Note on Computing Temperature from a Voltage Divider or Bridge Measurement

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 ^oK

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.

That means that we have to solve the equation above for the value of the sensor resistance.

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_aV_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 ^oK.

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.

Problem 1

In the voltage divider circuit, you have the following:

We know the input voltage, V_in is 15 volts;
Resistance, R_a. = 10,000W
The output voltage, V_out = 5 volts.

Determine the temperature in ^oC. (And please note the units!)

Question

Q1 The output voltage changes to 6 volts. Has the temperature gone up or gone down?

Problem 2

Determine the temperature in ^oC when the output voltage rises to 6 volts. When you input your answer use only one figure after the decimal point.