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# Thermistor Study Essay

A thermistor is a type of resistor whose resistance varies significantly with temperature, more so than in standard resistors. The word is a portmanteau of thermal and resistor. Thermistors are widely used as inrush current limiters, temperature sensors, self-resetting overcurrent protectors, and self-regulating heating elements.

Thermistors differ from resistance temperature detectors (RTD) in that the material used in a thermistor is generally a ceramic or polymer, while RTDs use pure metals. The temperature response is also different; RTDs are useful over larger temperature ranges, while thermistors typically achieve a higher precision within a limited temperature range, typically ?90 °C to 130 °C.[1]

Basic operation

Assuming, as a first-order approximation, that the relationship between resistance and temperature is linear, then:

:

\Delta R=k\Delta T \,

where

\Delta R = change in resistance
\Delta T = change in temperature
k = first-order temperature coefficient of resistance

Thermistors can be classified into two types, depending on the sign of k. If k is positive, the resistance increases with increasing temperature, and the device is called a positive temperature coefficient (PTC) thermistor, or posistor. If k is negative, the resistance decreases with increasing temperature, and the device is called a negative temperature coefficient (NTC) thermistor. Resistors that are not thermistors are designed to have ak as close to zero as possible, so that their resistance remains nearly constant over a wide temperature range.

Instead of the temperature coefficient k, sometimes the temperature coefficient of resistance \alpha_T (alpha sub T) is used. It is defined as[2]

\alpha_T = \frac{1}{R(T)} \frac{dR}{dT}.

This \alpha_T coefficient should not be confused with the a parameter below.

Steinhart–Hart equation

In practice, the linear approximation (above) works only over a small temperature range. For accurate temperature measurements, the resistance/temperature curve of the device must be described in more detail. The Steinhart–Hart equation is a widely used third-order approximation:

\frac{1}{T}=a+b\,\ln(R)+c\,\ln^3(R)

where a, b and c are called the Steinhart–Hart parameters, and must be specified for each device. T is the temperature in kelvin and R is the resistance in ohms. To give resistance as a function of temperature, the above can be rearranged into:

R=e^{{\left( x-{y \over 2} \right)}^{1\over 3}-{\left( x+{y \over 2} \right)}^{1\over 3}}

where

y={{a-{1\over T}}\over c} and x=\sqrt{{{{\left({b\over{3c}}\right)}^3}+{{y^2}\over 4}}}

The error in the Steinhart–Hart equation is generally less than 0.02 °C in the measurement of temperature over a 200 °C range.[3] As an example, typical values for a thermistor with a resistance of 3000 ? at room temperature (25 °C = 298.15 K) are:

a = 1.40 \times 10^{-3}

b = 2.37 \times 10^{-4}

c = 9.90 \times 10^{-8}

B or ? parameter equation

NTC thermistors can also be characterised with the B (or ?) parameter equation, which is essentially the Steinhart Hart equation with a = (1/T_{0}) – (1/B) \ln(R_{0}), b = 1/B and c = 0,

\frac{1}{T}=\frac{1}{T_0} + \frac{1}{B}\ln \left(\frac{R}{R_0}\right)

Where the temperatures are in kelvins and R0 is the resistance at temperature T0 (25 °C = 298.15 K). Solving for R yields:

R=R_0e^{B(\frac{1}{T} – \frac{1}{T_0})}

or, alternatively,

R=r_\infty e^{B/T}

where r_\infty=R_0 e^{-{B/T_0}}.

This can be solved for the temperature:

T={B\over { {\ln{(R / r_\infty)}}}}

The B-parameter equation can also be written as \ln R=B/T + \ln r_\infty. This can be used to convert the function of resistance vs. temperature of a thermistor into a linear function of \ln R vs. 1/T. The average slope of this function will then yield an estimate of the value of the B parameter.

Self-heating effects

When a current flows through a thermistor, it will generate heat which will raise the temperature of the thermistor above that of its environment. If the thermistor is being used to measure the temperature of the environment, this electrical heating may introduce a significant error if a correction is not made. Alternatively, this effect itself can be exploited. It can, for example, make a sensitive air-flow device employed in a sailplane rate-of-climb instrument, the electronic variometer, or serve as a timer for a relay as was formerly done in telephone exchanges.

The electrical power input to the thermistor is just:

P_E=IV\,

where I is current and V is the voltage drop across the thermistor. This power is converted to heat, and this heat energy is transferred to the surrounding environment. The rate of transfer is well described by Newton’s law of cooling:

P_T=K(T(R)-T_0)\,

where T(R) is the temperature of the thermistor as a function of its resistance R, T_0 is the temperature of the surroundings, and K is the dissipation constant, usually expressed in units of milliwatts per degree Celsius. At equilibrium, the two rates must be equal.

P_E=P_T\,

The current and voltage across the thermistor will depend on the particular circuit configuration. As a simple example, if the voltage across the thermistor is held fixed, then by Ohm’s Law we have I=V/R and the equilibrium equation can be solved for the ambient temperature as a function of the measured resistance of the thermistor:

T_0=T(R) -\frac{V^2}{KR}\,

The dissipation constant is a measure of the thermal connection of the thermistor to its surroundings. It is generally given for the thermistor in still air, and in well-stirred oil. Typical values for a small glass bead thermistor are 1.5 mW/°C in still air and 6.0 mW/°C in stirred oil. If the temperature of the environment is known beforehand, then a thermistor may be used to measure the value of the dissipation constant. For example, the thermistor may be used as a flow rate sensor, since the dissipation constant increases with the rate of flow of a fluid past the thermistor.

The power dissipated in a thermistor is typically maintained at a very low level to ensure insignificant temperature measurement error due to self heating. However, some thermistor applications depend upon significant “self heating” to raise the body temperature of the thermistor well above the ambient temperature so the sensor then detects even subtle changes in the thermal conductivity of the environment. Some of these applications include liquid level detection, liquid flow measurement and air flow measurement.[4]

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