Sensors and actuators: the error budget decides
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| error/ˈerə/ | The difference between the setpoint and the measured output, which a controller acts on. |
| measurand | The physical quantity a sensor is meant to measure. |
| platinum resistance thermometer | A temperature sensor whose platinum element changes resistance in a known way with temperature. |
| signal chain | The sequence of stages from sensor to digital value, such as amplifier, filter and converter. |
| error budget | A table that adds up each source of measurement error to check that the total meets the requirement. |
| ADC/ˌeɪ diː ˈsiː/ | An analogue-to-digital converter: it samples a voltage and turns it into a number. |
Only as good as the thermometer
- The temperature loop can hold the water only as well as the thermometer reads it. A sensor chain with an unknown error makes the whole design a guess.
- Choose the principle for the measurand (a platinum resistance thermometer, Pt100, for 0–100 °C water), follow the signal chain and write an error budget.
Scaling the chain
- A Pt100 changes 0.385 Ω/K: at 1 mA that is 0.385 mV/K, and a gain of 50 makes 19.25 mV/K.
- A 12-bit ADC over 3.3 V steps 0.806 mV, so one step is $\mathbf{0.042}$ K, with a rounding error of step$/\sqrt{12} = 0.012$ K.
Sensitivity after a gain of 50 on 0.385 mV/K (two decimals, mV/K)?
Độ khó 2/5 · 140 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
50 times 0.385.
Kelvin per step of a 12-bit ADC over 3.3 V (three decimals, K)?
Độ khó 2/5 · 140 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
0.806 mV over 19.25 mV/K.
The error budget at 40 °C
- Sensor class A: $\pm(0.15 + 0.002 \times 40) = \pm\mathbf{0.23}$ K. ADC 0.012 K; gain error 0.04 K; self-heating 0.05 K.
- RSS: $\sqrt{0.23^2 + 0.012^2 + 0.04^2 + 0.05^2} = \pm\mathbf{0.24}$ K.
- The sensor dominates: a 16-bit ADC would change nothing.
Which error term?
Place each number in the error budget or the actuator limits.
Class A tolerance at 40 degrees C (two decimals, K)?
Độ khó 2/5 · 140 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
0.15 + 0.002 x 40.
RSS of 0.23, 0.012, 0.04 and 0.05 K (two decimals, K)?
Độ khó 2/5 · 140 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
Root of the sum of squares.
To improve this budget, first:
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
The sensor term dominates.
The actuator's limits
- Volts set speed: 12 V gives at most 4.78 rad/s at the valve. Amps set torque: 5 A gives 10 N·m, twice the load.
- Power at full speed and load is about 30 W. The motor's 9.6 ms lag is the whole plant for the position loop, but instantaneous for the 2.3-hour tank.
For a motor drive the voltage limit sets the top speed.
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
Through the back emf.
Leads that read as heat
- A two-wire Pt100 measures its leads too. Two 0.2 Ω leads add 0.4 Ω, read as $0.4/0.385 = \mathbf{1.04}$ K: four times the whole error budget.
- A three-wire connection cancels equal leads, and a four-wire one carries the current and senses the voltage on separate pairs, so lead resistance drops out entirely.
Two-wire Pt100 with two 0.2 Ω leads: temperature error (two decimals, K)?
Độ khó 2/5 · 140 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
$0.4/0.385 = 1.04$ K.
A four-wire connection removes the lead resistance from a Pt100 reading.
Độ khó 1/5 · 100 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
Current flows on one pair and the voltage is sensed on the other, which carries almost no current.
Worked example: the self-heating term
- At 40 °C the Pt100 is 115.4 Ω, so 1 mA dissipates $I^2R = 0.115$ mW. In stirred water it sheds 2.5 mW per kelvin of rise.
- The sensor sits $0.115/2.5 = \mathbf{0.046}$ K warm: the 0.05 K in the budget. At 5 mA the power is 25 times larger and so is the error, 1.15 K, which is why the excitation is kept at 1 mA.
Pt100 at 115.4 Ω carrying 1 mA, dissipation 2.5 mW/K: self-heating error (three decimals, K)?
Độ khó 2/5 · 140 xu + phần thưởng nhỏ khi hoàn thành lần đầu tiên hôm nay
$10^{-6} \times 115.4 = 0.1154$ mW; $0.1154/2.5 = 0.046$ K.
Three measurement slips
- Adding independent errors linearly. Use RSS; and only for independent ones.
- Forgetting self-heating and lead resistance. They bias a resistance thermometer.
- Buying resolution the sensor cannot use. The biggest term decides.
Recap
- 19.25 mV/K into a 12-bit ADC: 0.042 K per step.
- RSS budget $\pm0.24$ K, dominated by the sensor.
- Volts limit speed, amps limit torque.