# 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

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

## 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? = $\frac{1}{\omega RC} = \frac{1}{2 \pi f RC} = t \frac{1}{2 \pi RC}$

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

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