# Fundamental Physics/Electronics/Series circuit

## Series circuit

Circuit that has components connected in series or adjacent to each other

## Series n resistors

${\displaystyle R_{t}=R-1+R_{2}+...+R_{n}}$

## Series n inductors

${\displaystyle L_{t}=L-1+L_{2}+...+L_{n}}$

## Series n capacitors

${\displaystyle {\frac {1}{C_{t}}}={\frac {1}{C_{1}}}+{\frac {1}{C_{2}}}+...+{\frac {1}{C_{n}}}}$

## Series RC

RC series
${\displaystyle V_{C}+V_{R}=0}$
${\displaystyle C{\frac {d}{dt}}v(t)+{\frac {v(t)}{R}}=0}$
${\displaystyle {\frac {d}{dt}}v(t)=-{\frac {1}{RC}}v(t)}$
${\displaystyle v(t)=Ae^{-{\frac {1}{RC}}t}}$

## Series RL

RL series
${\displaystyle V_{L}+V_{R}=0}$
${\displaystyle L{\frac {d}{dt}}i(t)+Ri(t)=0}$
${\displaystyle {\frac {d}{dt}}i(t)=-{\frac {R}{L}}i(t)}$
${\displaystyle i(t)=Ae^{-{\frac {R}{L}}t}}$

## Series LC

LC series
${\displaystyle V_{L}+V_{C}=0}$
${\displaystyle L{\frac {d}{dt}}i(t)+{\frac {1}{C}}\int i(t)dt=0}$
${\displaystyle {\frac {d^{2}}{dt^{2}}}i(t)=-{\frac {1}{LC}}i(t)}$
${\displaystyle i(t)=Ae^{\pm j{\sqrt {\frac {1}{LC}}}t}=Ae^{\pm j\omega t}=ASin\omega t}$
${\displaystyle \omega ={\sqrt {\frac {1}{LC}}}}$

## Series RLC

RLC series
${\displaystyle V_{L}+V_{C}+V_{R}=0}$
${\displaystyle L{\frac {d}{dt}}i(t)+{\frac {1}{C}}\int i(t)dt+Ri(t)=0}$
${\displaystyle {\frac {d^{2}}{dt^{2}}}i(t)=-{\frac {R}{L}}{\frac {d}{dt}}i(t)-{\frac {1}{LC}}i(t)=0}$
${\displaystyle {\frac {d^{2}}{dt^{2}}}i(t)=-2\alpha {\frac {d}{dt}}i(t)-\beta i(t)=0}$
${\displaystyle \alpha ={\frac {R}{2L}}}$
${\displaystyle \beta ={\frac {1}{LC}}}$

One real root . ${\displaystyle \alpha =\beta }$

${\displaystyle i(t)=Ae^{-\alpha t}=A(\alpha )}$

Two real roots . ${\displaystyle \alpha >\beta }$

${\displaystyle i(t)=Ae^{(\alpha \pm {\sqrt {\alpha -\beta }})t}}$

Two complex roots . ${\displaystyle \alpha <\beta }$

${\displaystyle i(t)=A(\alpha )e^{(\alpha \pm j{\sqrt {\beta -\alpha }})t}=A(\alpha )Sin\omega t}$
${\displaystyle \omega ={\sqrt {\beta -\alpha }}}$
${\displaystyle A(\alpha )=Ae^{\alpha t}}$