Electronics/Electronics Formulas/Series Circuits/Series RL

Circuit Configuration

 * RL_Series_Open-Closed.svg

Formula
The total Impedance of the circuit
 * $$Z = Z_R + Z_L$$
 * $$Z = R + j\omega L$$
 * $$Z = \frac{1}{R} (1 + j\omega T)$$
 * $$T = \frac{L}{R}$$

The 1st order Differential equation of the circuit in equilibrium
 * $$L \frac{di}{dt} + i R = 0$$
 * $$\frac{di}{dt} + i \frac{R}{L} = 0$$
 * $$\int \frac{di}{i} = -\frac{R}{L} \int dt$$
 * $$Ln i = -\frac{t}{T} + c$$

The root of the equation is a Exponential Decay function characterised the Natural Response of the circuit
 * $$i = Ae^(-\frac{t}{T})$$


 * $$Z = Z_R + Z_L$$
 * $$Z = R \angle 0 + \omega L \angle 90$$
 * $$Z = |Z| \angle \theta$$
 * $$Z = \sqrt{R^2 + (\omega L)^2} \angle \omega \frac{L}{R}$$

Phase Shift, Change in Frequency relate to change in RL value
 * $$Tan \theta = \omega \frac{L}{R} = 2 \pi f \frac{L}{R} = 2 \pi \frac{1}{t} \frac{L}{R}$$

Summary
Series RL is characterised by

1st ordered Differential equation of the circuit at equilibrium.
 * $$\frac{di}{dt} + \frac{1}{T} = 0$$

Natural Reponse of the circuit is the Exponential Decay
 * $$i = Ae^(-\frac{t}{T})$$