sensitivity and frequency-stability of capacitor-based oscillator circuit

I am constructing a capacitive transducer which changes capacitance as a function of changes in other physical properties. The capacitance is about 0.1~0.5 nF range, and I want to detect about 0.05% change in its value. Due to some other constraints, I need to detect this change by shift in its resonant frequency such as an LC resonator. Another constraint is that there is only one unknown capacitor, so any circuit that depends on closely-matched capacitors would not work.

My question is of the different tunable oscillator circuits that utilitizes capacitors to select oscillation frequency, which configuration(s) have the best frequency stability when the unknown capacitance does not change, and what limits their frequency stabilities?

And which circuit has the best sensitivity to capacitance change, i.e. largest df/dC (assuming other components do not change)?

• You want to detect changes in value? In other words, static long term drift isn't a problem, yes? How farg is the capacitive transducer to be located from its electronics and, what temperature might the "probe" be subject to? Commented Mar 26, 2022 at 9:00
• @Andyaka: Each measurement is done in less than an hour, and periodic calibration can be done. So long-term drift is not a problem. The transducer is separated from the electronics by stable and static coaxial cables of a few meters long, which will also add to the total capacitance. The probe is located in a cryogenic fridge with good temperature stability. Commented Mar 26, 2022 at 15:47

An LC oscillator is inherently less sensitive than an RC oscillator, due to the square root term. However this difference may be completely reversed by the difference in noise or drift performance between the two types.

The easiest way to make an LC oscillator that's insensitive to capacitor ratios is to make one with only one capacitor, a Hartley oscillator. Here's one I was playing with in LTSpice. I can post the .asc file if you don't want the work of re-entering it.

C1 is the transducer capacitor, all other capacitors are just 'large'. D1 is for clamping the amplitude to be very small, though there are other methods. This is obviously not a finished circuit, but it does oscillate and will give you a leg up if you want to investigate how a Hartley performs.

RC oscillators come in two types, relaxation and sine-wave. The former have more or less linear variation of frequency with capacitance, but are very noisy, due to voltage noise on the wideband input of the comparator. This can be mitigated with a clamped low bandwidth low noise amplifier before the comparator, in a method advocated by Oliver Collins.

Sine wave RC oscillators like Wein or state variable go back to lower capacitance to voltage sensitivity due to needing multiple timing capacitors. They are quieter than relaxation oscillators.

All of these oscillators will show a voltage and a temperature dependence. This can be mitigated by making the detector capacitance switchable between a low tempco reference capacitor (plastic film or NP0), and measuring the difference between calibration and detector capacitance. Obviously the switch to do this must have a very stable stray capacitance, which is ideally somewhat lower than the capacitors being measured.

An L.C oscillator is not the best as the frequency only changes as sqrt(C).
A 555-type oscillator can be quite stable, but you probably need a regulated supply (e.g. 5.0 V), not driven directly from a (variable) battery.

Alternatively, a CMOS oscillator using a chain of CD4009UB inverters, with the C across two of them (positive feedback), and a 3rd driven from the output and driving a R back to the input is very stable over temperature and voltage. It also has the advantage that it can be quite immune to parasitic capacitance at either end of the capacitor being measured.

What operating temperature range do you need ? Will you use a crystal-based timebase to measure the frequency ?

Here is a simple oscillator -- For the comparator #12, you can use 2 CMOS inverters in series (need to be the 'unbuffered' type), for buffer #20, use another 1 or 2 CMOS inverters/buffers. For the current source(s), you can use a resistor with some limitations on stability -- basically connect a resistor from the output of buffer #20 to node #13.

This type of oscillator is very insensitive to parasitic capacitances (e.g. from long wires etc.) around the capacitance you are trying to measure.

• I am thinking about using an oscilloscope to measure the frequency as a start, so yes, it's a crystal-based timebase. The operating temperature of the electronics is just room temperature. Can you point me to more resources about the CMOS oscillator, such as schematics, etc? I assume it will generate square waves? Commented Mar 26, 2022 at 15:48

Due to tolerance on parts 0.05% is better detected with more accurate parts such as a crystal. Using 500 ppm you can construct a VCXO and PLL to convert the change to voltage. But instead of a voltage control Varicap, you have a gap controlled capacitance.

If you need more design details, I can refer you to other answers.

Due to some other constraints, I need to detect this change by shift in its resonant frequency such as an LC resonator.

That's awfully specific. I don't think there's any reason to insist on doing it only that way. I'm sure integrating the capacitor into an explicit oscillator circuit will work, but it's not the only method possible. If you can drive an LC tank, you can do pretty much anything else as well.

One general approach that works well and is widely used is RC-to-digital conversion.

There's always a variable capacitor or resistor being measured, against, respectively a fixed resistor or a fixed capacitor.

The charge and discharge of the capacitor is timed, usually with picosecond effective resolution, and the RC constant is determined, and thus the value of the variable resistance or capacitance against the fixed complementary reference.

The RC-to-digital conversion can be done in two fashions:

• discretely, by using a time-to-digital converter that times discrete events, such as output transitions of an analog comparator that compares the capacitor voltage against a reference,

• continuously, by measuring the charge-discharge waveform using an ADC, and then fitting the coefficients of a model charge-discharge curve; the curve can be a simple exponential, or a more complex one based on the non-ideal behavior of the resistor, capacitor and the A/D converter itself.

• in a hybrid of the above, by measuring the charge/discharge waveform only in the vicinity of the reference voltage, and using this short time series of A/D samples to more accurately determine the exact time of the crossing of the threshold. This is typically used in high resolution multislope A/D converters, with resolutions up to 8.5 decimal digits (!).

The capacitor charging can be achieved:

• through a resistor from a fixed reference voltage, with variable charge current, or

• using a voltage reference and a fixed resistor to convert voltage into a constant current.

The current source and the capacitor form an integrator.

A variety of techniques can be used to cancel out the errors caused by capacitor's dielectric absorption and charge-to-voltage nonlinearity, as well as linear gain and offset errors. These are quite well represented in the "roll-your-own" high precision voltmeters from HP, Solartron, etc.

There's a multitude of discrete time-to-digital converters, from big chip design houses like TI (TDC7200 and TDC7201), as well as smaller, more specialized parts borne from ACAM's technology, later acquired by AMS, offered by https://www.pmt-fl.com/, e.g. TDC-GP family, or the PICOSTRAIN family.

The continuous time-to-digital converters are basically precision analog data acquisition systems with all the magic done at the digital end of things. The quality of the statistical models used in interpreting the signal has major impact on the resolution and accuracy of the conversion. Such approaches are relatively easy to simulate numerically and thus evaluate before putting together any hardware.