Practical Electronics/Series RC

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Series RC[edit]

Electric circuit of two components R and C connected in series

Circuit analysis[edit]

Circuit's Impedance[edit]

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 f RC} = \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[edit]

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
Ln I = -(\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 reponse of the circuit is an exponential decrease

Summary[edit]

Circuit Series RC
Configuration
RLC series circuit.png
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 = A e^(-\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