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

$Z=Z_{R}+Z_{C}$

Z=R+{\frac {1}{j\omega C}}={\frac {1}{j\omega C}}(1+j\omega T)

T=RC

In Polar coordinate

$Z=Z_{R}+Z_{C}$

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}}

Tan\theta ={\frac {1}{\omega RC}}={\frac {1}{2\pi fRC}}={\frac {t}{2\pi RC}}

The value of $\theta ,\omega ,f$ 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

\omega ={\frac {1}{Tan\theta RC}}

f={\frac {1}{2\pi Tan\theta RC}}

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

L{\frac {dI}{dt}}+IR=0

{\frac {dI}{dt}}=-{\frac {R}{L}}I

\int {\frac {dI}{I}}=-{\frac {R}{L}}\int dt

LnI=-({\frac {R}{L}})t+e^{c}

I=Ae^{-}({\frac {R}{L}})t

I=Ae^{-}({\frac {t}{T}})

A=e^{c}={\frac {V}{R}}

T=RC

The natural reponse of the circuit is an exponential decrease

Summary

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

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