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

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