RL series

RL Series

A circuit consists of 2 component Resistor and Inductor connected in series as shown below

At equilibrium, the net voltage in the circuit is zero

V_{L}+V_{R}=0

L{\frac {d}{dt}}i+iR=0

{\frac {d}{dt}}i=-{\frac {R}{L}}i

{\frac {di}{i}}=-{\frac {R}{L}}dt

\int {\frac {di}{i}}=-{\frac {R}{L}}\int dt

Lni=-{\frac {R}{L}}t+c

i=e^{-{\frac {R}{L}}+c}=e^{c}e^{-{\frac {R}{L}}t}=Ae^{-{\frac {1}{T}}t}

T={\frac {L}{R}}

At equilibrium, the total impedance in the circuit is zero

Z_{t}=Z_{L}+Z_{R}

Z_{t}=j\omega L+R

Z_{t}=j\omega {\frac {L}{R}}+1=j\omega T+1

Z_{t}=j\omega {\frac {L}{R}}+1=j\omega T+1={\frac {v}{i}}

T={\frac {L}{R}}

At equilibrium, the total impedance in the circuit is zero

Z_{t}=Z_{L}+Z_{R}=0

Z_{t}=\omega L\angle 90+R\angle 0

Z_{t}={\sqrt {(\omega L)^{2}+R^{2}}}\angle \omega T

T={\frac {L}{R}}

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