Designing Charge Amplifier

[mecs]

Hi community,
I'm trying to simulate a charge amplifier, which is connected to a piezoelectric sensor that will be measuring vibrations. Later down the line, it will be connected to a Low Pass filter and an ADC, so only positive voltages. Therefore, a Vref = Vcc/2 will be used in the positive terminal of the amp.
(This post only focuses on the charge amp and piezo part).
The OpAmp that I will be using is OPA330 https://www.ti.com/lit/ds/symlink/opa330.pdf
Frequency range: 100 kHz - 1.5 MHz.
Sensitivity of sensor: unknown (until the manufacturer answers the email).
I'm using LTSpice

I have two main questions:
1) Is it okay to simulate the piezo sensor as an AC voltage source with a capacitor in series (The value of which is equal to the electrical capacitance of the sensor)?
I've also come across this paper from TI https://www.ti.com/lit/an/slyt369/slyt369.pdf which uses a transformer to accurately simulate the sensitivity of the piezo.

2) I've done countless variations of values, by trial and error or by using equations, yet I feel like I can't find any values that correctly show the results.
The main source of information is: https://www.ti.com/lit/an/sboa287/sboa287.pdf
C_f = 1/(2*pi*R_f*f_low) = 159 pF
  - I'll choose an R_f of 100 MOhm and an f_low of 10 Hz. (I selected 10 Hz because if not, C_f would be very low).
R_in = 1/(2*pi*C_sens*f_high) = 107 Ohm
  - I find this value pretty low.
I'm also confused, because other sources establish the gain as charge Q divided by Cf [(1/Cf)], therefore Cf does not depend on the first equation.

These values look wrong to me, and the simulations don't seem to look nice or change a lot when playing with the values. Where should I get my values from?
Another observation is that the output should oscillate at 0.9 V, yet it's 20 mV lower. This might be due to R_f having a high value.



The transformer setup should give a charge of 0.5 pC/G. Value from:  https://www.ti.com/lit/an/slyt369/slyt369.pdf

Thanks in advance. Any advice is greatly appreciated.

--------------
Some information about the requirements:
It has to be able to properly measure frequencies between 100 kHz and 1.5 MHz.
The circuit will be powered by 1.8 V.

Here's the information about the piezo sensor:
Sensitivity is not given in the information sheet
Electrical capacitance 0.992 nF
Coupling factor: kp 0.62 ; kt 0.47
Resonant frequency (thickness) 8000 kHz
Resonant frequency (radial) 400 kHz
Outer diameter 5 mm, thickness 0.25 mm

[David Hess]

1) Is it okay to simulate the piezo sensor as an AC voltage source with a capacitor in series (The value of which is equal to the electrical capacitance of the sensor)?

Sure, that works fine.

2) I've done countless variations of values, by trial and error or by using equations, yet I feel like I can't find any values that correctly show the results.

...

R_in = 1/(2*pi*C_sens*f_high) = 107 Ohm
  - I find this value pretty low.
I'm also confused, because other sources establish the gain as charge Q divided by Cf [(1/Cf)], therefore Cf does not depend on the first equation.

The feedback capacitance controls the high frequency gain.  The high value feedback resistor controls the DC gain and something is required to provide the DC bias current into the inverting input; there are other circuit configurations which handle this in a different way.

The input resistance is needed because as frequency increases, the sensor capacitance adds phase lag between the output and inverting input, which will eventually result in oscillation or at least large amounts of high frequency noise.  The input resistance adds phase lead to maintain stability so at high frequencies.  Its value can be derived from a bode plot.  Study operational amplifier differentiators to get a better understanding of why the series resistance is necessary.

[MarkT]

Piezo sensors act more like current sources with capacitance.  In particular they can develop large voltages in some situations into high impedance loads, so you may want to consider protection circuitry, and carefully chose the load impedance.

[jwet]

A charge amp is an unusual circuit.  They are common in photodiode circuits where most writing has been done.

The "gain" of a charge amp is 1/feedback capacitor value.  The R across the C is to provide some DC bias to the amp and discharging the C with a long time constant.  Its often best to think about Q amps in the time domain as a pure integrator with a leaky cap.  What you're doing is taking the input charge (coulombs= I x t) and depositing it on the feedback C, creating an output pulse proportional to the charge deposited.  If you deposit a picocoulomb of charge on the input, the feedback cap will end up with this on it, it will discharge through the 100M.  If the cap was 10 pF, you'd get a tenth of a volt that would be decay with an RC of 1 mS.  With a piezo sensor, you probably want to put pair of diodes clamps on the input.

The capacitance of your source matters too, looking into a charge amp, the noise gain is 1+ Cin/Cf, this also goes up with frequency and gives rise to "noise peaking"- having noise rise at higher frequencies. "Photodiode Amplifiers" by Graeme covers this somewhat.  There is a tradeoff with gain and noise.  I've done a little with charge amps for vibration but mostly photodiodes but this sounds different yet.  There are sparse references for Q amps.  Good Luck.

One of the other oddities is that you'd like to make the input node at the negative amp input as a virtual ground and an "infinite" charge sink (a BIG cap).  The AC input impedance is a strong function of the GBW of the Op-Amp and will often require a much faster op-amp that you would expect.  Any charge desposted that isn't collected instantly is lost- time domain thinking again.

Hope this makes sense.  Challenging circuits.

Good luck.

[mecs]

Well, turns out I was using an OpAmp that wasn't suited for my frequency range. The values I got from the calculations seem to work correctly.
Thanks for all the help and suggestions!