Electricity/Electric circuit/LC circuit
LC circuit
Circuit at equilibrium
v
L
+
v
C
=
0
{\displaystyle v_{L}+v_{C}=0}
L
d
d
t
i
+
1
C
∫
i
d
t
=
0
{\displaystyle L{\frac {d}{dt}}i+{\frac {1}{C}}\int idt=0}
d
2
d
t
2
i
=
−
1
T
i
{\displaystyle {\frac {d^{2}}{dt^{2}}}i=-{\frac {1}{T}}i}
i
=
A
e
±
j
1
T
t
=
A
e
±
j
ω
t
=
A
S
i
n
ω
t
{\displaystyle i=Ae^{\pm j{\sqrt {\frac {1}{T}}}t}=Ae^{\pm j\omega t}=ASin\omega t}
ω
=
1
T
{\displaystyle \omega ={\sqrt {\frac {1}{T}}}}
T
=
L
C
{\displaystyle T=LC}
Circuit at resonance
Z
L
+
Z
C
=
0
{\displaystyle Z_{L}+Z_{C}=0}
j
ω
L
+
1
j
ω
C
=
0
{\displaystyle j\omega L+{\frac {1}{j\omega C}}=0}
ω
=
±
1
T
{\displaystyle \omega =\pm {\sqrt {\frac {1}{T}}}}
T
=
L
C
{\displaystyle T=LC}
V
L
+
V
C
=
0
{\displaystyle V_{L}+V_{C}=0}
v
(
θ
)
=
A
S
i
n
(
ω
+
2
π
)
−
A
S
i
n
(
ω
−
2
π
)
{\displaystyle v(\theta )=ASin(\omega +2\pi )-ASin(\omega -2\pi )}
