Errors in Thermistor Thermometers

The most significant errors affecting thermistor thermometers are:

Errors in Thermistor Thermometers due to Self-heating

In order to measure a resistance, a current must be passed through the thermistor, which leads to Joule heating and a small temperature increase called the self-heating error. The error, ∆T, is proportional to the power dissipated and the thermal resistance between the thermistor and its environment:

Equation 1

Where I and V are the sensing current and the voltage across the thermistor, respectively ρ is the thermal resistance between the thermistor and its surroundings.

The thermal resistance depends in part on the dimensions of the thermistor and the material from which it is made, as well as the thermal conductivity and, if fluid, the velocity and viscosity of the medium in which the thermistor is immersed. Self-heating can be a serious problem for measurements made over a wide temperature range.

If a constant sensing current is used, the power dissipated (I2R) at low temperatures becomes large, and if constant voltage excitation is used, the power at high temperatures (V2/R) becomes a problem. The thermal resistance is often expressed as the dissipation constant, which is the reciprocal of the thermal resistance.

The dissipation constant measures the power required to raise the thermistor temperature 1 °C and is usually reported in units of mW/°C. Typical values of the dissipation constants range from 0.05 to 20 mW/°C in stirred fluids, but are very dependent on the environment so may vary as much as 10–100 times between still air and stirred water. For a 10 kΩ thermistor with a dissipation constant of 10 mW/°C driven with a sensing current of 100 μA, the self-heating error is 0.01 °C.

In many applications, the thermal resistance may fluctuate due to turbulence in the air or stirred fluids. This causes the self-heating error to fluctuate and can limit the resolution of thermistors, when used, for example, in precision temperature controllers or differential thermometers.

Errors in Thermistor Thermometers due to Stray Thermal Influences

The generally high thermal resistance between the thermistor and the environment means that thermistors are prone to stray thermal influences from either infrared radiation or heat conducted along the lead wires to the thermistor.

This effect may be exacerbated by the poor thermal design of many commercial thermistor probes. The “stem conduction” or“immersion errors” tend to fall exponentially with immersion depth, and a useful rule of thumb is that the error is reduced by a factor of 100 for each 10 diameters of immersion.

For example, to measure temperatures near 120 °C (100 °C different from ambient temperature) with accuracy of 0.01 °C, a stainless-steel sheathed probe may need to be immersed as much as 20 diameters.

Stray thermal effects are a particular problem when thermistors, and other contact thermometers, are used to measure air and surface temperatures. For air temperature measurements, the leads should be long and thermally anchored to an object at a temperature as close as practical to that of the air, so that the leads adjacent to the thermistor also come to thermal equilibrium with the air and do not provide a source of heating or cooling.

Polished-metal cylinders can be used asradiation shields to reduce the influence of infrared radiation sources such as incandescent lamps, sunlight, warm electronics, heaters, and people.

With surface temperatures, a sufficient length of the leads must be thermally anchored to the surface to ensure that the thermistor is at the surface temperature. Radiation shielding is often required.

Errors in Thermistor Thermometers due to Settling Time

One of the major advantages of thermistors is their small size and fast response time. The responses times are normally characterized by the 1/e time constant, which is the constant, τ in the model of the temperature error, ∆T, due to a step change in temperature:

Equation 2

Where t is the time following the step change ∆T step is the step change to which the thermistor is responding.

This model suggests that the error is reduced by a factor of 100 for each five time constants. For example, a thermistor with a time constant of 100 ms will have settled to within 1% of a step change within 500 ms.

The time constants for thermistors range from about 20 ms for ultra-small beads to more than 200 s for large disk types and are very strongly correlated with size and thermal resistance.

Bead thermistors with environmental protection such as glass or epoxy coatings, or a stainless steel sheath, typically have time constants in the range of 1 – 20 s.

Often, larger and more complex probes can exhibit settling behavior that is more complex than that suggested by Equation 2 and may be characterized by two or more time constants. For these probes, the settling behavior may be more usefully characterized by the 10%–90% settling time, and many industrial and military applications use such specifications.

As with the dissipation constants, the time constants are very strongly dependent on the medium into which the thermistors are immersed. A thermistor in a well-stirred low-viscosity fluid may be 10 – 100 times faster than the same thermistor in still air.

Errors in Thermistor Thermometers due to Stability

There are several effects causing instability in thermistors, including mechanical cracking of the thermistor body with temperature cycling, drift at high temperatures due to ingress of atmospheric gases, changes in the atomic and crystallographic structure, and changing contact resistance between the leads and the thermistor body. The most stable thermistors are bead thermistors encapsulated in glass, closely followed by epoxy encapsulated beads.

Within the range −20 °C to 60 °C, selected and pre-aged thermistors are often stable to better than a few tenths of a millikelvin per year, and thermistors used as high as 120 °C can be stable to a few millikelvin.

Thermistors with R(25 °C) resistances in the range 2 – 10 kΩ appear to be the most stable.

Errors in Thermistor Thermometers due to Resistor Stability

Fig. 1

A thermistor-based thermometer is only as stable as the other components in the instrument. The effect of the instability of the other resistors in Figure 1, for example, can be inferred from the zero-output condition, so that the apparent change in the thermistor resistance due to changes in the other three resistors is

Equation 3

and hence, the apparent change in measured temperature caused by changes in the resistances is

Equation 4

The effects of resistor drift on the observed temperature can be estimated using this equation. If the changes in the resistors are due to changes in the ambient temperature, Ta, then the apparent temperature change is

Equation 5

Where αr , αRf , and αRin are the temperature coefficients of the three resistors.

Equation 5 shows that the effects of ambient temperature are reduced by the ratio of the temperature coefficients.

Good-quality resistors have temperature coefficients of 0.0005%/°C or less, being about 1/10,000 that of thermistors. Hence a 10 °C change in the temperature of the circuit will cause an apparent change of about 1 mK in the measured temperature.

Errors in Thermistor Thermometers due to Lead Resistances

The high sensitivity and high resistance of thermistors mean that for many measurements, a two-wire resistance measurement is satisfactory and provides a useful simplification.

However, the lead resistances are often forgotten and can become a problem at higher temperatures when the thermistor resistance is low. The error due to a lead resistance, RL, is

Equation 6

The error increases very rapidly with temperature due to the falling resistance of the thermistor and the T2 temperature dependence.

For a 10 kΩ thermistor with a β value of 3600 K and a 1 Ω lead resistance, the error at 25 °C (∼300 K) is 2.5 mK, while at 50 °C, the error increases to 7.4 mK. Where lead resistances are a problem, a four-wire measurement should be used.

Errors in Thermistor Thermometers due to Offset Voltages

Typically, a thermistor resistance measurement involves a voltage measurement, the accuracy of which may be limited by the resolution of a voltmeter, or the input-offset voltage and input bias currents of an operational amplifier, and/or thermoelectric effects.

The error in a temperature measurement due to an error, ∆V, in the voltage measurement is

Equation 7

Where, I0 is the sensing current.

For a 10 kΩ thermistor with a β value of 3600 K, a 100 μV error, and a sensing current of 100 μA, the error at 25 °C (∼300 K) is 2.5 mK.

Errors in Thermistor Thermometers due to Insulation Resistance

At low temperatures, the thermistor resistance becomes very large, often greater than 10 MΩ. In such applications, care should be given to the insulation on the connecting leads to ensure that the insulation resistance does not shunt the measuring current. Some pigmented plastics can be very poor in this respect. The temperature error due to an insulation resistance Rins shunting the thermistor is

errors in thermistor thermometers
Equation 8

Which increases in proportion to the thermistor’s resistance (low temperatures). A 10 kΩ thermistor shunted by 10 MΩ will exhibit an error of 25 mK.

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