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Circuit DiagramAnalysisAlthough this is a very simple and well known circuit some interesting questions here and results here that we derive here is
Why are those important? If we know how soon the output catches up to a ramp and what is the voltage lag we can determine the RC values that give the desired response to a ramp input encountered so often in circuits. Let $V_i=Mt$ where M is the ramp slope and t is time so we have:$$V_iV_{out}=RC{dV_{out} }/{dt}$$ $$Mt=RC{dV_{out} }/{dt}+V_{out}$$ Solving this differential equation for $V_{out}$ by multiplying with $e^{t/{RC}}$ and integrating we get:$$V_{out}=MtMRC+A_oe^{t/{RC}}$$ where $A_o$ is the integration constant which can be solved by the initial condition at t=0 $V_{out}=0$ Thus $A_o=MRC$ So we have: $$V_{out}=Mt+MRC(e^{t/{RC}}1)$$ So now lets look at $V_{out}$ after certain number of time constants say K i.e. $t=KRC$ so we have:$$V_{out}=MKRC+MRC(e^{K}1)$$ Here we can write $V_i=MKRC$ therefore we have:$$V_iV_{out}=MRC(1e^{K})$$ Therefore the output would start tracking after a couple of time constants and the difference (voltage lag) between the output and the input is approximately MRC. Amplitude limited rampAn amplitude limited ramp is given as$$\table V_i, = ,Mt , t<t_F; ,=, Mt_F, t≥t_F$$ To make a good ramp follower RC circuit the time $t_F≫RC$ so that it starts following the ramp much before the ramp reaches its limitReferences       


Copyright 2018 Milind Gupta 