Temperature Surges in Current-Limiting Circuit Devices

Fowler, A. C.; Frigaard, I.A.; Howison, Sam

doi:10.1137/0152058

Cited by 65 publications

(49 citation statements)

References 13 publications

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“…The above quantity, for these values, is positive (S q M 2 ∼ 0.4286). Therefore, equation (3.7) can be written aṡ 13) which has as a solution…”

Section: C)mentioning

confidence: 99%

Estimates of Blow-Up Time for a Non-Local Reactive–convective Problem Modelling Ohmic Heating of Foods

Nikolopoulos¹,

Tzanetis²

2006

Proceedings of the Edinburgh Mathematical Society

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In this work, we estimate the blow-up time for the non-local hyperbolic equation of ohmic type, ut + ux = λf (u)/( 1 0 f (u) dx) 2 , together with initial and boundary conditions. It is known that, for f (s), −f (s) positive and ∞ 0 f (s) ds < ∞, there exists a critical value of the parameter λ > 0, say λ * , such that for λ > λ * there is no stationary solution and the solution u(x, t) blows up globally in finite time t * , while for λ λ * there exist stationary solutions. Moreover, the solution u(x, t) also blows up for large enough initial data and λ λ * . Thus, estimates for t * were found either for λ greater than the critical value λ * and fixed initial data u 0 (x) 0, or for u 0 (x) greater than the greatest steadystate solution (denoted by w 2 w * ) and fixed λ λ * . The estimates are obtained by comparison, by asymptotic and by numerical methods. Finally, amongst the other results, for given λ, λ * and 0 < λ − λ * 1, estimates of the following form were found: upper bound + c 1 ln[c 2 (λ − λ * ) −1 ]; lower bound c 3 (λ − λ * ) −1/2 ; asymptotic estimate t * ∼ c 4 (λ − λ * ) −1/2 for f (s) = e −s . Moreover, for 0 < λ λ * and given initial data u 0 (x) greater than the greatest steady-state solution w 2 (x), we have upper estimates: either c 5 ln(c 6 A −1 0 + 1) or + c 7 ln(c 8 ζ −1 ), where A 0 , ζ measure, in some sense, the difference u 0 − w 2 (if u 0 → w 2 +, then A 0 , ζ → 0+). c i > 0 are some constants and 0 < 1, 0 < A 0 , ζ. Some numerical results are also given.

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“…The above quantity, for these values, is positive (S q M 2 ∼ 0.4286). Therefore, equation (3.7) can be written aṡ 13) which has as a solution…”

Section: C)mentioning

confidence: 99%

Estimates of Blow-Up Time for a Non-Local Reactive–convective Problem Modelling Ohmic Heating of Foods

Nikolopoulos¹,

Tzanetis²

2006

Proceedings of the Edinburgh Mathematical Society

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“…A form of the dependence of the (dimensionless) resistivity f on temperature for an NTC-thermistor can be f (s) = e −s or f (s) = 1 (1+s) p for p ≥ 2. A derivation of the time evolution model that describes the temperature profile of a thermistor can be found in [5] and [13]. The resulting parabolic equation is non-local and has the form…”

Section: Numerical Solution Of a Non-local Elliptic Problem 769mentioning

confidence: 99%

“…In particular, if the electrical resistivity increases with increasing temperature, then f is an increasing function and the thermistor device is called Positive Temperature Coefficient thermistor (PTC-thermistor). We refer the reader to [5] for an application of a PTC-thermistor made of a ceramic material for which the resistivity increases rapidly with temperature. Also, if the electrical resistivity decreases with increasing temperature, then f is a decreasing function and the thermistor device is called Negative Temperature Coefficient thermistor (NTCthermistor).…”

Section: Numerical Solution Of a Non-local Elliptic Problem 769mentioning

confidence: 99%

“…However, for a general resistivity function f the solution of the steady state problem is not available in analytic form. This restriction motivates us to consider the problem of constructing accurate numerical approximations of the solution of the non-local elliptic problems (BVR) and (BVD), which is also of independent practical interest from the point of view of applications (see, e.g., [5], [9]). …”

Section: Numerical Solution Of a Non-local Elliptic Problem 769mentioning

confidence: 99%

“…In the rest of the paper, we will refer to the boundary value problem (1) and (2) as problem (BVR), and to the boundary value problem (1) and (3) as problem (BVD). The equation (1) models the steady state temperature profile of a thermistor device with electrical resistivity (known also as specific electrical resistance) f (see [5], [13]). A thermistor is a thermo-electric device made of ceramic material whose…”

mentioning

confidence: 99%

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Numerical Solution of a Non-Local Elliptic Problem Modeling a Thermistor with a Finite Element and a Finite Volume Method

Nikolopoulos

Zouraris

2008

Progress in Industrial Mathematics at ECMI 2006

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Abstract. We consider the following non-local elliptic boundary value problem:where α and λ are positive constants and f is a function satisfyingThe solution of the equation represents the steady state of a thermistor device. The problem has a unique solution for a critical value λ * of the parameter λ, at least two solutions for λ < λ * and has no solution for λ > λ * . We apply a finite element and a finite volume method in order to find a numerical approximation of the solution of the problem from the space of continuous piecewise quadratic functions, for the case that λ < λ * and for the stable branch of the bifurcation diagram. A comparison of these two methods is made regarding their order of convergence for f (s) = e −s and f (s) = (1 + s) −2 . Also, for the same equation but with Dirichlet boundary conditions, a situation where the solution is unique for λ < λ * , a similar comparison of the finite element and the finite volume method is presented.

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Various methods to the thermistor problem with a bulk electrical conductivity

Kutluay

Bahadır

Özdeş

1999

Int. J. Numer. Meth. Engng.

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SUMMARYIn this paper, Explicit Finite Difference (EFD), Galerkin Finite Element (GFE) and Heat-Balance Integral (HBI) methods are applied to the one-dimensional thermistor problem with a bulk electrical conductivity to obtain its steady-state solutions. It is shown that EFD, GFE and HBI solutions exhibit the correct physical characteristic of the problem, and they are in very good agreement with the exact solution. The only marked difference is time to attain steady states.

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Temperature Surges in Current-Limiting Circuit Devices

Cited by 65 publications

References 13 publications

Estimates of Blow-Up Time for a Non-Local Reactive–convective Problem Modelling Ohmic Heating of Foods

Estimates of Blow-Up Time for a Non-Local Reactive–convective Problem Modelling Ohmic Heating of Foods

Numerical Solution of a Non-Local Elliptic Problem Modeling a Thermistor with a Finite Element and a Finite Volume Method

Various methods to the thermistor problem with a bulk electrical conductivity

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