WO2015097218A1

WO2015097218A1 - Heating element with adjustable temperature distribution

Info

Publication number: WO2015097218A1
Application number: PCT/EP2014/079156
Authority: WO
Inventors: Thomas Wittkowski
Original assignee: Iee International Electronics & Engineering S.A.
Priority date: 2013-12-23
Filing date: 2014-12-23
Publication date: 2015-07-02
Also published as: DE112014005889T5; LU92344B1

Abstract

The invention concerns a heating element (1) for generating heat when connected to an electrical power source comprising a first feedline (2) and a second feedline (2a) extending generally along opposite longitudinal sides of the heating element (1), each feedline (2, 2a) comprising an electrical contact point to an electrical power source, each feedline (2, 2a) distributing the electrical power supply along it, at least a layer of resistive material (3) being sandwiched between the two feedlines (2, 2a), its width (w) being delimited by the feedlines (2, 2a), an electrical current flowing from the first feedline (2) to the second feedline (2a) in operation through the layer of resistive material (3) and thereby dissipating heat, characterized in that the layer of resistive material width (w) varies in function of the length coordinate (x) of the heating element (1).

Description

HEATING ELEMENT WITH ADJUSTABLE TEMPERATURE DISTRIBUTION Technical field

[0001 ] The present invention generally relates to a heating element with adjustable temperature distribution. As a preferential but non limitative application such a heating element may be used for heating surfaces in automotive vehicles, i.e. compartments of aviation, rail, and road vehicles with the aim of warming the occupants. A special but non limitative focus is laid on heating of all kinds of surfaces such as door panels, steering wheels, arm rests, and several more in the cabin of automotive vehicles, especially when these surfaces are non planar surfaces. Background Art

[0002] According to the state of the art, heating elements for generating heat when connected to an electrical power source comprise a first feedline or first bus and a second feedline or second bus extending generally along opposite longitudinal sides of the heating element. Each feedline comprises an electrical contact point for the electrical power source to provide and distribute the electrical power supply along it. At least a layer of resistive material is arranged between and in electrical contact with the two feedlines. In operation an electrical current flows from the first feedline to the second feedline through the layer of resistive material and thereby dissipates heat to perform the heating of a surface on which the heating element is applied.

[0003] One or a plurality of these heating elements may form or may be grouped to form an electrically powered heater. These heating elements may be operated either in serial or in parallel by specific electrical supply circuits, thereby heating respective areas of an object to heat.

[0004] The design and configuration of planar heaters are subject to various restrictions and limitations from a technical point of view but also from commercial considerations. Even though heaters in parallel electrical connection allow a maximum of freedom in design, there are still severe limitations regarding their employment in planar heaters. [0005] Some of the strongest constraints in heater design and configuration are imposed by the availability of materials of suitable sheet resistance, long term stability and economic acceptance. In some cases a too high electrical resistivity of the feedlines may, at least to some extent, be compensated by higher layer thickness or increased line width. But often stability requirements as well as geometrical and commercial constraints limit the thickness and/or the width of the feedlines.

[0006] Another constraint comes from the geometry of the area which should be heated. This area may be a flat panel as well as a non-flat area forming an element with a complex form. This is the case for example for a steering wheel which typically exhibits a three-dimensional shape with different proportions of the heated area. It is desired that the heating has to perform a uniform temperature across the complete shape of such a piece.

[0007] As another example, there may be openings for door openers or switches which do not need to be heated. In all cases, for its integration the heater is to be positioned as close as possible to the outer surface of the piece to be heated in order to warrant shortest thermal response times and to minimize energy consumption.

[0008] Persisting challenges with these heaters are to compensate the nonuniform heating of complex shaped areas, i.e. in general non-rectangular heating areas, as well as the unavoidable potential drop across the heating element. These challenges are particularly critical if the heater needs to yield high power density.

[0009] If the required power density, i.e. the power per area unit in the heating area is high and the heating element extends over a large area, one very soon faces the situation where the voltage drop along the length of the heating element exhibits significantly different power densities at both ends of a rectangular heating element. This leads to a strongly uneven temperature distribution which is in contradiction to the requirement of maximum power density over the complete heated area and high homogeneity of the temperature distribution. This is particularly critical when a certain maximum temperature at the surface to be heated must not be exceeded. [0010] In order to obtain a unifornn heating of a complex shaped area it has been proposed to provide wide, low resistance feedlines and to distribute many small-sized, discrete rectangular heating elements over the complex shaped area. However this leads to a reduced power density across the complex shaped area at a given maximum temperature. Also the material consumption for feedlines and heated area is high because of the unavoidable safety margins in heater design and the overlap of layers of different material.

[001 1 ] It has further been proposed to apply a technique using an array of heating elements in which the voltage drop along the array is taken into account. Several heating elements of different geometric extensions or sheet resistances are operated so that the power density across the complete array of heating elements is quite uniform. However, this approach relies on a rectangular shape of each heating element.

[0012] Consequently a heating element that compensates the deviations from rectangular geometry in a complex shaped area as well as the voltage drop in order to adjust the temperature distribution does not exist so far.

[0013] Heating elements for complex shaped areas are small and rectangular in order to warrant a uniform temperature distribution over a small rectangular area or complex shaped area and are therefore submitted to a vastly non-uniform and non-adjustable heating temperature distribution.

Technical problem

[0014] It is an object of the present invention to provide an improved heating element preferably with adjustable temperature distribution.

General Description of the Invention

[0015] This aim is achieved by a heating element for generating heat when connected to an electrical power source, comprising a first feedline and a second feedline extending generally along opposite longitudinal sides of the heating element, each feedline comprising an electrical contact point for connecting said feedline to an electrical power source. When connected to the electrical power source, each feedline distributes the electrical power supplied along the extension of the feedline. At least a layer of resistive material being is arranged between the two feedlines and in electrical contact with said feedlines, a width of said layer of resistive material being delimited by the feedlines. In operation, an electrical current flows from the first feedline to the second feedline through the layer of resistive material and thereby dissipating heat. According to the invention the arrangement of said first and second feedlines, and accordingly said layer of resistive material, is such that said width (w) of the layer of resistive material varies along a longitudinal direction in function of the length coordinate of the heating element and in that at least one of said first and second feedlines has a feedline width which varies in the same way in function of the length coordinate of the heating element..

[0016] In a possible embodiment, the width (w) of the layer of resistive material varies in function of the length coordinate of the heating element, starting from a lateral side of the heating element, according to the following equation:

whereby w(x) is the width of the layer of resistive material at a length coordinate x of the heating element starting from the said lateral side, wO the width of the layer of resistive material at the said lateral side, I the total length of the heating element, b and a respective coefficients with b different from zero.

[0017] Accordingly, the feedline width of at least one of said first and second feedlines, preferably of both feedlines, varies in function of the length coordinate of the heating element, starting from a lateral side of the heating element, according to the following equation:

whereby wfl(x) is the feedline width at a length coordinate x of the heating element starting from the said lateral side, wflO the feedline width at the said lateral side, I the total length of the heating element, b and a respective coefficients with b different from zero.

[0018] Preferably, a is a real value equal to or greater than zero while coefficient b may adopt any real value greater than -1 which is different from zero, a may e.g. be equal to 1 or 2. [0019] In an embodiment in which a is equal to 1 , the heating element exhibits a difference in electric potential, i.e. voltage u(x) in function of the length coordinate of the heating element defined by:

6- Rsq wfl 6- Rsq wfl 6- Rsq wfl 6- Rsq wfl uO being the voltage at the lateral side with electrical contact points, Rsqfl being the sheet resistance of the feedlines and Rsq the sheet resistance of the layer of resistive material, w and wfl being w(x) and wfl(x) at x = 0, respectively.

[0020] In an embodiment where a is equal to 2, the heating element preferably presents a voltage u(x) in function of the length coordinate defined by:

V2 i RsqflArcTan [Vfc] u(x) = uOCosh Tanh

VfajRsqVwVwfl

uO being the voltage at the electric contact points, Rsqfl being the sheet resistance of the feedlines and Rsq the sheet resistance of the layer of resistive material, w and wfl being w(x) and wfl(x) at x = 0, respectively.

[0021 ] It will be noted that the person skilled in the art can easily deduce the power density as a function of the length coordinate from the voltage.

[0022] The feedlines are preferably made of a high conductance material such as copper or silver applied on a substrate, the layer of resistive material being applied on the substrate. The substrate may be a polymer film or a fabric.

[0023] Preferably, a dielectric protection layer or a double sided layer adhesive covers the feedlines and the layer of resistive material.

[0024] Preferably, the layer of resistive material is made of a material presenting a positive temperature coefficient of resistance.

[0025] The present invention also concerns a heater comprising one or more heating elements, characterized in that it comprises at least such a heating element.

[0026] Finally the present invention also concerns a method of producing such a heating element, the method comprising the steps of: applying a high conductive layer on a substrate at each of two opposite sides of the heating element, these high conductive layers forming respectively a feedline of the heating element to be produced, applying a resistive material layer on the substrate, whereby the layer of resistive material width varies in function of the length coordinate of the heating element.

[0027] Preferably, the feedline width varies in function of the length coordinate of the heating element.

[0028] Preferably, the conductive layers and the layer of resistive material are applied by printing.

Brief Description of the Drawings

[0029] Preferred embodiments of the invention will now be described, by way of example, with reference of the accompanying drawings, in which :

FIG.1 shows curves expressing the ratio of feedline and resistive layer widths on respective initial feedline and resistive layer widths in function of the ratio length on total length for a heating element in accordance with an embodiment of the invention;

Fig.2 is a schematic top view of a heating element with respective feedline and resistive layer widths in function of the length of the heating element in accordance with a first embodiment of the invention,

FIG.3 shows a curve expressing the relative power density of the heating element in accordance with the first embodiment of the invention in function of the heating element length;

Fig.4 is a schematic top view of a heating element with respective feedline and resistive layer widths in function of the length of the heating element in accordance with a second embodiment of the invention,

FIG.5 shows a curve expressing the relative power density of the heating element in accordance with the second embodiment of the invention in function of the heating element length. Description of Preferred Embodiments

[0030] In respect of Figs. 2 and 4 showing respectively the first and the second embodiments of the present invention, a heating element 1 for generating heat when connected to an electrical power source comprises a first feedline 2 and a second feedline 2a extending generally along opposite longitudinal sides of the heating element 1 . Each feedline 2, 2a has a feedline width wfl and comprises an electrical contact point for connecting said feedline to an electrical power source, each feedline 2, 2a distributing the electrical power supply along it.

[0031 ] At least a layer of resistive material 3 is arranged between the two feedlines 2, 2a and in electrical contact with the feedlines. The width w of the layer of resistive material is delimited by the feedlines 2, 2a. In operation to perform the heating, an electrical current flows from the first feedline 2 to the second feedline 2a through the layer of resistive material 3 and thereby dissipates heat. It is to consider that the layer of resistive material may be of various types. It may be an effective layer applied on a substrate according to any suitable adhering process but it may also be separate elements of resistive elements applied locally on a substrate, the application may for example be performed by printing a suitable material onto the substrate.

[0032] It is possible that such a heating element 1 be made from a flexible but essentially planar substrate such as a polymer film, preferably PET, PEN, PU or silicone, or alternatively from some kind of fabric for example.

[0033] Feedlines 2, 2a that provide and distribute the electrical power to the layer of resistive material 3 may be made from high conductance material such as copper or silver, preferably, that are applied and structured either in an additive or in a combination of additive and subtractive process steps. Preparation of the layer of resistive material is preferably carried out in a printing process yielding a low conductance print of sheet resistance Rsq.

[0034] Finally, a protection and electrically insulating layer may be added to the heating element 1 or to the complete heater consisting of several such heating elements 1 combined or not with known heating elements. This third layer protects the all heating element 1 against environmental damage and may be typically applied by a printing process or in a simple wet coating process. [0035] Such a heating element 1 generally may be a constitutive part of a heater possibly with one or more heating elements, for example a heater in the form of a heating foil and provides a solution to the problem of planar heating surmounting some of the above listed constraints and limitations. A heater, possibly a heating panel, may comprise more than one of the here claimed heating elements and may comprise other heating elements which may be heating elements already known from the state of the art.

[0036] According to the present invention, the layer of resistive material width w varies in function of the length coordinate x of the heating element 1 .

[0037] Advantageously, each feedline 2, 2a also presents a feedline width wfl which varies in function of the length coordinate x of the heating element 1 .

[0038] The first feedline width and the second feedline width may vary in the same manner along the longitudinal direction. Furthermore, the layer of resistive material width w and the feedline width wfl may vary in the same specified manner.

[0039] In Figs. 2 and 4 a lateral side of the layer of resistive material is referenced point zero of the longitudinal direction or length coordinate, a point at which the feedlines and the layer initial widths are respectively referenced wflO and wO. The electrical contact point of each feedline 2, 2a may be located at this lateral side and these electrical contact points of each feedline 2, 2a may be located close to each other, i.e. by being located at the same length of the heating element 1 this length being referenced 0. In Figs. 2 and 4 the feedlines 2, 2a extend laterally beyond the lateral side of the layer of resistive material 3, but this embodiment is not limitative.

[0040] It should be noted that the arrangement of the points of contacts disposed on the feedlines 2, 2a at the region of the lateral side of the layer of resistive material 3 permit to simplify the equations expressing the feedline and layer of resistive material widths wfl(x) and w(x) in function of the length coordinate x of the heating element 1 . However it will be appreciated that such an arrangement of the points of contact is not an essential feature of the heating element 1 in accordance of the present invention. At least one of the points of contact of respective feedlines 2, 2a may be located for example somewhere else on the heating element 1 . The points of contact for the first feedline 2 and the second feedline 2a may be located at different lengths on the heating element 1 .

[0041 ] The heating element 1 in accordance with the invention does not need to have a rectangular form and exhibits an adjustable temperature distribution. This is achieved by controlled variation of the resistive material layer width w(x) and advantageously simultaneously controlled variation of the feedline width wfl(x), both as a function of the length coordinate x of the heating element 1 . Even with the sole variation of the resistive material layer width w(x), the invention enables e.g. an improved uniformity of the temperature distribution of heated surfaces which have unfavorable geometrical shapes, i.e. most frequently a complex shape.

[0042] Figs 1 is related to all the embodiments while figures 2 and 3 are related to the first embodiment and Figs. 4 and 5 are related to a second embodiment of a heating element 1 in accordance to the invention.

[0043] For the first and second embodiments, as shown in Figs. 2 and 4, the layer of resistive material width w varies in function of a length coordinate x of the heating element 1 , starting from one of its lateral side, according to the following equation:

whereby w(x) is the layer of resistive material width at a length coordinate x of the heating element 1 starting from the said lateral side, wO the layer of resistive material width at the said lateral side, I the total length of the heating element 1 , b and a respective coefficients with b different from zero.

[0044] According to an advantageously feature of the present invention, the feedline width wfl varies in function of the length coordinate x of the heating element 1 , starting from a lateral side of the heating element 1 , according to the following equation:

Equ. 1 .2 whereby wfl(x) is the feedline width at a length coordinate x of the heating element 1 starting from the said lateral side, wflO the feedline width at the said lateral side, I the total length of the heating element 1 , b and a respective coefficients with b different from zero.

[0045] For the two equations, when b equals zero, there is no change for the layer of resistive material width w and the feedline width w in respect to the length of the heating element and theses widths remain the initial widths wflO and wO. Consequently for b equals zero the heating element 1 illustrated in Figs. 2 and 4 would remain unchanged compared to a rectangular heating element of the state of the art.

[0046] According to one embodiment of invention, the exponent coefficient a may be a real value equal to or greater than zero while coefficient b may adopt any real value greater than -1 which is different from zero. Preferentially a may be equal to 1 or 2.

[0047] Figure 1 shows how the ratios w/wO and wfl/wflO vary as a function of x/l, i.e. the length coordinate on the total length of the heating element 1 , and this for two arbitrary choices of coefficient b, i.e. b = 0.5 represented in dashdotted curves and b = -0.3 represented in full curves. Values of exponent coefficient a are arbitrarily chosen respectively 0.5, 1 , 2, and 5 at this figure.

[0048] When both feedlines and layer of resistive material widths wfl, w vary simultaneously, it is apparent from this figure that for positive b the feedlines 2, 2a and the heating area, i.e. the layer of resistive material 3 widen with growing x/l, as it may be seen in Fig.2. Conversely for negative b the feedlines 2, 2a and the heating area, i.e. the layer of resistive material 3, narrow with growing x/l as it may be seen in Fig.4.

[0049] When the exponent coefficient a equals 1 , a linear variation of the feedline width wfl and the layer width w in respect of the ratio length coordinate x on total length I is obtained. When exponent coefficient a differs from 1 , a heating element with curved borderlines is obtained.

[0050] The variation of the potential differences along the heating element as a function of length coordinate x and for b different from zero is provided in Eqs. 2 and 3 for the technically most important cases where exponent coefficient a equals 1 or 2. Starting from this, the power density as a function of x may be calculated directly from the voltage as a function of x: for a = 1

6- Rsq wfl 6- Rsq wfl 6- Rsq wfl 6- Rsq wfl

Eq. 2 and for a = 2

Eq. 3

[0051 ] As far as a thermal equilibrium is reached the temperature increase is proportional to the power density. In the following paragraphs examples will be disclosed illustrating how the power density and consequently the temperature distribution are adjusted along the length of the heating element.

[0052] An example according to the first embodiment of the present invention is described in respect of Figs. 2 and 3.

[0053] In this embodiment, it is intended to increase the width of the heated area and consequently the layer of resistive material width w by 25 % from wO = 0.1 m at a length coordinate x = 0 to w = 0.125 m at a length coordinate x = I. It is further intended to simultaneously reduce the power density in a more or less linear manner to 50 % at the lateral end of the heating element opposite to the contact point of the feedlines 2, 2a compared to its value at the contacted lateral side of the heating element 1 . A possible application is the compensation of linearly varying thermal isolation between the heating element 1 and the surface to be heated.

[0054] Figure 2 shows a top view of the heating element 1 , the heating element 1 being not in scale at this figure. The curved feedline width wfl increases according to the same functional relation as the layer width w. The example illustrated at this figure is based on the following geometry and material parameters:

• Feedline width at x = 0: wflO = 0.014 m

• Feedline sheet resistance Rsqfl = 0.04 Ohm

• Layer of resistive material width at x = 0: wO = 0.1 m

• Heating element length I = 0.4 m

• Layer of resistive material sheet resistance Rsq = 30 Ohm

• On board voltage U0 = 13.5 V

• a = 2

• b = 0.25

[0055] These parameters result in the targeted power density of approx. 600 W/m² at the contacted side of the heating element and the power density reduces in a linear fashion to 300 W/m² at the end of the heating element. Figure 3 shows the achieved power density distribution.

[0056] An example according to a second embodiment of the present invention is described in respect of Figs. 4 and 5.

[0057] In this embodiment, it is desired to achieve a uniform power density, i.e. a uniform temperature distribution across the complete area of a trapezoidal shaped heating element. The following geometrical and resistance values are given by way of example:

• Width of feedline at x = 0, wflO = 0.014 m

• Sheet resistance of feedline, Rsqfl = 0.04 Ohm

• Width of the heating area of the heating element at x = 0, wO = 0.1 m

• Length of the heating area of the heating element, I = 0.4 m

• Sheet resistance of heating area, Rsq = 30 Ohm

• On board voltage, U0 = 13.5 V [0058] In this example of the second embodiment it is desired to achieve a power density of approx. 600 W/m² over the complete heating area. Good uniformity is achieved with:

• a = 1 corresponding to a linear variation of the layer of resistive material width w and the feedline width wfl, x/l thus determining the trapezoidal shape, and

• b = -0.18 corresponding to a narrowing of the feedline width wfl and of the layer of resistive material width w, with growing x l.

[0059] Figure 4 shows the schematic topview of a heating element 1 according to this example. The initial layer width wO of 0.1 m of the layer of resistive material at x = 0 diminishes linearly down to 0.082 m at the end of the heating element 1 , i.e. at x = I. The initial width wflO of 1 .4 cm of the feedlines at x = 0 diminishes linearly down to 1 .15 cm at the end of the heating element 1 .

[0060] As already mentioned above, the heating element 1 may be produced by means of printing on a polymer film. It is clear that other printing substrates such as fabrics or the backside of some decor may be used as well. In the preferred implementation a screen printing technique, for example flat bed or rotary, may be employed and the preferred substrate is a thermally stabilized polyester film of a preferred thickness between 50 and 125 micron. Optionally the surface of its printing side is pre-treated with some thin, adhesion promoting layer.

[0061 ] Firstly the highly conductive print may be applied on a substrate. It consists e.g. of a polymer thick film (PTF) of typical thickness between 5 and 10 micron. It contains silver flakes and a polymer binder. Alternatively an ink containing silver or copper nanoparticles as the electroconductive component may be employed. The wet print may be dried in a conveyer belt oven for approximately 90 s at an air temperature of 145 °C. The feedlines are consequently formed on the substrate.

[0062] Secondly the low conductance heating print may be printed, whereby this print forms the layer of resistive material. It consists of a PTF of typical thickness between 5 and 10 micron. It contains carbon black particles and a polymer binder. Alternatively an ink containing graphite, carbon nanotubes or graphene may be used. As for the highly conductive print the low conductance print may be dried in a conveyer belt oven for approximately 90 seconds at an air temperature of 145 °C.

[0063] Of relevance is also the alternative use of an ink/print that exhibits a positive temperature coefficient of resistance or PTCR. Choosing geometry and materials with an implementation where the low conductance print is carried out as a PTCR leads to homogeneous heat-up across the complete heated area. If heated up, such a print helps to further homogenize the temperature distribution across the heating element and provides additional safety in a hypothetical overheating scenario.

[0064] Thirdly a dielectric protection layer may be applied with a typical thickness between 20 and 30 micron. The dielectric is a UV-reactive system that is cross-linked with a UV dose of approx. 1 J/cm². Depending on the integration of the heating element into a panel forming a heater it is often also possible to apply a double-sided adhesive on the printed heating element which takes over the function of a protection layer and makes the dielectric layer obsolete. The double sided adhesive simultaneously ensures fixation of the heating element and its adhesion to the panel or to the decor.

[0065] The resulting power density of the so produced heating element is shown in Fig. 5. The power density exhibits a minimum and maximum at both ends of the heating element. The power density at x = 0 is still 600 W/m², the relative variations, however, amount to only ± 6 % across the length of the heating element.

[0066] The generated distribution of the power density and the according temperature distribution meet the requirements with respect to magnitude, in average approx. 600 W/cm², and uniformity, with a deviation less than 6 %, in an excellent manner. At the same time the trapezoidal shape of the heating element fits perfectly to the geometry of the panel heater so that the power density of the panel heater at a given maximum temperature is maximized.

[0067] The heating element with adjustable temperature distribution in accordance to the invention is a key constitutive element when the complete area of a panel needs to be heated to a given maximum temperature of high uniformity across the panel. The panel may well comprise several heating elements of the here described type or combined with other types of heating elements which are already known from the state of the art. In this way the heating of panels that exhibit non-heated openings, for example door opener, switches, etc ., is easily achieved.

Legend

1 Heating element

2 Feedline

2a Feedline

3 Layer of resistive material

a Exponent coefficient

b Coefficient

I Total length

uO Difference in electric potential, i.e. voltage, at the electric contact point

Rsq Layer of resistive material sheet resistance

Rsqfl Feedlines sheet resistance

w Layer of resistive material width

wO Initial layer of resistive material width

wfl Feedline width

wflO Initial feedline width

x Length coordinate

Claims

1. Heating element (1 ) for generating heat when connected to an electrical power source, said heating element comprising a first feedline (2) and a second feedline (2a) extending generally along opposite longitudinal sides of the heating element (1 ), each of said first and second feedline (2, 2a) comprising an electrical contact point for connecting said feedline to an electrical power source, at least a layer of resistive material (3) being arranged between the two feedlines (2, 2a) and in electrical contact with said feedlines, wherein a width (w) of said layer of resistive material is generally delimited by said feedlines (2, 2a), and wherein, in operation, an electrical current flows from the first feedline (2) to the second feedline (2a) through said layer of resistive material (3), characterized in that the arrangement of said first and second feedlines is such that said width (w) of the layer of resistive material varies in function of the length coordinate (x) of the heating element (1 ) and that at least one of said first and second feedlines (2, 2a) has a feedline width (wfl) which varies in the same way in function of the length coordinate (x) of the heating element (1 ).

2. Heating element (1 ) according to claim 1 , in which both said first and second feedlines (2, 2a) have a feedline width (wfl) which varies in the same way in function of the length coordinate (x) of the heating element (1 ).

3. Heating element (1 ) as claimed in claim 1 or 2, in which the width (w) of the layer of resistive material varies in function of the length coordinate (x) of the heating element (1 ), starting from a lateral side of the heating element (1 ), according to the following equation:

and in which the feedline width (wfl) of at least one of said first and second feedlines varies in function of the length coordinate (x) of the heating element (1 ), starting from a lateral side of the heating element (1 ), according to the following equation:

whereby w(x) is the width of the layer of resistive material at a length coordinate x of the heating element (1 ) starting from the said lateral side, wO the width of the layer of resistive material at the said lateral side, wfl(x) is the feedline width at a length coordinate x of the heating element (1 ) starting from the said lateral side, wflO the feedline width at the said lateral side, I the total length of the heating element (1 ), b and a respective coefficients with b different from zero.

4. Heating element (1 ) as claimed in claim 3, in which a is a real value equal to or greater than zero while coefficient b may adopt any real value greater than -1 which is different from zero.

5. Heating element (1 ) as claimed in claim 4, in which a is equal to 1 or 2.

6. Heating element (1 ) as claimed in claim 5, in which, when a is equal to 1 , the heating element (1 ) exhibits a voltage u(x) in function of the length coordinate (x) of the heating element (1 ) defined by:

ij-^RsqV Vwfl ij^/RsqVwVwfl ij-^RsqV Vwfl ij-^RsqV Vwfl uO being the voltage at the lateral side presenting the electrical contact points, Rsqfl being the sheet resistance of the feedlines (2, 2a) and Rsq the sheet resistance of the layer of resistive material (3), w and wfl being w(x) and wfl(x) at x = 0, respectively.

7. Heating element (1 ) as claimed in claim 5, in which, when a is equal to 2, the heating element (1 ) exhibits a voltage u(x) in function of the length coordinate (x) defined by:

uO being the voltage at the electric contact points, Rsqfl being the sheet resistance of the feedlines (2, 2a) and Rsq the sheet resistance of the layer of resistive material (3), w and wfl being w(x) and wfl(x) at x = 0, respectively.

8. Heating element (1 ) as claimed in anyone of the preceding claims, whereby the feedlines (2, 2a) are made of a high conductance material such as copper or silver applied on a substrate, the layer of resistive material (3) being applied on the substrate between the two feedlines (2, 2a).

9. Heating element (1 ) as claimed in claim 8, in which a dielectric protection layer or a double sided layer adhesive covers the feedlines (2, 2a) and the layer of resistive material (3).

10. Heating element (1 ) as claimed in claim 8 or 9, in which the substrate is a polymer film or a fabric.

11. Heating element (1 ) as claimed in anyone of the preceding claims, in which the layer of resistive material (3) is made of a material presenting a positive temperature coefficient of resistance.

12. Heater comprising one or more heating elements (1 ), characterized in that it comprises at least one heating element (1 ) according to any one of the preceding claims.

13. Method of producing a heating element (1 ) according to any one of the preceding claims, the method comprising the steps of:

- applying a high conductive layer on a substrate at each of two opposite sides of the heating element, these high conductive layers forming respectively a feedline (2, 2a) of the heating element (1 ) to be produced,

- applying a resistive material layer on the substrate, whereby the layer of resistive material width (w) varies in function of the length coordinate (x) of the heating element (1 ) and whereby the feedline width (wfl) varies in the same way in function of the length coordinate (x) of the heating element (1 ).

14. Method according to claim 13, in which the conductive layers and the layer of resistive material are applied by printing.