Circuitos RLC

1 v + R L

t

ˆ 

vdt i(t0 ) + c

to

dv =0 dt

i(0+ ) = I 0 v(0+ ) = V 0

1dv 1 d2 v + v =0 C  2 + dt Rdt L

v = Aest

CAs 2 est +

1 R

Asest +

1 L

Asest = 0

Aest (Cs 2 + frac1Rs +

Cs 2 +

s1 =

s1 =

1

1 R

− 2RC  + 1

− 2RC 

s+

1 L

1

L

)

=0

   − 1

2

2RC 

   − − 1

2

2RC 

1 RC 

1 RC 

v1 = A1 es1t

v2 = A2 es2t

1dv1 1 Cd 2 v1 + + v1 = 0 dt2 Rdt L

1dv2 1 Cd 2 v2 + + v2 = 0 2 dt Rdt L

C 

1 d2 (v1 + v2 ) 1 d2 (v1 + v2 ) + + (v1 + v2 ) = 0 2 dt R dt L

v1 = A1 es1t + A2 es2t

√1

ω0 =

α=

LC 

1 2RC 

v1 = A1 es1t + A2 es2t

s1 =

s1 =

  − −   − − − α+

α2

ω02

α

α2

ω02

ω0 =

α=

α2 > ω 02 . α2 = ω02 . α2 < ω 02 .

√1

LC 

1 2RC 

α2 > ω 02 .

  − − −   −  −   −  α2

α

α2

ω02

<

ω02 < a

α+

α2

α = ω0 . LC  = 4 R2 C 2 . L = 4 R2 C.

v(t) = A1 e−αt + A2 te−αt

LC < 4R2 C 2 . L < 4R2 C.

v(t) = A1 es1t + A2 es2t

s1,2 =

  − ± − α

α2

ω02

ω02

<0

α2 > ω 02 .

s1,2 =

  − ± − α

 j =

ωd=

α2

√ −1

  −

s1,2 =

ω02

α2

−α ± jω

v (t) = e−αt A1 e−jω



ω02

 j

d

e±jωt = cos(ωt)

t

d

+ A2 te−jω

d

t



± sin(ωt)

v(t) = e−αt ((A1 + A2 ) cos(ωd t) + j (A1 + A2 ) sin(ωd t))

v(t) = e−αt (B1 cos(ωd t) + B2 sin(ωd t))

vn (0) = B 1

dv (0) = ωd B2 dt

− αB

1

dv (0) v(0) = dt RC 

ωd B2 = αB1

(0) i(0) − vRC  − C 

t

1 di L + Ri + dt C 

ˆ 

1 1 dv + v+ c dt R L

ˆ 

L

− i(0) C 

idt V c(t0 ) = 0

to

t

vdt i(t0 )+ = 0

to

1 d2 v di + + R i =0 dt2 dt C 

1dv 1 d2 i + v =0 C  2 + dt Rdt L

i(t) = A1 es1t + A2 tes2t

s1, 2 =

−

R 2L

   ± − R 2L

ω0 =

α=

2

1 L

√1

LC 

R 2L

i(t) = e−αt (A1 t + A2 )

i(t) = e−αt (B1 cos(ωd t) + B2 sin(ωd t))

ωd=

  − ω02

α2

vf (t) = V f 

vn (t) = Aes1t + Be s2t

v (t) = V f  + Aes1t + B es2t

dv = 0 + s1Aes1t + s2Bes2t dt