Thermal analysis of fuses with variable cross-section fuselinks

Electric Power Systems Research 92 (2012) 73–80

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Thermal analysis of fuses with variable cross-section fuselinks Gabriel Chiriac ∗ Gheorghe Asachi Technical University of Iasi, 21-23, Blvd. Dimitrie Mangeron, Iasi 700050, Romania

a r t i c l e

i n f o

Article history: Received 5 January 2012 Received in revised form 2 May 2012 Accepted 17 June 2012 Available online 17 July 2012 Keywords: Thermal analysis Fuselink Notch Thermal simulation

a b s t r a c t This paper describes a thermal model of fast fuse suitable for power electronic circuits. This model can be used to analyse the thermal behaviour of the fuselink during steady-state conditions when different types of notches are applied. Results from mathematical models have been compared with those from 3D thermal models using software based on the finite element method and from experimental tests. The temperature values from thermal simulations and experimental data are very similar those coming from mathematical models. © 2012 Elsevier B.V. All rights reserved.

1. Introduction

2. Thermal model

To prevent damage because of overcurrents, many electrical equipments use the fuse protection. This protection is to be found also in static electrical devices and for the equipments used on mobile applications, as in vehicles, for increasing safety. Moreover, for vehicle application, electric wires and the equipments are restricted as weight, which requires a better design of any component, including fuses [1,2]. According to [3] for the power converters it is important to estimate the harmonic power losses in fuses to assess the lifetime and the protection level on the converters. The temperature inside the circuit components is essential to be measured, because the temperature rise of the fuse element can be expressed in terms of temperature rise of particularly vulnerable circuit components [4]. If fuses are not installed correctly, overloaded accidents would easily occur [5]. Nowadays, to obtain a better characteristic curve, the fuselink has evolved towards more complicated geometric shapes, presenting restrictions at regular intervals along their length [6]. There are several mathematical approaches to model the operation of a fuse [7–9], in order to study the thermal behaviour of relatively simple fuse geometries without notches and with one single notch, respectively [10]. Other models based on finite element method have been reported in [11–14]. This study attempts to model the operation of fuse with variable cross-section fuselink used to protect power electrical installations in the case of steady-state conditions.

The aim of this study is to develop a mathematical model of a fuse with variable cross-section fuselink used for power semiconductor devices protection. A typical application is the protection of power converters to control the asynchronous three-phase motors from the trains. A block diagram is presented in Fig. 1. There are two power converters PSM-80, supplied with medium voltage (3 kV, direct current), through switch-disconnectors S1 and S2 . The protection to overcurrents is provided by fuses B1 . The inverters control the three-phase electric motors through the switch-disconnectors S3 . The protection to fault currents is provided by fuse-switchdisconnector K2 . The starting point is the power balance equation [15,16], Pc = Pt − Pr + Pa

The left term of the equation is the heating power from the current flow, Pc . It is in balance with the heat stored by temporal change of temperature Pt , the power removed from the element by thermal conduction Pr , and the thermal power dissipated to the surrounding area by the surface convection, Pa . For Pc , Pt , Pr and Pa , the following equations can be written: Pc = ()j2 (x, t)

0378-7796/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.epsr.2012.06.010

(2)

∂(x, t) ∂t

(3)

∂2 (x, t) ∂x2

(4)

Pt =  Pr = 

∗ Tel.: +40 232 278683; fax: +40 232 237627. E-mail address: [email protected]

(1)

Pa = k()

 lx  (x, t) − a S(x)

(5)

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G. Chiriac / Electric Power Systems Research 92 (2012) 73–80

Fig. 3. Fuselink with rectangular cross-section.

the following relation is obtained: ˛0 I 2 ϑ(x) 2·k ∂2 ϑ(x) + · − · ϑ(x) g ∂x2 4g 2 [a + (x/x1 ) · ((b/2) − a)]2 =−

Fig. 1. Power converter block diagram from a train supplied with 3 kVdc.

Thus, the differential equation of heating for fuselinks with variable cross-section is:



2

∂(x, t) ∂ (x, t) =  ·· + () · j2 (x, t) − ∂t ∂x2

k  · lx S(x)

(6)

· [(x, t) − a ]

2.1. Case 1. Notch with rhombic shape For the notches with the geometry as shown in Fig. 2, the crosssection and perimeter length are:



S(x) = 2 · g · a +

x · x1





b −a 2



+g ;



lx = 4 · a +

x · x1





b −a 2



+g

(7)

Taking into account that in the case of fuselinks the thickness is ranging from 0.02 mm to 0.3 mm and the width of the fuselink is bigger than 2 or 3 mm, during some computations the thickness can be neglected with respect to the width. Therefore, the perimeter length becomes:



lx = 4 · a +

x · x1

b

2



−a

(8)

Hence, the heating equation of the fuselink, in steady-state conditions, has the following expression: 

∂2 (x) I2 2 · k() + () · − · [(x) − a ] = 0 2 g ∂x2 4 · g 2 · [a + (x/x1 ) · ((b/2) − a)]

(9)

The electrical resistivity has a significant temperature variation and can be estimated through either a parabolic variation or a linear one. The experimental tests [17–19] concluded that the difference between these two types of variation is not so important. For the electrical resistivity a linear variation with the temperature has been considered,  = 0 [1 + ˛( − a )]

(10)

and with the notation: ϑ(x, t) = (x, t) − a

(11)

0 I 2

(12)

2 4g 2 [a + (x/x1 ) · ((b/2) − a)]

with the following limit conditions: ϑ(x) = ϑ1

x = x1 ;

and

x = 0;

dϑ(x) =0 dx

(13)

The value for the heating ϑ1 is about 102 ◦ C and has been established from experimental tests. In the case of fuselinks with notches, because the length of the notch is very small, a few millimetres or less, results that their lateral surface is very small. Therefore, the heat transfer of lateral surface can be neglected (k = 0). Considering this assumption, Eq. (12) becomes: ˛0 I 2 ϑ(x) ∂2 ϑ(x) + · ∂x2 4g 2 [a + (x/x1 ) · ((b/2) − a)]2 =−

0 I 2 4g 2 [a + (x/x1 ) · ((b/2) − a)]

(14)

2

and has the solution, ϑ(x) =



ϑ1 +

1 ˛

 r er1 ln(1+|x|c) − r er2 ln(1+|x|c) 2 1 r2 er1 ln(1+|x1 |c) − r1 er2 ln(1+|x1 |c)

−

1 ˛

(15)

2.2. Case 2. Notch with rectangular cross-section When the notches have a rectangular cross-section, as shown in Fig. 3, the expressions for perimeter length and cross-section are, S(x) = ag,

lx = 2(g + a)

(16)

Because the thickness has very small values compared to the width of the fuselink, during computations, the thickness can be neglected respect to the width. Hence, taking into account Eq. (11), the fuselink heating in steady-state conditions and considering a linear variation for the electric resistivity with the temperature, the following expression occurs: ∂2 ϑ(x) + ∂x2



˛0 I 2 2·k − g a2 g 2



· ϑ(x) = −

0 I 2 a2 g 2

(17)

with the same limit conditions from (13). Because the length of the notch is very small it results that their lateral surface is very small and the heat transfer of lateral surface can be neglected (k = 0). Therefore, Eq. (17) becomes: ˛0 I 2 0 I 2 ∂2 ϑ(x) + · ϑ(x) = − 2 2 ∂x2 a2 g 2 a g

(18)

and has the solution: Fig. 2. Fuselink with rhombic cross-section.

ϑ(x) = A sin rx + B cos rx −

1 ˛

(19)

G. Chiriac / Electric Power Systems Research 92 (2012) 73–80

75

Table 1 Relative errors calculations for notch with circular shape for different approximations of heating solution. Geometric parameters [mm]

4

◦

ϑ x [ C] ϑ x6 [◦ C] ϑexp [◦ C] ␧ x4 [%] ␧ x6 [%]

Fig. 4. Fuselink with circular cross-section.

Current [A]

a = 1; x1 = 2

a = 1.5; x1 = 1.5

a = 2; x1 = 1

40

63

100

114.07 111.37 110.5 −3.23 −0.78

104.99 104.4 102.9 −2.03 −1.45

102.77 102.6 101.2 −1.55 −1.38

106.76 105.7 104.9 −1.77 −0.76

114.07 111.37 110.5 −3.23 −0.78

134.11 126.5 128.1 −4.69 1.24

where A and B are integration constants and r is the root of the characteristic equation: r=

I ag



˛0 

(20)

Taking into account the limit conditions (13) the following solution has been obtained:







cos ((I/ag)



ϑ(x) = ϑ1

˛0 /)x

cos ((I/ag)



⎡ +

1 ˛

˛0 /)x1



 − 1⎦

cos ((I/ag)

⎣



cos ((I/ag)

˛0 /)x

⎤ (21)

˛0 /)x1

2.3. Case 3. Notch with circular shape For the notches with circular shape as shown in Fig. 4, the expressions of cross-section and perimeter length are,



S(x) = 2g r + a −



r 2 − x2 ,



lx = 4 g + r + a −



r 2 − x2 (22)

Taking into account the same assumptions as in previous cases, the general equation (6) for circular notch becomes,



r+a−

r 2 − x2

2 d2 ϑ(x) dx2

+

˛0 I 2 0 I 2 ϑ(x) = − 4g 2 4g 2

(23)

and has the same limit conditions from (13). After computations, the solution is, ϑ(x) =

+

+

rest of the transversal section of fuselink and increases of the current density too. When the current through fuselink is increasing from 40 A to 100 A, Fig. 7, because of the increasing current density, the maximum of the heating will increase too (from 106.5 to 132.1 ◦ C). For the fuselink with rectangular notch shape, when the parameter a is decreasing from 4 mm to 1 mm, it is to observe, as in the precedent case, an increasing of the maximum of the heating (from 108.2 to 231.3 ◦ C, Fig. 8). Also, the increase of x1 from 1 mm to 3 mm implies an increase of the surface of the reduction section of the notch and from this an increase of the maximum value for heating from 113.2 to 227.6 ◦ C, Fig. 9. As it is to expect, the increasing current by fuselink, from 40 A to 75 A, will give, inevitably, an increasing for the maximum heating, from 146 to 312.32 ◦ C, Fig. 10. Compared with the rhombic notch shape, in the same overall size for fuselink, for the rectangular notch shape the heating is higher for the same current variation, which is to explain by the smaller section of the notch area. For the notch with circular shape, to maintain a constant width of the fuselink, an increasing of the parameter a will imply a reduction of the notch radius (which is equal with parameter x1 ) and vice versa. For a reduction of the notch radius from 2.5 mm to 1.5 mm the maximum heating is reduced from 192.9 to 104.9 ◦ C, Fig. 11, having the same explanations as in the above-mentioned cases. Inevitably, a current increasing from 40 A to 100 A gives an increasing of the maximum heating from 106.7 to 134.1 ◦ C, Fig. 12. In order to limit the complexity of the computations, the heating solution (24) includes the terms up to the 4th order. With the aim to increase the accuracy of the heating values, the computations with the terms

2r 3 (2a2 ϑ1 − x12 )[12(r + a − 1) + m] − nx14 (r + a) 2r 3 (2a2 − x12 )[12(r + a − 1) + m] + mx14 (r + a) 2r 3 [12(r + a − 1) + m]{2r 3 (ϑ1 − 1)[12(r + a − 1) + m] − x12 (n + mϑ1 )(r + a)}

{x12 (r

+ a) − 2r 3 [12(r + a − 1) + m]}{2r 3 (2a2 − x12 )[12(r + a − 1) + m] + mx14 (r + a)} (r + a){2r 3 (ϑ1 − 1)[12(r + a − 1) + m] − x12 (n + mϑ1 )(r + a)}

{2r 3 [12(r

+ a − 1) + m] − x12 (r + a)}{2r 3 (2a2 − x12 )[12(r + a − 1) + m] + mx14 (r + a)}

x2

x4

(24)

3. Discussion of the results Further on, the influence of the geometrical and physical parameters upon fuselink temperature is analyzed [20]. The fuselink with rhombic notch shape has been analyzed for the variation of a in Fig. 5, x1 in Fig. 6 and for current in Fig. 7. It is to see an increasing of the maximum heating of the fuselink (from 106.7 to 129.1 ◦ C, Fig. 5) when the parameter a is decreasing from 2 mm to 0.5 mm, which is explained by the reduction of the fuselink surface and, implicitly, an increasing of the current density. When the parameter x1 increases from 1 mm to 3 mm, Fig. 6, it is to observe an increasing of the maximum of the heating (from 103.2 to 113.4 ◦ C), which is explained by increasing of the surface of the reduction zone compared with the

up to the 6th order have been done. Related to the experimental values in the middle of the notch, actually the maximum values, the relative errors in the case with terms up to the 4th order and up to the 6th order, have been computed. These values are reported in Table 1. The relative errors (ε x4 , ε x6 ) have been calculated for different geometric parameters (a and x1 ) and at different current values. It results a maximum error of −3.23% when the parameters have the values of a = 1 mm and x1 = 2 mm, and in the case when the current has been varied there is a maximum error of −4.69%. Of course, the higher values belong to the situation when the heating solution included the terms up to the 4th order.

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G. Chiriac / Electric Power Systems Research 92 (2012) 73–80

135 130 125

ϑ[°C]

120 115 110 105 100 -3

-2

-1

0

1

2

3

x[mm ] ϑ1calc

ϑ2calc

ϑ3calc

ϑ1sim

ϑ2sim

ϑ3sim

ϑ1exp

ϑ2exp

ϑ3exp

Fig. 5. Comparison between computed (ϑ1calc , ϑ2calc , ϑ3calc ), simulated (ϑ1sim , ϑ2sim , ϑ3sim ) and experimental (ϑ1exp , ϑ2exp , ϑ3exp ) values of the heating in the case of fuselink with rhombic notch for parameter a variation (a = 0.5, 1, and 2 mm) with x1 = 3 mm and I = 63 A.

116 114 112

ϑ[°C]

110 108 106 104 102 100 -3

-2

-1

0

1

2

3

x[mm ] ϑ1calc

ϑ2calc

ϑ3calc

ϑ1sim

ϑ3sim

ϑ2sim

ϑ1exp

ϑ2exp

ϑ3exp

Fig. 6. Comparison between computed (ϑ1calc , ϑ2calc , ϑ3calc ), simulated (ϑ1sim , ϑ2sim , ϑ3sim ) and experimental (ϑ1exp , ϑ2exp , ϑ3exp ) values of the heating in the case of fuselink with rhombic notch for parameter x1 variation (x1 = 1, 2, and 3 mm) with a = 1 mm and I = 63 A.

135 130 125

ϑ[°C]

120 115 110 105 100 -3

-2

-1

0

1

2

3

x[mm ] ϑ1calc

ϑ2calc

ϑ3calc

ϑ1sim

ϑ2sim

ϑ3sim

ϑ1exp

ϑ2exp

ϑ3exp

Fig. 7. Comparison between computed (ϑ1calc , ϑ2calc , ϑ3calc ), simulated (ϑ1sim , ϑ2sim , ϑ3sim ) and experimental (ϑ1exp , ϑ2exp , ϑ3exp ) values of the heating in the case of fuselink with rhombic notch for current variation (I = 40, 63, and 100 A) with a = 1 mm and x1 = 3 mm.

G. Chiriac / Electric Power Systems Research 92 (2012) 73–80

77

240 220 200

ϑ[°C]

180 160 140 120 100 -3

-2

-1

0

1

2

3

x[mm ] ϑ1calc

ϑ2calc

ϑ3calc

ϑ3sim

ϑ2sim

ϑ1sim

ϑ1exp

ϑ2exp

ϑ3exp

Fig. 8. Comparison between computed (ϑ1calc , ϑ2calc , ϑ3calc ), simulated (ϑ1sim , ϑ2sim , ϑ3sim ) and experimental (ϑ1exp , ϑ2exp , ϑ3exp ) values of the heating in the case of fuselink with rectangular cross-section notch for parameter a variation (a = 1, 2, and 4 mm) with x1 = 3 mm and I = 63 A.

240 220 200

ϑ[°C]

180 160 140 120 100 -3

-2

-1

0

1

2

3

x[mm ] ϑ1calc

ϑ2calc

ϑ3calc

ϑ3sim

ϑ2sim

ϑ1sim

ϑ1exp

ϑ2exp

ϑ3exp

Fig. 9. Comparison between computed (ϑ1calc , ϑ2calc , ϑ3calc ), simulated (ϑ1sim , ϑ2sim , ϑ3sim ) and experimental (ϑ1exp , ϑ2exp , ϑ3exp ) values of the heating in the case of fuselink with rectangular cross-section notch for parameter x1 variation (x1 = 1, 2, and 3 mm) with a = 1 mm and I = 63 A.

350 300

ϑ[°C]

250 200 150 100 -2

-3

-1

0

1

2

3

x[mm ] ϑ1calc

ϑ2calc

ϑ3calc

ϑ1sim

ϑ2sim

ϑ3sim

ϑ1exp

ϑ2exp

ϑ3exp

Fig. 10. Comparison between computed (ϑ1calc , ϑ2calc , ϑ3calc ), simulated (ϑ1sim , ϑ2sim , ϑ3sim ) and experimental (ϑ1exp , ϑ2exp , ϑ3exp ) values of the heating in the case of fuselink with rectangular cross-section notch for current variation (I = 40, 63, and 75 A) with a = 1 mm and x1 = 3 mm.

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G. Chiriac / Electric Power Systems Research 92 (2012) 73–80

200 190 180 170

ϑ[°C]

160 150 140 130 120 110 100 -3

-2

-1

0

1

2

3

x[mm ] ϑ1calc

ϑ2calc

ϑ3calc

ϑ1sim

ϑ2sim

ϑ3sim

ϑ1exp

ϑ2exp

ϑ3exp

Fig. 11. Comparison between computed (ϑ1calc , ϑ2calc , ϑ3calc ), simulated (ϑ1sim , ϑ2sim , ϑ3sim ) and experimental (ϑ1exp , ϑ2exp , ϑ3exp ) values of the heating in the case of fuselink with circular notch for parameters a and x1 variation (a = 0.5 mm and x1 = 2.5 mm; a = 1 mm and x1 = 2 mm; and a = 1.5 mm and x1 = 1.5 mm) with I = 63 A.

4. Thermal simulations

Hence, the thermal load has a spatial distribution depending on the notch shape. It was considered the convection condition with a uniform spatial variation like boundary condition for the outer boundaries such as constant cross-section on both lateral sides of the notches [21]. An ambient temperature of 25 ◦ C has been considered. The temperature distributions for all three different types of notches are presented in Figs. 13–15. The heating variations under the influence of different geometrical and physical parameters are shown in comparison with the results from mathematical models, from Figs. 5–12. From all the simulation results, it can be noticed that the temperature values in steady-state conditions for the thermal models are very similar to those from the mathematical models. The differences appear in the 3D thermal simulations because it was considered the geometrical model not only in the notch area but also for a part of the constant fuselink, symmetrically, for the both sides of the notch. These areas, with a bigger cross-section compared with the one of the notch, are in fact heatsinks, having the effect of a decreasing for the maximum of the heating and an increasing for the extreme thermal values of the notch. The bigger cross-sections have a thermal influence due to a thermal flux which cannot be neglected.

In order to obtain the temperature distribution in steadystate conditions, a 3D thermal model of three types of fuselinks have been developed using specific software, the Pro-ENGINEER, an integrated thermal design tool. For all thermal simulations a 3D finite element Pro-MECHANICA software has been used. The thermal models include the rhombic notch, a rectangular cross-section notch and a circular notch in concordance with the mathematical models which have been established in the previous section. As general oversize, all the fuselinks have a thickness of 0.1 mm, width by 6 mm and a length of 12 mm, except the fuselink with rectangular notch where the total fuselink length is 18 mm. The fuses to protect power converters have the fuselinks made from silver with the following material parameters:  = 10.5 × 103 kg/m3 ,  = 1.63 × 10−8 m,  = 418 W/m ◦ C, ˛ = 4.1 × 10−3 1/◦ C, c = 232 J/kg ◦ C. The thermal load has been applied on component with a volume distribution using the wellknown expression, p = j2 =

I 2 S 2 (x)

(25)

140 135 130

ϑ[°C]

125 120 115 110 105 100 -2

-1,5

-1

-0,5

0

0,5

1

1,5

2

x[mm ] ϑ1calc

ϑ2calc

ϑ3calc

ϑ1sim

ϑ2sim

ϑ3sim

ϑ1exp

ϑ2exp

ϑ3exp

Fig. 12. Comparison between computed (ϑ1calc , ϑ2calc , ϑ3calc ), simulated (ϑ1sim , ϑ2sim , ϑ3sim ) and experimental (ϑ1exp , ϑ2exp , ϑ3exp ) values of the heating in the case of fuselink with circular notch for current variation (I = 40, 63, and 100 A) with a = 1 mm and x1 = 2 mm.

G. Chiriac / Electric Power Systems Research 92 (2012) 73–80

Fig. 13. Temperature distribution in the case of fuselink with rhombic notch (a = 1 mm, x1 = 3 mm, and I = 63 A).

79

secondary side of the current source flows through the fuselink F. The current value is measured by an ammeter A, through a current transformer CT. Due to technological reasons and small dimensions of the notches, the thermocouples Th, type K, have been mounted only in the middle of the notches (x = 0). The small voltage signal provided by thermocouple was the input data for a data acquisition board type PC-LPM-16 which can be programmed with LabVIEW software. The sampling rate was 50 kS/s and the analogue inputs have a resolution on 12 bits. The comparisons between simulation, the calculated and experimental results are presented from Figs. 5–12. The experimental temperature values are smaller than the computation results. This is because during the experimental tests, the fuse was mounted on its fuse-carrier and there are conductors to connect the fuse with electric circuit. Because of their volume and thermal capacity, these entire components act like real heatsinks for the fuselink, resulting in an important heat dissipation rate. On the other hand, the differences between the temperature values resulting from experimental tests and those obtained during simulations are due to various factors: measurement errors, thermal model simplifications and mounting test conditions. The thermal model has not included different types of conductors from the fuse to other devices to be protected like power semiconductors. Nevertheless, the maximum difference between the mathematical models, simulation results and experimental data is less than 5 ◦ C. 5. Conclusion

Fig. 14. Temperature distribution in the case of fuselink with rectangular crosssection notch (a = 2 mm, x1 = 3 mm, and I = 63 A).

Fig. 15. Temperature distribution in the case of fuselink with circular notch (a = 1 mm, x1 = 2 mm, and I = 63 A).

Having the opportunity to simulate the thermal processes of the fuselink in relation to the fuse design enables new features for the optimization of fast fuses. On the other hand, new fuse designs can be evaluated for their thermal behaviour. This will have a great impact on the development and test costs of new fuses. Extending the thermal models for the specific applications enables the user of fast fuses to choose the right ratings, to evaluate critical load cycles and to identify potential overload capacities. From the outlined results, in steady-state conditions, the fuse link with circular and/or rhombic shape of the notch has lower values for the maximum temperature of the notches respect to the fuse link with rectangular shape of the notch. Therefore, a fuse with circular and/or rhombic shape of the notches will be proper to be used in different industrial applications. Appendix A. A.1. Demonstration of Eq. (15)

To validate the thermal model some experimental tests have been developed. An electric circuit diagram used for experimental tests is shown in Fig. 16. The main switch K supplies the auto-transformer ATR which adjusts the input voltage for the high current source CS. The high value current provided by the

Eq. (14) can be written as follows (1 + xc)2 ·

∂2 ϑ(x) + d · ϑ(x) = −e ∂x2

(A1)

With the substitution, K

~

A ATR

(1 + xc) = ey

CS CT

PC/DAQ F

Th

(A2)

Eq. (A1) becomes: d ∂ϑ(y) e ∂2 ϑ(y) + 2 ϑ(y) = − 2 − ∂y ∂y2 c c

(A3)

with the solution:

Fig. 16. Experimental set-up.

ϑ(y) = Aer1 y + Ber2 y −

1 ˛

(A4)

80

G. Chiriac / Electric Power Systems Research 92 (2012) 73–80

where A and B are integration constants and r1 , r2 are the roots of the characteristic equation:



r1,2

1 = ± 2

˛0 x12 I 2 1 − 2 4 4g 2 ((b/2) − a)

(A5)

Taking into account the limit conditions (13) and the substitution (A2), the solution is given by Eq. (15). A.2. Demonstration of Eq. (24) The analytical solution of the differential equation (23) with variable coefficient can be approximated by a polynomial form with the general relation, ϑ(x) =

∞ 

ak · xk

(A6)

k=0

On the other hand, the variable coefficient of the first term of the differential equation includes a square root expression which can be transformed using the Newton binomial formula into,



r 2 − x2 = r ·

=r·

1−

x2 r2

1/2

∞  (−1)k · (1/2) · ((1/2) − 1) . . . ((1/2) − k + 1·) x2k

·

k!

r 2k

k=0

(A7)

Therefore, taking into account the expressions (A6) and (A7), Eq. (23) becomes,



r+a−r·

∞  (−1)k · 12 · ((1/2) − 1) . . . ((1/2) − k + 1·) x2k

k! k=0

×

∞  k=2

∞ 

k(k − 1)ak · xk−2 + m

ak · xk = −n

·

2

r 2k

(A8)

k=0

In order to limit the complexity of the computations, the solution (A6) will consider the terms up to the 4th order. Considering the initial condition at x = 0, with (dϑ(x)/dx) = 0, it results from relations of coefficients identification that odd coefficients of the polynomial approximate solution (a1 , a3 , . . ., a2k+1 ) are equal to zero. Taking into account the limit condition at x = x1 , ϑ(x1 ) = ϑ1 , it result the expressions for the even coefficients (a0 , a2 and a4 ). Hence, the solution is given by Eq. (24). List of symbols j current density  electrical resistivity  material density  specific heat thermal conductivity   temperature ambient temperature a S(x) cross-section of the fuselink coefficient of electrical resistivity variation with temper˛ ature k convection coefficient g thickness of the fuselink

a b r c d e m n

distance between the peak of the notch and the nearest edge of the fuselink width of the fuselink radius of the notch in the case of fuselink with circular notch by definition c = ((b/2) − a)(1/ax1 ) by definition d = ˛0 I 2 /4g 2 a2 by definition e = 0 I 2 /4g 2 a2 by definition m = ˛0 I 2 /4g 2 by definition n = 0 I 2 /4g 2 .

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