SaM - MEMORIZE (RTD Sensor)

List of things to memorize:

SaM - RTD Sensor

• SaM - Definition of RTD Sensors • Resistive Temperatrure Detector Sensors • TCR (Temperature Coefficient of Resitance)
• SaM - Calendar Van Dusen Equation
• SaM ~ Real World Example • Standard RTD Sensor • PT100 Sensor
• SaM - Accuracy of the Complete Measurement System
• SaM - Remove the Offset in the Measurement System
• SaM - RTD Example • Linearized Uncertainty
• SaM ~ Solution Based on a Resistive Bridge and an RTD (Skipped)

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SaM - Definition of RTD Sensors • Resistive Temperatrure Detector Sensors • TCR (Temperature Coefficient of Resitance)

• RTD (Resistance Temperature Detectors)
• TCR (Temperature Coefficient of Resistance) formula:${\alpha }_R=\frac{dR}{dT}\frac{1}{R}$
  • PTC: Positive Temperature Coefficent (${\alpha }_R>0$).
  • NTC: Negative Temperature Coefficent (${\alpha }_R<0$).
• TCR Definition for RTDs:$\text{TCR}=\frac{R_{100}-R_0}{100°C\cdot R_0}$
• Real World Measure:
  • The usual sensitivity of these sensors is: $\alpha \simeq +10^{-3}°{\text{C}}^{-1}$
  • We have seen two tyes of RTDs:
    • Thin Films:
      
    • Wirewound:
      
• Terminology:
  • $R_{100}$ is the resistance the sensor assumes at $100°\text{C}$.
  • $R_0$ is the resistance the sensor assumes at $0°\text{C}$.

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SaM - Calendar Van Dusen Equation

• Formula:$R(T)=R_0\cdot \left[1+AT+B{T}^2+C{T}^3\cdot (T-100°)\right]$
• Simplified formula:$R(T)=R_0\cdot \left[1+AT+B{T}^2\right]$
• More accurate sensitivity:$\alpha (\overline{T})={\frac{R_{0}A+2R_{0}BT}{R_0}|}_{T=\overline{T}}$
• Find the maximum error for using the Calendar Van Dusen Equation:
  1. Define the formula $f$: $R=f(T)$:$R(T)=R_0\cdot \left[1+AT+B{T}^2\right]$
  2. Find the inverse formula $T=f^{-1}(R)$.
  3. Find the maximum error in the Worst Case (WC) possible:$\Delta T_{WC}=\frac{dT}{dA}\Delta A+\frac{dT}{dB}\Delta B+\frac{dT}{d{R}_0}\Delta R_0+\frac{dT}{dR}\Delta R$Where:
     • $\frac{dT}{dR}\Delta R$ is the maximum measurement error and depends on the circuit used, not just on the sensor, so we forget about it.
     • $\Delta A,\dots ,\Delta R$ are the maxium possible variations, since we took the worst case possible.
• Real World Measure:
  • For the PT100 sensor:
    • $A=3.9\times 10^{-3}{\text{°C}}^{-1}$
    • $B=-5.8\times 10^{-7}{\text{°C}}^{-2}$
    • $C=-4.2\times 10^{-12}{\text{°C}}^{-4}$
    • $\frac{\Delta R_0}{R_0}=6\cdot 10^{-4}$, or: $R_0=100\text{ohm}\pm 0.06\text{ohm}$.
  • For “Class A Devices” the maximum error is $0.15°\text{C}+0.002T$.

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SaM ~ Real World Example • Standard RTD Sensor • PT100 Sensor

• This RTD sensor is made of platinum, it is a standard material very stable, reproducible and resist to oxidation
• It is called PT100 since:
  • It is made of platinum, “PT” stands for Platinum.
  • Its nominal resistance $(R_0)$ value assumed at $0°\text{C}$ is $100\text{ohm}$.
• Calendar Van Dusen coefficients:
  • $A=3.9\times 10^{-3}{\text{°C}}^{-1}$
  • $B=-5.8\times 10^{-7}{\text{°C}}^{-2}$
  • $C=-4.2\times 10^{-12}{\text{°C}}^{-4}$
• Self-heating effect: $\simeq 0.5\frac{\text{°C}}{\text{mW}}$.
• Usual temperature range of these sensors: $[-200°\text{C}÷850°\text{C}]$.
  ⇒ Usual resistance value range: $[22\text{ohm}÷431\text{ohm}]$.

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SaM - Accuracy of the Complete Measurement System

• Complete Measurement system:
  
• Simple example using the PT100 sensor:
  
• Accuracy of the complete system:$\alpha \triangleq \frac{d{V}_A}{dT}=I_{DC}\cdot A_0\cdot A\cdot R_0$Where:
  • $R_0$ : nominal resistance of the PT100 sensor.
  • $A$ : Calendar Van Dusen coefficient.
  • $A_0$ : open loop gain of the operational amplifier.
  • $I_{DC}$ : current source.
    A higher current improves the SNR (Signal to Noise Ratio), however it can create a self-heating problem.

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SaM - Remove the Offset in the Measurement System

• Linear sensor (~Ex.: RTD) equation, taken within $T\in \left[T_{MIN}÷T_{MAX}\right]$:
  
• Simplest read-out electronics:
  
• Output formula:$V_0=V_{DC}\frac{R_{REF}}{R(T)+R_{REF}}$
• Output graph:
  
• Formulas:$\begin{array}{l}{V}_{0_{MIN}}=V_{DC}\frac{R_{REF}}{R_{MAX}+R_{REF}}\\ V_{0_{MAX}}=V_{DC}\frac{R_{REF}}{R_{MIN}+R_{REF}}\\ V_{REF}=V_{DC}\frac{R_{REF}}{R(T_{REF})+R_{REF}}\end{array}$Where:
  • $T_{REF}$ is taken as $\frac{T_{MAX}-T_{MIN}}{2}$.
  • $V_{REF}$ can be seen as the offset of the system.
  • While $\left[V_{0_{MIN}},V_{0_{MAX}}\right]$ is its dynamic range.
  • The information given by the sensor lies all in this small range that we will call $\Delta V$.
• Problem: we don’t fully utlize the range given by the acquisition system (A/D).
  Solution: compensate the offset, so bring the output signal to have a $V_{REF}=0$:
  
• Using a balanced bridge:
  
• Problem: we might need to consider the wires resistances.
• Problem: we have corrected the offset, but we cannot correct the drift, which is usually taken as:$V_{\text{drift}}=\frac{1}{10}{V}_{\text{offset}}(@50°\text{C})$

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SaM - RTD Example • Linearized Uncertainty

• Circuit:
  
• Output formula:$V_{RTD}=A_0\left[\left(V_{REF}-V_{io}-\frac{V_{out}}{G_0}\right)\frac{R(T)}{R_{REF}}-V_{id}\right]$
• The complete formula for this RTD circuit depends on:
  1. $V_{REF}$
  2. $V_{io}$
  3. $V_{id}$
  4. $V_{out}$
  5. $V_{RTD}$
  6. $R_0$
  7. $A$
• Uncertanty formula:$u(T)=\left|\frac{\partial T}{\partial V_{REF}}\right|u(V_{REF})+\left|\frac{\partial T}{\partial V_{io}}\right|u(V_{io})+\dots +\left|\frac{\partial T}{\partial A}\right|u(A)$
• Uncertanty simplified:$u(T)\le \frac{R_{REF}}{R_0{A}_{0}A{V}_{REF}}\cdot 10\mu \text{V}$
• Terminology:
  • $A_0$ : open loop gain of the differential amplifier.
  • $V_{io}$ : voltage input offset.
  • $V_{out}$ : output of the operational amplifier $(G_0)$.
  • $G_0$ : open loop gain of the operational amplifier.
  • $R(T)$ : RTD sensor.
  • $u(V_{REF})$ : “drift of the reference voltage”.
  • $u(V_{io})$ : “drift offset” for the PT100 it is about $\simeq 10\mu \text{V}$.IMPORTANTE
  • $u(V_{id})$ : “drift offset”.
  • $u(V_{RTD})$ or $V_{LSB}$ : “Voltage at the Least Significant Bit”.

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SaM ~ Solution Based on a Resistive Bridge and an RTD

(Skipped)