Using an analog filter to inject noise
Changing your circuit design by reducing your filter's resistor values will increase the noiseless bits in the circuit.
By Bonnie Baker -- EDN, July 23, 2009
Sometimes things just don’t make sense! For instance, your RC filter or amplifier’s lowpass filter at the input of a delta-sigma ADC can produce a noisier digital output. Didn’t you design the filter to reduce noise so that you’d get more instead of fewer noiseless bits from your converter? It is as easy to eliminate higher-frequency noise with an analog lowpass filter as it is to inject noise into the frequency band below the corner frequency of your filter. If your filter produces noise in the frequency band of interest, your conversion output results will be noisier than you might expect.
If you change your circuit design by reducing your filter’s resistor values, you will increase the noiseless bits in the circuit. For example, the delta-sigma ADC in Figure 1 uses a lowpass filter to reduce noise above the converter’s output data rate, FD. With this filter, use the output data rate of the delta-sigma converter to select the resistor and capacitor values in this circuit. You can use the formula FD=1/(2πRFLT×CFLT) to calculate the values of RFLT (filter resistance) and CFLT (filter capacitance). This filter reduces noise by targeting the sampling frequency of the converter as the combination of RFLT/2 and CFLT goes to work (Reference 1).
The missing detail in this design formula is resistor noise. There is no such thing as a noiseless resistor. The ideal resistor noise is 
where k is Boltzmann’s constant (1.38×
10–23×JK–1), T is the temperature in Kelvin, R is the nominal resistance in ohms at 25°C, and B is the bandwidth of interest in hertz.
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Now, let’s make sense of this resistor-noise formula. The noise that you inject into your circuit up to the output data rate is equal to the resistor noise. To determine the maximum allowable resistance value in this circuit, use the following equation:
where ER is the specified effective resolution from the ADC manufacturer’s data sheet. Figure 1 illustrates the characteristics of this formula. If you operate your 23-bit-effective-resolution delta-sigma converter at a data rate of 200 Hz, the maximum value of the filter’s resistance is 4.297 kΩ or less, and RFLT/2 is 2.148 kΩ or less.
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Bonnie,
I see Martin's point but I don't understand your response. Even if you take his 20pF analysis as the reset noise, the external RC thermal noise should follow the same formulas, shouldn't they? Can you show experimental data and also perhaps go deeper in your explanations?
Wayne - 2009-12-12 01:22:00 PST -
Thank you Bonnie, that was a very interesting article. I always enjoy reading your columns.
yerpa58 - 2009-23-11 17:34:00 PST -
So I thought about this for a while, and I believe the problem with the analysis is that the band-limiting of the delta sigma ADC's digital filter occurs AFTER the sampling process. At the ADC's front end, there is (generally and simplistically) a switch that samples the input onto a capacitor. No different from any other direct-sampling ADC (SAR, pipeline). If you consider the voltage on the sample cap after each discrete sample, the RMS noise of a number of samples is SQRT(KT/C). So any ADC with a 10pF sample cap will have about 20uV RMS noise, regardless of the resistance at the input, and assuming no other noise sources. (check the math - silly Windows calculator!!) This shows up directly in the output of a SAR or pipeline ADC, but will be reduced greatly in a delta sigma ADC. Is there a particular ADC you were considering?
Martin - 2009-7-8 08:39:00 PDT -
Martin,
I can see where you are coming from, however at the frequency range below the output data rate of the converter, the noise from the resistor will go straight into the converter as if it were part of the signal.
Bonnie Baker - 2009-4-8 15:31:00 PDT -
Isn't the thermal noise of an ADC determined only by the sample capacitor value - SQRT(kT/C)? Resistance drops out of the equation. Large resistors in series with the input make no difference unless the input to the ADC has an active buffer or unless other effects surface - incomplete settling during acquisition or changing leakage current masquerading as voltage noise.
Martin - 2009-4-8 12:56:00 PDT
