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Original question:

MSP430FR2355: in slaa904 file,How to obtain the formula for R1

CIRCUIT060002: Calculation optimum voltage divider resistor R1 for an NTC

Daniel Ka
Intellectual 627 points
Part Number: CIRCUIT060002

Tool/software:

Hello,

this is a follow-up question to this posting.

An NTC is usually connected to an Op Amp or single-ended ADC through a voltage divider:

In both cases, the resistor R1 of the voltage divider can be chosen in order to optimize the linearity of the output for a given temperature range TMIN to TMAX. The formula is given in the datasheets and uses the respective NTC resistances RMIN (at temperature TMIN)  and RMAX (at temperature TMAX):

R1 = sqrt( RMIN * RMAX)

I wonder how to derive at this formula? The original posting does not answer this question or rather I don't understand the answer.

The original posting refers to a formula of the NTC resistor as below:

It is not explained, why R1 in this formula should be the same as R1 in the circuit.

As you know, the formula of the NTC resistance usually includes a certain resistance Rx and temperature Tx where the NTC is calibrated to, e.g.,  Rx=10k at Tx=25°C, and a BETA value. These three values are usually provided in the specification by the  manufacturer. So the general NTC resistance formula is:

RNTC(T) = Rx * exp(BETA(1/T-1/Tx)

I lack to see why the calibration resistance Rx inevitably corresponds to the resistance R1 of the voltage divider.

However, I know that I can calculate a (geometrical) mean temperature Tm for the range TMIN to TMAX respecting the reciprocal nature of the temperature in the NTC resistance formula:

1/Tm = 1/2 (1/TMIN + 1/TMAX) => Tm = 2* TMIN*TMAX/(TMIN+TMAX)

To calculate the corresponding mean NTC resistance Rm, I use this NTC resistance formula with the specifics for the NTC factored out into constant C:

RNTC(T) = C * exp(BETA/T) with the constant C including the specification of the NTC: C = Rx * exp(-BETA/Tx)

Inserting Tm gives: Rm = RNTC(Tm) = C*exp(1/2 * BETA * (1/TMIN + 1/TMAX))

= C* sqrt(exp(BETA/TMIN)) * sqrt(exp(BETA/TMAX))

=  sqrt( C * exp(BETA/TMIN)) * sqrt( C * exp(BETA/TMAX))

= sqrt (RMIN * RMAX)

So I understand how the mean resistance Rm is calculated but I don't see why the linearity is improved by setting the resistor of the voltage divider R1 to Rm.

Can you please help?

Thanks.
Daniel

  • 0
    TI__Guru**** 165236 points

    Daniel,

    Daniel Ka said:
    So I understand how the mean resistance Rm is calculated but I don't see why the linearity is improved by setting the resistor of the voltage divider R1 to Rm.

    This method for R1 is not for linearity, it gives the highest passive voltage gain (per change in temperature).

    R1 at zero is no gain, R1 at infinity is no gain. R1 at NTC resistance at average temperature is highest gain.

  • 0 in reply to Ron Michallick
    Intellectual 627 points

    Dear  ,

    Thank you for your immediate response. You state that this method for calculating R1 is "not for linearity".

    I'd like to point out that your statement is contrary to what TI's own design specification asserts, see for example SBOA323A from "CIRCUIT060002
    Temperature sensing with NTC thermistor circuit":

    Could you please add you perspective on this apparent contradiction?

    Please note that I understand that the response of RNTC(T) is highly non-linear. So no matter what you do regarding resistor R1 (or any other more complex resistor network), you can only minimize the error of the response compared to a linear approximation over a limited temperature range, but you cannot make the response truly linear.

    Cheers
    Daniel

  • 0 in reply to Daniel Ka
    TI__Guru**** 165236 points

    Daniel.

    Daniel Ka said:
    Please note that I understand that the response of RNTC(T) is highly non-linear.

    Noted. You also understand that a resistor (Bot & Top) voltage divider. Vout = VIN * B / (B+T) has an inverse element control that has very limited ability to correct a highly non linear input. The op amp U1 does a linear transfer Vout = m * Vin +b (a line)

    SBOA323A is a linear in the fact that it isn't digital. The result curves look linear until I use paint to add a pixel line. Clearly green curve and black line diverge.  

    Is this a purely theoretical discussion or is there a design need motivator? 

    In later, I would like to know if NTC in on board or off board? What is the temperature range and accuracy need / would like? Where does the output voltage go / how is it used?

    I can help with a resistance to voltage solution. So I need to know resistance range. You and NTC provider can do the non linearity conversions.