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

$L\frac{dI}{dt}+IR=0$

$\frac{dI}{dt}=-I\frac{R}{L}$

$\int \frac{1}{I}dI=-\int \frac{R}{L}dt$

$lnI=(-\frac{R}{L}t+c)$

$I=e^{-\frac{R}{L}t+c}=e^c\times e^{-\frac{R}{L}t}$

$I=A{e}^{-\frac{t}{T}}$

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

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

$\omega =\frac{1}{Tan\theta RC}$

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

$t=2\pi Tan\theta RC$

Template:BookCat