Thermistors - a full of formulas

The explanations found in the documents provided by NTC Thermistor companies are more or less the same and usually a full of mathematics formulas.

I have used the following materials:

1) Mitsubishi Materials (not available any more)
2) wiki <Thermistor>
3) wiki <Temperature coefficient>
4) EPCOS - NTC Thermistors, General technical information
5) NTC Thermistors (Bowthorpe Thermometrics, etc)
6) NTC Thermistor theory (BetaTHERM Sensors)


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1) Mitsubishi Materials (not available any more)

My additional explanations and comments are made in bold Italic.
■ Resistance - temperature characteristic
The resistance and temperature characteristics of a thermistor can be approximated by equation 1.

(eq1) R=R_o exp {B(1/T -1 /T_o)}

R	: resistance at absolute temperature T(K)
Ro	: resistance at absolute temperature To(K)
B	: B value
*T(K) = t(ºC) + 273.15   The B value for the thermistor characteristics is not fixed, but can vary by as much as 5K/ºC according to the material composition. Therefore equation 1 may yield different results from actual values if applied over a wide temperature range.

By taking the B value in equation 1 as a function of temperature, as shown in equation 2, the difference with the actual value can be minimized.

(eq2) B _T = CT² + DT + E

C, D, and E are constants. 

The B value distribution caused by manufacturing conditions will change the constant E, but will have no effect on constants C or D. This means, when taking into account the distribution of B value, it is enough to do it with the constant E only.

Calculation for constants C, D and E
Using equations 3~6, constants C, D and E can be determined through four temperature and resistance value data points (T₀, R₀). (T₁, R₁). (T₂, R₂) and (T₃, R₃).
With equation 3, B₁, B₂ and B₃, can be determined from the resistance values for To and T₁, T₂, T₃ and then substituted into the equations below. 


Sorry, the equations 3~6 are no more available now. 


 Example

Using a resistance-temperature characteristic chart, the resistance value over the range of 10ºC~30ºC is sought for a thermistor with a resistance of 5kΩ and a B value deflection of 50K at 25ºC.

Process

(1) Determine the constants C, D and E from the resistance-temperature chart.

T_o = 25+273.15, T₁ = 10+273.15, T₂ = 20+273.15, T₃ = 30+273.15  

(2) B_T =  CT² + TD + E + 50 ; substitute the value into equation and solve for B_T

(3) R= 5exp {B_T (1/T - 1/298.15)} ; substitute the values into equation and solve for R

 *T : 10+273.15 ~ 30+273.15


Results of plotting the resistance temperature ■ Characteristics are shown figure 1 (not available any more)

The vertical axis is the base 10 log measure, i.e. log10R/R25 while the horizontal axis is 1/T (reciprocal of the temperature). This corresponds with (eq1) R=R_o exp {B(1/T-1/T_o)}. I.e. log10R/R25 = B(1/T-1/T25). Please note the linear relation - straight lines in figure 1.

■ Resistance temperature coefficient

The resistance-temperature coefficient (α) is defined as the rate of change of the zero-power resistance associated with a temperature variation of 1ºC at any given temperature.
The relationship between the resistance-temperature coefficient (α) and the B value can be obtained by differentiating equation 1 above.

1/R x dR/dT is a part of the definition of (α).

can be obtained by differentiating equation 1 above. - How?

(eq1) R = R_o exp {B(1/T - 1/T_o)} -----> ln R/R_{o =} ln {B(1/T-1/T_o)}
 

By differentiating the both sides - with respect to T or simply differentiating the both sides

R/R_{o  =}  B(1/T - 1/T_o)

(1/T - 1/T_o) _{is simplified to or regard as} d(1/T) or 1/dT, which is -1/T²

R/R_{o  =}  -B /T²

d/dT (R/R_o) is considered as (1/R) x (dR/dT) since R is Resistance at T and R_o means Resistance at T_o

According to the definition of (α) --- 1/R x dR/dT

Or simply d/dT (R/R_o) is dR. dR/dT has a unit of Ohm so should be x 1/R to make a ratio (α), which has no unit.

A negative value signifies that the rated zero-power resistance decreases.


■ Heat dissipation constant (JIS C2570-1)
The dissipation constant (δ) indicates the power necessary for increasing the temperature of the thermistor element by 1ºC through self-heating in a heat equilibrium.
Applying a voltage to a thermistor will cause an electric current to flow, leading to a temperature rise in the thermistor. This " intrinsic heating " process is subject to the following relationship among the thermistor temperature T₁, ambient temperature T₂, and consumed power P. 

Measuring conditions for all parts in this catalog are as follows:

(1) Room temp is 25ºC (2) Axial and radial leaded parts were measured in their shipping condition. ■ Maximum power dissipation (JIS C2570-1) The power rating is the maximum power for a continuous load at the rated temperature. For parts in this catalog, the value is calculated from the following formula using 25ºC as the ambient temperature. (formula) Rated power=heat dissipation constant × (maximum operating temperature-25ºC) In the detail specification, it is likely to write by "Power rating"that is a past name. ■ Permissible operating power Definition : The power to reach the maximum operating temperature through self heating when using a thermistor for temperature compensation or as a temperature sensor. (No JIS definition exists.) The permissible operating power, when t ºC is the permissible temperature rise, can be calculated using the following formula.

Permissible operating power= t*heat dissipation constant ■ Thermal time constant by ambient temperature change (JIS C2570-1) A constant expressed as the time for the temperature at the electrodes of a thermistor, with no load applied, to change to 63.2% of the difference between their initial and final temperatures, during a sudden change in the surrounding temperature. When the surrounding temperature of the thermistor changes from T1 to T2, the relation between the elapsed time t and the thermistor's temperature T can then be expressed by the following equation. T =  (T1 - T2) exp(-t/τ) + T2         (T2  -T1){1 - exp(-t/τ)} + T1

The constant t is called the heat dissipation constant.

If t = τ, the equation becomes : (T - T1) / (T2-T1) = 0.632

In other words, the above definition states that the thermal time constant is the time it takes for the temperature of the thermistor to change by 63.2% of its initial temperature difference.
The rate of change of the thermistor temperature versus time is shown in table 1. (not available any more)
Measuring conditions for parts in this catalog are as follows:

(1) Part is moved from a 50ºC environment to a still air 25ºC environment until the temperature of the thermistor reaches 34.2ºC.

(2) Axial and radial leaded parts are measured in their shipping form

Please note, the thermal dissipation constant and thermal time constant will vary according to environment and mounting conditions.

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