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RCdv(t)

dt+ v(t) = vs (t)

L

R

di(t)

dt+ i (t) = is (t)

RCdv(t)

dt+ v(t) = vs (t)

L

R

di(t)

dt+ i (t) = is (t)

dy(t)

dt+ y(t) = x(t)

y(t) = unknown variable =v(t) for the capacitive case

i (t) for the inductive case

x(t) = forcing function =vS (t) for the capacitive case

iS (t) for the inductive case

= time constant =RC for the capacitive case

L /R for the inductive case

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–

–

dy(t)

dt+ y(t) = x(t)

dy* (t)

dt+ y* (t) = 0

dy** (t)

dt+ y** (t) = x(t)

y(t) = y* (t) + y** (t)

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dy* (t)

dt+ y* (t) = 0

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y* (t) yHA (t) yN (t)

dy(t)

dt+ y(t) = x(t)

dy(t)

dt+ y(t) = 0

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y** (t) yP (t) yF (t)

y(t) = yHA (t) + yP (t) yN (t) + yF (t)

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is=I

v = LdiLdt

= Ld (constant)

dt= 0

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is=I

Ldi(t)

dt+ Ri(t) = 0 for t 0

Ldi(t)

dt+ Ri(t) = 0 L

di

dtdt + R i dt = 0

Ldi = R i dt di

i=

R

Ldt

di

ii ( 0)

i ( t )

=R

Ldt

0

t

lni (t)

i (0)=

R

Lt i (t) = i(0)e ( R /L ) t with t 0

i(0) = i (0 ) = I 0

i(t) = I 0 e ( R /L ) t with t 0

i(t) = I 0 e ( R /L ) t with t 0

v(t) = R i (t) = RI 0 e ( R /L ) t with t 0

v(0 ) = LdiLdt t=0

= LdI 0

dt= 0 with t < 0

v(0+) = Ldi

dt t=0

= Ld

dtI 0 e ( R /L ) t( )

t=0= RI 0 with t 0

v(t) = R i (t) = RI 0 e ( R /L ) t with t 0

v(0+) = R iR (0+) = RI 0

Is=I

v(0 ) = R iR (0 ) = R 0 = 0

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p(t) = i (t)v(t) = I 0 e ( R /L ) t I 0R e ( R /L ) t= I 0

2R e 2( R /L ) t for t 0

w(t) = p(t)dt0

t

= I 0

2R e ( 2R /L ) t dt0

t

= I 0

2Re ( 2R /L ) t

2R /L

0

t

=I 0

2R

2R /Le ( 2R /L ) t[ ]0

t=

=1

2LI 0

2 1 e ( 2R /L ) t( ) for t 0

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i I 0 =I 0 t

di

dt t=0

=d

dtI oe

( R /L ) t( )t=0= I 0

R

L=

I 0

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i(t) = I 0et / with t 0

=L

R

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i = CdvCdt

= Cd (constant)

dt= 0

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v(0 ) = V0

Cdv

dt+v

R= 0 for t 0

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v(0 ) = V0 = v(0+)

Cdv

dt+v

R= 0 for t 0

v(t) = v(0+)etRC = V0e

tRC for t 0

v(t) = v(0+)etRC = V0e

tRC for t 0

i(t) =v(t)

R=

V0

Re

tRC for t 0

i(0 ) = 0; i (0+) =V0

R

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p(t) = i (t)v(t) =V0

2

Re

2tRC for t 0

w(t) = p(t)dt0

t

=1

2CV0

2 1 e2tRC( ) for t 0

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= RC

dy(t)

dt+ y(t) = 0

dy(t)

dt+ y(t) = 0

dy(t)

dt= y(t)

y(t) = A est

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dy(t)

dt+ y(t) = 0 ( s+1)Aest = 0 s+1 = 0

Aest 0

y(t) = Aest

s+1= 0 s =1

y(t) = Aest t 0 y(0) = A

• y(t) = y(0)e t /

y( ) = y(0)e 1=1

ey(0) 0.37y(0)

dy(t)

dt t=o

=1y(0)e t /

t=0=

y(0)

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dy(t)

dt+ y(t) = x(t)

1et /

et / dy(t)

dt+et /

y(t) =et /

x(t) d

dtet / y(t)( ) =

et /

x(t)

d

dtet / y(t)( )

0

t

dt =et /

x(t)0

t

dt

d

dtet / y(t)( ) =

et /x(t) d

dtet / y(t)( )

0

t

dt =et /

x(t)0

t

dt

d et / y(t)( )0

t

=1

x(t)et /0

t

dt et / y(t) e0y(0) =1

x(t)et /0

t

dt

y(t)et / y(0) =1

x(t)et /0

t

dt y(t) = y(0)e t /+

1e t / x(t)et /

0

t

dt

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ycomplete = ynatural + y forced

ycomplete = y(t)

ynatural = y(0)et /

y forced =1e t / x(t)et /

0

t

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x(t) = XS (with Xs being a constant)

x(t) =0 for t < 0

XS for t 0

= XS u(t)

x(t)

Xs

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dy(t)

dt+ y(t) = XS for t 0

y forced =1e t / XSe

t /

0

t

dt =XS e t / et /

0

t

dt =XS e t / et /[ ]0

t for t 0

y forced = XS 1 e t /( ) for t 0

ynatural = y(0)e t / for t 0

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y(t) = ynatural + y forced = y(0)e t /+ XS (1 e t / ) =

= y(0)e t /+ XS XSe

t / for t 0

y( ) = y(0)e + XS XSe = XS

y(t) = y(0)e t / XSet /+ XS = [ y(0) y( )]e t /

+ y( ) for t 0

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y(t) = [ y(0) y( )]e t /+ y( ) for t 0 (with y( ) = XS )

t 0

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y(t) = [ y(0) y( )]e t /+ y( ) for t 0

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y(t) = [ y(0) y( )]e t /+ y( ) for t 0 (with y( ) = XS )

ytransient (t) = [ y(0) y( )]e t / for t 0 (with y( ) = XS )

y(t ) = [ y(0) y( )]e /+ y( ) for t 0 (with y( ) = XS )

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y(t) = ytransient + ysteady state for t 0

ytransient = [ y(0) y( )]e t /

ysteady state = y( )

for t 0

ytransient = y(0)e t / y( )e t / for t 0

ysteady state = y( ) = XS

VS = Ri(t) + Ldi (t)

dt for t 0 VSu(t) = Ri(t) + L

di (t)

dt

i(t) =VS

R+ I 0

VS

R

e t / for t 0

i(t) =VS

R+ I 0

VS

R

e t / for t 0

i(t) =VS

R+ I 0

VS

R

e t / for t 0

v(t) = Ldi (t)

dt= (VS I 0R)e t / for t 0

CdvCdt

+vCR= I S for t 0 C

dvCdt

+vCR= I Su(t)

vC (t) = I SR + (V0 I SR)e t / for t 0

iC (t) = CdvCdt

= (I SV0

R)e t / for t 0

vC (t) = I SR + (V0 I SR)e t / for t 0

iC (t) = CdvCdt

= (I SV0

R)e t / for t 0

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iaux =vaux10K

7i + i

i =vaux20K

iaux =vaux10K

6vaux20K

= vaux (0.1 10 3 0.3 10 3)

vauxiaux

= 5K

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iC = i R i

vO (t) = Ri(t)

i(t) = Cdv I (t)

dt

vO (t) = RCdv I (t)

dt

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iC = i R i

i(t) =v I (t)

R

i(t) = CdvO (t)

dt

vO (t) =1

RCvI (t)dt + vO (0)

0

t

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vO (t) =VI

RCt + vO (0)

dvO (t)

dt=

VI

RC

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vO (t) =1

RCvI (t)dt + vO (0)

0

t

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