# 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:Solving this differential equation for

`$V_{out}$`

by multiplying with `$e^{t/{RC}}$`

and integrating we get: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:

So now lets look at

`$V_{out}$`

after certain number of time constants say K i.e. `$t=KRC$`

so we have:Here we can write

`$V_i=MKRC$`

therefore we have: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 asTo 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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