Mạch điện RLC
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Ở trạng thái cân bằng
[
edit
]
Với R khác không
v
L
+
v
c
+
v
R
=
0
{\displaystyle v_{L}+v_{c}+v_{R}=0}
L
d
d
t
i
+
1
C
∫
i
d
t
+
i
R
=
0
{\displaystyle L{\frac {d}{dt}}i+{\frac {1}{C}}\int idt+iR=0}
d
2
d
t
2
i
+
R
L
d
d
t
i
+
1
L
C
i
=
0
{\displaystyle {\frac {d^{2}}{dt^{2}}}i+{\frac {R}{L}}{\frac {d}{dt}}i+{\frac {1}{LC}}i=0}
d
2
d
t
2
i
=
−
2
α
d
d
t
i
−
β
i
{\displaystyle {\frac {d^{2}}{dt^{2}}}i=-2\alpha {\frac {d}{dt}}i-\beta i}
Phương trình trên có nghiệm
Một nghiệm số thực khi
α
=
β
{\displaystyle \alpha =\beta }
.
i
=
A
e
−
α
t
=
A
(
α
)
{\displaystyle i=Ae^{-\alpha t}=A(\alpha )}
Hai nghiệm số thực khi
α
>
β
{\displaystyle \alpha >\beta }
.
i
=
A
e
(
−
α
±
λ
)
t
=
A
e
−
α
t
e
λ
t
+
A
e
−
α
t
e
−
λ
t
=
A
(
α
)
e
λ
t
+
A
(
α
)
e
−
λ
t
{\displaystyle i=Ae^{(-\alpha \pm \lambda )t}=Ae^{-\alpha t}e^{\lambda t}+Ae^{-\alpha t}e^{-\lambda t}=A(\alpha )e^{\lambda t}+A(\alpha )e^{-\lambda t}}
Hai nghiệm số phức khi
α
<
β
{\displaystyle \alpha <\beta }
.
i
=
A
e
(
−
α
±
j
ω
)
t
=
A
e
−
α
t
e
±
j
ω
t
=
A
(
α
)
sin
ω
t
{\displaystyle i=Ae^{(-\alpha \pm j\omega )t}=Ae^{-\alpha t}e^{\pm j\omega t}=A(\alpha )\sin \omega t}
Với
β
=
1
T
=
1
L
C
{\displaystyle \beta ={\frac {1}{T}}={\frac {1}{LC}}}
α
=
β
γ
=
R
2
L
{\displaystyle \alpha =\beta \gamma ={\frac {R}{2L}}}
T
=
L
C
{\displaystyle T=LC}
γ
=
R
C
{\displaystyle \gamma =RC}
A
(
α
)
=
A
e
−
α
t
{\displaystyle A(\alpha )=Ae^{-\alpha t}}
ω
=
β
−
α
{\displaystyle \omega ={\sqrt {\beta -\alpha }}}
λ
=
α
−
β
{\displaystyle \lambda ={\sqrt {\alpha -\beta }}}
Với R =0
i
″
(
t
)
=
−
β
i
(
t
)
{\displaystyle i^{''}(t)=-\beta i(t)}
i
(
t
)
=
A
e
±
j
ω
t
=
A
s
i
n
ω
t
{\displaystyle i(t)=Ae^{\pm j\omega t}=Asin\omega t}
ω
=
β
{\displaystyle \omega ={\sqrt {\beta }}}
Với R,C =0
∇
E
(
t
)
=
−
ω
E
(
t
)
{\displaystyle \nabla E(t)=-\omega E(t)}
∇
B
(
t
)
=
−
ω
B
(
t
)
{\displaystyle \nabla B(t)=-\omega B(t)}
E
(
t
)
=
A
sin
ω
t
{\displaystyle E(t)=A\sin \omega t}
B
(
t
)
=
A
sin
ω
t
{\displaystyle B(t)=A\sin \omega t}
ω
=
λ
f
=
1
T
=
C
{\displaystyle \omega =\lambda f={\sqrt {\frac {1}{T}}}=C}
T
=
μ
ϵ
{\displaystyle T=\mu \epsilon }
Với L =0
v
C
+
v
R
=
0
{\displaystyle v_{C}+v_{R}=0}
C
d
d
t
v
(
t
)
+
v
(
t
)
R
=
0
{\displaystyle C{\frac {d}{dt}}v(t)+{\frac {v(t)}{R}}=0}
d
d
t
v
(
t
)
=
−
1
T
v
(
t
)
{\displaystyle {\frac {d}{dt}}v(t)=-{\frac {1}{T}}v(t)}
v
(
t
)
=
A
e
−
t
T
{\displaystyle v(t)=Ae^{-{\frac {t}{T}}}}
T
=
R
C
{\displaystyle T=RC}
Với C =0
v
L
+
v
R
=
0
{\displaystyle v_{L}+v_{R}=0}
L
d
d
t
i
(
t
)
+
i
(
t
)
R
=
0
{\displaystyle L{\frac {d}{dt}}i(t)+i(t)R=0}
d
d
t
i
(
t
)
=
−
1
T
i
(
t
)
{\displaystyle {\frac {d}{dt}}i(t)=-{\frac {1}{T}}i(t)}
i
(
t
)
=
A
e
−
t
T
{\displaystyle i(t)=Ae^{-{\frac {t}{T}}}}
T
=
L
R
{\displaystyle T={\frac {L}{R}}}
Ở trạng thái đồng bộ
[
edit
]
Với R khác không
Z
C
=
−
Z
L
{\displaystyle Z_{C}=-Z_{L}}
.
Z
t
=
R
{\displaystyle Z_{t}=R}
,
i
(
ω
=
0
)
=
0
{\displaystyle i(\omega =0)=0}
i
(
ω
=
ω
o
)
=
v
R
{\displaystyle i(\omega =\omega _{o})={\frac {v}{R}}}
i
(
ω
=
0
)
=
0
{\displaystyle i(\omega =0)=0}
ω
o
=
±
j
1
T
o
{\displaystyle \omega _{o}=\pm j{\sqrt {\frac {1}{T_{o}}}}}
T
o
=
L
C
{\displaystyle T_{o}=LC}
Với R =0
Z
C
=
−
Z
L
{\displaystyle Z_{C}=-Z_{L}}
.
V
C
=
−
V
L
{\displaystyle V_{C}=-V_{L}}
,
i
(
t
)
=
A
sin
(
ω
o
t
+
2
π
)
−
A
sin
(
ω
o
t
−
2
π
)
{\displaystyle i(t)=A\sin(\omega _{o}t+2\pi )-A\sin(\omega _{o}t-2\pi )}
ω
o
=
±
j
1
T
o
{\displaystyle \omega _{o}=\pm j{\sqrt {\frac {1}{T_{o}}}}}
T
o
=
L
C
{\displaystyle T_{o}=LC}
Với R,C =0
∇
E
(
t
)
=
−
ω
o
E
(
t
)
{\displaystyle \nabla E(t)=-\omega _{o}E(t)}
∇
B
(
t
)
=
−
ω
o
B
(
t
)
{\displaystyle \nabla B(t)=-\omega _{o}B(t)}
E
(
t
)
=
A
sin
ω
o
t
{\displaystyle E(t)=A\sin \omega _{o}t}
B
(
t
)
=
A
sin
ω
o
t
{\displaystyle B(t)=A\sin \omega _{o}t}
ω
o
=
λ
o
f
o
=
1
T
o
=
C
{\displaystyle \omega _{o}=\lambda _{o}f_{o}={\sqrt {\frac {1}{T_{o}}}}=C}
T
o
=
μ
o
ϵ
o
{\displaystyle T_{o}=\mu _{o}\epsilon _{o}}
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