By Denny Wong | Dec 17, 2013
Previously we discussed how accuracy and precision can mean different things depending on the context and definition that you’re using. Here’s an example to show how you might encounter these terms in an electronic product design, and how to ensure your design meets these specifications.
Measuring Ambient Temperature
Recently we were involved in a project where a device was required to measure the ambient temperature to an accuracy of ±0.3°C in the range 5°C to 60°C. After reading the previous post you’ll understand this means that the measurement is to have a combined trueness, reproducibility, and a numeric representation with a resolution of ±0.3°C or better. The circuit below shows a NTC thermistor and Analog-to-Digital converter (A/D) used to measure the ambient temperature.
The voltge at Vin and its relationship to the A/D counts is given by the equation:
(1)
Where RT is the resistance of the thermistor, RB is the resistance of the bias resistor, Vref is the thermistor bias voltage and A/D reference voltage, N is the A/D number of bits, and ADC_CNTS is the A/D count value. Solving equation (1) for RT yields:
(2)
From the thermistor data sheet, the thermistor resistance has the following relationship with temperature:
(3)
Where RT is the thermistor resistance, R0 is the thermistor resistance at 25°C, B is a constant, T0 is equal to 298.15K, and T is the ambient temperature in Kelvin. Equating equations (2) and (3) and solving for T yields:
(4)
Equation (4) is the closed form transfer function from A/D counts to ambient temperature. The A/D employed in the implementation is one that is integrated into a microcontroller (we used a Texas Instruments Mixed Signal Microcontroller).
Parameter values for the circuit:
| Bias resistor (RB) | 10,000Ω ±0.5% |
| Thermistor resistance @ 25°C (R0) | 10,000Ω ±0.5% |
| B constant | 3380K ±0.7% |
| Thermistor Dissipation Constant @ 25°C | 1mW/°C |
| A/D number of bits (N) | 10 bits |
| A/D Total Unadjusted Error | ±2 LSBs |
Worst Case Resolution
The worst case resolution for this response occurs at the high end of the temperature range, i.e. at 60°C ambient temperature. A/D count values of 239 and 240, i.e. one A/D count change, yields temperatures of 59.900°C and 59.721°C respectively from equation (4) and is a 0.18°C change. This resolution satisfies the requirement to measure to ±0.3°C but higher resolution is recommended.
The microcontroller A/D has a total unadjusted error specification of ±2 LSBs or ±3 A/D counts. This is a comprehensive error specification and includes integral and differential nonlinearity errors, gain and offset errors. The worst case error occurs at the lower end of the temperature range, i.e. at 5°C. For the A/D count values of 710 and 707, i.e. -3 counts of error, equation (4) yields temperatures of 4.910°C and 5.226°C respectively, and is a +0.32°C error. Adding the worst case component tolerances to equation (4) for the thermistor resistance, B-constant, and bias resistor, a temperature of 5.582°C is computed and is a +0.67°C error. This amount of error does not satisfy the accuracy requirement of ±0.3°C and calibration is required to meet the required trueness. The thermistor dissipation constant is 1mW/°C @ 25°C and thermistor self-heating can add an additional 0.23°C error.
The noise and distortion data for this microcontroller A/D is not published, so the precision cannot be determined by analysis. The noise, distortion, and resulting circuit precision can be determined during validation testing. The thermal noise voltage, en, is given by
(5)
In Conclusion
Where k is Boltzmann’s constant, T is temperature in Kelvin, R is the resistance, and Δf is the noise bandwidth. At 297.15K and a bandwidth of 5Hz, the thermal noise from the thermistor and bias resistor is 40.6nV and is not a significant source of variability.
It is worth noting to satisfy the accuracy requirements, the combination of the maximum trueness error and the precision component are required. Trueness may be improved by calibration and precision may be improved by filtering and oversampling techniques.
To summarize what we’ve explained: trueness is the closeness to the true value, and precision is the repeatability of a measurement or reproducibility from a system (and also the meaning of resolution when representing a quantity). Accuracy refers to both trueness and precision. For colloquial use, accuracy and precision are often used interchangeably, but as an electronics designer don’t let that throw you off.
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