Microcontroller Solves Complex Temperature Polynomial Equations

To illustrate, we will use the negative-temperature-coefficient (NTC) LM103 thermistor, which has the resistance at three different temperatures shown in Table 1.

With these three measurements, the temperature is converted to the Kelvin scale and substituted into three Steinhart-Hart equations:

Solving these three equations yields the coefficients a, b, and c (Table 2).

Substituting the coefficients in the Steinhart-Hart equation, we get the thermistor equation:

or:

The challenge for the PIC 16F887 microcontroller is to use Equation 4 to get true temperature readings. This project consumes 1227 words of the microcontroller’s memory. Input AN0 is declared as analog, and the rest of the I/O lines are declared digital outputs. The analog-to-digital converter (ADC) is configured for 10 bits, with a sampling time of 50 µs.

The thermistor’s voltage reading is stored in binary format in the variable “volt” with the instruction ADCIN 0, volt. Then, this reading is multiplied by 48,828 to convert it to decimal, and it is stored in a new variable called “v1.” To obtain each decimal digit from variable “v1,” the commands DIG3, DIG2, and DIG1 are used to store these readings in three variables called dig3, dig2, and dig, respectively.

To achieve that, we store in a variable called “dif” the power’s supply voltage minus the thermistor’s voltage. Then we multiply the thermistor’s voltage by the 10-kΩ fixed resistor, and we apply DIV32 to divide this last product by “dif,” resulting in the thermistor’s resistance value.

Fourth, we now compute the base 2 logarithm from the thermistor’s resistance with Equations 7, 8, and 9:

where a₀, a₁, and a₂ are the next-digit mantissa in base 2.

Next, we find the characteristic a₀ or logarithm’s integer with the function NCD, which delivers the most significant bit (MSB) of a number. If we decrement it by 1, we get the base-2 logarithm characteristic. We know, for example, that the base-2 logarithm for number 47 is 5.554 using four significant digits. Thus, NCD (47) = 6, where 47 in binary is 00101111, and the MSB is in the sixth position from left to right. If we decrement it by 1, we get number 5, which is the logarithm’s characteristic.

To get the logarithm, we need the Mantissa (fractions part). To compute it, we need the result of the successive divisions “wu” by 2 (). Nonetheless this operation or process cannot be applied directly, since dividing 47/2 results in 23.5, and we lose the fractions part. (PIC Basic Pro does not work with fractions.) Therefore, to keep the fractions part, we create a special subroutine that performs the following series:

47/2 = 23.5, 23.5/2 = 11.75, 11.75/2 = 5.875, 5.875/2 = 2.9375, 2.9375/2 = 1.46875

When dividing 47/2 = 23, with a remainder of 1, this “1” appears in the first division of the five that will be performed. Therefore, the fraction 1/2⁵ = 0.03125. We cannot work with fractions, though, so instead of dividing by 1 by 2ⁿ, we take the number 10,000 as a numerator, and we create a subroutine that computes the denominator depending on the number of division where appears a remainder of 1.

Then, we sum all the results of the divisions. To continue with this example, we have the following five steps:

• 47/2 = 23, with a remainder of 1; this is the first division of 5, 10,000/2⁵ = 312.5 = 312

• 23/2 = 11, with a remainder of 1; this is the second division of 4, 10,000/2⁴ = 625

• 11/2 = 5, with a remainder of 1; 10,000/2³ =  1250

• 5/2 = 2, with a remainder of 1; 10,000/2² = 2500

• 2/2=1, with a remainder of 0; there is no portion to add

In this case the variable “Ja” is equal to 4687, and it still needs to be incremented by one. Therefore, we divide “ja” by 1000 and then we add up 1000. Therefore, Ja = 1468, obtaining approximately the original quantity mentioned above, multiplied by 1000 with four significant digits.

Once we have obtained M, we need to get the mantissa using the process described in Figure 1. The division by 2^an is not complicated because we are working in base 2, and the division is performed by 2⁰ or 2¹. When raising to the square power, we must account for where we will multiply M_a by M_a, and then we use that division by 1000 with DIV32 because the result is greater than 16 bits.

After this process is performed a required number of times, we need to convert a_n to a decimal base. To achieve this, we multiply by the respective weight, 5, 25, 125, and so on sequentially. And because we cannot work with fractions, we multiply by 5000, 2500, etc.

Finally, we add all the elements to the Mantissa a_n in decimal format and divide the sum of all of them by 10. We sum a₀ × 1000 and we save it in variable “l2,” so “l2” stores the value of the logarithm in base 2 of the thermistor’s impedance. The last step to obtain the natural logarithm is to apply the equation shown below to change its base:

log₂x (ln2) = ln x

We know that ln(2)= 0.6931, so we just need to multiply “l2” by 6931 and then divide it by 10,000 using the DIV32 command to store it in variable “lm,” which is the natural logarithm of the resistor multiplied by 1000.

To drive the common pin of the LCD, a for-next loop enables the output RC2 to be “1” (Fig. 2).

Then the subroutine “D” is invoked to convert the decimal data to a seven-segment code. The LOOKUP function decodes the decimal value of digit3 to a seven-segment format that is stored in variable pattern3. Now the value of RC2 (stored as a byte in variable L) is XORED with pattern3 to drive the LCD in phase.

This process is repeated with the other two digits while a 10-ms pause is added. Then, RC2 is turned off and stored as a byte in variable L. Subroutine D is invoked again and the process is repeated with the phase output (RC2) changed for another 10 ms. After every LOOKUP function, every pin is assigned to the corresponding port’s pin dedicated to drive each LCD segment. Total time required is 40 ms (Fig. 3).

Reference

Computing Logarithms Digit-by-Digit, Mayer Goldberg, BRICS RS-04-17. ISSN: 0909-0878