# Practical Electronics/Series RC

## Series RC

Electric 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 reponse of the circuit is an exponential decrease

## Summary

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