AC analysis of a circuit including variable resistors

I have a problem concerning the AC analysis of a simple circuit, containing variable resistors (each variable resistor is a pressure sensor). I am sure this is quite simple, but I have never worked on variable networks and I cannot find answers by searching on the web. Goal

R1, R2, R3, and R4 are the variable resistors, each one corresponding to a pressure sensor. Their resistance varies with the applied pressure, which is a stochastic process (trackinf of the hand pressure). My goal is to track in 1D the pressure along the sensor, thanks to the multiple pressure areas.

I know the circuit is bad, poor dynamics, etc. but I need a very compact and cheap circuit that can find the value of the resistors R1 to R4 by measuring output voltages via an ADC. The simplest way I have found is this one, where I put equal resistors R between the ground/power voltage and each variable resistor. By doing this way, I can measure the output voltages V0 to V4 and find the resistors by a quick circuit analysis.

As a result, I just want to measure the pressure with not so much noise to be able to conclude if the work I am doing require more investment or if this is useless to spend more time on it.

Problem

The resistors R1, R2, R3 and R4 are variable. I want to filter the output voltages (V0 to V4) as they don't see variation in voltage higher than the nyquist frequency of my ADC converter. I can sample at 50kHz, but sending my data to the computer take time and I need to filter out frequencies above 500 Hz. For this purpose, I would like to use a passive filter to remove frequencies above 500Hz and then use a digital filter to remove everything above 500Hz.

So I have two ways of doing this passive filter:

1. Finding the matching R and C that will reduce the impact of the variable resistors and filter close to 500Hz in any case. I think it is problematic as R1 to R4 are time-varying and are a stochastic process;
2. Finding a way to reduce at max the output impedance of each output and use a RC filter in cascade with each output. I would like to not use an Op Amp;

AC Analysis of the circuit

Can I consider the circuit as multiple two-ports networks in cascade (simple RC filter)?

Vd is my power voltage, all resistors R and capacitor C are equal.

I thought it will be possible to model the circuit like this, since the charge of the capacitor provoked the transient. Note that I didn't include V3 and V4 to clear the scheme, but I obviously should. Do you have any idea on how I am supposed to perform the AC analysis? How can I find the bandwidth of each output Vx in response to the change in pressure at each pressure sensor Rx?

How do I tell Spice that, say, R4 changes between resistance Rlow and Rhigh in a square wave that ramps up in frequency from 10 Hz to 10 MHz? Thank you so much for your help!

• You want "to know the transfer function of the output voltage V1 when a a change occurs on the resistor R1". What does this mean - "when a change occurs"? After the change? A transfer function applies to steady-state conditions only. – LvW May 14 '14 at 20:28
• Does your simulator do monte carlo analysis? – Andy aka May 14 '14 at 20:38
• Hi, thank you for your answer. I have edited my post, hopefully it's clearer. I want to filter out rapid changes of the resistor values (R1 to R4 are variable, with the pressure). So, I was looking for a way to find the C and R values (look on the first scheme) that filter out these rapid changes. However, I don't know how to start to analyze the circuit and determine the values of R and C as the resistors R1 to R4 are variable. I prefer to find an "analytic" method if possible as I would have to explain my approach. – jujux789 May 14 '14 at 20:53

If you are building an anti alias filter get rid of the components called R - they are not helping and are also attenuating the signal across the whole of the baseband - you should consider several cascaded sections made like this: - simulate this circuit – Schematic created using CircuitLab

Pick a value of R and C and do an AC analysis and, if the cut off frequencies are still too high make R bigger until they are not.

BTW it's not a great way of designing an anti alias filter and, it won't be as good as a couple of 2nd order sallen-key filters made with op-amps but nonetheless it should suffice.

• Thank you for your answer. Even if it is attenuating the signal, I need the resistors R to compute my variable resistors faster in my microcontroller. I am going to try like you propose, thanks again. – jujux789 May 14 '14 at 21:21
• @jujux789 I have to say that made no sense to me whatsoever !! Why use a microcontroller to compute values of variable resistors (faster or not)? Why on earth should they be variable? Are you changing the sampling speed of your ADC dynamically? – Andy aka May 14 '14 at 21:27
• I am using an array of pressure sensors and I need a very compact design for my purpose. It is the way I reached the faster sampling by using the less space. If I remove the resistors R, I loose resolution and I canont be sure of the value of the variable resistor, as two parralel one can be grounded. It is my thought, I am maybe wrong. – jujux789 May 14 '14 at 21:30
• Why don't you explain in your question what you are trying to accomplish and let the supposedly clever folk on EE decide what options are best. – Andy aka May 15 '14 at 7:19
• I am sorry I didn't think it was so important, but when I have seen your response, I was definitely wrong, I am going to complete the question. – jujux789 May 15 '14 at 15:43

I have a problem concerning the AC analysis of a simple circuit, containing variable resistors.

If the resistors are changing with time, the system is not time invariant.

If the resistance rate of change is very slow compared to the rate of change of the signals of interest, the system is quasi-time invariant and within limits, LTI frequency analysis techniques can be used.

However, if the resistance rate of change is not very slow, standard AC frequency response analysis, which assumes the system is LTI, is no longer applicable.

See, for example, Time Variant Filtering for some considerations.

• It is exactly the problem I thought I will have... However, the change in resistor is stochastic as it depends on the variation of pressure (exerced by the hand). How can I do in that case? – jujux789 May 14 '14 at 21:24