# Electronics/RL transient For a series RL of one resistor connected with one inductor in a closed loop

## Circuit Impedance

In Polar Form Z/_θ

$Z=Z_{R}+Z_{L}$  = R/_0 + ω L/_90
Z = |Z|/_θ = ${\sqrt {R^{2}+(\omega L)^{2}}}$ /_Tan-1$\omega {\frac {L}{R}}$

In Complex Form Z(jω)

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

## Differential Equation of circuit at equilibrium

$L{\frac {dI}{dt}}+IR=0$
${\frac {dI}{dt}}=-I{\frac {R}{L}}$
$\int {\frac {1}{I}}dI=-\int {\frac {L}{R}}dt$
$lnI=(-{\frac {L}{R}}+c)$
$I=e^{(}-{\frac {L}{R}}t+c)=e^{c}+e^{(}-{\frac {L}{R}}t$
$I=Ae^{-}({\frac {t}{T}})$

## Time Constant

$T={\frac {L}{R}}$
t I(t) % Io
0 A = eC = Io 100%
R/L .63 Io 60% Io
2 R/L Io
3 R/L Io
4 R/L Io
5 R/L .01 Io 10% Io

## Angle Difference between Voltage and Current

Voltage leads Current at an angle ? When a determining process is necessary many problems arise in a diagram. We need to expend on one process for the determing factor in this type of formulae

Tan? = ${\frac {1}{\omega RC}}={\frac {1}{2\pi fRC}}=t{\frac {1}{2\pi RC}}$

Change the value of R and L will change the value Angle Difference, Angular Frquency, Frequency, Time

$\omega ={\frac {1}{Tan\theta RC}}$
$f={\frac {1}{2\pi Tan\theta RC}}$
$t=2\pi Tan\theta RC$