Electronics Handbook/Components/RL Network

Series RL

Circuit of R and L connected in series

Differential Equation of the circuit

At equilibrium the total voltage is zero

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

$L{\frac {dI}{dt}}+IR=0$

${\frac {dI}{dt}}=-I{\frac {R}{L}}$

$\int {\frac {1}{I}}dI=-\int {\frac {L}{R}}dt$

ln I = $(-{\frac {L}{R}}t+c)$

I = $Ae^{-}{\frac {t}{T}}$

Với

$T={\frac {L}{R}}$

A = e^c

The natural response of the circuit is Exponential Response

Circuit Impedance

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

$Z=R+j\omega L$

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

$T={\frac {L}{R}}$

Summary

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