FR2690033A1 - Linear aerial for underwater detection - includes groups of sensors with weighting applied to outputs in order to minimise noise effects - Google Patents

Abstract

The aerial consists of sensors (H-B to H+B) distributed with a constant step spacing in a linear array. Together these sensors form an assembly or 'macro-sensor'. The response of separate macro-sensors (MCA,MCB) is then weighted by using a selective cabling system combining series and parallel elements. The configuration is chosen in order to approach the theoretically determined weighting law. USE/ADVANTAGE - Sensor may be used in towed underwater probe for prospecting, with effects of parasitic noise propagating within structure being minimised. Esp. mechanically induced noise is minimised.

Description

Method of producing a linear antenna
towed and antenna made by this method.

 The invention relates to a method for producing a towed antenna and more particularly to a production method for obtaining a rejection of certain types of noise on these antennas, in particular mechanical noise and flow noise.

 The invention also relates to an antenna made according to this method.

 A towed linear antenna is generally formed of a string of acoustic sensors, for example hydrophones, placed inside a cylindrical structure. It is this structure that is towed for purposes, for example, underwater prospecting.

 One of the important parameters of such an antenna is the directivity parameter. The directivity is a function of the ratio (X / L) with X length of the wave to be detected and L length of the antenna.

 To use the antenna in a wide band of frequencies, typically several octaves, the signals of the sensors are processed by antenna sections of different lengths.

 An antenna of given length is assigned to an octave of frequencies, for example f to 2f, and an antenna of double length to octave f / 2 to f; f being a given frequency value.

In a particular embodiment, these two antennas are imbricated by using for the second antenna (octave f / 2 to f) a portion of the sensors of the first antenna. Physically the antenna consists of a central section comprising (4N + 1) equidistant sensors, at pitch Po depending on the minimum wavelength of the octave considered (f at 2 wire. of this central section for each octave "k" lower 2N sensors at step 2kPo (with k = 1,2,3 ...)
In reality, most often, parallel sensor groups are formed which will be referred to in the following as macrosensors. These "macrosensors" are also called "clusters" in the English terminology. The number of sensors in each grouping is not identical but the intercapteur pitch becomes constant and equal to Pg.

In general, the further one moves away from the center of the antenna on both sides, the greater the number of sensors in each grouping.

 The invention relates to an antenna of this type, comprising sensor groups.

 The towed antennas are subject to a number of parasitic phenomena. In particular, it is known that the towing of the antenna is the cause of noises of various kinds. These noises come mainly from the towing cable (mechanical noise) and from the flow noise.

 To reduce the first source of noise, it is known to insert at the head of the antenna a damper section as described, for example, in French Patent Application No. 2,626,051, published February 9, 1990.

 The invention proposes to overcome the drawbacks of the known art, in particular, the disadvantages caused by mechanical noise or flow.

 - The frequency bands of the recalled noises are located in the lower part of the listening spectrum. As it was recalled, the further away from the center of the antenna, the groups comprise a large number of sensors. However, it is precisely the longest antenna sections that are intended to listen for the lowest frequencies, that is to say those that include the most macrosensors.

 The theory also shows that a weighting of the hydrophones within a group makes it possible to improve the signal-to-noise ratio.

 The invention takes advantage of these features.

 It proposes a method of producing antennas making it possible to simply obtain a weighting of the sensor groups by performing selective serial-parallel wiring of these.

 The subject of the invention is therefore a method for producing a towed linear antenna consisting of distributed sensors, said antenna comprising at least one assembly of sensor groups in which weighting is carried out according to a given law, each group comprising, at minus one sensor; characterized in that said weighting is performed by selective serial wiring of sensors of said assembly, the other sensors being wired in parallel so as to obtain a weighting curve approaching said determined weighting law.

 The invention also relates to an antenna thus produced.

The invention will be better understood and other features and advantages will appear on reading the description which follows with reference to the appended figures and among which
FIGS. 1 and 2 schematically illustrate antennas
according to the known art,
FIGS. 3 and 4 are diagrams illustrating the effect of the
speed of the noise on the grouping gain according to the
frequency, for a given antenna structure,
FIG. 5 is a diagram illustrating the group gain
a macrosensor as a function of frequency without weighting
FIGS. 6 and 7 illustrate the effect of two weighting laws
distinct on the grouping gain of that same
macrocapteur,
FIG. 8 illustrates, by a diagram, a particular aspect of the
weighting method according to the invention,
FIGS. 9 and 10 schematically illustrate two
macrosensors made according to the method of the invention,
FIG. 11 is a diagram illustrating the weighting
associated with the macrosensor of Figure 10,
- Figure 12 illustrates more fully an example of
realization of a macrosensor according to the invention that can be used in the
frequency range 40-80Hz,
FIGS. 13 and 14 are diagrams for
compare the macrosensor of Figure 12 with a macro-sensor
equivalent of the known art,
- Figure 15 illustrates more fully an example of
realization of a macrosensor according to the invention that can be used in the
frequency range 10-40Hz,
FIGS. 16 and 17 are diagrams for
compare the macrosensor of Figure 15 with a macro-sensor
equivalent of the known art.

 The configuration of a linear antenna towed according to the known art will first be recalled briefly.

 Figure 1 schematically illustrates a linear antenna of the type called "nested". The central section usually includes (4N + 1) sensors. For the sake of simplification, N = 1 was chosen. The central section intended to process the octave of frequencies f at 2f therefore comprises five sensors: H 2, H 1, Ho, H 1, H 2, all equidistant. The inter-sensor pitch is Po dependent on the minimum wavelength to be sensed.

In the lower part of the figure is shown an X'X axis graduated in steps Pg.

 This central section forms the antenna referenced A.

 On both sides of this section, N sensors were added at 2Po, which is a total of 2N. In the example N = 1 and the added sensors are H 3 and H 3. We take one sensor out of two of the central section to form, with the side sensors a homothetic antenna of the first step 2Po and double length. This "sub-antenna", B in the figure, is intended to treat the octave f to f / 2.

Most often, the configuration of a linear antenna is not the one just described. It is formed of sensors all at Po pitch and is grouped in parallel, selectively, sensors to form macrosensors. Such an arrangement is illustrated in FIG. 2. This allows certain advantages, in particular a group gain.
The antenna illustrated in Figure 2 comprises three sub-antennas A, B, C to cover three octaves.

 We have chosen, as before, N = 1. The central section Tc thus comprises (4N + 1) = 5 sensors and constitutes the antenna A.

 The antenna B "takes" one sensor out of two of the central section as before. However, the lateral parts of this sub-antenna are constituted by groups of sensors or macrosensors each comprising three individual sensors. These macrosensors are labeled MC11 and MC2 + 1.

 Finally, the sub-antenna C consists of the central sensor of the section Tc, the macrosensors MC11 and MC2 2 as well as two additional macro-sensors: MC2 1 and MC2 + 1. These last two each comprise five sensors.

The set is perfectly symmetrical with respect to the abscissa 0 (X'X axis) and comprises a total of 21 sensors, all in pitch Po
We will now recall the types of noise encountered on towed linear antennas. These noises are of different origins and natures. By adopting special measures, we can hope to increase the ratio "useful signal on noise" which will be called in the following: [S i B]. However, this potential gain essentially depends on the characteristics of coherence, isotropy and homogeneity of the different noises.

Table 1 at the end of this description summarizes the main types of noise, their nature and characteristics, and the potential [S / B] gains.
In what follows, we will focus essentially on two types of noise whose respective amplitudes are particularly high, that is to say the mechanical noise and the flow noise, and whose spectra are of different frequency media . Indeed, experience and modeling have shown that mechanical noise is essentially troublesome for frequencies below 80 Hz, while flow noise is harmful between 80 - 160 Hz. This is only true, however, if noise mechanical is reduced by the addition of dampers.

In the opposite case, the mechanical noise is predominant up to 300 Hz. Provisions for reducing the mechanical noise are known, for example those described in the aforementioned French patent application.

 In the case of flow noise, the inter-sensor pitch must be greater than the spatial correlation length of the parietal excitation, which imposes a low number of sensors for a given grouping length. The length lc of correlation is given by the relation: lc = 0.7.v / f, with v towing speed (m / s) and f minimum working frequency.

Mechanical noise is a correlated noise, propagated in a particular mode that is not very dispersive up to 160 Hz. We can therefore plot the spatial Fourier transform in wave numbers for a given celerity: C. Celerity C must be understood as the velocity of propagation of the noise inside the structure of the antenna. This transform naturally depends on the spatial distribution of the sensors (length of the macrosensors, sampling that is to say the not-intercapteur pitch). Typically, a length-wise sampler I, sampled in a step d and composed of sensors wired in parallel has a "frequency directivity" such that the zeros are at nC / I (n being the number of the secondary lobe). 2n + 1) C / 21 (1) the level of the first and second secondary lobe -13 and -18dB .the image lobe is at CId (2)
Relation (1) is interesting because it shows that if we want a rejection of noise, of the order of +20 dB in a band 20 - 40 Hz, it is necessary that the following relation be verified 30> (2n + 1) C / 21 with n = 2 and C = 100 m / s, ie 1> 8,33 m, value independent of the sampling.

Equation (2), likewise, shows that if it is desired to reject the image lobe beyond 160 Hz, then C / d> 1 60 is required, ie d <C / 1 60 - 0.625 m.

 It should be noted that the higher the speed C is, the less effective is a macrosensor since the spectrum will "spread" towards the high frequencies. The first lobes of rejection are less efficient in terms of levels and then go up in the band of work.

 Figures 3 and 4 illustrate the effect of celerity, for the same macrosensor configuration. The number N of the sensor chosen in the example is 8 step sensors 0.52 m for the 4080 Hz band.

 FIG. 3 illustrates the group gain (in dB) as a function of the frequency F (Hz) for a speed of propagation of the so-called sheath mode C = 100 m / s.

 This figure shows two curves: C1 and C2. Curve C1 is a theoretical curve that does not take into account the damping provided by the structure of the antenna. The curve C'1 takes into account and represents the slow mode said "dissipative".

 FIG. 4 illustrates the group gain as a function of the frequency under the same conditions (number of sensors, etc.) but for C = 200 m / s. The curves C2 and C'2 represent, as before, the non-dissipative mode and the dissipative mode.

 It can be seen from these curves that the average rejection is -18 dB (2nd secondary lobe) for C = 100 m / s and -13 dB (1st secondary lobe) for C = 200 m / s.

 On the other hand, stimulations have shown that the speed of the first guided mode, by a linear antenna of 85 cm diameter is of the order of 100 m / s and, for a linear antenna of diameter 45 cm of the order of 160 m / s. In other words, the efficiency of a macro-sensor depends on the diameter of the cylindrical structure of the antenna.

 - The macrosensors, in the configuration that has just been recalled, therefore have a number of limitations.

 The theory shows that it is possible to improve, in particular the above-mentioned ratio [S i B] by applying amplitude weightings along the macro-sensor so that the spatial transform thereof has a directivity more adapted to the intended application.

 Indeed, we seek a maximum rejection in a given frequency band (often low frequency).

 To illustrate the weighting method, we chose a macro-sensor of 9 individual 0.52 m pitch sensors adapted for the 40-80 Hz band. We chose two types of weighting: the so-called "cosine sur-remonté" weighting and the "triangle" weighting.

 With or without weighting, spatial Fourier transform curves were calculated for non-dissipative and dissipative slow modes; in any case for a gain mode celerity C = 100 m / s.

 Figure 5 is a diagram illustrating the case for which there is no weighting. The second secondary lobe is at -18 dB at 55 Hz.

The curve C3 illustrates the non-dissipative mode and the curve C'3 the dissipative mode, which is a mode taking into account the real conditions. The choice of an up-winded cosine weighting (Figure 6) or a triangular law (Figure 7) makes it possible to reach -25 dB or more at the same frequency. It should be noted that this weighting does not affect the acoustic performance because the sensor remains small in front of the acoustic wavelength. The corresponding curves are C4 and C'4 (Figure 6) and C5 and C'5 (Figure 7).

Naturally, the grouping gain can be improved by applying Chebyshev weights or other apodization techniques.

 Conventionally, the obtaining of weights, as just described, however require the use of complex electronic circuits associated with the sensors.

 On the contrary, the invention proposes a method of making it possible to very simply obtain a weighting according to a given template.

 The weighting is achieved by performing selective serial-parallel wiring of the sensors within a macrosensor. All the sensors have, in a preferred variant of the invention, the same characteristics of sensitivity (Sh) and impedance (Zh). A macrosensor is thus made composed of v groups of sensors wired in parallel.

Inside a group, the sensors are wired in series.

 The process will now be detailed. It is known that for a group composed of n hydrophones in series, the delivered signal is equal to the vector sum of the signals delivered by each of the transducers and that the impedance is the sum of the impedances.

(3 bis)
On montre simplement par récurrence que le signal délivré par un macrocapteur composé de v groupes tels que décrit ci-dessus câblés en parallèle, vaut

For a group i of hydrophones in series, the output signal obeys the relation S C1 = zsk (for hydrophones) (3)
k = 1, no relation in which Sik is the signal generated by the noise for any hydrophone k with 1 <k <nor the impedance is given by the relation

(3a)
It is simply shown by recurrence that the signal delivered by a macro-sensor composed of v groups as described above wired in parallel is worth

le poids du groupe i comme étant égal au produit multiple du nombre de capteurs composant les autres groupes.We notice

the weight of the group i being equal to the product multiple of the number of sensors composing the other groups.

 According to a preferred variant of the method according to the invention, for a predefined weighting law it will approach the best of this law using a graphical method that will now be described.

 FIG. 8 illustrates the method for approaching an optimal weighting law by performing selective serial-parallel wiring of the sensors within the macrosensors.

 FIG. 8 shows the optimum voltage weighting curve C6 to be obtained as a function of the position of the sensors.

The process has been illustrated by a so-called "cosine surrendered" curve. This is a portion of the cosine function. The weighting obtained by the serial-parallel wiring will result in a staircase curve C'6
Each "step" corresponds to a group i of nj sensors connected in series.

It appears that each area delimited by n1 sensor and by its height Pi is
## EQU1 ## where ## EQU1 ## is a constant term for any area.

 Each area has a constant value, heavy weight for few sensors, low weight for many sensors. One can reduce the height of an area by two by wiring in series sensors positioned symmetrically on either side of the maximum (which amounts to doubling the number of sensors by two for this group).

In FIG. 8, the areas A1 to As and the weights P1 to
P5. In view of the foregoing it appears that it is impossible to strictly follow a continuous curve. This curve is approached by a succession of high and short trays (qualifier referring to the weight and number of sensors respectively) or flat and long.

 The relation (3) can also be used to express the signal delivered by a sensor under the effect of a useful acoustic signal. In this case, however, all the sensors deliver the same signal, proportional to the sensitivity Sh that has been assumed constant from one sensor to the other.

The macrosensor is omnidirectional. Its length was chosen to be less than k / 4, where X is the wavelength of the acoustic signal heard.

 In the following will be used the index "a" to indicate that it is the acoustic signal.

It follows that the signal delivered by a group i of n1 sensors in series obeys the relation S1a = nI.Sh (9)
It can be noted that

The relation (4) becomes, using the relation (3)

a pour poids principal le terme provenant du groupe central , présentant le poids le plus fort (et donc un nombre de capteurs minimum : 1 ou 2 par exemple). La prise en compte des autres groupes modifie peu la valeur de cette suite, autrement dit: Z est sensiblement constant quelque soit le nombre de groupes et dépend simplement du nombre de groupes à faible nombre de capteurs. Sa dépend directement du nombre de groupes générant le signal total.The following

its main weight is the term coming from the central group, presenting the strongest weight (and therefore a minimum number of sensors: 1 or 2 for example). Taking into account the other groups does not modify the value of this sequence, in other words: Z is substantially constant regardless of the number of groups and simply depends on the number of groups with a low number of sensors. It depends directly on the number of groups generating the total signal.

 To fix the ideas and to illustrate more completely the method of the invention, we will consider a macro-sensor as shown schematically in Figure 9. There are seven groups of sensors referenced G-3 to G3. The H-4 and H-3 sensors are in series; H3 and H4 are in series; H2 to H2 are isolated. The weight of each sensor H-2 at H2vautP1 = li is PO = 2.1.1.1.1.2 = 4 (relation (6)).

joi
Group G3 composed of H3 and H4 sensors, presents the weight
P3 = 2.1.1.1.1.1 = 2
The sensors of groups G 3 and G 3 are therefore weighted at 0.5 with respect to the sensors H-2 to H2.

(relation (10))
The acoustic signal and the impedance of this macrosensor are respectively: Sa = ZSh and Z z = Lh (relations (12) and (13) or Sa = 1,167 Sh.

6 6
The macrosensor thus defined can be extended by adding on either side other groups of sensors.

Figure 10 illustrates such an embodiment. The central zone is constituted by the macrosensor of FIG. 9, labeled MC11, and a group of four sensors, in series on either side of the central zone, is added. The whole forms a new macrosensor that we will call
MC2.

There are therefore nine groups: seven for MCo and two lateral groups G 4 and G4.

The weights are as follows
pO = p1 = p2 = 64
P3 = 32
P4 = 16
We perform a weighting according to the "triangle" law.

 FIG. 11 illustrates the weighting curve C7 obtained.

(relation (10)) et l'impédance
Zh
6,5
On constate que l'impédance de MC2 est peu différente de celle que l'on avait calculé par MC1. On peut donc utiliser les mêmes organes de traitement électronique des signaux, en particulier les mêmes amplificateurs, ce qui est avantageux.The acoustic signal of the macro-sensor MC2, as illustrated in FIG. 10, is Sa = -9Sh = 1.385Sh
6.5

(relation (10)) and impedance
zh
6.5
It is found that the impedance of MC2 is not very different from that calculated by MC1. It is therefore possible to use the same electronic signal processing devices, in particular the same amplifiers, which is advantageous.

 To further illustrate the invention, a linear antenna has been made according to the method just described.

The assumptions and constraints that prevailed were as follows
(a) The sensors are equidistant along the linear antenna.

 (b) The sensors must be usable according to the "octave band" method.

 (c) The noise to be rejected is the mechanical frequency spectrum noise between 10 and 80 Hz and the frequency spectrum flow noise between 80 and 160 Hz.

 The "useful" signals, ie the acoustic signals to be detected, ranged from high frequencies: 1280 Hz at very low frequencies, in the range precisely covered by parasitic noise frequencies.

 The antenna included several sections intended to cover the various frequency bands: bands 640 to 1280 Hz, band 320 at 640 Hz, band 80 at 160 Hz, band 40 at 80 Hz and band 20 at 40 Hz.

 In the following, we will only be interested in very low frequencies, frequencies common to those of noise signals of mechanical origin or flow. It is however useful to indicate that the constraint (a) mentioned above requires that the macrosensors consist of individual sensors with a pitch of 0.52 m.

 For the band 80 to 160 Hz, four sensors in parallel can be implemented. The length of the macrosensor is less than a quarter of the wavelength at 160 Hz.

 With regard to the 40 to 80 Hz band, a solution has eight wired sensors as illustrated in FIG. 12 can be implemented.

The central part uses the previously described macrosensor for the band 80-160 Hz (four sensors in parallel) and spotted MCA
Two side groups of two sensors, HA, Hg, HA and H g, wired in series, were added. The interconnect pitch for all these sensors is 0.52 m.

 By convention, there is shown a "+" link in solid line and a link "-" in dotted line to differentiate the wiring order of the two output terminals of the sensors. Figure 12 explains the exact wiring of the set of sensors, generating an output signal Sa.

 In the lower part of Figure 12, an insert indicates the different weightings associated with the sensors.

 This MCB macro-sensor set will be called.

 The gain obtained in this band compared to a solution according to the known art (without weighting) is of the order of 10 dB on the mechanical noise.

 The diagrams of FIGS. 13 and 14 (gain of grouping Gg as a function of frequency make it possible to compare a known art antenna (diagram of FIG. 13: sensors all in parallel) and the antenna according to FIG. invention (diagram of Figure 14).

 The curves C7 (FIG. 13) and Cg (FIG. 14) illustrate the non-dissipative mode and the curves C'7 (FIG. 13) and C'8 (FIG. 14) the dissipative mode. The speed is C = 160 m / s.

 The possible wirings for the 20-40 Hz band have many degrees of freedom since the number of sensors can be large. There is shown in Figure 15 an example of possible embodiment among others.

 The previously defined MCB macrosensor (FIG. 12) is used to process the 40-80 Hz band. Two lateral groups of four sensors, respectively HC to HF and Hc to HF, are added.

 The HC and HD sensors as well as the HE and HF sensors are wired in series, two by two. It is the same for the symmetrical branch Hc to F.

 The output signal of the set is S'a.

 The weighting associated with each sensor is indicated, as previously, in the bottom inset of FIG.

Figures 16 and 17 compare the gain provided by an antenna according to the invention (Figure 7) compared to an antenna of the prior art (Figure 16: sixteen sensors in parallel). The curves C9 and
C10 illustrate the non-dissipative mode and the curves C'g and C'10 the dissipative mode. The speed is C = 160 m / s.

We see that the rejection is very important at 30 Hz: about 30 dB is an improvement of the order of 10 dB. This frequency corresponds to the harmonic known as the 'harmonic of
Strouhal ", which corresponds to a frequency due to the towing cable.

On these non-limiting examples, it is clearly seen that the method of the invention makes it possible to simply perform the weighting of a macro-sensor by serial-parallel wiring. It also has the advantage of minimizing the variations of the impedance of the macrosensor while increasing its acoustic sensitivity.

<Tb>

<SEP> Noise <SEP> | <SEP> Nature <SEP> Gain <SEP> potential <SEP> in <SEP> [S / B]
<tb> electrical <SEP> and <SEP> - <SEP> inconsistent <SEP> 10 <SEP> log <SEP> N
<tb> thermal <SEP> - <SEP> homogeneous <SEP> N <SEP>: <SEP> number <SEP> of <SEP> sensors
<tb><SEP> SEP <SEP> - <SEP> consistent <SEP><SEP><SEP> directivity <SEP> - <SEP> 10 <SEP> log
<tb><SEP> - <SEP> homogeneous <SEP> N
<tb><SEP> - <SEP> isotropic
<tb> radiated <SEP> of the <SEP> - <SEP> consistent <SEP> directivity <SEP> of <SEP> the antenna
<tb> carrier <SEP> - <SEP> homogeneous
<tb><SEP> - <SEP> directive
<tb> mechanical <SEP> - <SEP> consistent <SEP> SEP <SEP> filtering <SEP> spectrum <SEP> spatial
<tb><SEP> - <SEP> no <SEP> homogeneous <SEP> time <SEP> of <SEP> correlation
<tb> flow <SEP> - <SEP> little <SEP> consistent <SEP> .. <SEP>
<Tb>

<SEP> - <SEP> homogeneous
<Tb>
Table 1

Claims (9)

 1. A method for producing a towed linear antenna consisting of distributed sensors, said antenna comprising at least one assembly of sensor groups (MCB) in which weighting is carried out according to a specific law, each group comprising at least one sensor ; characterized in that said weighting is performed by selective wiring in series (H ~ BH ~ A and HA-HB) of sensors of said assembly, the other sensors (MCB) being wired in parallel so as to obtain a weighting curve (C '6> approaching said determined weighting law (C6).  2. Method according to claim 1; characterized in that the intercapturing pitch is constant.  3. Method according to claim 2; characterized in that the weight Pi of a group i of sensors is equal to the product multiple of the number of sensors composing the other groups.  4. The method of claim 3; characterized in that the weighting curve (C'6) of a sensor assembly is obtained by graphical construction around the curve of said determined weighting law (C6>; in that the weighting curve obtained by said selective wiring is a staircase curve, each stair step having a height equal to one of the weights Pi associated with the groups and a width equal to the number n of sensors comprising this group, and in that the areas defined by the it produces a number of sensors of a group by its weight Pi is constant and obeys the relation ll n, where v is the total number of groups of sensors in series, these groups being wired in parallel to form said assembly.  5. Method according to any one of claims 1 to 4; characterized in that said weighting makes it possible to attenuate unwanted noise signals propagating inside the structure of the antenna in a determined frequency range.  6. The method of claim 5; characterized in that said range comprises the signals corresponding to noises of mechanical origin whose frequencies are between 10 and 160 Hz.  7. Linear antenna towed; characterized in that it is carried out according to the method of any one of claims 1 to 6.  Antenna according to claim 7; characterized in that said sensors are hydrophones converting acoustic pressure variations into electrical signals, these hydrophones all having identical characteristics in sensitivity Sh; and in that the electrical signal Sa supplied by an assembly (MCB) of sensor groups obeys the relationship

 relation in which v is the total number of groups of an assembly and n the number of sensors of a group i of said assembly.  9. Antenna according to the rendication 7; characterized in that said sensors are hydrophones converting acoustic pressure variations into electrical signals, these hydrophones all having identical characteristics in terms of impedance Zh; and in that the impedance Z of an assembly (MCB) of sensor groups obeys the relationship

 relationship in which v is the total number of sensors of an assembly and n the number of sensors of a group i of said assembly.