[<< wikibooks] Electronics Handbook/Circuits/Summary of RLC Series

== Normal ==
RL and RC both pocess the same character
Differential equation of first ordered in the form
d
d
t
f
(
t
)
+
1
T
=
0
{\displaystyle {\frac {d}{dt}}f(t)+{\frac {1}{T}}=0}
Which can only has one real root in the form
f
(
t
)
=
A
e
(
−
t
T
)
{\displaystyle f(t)=Ae^{(}-{\frac {t}{T}})}
A
=
e
c
{\displaystyle A=e^{c}}
LC and RLC both pocess the same character
Differential equation of second ordered in the form
d
2
d
t
2
f
(
t
)
+
α
d
d
t
f
(
t
)
+
β
=
0
{\displaystyle {\frac {d^{2}}{dt^{2}}}f(t)+\alpha {\frac {d}{dt}}f(t)+\beta =0}
Which can only has one real root in the form
f
(
t
)
=
A
(
e
ω
t
+
e
−
ω
t
)
{\displaystyle f(t)=A(e^{\omega }t+e^{-}\omega t)}
A
=
e
α
t
{\displaystyle A=e^{\alpha }t}
ω
=
α
2
−
β
2
{\displaystyle \omega ={\sqrt {\alpha ^{2}-\beta ^{2}}}}
== Resonance ==
At resonance
LC will generate Oscillation of Standing Sin Wave of frequency
ω
=
1
L
C
{\displaystyle \omega ={\sqrt {\frac {1}{LC}}}}
RLC will act as Resonant Tuned Selected Bandpass Filter
ω
=
1
L
C
a
t
I
=
V
R
{\displaystyle \omega ={\sqrt {\frac {1}{LC}}}atI={\frac {V}{R}}}
δ
ω
=
ω
2
−
ω
1
a
t
I
=
V
2
R
{\displaystyle \delta \omega =\omega _{2}-\omega _{1}atI={\frac {V}{2R}}}