the_integrator

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 the_integrator [2015/05/25 19:29]admin created the_integrator [2015/05/25 20:06] (current)admin 2015/05/25 20:06 admin 2015/05/25 19:29 admin created 2015/05/25 20:06 admin 2015/05/25 19:29 admin created Line 15: Line 15: which is the same current that flows into the capacitor. which is the same current that flows into the capacitor. - $i=\frac{v_{IN}}{R}=C\frac{dV_C}{dt}$ + $i=\frac{v_{IN}}{R}=C\frac{dv_C}{dt}$ + + Notice that $v_C=-v_{OUT}$. + + $i=\frac{v_{IN}}{R}=-C\frac{dv_{OUT}}{dt}$ + + Now we just have to get the expression into the final form. + + $\frac{-v_{IN}}{RC}dt=dv_{OUT}$ + + $\frac{-1}{RC}\int_{t_0}^t{v_{IN}dt}=\int_{v_0}^v{dv_{OUT}}$ + + $\frac{-1}{RC}\int_{t_0}^t{v_{IN}dt}+v_{0}=v_{OUT}$