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RLC Series

An RLC series circuit consists of a resistor, inductor, and capacitor connected in series:

By Kirchhoff's voltage law the differential equation for the circuit is:

${\displaystyle L{\frac {dI}{dt}}+IR+{\frac {1}{C}}\int Idt=V(t)}$

or

${\displaystyle L{\frac {d^{2}I}{dt^{2}}}+R{\frac {dI}{dt}}+{\frac {I}{C}}={\frac {dV}{dt}}}$

Leading to:

${\displaystyle s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}=0}$

${\displaystyle s=-\alpha }$ ± ${\displaystyle {\sqrt {\alpha ^{2}-\beta ^{2}}}}$

with

${\displaystyle \alpha ={\frac {R}{2L}}}$ and ${\displaystyle \beta ={\sqrt {\frac {1}{LC}}}}$

There are three cases to consider, each giving different circuit behavior, ${\displaystyle \alpha ^{2}=\beta ^{2},\alpha ^{2}>\beta ^{2},or\alpha ^{2}<\beta ^{2}}$ .

${\displaystyle \alpha ^{2}=\beta ^{2}}$ .

${\displaystyle {\frac {R}{2L}}}$ = ${\displaystyle {\sqrt {\frac {1}{LC}}}}$

${\displaystyle R=2{\sqrt {\frac {L}{C}}}}$

Equation above has only one real root

s = -α = ${\displaystyle {\frac {R}{2L}}}$

${\displaystyle I=Ae^{(-{\frac {R}{2L}})t}}$

${\displaystyle \alpha ^{2}>\beta ^{2}}$ ,

${\displaystyle {\frac {R}{2L}}}$ > ${\displaystyle {\sqrt {\frac {1}{LC}}}}$

${\displaystyle R>2{\sqrt {\frac {L}{C}}}}$

Equation above has only two real roots

${\displaystyle s=-\alpha }$ ± ${\displaystyle {\sqrt {\alpha ^{2}-\beta ^{2}}}}$

${\displaystyle I=e^{(}-\alpha +{\sqrt {\alpha ^{2}-\beta ^{2}}})t+e^{-}(\alpha +{\sqrt {\alpha ^{2}-\beta ^{2}}})t}$

${\displaystyle I=e^{(}-\alpha )e({\sqrt {\alpha ^{2}-\beta ^{2}}})t-e^{-}({\sqrt {\alpha ^{2}-\beta ^{2}}})t}$

${\displaystyle \alpha ^{2}<\beta ^{2}}$ .

${\displaystyle R<2{\sqrt {\frac {L}{C}}}}$

Equation above has only two complex roots

${\displaystyle s=-\alpha }$ + j${\displaystyle {\sqrt {\beta ^{2}-\alpha ^{2}}}}$

${\displaystyle s=-\alpha }$ - j${\displaystyle {\sqrt {\beta ^{2}-\alpha ^{2}}}}$

${\displaystyle I=e^{j}(-\alpha +{\sqrt {\beta ^{2}-\alpha ^{2}}})t+e^{j}(-\alpha +{\sqrt {\beta ^{2}-\alpha ^{2}}})t}$

Circuit Analysis

Further Reading

1. RCL circuit analysed in the time domain
2. RCL circuit analysed in the frequency domain