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

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

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

$Z=R\angle 0+\omega L\angle 90$

$Z=|Z|\angle \theta$

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

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

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