Electronics/RL transient

From Wikibooks, open books for an open world
Jump to: navigation, search
RL Series Open-Closed.svg

For a series RL of one resistor connected with one inductor in a closed loop

Circuit Impedance[edit]

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[edit]

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[edit]

 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[edit]

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