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 an angle θ

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

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

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

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

$t = 2\pi Tan\theta RC$