Practical Electronics/Series RC

Series RC
A circuit of two components R and C connected in series
 * [[Image:RC switch.svg|200px]]

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
 * $$Z = Z_R + Z_C$$
 * $$Z = R \angle 0 + \frac{1}{\omega C} \angle -90 = |Z| \angle \theta = \sqrt{R^2 + (\frac{1}{\omega C})^2} \angle Tan^-1 \frac{1}{\omega RC}$$


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

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
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 response of the circuit is an exponential decrease