# Electronics/Electronics Formulas/Series Circuits/Series RL

## Formula

The total Impedance of the circuit

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

The 1st order Differential equation of the circuit in equilirium

${\displaystyle L{\frac {di}{dt}}+iR=0}$
${\displaystyle {\frac {di}{dt}}+i{\frac {R}{L}}=0}$
${\displaystyle \int {\frac {di}{i}}=-{\frac {R}{L}}\int dt}$
${\displaystyle Lni=-{\frac {t}{T}}+c}$

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

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

${\displaystyle Z=Z_{R}+Z_{L}}$
${\displaystyle Z=R\angle 0+\omega L\angle 90}$
${\displaystyle Z=|Z|\angle \theta }$
${\displaystyle Z={\sqrt {R^{2}+(\omega L)^{2}}}\angle \omega {\frac {L}{R}}}$

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

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

## Summary

Series RL is characterised by

1st ordered Differential equation of the circuit at equilibrium.

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

Natural Reponse of the circuit is the Exponential Decay

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