Original measurement pattern for an electrical process tomography system

Abstract

The invention relates to an electrical process tomography system, comprising a substrate (3) and a device including: - electrodes (E1,…,E10) connected to the substrate (3), - a power supply and a voltage sensor arranged to be connected respectively to a doublet and a pair of electrodes, - a control unit for controlling the power supply to adjacent electrode doublets and the voltage measurement of electrode pairs, and - a computer for mapping the electrical conductivity of the substrate (3). A section of the substrate (3), i.e., the intersection of a plane with the substrate (3), is divided into angular sectors (SA1,…,SA4) by straight lines (D1, D2) intersecting at a point (C). The electrodes (E1,…,E10) are connected to the section of the substrate (3) such that two opposite angular sectors are either either electrode-free or electrode-equipped. Each pair of electrodes is formed from two electrodes of opposite angular sectors.[Fig. 4].

Description

Title of the invention: Original measurement pattern for an electrical process tomography system

[0001] The field of the invention relates to electrical process tomography (also known by the English acronym EPT for "electrical process tomography") of a substrate.

[0002] Electrical process tomography is an imaging technique that involves determining the electrical properties of a substrate from voltage measurements of pairs of electrodes arranged at the periphery of the substrate. The voltage measurements are used to map the electrical charge density or electrical conductivity of the substrate.

[0003] Electrical process tomography covers different processes, including electrical impedance tomography (also known by the acronym EIT for "electrical impedance tomography") and electrical resistance tomography (also known by the acronym ERT for "electrical resistance tomography"). Electrical resistance tomography is particularly suitable when the real part of the substrate impedance predominates over the complex part.

[0004] Electrical impedance tomography (EIT) has applications in the medical field. For example, international application WO 2020 / 094628 A1 describes a smart textile formed from a conductive fabric on which electrodes are placed to detect joint movements in the context of the prevention of musculoskeletal disorders.

[0005] It is also known to use TIE in an industrial context to analyze a liquid substrate, for example a fluid circulating in a pipeline. Furthermore, TIE can be applied to a solid material to know its structure and visualize the way in which the different zones of the material conduct or oppose the passage of current in order to detect anomalies such as cracks.

[0006] The general principle of TIE is to place electrodes at the periphery of the substrate to be analyzed, to supply doublets of electrodes with a current or voltage source and to measure the voltage between the two electrodes of different pairs of electrodes. Several processes have thus been developed on this principle and are differentiated by their injection pattern - or "injection pattern" in the English literature -, that is to say the way in which the doublets of electrodes to be stimulated are formed.

[0007] The most widely used injection pattern is the adjacent pattern - or "adjacent pattern" - described by DC Barber and BH Brown in the article "Applied potential tomography" (Journal of Physics E: Scientific Instruments, vol. 17, n°9, 1984) and which consists of supplying power to each doublet of electrodes formed of two neighboring electrodes and then measuring the voltage at the level of each pair of electrodes also formed of two neighboring electrodes.

[0008] However, such an injection pattern provides little information about the center of the substrate since the stimulation and measurement are carried out on the edge thereof.

[0009] Other processes have been proposed to fill this gap and are notably discussed by M. Haingartner et al. in the article “Improved current injection pattern for the detection of delaminations in carbon fiber reinforced polymer plates using electrical impedance tomography” (Structural Health Monitoring, vol. 20, no. 5, 2021, pp. 2747-2757). The substrate to be analyzed is a carbon fiber reinforced polymer plate (better known by the acronym CFRP for “carbon filter reinforced polymer”) whose rectangular shape lends itself to various injection patterns. This publication recalls that, in addition to the adjacent pattern, it is possible to use an opposite pattern - or “opposite pattern” - in which each doublet of electrodes is formed of two electrodes symmetrical with respect to a vertical or horizontal axis or with respect to the center of the plate.A diagonal pattern, in which each electrode doublet is formed by two electrodes symmetrical about a diagonal of the substrate, is also mentioned. As with the adjacent pattern, these two injection patterns are intended to measure the voltage of neighboring pairs of electrodes.

[0010] The opposite pattern and the diagonal pattern make it possible to know the electrical properties of the center of the material. However, these injection patterns remain sensitive to edge conditions and are therefore poorly suited to heterogeneous materials such as composite materials. In addition, successive stimulations of doublets of distant electrodes on the material create opposite directions of current propagation which are likely to compensate each other and therefore alter the voltage measurements.

[0011] More comprehensive injection patterns have been developed. B. Loyola et al. proposed, in the article “Detection of spatially distributed damage in fiber-reinforced polymer composites” (Structural Health Monitoring, vol. 12, no. 3, 2013, pp. 225-239), to combine the opposite pattern and the diagonal pattern for anisotropic materials. Furthermore, the article by M. Haingartner et al. (2021) cited above describes a star pattern - or “star pattern” - which consists of stimulating all possible doublets of electrodes, except neighboring electrodes on the same side of the plate.

[0012] These injection patterns have the disadvantage of having a high time cost and requiring significant computing resources.

[0013] Another possible solution to obtain information about the center of the substrate consists of adapting the measurement pattern rather than the injection pattern. In this respect, M. Aller et al. describe, in the article "Study and comparison of different Machine Learning-based approaches to solve the inverse problem in Electrical Impedance Tomographies" (Neural Computing and Applications, vol. 35, n°7, 2022, pp. 5465-5477), an opposite measurement pattern - also called "opposite pattern" - in which each pair of electrodes is formed by two electrodes symmetrical with respect to the center of a substrate with a circular outline. The injection pattern used is the adjacent pattern proposed by DC Barber and BH Brown (1984).

[0014] The present invention improves the situation.

[0015] In this respect, the invention relates to an electrical process tomography system, comprising a device and a substrate.

[0016] The device comprises: - a plurality of electrodes connected to a portion of the substrate, - an electrical power source arranged to be connected to a pair of electrodes of the plurality of electrodes, - a voltage sensor arranged to be connected to a pair of electrodes of the plurality of electrodes, - a control unit arranged to control the successive supply by the electrical power source of doublets of electrodes each formed of two neighboring electrodes, and to control, each time a doublet of electrodes is supplied, respective voltage measurements by the voltage sensor, each between two electrodes of pairs of electrodes associated according to a measurement pattern, and - a calculator designed to map, from voltage measurements, the electrical charge density or electrical conductivity of the portion of the substrate.

[0017] A section of the substrate, defined by the intersection of a plane with the portion of the substrate, is divided into angular sectors by a set of intersecting lines at the same point of the section of the substrate, two angular sectors delimited by the same two lines forming opposite angular sectors.

[0018] The plurality of electrodes is connected to the portion of the substrate such that, by projection onto the section of the substrate, two opposite angular sectors are both devoid of electrodes or both provided with at least two electrodes.

[0019] The measurement pattern meets the following two conditions: • each pair of electrodes of the measurement pattern is formed of two electrodes of opposite angular sectors, and • two pairs of electrodes of the measurement pattern associated with the same angular sectors are such that the two segments each connecting, by projection onto the section of the substrate, the two respective electrodes of the two pairs of electrodes of the measurement pattern are not intersecting.

[0020] In one or more embodiments, the plurality of electrodes are projectingly connected substantially to the periphery of the substrate section.

[0021] In one or more embodiments, the plurality of electrodes is connected by projection to a closed line having a direction of travel of the section of the substrate and two neighboring electrodes are two consecutive electrodes of the closed line following the direction of travel.

[0022] In one or more embodiments, the computer is arranged to map the electrical charge density or electrical conductivity of the portion of the substrate by solving an inverse problem whose inputs are the voltage measurements.

[0023] In one or more embodiments, the electrical power source is a voltage source or a current source.

[0024] The invention also relates to a method of electrical process tomography of a substrate, comprising the following operations: - connecting a plurality of electrodes to a portion of the substrate, - successively supplying doublets of electrodes, each formed from two neighboring electrodes, with an electrical power source, - for each supply of a doublet of electrodes, carry out respective voltage measurements by a voltage sensor, each between two electrodes of pairs of electrodes associated according to a measurement pattern, and - map by a calculator, from the voltage measurements, the electrical charge density or the electrical conductivity of the portion of the substrate.

[0025] A section of the substrate, defined by the intersection of a plane with the portion of the substrate, is divided into angular sectors by a set of intersecting lines at the same point of the section of the substrate, two angular sectors delimited by the same two lines forming opposite angular sectors.

[0026] The plurality of electrodes is connected to the portion of the substrate such that, by projection onto the section of the substrate, two opposite angular sectors are both devoid of electrodes or both provided with at least two electrodes.

[0027] The measurement pattern meets the following two conditions: • each pair of electrodes of the measurement pattern is formed of two electrodes of opposite angular sectors, and • two pairs of electrodes of the measurement pattern associated with the same angular sectors are such that the two segments each connecting, by projection onto the section of the substrate, the two respective electrodes of the two pairs of electrodes of the measurement pattern are not intersecting.

[0028] In one or more embodiments, the electrical power source powers all possible pairs of neighboring electrodes.

[0029] Advantageously, the plurality of electrodes is connected to the portion of the substrate so that, by projection onto the section of the substrate, two opposite angular sectors are provided with the same number of electrodes. For each supply of a doublet of electrodes, opposite angular sectors are provided with a bijection between their respective electrodes, and the measurement pattern is such that each pair of electrodes is formed of two electrodes respectively antecedent and image by the bijection associated with the corresponding opposite angular sectors.

[0030] The bijection associated with two opposite angular sectors is for example invariant from the supply of one doublet of electrodes to another.

[0031] In one or more embodiments, the voltage sensor performs respective voltage measurements, each between the two electrodes of all possible pairs of associated electrodes according to the measurement pattern.

[0032] Other characteristics, details and advantages will appear on reading the detailed description below, and on analyzing the attached drawings in which:

[0033] [Fig.l] illustrates an electrical process tomography system according to the invention;

[0034] [Fig.2] illustrates a method of tomography by electrical process of a substrate according to the invention;

[0035] [Fig.3] illustrates an operation of positioning the electrodes on the substrate of the method of [Fig.2]; and

[0036] [Fig.4] illustrates a variant of the measuring pattern of the invention.

[0037] [Fig.l] illustrates an electrical process tomography system comprising a device 1 and a substrate 3.

[0038] The device 1 is arranged to apply electrical process tomography to the substrate 3 in order to determine its electrical properties. In the remainder of the description, it is considered, without loss of generality, that the device 1 is arranged to implement electrical impedance tomography, hereinafter referred to by the acronym TIE. However, the device 1 may also be adapted to implement electrical resistance tomography.

[0039] The substrate 3 can be formed from any type of material, liquid or solid, homogeneous or heterogeneous.

[0040] For example, the substrate 3 is a fluid circulating in a pipeline or a liquid stored in an industrial tank.

[0041] The substrate 3 may also be a composite material, i.e. a material formed from a matrix and a reinforcement. The matrix may be organic, plastic, metal, ceramic, graphite or a combination thereof. In an organic matrix composite, the matrix - then called resin - is for example thermosetting: epoxy resin, polyester or polyimide. The organic matrix may also be thermoplastic: polypropylene (PP), polyamide (PA) or polyetheretherketone (PEEK). The reinforcement may include carbon fibers, glass fibers, aramid fibers or plant fibers. The reinforcement may also include fillers, for example metal fillers, carbon fillers, silica fillers or polymer fillers.

[0042] Typically, the substrate 3 is a carbon / carbon composite - or C / C - formed from a graphite matrix reinforced with carbon fibers or a carbon fiber reinforced polymer - or CFRP for "carbon filter reinforced polymer".

[0043] The TIE makes it possible to obtain a distribution of the electrical charge density or the electrical conductivity of a portion of the substrate 3. In the particular case where the substrate 3 is a multilayer material, the portion of the substrate 3 can be a layer of the material.

[0044] TIE can be used to detect anomalies - usually cracks, but also impact or delamination - that change the way the substrate 3 conducts current.

[0045] As illustrated in [Fig.l], the device 1 comprises a plurality of electrodes, an electrical power source 5, a voltage sensor 7, a control unit 9 and a computer 11.

[0046] The electrodes of the plurality of electrodes are arranged to be connected to a portion of the substrate 3 whose electrical properties are to be known.

[0047] When the substrate 3 is liquid, the electrodes are positioned on the container, for example a pipe, a tank or a reservoir.

[0048] When the substrate 3 is solid, the electrodes are for example directly deposited on its outer surface. In a multilayer material, the electrodes can be deposited on one of the layers of the material or even be in contact with several - or even all - of the layers. The electrodes can in particular pierce the material to be in contact with one or more layers.

[0049] The positions of the electrodes delimit the portion - two-dimensional or three-dimensional - for which the distribution of the electrical charge density or the electrical conductivity is to be determined. Furthermore, the positioning of the electrodes can be characterized by a projection - typically an orthogonal projection - onto the intersection of a plane with the portion of the substrate 3. Such an intersection is called a "section" in the remainder of the description. It can be considered that the electrodes are connected by projection to the section of the substrate 3.

[0050] The plurality of electrodes may comprise an even number of electrodes. For example, in the embodiment shown in [Fig.l], the plurality of electrodes comprises eight electrodes respectively referenced E1, E2, E3, E4, E5, E6, E7 and E8. Of course, the plurality of electrodes may comprise more or less than eight electrodes. For example, it is known to use sixteen electrodes, as proposed, among others, by G. Pugach et al. in the article “Neural learning of the topographic tactile sensory information of an artificial skin through a self-organizing map” (Advanced Robotics, vol. 29, no. 21, 2015, pp. 1393-1409), to implement a TIE. Furthermore, the plurality of electrodes may also comprise an odd number of electrodes.

[0051] The source 5 is arranged to be connected to a doublet of electrodes of the plurality of electrodes. More particularly, the source 5 is arranged to successively supply doublets of associated electrodes according to a predetermined injection pattern. In the context of the TIE, the stimulation of doublets of electrodes aims to make the current flow within the substrate 3 to observe the manner in which the different zones of the portion of the substrate 3 conduct or oppose the passage of the current.

[0052] In the context of the invention, the injection pattern used is that proposed by DC Barber and BH Brown (1984): each doublet of electrodes is formed of two neighboring electrodes. The adjacent pattern has the advantage, unlike the opposite pattern and the diagonal pattern, of not creating opposite directions of current propagation which are likely to compensate each other and therefore to alter subsequent voltage measurements.

[0053] In the example of [Fig.l], the source 5 is thus connected to the doublet of neighboring electrodes E1 and E2.

[0054] The source 5 may be a voltage source, in which case the source 5 applies a voltage between the two electrodes of the doublet of electrodes. Alternatively, the source 5 is a current source, in which case the source 5 generates a current which flows from one electrode to the other of the doublet of electrodes.

[0055] The sensor 7 is arranged to be connected to a pair of electrodes of the plurality of electrodes. More particularly, the sensor 7 is arranged to measure the voltage respectively between two electrodes of several pairs of electrodes, the two electrodes of each pair of electrodes being associated according to a predetermined measurement pattern. In the context of the TIE, the voltage measurements, carried out for each supply of a doublet of electrodes, are intended to be used by the computer 11 to map the electrical charge density or the electrical conductivity of the portion of the substrate 3 to which the plurality of electrodes is connected.

[0056] The details of the measurement pattern which is the subject of the present invention will be detailed in the remainder of the description, in particular with reference to [Fig.2] and [Fig.3].

[0057] In the example illustrated in [Fig.l], the sensor 7 is connected to the pair of electrodes E3 and E5.

[0058] The sensor 7 is for example a measuring instrument of the voltmeter type.

[0059] The control unit 9 is arranged to control both the source 5 and the sensor 7. More particularly, the control unit 9 is arranged to control the successive supply, by the source 5, of doublets of neighboring electrodes and to control, each time a doublet of electrodes is supplied, voltage measurements. Each voltage measurement is carried out between two electrodes associated according to the measurement pattern and together forming a pair of electrodes.

[0060] The control unit 9 comprises a memory 13 and a processor 15.

[0061] The memory 13 is arranged to store instructions whose implementation, by the processor 15, results in the operation of the control unit 9.

[0062] The memory 13 can also store the information and measurements necessary for the computer 11 to map the electrical charge density or the electrical conductivity of the portion of the substrate 3. For example, the memory 13 can keep a register in which are matched, on the one hand, identification data of the electrode doublets and, on the other hand, the voltage measurements.

[0063] The identification data of a doublet of electrodes comprise the respective positions of the two electrodes of the doublet of electrodes in a reference frame of the substrate 3 as well as the value of the voltage or the intensity of the current applied by the source 5 to the doublet of electrodes. The voltage measurements associated with a doublet of electrodes correspond to the measurements made by the sensor 7 between two electrodes of several pairs of electrodes. Each voltage measurement is accompanied by the respective positions of the two electrodes of the corresponding pair of electrodes in the reference frame of the substrate 3.

[0064] Of course, it is also possible that the measurements are only known to the computer 11 and therefore do not pass through the control unit 9. Furthermore, the positions of the two electrodes of the pair of electrodes to which each voltage measurement corresponds can be deduced from the order in which the voltage measurements are carried out.

[0065] The memory 13 may designate any data storage medium arranged to receive and store digital data, for example a hard disk, a solid-state drive (better known by the English acronym SSD for “solid-state drive”) or more generally any computer hardware allowing the storage of data on flash memory. The memory 13 may also be a random access memory or a magneto-optical disk. A combination of several storage media may also be envisaged.

[0066] The processor 15 can be produced in any known manner, for example in the form of a microprocessor, a programmable logic circuit (better known by the English acronym PLD for “Programmable Logical Device”) or a dedicated chip of the FPGA type (English acronym for “Field Programmable Gate Array”) or SoC (English acronym for “System on Chip”), a grid of computing resources, a microcontroller or any other specific form having the computing power required to implement the electrical process tomography method described below. One or more of these elements can also be implemented in the form of specialized electronic circuits of the ASIC type (acronym for “Application-Specific Integrated Circuit”). A combination of processors and electronic circuits can also be considered.

[0067] Finally, the computer 11 is arranged to map, from the voltage measurements, the electrical charge density or the electrical conductivity of the portion of the substrate 3.

[0068] It is known to those skilled in the art that the distribution of the electrical charge density or of the electrical conductivity can be obtained by solving an inverse problem whose inputs are the voltage measurements.

[0069] To do this, the computer 11 can communicate with the control unit 9 to receive the necessary information and measurements. In the example of [Fig.l], the control unit 9 and the computer 11 are distinct from each other. However, such a distinction is functional and it is understood that the computer 11 can be an integral part of the control unit 9, in which case the control unit 9 and the computer 11 form a single device.

[0070] A method of electrical process tomography of a substrate - here the substrate 3 - will now be described with reference to [Fig.2]. More particularly, the method detailed below is a TIE method. However, electrical resistance tomography can be implemented in the same way.

[0071] During an operation 200, the plurality of electrodes is connected to a portion of the substrate 3 of which the electrical properties are to be known, and more precisely the distribution of the electrical charge density or the electrical conductivity.

[0072] The positioning of the electrodes is illustrated in [Fig. 3] which shows a section of the substrate 3 to which the plurality of electrodes of the device 1 are connected. Indeed, as mentioned previously, the position of each electrode is projected onto a section of the substrate 3 corresponding to the intersection of a plane with the portion of the substrate 3.

[0073] In the example discussed here, the plurality of electrodes comprises ten electrodes respectively referenced E1, E2, E3, E4, E5, E6, E7, E8, E9 and E10.

[0074] First of all, the section of the substrate 3 is divided into angular sectors defined by a set of intersecting lines at a point of the section of the substrate 3. In the example of [Fig.3], four angular sectors SA1, SA2, SA3 and SA4 are defined by two intersecting lines DI and D2 at a point C of the section of the substrate 3. Generally speaking, a number p of intersecting lines at the same point define a number 2p of angular sectors.

[0075] It should be noted that the center C is not necessarily a geometric center of the section of the substrate 3. Similarly, the lines can be any provided that they are intersecting at the same point of the section of the substrate 3.

[0076] In the remainder of the description, we call "opposite angular sectors" two angular sectors delimited by the same two straight lines. In other words, two opposite angular sectors are symmetrical with respect to the intersection - therefore the center C - of the straight lines. In the case of [Fig.3], the angular sectors SA1 and SA3 are opposite, as are the angular sectors SA2 and SA4.

[0077] The electrodes are then connected to the section of the substrate 3 so that two opposite angular sectors are both devoid of electrodes or both provided with at least one electrode. In other words, if at least one electrode is connected to an angular sector, then at least one electrode is connected to the opposite angular sector; and if no electrode is connected to an angular sector, then no electrode is connected to the opposite angular sector.

[0078] In a variant of the measurement pattern discussed below with reference to [Fig. 4], the electrodes are connected to the section of the substrate 3 so that two opposite angular sectors are both devoid of electrodes or both provided with at least two - and no longer one - electrodes.

[0079] The positioning of the electrodes illustrated in [Fig. 3] is compatible with this variant given that the opposite angular sectors SA1 and S A3 are both provided with at least two electrodes, in this case three electrodes E1, E2 and E3 for the angular sector SA1 and three electrodes E6, E7 and E8 for the angular sector SA3, and that the same is true for the opposite angular sectors SA2 and S A4, in this case two electrodes E4 and E5 for the angular sector SA2 and two electrodes E9 and E10 for the angular sector S A4.

[0080] By “electrode connected to an angular sector”, it is meant here that the projection of the electrode onto the section of the substrate 3 is located in this angular sector.

[0081] For example, the number of electrodes connected to an angular sector is equal to the number of electrodes connected to the opposite angular sector. Thus, in [Fig.3], three electrodes E1, E2 and E3 are connected to the angular sector SA1 and three electrodes E6, E7 and E8 are connected to the opposite angular sector SA3. Similarly, two electrodes E4 and E5 are connected to the angular sector SA2 and two electrodes E9 and E10 are connected to the opposite angular sector SA4.

[0082] Furthermore, the electrodes can be placed on a closed line L with a direction of travel. The closed line L passes through each of the angular sectors and delimits a closed zone of the section of the substrate 3 comprising the center C. The closed line L makes it possible to define neighboring electrodes which are then two consecutive electrodes of the closed line L following the direction of travel. In [Fig. 3], the direction of travel is indicated by arrows. For example, the electrodes E3 and E4 are neighboring electrodes, as are the electrodes E6 and E7 or the electrodes ElOetEL

[0083] It should be noted that the electrodes are not necessarily positioned at the periphery of the section of the substrate 3.

[0084] As explained previously, the substrate 3 may be liquid or solid. In the case of a liquid substrate, the electrodes are placed on the container. In the case of a solid substrate, the electrodes are for example directly arranged on the outer surface of the substrate 3. In the case of a multilayer material, the electrodes may be arranged on one of the layers or be in contact with several layers.

[0085] The electrodes are positioned so as to delimit the portion of the substrate 3 for which the distribution of the electrical charge density or the electrical conductivity is to be determined. The electrodes are coplanar when the portion of the substrate 3 is a plane, for example the flat outer surface of a material or a layer of a multilayer material. In such a case, the section of the substrate 3 may coincide with the portion of the substrate 3. However, the electrodes may also delimit a three-dimensional portion of the substrate 3, in which case the electrodes are not coplanar, hence the interest in considering a section of the substrate 3 onto which the positions of the electrodes are projected.

[0086] Once the plurality of electrodes are connected to the portion of the substrate 3, the control unit 9 implements a successive supply sequence of several doublets of electrodes.

[0087] To do this, an injection pattern, i.e. the way in which the electrode doublets to be supplied are formed, is previously defined. As mentioned previously, the injection pattern used here is the adjacent pattern. In other words, an electrode doublet is formed of two neighboring electrodes. The neighborhood of an electrode typically comprises the two closest electrodes located on either side of the electrode considered. As explained previously, it is also possible to use the closed line L to define the neighborhood of an electrode.

[0088] The control unit 9 is configured with the power supply sequence, i.e. the ordered list of electrode doublets to be powered successively. Advantageously, for a number N of electrodes, it is possible to power N different neighboring electrode doublets.

[0089] In the example illustrated in [Fig.3], the possible neighboring electrode doublets are: E1-E2, E2-E3,..., E9-E10 and E10-E1. It is nevertheless possible to supply only a part of all the possible neighboring electrode doublets.

[0090] A variable i corresponding to the identifier of a doublet of electrodes is initialized to 1. This variable makes it possible to implement, in the method illustrated in [Fig.2], a loop in which the doublets of electrodes are successively supplied according to the supply sequence.

[0091] During an operation 210, the control unit 9 controls the supply, by the source 5, of the doublet of electrodes corresponding to the current value of the variable i.

[0092] In the case of [Fig.3], the doublet of electrodes corresponding to the value 1 of the variable i is for example the doublet of electrodes formed by the neighboring electrodes El and E2.

[0093] Once the doublet of electrodes is supplied by the source 5, the control unit 9 implements a measurement sequence between the two electrodes of several pairs of electrodes.

[0094] To do this, a measurement pattern, i.e. the way in which the pairs of electrodes at the terminals of which the voltage is to be measured are formed, is previously defined. The measurement pattern which is the subject of the present invention has the particularity that each pair of electrodes is formed of two electrodes of opposite angular sectors.

[0095] The proposed measurement pattern makes it possible to collect information on the center of the section of the substrate 3 by measuring voltage - and not by injecting current - between electrodes distant from each other. Conversely, the opposite pattern and the diagonal pattern certainly make it possible to obtain information on the center of the substrate, but by injecting current - the voltage measurement is then carried out between neighboring electrodes -, which implies, as explained previously, creating opposite directions of propagation of the current.

[0096] [Fig.4] illustrates a variation of the measurement pattern that allows for more uniform coverage form the section of the substrate 3.

[0097] As mentioned previously, this variant is part of a context in which the electrodes are connected to the section of the substrate 3 so that two opposite angular sectors are both devoid of electrodes or both provided with at least two electrodes.

[0098] In this variant, the condition according to which each pair of electrodes is formed of two electrodes of opposite angular sectors is accompanied by an additional condition: two pairs of electrodes associated with the same angular sectors are such that the two segments each connecting, by projection onto the section of the substrate 3, the two respective electrodes of these two pairs of electrodes are not intersecting.

[0099] This additional condition is fulfilled for any pair of a first and a second pair of electrodes to the extent that these first and second pairs of electrodes are associated with the same angular sectors, that is to say when the opposite angular sectors to which the two electrodes of the first pair of electrodes are respectively connected are the same opposite angular sectors as those to which the two electrodes of the second pair of electrodes are respectively connected.

[0100] The expression “by projection onto the section of the substrate 3” means here that the segment connecting the two electrodes of a pair of electrodes corresponds to the segment included in the section of the substrate 3 and connecting the respective orthogonal projections of these two electrodes onto the section of the substrate 3.

[0101] The positioning of the electrodes E1 to E10 in [Fig.4] is the same as in [Fig.3]. However, the closed line L, the straight lines DI and D2, the center C and the angular sectors SA1, SA2, SA3 and SA4 are not indicated in [Fig.4] to improve clarity.

[0102] In [Fig.4], the two electrodes of each pair of electrodes are connected by a segment. As mentioned above, it appears that two segments each connecting, by projection onto the section of the substrate 3, the two respective electrodes of two pairs of electrodes of the same opposite angular sectors are not intersecting.

[0103] In particular, the pair of electrodes E1-E8 and the pair of electrodes E3-E6 are formed from two electrodes of the same angular sectors, namely the opposite angular sectors SA1 and SA3. The segments respectively connecting the two electrodes of the pair of electrodes E1-E8 and the two electrodes of the pair of electrodes E3-E6 are not intersecting.

[0104] Conversely, the pair of electrodes E2-E7 and the pair of electrodes E5-E9 are formed of two electrodes of distinct angular sectors, namely the opposite angular sectors SA1 and SA3 for the pair of electrodes E2-E7 and the opposite angular sectors SA2 and SA4 for the pair of electrodes E5-E9. The segments respectively connecting the two electrodes of the pair of electrodes E2-E7 and the two electrodes of the pair of electrodes E5-E9 may be intersecting, which is also the case here.

[0105] Such a variant of the measurement pattern makes it possible, as illustrated in [Fig.4], to produce a grid of the section of the substrate 3, and therefore to know the electrical properties of the section of the substrate 3 both in the center and around the latter.

[0106] On the contrary, the opposite pattern described in the article by M. Aller et al. (2022) necessarily implies that two segments connecting the two respective electrodes of any two pairs of electrodes are intersecting since any pair of electrodes is formed by two electrodes symmetrical with respect to the center of the substrate. Thus, the electrical properties are much less known at the periphery of the substrate than at its center since the segments connecting the two respective electrodes of all pairs of electrodes are intersecting at a single point: the center of the substrate. Generally speaking, it is not known from the prior art to have non-intersecting segments between opposite angular sectors; the article by M. Aller et al. (2022) just proposes to combine the adjacent injection pattern with the opposite measurement pattern without such considerations being discussed.

[0107] The control unit 9 is configured with the measurement sequence, i.e. the ordered list of pairs of electrodes across which the voltage is to be measured. Typically, for an even number N of electrodes, it is possible to form N / 2 pairs of electrodes so that each electrode belongs to only one pair of electrodes. In general, and for an odd number of electrodes in particular, one or more electrodes may not be used in the measurement sequence.

[0108] A variable j corresponding to the identifier of a pair of electrodes is initialized to 1. This variable makes it possible to implement, in the method illustrated in [Fig.2], a loop in which the respective voltage between the two electrodes of pairs of electrodes is measured according to the measurement sequence.

[0109] During an operation 220, the control unit 9 commands the measurement of a voltage, by the sensor 7, between the two electrodes of the pair of electrodes corresponding to the current value of the variable j.

[0110] In the example illustrated in [Fig.3], the pair of electrodes corresponding to the value 1 of the variable j is for example the pair of electrodes formed by the electrodes E3 and E7. Indeed, the electrode E3 is connected to the angular sector SA1, while the electrode E7 is connected to the opposite angular sector SA3.

[0111] In the example illustrated in [Fig.4], the pair of electrodes corresponding to the value 1 of the variable j is for example the pair of electrodes formed by the electrodes E4 and E10.

[0112] Once the voltage measurement has been carried out by the sensor 7, the measured value is transmitted to the computer 11. This value can also be stored in the memory 13 of the control unit 9. Alternatively, the measured values ​​can only be transmitted to the computer 11 when all the measurements have been carried out.

[0113] During an operation 230, the control unit 9 determines whether the sensor 7 has measured the value of the voltage between the two electrodes of all the pairs of electrodes in the measurement sequence.

[0114] If the measurement sequence is not completed, the variable j is incremented during an operation 240. The operations 220 and 230 are then repeated until the measurement sequence, provided in response to the supply of the doublet of electrodes corresponding to the current value of the variable i, is completed.

[0115] Conversely, if the measurement sequence is finished, the control unit 9 determines, during an operation 250, whether the source 5 has supplied all the electrode doublets of the supply sequence.

[0116] If the feeding sequence is not completed, the variable i is incremented during an operation 260. The operations 210, 220, 230, 240 and 250 are then repeated until the feeding sequence is completed. In other words, the control unit 9 controls the feeding, by the source 5, of a new doublet of electrodes corresponding to the incremented value of the variable i and a new measurement sequence is carried out.

[0117] It should be noted that the measurement sequence may be different from one supply of a doublet of electrodes to another. For example, in the case where opposite angular sectors are provided with the same number of electrodes - this number being zero for angular sectors without electrodes -, opposite angular sectors are provided with a bijection between their respective electrodes for each doublet of electrodes of the supply sequence. Thus, when a doublet of electrodes of the supply sequence is supplied, the measurement pattern is such that each pair of electrodes is formed of two electrodes respectively antecedent and image by the bijection associated with the corresponding opposite angular sectors.

[0118] By way of example, still with reference to [Fig. 3], the control unit 9 can be configured so that, when the pair of neighboring electrodes E6-E7 is powered, a bijection is constructed between the opposite angular sectors SA2 and SA4 and associates the electrodes E4 and E5 respectively with the electrodes E9 and E10. Consequently, the measurement sequence provided, for the power supply of the pair of neighboring electrodes E6-E7, consists of measuring the voltage between the two electrodes of the pair of electrodes E4-E9 and the voltage between the two electrodes of the pair of electrodes E5-E10.

[0119] Furthermore, this measurement sequence can be modified for the supply of the doublet of neighboring electrodes along E7-E8. Indeed, a new bijection can be constructed between the opposite angular sectors SA2 and SA4 and this time associate the electrodes E4 and E5 respectively with the electrodes E10 and E9. The planned measurement sequence, for the supply of the doublet of neighboring electrodes E7-E8, then consists of measuring the voltage between the two electrodes of the pair of electrodes E4-E10 and the voltage between the two electrodes of the pair of electrodes E5-E9.

[0120] Alternatively, the bijection associated with two opposite angular sectors is invariant from the supply of one doublet of electrodes to another.

[0121] The measurement sequence can also be adapted so that none of the pairs of electrodes across which the voltage is measured includes one of the two electrodes of the powered electrode doublet.

[0122] Reference is again made below to [Fig. 4]. The variant of the measurement pattern illustrated in this figure offers less freedom in configuring a measurement sequence because of the additional condition discussed above.

[0123] Typically, a measurement sequence provides for the voltage to be measured by the sensor 7 between the two electrodes of all possible pairs of electrodes associated according to the variant of the measurement pattern. In particular, when two opposite angular sectors are provided with the same number of electrodes, there is then a unique way to form the electrode pairs. As such, [Fig.4] can be considered to illustrate a typical measurement sequence in a configuration in which each angular sector is provided with the same number of electrodes as the opposite angular sector and in which the voltage is measured between all possible pairs of electrodes: E1-E8, E2-E7, E3-E6, E4-E10, E5-E9.

[0124] Furthermore, it is also possible to adapt the measurement sequence so that none of the pairs of electrodes at the terminals of which the voltage is measured includes one of the two electrodes of the powered doublet of electrodes.

[0125] In the case illustrated in [Fig.2], it is planned to successively supply the N possible neighboring electrode doublets - that is, ten electrode doublets in the example of [Fig.3] - and to measure the voltage, for each neighboring electrode doublet supplied, between the two electrodes of N / 2 pairs of electrodes associated according to the measurement pattern - that is, five pairs of electrodes in the example of [Fig.3]. Advantageously, the N / 2 pairs of electrodes are formed so that each electrode belongs to only one pair of electrodes. The number of voltage measurements carried out in total is therefore N2 / 2 - that is, fifty measurements in the example of [Fig.3]

[0126] As explained previously, it is also possible, for each powered electrode pair, to exclude from the measurements the two electrodes forming the powered electrode pair, which leaves (N-4) / 2 pairs of electrodes - or three pairs of electrodes in the example of [Fig. 3] - also formed so that each electrode belongs to only one pair of electrodes. The number of voltage measurements carried out in total is then N(N-4) / 2 - or thirty measurements in the example of [Fig. 3].

[0127] The same considerations as to the number of measures apply to the variant of the measure pattern illustrated in [Fig.4].

[0128] The proposed TIE method, and more precisely the measurement pattern, makes it possible to significantly reduce the number of voltage measurements compared to the combination of the opposite pattern and the diagonal pattern described by B. Loyola et al. (2013) or the star pattern of M. Haingartner et al (2021).

[0129] Finally, during an operation 270, the computer 11 maps, from the voltage measurements, the electrical charge density or the electrical conductivity of the portion of the substrate 3. Generally, the distribution of the electrical charge density is more suitable for liquid substrates, while the distribution of the electrical conductivity is more suitable for solid substrates.

[0130] In the field of TIE, it is known to use the theory of inverse problems to know the electrical properties of a substrate. The principle of this approach is to determine the cause - here the distribution of the charge density electrical or electrical conductivity - from the effects - here the voltage measurements between the two electrodes of the electrode pairs. Indeed, each change in the distribution of electrical conductivity leads to changes in the distribution of the potential and, consequently, to changes in the voltage between the electrodes.

[0131] A TIE inverse problem is a nonlinear and ill-posed problem. In particular, this inverse problem does not depend continuously on the voltage measurements. As recalled by SR Kupis in the publication “Methods for the Electrical I mpedance Tomography Inverse Problem: Deep Learning and Regularization with IV avelets” (Ail Theses, 2021), it is possible to use a Gauss-Newton algorithm of the MIRGN type (acronym for “modified iteratively regularized Gauss-Newton”) or a Tikhonov regularization to solve a TIE inverse problem.

[0132] Furthermore, there are currently solvers capable of reconstructing the distribution of electric charge density or electrical conductivity from voltage measurements. For example, the free software EIDORS running under MATLAB (registered trademark) can be used to solve an inverse problem.

Claims

1. Claims Electrical process tomography system, comprising a device (1) and a substrate (3), wherein the device (1) comprises: - a plurality of electrodes (El,.. .,EN) connected to a portion of the substrate (3), - an electrical power source (5) arranged to be connected to a doublet of electrodes of said plurality of electrodes (El,.. .,EN), - a voltage sensor (7) arranged to be connected to a pair of electrodes of said plurality of electrodes (El,.. .,EN), - a control unit (9) arranged to control the successive supply by the electrical power source (5) of doublets of electrodes each formed by two neighboring electrodes, and to control, each time a doublet of electrodes is supplied, respective voltage measurements by the voltage sensor (7), each between two electrodes of pairs of electrodes associated according to a measurement pattern, and - a calculator (11) arranged to map, from the voltage measurements, the electrical charge density or the electrical conductivity of the portion of the substrate (3), in which a section of the substrate (3), defined by the intersection of a plane with the portion of the substrate (3), is divided into angular sectors (SA1, SA2, SA3, SA4) by a set of straight lines (D1, D2) intersecting at the same point (C) of said section of the substrate (3), two angular sectors delimited by the same two straight lines forming opposite angular sectors, the plurality of electrodes (El,.. .,EN) being connected to the portion of the substrate (3) so that, by projection onto the section of the substrate (3), two opposite angular sectors are both devoid of electrodes or both provided with at least two electrodes, said system being characterized in that the measurement pattern meets the following two conditions: • each pair of electrodes of the measurement pattern is formed of two electrodes of opposite angular sectors, and • two pairs of electrodes of the measurement pattern associated with the same angular sectors are such that the two segments each connect, by projection onto the section of the substrate (3), the two respective electrodes said two pairs of electrodes of the measuring pattern are not intersecting.

2. System according to claim 1, characterized in that the plurality of electrodes (El,.. .,EN) is connected by projection substantially to the periphery of the section of the substrate (3).

3. System according to claim 1 or 2, characterized in that the plurality of electrodes (El,.. .,EN) is connected by projection to a closed line (L) provided with a direction of travel of the section of the substrate (3) and in that two neighboring electrodes are two consecutive electrodes of the closed line (L) following the direction of travel.

4. System according to one of the preceding claims, characterized in that the calculator (11) is arranged to map the electrical charge density or the electrical conductivity of the portion of the substrate (3) by solving an inverse problem whose inputs are the voltage measurements.

5. System according to one of the preceding claims, characterized in that the electrical power source (5) is a voltage source or a current source.

6. Method of electrical process tomography of a substrate (3), comprising the following operations: - connecting (200) a plurality of electrodes (El,.. .,EN) to a portion of the substrate (3), - successively supplying (210) by an electrical power source (5) doublets of electrodes each formed by two neighboring electrodes, - for each supply of a doublet of electrodes, carrying out respective voltage measurements (220) by a voltage sensor (7), each between two electrodes of pairs of electrodes associated according to a measurement pattern, and - mapping (270) by a computer (11), from the voltage measurements, the electrical charge density or the electrical conductivity of the portion of the substrate (3), in which a section of the substrate (3), defined by the intersection of a plane with the portion of the substrate (3), is divided into angular sectors (SA1, SA2, SA3, SA4) by a set of straight lines (Dl, D2) intersecting at the same point (C) of said section of the substrate (3), two angular sectors delimited by the same two straight lines forming angular sectors opposites,. the plurality of electrodes (El,.. .,EN) being connected to the portion of the substrate (3) so that, by projection onto the section of the substrate (3), two opposite angular sectors are both devoid of electrodes or both provided with at least two electrodes, said method being characterized in that the measurement pattern meets the following two conditions: • each pair of electrodes of the measurement pattern is formed of two electrodes of opposite angular sectors, and • two pairs of electrodes of the measurement pattern associated with the same angular sectors are such that the two segments each connecting, by projection onto the section of the substrate (3), the two respective electrodes of said two pairs of electrodes of the measurement pattern are not intersecting.

7. Method according to claim 6, characterized in that the electrical power source (5) supplies all possible doublets of neighboring electrodes.

8. Method according to claim 6 or 7, characterized in that the plurality of electrodes (El,.. .,EN) is connected to the portion of the substrate (3) so that, by projection onto the section of the substrate (3), two opposite angular sectors are provided with the same number of electrodes, in that, for each supply of a doublet of electrodes, opposite angular sectors are provided with a bijection between their respective electrodes, and in that the measurement pattern is such that each pair of electrodes is formed of two electrodes respectively antecedent and image by the bijection associated with the corresponding opposite angular sectors.

9. Method according to claim 8, characterized in that the bijection associated with two opposite angular sectors is invariant from the supply of one doublet of electrodes to another.

10. Method according to claim 8 or 9, characterized in that the voltage sensor (7) carries out respective voltage measurements, each between the two electrodes of all possible pairs of electrodes associated according to the measurement pattern.

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