Electronics Handbook/Circuits/RC Series

Circuit's Configuration

 * [[Image:RC switch.svg|200px]]

Circuit's Impedance
In Polar coordinate
 * $$Z = Z_R + Z_C $$
 * Z = R /_0 + ( 1 / ωC ) /_ - 90
 * Z = = |Z|/_θ = $$\sqrt{R^2 + (\frac{1}{\omega C})^2}$$ /_ Tan-1 $$\frac{1}{\omega RC}$$

In Rectangular coordinate
 * $$Z = Z_R + Z_C $$
 * Z = $$ R + \frac{1}{j\omega C} = \frac{1 + j\omega RC}{j\omega C}$$
 * $$Z = \frac{1}{X_C} (1 + j\omega T$$)

Phase Angle Difference Between Voltage and Current
''There is a difference in angle Between Voltage and Current. Current leads Voltage by an angle θ''
 * $$Tan\theta = \frac{1}{\omega RC} = \frac{1}{f} \frac{1}{2\pi RC} = t \frac{1}{2\pi RC}$$

''The difference in angle between Voltage and Current relates to the value of R, C and the Angular of Frequency ω which also relates to f and t. Therefore when change the value of R or C, the angle difference will be changed and so are ω , f , t ''
 * $$\omega= \frac{1}{RC} \frac{1}{Tan\theta}$$
 * $$f = \frac{1}{2\pi} \frac{1}{RC} \frac{1}{Tan\theta}$$
 * $$t = 2\pi RC Tan\theta$$

Natural Response

 * $$C \frac{dV}{dt} + \frac{V}{R} = 0$$
 * $$\frac{dV}{dt} = - \frac{1}{RC} V$$
 * $$\frac{1}{V} dV = - \frac{1}{RC} dt$$
 * $$\int \frac{1}{V} dV = - \int \frac{1}{RC} dt$$
 * ln V = $$ - \frac{1}{RC} + C$$
 * $$ V = e^- (\frac{1}{RC}) t + C$$
 * $$ V = A e^- (\frac{1}{T}) t $$
 * $$A = e^C$$
 * T = RC

Summary
In summary, RL series circuit has a first order differential equation of voltage
 * $$\frac{d}{dt}f(t) + \frac{t}{T} = 0$$

Which has one real root
 * $$V(t) = Ae^\frac{-t}{T}$$
 * $$A = e^c$$