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Due to their simplicity and operating mode, magnetic loops are one of the most used traffic sensors in Intelligent Transportation Systems (ITS). However, at this moment, their potential is not being fully exploited, as neither the speed nor the length of the vehicles can be surely ascertained with the use of a single magnetic loop. In this way, the vast majority of them are only being used to count vehicles on urban and interurban roads. For this reason, in order to contribute to the development of new traffic sensors and make roads safer, this paper introduces a theoretical study to explain the design and peculiarities of the innovative double loops, how to calculate their magnetic field and three different methods to calculate their inductance. Finally, the different inductance values obtained by these three methods will be analyzed and compared with experimental measurements carried out by our research group in order to know which method is more accurate and if all of them are equally reliable.

Magnetic loops are the most common sensors on roads around the world since they are an affordable and highly developed technology with a simple operation that is not affected by environmental conditions [

Their operation is straightforward, since it is based on the impedance variation that is recorded in the magnetic loops during the passage of vehicles over them, and as shown in Figure

A magnetic loop formed by a wire with one or more turns superficially buried in the pavement.

A cable that links the magnetic loop with the control booth, which is also buried in the pavement.

An electronic unit located in the control booth that contains an oscillator and amplifiers to excite the inductive loop.

Magnetic loop system scheme.

In order to have a better understanding of how they work, there are many publications and bibliography [

The electronic unit together with the magnetic loop forms an oscillator circuit.

The current which passes through the loop produces a magnetic field

This magnetic field ^{2}).

The result is that the inductance of a common single loop

Magnetic loop operation mode.

As seen in (

Magnetic profiles. (a) Car. (b) Bus.

To estimate the vehicle speed and classify them, nowadays it is necessary to use two single loops, since a single one is not able to get all the necessary parameters to do it. After analyzing the magnetic profile, there are two unknown data (vehicle length and vehicle speed) with the variation of a single parameter (inductance or oscillation frequency).

For that reason, there are usually two loops per lane separated by a certain distance. In this manner, the passage of a vehicle over the first loop is recorded in the detector, and after a short interval of time, the vehicle passes again over the second loop where it is also recorded [

Nevertheless, with the use of the double loops this problem would be solved and it would only be necessary to use one loop to find out all the previous data, since they have a simpler, more compact and more economical electronics. Moreover, having a single signal instead of two would facilitate the implementation of the measurement system.

Therefore, our work will aim to present and describe the characteristics of the double loop, to offer different methods to calculate its inductance, to verify which one provides better results and to improve the functional characteristics of the popular single loops, which despite being the most installed sensor on the roads, they are actually only dedicated to count vehicles. The presentation of the new vehicle magnetic profiles, the parameters that can be extracted with them and the advantages offered over the conventional loop will be the subject of the following paper.

The design, shape, and construction of a rectangular or circular single loop are well-known worldwide [

A double loop is no more than the union of two rectangular loops, which can have different dimensions and turns (not to be confused with two single loops spaced at a certain distance as described above). How to implement this type of loop can be quite varied, but a classic way to proceed would be to build the outer loop and then a smaller inner loop located at one extreme. However, another way to build it could be to construct a small loop and then put another small one next to the first one. In all cases, the direction of the current in each loop can be chosen with the aim of generating different types of configurations. At any rate, in our study we will present a general theoretical analysis capable of simulating any type of design. Therefore, in order to analyze this type of loop, the space will be divided into three sections as shown in Figure

The first section,

The second section,

The third section,

Double loop presented in three sections.

In this way, point

After obtaining the expression that describes the magnetic field generated by a double loop, the next step was to verify that the theoretical results coincided with the experimental ones. For this reason, a double loop was constructed in the laboratory by our research team (Group of Traffic Control System, ITACA Institute, Universitat Politècnica de València, Spain). It was implemented with an external loop of

Outline of the double loop constructed.

With these conditions, we only were interested in the component of the magnetic field perpendicular to the surface of the loop

Conductor type is tinned copper wire conductors individually insulated with polyvinyl chloride with a cross section of

Number of turns is 4 exterior turns

Dimensions is

With these values, the parameters of the loop according to the nomenclature used in Figures

Frequency of the signal applied to the loop is

Signal type is rectangular.

Current intensity through the loop

Height above the plane of the loop is

Position is along the

With the above-mentioned characteristics, the theoretical calculation in the region of the measurements was carried out by applying and programming the expression obtained in (

Exposure Level Tester ELT-400.

This device contains a series of turns with a diameter of

ELT-400 configuration.

Selected frequency range:

Reading range:

RMS signal value.

After collecting and processing all the information, the comparison between the calculated and measured values of the magnetic field as well as the tolerance of the measuring instrument (

Calculated and measured values of the magnetic field

It can be observed that the differences between the measured and calculated magnetic field values are, except in specific points, within the tolerance range of the instrument. The difference between the theoretical and measured values within the contour of the loop is below 20% of the reading and the mean value is below 8%. Therefore, it can be concluded that the theoretical model for double loops developed in this paper predicts with a good precision the behavior of the magnetic field. In addition, different types of tests were also carried out with other types of loops, both single and double, varying the type and amplitude of the applied current, and very similar results were obtained.

Throughout time, various methods to calculate the inductance of magnetic loops according to different geometric configurations have been proposed [

However, the development of computer systems has allowed to implement numerical methods that make use of the intrinsic definition of the physical process of magnetic induction. In this regard, it would be convenient to include our previously presented work [

In this way, this time we will present three methods to calculate the inductance of the innovative double loops, since there are no studies about them. Therefore, these methods will be deeply analyzed and compared in order to know which method is more effective and if all of them are equally good. These three methods are as follows:

(A) Electromagnetic Analysis Method

(B) Numerical Integration Method

(C) Mills and Grover’s Method

The procedure starts by making use of the flux calculation, since all the loops work in the same way. However, in this case, the magnetic field and the flux will present some peculiarities. The magnetic flux will be obtained as the integral of the product of the magnetic field by the differential surface all along the entire surface of the loop:

In this way, if we defined

On the other hand, due to the abrupt change that appears in the

For the purpose of calculation mentioned, it is assumed an assembly in which the loops are stacked vertically. First, the larger

The different flux components mentioned above will be represented by the terms

With this nomenclature, the inductance of the double loop would be given by the following:

In

In

In

The values used in (

When working with double loops, the self-inductance of the

where

It must be noted that the above-mentioned parameters

Features of two ideal parallel conductors of no straight section for measuring mutual induction.

In addition, it is know that the external inductance of a pair of parallel conductors with the dimensions shown in Figure

Features of two parallel conductors to measure the mutual induction.

As it can be deduced from this expression, the external inductance of a rectangular loop with one turn is equal to the mutual inductance of two identical coaxial rectangular loops separated by a distance equal to the radius of the conductor. In this way, the mutual inductance of two parallel rectangular loops as shown in Figure

Geometry for calculating the mutual inductance between two parallel and coaxial rectangular loops with the same dimensions.

Therefore, the mutual inductance between the two rectangular loops with the same dimensions as shown in Figure

On the other hand, to calculate the mutual inductance between two parallel loops with different dimensions, Grover’s equations must be used again, since they provide the mutual inductance between two parallel straight conductors as shown in Figure

Disposition of two parallel straight conductors.

According to Grover’s formula, the mutual inductance between two parallel conductors with sizes

In any case, from all these expressions it is finally possible to obtain the inductance between two parallel rectangular loops with different dimensions as shown in Figure

Disposition of two parallel loops of different dimensions.

Therefore, the mutual inductance between the two parallel loops is finally obtained as a sum of the mutual inductances of the parallel conductors where the terms

Thus, once all methods have been presented, the inductance values obtained are going to be compared to each other to verify the similarity between them and know which the best methods are and which are not.

After making the description and performing the analysis of the three methods, the first thing that stands out is that there are two methods that consider the spacing between the turns

To check all of the above, various test related to the inductance values of the different methods and types of loops were performed and are presented below. In fact, it was studied the effect of increasing the number of turns in a single and in a double loop and the effect of changing the dimensions of a double loop.

For the purpose of this study, a single

As mentioned, in the ideal case it would be expected for the three methods to provide similar results, but it is clearly observed in Figure

Inductance values. (a) When increasing the number of turns in a single loop. (b) When increasing the number of turns in a double loop. (c) When increasing the length of “a” in a double loop. (d) When increasing the length of “d” in a double loop.

This means that the more turns the loops have, the more different the results are, because the error of not considering separation between turns increases for each turn.

The next test was the same as the previous one but instead of working with a single loop, we will work with a double loop. This time, it was a double

The result obtained was really similar to the previous one. As it can be seen in Figure

Once we had analyzed how the number of turns affects the inductance value, in the two remaining experiments we analyzed what happens when varying the dimensions of the loops. For this purpose, we worked with a double loop formed by

In Figure

On the other hand, after analyzing what happens when the length of

In this test, it was seen how two methods followed the same trend again, providing almost the same results, but the other one did not behave properly. However, in Figure

Therefore, after analyzing the previous results, we could conclude that of the three methods proposed, the electromagnetic method could be useful for very thin conductors with little separation between them and for complex geometries in which Mills and Grover can not be used, as it only serves for parallel and perpendicular conductors. When it was tried to increase the turns of the loop, it was the only one that differed from the rest. On the other hand, when the length of the loop was increased, it was also evident that this method did not behave correctly. For this reason, we can affirm that although it is a valid method and can give us an approximation if the separation between turns is big, it should be only used as a reliable source if that separation is minimal.

Regarding the two remaining methods, it must be noted that both offer good and similar results as it can be seen in Figures

This article is aimed to be a presentation of the double loop, where geometry, construction, operating mode and three possible ways to calculate its inductance have been explained. After presenting these three above-mentioned methods, an analysis has clarified that if precision is required, Mills and Grover’s method or the numerical integration method must be used, as they both take into account the separation between turns, although we recommend to choose the first one because of its low computational cost.

In future papers, we will focus more closely on the advantages offered by using this type of loops, what will help to understand the need and importance of this paper. It will focus on the new vehicle magnetic profiles, the parameters that can be extracted from them and the benefits of using them in comparison with the conventional loops.

The reality is that magnetic loops, despite being from the eighties, are still the most used technology to capture data from traffic. For that reason, we must improve the existing infrastructure and provide this sensor with greater potential and reliability.

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.