Electronics/Electronics Formulas/Series Circuits/Series RL

Circuit Configuration

Formula

The total Impedance of the circuit

$Z=Z_R+Z_L$

$Z=R+j\omega L$

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

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

The 1st order Differential equation of the circuit in equilibrium

$L\frac{di}{dt}+iR=0$

$\frac{di}{dt}+i\frac{R}{L}=0$

$\int \frac{di}{i}=-\frac{R}{L}\int dt$

$Lni=-\frac{t}{T}+c$

The root of the equation is a Exponential Decay function characterised the Natural Response of the circuit

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

$Z=Z_R+Z_L$

$Z=R\mathrm{\angle }0+\omega L\mathrm{\angle }90$

$Z=|Z|\mathrm{\angle }\theta$

$Z=\sqrt{R^2+(\omega L{)}^2}\mathrm{\angle }\omega \frac{L}{R}$

Phase Shift , Change in Frequency relate to change in RL value

$Tan\theta =\omega \frac{L}{R}=2\pi f\frac{L}{R}=2\pi \frac{1}{t}\frac{L}{R}$

Summary

Series RL is characterised by

1st ordered Differential equation of the circuit at equilibrium.

$\frac{di}{dt}+\frac{1}{T}=0$

Natural Reponse of the circuit is the Exponential Decay

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