How to relate the time constants, opamp bandwidth and settling time?

[AMSA84]

Hello guys,

Is someone here that could explain me how to relate the parameters mentioned in the topic, when designing an opamp? For example, design the opamp to have a very small error in the settling time and relate that to how many time constants are needed for that small error, say 0.01%? I think it is related to the opamp bandwidth as well.

Regards.

 

[KlausST]

Hi,

how many time constants are needed for that small error, say 0.01%?

9 time constants:

0.01% = 0.0001 = 1 x 10^-4

(1/e)^9 = 0.000123 = 0.0123% = about 0.01%

e = 2.7183
1/e = 0.3679

0.3679 ^9 = 0.000123

how to come to 9?
log(0.0001) / log(1/e) = 9.2 --> 9.2 time constants (info: log base 10)
= -ln(0.0001) = 9.21 (info: ln = log base e)

Klaus

 

[AMSA84]

Thanks for your response.

Klaus, those calculations are assuming that the opamp has a dominant pole? At the end of the day is a simple 1st order system right? So you're assuming the first ordem system equation 1/(Ts+1) where the inverse Laplace when applied a step unity is simply y(t)=1-e^-(t/T)?

If so, where and why are you getting the 1/e? what you mean is 1-e^-(t/T) = 1 - (1/e^(t/T)) ? What about the voltage, you don't care? Have you normalized it? The expression if y(t) = V (1-e^-(t/T)). What did you do to the V?

Finally, why are you diving both logs? How do you get there?

Regards.

- - - Updated - - -

By the way Klaus, both calculations in my calculator gives 9.21.

 

[KlausST]

Hi,

You speak of time constant.
I assumed you meant tau, like R x C.., a low pass filter first order
Did you talk of any other?

If tau, then after one time constant the error is 1/e = 1/2.7 = 0.37
After two time constants the error is 0.37 ^ 2
After three time constants 0.37 ^ 3
And so on

With a time constant the error is independent of the absolute step size..

With a real opamp there are for sure other influences. Like limited dV/dt. Or riniging at the border of stability...

Klaus

 

[AMSA84]

Thanks for you comment KlausST.

Well, I spoke about the time constants but I was wondering if you can relate that to the opamp BW because the opamp BW can influence the settling time of your opamp. So I was wondering if there is any way to link both, the BW and the time constants. Basicaly how many time constants do I need to get a certain amount of error for a certain BW, that is if I apply a square wave at the input of the opamp in unity gain mode, I would like the opamp to settle in half period, right, before the input changes again.

I imagine if we apply a certain square wave at the input with certain characteristics and if we vary the bandwidth of the amplifier I am sure we would see some differences in the settling time and error no?

Regards.

 

[KlausST]

Hi,

Basicaly how many time constants do I need to get a certain amount of error for a certain BW, that is if I apply a square wave at the input of the opamp in unity gain mode, I would like the opamp to settle in half period, right, before the input changes again.

I already answered this if it a first order bandwidth limitation.
If it is a higher order I can't answer this. I doubt somewhere else can...but maybe. The problem is that higher order filters have a "q" factor, but it is unknown here. The "q" factor influences settling time.

Klaus