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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
•
•
–
–
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)
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
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
•
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
•
•
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
•
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
•
•
•
ycomplete = ynatural + y forced
ycomplete = y(t)
ynatural = y(0)et /
y forced =1e t / x(t)et /
0
t
•
•
•
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
•
•
•
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
•
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 )
•
•
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
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