Determining
Temperature From A Thermistor Voltage Divider Measurement
If you use a thermistor in a voltage divider, the output voltage depends upon
the temperature. In the
lesson on thermistor temperature sensors, the resistance of a thermistor is
given by:
1/T = A + B*ln(R) +
C*(ln(R))^{3} R in W,
T in ^{o}K
Now, the thermistor appears
in a voltage divider. That voltage divider might be half of a bridge
circuit. In any event, the circuit looks like a typical voltage divider 
like the one shown below.
We have an expression for
the output voltage of the voltage divider  given below.
V_{out}
= V_{in} R_{b}/( R_{a} + R_{b})
In this circuit, we have the
following:

The sensor is R_{a}
 and the value of the sensor resistance is temperature dependent and unknown.

We know the input voltage, V_{in}.

We know the resistance, R_{b}.

We measure the output voltage,
V_{out}.
Given a voltage measurement for the output voltage, we need to know how to
calculate the temperature. We do that in two steps.

First we find the value of the
resistance of the sensor.

Secondly, once the sensor
resistance is computed, we need to compute the temperature.
First, we solve for the value of the resistance using the expression above.
V_{out}
= V_{in} R_{b}/( R_{a} + R_{b})
Multiply through by the
denominator of the right hand side, to obtain:
( R_{a}
+ R_{b})V_{out} = V_{in} R_{b}
Then, isolate the terms with
the sensor resistance:
R_{a}V_{out}
= R_{b}(V_{in}  V_{out})
Then, solve for the sensor
resistance:
R_{a}
= (R_{a})(V_{in}  V_{out})/(V_{out})
Once you have the sensor
resistance, then you can use the first expression above to calculate the
temperature. That calculation yields the following expression for the
measured temperature. (And, don't forget that this is the absolute
temperature in ^{o}K.
T_{meas}
= 1/[A + B*ln(R_{a}) + C*(ln(R_{a}))^{3}]
Now, you could insert the
expression for the sensor resistance into the expression for the measured
temperature. However, when programming this calculation, you should
probably first calculate the sensor resistance, and then have second calculation
to calculate the temperature from the calculated resistance. Although that
is two calculations, that may well be the simplest way to implement the
calculations above.
