Electronics Handbook/Components/RC Network

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

Differential Equation of the circuit
At Equilibrium the total voltage is zero
 * $$V_C + V_R = 0$$
 * $$C \frac{dV}{dt} + \frac{V}{R} = 0$$
 * $$\frac{dV}{dt} = - \frac{1}{RC} V$$
 * $$\frac{1}{V} dV = - \frac{1}{RC} dt$$
 * $$\int \frac{1}{V} dV = - \int \frac{1}{RC} dt$$
 * ln V = $$ - \frac{1}{RC} + C$$
 * $$ V = e^- (\frac{1}{RC}) t + C$$
 * $$ V = Ae^- (\frac{t}{T})$$

With
 * T = RC
 * A = ec

Circuit Impedance

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