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

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

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

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

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

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

$I=Ae^{-{\frac {t}{T}}}$

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

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

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

$t=2\pi Tan\theta RC$