A Question about SC Integrators

[naalald]

Hi,
In this paper, when modeling the Opamp SR and BW, it's said "the unity gain frequency of the integrator loop-gain " (pp. 356, part V. B).

P. Malcovati et al., "Behavioral Modeling of Switched-Capacitor Sigma-Delta Modulators," IEEE Trans on Circuits Syst. I, vol. 50, no. 3, pp. 352-364, Mar. 2003.

The paper is here:


The question is, what is the relation between the GBW (unity gain frequency) of the integrator loop-gain and the GBW (unity gain frequency) of the Opamp?
Is this right?



In which β is the feedback factor of the integrator.
If not, what is the relation? Why?

 

[LvW]

I don´t think that the formula given by you is right since β for the integrator is frequency dependent.
I have another approach:
There is, indeed, a frequency (rather low) for which the loop gain of the integrator circuit is "1" (0 dB) - and this frequnency is called by the author of the paper "unity gain frequency of the integrator loop gain". I call this frequency wi.
In a BODE diagram this frequency wi can be found at the crossing of the finite maximum opamp gain Amax and the ideal integrating curve which intersects the 0 dB line at ωo=1/Ti (Ti=integrate time constant).
A simple geometric relation shows that the following ratios are equal:

opamp_GBW/wo=opamp_wg/wi (opamp_wg=corner frequency of single pole opamp).
This can be modified to wi=1/(Amax*Ti)

I think - in the context of the article - this result makes sense, since wi (called by the author GBW of the integrator open loop gain) is a very low frequency leading to a rather large time constant in the formulas of the paper.

 

[safwatonline]

is it true that the beta is frequency dependent?
i thought that for a good SC amplifier/integrator we can consider neglect the frequency dependency given a good settling of the OTA

 

[naalald]

I don´t think that the formula given by you is right since β for the integrator is frequency dependent.
I have another approach:
There is, indeed, a frequency (rather low) for which the loop gain of the integrator circuit is "1" (0 dB) - and this frequnency is called by the author of the paper "unity gain frequency of the integrator loop gain". I call this frequency wi.
In a BODE diagram this frequency wi can be found at the crossing of the finite maximum opamp gain Amax and the ideal integrating curve which intersects the 0 dB line at ωo=1/Ti (Ti=integrate time constant).
A simple geometric relation shows that the following ratios are equal:

opamp_GBW/wo=opamp_wg/wi (opamp_wg=corner frequency of single pole opamp).
This can be modified to wi=1/(Amax*Ti)

I think - in the context of the article - this result makes sense, since wi (called by the author GBW of the integrator open loop gain) is a very low frequency leading to a rather large time constant in the formulas of the paper.

Hi, Thanks for your answer,
But,
1- β=Cf/(Cs+Cf) in which Cf is the feedback capacitor and Cs is the sampling capacitor, how is it frequency dependent?
2- The paper says 2*pi*wi=1/Ti (wi the unity gain frequency of the integrator loop-gain and Ti is the integrator time constant) and there is no Amax?
3- You didn't mention the direct relation between opamp GBW and the integrator open loop-gain GBW (unity gain frequency)?

 

[LvW]

is it true that the beta is frequency dependent?
i thought that for a good SC amplifier/integrator we can consider neglect the frequency dependency given a good settling of the OTA

In the contribution of naalald β is the feedback factor which includes a capacitor !
(Remember, β is not always the symbol for current gain)

Added after 11 minutes:

But,
1- β=Cf/(Cs+Cf) in which Cf is the feedback capacitor and Cs is the sampling capacitor, how is it frequency dependent?
2- The paper says 2*pi*wi=1/Ti (wi the unity gain frequency of the integrator loop-gain and Ti is the integrator time constant) and there is no Amax?
3- You didn't mention the direct relation between opamp GBW and the integrator open loop-gain GBW (unity gain frequency)?

To 1) OK, if you define β only for one switching intervall, it is constant; however, this seems to be not the real problem, does it ?
To 2) I don´t know how to answer, because I don´t understand the question. I remember the whole stuff we are talking about is connected with nonideal opamp properties. And the finite dc gain Amax is the most important non-ideal parameter.
To 3) Look at the equation with the two ratios. This equation relates the opamp GBW and the "loop gain GBW" (bad expression, therefore my symbol: wi).

 

[safwatonline]

is it true that the beta is frequency dependent?
i thought that for a good SC amplifier/integrator we can consider neglect the frequency dependency given a good settling of the OTA

In the contribution of naalald β is the feedback factor which includes a capacitor !
(Remember, β is not always the symbol for current gain)

what i am saying is that given a good settling of the OTA, the voltage at the capacitors are not frequency dependent (i.e. we always settle to the right value regardless the input frequency) and hence the frequency dependency comes the phase delay between two samples taken at the feedback capacitor (integrating capacitor), so to be short we can neglect the frequency dependency of the beta factor.
p.s. it is nice to know the beta is not always the current gain !

 

[LvW]

what i am saying is that given a good settling of the OTA, the voltage at the capacitors are not frequency dependent (i.e. we always settle to the right value regardless the input frequency) and hence the frequency dependency comes the phase delay between two samples taken at the feedback capacitor (integrating capacitor), so to be short we can neglect the frequency dependency of the beta factor.

Sorry, it seems that I misunderstood your first reply. Now I see what you have ment.
On the other hand, it was my fault to think of an analog integrator with frequency dependent feedback. But in the present case there is an S/C integrator and the feedback action is developed only within one switching period with a constant feedback path which has to be defined in the time rather than in the frequency domain.
Regards
LvW

 

[jiangxb]

this equation is right, and the feedback factor is independent on frequency.

 

[ricklin]

I think there's 2 concepts here:

1 is for settling issue, this demonstrate how the loop act as an one pole integrator, and the equation given by naalald do descript the transfer function in settling action. We need enough GBW at this time to make sure the loop settle to enough resolution that we want.

2 is for integration action, in this part we consider the analog parts as ideal and study the frequency response of input signal in Z domain, and it do have another unity loop gain. However it almost has nothing to do with the GBW in 1.