# RL series

## RL Series

A circuit consists of 2 component Resistor and Inductor connected in series as shown below

## Chracteristics

### Time Domain

At equilibrium, the net voltage in the circuit is zero

${\displaystyle V_{L}+V_{R}=0}$
${\displaystyle L{\frac {d}{dt}}i+iR=0}$
${\displaystyle {\frac {d}{dt}}i=-{\frac {R}{L}}i}$
${\displaystyle {\frac {di}{i}}=-{\frac {R}{L}}dt}$
${\displaystyle \int {\frac {di}{i}}=-{\frac {R}{L}}\int dt}$
${\displaystyle Lni=-{\frac {R}{L}}t+c}$
${\displaystyle i=e^{-{\frac {R}{L}}+c}=e^{c}e^{-{\frac {R}{L}}t}=Ae^{-{\frac {1}{T}}t}}$
${\displaystyle T={\frac {L}{R}}}$

### Frequency Domain

At equilibrium, the total impedance in the circuit is zero

${\displaystyle Z_{t}=Z_{L}+Z_{R}}$
${\displaystyle Z_{t}=j\omega L+R}$
${\displaystyle Z_{t}=j\omega {\frac {L}{R}}+1=j\omega T+1}$
${\displaystyle Z_{t}=j\omega {\frac {L}{R}}+1=j\omega T+1={\frac {v}{i}}}$
${\displaystyle T={\frac {L}{R}}}$

### Phasor Domain

At equilibrium, the total impedance in the circuit is zero

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