RC series

RC Series

A circuit consists of 2 component Resistor and Capacitor connected in series as shown below

At equilibrium, the net voltage in the circuit is zero

V_{C}+V_{R}=0

C{\frac {d}{dt}}v+{\frac {v}{R}}=0

{\frac {d}{dt}}v=-{\frac {1}{RC}}i

{\frac {dv}{v}}=-{\frac {1}{RC}}dt

\int {\frac {dv}{v}}=-{\frac {1}{RC}}\int dt

Lnv=-{\frac {1}{RC}}t+c

v=e^{-{\frac {1}{RC}}+c}=e^{c}e^{-{\frac {1}{RC}}t}=Ae^{-{\frac {1}{T}}t}

T=RC

At equilibrium, the total impedance in the circuit is zero

Z_{t}=Z_{C}+Z_{R}=0

Z_{t}={\frac {1}{j\omega C}}+R

Z_{t}={\frac {1+j\omega RC}{j\omega C}}

Z_{t}={\frac {1+j\omega RC}{j\omega C}}={\frac {v}{i}}

At equilibrium, the total impedance in the circuit is zero

Z_{t}=Z_{C}+Z_{R}=0

Z_{t}={\frac {1}{\omega C}}\angle -90+R\angle 0

Z_{t}={\sqrt {R^{2}+{\frac {1}{(\omega C}})^{2}}}\angle {\frac {1}{\omega RC}}

Z_{t}={\sqrt {R^{2}+{\frac {1}{(\omega C}})^{2}}}\angle {\frac {1}{\omega RC}}={\frac {v}{i}}

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