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

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

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

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	$V(t)=A{e}^{(}-\frac{t}{T})$
A	$A=IR$