Electricity/Electric circuit/RLC circuit
RLC circuit
RLC series
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
=
−
2
α
d
d
t
i
−
β
i
{\displaystyle {\frac {d^{2}}{dt^{2}}}i=-2\alpha {\frac {d}{dt}}i-\beta i}
i
=
A
e
(
−
α
±
j
β
−
α
)
t
=
A
e
−
α
t
e
±
j
ω
t
=
A
(
α
)
S
i
n
ω
t
{\displaystyle i=Ae^{(-\alpha \pm j{\sqrt {\beta -\alpha }})t}=Ae^{-\alpha t}e^{\pm j\omega t}=A(\alpha )Sin\omega t}
ω
=
β
−
α
{\displaystyle \omega ={\sqrt {\beta -\alpha }}}
A
(
α
)
=
A
e
−
α
t
{\displaystyle A(\alpha )=Ae^{-\alpha t}}
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}
2 port LC series-R
2 port R-LC series
