RC circuit

A circuit of 2 components are connected in series namely resistor and capacitor

Circuit	Configuration	Formula
RC series		${\displaystyle V_{C}+V_{R}=0}$ ${\displaystyle C{\frac {dv}{dt}}+{\frac {v}{R}}=0}$ ${\displaystyle {\frac {dv}{dt}}=-{\frac {1}{T}}v}$ ${\displaystyle T=RC}$ ${\displaystyle \int {\frac {dv}{v}}=-{\frac {1}{T}}\int dt}$ ${\displaystyle Lnv=-{\frac {1}{T}}+c}$ ${\displaystyle v=e^{-{\frac {1}{T}}t+c}=Ae^{-{\frac {1}{T}}t}}$
Low pass filter		${\displaystyle {\frac {v_{o}}{v_{i}}}={\frac {\frac {1}{j\omega C}}{R+{\frac {1}{j\omega C}}}}={\frac {1}{j\omega RC+1}}={\frac {1}{1+j\omega T}}}$ ${\displaystyle T=RC}$ ${\displaystyle \omega _{o}={\frac {1}{T}}={\frac {1}{RC}}=2\pi f}$ ${\displaystyle v_{o}(\omega =0)=v_{i}}$ ${\displaystyle v_{o}(\omega =\omega _{o})={\frac {v_{i}}{2}}}$ ${\displaystyle v_{o}(\omega =00)=0}$
High pass filter		${\displaystyle {\frac {v_{o}}{v_{i}}}={\frac {R}{R+{\frac {1}{j\omega C}}}}={\frac {j\omega CR}{1+j\omega CR}}={\frac {j\omega T}{1+j\omega T}}}$ ${\displaystyle T=RC}$ ${\displaystyle \omega _{o}={\frac {1}{T}}={\frac {1}{RC}}}$ ${\displaystyle v_{o}(\omega =0)=0}$ ${\displaystyle v_{o}(\omega =\omega _{o})={\frac {v_{i}}{2}}}$ ${\displaystyle v_{o}(\omega =00)=v_{i}}$