1LimitationsofresistordividersforRTDmeasurement(orwhytheysuck)
ConsideracircuitwithRTDinlowerlegofresistordivider
Weareinterestedinmaximizingchangein
maximizethepartialderivative
theboundary.ForPT1000,
Since R2 isinthedenominatorofthefunctionsabove,highervaluesof R2 (warmer temperatures)willyieldsmallervaluesof ∂Vout ∂R2 (thereforesmallerprecision).Clearly, precisionischangingwithtemperatureanditisgettingworseforhighertemperatures.Wewanttoknowwhatmaybethebestattainableprecision,sowetryto maximizethefirstfunctionandfindvalueof
:
Vout(R1 ,R2 )= R2 R1 + R2 × Vin (1)
Vout
∂Vout ∂R2 (R1 ,R2 )= R1 (R1 + R2 )2 (2)
perchangein R2 .Wewantto
800Ω ≤ R2 ≤ 1500Ω (3)
∂Vout ∂R2 tothetwoendpoints,eachisafunction
1
∂Vout ∂R2 (R1 )morebetter = R1 (R1 +800)2 (4) Or ∂Vout ∂R2 (R1 )worse = R1 (R1 +1500)2 (5)
Thisfunctionhas nolocalmaximaorminima,sowemustlookformaximaat
Thisreducesoursearchofmaximum
of R
:
R1 thatyieldsmaximum ∂V
∂R2
d dR1 R1 (R1 +800)2 =0 (6) 800 R1 (R1 +800)3 =0 R1 =800 (7) 1
out
Sothemaximumsensitivitypossibleisat 50 ◦ C with R1 =800Ω: ∂Vout ∂R2 (800)= R1 (R1 +800)2 =0.0003125V/Ω (8) Ifwechoosetocenteraround 0 ◦ C,thefunctionbecomes: ∂Vout ∂R2 (1000)= R1 (R1 +1000)2 =0.00025V/Ω=0.25mV/Ω (9) Itgetsworseafterthat. Translatingitto mV/°C: (1573.30 803.10) 150 ( 50) =3 851Ω/◦ C (10) 0.25mV/Ω × 3.81Ω/◦ C=0.96275mV/◦ C (11) Plugginintooriginalequation Vout(R2 )= R2 1000+ R2 × Vin (12) At 0 ◦ C with Vin =5V Vout(1000)= 1000 1000+1000 × 5V=2 5V (13) At 150 ◦ C Vout(1573.30)= 1573.30 1573 30+1000 × 5V=3.05697V (14) At 50 ◦ C Vout(803.10)= 803 10 803 10+1000 × 5V=2.22698V (15) 2