Electronics/RL transient


 * RL_Series_Open-Closed.svg

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

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

Time Constant

 * $$ T = \frac{L}{R}$$
 * {| class="wikitable" width=50%

! 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
 * }
 * 3 R/L || Io ||
 * 4 R/L || Io ||
 * 5 R/L || .01 Io || 10% 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
 * $$\omega = \frac{1}{Tan\theta RC}$$
 * $$f = \frac{1}{2\pi Tan\theta RC}$$
 * $$t = 2\pi Tan\theta RC$$