Electronics Handbook/Components/RC Network

RC Series

Circuit of R and C connected in series

Differential Equation of the circuit

At Equilibrium the total voltage is zero

$V_{C}+V_{R}=0$

$C{\frac {dV}{dt}}+{\frac {V}{R}}=0$

${\frac {dV}{dt}}=-{\frac {1}{RC}}V$

${\frac {1}{V}}dV=-{\frac {1}{RC}}dt$

$\int {\frac {1}{V}}dV=-\int {\frac {1}{RC}}dt$

ln V = $-{\frac {1}{RC}}+C$

$V=e^{-}({\frac {1}{RC}})t+C$

$V=Ae^{-}({\frac {t}{T}})$

With

T = RC

A = e^c

Circuit Impedance

$Z=Z_{R}+Z_{C}$

$Z=R+{\frac {1}{j\omega C}}$

$Z={\frac {1}{j\omega C}}(1+j\omega T)$

T = RC

Summary

Circuit	Series RC
Configuration
Impedance	$Z=R+j\omega C$ $Z={\frac {1}{j\omega C}}(1+j\omega T)$
T	$T=RC$
Differential Equation	$C{\frac {dV}{dt}}+{\frac {V}{R}}=0$
General Form of Differential Equation	${\frac {dI}{dt}}+{\frac {1}{T}}=0$
Natural Response	$V(t)=Ae^{(}-{\frac {t}{T}})$
A	$A=IR$