# Electronics/Electronics Formulas/Series Circuits/Series RL

## 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 equilirium

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

The root of the equation is a Exponential Decay function chracterised 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

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