Electronics/Electronics Formulas/Series Circuits/Series RL

The total Impedance of the circuit

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

${\displaystyle Z=R+j\omega L}$

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

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

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

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

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

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

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

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

The 1st order Differential equation of the circuit in equilibrium

${\displaystyle v_{L}+v_{R}=0}$

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

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

${\displaystyle si+{\frac {1}{T}}i=0}$

${\displaystyle s=-{\frac {1}{T}}=-\alpha }$

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

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

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

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