Nonlinearity temperature coefficients

Thermistors are temperature-dependent resistances, normally constructed from metal oxides. The resistance change with temperature is high compared with the metallic resistances, and is usually negative the resistance decreases with temperature increase. The temperature characteristics are highly nonlinear. Such thermistors, having a negative temperature coefficient, are called NTC thermistors. Some thermistors have a positive temperature coefficient (PTC), but they are not in common use for temperature measurement. 

It is possible, in some situations, that two different phenomena which proceed at different rates with different temperature coefficients or activation energies will affect the physical properties. In such complex cases, it is not expect to obtain a linear relation between the logarithm of life and reciprocal absolute temperature. If one obtains a nonlinear curve, however, it may he possible to identify the reaction causing the nonlinearity and correct for it. When one can make such a correction, one obtains a linear relationship. 

The semiconducting properties of the compounds of the SbSI type (see Table XXVIII) were predicted by Mooser and Pearson in 1958 228). They were first confirmed for SbSI, for which photoconductivity was found in 1960 243). The breakthrough was the observation of fer-roelectricity in this material 117) and other SbSI type compounds 244 see Table XXIX), in addition to phase transitions 184), nonlinear optical behavior 156), piezoelectric behavior 44), and electromechanical 183) and other properties. These photoconductors exhibit abnormally large temperature-coefficients for their band gaps they are strongly piezoelectric. Some are ferroelectric (see Table XXIX). They have anomalous electrooptic and optomechanical properties, namely, elongation or contraction under illumination. As already mentioned, these fields cannot be treated in any detail in this review for those interested in ferroelectricity, review articles 224, 352) are mentioned. The heat capacity of SbSI has been measured from - 180 to -l- 40°C and, from these data, the excess entropy of the ferro-paraelectric transition... 

A method is described for fitting the Cole-Cole phenomenological equation to isochronal mechanical relaxation scans. The basic parameters in the equation are the unrelaxed and relaxed moduli, a width parameter and the central relaxation time. The first three are given linear temperature coefficients and the latter can have WLF or Arrhenius behavior. A set of these parameters is determined for each relaxation in the specimen by means of nonlinear least squares optimization of the fit of the equation to the data. An interactive front-end is present in the fitting routine to aid in initial parameter estimation for the iterative fitting process. The use of the determined parameters in assisting in the interpretation of relaxation processes is discussed. 

This model was fitted to the data of all three temperature levels, 375, 400, and 425°C, simultaneously using nonlinear least squares. The parameters were required to be exponentially dependent upon temperature. Part of the results of this analysis (K6) are reported in Fig. 6. Note the positive temperature coefficient of this nitric oxide adsorption constant, indicating an endothermic adsorption. Such behavior appears physically unrealistic if NO is not dissociated and if the confidence interval on this slope is relatively small. Ayen and Peters rejected this model also. 

The Horiuti group treats the temperature coefficient of the rate differently from the way it is usually treated in TST. They clearly identify E as the experimentally observed activation energy, but according to TST [cf. Eq. (5)] the (E — RT) quantity of Eq. (52) is the enthalpy of activation. The RT term in Eq. (5) arises because the assumption that the Arrhenius plot is linear is equivalent to the assumption that the preexponential factor A of the Arrhenius equation is constant, whereas, according to TST, A always contains the factor (kT/h). In addition, the partition function factors of Table I are also part of A, and most of them are functions of T. Since the Horiuti group takes this temperature dependency of the preexponential factor into account, the factor exp[(5/2)(vi -I- V2)] (where 5/2 is replaced by 3 for nonlinear molecules) arises. 

How can this be No additional gas was added to the water. The answer lies in the nonlinear temperature effect on the Bunsen solubility coefficient (Figure 6.1). Because of the concave nature of the curves relating the Bunsen solubility coefficient to temperature, the result of this type of postequilibration temperature change is always supersaturation. 

These equations look innocuous, but they are highly nonlinear equations whose solution is almost always obtainable only numerically. The nonlinear terms are in the rate r (Ca, T), which contains polynomials in Ca, especially the very nonlinear temperature dependence of the rate coefficient k(T). For first-order kinetics this is... 

Thermistors Thermistors are nonlinear temperature-dependent resistors, and normally only the materials with negative temperature coefficient of resistance (NTC type) are used. The resistance is related to temperature as... 

In order to compare calculated and experimentally observed phase portraits it is necessary to know very exactly all the coefficients of the describing nonlinear differential Equation 14.3. Therefore, different methods of determination of the nonlinear coefficient in the Duffing equation have been compared. In the paraelectric phase the value of the nonlinear dielectric coefficient B is determined by measuring the shift of the resonance frequency in dependence on the amplitude of the excitation ( [1], [5]). In the ferroelectric phase three different methods are used in order to determine B. Firstly, the coefficient B is calculated in the framework of the Landau theory from the coefficient of the high temperature phase (e.g. [4]). This means B = const, and B has the same values above and below the phase transition. Secondly, the shift of the resonance frequency of the resonator in the ferroelectric phase as a function of the driving field is used in order to determine the coefficient B. The amplitude of the exciting field is smaller than the coercive field and does not produce polarization reversal during the measurements of the shift of the resonance frequency. In the third method the coefficient B was determined by the values of the spontaneous polarization... 

Such estimations for H2 are single-point extrapolations of nonlinear functions, and therefore should be used with caution. In both cases estimation error is introduced due to the nonlinear temperature dependence of the activity coefficient. 

As explained in Appendix M, you can estimate the values of the coefficients in Eq. (3.32) by the method of least squares. We look at another way, a graphical method. Over very wide temperature intervals experimental data will not prove to be exactly linear as indicated by Eq. (3.31), but have a slight tendency to curve. This curvature can be straightened out by using a special plot known as a Cox chart. The In or logic of the vapor pressure of a compound is plotted against a special nonlinear temperature scale constructed from the vapor-pressure data for water (called a reference substance). 

In practice, polynomial approximations including linear and parabolic terms are used to calculate the temperature dependency with sufficient accuracy. The following equations describe the temperature dependency of the output signal Voul n and include the sensitivity S, signal offset Vorr, the nonlinearity of the characteristic curve, as well as their respective temperature coefficients TCxi ... 

MEMS devices have to meet certain criteria with respect to their functional parameters, for example, a scale factor, offset of the output value, temperature coefficient, nonlinearity, hysteresis, noise, resolution, and cross sensitivities, which characterize the system s performance. In addition, we are interested in the reliability, yield, and cost of the devices. The set of functional parameters depends on a set of model parameters, consisting of processing, material, and geometrical parameters. All model parameters act as input parameters for the design procedure as well as for the manufacturing process. Material parameters are influenced by... 

Later we will see that the nonlinearity is not a problem in use. It is not possible here to go into all the details of NTC theory for temperature coefficient alpha, beta value, self-heating effects, thermal time constant, thermal dissipation constant, stability, and reliability. Good information can be found elsewhere [1, 2] and in NTC manufacturers handbooks, catalogs, or web pages [3-10]. 

Lubricants have a nonlinear viscosity coefficient against temperature. At very low or very high temperature this nonlinearity can create significant measurement failures. A careful analysis will show this effect within the transmission chain. Sticking is the worst case. It can happen that frozen water makes a short cut in torque sensors. In most cases damping factors are based on the electrical filter circuits (Fig. 7.12.13). 

Resistive materials used in thermometry include platinum, copper, nickel, rhodium-iron, and certain semiconductors known as thermistors. Sensors made from platinum wires are called platinum resistance thermometers (PRTs) and, though expensive, are widely used. They have excellent stability and the potential for high-precision measurement. The temperature range of operation is from -260 to 1000°C. Other resistance thermometers are less expensive than PRTs and are useful in certain situations. Copper has a fairly linear resistance-temperature relationship, but its upper temperature limit is only about 150°C, and because of its low resistance, special measurements may be required. Nickel has an upper temperature limit of about 300°C, but it oxidizes easily at high temperature and is quite nonlinear. Rhodium-iron resistors are used in cryogenic temperature measurements below the range of platinum resistors [11]. Generally, these materials (except thermistors) have a positive temperature coefficient of resistance—the resistance increases with temperature. 

We may now identify some of the nonlinear viscoelastic parameters that (as mentioned at the start of this paper) are critical to adhesion. The most important property is the yield strength, Oy. After that, there is the property of elongational strengthening, i.e., the increase of force with rate of pulling on the end of a rod or fibre of the polymer and also the temperature coefficient of Oy. 

Intermediate-valent YbCuAl shows the well-known nonlinear temperature dependence (above 50 K) with a strong saturation. The temperature dependence of the Hall coefficient shows a discontinuity at aroimd 40 K. This peculiar behavior is attributed to an extraordinary contribution to the Hall coefficient (Cattaneo 1985, 1986). High-pressure resistivity measurements were also performed on this material (Mignot and Wittig 1981, 1982). 

In order to model the transport phenomena in polymeric materials, Lefebvre et derived a nonlinear diffusion coefficient based on the concept of free volume. According to this theory, the diffusion coefficient D for a polymeric material above its glass transition temperature is given by... 

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