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

• $Z=Z_R+Z_C$

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

$T=RC$

In Polar coordinate

• $Z=Z_R+Z_C$

$Z=R\mathrm{\angle }0+\frac{1}{\omega C}\mathrm{\angle }-90=|Z|\mathrm{\angle }\theta =\sqrt{R^2+(\frac{1}{\omega C}{)}^2}\mathrm{\angle }Ta{n}^{-}1\frac{1}{\omega RC}$

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

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

$L\frac{di}{dt}+iR=0$

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

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

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

$i=A{e}^{-}(\frac{R}{L})t$

$i=A{e}^{-}(\frac{t}{T})$

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

$T=RC$

The natural response of the circuit is an exponential decrease