Electronics/Electronics Formulas/Series Circuits/Series RL

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Circuit Configuration[edit]

RL Series Open-Closed.svg

Formula[edit]

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 equilirium

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 chracterised 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[edit]

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})