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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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![Chapter 1, Solution 1 C] = 103.84 mC C] = 198.65 mC C] = 3 ... · Chapter 1, Solution 3 (a) ³ q(t) i(t)dt q(0) (3t 1) C (b) ³ q(t) (2t s) dt q(v). 2 (t 5t) mC (c) q(t) 20 cos 10t](https://img.pdfslide.us/doc/110x75/5fe345d4aab5001e855e90e5/chapter-1-solution-1-c-10384-mc-c-19865-mc-c-3-chapter-1-solution.jpg)
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