Practical Electronics/Parallel RLC
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Practical Electronics
Summary
Network
Symbol
Series RLC
Parallel RLC
Network
Impedance
Z
Z
=
(
j
ω
)
2
+
(
j
ω
)
R
L
+
1
L
C
{\displaystyle Z=(j\omega )^{2}+(j\omega ){\frac {R}{L}}+{\frac {1}{LC}}}
Z
=
(
j
ω
)
2
+
(
j
ω
)
1
R
C
+
1
L
C
{\displaystyle Z=(j\omega )^{2}+(j\omega ){\frac {1}{RC}}+{\frac {1}{LC}}}
Roots
λ
λ =
−
ζ
±
ζ
2
−
ω
o
2
{\displaystyle -\zeta \pm {\sqrt {\zeta ^{2}-\omega _{o}^{2}}}}
λ =
−
ζ
±
ζ
2
−
ω
o
2
{\displaystyle -\zeta \pm {\sqrt {\zeta ^{2}-\omega _{o}^{2}}}}
I(t)
Ae
λ
1
t
+ Be
λ
2
t
Ae
λ
1
t
+ Be
λ
2
t
Ae
λ
1
t
+ Be
λ
2
t
Damping Factor
ζ
{\displaystyle \zeta }
ζ
=
R
2
L
{\displaystyle \zeta ={R \over 2L}}
ζ
=
1
2
R
C
{\displaystyle \zeta ={1 \over 2RC}}
Resonant Frequency
ω
o
{\displaystyle \omega _{o}}
ω
o
=
1
L
C
{\displaystyle \omega _{o}={1 \over {\sqrt {LC}}}}
ω
o
=
1
L
C
{\displaystyle \omega _{o}={1 \over {\sqrt {LC}}}}
Band Width
Δ
ω
=
2
ζ
{\displaystyle \Delta \omega =2\zeta }
R
L
{\displaystyle {R \over L}}
1
C
R
{\displaystyle {1 \over CR}}
Quality factor
Q
=
ω
o
Δ
ω
=
ω
o
2
ζ
{\displaystyle Q={\omega _{o} \over \Delta \omega }={\omega _{o} \over 2\zeta }}
Q
=
L
R
L
C
=
1
R
L
C
{\displaystyle Q={L \over R{\sqrt {LC}}}={1 \over R}{\sqrt {L \over C}}}
Q
=
C
R
L
C
=
R
C
L
{\displaystyle Q={CR \over {\sqrt {LC}}}={R}{\sqrt {C \over L}}}
