SaM - Calendar Van Dusen Equation

Remeber:

• The Calendar Van Dusen equation simplifies the relationship between the resistance of an RTD to its temperature:$R(T)=R_0\cdot \left[1+AT+B{T}^2+C{T}^3\cdot (T-100°)\right]$Where:
  • $T$ is the temperature expressed in Celsius [°C], and it need to be $T>0$°C
  • $R(T)$ is the resistance of the RTD at temperature $T$,
  • $R_0$​ is the resistance of the RTD at the reference temperature (usually $0$°C).
  • $A$, $B$, and $C$ are coefficients that depend on the specific RTD.
• Also in the previus formula we have considered $T>0$°C, while if $T<0$°C we need to discard the $C$ coefficient:$R(T)=R_0\cdot \left[1+AT+B{T}^2\right]$
• For the PT100 Sensor we have seen that the usual values are:
  • $A=3.9\times 10^{-3}{\text{°C}}^{-1}$
  • $B=-5.8\times 10^{-7}{\text{°C}}^{-2}$
  • $C=-4.2\times 10^{-12}{\text{°C}}^{-4}$
  • NOTE: We can also see the coefficent temperature $A$ as its sensitivity, but if we want a more accurate sensitivity (for example if we want a more precise accuracy at higher temperature $\overline{T}$) we might want to use the following formula:$\alpha (\overline{T})={\frac{R_{0}A+2R_{0}BT}{R_0}|}_{T=\overline{T}}$

• For RTD sensors also we have seen the TCR (Temperature Coefficient of Resistance), we can define as:$\text{TCR}=\frac{R_{100}-R_0}{100°C\cdot R_0}$Where:
  • $R_{100}$ is the resistance the sensor assumes at $100°\text{C}$
  • $R_0$ is the resistance the sensor assumes at $0°\text{C}$

• We have also seen the maximum error for using the Calendar Van Dusen Equation.
  Here’s how to find it:

1. Define the formula $f$: $R=f(T)$.
   Which is the calander van dausen equation, so again:$R(T)=R_0\cdot \left[1+AT+B{T}^2\right]$(We didn’t write the $C$ coefficient since it is really small, so we can ommit it for simplicity)
2. Define the inverse formula $T=f^{-1}(R)$.
3. Find the Maximum Error in the Worst Case Possible:$\Delta T_{WC}=\frac{dT}{dA}\Delta A+\frac{dT}{dB}\Delta B+\frac{dT}{d{R}_0}\Delta R_0+\frac{dT}{dR}\Delta R$Note that:

• $\frac{dT}{dR}\Delta R$ 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.

4. Finally we perform the derivatives and find the formula for the maximum error.
5. To have a proper value we need to know the values $\frac{\Delta R_0}{R_0}$ and $\frac{\Delta A}{A}$, which are the relative variations.

• For the PT100 sensor we know that: $\frac{\Delta R_0}{R_0}=6\cdot 10^{-4}$, or: $R_0=100\text{ohm}\pm 0.06\text{ohm}$.
• Also for Class A Devices the Maximum Error is $0.15°\text{C}+0.002T$.
  So it is important to note that even if a little, it depends on the temperature.

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Memory Card

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Index

• Defintion of the Calendar Van Dusen Relationship
• Linearization
• Errors and Accuracy of RTDs

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Defintion of the Calendar Van Dusen Relationship

- There is an error:$\text{TCR}=\frac{R_{100}-R_0}{100°C\cdot R_0}$

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Linearization

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Errors and Accuracy of RTDs