The subject of the theoretical analysis presented in this paper is an analytical approach to the temperature estimation, as an inverse problem, for different thermistors – linear resistances structures: series and parallel ones, by the STFT - Special Trans Functions Theory (S.M. Perovich). The mathematical formulae genesis of both cases is given. Some numerical and graphical simulations in MATHEMATICA program have been realized. The estimated temperature intervals for strongly determined values of the equivalent resistances of the nonlinear structures are given, as well.

The main subject of the theoretical analysis presented here is an analytical approach to the thermistors – linear resistors series and parallel combinations analysis, preferably oriented to the ambient temperature estimation. This analytical approach is based on the Special Trans Functions Theory. Accordingly, within this section few introductory observations are given.

Thermistors are solid temperature sensors that behave like temperature-sensitive electrical resistors. They are type of resistors with resistance which is proportional to the temperature. Though, their name is a contraction of “thermal” and “resistors”. Thermistors are widely used as inrush current limiters, temperature sensors, self-resetting over current protectors, and self-regulating heating elements. There are basically two broad types, negative temperature coefficient (NTC), used mostly in the temperature sensing and positive temperature coefficient (PTC), used mostly in electric current control. Because they are very small, thermistors are used inside many other devices as temperature sensing and correction elements, as well as, in specialty temperature sensing probes for science and technology.1

In this paper some different thermistors-linear resistances structures have been employed in unknown ambient temperature intervals estimation analytically, for given boundaries of the equivalent resistances of above noted structures, by the application of the STF Theory.3–9

Let us consider, now, both nonlinear resistances structures in more details.

Within this section, the problem of the ambient temperature estimation analytically, for the series case combination of K posistors (linear resistors) and M thermistors shall be presented.

Namely, for most linear resistors (posistors) the following empirical equation holds very closely:

\begin{equation*}R_m = R_{0m} \left( {1 + \alpha _m \left( {T - T_o } \right)} \right)\end{equation*}
$Rm=R0m1+αmT−To$

where, Rm is the resistance at temperature T; R0m is the resistance at the reference temperatureT0; T is the temperature, and αm is the constant depending on material. Consequently, for K different posistors we have for equivalent resistance the following formulae

$$R_K = A_K + B_K T$$
$RK=AK+BKT$
(1)

where

\begin{equation*}A_K = \sum\limits_{m = 0}^K {R_{om} } \left( {1 - \alpha _m T_o } \right);\quad {B_K = \sum\limits_{m = 0}^K {R_{om} \alpha _m } }.\end{equation*}
$AK=∑m=0KRom1−αmTo;BK=∑m=0KRomαm.$

The thermistor is a thermally sensitive variable resistor made of a ceramic like semi conducting material. Unlike metals, thermistors respond negatively to temperature. As the temperature rises, the thermistor resistance decreases. The temperature-resistance function for a thermistor is given by the relationship:

$${\rm R_T = R_\infty \exp \left( {\frac{\beta }{T}} \right)}$$
$RT=R∞expβT$
(2)

Where, RT is the resistance at temperature T, in [0K]; R is the resistance at temperature T  = 00K, and β is a constant. The constant β depends on the type of the thermistor.

Thermistors are useful for compensating electrical circuits for changing ambient temperature largely because of the negative temperature characteristics possessed by most electrical components. Of course, environment (heat transfer condition and heat dynamics) is a major factor in an actual application, i.e. in nonlinear resistance structures analysis, because it is more important in sense of the physical aspects of the considering problem. On the other hand the high sensitivity possessed by thermistors permits application of very simple electrical circuits for measurement of temperature.

The nonlinear functional equations for thermistors-linear resistors structure, by equations (1) and (2), for K resistors and M thermistors, take the form:

$$R_u = A_K + B_K T + M \cdot R_\infty \exp \left( {\frac{\beta }{T}} \right).$$
$Ru=AK+BKT+M·R∞expβT.$
(3)

After some simple modifications, from equation (3), we have the following nonlinear functional equation:

$$Ye^{ - Y} - DY = Ce^{ - Y} ;\quad {Y = \frac{\beta }{T}}$$
$Ye−Y−DY=Ce−Y;Y=βT$
(4)

where,

\begin{equation*}C = \frac{{\beta B_K }}{{R_u - A_K }};\quad {D = \frac{{MR_\infty }}{{R_u - A_K }}}.\end{equation*}
$C=βBKRu−AK;D=MR∞Ru−AK.$

The nonlinear functional equation (4) has the analytical closed-form solutions under the condition:1 > C + Dexp (1). Let us note that the nonlinear functional equation (4) is analytically solvable by application of the Special Trans Function Theory.3–10 We remark that equation (4) has two real solutions (Y < 1, T > β and Y > 1, T < β), but the subject matter of this paper is the first solution (Y < 1, T > β). The second solution, Y > 1, T < β, will be subject of one of the fore coming papers. Consequently, for Y < 1, T > β, the analytical closed-form solution of the equation (4) takes the form:

$${\rm Y = \mathop {\lim }\limits_{x \to \infty } \left[ {\ln \left( {\frac{{F(C,D,x + 1)}}{{F(C,D,x)}}} \right)} \right]}$$
$Y=limx→∞lnF(C,D,x+1)F(C,D,x)$
(5)

where,

$${\rm F(C,D,x) = (D\varphi (C,D,x) - \varphi (C,D,x - 1))}$$
$F(C,D,x)=(Dϕ(C,D,x)−ϕ(C,D,x−1))$
(6)

and,

$${\rm \varphi (C,D,x) = \sum\limits_{n = 0}^{\left[ x \right]} {n!} \sum\limits_{k = 0}^n {\frac{{( - 1)^k C^k (x - n)^k }}{{D^n k!k!(n - k)!}}}}$$
$ϕ(C,D,x)=∑n=0xn!∑k=0n(−1)kCk(x−n)kDnk!k!(n−k)!$
(7)

where [x] denotes the greatest integer less than or equal to x.

This section is organized as follows: an overview of the analytical closed form solutions is firstly presented, so the reader may inspect the formulae without previous rigorous concerning the derivation. The detailed derivation of the formulae is then presented. The outline of the derivation is based on the fact that the Special Trans Function Theory can be applied for arbitrary Thermistors - Linear Resistances Structures (TLRS) in a straightforward manner:

• determining the suitable equation for identification (EQID- equation for identification) of temperature nonlinear functional equations for TLRS in form of a partial differential equation;

• finding the analytical closed form solution to this chosen EQID;

• predicting the asymptotic solution of partial differential equation for identification, and finally,

• choosing the optimal equalization between unique solution and asymptotic solution.

As it was noted previously, the nonlinear functional equation for some nonlinear resistances structures takes the form (4). This nonlinear functional equation for Y < 1, T > β has the analytical closed-form solution in the form:

$${\rm Y = trans_T (C,D)}$$
$Y= trans T(C,D)$
(8)

where, transT(C, D) is a new special tran function defined as:

\begin{equation*}{\rm trans_T (C,D) = \mathop {\lim }\limits_{x \to \infty } \left[ {\ln \left( {\frac{{F(C,D,x + 1)}}{{F(C,D,x)}}} \right)} \right].}\end{equation*}
$trans T(C,D)=limx→∞lnF(C,D,x+1)F(C,D,x).$

Or, more explicitly equation (8) takes the form

\begin{equation*}Y = trans_T (\alpha _m,R_{om},\beta,R_\infty,M),\quad {m = 1,...K}.\end{equation*}
$Y=transT(αm,Rom,β,R∞,M),m=1,...K.$

The practical numerical model for calculation of the transcendental number Y takes the form:

$$\langle Y\rangle _{\left[ G \right]} = \langle trans_T (C,D)\rangle _G = \left\langle {\ln \left( {\frac{{F(C,D,x + 1)}}{{F(C,D,x)}}} \right)} \right\rangle _{\left[ G \right]}$$
$⟨Y⟩G=⟨transT(C,D)⟩G=lnF(C,D,x+1)F(C,D,x)G$
(9)

where, 〈Y〉[P] denotes the numerical value of number Y given with [G] accurate digits where the error function G is defined like: G = |Ye−Y − DY − Ce−Y|, and it satisfies the inequality: Ggm. Here gm is an arbitrary small and positive, real number. Because the 〈Y〉[P] denotes the numerical value of number Y given with [G] accurate digits, formulae (9) represents the numerical structures for transcendental number Y. The choice of x controls the number of accurate digits for the Y, and satisfies the error criteria. On the other hand, the number of accurate digits in the numerical structure of Y, is practically, determined by physical requirements for measurement ambient temperature level.

The transcendental equation (4) can be identified with a differential equation of type:

$${\rm F'( {C,D,x - 1}) - DF'\left( {C,D,x} \right) - CF(C,D,x - 1) = 0.}$$
$F′(C,D,x−1)− DF ′C,D,x− CF (C,D,x−1)=0.$
(10)

The differential equation (10) is analytically solvable using a Laplace transform. The Laplace transform to the differential equation (10) takes the form:

\begin{equation*}{\rm \Phi (C,D,s) = \frac{{(e^{ - s} - D)F(0)}}{{se^{ - s} - Ds - Ce^{ - s} }}.}\end{equation*}
$Φ(C,D,s)=(e−s−D)F(0) se −s− Ds − Ce −s.$

After a simple modification, the above equation takes the form:

\begin{equation*}{\rm \Phi (C,D,s) = \frac{{(D - e^{ - s})F(0)}}{{Ds}}\sum\limits_{n = 0}^\infty {\frac{{e^{ - ns} }}{{D^n }}} \sum\limits_{k = 0}^n {\frac{{( - 1)^k C^k n!}}{{s^k (n - k)!k!}}}.}\end{equation*}
$Φ(C,D,s)=(D−e−s)F(0) Ds ∑n=0∞e− ns Dn∑k=0n(−1)kCkn!sk(n−k)!k!.$

By inverse Laplace transform approach,2 from above equation, we have:

\begin{equation*}{\rm F(C,D,x) = \left( {\frac{{F(0)}}{D}} \right)\left( {D\varphi (x) - \varphi (x - 1)} \right)}\end{equation*}
$F(C,D,x)=F(0)DDϕ(x)−ϕ(x−1)$

where,

\begin{equation*}{\rm \varphi (C,D,x) = \sum\limits_{n = 0}^{\left[ x \right]} {\frac{{n!}}{{D^n }}} \sum\limits_{k = 0}^{k = n} {\frac{{( - 1)^k C^k \left( {x - n} \right)^k }}{{(n - k)!k!k!}}}.}\end{equation*}
$ϕ(C,D,x)=∑n=0xn!Dn∑k=0k=n(−1)kCkx−nk(n−k)!k!k!.$

The above series is the unique analytical solution of the differential equation (10). Or, more explicitly, equation (10) takes the form:

$${\rm \varphi (C,D,x) = \sum\limits_{n = 0}^\infty {\frac{{n!}}{{D^n }}} \sum\limits_{k = 0}^n {\frac{{( - 1)^k C^k (x - n)^k }}{{k!k!(n - k)!}}} H(x - n)}$$
$ϕ(C,D,x)=∑n=0∞n!Dn∑k=0n(−1)kCk(x−n)kk!k!(n−k)!H(x−n)$
(11)

where, H(xn) is the Heaviside's unit function defined as:

${\rm H(x - n) = \left\{ {\begin{array}{l@{\quad}ll}1 & {\rm for} & {\rm x > n} \\0 & {\rm for} & {\rm x < n.}\end{array}} \right.}$
$H(x−n)=1 for x>n0 for x⁠.

On the other hand, the asymptotic solution of the form:

$${\rm F_{as} (C,D,x) = F_{aso} \exp (x)}$$
$F as (C,D,x)=F aso exp(x)$
(12)

satisfies the differential equation (10), under the condition that Y satisfies the equation (4). According to the unique solution principle and function theory, we have:

$$\mathop {\lim }\limits_{x \to \infty } F_{as} (C,D,x) = \mathop {\lim }\limits_{x \to \infty } \left( {F(C,D,x)} \right).$$
$limx→∞Fas(C,D,x)=limx→∞F(C,D,x).$
(13)

From the prevision section it is clear that by applying the unique solution principle, we have that, analytical solution (6) is convergent to the asymptotic solution (12). Analogically, we have:

$${\rm Y = \mathop {\lim }\limits_{N \to \infty } \left\{ {\ln \left[ {\frac{{F(C,D,N + 1)}}{{F(C,D,N)}}} \right]} \right\}}$$
$Y=limN→∞lnF(C,D,N+1)F(C,D,N)$
(14)

where,

$$F(C,D,N) = \left( {\frac{{F(0)}}{D}} \right)(D\varphi (C,D,N) - \varphi (C,D,N - 1))$$
$F(C,D,N)=F(0)D(Dϕ(C,D,N)−ϕ(C,D,N−1))$
(15)

and,

$$\varphi (C,D,N) = \sum\limits_{n = 0}^N {\frac{{n!}}{{D^n }}} \sum\limits_{k = 0}^n {\frac{{( - 1)^k C^k (N - n)^k }}{{k!k!(n - k)!}}}$$
$ϕ(C,D,N)=∑n=0Nn!Dn∑k=0n(−1)kCk(N−n)kk!k!(n−k)!$
(16)

where, N is a positive integer, when the error function defined as:

\begin{equation*}G\mathop = \limits^ \wedge \left&#x007c; {Ye^{ - Y} - DY - Ce^{ - Y} } \right&#x007c;\end{equation*}
$G=∧Ye−Y−DY−Ce−Y$

satisfies inequality: Ggm, where, gm is a real, positive arbitrary small number. By expression (14), we introduce the special trans function in form:

\begin{equation*}{\rm Y = trans_T (C,D) = \mathop {\lim }\limits_{x \to \infty } \left[ {\ln \left( {\frac{{F(C,D,N)}}{{F(C,D,N - 1)}}} \right)} \right]}\end{equation*}
$Y= trans T(C,D)=limx→∞lnF(C,D,N)F(C,D,N−1)$

or

\begin{equation*}Y = trans_T (C,D) = \mathop {\lim }\limits_{x \to \infty } \left[ {\ln \left( {\frac{{D\varphi (C,D,N) - \varphi (C,D,N - 1)}}{{D\varphi (C,D,N - 1) - \varphi (C,D,N - 2)}}} \right)} \right].\end{equation*}
$Y=transT(C,D)=limx→∞lnDϕ(C,D,N)−ϕ(C,D,N−1)Dϕ(C,D,N−1)−ϕ(C,D,N−2).$

More explicitly we have

\begin{equation*}Y = \mathop {\lim }\limits_{x \to \infty } \left[ {\ln \left( {\frac{{D\sum\limits_{n = 0}^N {\frac{{n!}}{{D^n }}} \sum\limits_{k = 0}^n {\frac{{( - 1)^k C^k (N - n)^k }}{{k!k!(n - k)!}}} - \sum\limits_{n = 0}^{N - 1} {\frac{{n!}}{{D^n }}} \sum\limits_{k = 0}^n {\frac{{( - 1)^k C^k (N - 1 - n)^k }}{{k!k!(n - k)!}}} }}{{D\sum\limits_{n = 0}^{N - 1} {\frac{{n!}}{{D^n }}} \sum\limits_{k = 0}^n {\frac{{( - 1)^k C^k (N - 1 - n)^k }}{{k!k!(n - k)!}}} - \sum\limits_{n = 0}^{N - 2} {\frac{{n!}}{{D^n }}} \sum\limits_{k = 0}^n {\frac{{( - 1)^k C^k (N - 2 - n)^k }}{{k!k!(n - k)!}}} }}} \right)} \right]\end{equation*}
$Y=limx→∞lnD∑n=0Nn!Dn∑k=0n(−1)kCk(N−n)kk!k!(n−k)!−∑n=0N−1n!Dn∑k=0n(−1)kCk(N−1−n)kk!k!(n−k)!D∑n=0N−1n!Dn∑k=0n(−1)kCk(N−1−n)kk!k!(n−k)!−∑n=0N−2n!Dn∑k=0n(−1)kCk(N−2−n)kk!k!(n−k)!$

In this way, the formula for unknown ambient temperature takes the form:

$${\rm T = \frac{\beta }{Y} = \frac{\beta }{{trans_T (C,D)}}.}$$
$T=βY=β trans T(C,D).$
(17)

Or, form

\begin{equation*}T = \frac{\beta }{{\mathop {\lim }\limits_{x \to \infty } \left[ {\ln \left( {\frac{{D\sum\limits_{n = 0}^N {\frac{{n!}}{{D^n }}} \sum\limits_{k = 0}^n {\frac{{( - 1)^k C^k (N - n)^k }}{{k!k!(n - k)!}}} - \sum\limits_{n = 0}^{N - 1} {\frac{{n!}}{{D^n }}} \sum\limits_{k = 0}^n {\frac{{( - 1)^k C^k (N - 1 - n)^k }}{{k!k!(n - k)!}}} }}{{D\sum\limits_{n = 0}^{N - 1} {\frac{{n!}}{{D^n }}} \sum\limits_{k = 0}^n {\frac{{( - 1)^k C^k (N - 1 - n)^k }}{{k!k!(n - k)!}}} - \sum\limits_{n = 0}^{N - 2} {\frac{{n!}}{{D^n }}} \sum\limits_{k = 0}^n {\frac{{( - 1)^k C^k (N - 2 - n)^k }}{{k!k!(n - k)!}}} }}} \right)} \right]}}\end{equation*}
$T=βlimx→∞lnD∑n=0Nn!Dn∑k=0n(−1)kCk(N−n)kk!k!(n−k)!−∑n=0N−1n!Dn∑k=0n(−1)kCk(N−1−n)kk!k!(n−k)!D∑n=0N−1n!Dn∑k=0n(−1)kCk(N−1−n)kk!k!(n−k)!−∑n=0N−2n!Dn∑k=0n(−1)kCk(N−2−n)kk!k!(n−k)!$

This is an analytical closed-form solution to the transcendental equation (4), for T > β. For practical calculation the above equation takes the form

$$\left\langle T \right\rangle _{\left[ G \right]} = \LEFTRIGHT{\langle}{\rangle}{{\frac{\beta }{{\ln \left( {\frac{{D\sum\limits_{n = 0}^N {\frac{{n!}}{{D^n }}} \sum\limits_{k = 0}^n {\frac{{( - 1)^k C^k (N - n)^k }}{{k!k!(n - k)!}}} - \sum\limits_{n = 0}^{N - 1} {\frac{{n!}}{{D^n }}} \sum\limits_{k = 0}^n {\frac{{( - 1)^k C^k (N - 1 - n)^k }}{{k!k!(n - k)!}}} }}{{D\sum\limits_{n = 0}^{N - 1} {\frac{{n!}}{{D^n }}} \sum\limits_{k = 0}^n {\frac{{( - 1)^k C^k (N - 1 - n)^k }}{{k!k!(n - k)!}}} - \sum\limits_{n = 0}^{N - 2} {\frac{{n!}}{{D^n }}} \sum\limits_{k = 0}^n {\frac{{( - 1)^k C^k (N - 2 - n)^k }}{{k!k!(n - k)!}}} }}} \right)}}}}.$$
$TG=βlnD∑n=0Nn!Dn∑k=0n(−1)kCk(N−n)kk!k!(n−k)!−∑n=0N−1n!Dn∑k=0n(−1)kCk(N−1−n)kk!k!(n−k)!D∑n=0N−1n!Dn∑k=0n(−1)kCk(N−1−n)kk!k!(n−k)!−∑n=0N−2n!Dn∑k=0n(−1)kCk(N−2−n)kk!k!(n−k)!.$
(18)

Numerical results obtained from above formulae are obtained with high accuracy, controlled by integer N.

In the following section an appropriate numerical example has been given.

Within this section, we used a series resistance combination of K linear resistors (posistors) and M thermistors in aim to determine temperature interval for determined variations of equivalent resistance (Figure 1).

FIG. 1.

The series of K linear resistances and M thermistors

FIG. 1.

The series of K linear resistances and M thermistors

Close modal

For given all relevant circuit parameters, by the application of previously proposed and STF Theory we are in position to determine ambient temperature with high accuracy. Obtained numerical results by MATHEMATICA program are given in Table I. We remark that different values of integer K and M simulated the different parameters of thermistors and posistors. It is obvious nonsensitiv presentation but in sense of global analysis it is satisfactory. Of course, for posistors it is irrelevant because these resistances temperature functions are linear functions.

Table I.

The simulation results for the linear thermometer combination

Ru [Ω]YT [K]G
7000 7.018223 569.945 7.569 E-08
7100 7.033041 568.744 7.371 E-08
7200 7.047633 567.566 7.178 E-08
7300 7.062009 566.411 6.993 E-08
7400 7.076201 565.275 6.814 E-08
7500 7.090186 564.160 6.649 E-08
7600 7.103979 563.065 6.482 E-08
7700 7.117597 561.987 6.323 E-08
7800 7.131008 560.930 6.171 E-08
7900 7.144349 559.883 6.029 E-08
8000 7.157331 558.868 5.886 E-08
Ru [Ω]YT [K]G
7000 7.018223 569.945 7.569 E-08
7100 7.033041 568.744 7.371 E-08
7200 7.047633 567.566 7.178 E-08
7300 7.062009 566.411 6.993 E-08
7400 7.076201 565.275 6.814 E-08
7500 7.090186 564.160 6.649 E-08
7600 7.103979 563.065 6.482 E-08
7700 7.117597 561.987 6.323 E-08
7800 7.131008 560.930 6.171 E-08
7900 7.144349 559.883 6.029 E-08
8000 7.157331 558.868 5.886 E-08

In the following table (Table II), some results of the numerical analysis are presented. These results show that, in this case, for relatively small ambient temperature variations, we have relatively large equivalent resistance changes. This is interesting, since it shows that the considered resistors structure can be used for temperature measurement.11 On the other hand, the subject of the theoretical analysis in this paper is the case when the equivalent resistance of the series combination of posistor and thermistor is changing due to the temperature increasing less than the resistances of posistor and thermistor being taken into account separately. Accordingly, the thermistors are commonly suitable for compensation of the environment temperature changes and its impacts to the signals in electrical circuits.12 In other words, thermistors are useful as elements of electrical circuits for ambient temperature variations compensation that is mostly implied by the negative thermometers’ temperature coefficients.

Table II.

The simulation results for the combination of two linear resistances and three thermistors

K
(for M=2)2468101214
ΔT [K] 10.903 11.329 11.798 12.330 12.970 13.876 16.001

10 12 14
(for K=3)
ΔT [K] 11.965 14.689 16.737 18.461 19.989 21.382 22.676
K
(for M=2)2468101214
ΔT [K] 10.903 11.329 11.798 12.330 12.970 13.876 16.001

10 12 14
(for K=3)
ΔT [K] 11.965 14.689 16.737 18.461 19.989 21.382 22.676

Also, some graphical presentations of evaluated ambient temperature(s) for nonlinear resistance structure of two thermistors and three posistors, for different values of α and β coefficients are given in Figures 2, 3, and 4.

FIG. 2.

Functional presentation of the form T=f (Ru)

FIG. 2.

Functional presentation of the form T=f (Ru)

Close modal
FIG. 3.

The evaluated temperatures for different coefficients α

FIG. 3.

The evaluated temperatures for different coefficients α

Close modal
FIG. 4.

The evaluated temperatures for different coefficients β

FIG. 4.

The evaluated temperatures for different coefficients β

Close modal

Analogy to the previous notation (1) let us present the linear resistance (posistor) in the parallel combination as:

\begin{equation*}{\rm R = A + BT}\end{equation*}
$R=A+ BT$

where, A = R0 − R0 · αT0, B = α · R0, and R is the resistance at temperature T; R0 is the resistance at the reference temperature; T is the temperature, and α is the constant depending on material.

The temperature dependence for a thermistor is given, in the same manner as in the previous section (2):

\begin{equation*}{\rm R_T = R_\infty \exp \left( {\frac{\beta }{T}} \right)}\end{equation*}
$RT=R∞expβT$

where, RT is the resistance at temperature T, in [0K]; R is the resistance at temperature T  = 00K, and β is a constant. As we noted in section II, the constant β depends on the thermistor type.

The parallel thermistor - linear resistance (posistor) combination, by two latter equations takes the form:

$${\rm Ru = \frac{{\left( {A + BT} \right) \cdot R_\infty \exp \left( {\frac{\beta }{T}} \right)}}{{A + BT + R_\infty \exp \left( {\frac{\beta }{T}} \right)}}.}$$
$Ru =A+ BT ·R∞expβTA+ BT +R∞expβT.$
(19)

After a simple modification, from equation (19), we have the following nonlinear functional equation:

$${\rm \Psi e^{ - \Psi } + a\Psi - b = 0;\quad {\Psi = \frac{\beta }{T}} + \frac{{\beta B}}{A}}$$
$Ψe−Ψ+aΨ−b=0;Ψ=βT+βBA$
(20)

where, |${\rm a = R_\infty ( {1 - \frac{A}{{R_u }}}) \cdot e^{ - \frac{{\beta B}}{A}} ;\, {b = \frac{{R_\infty \beta B}}{A}} \cdot e^{ - \frac{{\beta B}}{A}}}$|$a=R∞(1−ARu)·e−βBA;b=R∞βBA·e−βBA$. We remark that equation (20) is an inverse problem.

Let us note that the nonlinear functional equation (20) is analytically solvable by application of the Special Trans Function Theory (STFT).3–10 The analytical closed-form solution of the equation (20) takes the form:

$${\rm \Psi = \mathop {\lim }\limits_{x \to \infty } \left[ {\ln \left( {\frac{{\varphi (a,b,x + 1)}}{{\varphi (a,b,x)}}} \right)} \right]}$$
$Ψ=limx→∞lnϕ(a,b,x+1)ϕ(a,b,x)$
(21)

where,

$${\rm \varphi (a,b,x) = \frac{1}{a}(\varphi _0 (a,b,x - 1) - \varphi _0 (a,b,x))}$$
$ϕ(a,b,x)=1a(ϕ0(a,b,x−1)−ϕ0(a,b,x))$
(22)

and,

$${\rm \varphi _0 (a,b,x) = \sum\limits_{n = 0}^{\left[ x \right]} {\sum\limits_{k = 0}^n {\frac{{n!b^{n - k} \left( { - 1} \right)^k \left( {x - k} \right)^{n + k} }}{{\left( {n - k} \right)!k!a^{n + k} \left( {n + k} \right)!}}} }}$$
$ϕ0(a,b,x)=∑n=0x∑k=0nn!bn−k−1kx−kn+kn−k!k!an+kn+k!$
(23)

where, [x] denotes the greatest integer less than or equal to x.

The nonlinear functional equation for simple parallel resistances combination takes the form (20). This nonlinear functional equation has the analytical closed-form solution in the form:

$$\Psi = tran_{PR} (a,b)$$
$Ψ=tranPR(a,b)$
(24)

where, tranPR(a, b) is a new special tran function defined as:

\begin{equation*}{\rm tran_{PR} (a,b) = \mathop {\lim }\limits_{x \to \infty } \left[ {\ln \left( {\frac{{\varphi (a,b,x + 1)}}{{\varphi (a,b,x)}}} \right)} \right].}\end{equation*}
$tran PR (a,b)=limx→∞lnϕ(a,b,x+1)ϕ(a,b,x).$

The practical numerical model for calculation of the transcendental number Ψ, takes the form:

$${\rm \langle \Psi \rangle _{\left[ G \right]} = \langle \ln \left( {\frac{{\varphi (a,b,x + 1)}}{{\varphi (a,b,x)}}} \right)\rangle _{\left[ G \right]}}$$
$⟨Ψ⟩G=⟨lnϕ(a,b,x+1)ϕ(a,b,x)⟩G$
(25)

where, 〈Ψ〉[G] is a value of the transcendental number Ψ given with [G] accurate digits, and G is error function, defined as:

\begin{equation*}{\rm G = \left&#x007c; {\Psi e^{ - \Psi } + a\Psi - b} \right&#x007c; \le \varepsilon}\end{equation*}
$G=Ψe−Ψ+aΨ−b≤ɛ$

where, ɛ is an arbitrary small positive real number.

Consequently, formulae (25) and (23) represent the numerical structure of transcendental number ψ. The choice of x controls the number of accurate digits of ψ. On the other hand, the number of accurate digits in the numerical structure of ψ, is practically, determined by physical requirements for measurement ambient temperature level.

The transcendental equation (20) can be identified with a differential equation of type:

$${\rm \varphi '\left( {x - 1} \right) + a\varphi '\left( x \right) - b\varphi (x) = 0.}$$
$ϕ′x−1+aϕ′x−bϕ(x)=0.$
(26)

The differential equation (26) is analytically solvable using a Laplace transform. Thus, the Laplace transform to the differential equation (26) takes the form:

\begin{equation*}{\rm \Phi \left( s \right) = \Phi _0 \left( s \right)\left( {e^{ - s} + a} \right){\rm, where}}\end{equation*}
$Φs=Φ0se−s+a, where$
\begin{equation*}{\rm \Phi _0 (s) = \frac{1}{{as}}\sum\limits_{n = 0}^\infty {\sum\limits_{k = 0}^n {\frac{{n!b^{n - k} \left( { - 1} \right)^k e^{ - ks} }}{{\left( {n - k} \right)!k!a^n s^{n - k} }}} }.}\end{equation*}
$Φ0(s)=1 as ∑n=0∞∑k=0nn!bn−k−1ke− ks n−k!k!ansn−k.$

From obtained Laplace transform, in real domain, by inverse Laplace transform, we have:

\begin{equation*}{\rm \varphi _0 \left( x \right) = \frac{1}{a}\sum\limits_{n = 0}^{\left[ x \right]} {\sum\limits_{k = 0}^n {\frac{{n!b^{n - k} \left( { - 1} \right)^k \left( {x - k} \right)^{n - k} }}{{\left( {n - k} \right)!k!a^n \left( {n - k} \right)!}}} }}\end{equation*}
$ϕ0x=1a∑n=0x∑k=0nn!bn−k−1kx−kn−kn−k!k!ann−k!$

and,

$${\rm \Psi = \mathop {\lim }\limits_{x \to \infty } \left( {\ln \left( {\frac{{\varphi \left( {x + 1} \right)}}{{\varphi \left( x \right)}}} \right)} \right)}$$
$Ψ=limx→∞lnϕx+1ϕx$
(27)

where

$${\rm \varphi (a,b,x) = \frac{1}{a}(\varphi _0 (a,b,x - 1) + a\varphi _0 (a,b,x)).}$$
$ϕ(a,b,x)=1a(ϕ0(a,b,x−1)+aϕ0(a,b,x)).$
(28)

By applying the unique solution principle, we have that, for large x, analytical solution (21) is convergent to the asymptotic solution of the form φas(x) = φasoexp (ψx). Analogically, for x = N and equation (27), we have:

$$\Psi = \mathop {\lim }\limits_{N \to \infty } \left\{ {\ln \left[ {\frac{{\varphi (a,b,N + 1)}}{{\varphi (a,b,N)}}} \right]} \right\}$$
$Ψ=limN→∞lnϕ(a,b,N+1)ϕ(a,b,N)$
(29)

where,

$${\rm \varphi (a,b,N) = \frac{1}{a}\sum\limits_{n = 0}^{\left[ N \right]} {\sum\limits_{k = 0}^n {\frac{{n!b^{n - k} \left( { - 1} \right)^k \left( {N - k} \right)^{n - k} }}{{\left( {n - k} \right)!k!a^n \left( {n - k} \right)!}}} }}$$
$ϕ(a,b,N)=1a∑n=0N∑k=0nn!bn−k−1kN−kn−kn−k!k!ann−k!$
(30)

N is a positive integer, when the error function defined as:

$$G\mathop = \limits^ \wedge \left&#x007c; {\Psi e^{ - \Psi } + a\Psi - b} \right&#x007c;$$
$G=∧Ψe−Ψ+aΨ−b$
(31)

satisfies the inequality: G(a, b, x) ⩽ gm, where gm is a real, positive arbitrary small number.

In this way, the formula for ambient temperature takes the form:

$${\rm T = \frac{\beta }{{tran_{PR} \left( {a,b} \right) - \frac{{\beta B}}{A}}}.}$$
$T=β tran PR a,b−βBA.$
(32)

Finally, for practical calculation equation (32) takes the form

$$\left\langle T \right\rangle _{\left[ G \right]} = \left\langle {\frac{\beta }{{tran_{PR} \left( {a,b} \right) - \frac{{\beta B}}{A}}}} \right\rangle _{\left[ G \right]}$$
$TG=βtranPRa,b−βBAG$
(33)

where 〈T[G] is value of temperature T given with [G] accurate digits. We remark that obtained temperature values are obtained with high accuracy, i.e., [G] is controlled by the integer N. More explicitly equation (33) takes the form

\begin{equation*}{\rm \left\langle T \right\rangle _{\left[ G \right]} = \LEFTRIGHT{\langle}{\rangle}{{\frac{\beta }{{\ln \left( {\frac{1}{a}\sum\limits_{n = 0}^{\left[ {N + 1} \right]} {\sum\limits_{k = 0}^n {\frac{{n!b^{n - k} \left( { - 1} \right)^k \left( {N + 1 - k} \right)^{n - k} }}{{\left( {n - k} \right)!k!a^n \left( {n - k} \right)!}}} } } \right) - \ln \left( {\frac{1}{a}\sum\limits_{n = 0}^{\left[ N \right]} {\sum\limits_{k = 0}^n {\frac{{n!b^{n - k} \left( { - 1} \right)^k \left( {N - k} \right)^{n - k} }}{{\left( {n - k} \right)!k!a^n \left( {n - k} \right)!}}} } } \right) - \frac{{\beta B}}{A}}}}} _{\left[ G \right]}.}\end{equation*}
$TG=βln1a∑n=0N+1∑k=0nn!bn−k−1kN+1−kn−kn−k!k!ann−k!−ln1a∑n=0N∑k=0nn!bn−k−1kN−kn−kn−k!k!ann−k!−βBAG.$

On the other hand, the equation (32) is the analytical closed-form solution of the transcendental equation (20).

In this section, by the application of Special Trans Function Theory, we employed a numerical example for the parallel structure of one thermistor and one posistor (Figure 5), while the Ru variation has been given, as well as, the others, relevant for the circuit parameters.

FIG. 5.

The parallel structure of posistor - thermistor

FIG. 5.

The parallel structure of posistor - thermistor

Close modal

By MATHEMATICA program, on the base of equation (33), we determined ambient temperature, as in the previous case of the series structures of posistors – thermistores, with high precision. The obtained numerical results are given in Table III.

Table III.

The numerical simulation results for the parallel structure of posistor – thermistor

Ru [Ω]ΨT [K]G
500 22.827518 799.495 2.486 E-09
600 22.958546 768.157 1.830 E-09
700 22.930375 747.925 1.965 E-09
800 22.909756 733.780 2.065 E-09
900 22.894007 723.332 2.143 E-09
1000 22.881585 715.298 2.206 E-09
1100 22.871534 708.927 2.257 E-09
1200 22.863236 703.752 2.299 E-09
1300 22.856267 699.464 2.335 E-09
1400 22.850332 695.853 2.366 E-09
1500 22.845217 692.771 2.393 E-09
Ru [Ω]ΨT [K]G
500 22.827518 799.495 2.486 E-09
600 22.958546 768.157 1.830 E-09
700 22.930375 747.925 1.965 E-09
800 22.909756 733.780 2.065 E-09
900 22.894007 723.332 2.143 E-09
1000 22.881585 715.298 2.206 E-09
1100 22.871534 708.927 2.257 E-09
1200 22.863236 703.752 2.299 E-09
1300 22.856267 699.464 2.335 E-09
1400 22.850332 695.853 2.366 E-09
1500 22.845217 692.771 2.393 E-09

In the next table (Table IV), some results of additional numerical analysis are presented. These results show that, in this case, for relatively small ambient temperature relative variations (expressed in %), we have relatively large equivalent resistance changes. The obtain results show the possibility of temperature measurement by this structure, with high accuracy, taking into account the accuracy of resistances measurements in the circuit, or the accuracy of the mathematical models of thermistor and posistor resistances. But, more rigorous analysis of this structure applicability in measuring temperature, in real conditions, shall be the subject of the forthcoming investigations.

Table IV.

Temperature changes (ΔT) due to changes of the equivalent resistance (ΔRu) of the paralel thermistor-posistor combination

Ru|${\rm \Delta R_{ui} = R_u + \Delta _i,i} = \overline {1,3}$|$ΔR ui =Ru+Δi,i=1,3¯$
Δ1 = 10%Δ2 = 15%Δ3 = 20%
[Ω]   TT10%T15%T20%ΔT10%ΔT15%ΔT20%
500 550 575 600 799 781 774 768 0.18 0.25 0.31
750 825 863 900 740 730 726 723 0.10 0.14 0.17
1000 1100 1150 1200 715 708 706 703 0.07 0.09 0.12
2500 2750 2875 3000 676 674 673 672 0.02 0.03 0.04
5000 5500 5750 6000 664 663 663 662 0.01 0.01 0.02
7500 8250 8625 9000 660.964 660.294 660.004 659.738 0.06 0.09 0.01
10000 11000 11500 12000 659.127 658.629 658.413 658.216 0.005 0.007 0.009
Ru|${\rm \Delta R_{ui} = R_u + \Delta _i,i} = \overline {1,3}$|$ΔR ui =Ru+Δi,i=1,3¯$
Δ1 = 10%Δ2 = 15%Δ3 = 20%
[Ω]   TT10%T15%T20%ΔT10%ΔT15%ΔT20%
500 550 575 600 799 781 774 768 0.18 0.25 0.31
750 825 863 900 740 730 726 723 0.10 0.14 0.17
1000 1100 1150 1200 715 708 706 703 0.07 0.09 0.12
2500 2750 2875 3000 676 674 673 672 0.02 0.03 0.04
5000 5500 5750 6000 664 663 663 662 0.01 0.01 0.02
7500 8250 8625 9000 660.964 660.294 660.004 659.738 0.06 0.09 0.01
10000 11000 11500 12000 659.127 658.629 658.413 658.216 0.005 0.007 0.009

Some graphical presentations for various parameter β and different intervals of Ru, are shown in Figure 6 and Figure 7, below.

FIG. 6.

The evaluated temperatures for different coefficient β and Ru intervals

FIG. 6.

The evaluated temperatures for different coefficient β and Ru intervals

Close modal
FIG. 7.

The evaluated temperatures for different coefficient β and Ru intervals

FIG. 7.

The evaluated temperatures for different coefficient β and Ru intervals

Close modal

Within the common, classical analysis, the ambient temperature from nonlinear functional equations (4) and (20) are solvable by application of the approximation methods or various iterative procedures available in conventional numerical computation.

In this paper the thermistors – posistors (linear resistors) series and parallel combinations ambient temperatures are estimated analytically, in the closed form, by the application of the Special Trans Functions Theory (S.M. Perovich). In such way, the temperature is analytically obtainable by genesis of a new special tran function transT(C, D), for the series resistors combination, i.e. by genesis of a new special tran function tranPR(a, b), for the parallel resistances combination. The numerical results obtained by application both special tran functions are of high accuracy. The STFT is the consistent theory applicable to the nonlinear functional equations for the nonlinear resistors structures analysis in the ambient temperature domain. In other words, the STFT is a novel and consistent theory for the evaluation of the ambient temperature for nonlinear- series and simple parallel resistors combinations.

Finally, it should be state that in the paper proposed STFT, should become a standard analytical method to the nonlinear functional equation for the ambient temperature intervals estimation (computation) for given intervals of equivalent resistances, to the broad temperature domain. Also, from the practical aspect, the presented method might be considered as a mathematical tool for compensation of ambient temperature impact to the signals in electrical circuits.

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