Practical Electronics/Series RC

Series RC

A circuit of two components R and C connected in series

Circuit analysis

Circuit's Impedance

In Rectangular coordinate

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

In Polar coordinate

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

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

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

Circuit's Response

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

${\displaystyle L{\frac {di}{dt}}+iR=0}$
${\displaystyle {\frac {di}{dt}}=-{\frac {R}{L}}i}$
${\displaystyle \int {\frac {di}{i}}=-{\frac {R}{L}}\int dt}$
${\displaystyle Lni=-({\frac {R}{L}})t+e^{c}}$
${\displaystyle i=Ae^{-}({\frac {R}{L}})t}$
${\displaystyle i=Ae^{-}({\frac {t}{T}})}$
${\displaystyle A=e^{c}={\frac {V}{R}}}$
${\displaystyle T=RC}$

The natural response of the circuit is an exponential decrease