# Fundamental Physics/Electronics/Series circuit

## Series circuit

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

## Series n resistors $R_{t}=R-1+R_{2}+...+R_{n}$ ## Series n inductors $L_{t}=L-1+L_{2}+...+L_{n}$ ## Series n capacitors ${\frac {1}{C_{t}}}={\frac {1}{C_{1}}}+{\frac {1}{C_{2}}}+...+{\frac {1}{C_{n}}}$ ## Series RC

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

$V_{L}+V_{R}=0$ $L{\frac {d}{dt}}i(t)+Ri(t)=0$ ${\frac {d}{dt}}i(t)=-{\frac {R}{L}}i(t)$ $i(t)=Ae^{-{\frac {R}{L}}t}$ ## Series LC

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

$V_{L}+V_{C}+V_{R}=0$ $L{\frac {d}{dt}}i(t)+{\frac {1}{C}}\int i(t)dt+Ri(t)=0$ ${\frac {d^{2}}{dt^{2}}}i(t)=-{\frac {R}{L}}{\frac {d}{dt}}i(t)-{\frac {1}{LC}}i(t)=0$ ${\frac {d^{2}}{dt^{2}}}i(t)=-2\alpha {\frac {d}{dt}}i(t)-\beta i(t)=0$ $\alpha ={\frac {R}{2L}}$ $\beta ={\frac {1}{LC}}$ One real root . $\alpha =\beta$ $i(t)=Ae^{-\alpha t}=A(\alpha )$ Two real roots . $\alpha >\beta$ $i(t)=Ae^{(\alpha \pm {\sqrt {\alpha -\beta }})t}$ Two complex roots . $\alpha <\beta$ $i(t)=A(\alpha )e^{(\alpha \pm j{\sqrt {\beta -\alpha }})t}=A(\alpha )Sin\omega t$ $\omega ={\sqrt {\beta -\alpha }}$ $A(\alpha )=Ae^{\alpha t}$ 