[<< 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}}}