Hello all,
Can anyone please help me on how to approach the following two problems in physics:
1. The resistance R(Θ) of a thermister at temperature Θ (Kelvin) is given by R(Θ) = α e(β/ Θ). Given that the resistance of the thermister at the ice point (Θ = 273.15 Kelvin) is 9 Kohm and the resistance at the steam point (Θ = 373.15 Kelvin) is 0.50 Kohm:
a. Find the resistance at Θ = 298.15 Kelvin
b. Find the standard deviation of the resistance R(Θ) at Θ = 298.15 Kelvin, if the standard deviation of the temperature Θ is 1.0 Kelvin. You may assume that the standard deviation of α is 1*10-2 ohms and the standard deviation of β is 0, and that all random variations are uncorrelated.
2. A temperature measurement system consists of a thermocouple sensor and an amplifier. The characteristic of the thermocouple sensor is E=aT. The characteristic of the amplifier is V = kE+b. Calculate the mean and standard deviation of the output voltage V for the system given the system parameters below:
a = 0.030 σa = 1*10-2
k = 25 σk = 1
b = 2.1 σb = 0
T=5.0 σT = 0
I just want to know how do you go about solving these kind of problems? I would appreciate if one could solve one of these and show me or point out to any good links.
Please Help!
You could use the relationship between range and standard deviation to solve this problem.
see http://stats.stackexchange.com/questions/69575/relationship-between-the-range-and-the-standard-deviation
Once you know how range depends on SD you can work out the range of the R(Theta)
from range of R(theta) work out, SD of R(Theta)
Hi Sourangsu,
Contrary to Loopinder's reply, I am of the opinion that both your queries are of the type that are typically encountered in instrumentation, where standard deviation of elemental quantities result in corresponding standard deviation of measurements.
Both can be solved in terms of small differentials (the task being marginally simple if the differentials are uncorrelated, as you specifically mention for the first query). Let me try out both of them for you.
- Using the R(Θ) data provided, compute the values of α and β.
- Converting R(Θ) = α e(β/ Θ) to logarithmic, we have ln R = ln α + β/ Θ
- In differential form, the logarithmic relation becomes dR/R = dα/α + dβ/Θ - (β/Θ2).dΘ
- Squaring both sides, and converting to expectations of square-of-differentials (while assuming uncorrelated differentials for all variables), we get σR2 = R2. [σα2/α2 + σβ2/Θ2 + (β2/Θ4).σΘ2]
In the second problem, V = kaT+b, so by substitution of values at T = 5.0, you can obtain the mean V. For standard deviation, proceed as follows:
- dV = aT.dK + kT.da + ka.dT + db in differential form.
- Squaring both sides, and converting to expectations of square-of-differentials (while assuming uncorrelated differentials for all variables), σV2 = a2T2.σk2 + k2T2.σa2 + k2a2.σT2 + σb2
Hope that helps ? Those two should clarify the general approach to most such problems !
-Sanjay
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