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Maximum dV by dT a RC filter can follow


Circuit Diagram



Analysis

Although this is a very simple and well known circuit some interesting questions here and results here that we derive here is
  • What happens when $V_i$ is a ramp rather than a step.
  • How does the output voltage behave different ramp slopes




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:
$$V_i-V_{out}=RC{dV_{out} }/{dt}$$
$$Mt=RC{dV_{out} }/{dt}+V_{out}$$

Solving this differential equation for $V_{out}$ by multiplying with $e^{t/{RC}}$ and integrating we get:
$$V_{out}=Mt-MRC+A_oe^{-t/{RC}}$$
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:
$$V_i-V_{out}=MRC(1-e^{-K})$$

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 ramp

An 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 limit

References

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Copyright 2018 Milind Gupta