Tunning an RC Integrator

[mordak]

Hi,

I need to use integrators in my circuit, and due to the process variations the final value of resistors and capacitors would be off by some value. So I am planning to use bank of capacitors and then tune the circuit later (like the below picture). My question is about the on resistance of those switches in series with the capacitors. The on resistance would add a zero to the transfer function and also move the pole, which would result in changing the behavior of the circuit. I was wondering how I can analytically find size of switches rather than running several simulations. I know that if it was a switched cap circuit, we could find the size based on the time constant we are looking for, but for RC integrator I am not sure. Any help would be appreciated!

 

[LvW]

The series resistance of the switch in principle will change the integrator into a PI characteristic (proportional part Rswitch/Rin) - however, for a "good" resistor ratio (small) the influence on the time constant of the integration process should be negligible. Don`t forget that also the finite and frequency-dependent opamp`s open loop gain disturbs the ideal characteristic.

 

[mordak]

however, for a "good" resistor ratio (small) the influence on the time constant of the integration process should be negligible.

Thanks for your response. That's my question, how we find a good resistor? the problem is that to have small on resistance, one should increase the size of transistor, which results in huge transistors and also parasitic caps. I guess if we increase size of switch, we should consider its parasitic cap as well, yeah?
Like in my case, I have an 9K resistor with 50PF capacitor, I run simulation and saw that even 60 ohm on resistance can affect the behavior of the circuit (I didn't consider parasitic cap, just on resistance, and frequency of operation is 13MHz). BTW, do you happen to know any paper analyzing such effects?

 

[LvW]

I didn`t realize that you are going to design an integrated circuit. In this case, it is certainly very important to find the best trade-off between on-resistance and transistor size.
Did you consider already the other alternative: Tuning of the resistor ?
More than that, what is your primary goal: To meet the time constant (cross-over frequenncy in the BODE diagram) or to achieve the "best" integrating feature?
What about your amplifier? Which data (input/output resistor, gain, bandwidth)?

Remark: "Tuning of the resistor" means: Realization using the ohmic part of a FET with external voltage control.

 

[mordak]

Yes my goal is to design an integrator, and it's supposed to be used in a continuous time delta sigma ADC, so basically 1/(RC) would define the coefficients of the converter. My whole ADC works perfect with all real components, and I just added those switches in series with capacitors to tune the integrator and saw that it affects the result. Technically having a small resistor in series with capacitor can even help to compensate the finite gain bandwidth of the amplifier, and that is the reason they put a small resistor in the feedback branch in an RC integrator, however, that's a pure resistor and doesn't have any parasitic cap.

Tuning resistors turned out not to be a good idea, since it has more negative effects on the transfer function, as I read in several theses.

 

[LvW]

Mordak, are you required to use the shown circuit design (with opamps and capacitive feedback) - or are you free to try another concept?
Did you hear about OTA designs for integrating functions? Such a design has the capability to control some paraeters with an externally applied current.

 

[mordak]

Mordak, are you required to use the shown circuit design (with opamps and capacitive feedback) - or are you free to try another concept?
Did you hear about OTA designs for integrating functions? Such a design has the capability to control some paraeters with an externally applied current.

I guess you're talking about the Gm-C filters, actually they are not quite proper structures in some cases, like the linearity problem, though adding a resistor inside the structure would improve linearity, still input of the GM-C has wide swing unlike the RC integrator which has the virtual ground at the input.
Actually I have seen in different papers and theses they use the structure I showed in the first post to tune the integrator, however, surprisingly no one mentioned anything about the on resistance of the switch and its negative impacts. So I'm sure there should be a solution to the problem, and analytical study of that.

 

[LvW]

So I'm sure there should be a solution to the problem, and analytical study of that.

"solution to the problem.." - I think, this sounds a bit too optimistic.
For my opinion, you must view this effect in conjunction with other non-idealities (mentioned already in my post#4) like opamp´s input and output resistances, open-loop gain, phase shift, parasitic elements,...
May be that some of these effects overshadow the influence of switch resistance?

 

[mordak]

"solution to the problem.." - I think, this sounds a bit too optimistic.
For my opinion, you must view this effect in conjunction with other non-idealities (mentioned already in my post#4) like opamp´s input and output resistances, open-loop gain, phase shift, parasitic elements,...
May be that some of these effects overshadow the influence of switch resistance?

I guess since finite gain bandwidth of the opamp would add a pole to the transfer function of the integrator (compared to the ideal integrator), placing a small resistor in series with the capacitor can help adding a zero which would somehow cancel effect of the finite gain bandwidth. I tried to use the resistance of those switches as the small resistor I mentioned before, now I'm running a long simulation to see if that works, crossing my fingers