# Practical Electronics/Series RL

## Series RC

Electric circuit of two components R and L connected in series

## Circuit analysis

### Circuit's Impedance

In Rectangular coordinate

• ${\displaystyle Z=Z_{R}+Z_{L}}$
${\displaystyle Z=R+\omega L={\frac {1}{R}}(1+j\omega T)}$
${\displaystyle T={\frac {L}{R}}}$

In Polar coordinate

• ${\displaystyle Z=Z_{R}+Z_{L}}$
${\displaystyle Z=R\angle 0+\omega L\angle 90=|Z|\angle \theta ={\sqrt {R^{2}+(\omega L)^{2}}}\angle Tan^{-}1\omega {\frac {L}{R}}}$
${\displaystyle Tan\theta =\omega {\frac {L}{R}}=2\pi f{\frac {L}{R}}={\frac {t}{2\pi {\frac {L}{R}}}}}$

The value of ${\displaystyle \theta ,\omega ,f}$ depend on the value of R and L . Therefore when the value of R or L changed the value of Phase angle difference between Current and Voltage , Frequency , and Angular of Frequency also change

${\displaystyle \omega ={\frac {R}{L}}Tan\theta }$
${\displaystyle f={\frac {1}{2\pi }}{\frac {R}{L}}Tan\theta }$
${\displaystyle t=2\pi {\frac {L}{R}}{\frac {1}{Tan\theta }}}$

### Circuit's Response

Natural Response of the cicuit can be obtained by setting the differential equation of the circuit to zero

${\displaystyle C{\frac {dV}{dt}}+{\frac {V}{R}}=0}$
${\displaystyle {\frac {dV}{dt}}=-{\frac {1}{RC}}V}$
${\displaystyle \int {\frac {dV}{V}}=-{\frac {1}{RC}}\int dt}$
${\displaystyle LnV=-({\frac {1}{RC}})t+e^{c}}$
${\displaystyle V=Ae^{-}({\frac {1}{RC}})t}$
${\displaystyle V=Ae^{-}({\frac {t}{T}})}$
${\displaystyle V=e^{c}=IR}$
${\displaystyle T={\frac {L}{R}}}$

The natural reponse of the circuit is an exponential decrease

## Summary

Circuit Series RL
Configuration
Impedance ${\displaystyle Z=Z_{R}+Z_{L}=R+j\omega L={\frac {1}{R}}(1+j\omega T)}$
${\displaystyle T={\frac {L}{R}}}$
Diferenial Equation ${\displaystyle C{\frac {dV}{dt}}+{\frac {V}{R}}=0}$
Root of the equation ${\displaystyle V=Ae^{(}-{\frac {t}{T}})}$
${\displaystyle Z\angle \theta }$ ${\displaystyle {\sqrt {R^{2}+(\omega L)^{2}}}\angle Tan^{-}1\omega {\frac {L}{C}}}$
Phase Angle Difference between Voltage and Current ${\displaystyle Tan\theta =\omega {\frac {L}{R}}}$
${\displaystyle \omega }$ ${\displaystyle \omega ={\frac {1}{Tan\theta }}{\frac {L}{R}}}$
${\displaystyle f}$ ${\displaystyle \omega ={\frac {Tan\theta }{2\pi }}{\frac {L}{R}}}$
${\displaystyle t}$ ${\displaystyle t={\frac {2\pi }{Tan\theta }}{\frac {R}{L}}}$