) ( )

eq 1 2

eq 0 1 0 2 0 0

N

N

C C C C

v t v t v t v t

= + + += = = =

…

…

( )( ) ( ) ( ) ( )

11 1 1eq 1 2

eq 1 0 2 0 00N

N

L L L L

i i t i t i t

−− − −= + + += + + +

…

…

Circuitos divisores de tensão e corrente com capaci tores e indutores

Tensão ( )( ) ( ) ( ) ( ){ } ( )

11 1 0 2 0 0

1 1 01 1 11 2

s N

N

C v t v t v t v tv t v t

C C C

−

− − −

− + + + = ++ + +

…

… ( ) ( )1

11 2

s

N

L v tv t

L L L=

+ + +…

Corrente ( ) ( )11

1 2

s

N

C i ti t

C C C=

+ + +… ( )

( ) ( ) ( ) ( ){ } ( )1

1 1 0 2 0 0

1 1 01 1 11 2

s N

N

L i t i t i t i ti t i t

L L L

−

− − −

− + + + = ++ + +

…

…

Resposta completa de circuitos RC e RL

( )Si t R C ( )v t

+

−

( ) ( )0 0qualquerSi t v t V=

( ) ( ) ( )1

S

dv tC v t i t

dt R+ =

( ) ( ) ( )0

0

1 1 1

0

tt x t tSRC RC RC

t

i xv t e e dx V e

C

− − −= +∫

+_( )Sv t

( )i t R

L

( ) ( )0 0qualquerSv t i t I=

( ) ( ) ( )S

di tRi t L v t

dt+ =

( ) ( ) ( )0

00

R R Rtt x t t

SL L L

t

v xi t e e dx I e

L

− − −= +∫

Integrais indefinidas 1

, 11

nn x

x dx n nn

+= ∈ ≠ −

+∫ ℤ u dv u v v du= −∫ ∫

1lndx x

x=∫ ( ) ( ) ( ) ( ) ( ) ( )1 2 3 1

nn n n n nf g dx f g f g f g g f dx− − −′ ′′= − + − −∫ ∫…

( ) ( )1cos senax dx ax

a=∫ ( ) ( )1

sen cosax dx axa

−=∫

( ) ( )2 sen 2cos

2 4

axxax dx

a= +∫ ( ) ( )2 sen 2

sen2 4

axxax dx

a= −∫

axax e

e dxa

=∫ ln

ln 0, 1ln ln

x a xx x a e a

a dx e dx a aa a

= = = > ≠∫ ∫

1axax e

xe dx xa a = − ∫ ( ) ( )2 3

ln1 1! 2 2! 3 3!

ax ax axe axdx x

x= + + + +

× × ×∫ …

2 22

2 2axax e x

x e dx xa a a

= − + ∫ ( ) 1 111

ax ax ax

n n n

e e a edx dx

nx n x x− −−= +

−−∫ ∫

1n ax

n ax n axx e nx e dx x e dx

a a−= −∫ ∫ ( ) ( ) ( )2 2

cos cos senax

ax ee bx dx a bx b bx

a b= + +∫

ln 1ln

ax axax e x e

e xdx dxa a x

= −∫ ∫ ( ) ( ) ( )2 2sen sen cos

axax e

e bx dx a bx b bxa b

= − +∫