Forschungsverbund Medientechnik Südwest

Transcription

1 Institut für Höchstfrequenztechnik und Elektronik (IHE) Universität Karlsruhe (TH) Institutsleiter: Prof. Dr.-Ing. W. Wiesbeck IHE Karlsruhe Forschungsverbund Medientechnik Südwest Projekt II.1 Wellenausbreitung und Planung digitaler Netze Teilprojekt II.1.1 (Universität Karlsruhe, IHE) Wellenausbreitungsmodelle für digitale Sendernetze Wave Propagation Modelling in Tunnels for Digital Radio Systems (DAB, DVB, DMB, UMTS) A fast method to characterise the propagation behaviour and The influence of exterior antenna positions

2 Index Abstract Chapter 1: Introduction Radio propagation prediction models 2 Chapter 2: The existing simulation program IHE-TUNNEL5 2.1 Functionality of IHE-TUNNEL Program flow chart of IHE-TUNNEL Sending algorithms of IHE-TUNNEL Methods of ray tracing in IHE-TUNNEL The input files The input file "geometry.ini" The input file "settings.ini" The input file "vehicles.ini" Modification of the program call 25 Chapter 3: Total power-flow method Description of the method Prerequisites The power-trace method Automatic insertion of plane receivers Space grids Influence of the space grids on the computing time Comparison between the power-trace method and the new total power-flow method Dependence of the results on the number of rays Influence of the cross-section area on the wave propagation Influence of the tunnel shape on the wave propagation Influence of the permittivity on the wave propagation Verification with measurements 71 Chapter 4: Position of the antenna outside the tunnel Different locations of the antenna outside the tunnel Coupling of emitted rays into the tunnel A first proposal to the problem of fitting the sending range for tunnels with rectangular cross-section A first proposal to the problem of fitting the sending range for tunnels with elliptic cross-section Determination of the exact sending range The new program flow chart Influence of different antenna parameters on the power flow Influence of the height of the transmitter on the power-flow Influence of the distance between the trasmitter and the tunnel trance on the power-flow 101

3 Conclusions Influence of the position of the transmitter on the power flow Study of the coupling of the rays into the tunnel with the method of the Image Theory 105 Bibliography

4 Abstract Characterization of radiowave propagation in tunnels has important applications for mobile and personal communications. This work focuses on extending an existing simulation program, called IHE-TUNNEL, whose purpose consists in calculating the radiowave propagation in tunnels with an arbitrary geometry. The calculation of the radiowave propagation bases on an optical ray tracing with discrete rays. Rays are launched from the transmitting antenna location thanks to a sending algorithm. Three different sending algorithms were already implemented in IHE-TUNNEL: stochastic, deterministic and a third one that determines the sending angle according to the Image Theory. As regards the ray tracing, four methods were already implemented in the program: power-trace method, electric field-trace method, identification of multiple rays and Image Theory. IHE-TUNNEL is able to carry out many tasks, such as coherent and incoherent point-wise or plane analyses with moving vehicles. However, that takes hours and even a day. Thus, an acceleration of the program is desirable and that can be realized by implementing the new total power-flow method that allows, as rapid as possible, to obtain a quick first characterization of the radio propagation in tunnels. Another aim of this work is to study the influence of the diverse parameters that describe a tunnel on the wave propagation. After that, a verification of the powerflow method is made with the aid of measurements coming from Spain. A limitation of the existing program IHE-TUNNEL is the fact that the transmitting antenna can be located only inside the tunnel. In reality, the transmitting antenna is mostly installed outside the tunnel. For this reason, another objective of this work resides in extending the application area of the program in order to make possible the simulations of the situations in which the antenna is situated outside the tunnel.

5 In this way, the coupling of the emitted rays into the tunnel is analysed and two solutions to the problem of fitting the sending range to the tunnel entrance are implemented. Finally, the influence of the different antenna parameters on the power-flow analysed when the antenna stands out of the tunnel. is

6 Chapter 1: Introduction Mobile communications are developing rapidly and their expansion forces mobile operators to provide a high quality of service at any time and situation. Achieving such versatile communication services heavily relies on the radiowave propagation knowledge that can minimize infrastructure costs. Therefore, extensive studies have been undertaken both theoretically and experimentally in various environments, like e.g., propagation in rural areas. Studies related to determine urban propagation mechanisms have received even much greater attention since the vast majority of communications take place in the centres of population. Recently, propagation research has turned to concentrate on the characterization of radio propagation in indoor environments such as houses, office buildings and factories. Tunnels can be often seen in metropolitan cities and mountainous areas. The confined spaces of tunnels considerably constitute special radio propagation environments and have their own unique propagation characteristics. Propagation losses, for example, become even lower than in free space because of the waveguide effect. Moreover there are vehicles, that go along this space and influence the radio propagation channel. Consequently, new prediction models are required because radio propagation in tunnels cannot be computed using outdoor/indoor propagation models. 1.1 Radio propagation prediction models Radio propagation prediction methods for tunnels can be divided into three main categories: waveguide, empirical and ray launching models. The waveguide model represents an analytical approach and it is applicable only for very simple tunnel geometries based on orthogonal coordinates. Conversely, empirical models represent 1

7 Chapter 1: Introduction a practical approach for tunnels of any geometry. The prediction made by empirical models always requires time consuming measurements in order to tune the propagation model. The last approach is a ray launching technique that is superior than the before mentioned ones in tunnel coverage predictions because it does not limit the tunnel geometry, and the number of measurements can be minimized. In ray launching models, the rays are sent in all directions and followed until a determined condition is satisfied. That is an attempt to find all the rays that reach the receiver. This method functions for complex geometries too: the reflected ray is simply calculated with help of the normal surface at the reflection point. Diffraction or scattering can be modelled so that new rays are sent in determined directions. A maximum attenuation or a maximum number of reflections can be chosen as termination criterion for the following of the rays. The disadvantage of ray launching is the fact that it is not sure that all rays, which have a contribution to the total propagating power are found. The last question to be answered in the ray launching method is whether ray tubes or discrete rays should be used. A ray tube is defined by means of its central ray, which determines the propagating direction, and its limiting rays, which delimit the ray tube. The higher time requirement and the problem at curved surfaces (see Fig. 1.1-a) are the main and maybe more critical disadvantages of ray tubes. On the other hand, it is very easy to decide if a receiver was hit by a ray. For that purpose it must be tested if this receiver is situated inside the ray tube. Another disadvantage is the fact that plane receivers can not be used, but the plane analysis must be composed of a series of points. 2

8 Chapter 1: Introduction R T Fig. 1.1-a: The problem of ray tubes at curved surfaces If discrete rays are used, the ray tracing becomes much simpler. But there are problems too. Since the receivers are ideal dots, it is difficult to decide if a particular ray hits a receiver. These receivers must be converted to spheres with a finite radius (see Fig. 1.1-b), but this radius is not known. If this radius is selected too small, too few rays hit the receiver. But if it is selected too large, many identical rays arrive at the receiver. R T Fig. 1.1-b: Ray tracing with discrete rays This issue was treated in [Sch98] and a normalization technique was developed, which allows the use of discrete rays. 3

9 Chapter 2: The existing simulation program IHE-TUNNEL The starting point of this work is an existing simulation program, called IHE- TUNNEL. This program was written in the programming language C++ on UNIX. Its application is to calculate the radiowave propagation in almost any tunnel. This means that the tunnel geometry is arbitrary. The most commonly used tunnel shapes in road construction are the elliptic and the rectangular cross-sections (see Fig. 2-a and Fig. 2-b). The rectangular cross-section is especially used in metropolitan cities or short tunnels. Elliptic tunnels are mostly constructed in mountainous areas. Ventilation conductions are normally placed in long tunnels at the upper part as an additional ceiling (in Fig 2-b the broken line). Fig 2-a: Rectangular cross-section Fig 2-b: Elliptic cross-section Regarding the trajectory, the tunnel can be composed of any combination of straight and/or curved sections (see Fig. 2-c). Fig. 2-c: A tunnel trajectory 4

10 Chapter 2: The existing simulation program IHE-TUNNEL The main features of IHE-TUNNEL are: A transmitting antenna can be located anywhere inside the tunnel. The program is able to deal with isotropic and directional radiation patterns. Single spherical receivers and plane receivers can be inserted. A single spherical receiver allows to obtain the electric field strength or the power at a specific point. An integral calculation on the total cross-section can be required, too. Such plane analysis can be reached by distributing single spherical receivers as a matrix, but this approach is very time-consuming with respect to the calculation of the intersection points. The reason for employing plane receivers is a shorter calculating time. Such receivers, specially plane receivers, provide a field strength prognosis that is useful to determine the propagation loss along the tunnel (coherent and incoherent addition). The signal is not continuous, but rather it consists of pulses. Many single pulses are registered at the receiver because of the multipath propagation and the different propagating time. IHE-TUNNEL is capable of calculating the delay and Doppler spread. Delay spread is a measurement of the frequency selectivity on radio propagation channel between transmitter and receiver. Time domain and frequency domain are related by the Fourier transformation, i.e. a time change in radio propagation channel implies a frequency shift of the received signal. Therefore, the properties of a radio propagation channel depend not only on the signal frequency but also on the time. For example, the moving vehicles cause that new propagation paths arise or old paths disappear. Doppler spread is a measure of the time selectivity that allows to estimate how long the properties of the channel are constant. 5

11 Chapter 2: The existing simulation program IHE-TUNNEL IHE-TUNNEL enables not only to introduce cars and lorries but also to simulate their movement. Both kind of vehicles are modelled as cubes whose walls are taken as ideal conductor, except the underbody that is perfectly absorbing. The study of the behaviour of the tunnel at various time instants is possible thanks to the automatic generation of time series. So far the geometry of a tunnel and the principal attributes of IHE-TUNNEL have been explained. The functionality of IHE-TUNNEL is introduced in the next section. 2.1 Functionality of IHE-TUNNEL Figure 2.1-a illustrates how the program works. In this example, the tunnel consists of a straight and a curved section. A lorry and two cars have been inserted. There is a transmitter that is always inside the tunnel. Rays are stochastically emitted from the transmitting antenna location thanks to a Monte Carlo method. By means of a Monte Carlo method, every ray direction gets the same probability. In other words, the probability density function is uniform in space. Each ray interacts with every possible obstacle inside the tunnel, for instance, with the two cars, the lorry, the four walls of the tunnel and the single spherical receivers and/or plane receivers. Additionally, it is possible to employ other sending algorithms that will be seen in section

12 Chapter 2: The existing simulation program IHE-TUNNEL Tunnel entrance CAR Single spherical receivers Plane receiver (Plane analysis) LORRY Tunnel exit Transmitter CAR Fig. 2.1-a: Functionality of IHE-TUNNEL The ray launching technique has the problem that identical rays can be registered many times due to the finite size of the receiver (see Fig. 2.1-b). The identical rays that belong to the same physical path lead to wrong results. Under the condition that every relevant propagation path is found at least once, the received electric field strength or power should be independent on the number of emitted rays and on the size of the receiver by means of a normalization. As a solution, two possibilities are developed on the program IHE-TUNNEL: Detection of identical arriving rays and ray density normalization. Both procedures will be explained in section 2.2. Transmitter Receiver 1 Receiver 2 Fig. 2.1-b: Problem of identical rays In conclusion, a Monte Carlo sending algorithm is applied concurrently with a ray launching and ray density normalization. 7

13 Chapter 2: The existing simulation program IHE-TUNNEL Program flow chart of IHE-TUNNEL Figure a illustrates the program flow chart of IHE-TUNNEL. Firstly, the input files are read and the input data are verified. Afterwards, the internal calculations of the tunnel configuration are carried out and following, the ray tracing is started. During and after the ray tracing, the rays arriving at receivers are saved. If a motion takes place, the new positions of transmitter, receivers and vehicles must be calculated and a new ray tracing will be started. input files reading ray tracing input data verification data saving internal calculations motion end of program Fig a: Program flow chart of IHE-TUNNEL Sending algorithms of IHE-TUNNEL Before the ray is initialized, the sending angle must be calculated. Three different sending algorithms were implemented, which can be selected in the input files. A sending algorithm is based on stochastic calculation, another one is deterministic and a third one determines the sending angle according to the Image Theory. 8

14 Chapter 2: The existing simulation program IHE-TUNNEL Monte-Carlo methods are often used in problems that are based on stochastic principles. By means of a Monte Carlo method, every ray direction gets the same probability. In other words, the probability density function is uniform in space. Otherwise, the strategy of the deterministic sending algorithm [Vid95] is: starting with a "circle" of triangles with the peak at the centre, to connect new rings of triangles in order to cover the area of the circle. Since the areas must always remain roughly similar, the number of triangles in a ring is duplicated when their area exceeds a determined magnitude. The last method that was implemented bases on Image Theory. This sending algorithm calculates the exact angles θ and φ which cause the ray to hit the receiver. Contrary to the usual strategy of reflecting the sender at the walls, here the receivers are reflected. The maximum number of reflections is used as final criterion. 2.2 Methods of ray tracing in IHE-TUNNEL The following four methods of ray tracing were already implemented in the program: 1. Power-trace method. 2. Electric field-trace method. 3. Identification of multiple rays. 4. Image Theory. The power-trace method will be explained afterwards in detail because one of the goals of this work is based on it. In the electric field-trace method, rays are emitted with an initial electric field strength. The arising propagation loss is calculated. Identic rays can be registered 9

15 Chapter 2: The existing simulation program IHE-TUNNEL several times because of the finite dimensions of the receiver. That means, that multiple rays, which belong to the same propagation path, hit the receiver. Consequently, wrong results are obtained without normalization. This normalization is not an attempt to avoid multiple rays, but a suitable procedure to get right results. Provided that each propagation path was found at least once, the power or field strength must be independent of the number of sent rays. A ray density normalization is used in this method. The program calculates the ray density n d for each ray, that is to say, the number of rays per unit of area. If a ray hits a receiver, the number of rays, which theoretically hit the receiver, is calculated. For a number N of sent rays and a propagation without curved surfaces, the following ray density must be considered: n d N 4πr = (2.1) 2 where r stands for the covered distance. If the rays are not distributed in all space directions, but in a limited solid angle 0 <δφ<360 and 0 <δθ<180 (the centre of the selected sending range is always the tunnel direction, that is to say, θ=90 and φ=0 ), the ray density is calculated by means of: N n d = πr δφ δθ sin 2 (2.2) The rays are uniformly distributed in φ (therefore 360 /δφ). Otherwise, the ray distribution in θ follows a sine curve and the probability density is: 10

16 Chapter 2: The existing simulation program IHE-TUNNEL θ p( θ)= sin 2 (2.3) The sending range (δθ) in program IHE-TUNNEL is symmetrical, hence the integration limits for the probability calculation are ( 90 δθ ) and ( 90 + δθ ) δθ 2 sinθ cosθ δθ d θ = = sin (2.4) δθ δθ 2 90 δθ 2 Another normalization technique consists in attempting to identify the multiple rays of a propagation path and combining them in order to obtain a representative ray for such propagation path. The normalization problem can be solved if the multiple rays are searched in the output files and than deleted. But first, it must be clarified, how the rays that belong to a same propagation path can be identified. Three different criteria are necessary. Firstly, the number of reflections is considered. If this number is different, the rays have reached the receiver by also different propagation paths. Secondly, the propagation time of the rays is compared. At last, the sending angle is calculated. If two rays have the same number of reflections and their propagation times and sending angles differ only in a very low percentage, it can be said, that a multiple ray was found. The last method that was already implemented is based on Image Theory. The transmitter position is mirrored at every wall and these mirror images are directly linked with the receivers (see Fig. 2.2-a). Afterwards, the higher order mirror images are calculated. But contrary to the usual strategy of reflecting the sender at the walls, the receivers are reflected in existing program IHE-TUNNEL. The great 11

17 Chapter 2: The existing simulation program IHE-TUNNEL advantage of the Image Theory is the fact that only the rays that reach exactly the receiver are calculated. However, this technique is restricted to empty straight tunnels of rectangular cross-section. 1 Rx Tx 1 Tx 2 Tx 2 Fig. 2.2-a: Image Theory Principle To sum up, only the power-trace method and the field-trace method can be applied in any situation or tunnel geometry. Therefore, these methods will be from now on taken into account and in particular the power-trace method. 12

18 Chapter 3: Total power-flow method As already has been seen the existing program IHE-TUNNEL is able to carry out many tasks, e.g. coherent and incoherent pointwise analyses with moving vehicles. However, that takes hours and even a day. An aim of this work is to study the influence of the diverse parameters that describe a tunnel on the wave propagation. That can be realized thanks to the total power-flow method that allows, as rapid as possible, to get a quick first characterization of the radio propagation in tunnels. 3.1 Description of the method Each ray in the tunnel represents a locally plane wave. Plane waves are reflected in the tunnel based on Fresnel s reflection formulas: R = ε rtot cos θ i ε rtot sin2 θ i ε rtot cos θ i + ε rtot sin 2 R = cos θ i ε rtot sin 2 θi θ i cos θ i + ε rtot sin 2 θ i (3.1) Where ε rtot is the total permittivity: ε rtot =ε r j σ ωε o (3.2) If the ray does not hit a wall and reflect, it sustains all the power that it had at the moment of the emission. The propagating power can be obtained by calculating separately a Poynting vector for each plane wave. The time-averaged Poynting vector is: S av = 1 2 (E H ) (3.3) 13

19 Chapter 3: Total power-flow method and for a single ray i, that has reflected n times in tunnel walls, it is: S av,i = 1 2 (E H ) = k 2ωµ E i 2 (3.4) where Ei = [ R1] [ R2]... [ Rn 1] [ Rn] E0 and R k the surface k. [ ] is the respective dyad of reflection in As the figure 3.1-a shows, the idea of the total power-flow method consists in an incoherent power addition of the rays that reach the plane of analysis, by means of inserting a string of plane receivers. These plane receivers do not have any radiation pattern. In other words, they are fictitious surfaces. Plane receivers N 2 Tx for thepower E analysis i i = 1 Fig. 3.1-a: Total power-flow method The total time average propagating power through the tunnel cross-section is computed by adding together all Power/Energy contributions from every passing ray: P = Sav ds = Pi (3.5) S i 14

20 Chapter 3: Total power-flow method where P i is the Power/Energy content of each ray through the cross-section. After briefly explaining the power-flow method, the prerequisites will be taken into account in order to implement the power-flow method. 3.2 Prerequisites The power-trace method, the use of plane receivers and a technique that allows to reduce computing time costs will be considered as prerequisites The power-trace method The total power-flow method is based on the power-trace method. The idea of the power-trace method is not to regard the electric field strength of the rays, but their energy or power. The power of a ray must remain constant along the total propagation path. The only losses are caused by the reflections. As Figure a shows, the longer the distance between transmitter and receiver, the fewer rays reach the receiver and the lower the absolute value of the power addition becomes. Fig a: The power of a ray remains constant along the total propagation path The total power at the receiver makes sense only after normalizing and adding up every arriving ray. A correction coefficient must be applied both for coherent and incoherent addition. 15

21 Chapter 3: Total power-flow method The calculation of the coherent coefficient requires the use of ray densities, that is to say, the number of rays per unit of area. Moreover, the focussing phenomena must be taken into consideration. On the one hand, the incident ray at each reflection is simply reflected in the tangential plane and the reflection factors are calculated for such angle of incidence according to (3.1). On the other hand, the ray tubes are focussed or defocused owing to the reflection at curved surfaces. This effect provokes the change of the principal radii of curvature of the ray tubes and their directions, too. These calculations are very time-consuming [Sch98]. The new total power-flow method follows the idea of the existing power-trace method but it saves calculations in order to get a shorter calculation time. The main attributes of the new total power-flow method are: The correction coefficients always remain constant and they do not depend on the ray density. Therefore, neither the ray tubes nor the radii of curvature are necessary. The phases of the electric field strength are neither required because the rays that reach the plane of analysis are added incoherently in the total power-flow method. An automatic insertion of plane receivers has been developed, that is to say, an arbitrary quantity of equidistant plane receivers can be placed into the tunnel. A technique based on the use of space grids allows to reduce computing time during the ray tracing. The last two characteristics will be explained in the next two sections because they have been specifically developed in order to accelerate calculations in the total power-flow method. 16

22 Chapter 3: Total power-flow method Automatic insertion of plane receivers Plane receivers are suitable as a means to calculate the power-flow in the new total power-flow method. These plane receivers do not include any radiation pattern because the power-flow can be determined from the number of receiving rays inside each plane receiver. Therefore, plane receivers are imaginary surfaces. Thanks to the automatic insertion of plane receivers, the user can insert easily an arbitrary amount of plane receivers between the desired limits. Plane receivers could be inserted in the early version of IHE-TUNNEL too, but each plane receiver had to be manually introduced in the input file. That could be a possible source of mistakes and a time-consuming typing of the input information. Now, the process of typing the input data is faster and safer. By means of the keyword REPEAT N a b in the file that describes the tunnel geometry, the positions of sender, receivers and vehicles (filename.geo), N plane receivers can be automatically inserted from a to b. Concerning calculation celerity, the space grid will be presented in next section Space grids In this work, a stochastic sending algorithm is applied together with ray launching and ray density normalization. Figure a shows the program flow chart of the ray tracing. Firstly, the direction of emission is determinated by electing both angles Θ and Φ. Secondly, the ray is initialized to its initial value and emitted. Thirdly, the calculation of the next intersection point takes place. If the ray intersects an obstacle, i.e. the ray will be reflected in it. If the ray passes through a receiver, its data will be saved. In both cases, a test is carried out in order to know whether the maximum attenuation or the maximum number of reflections for a single ray is exceeded. If the test is positive, the next ray will be emitted. In the opposite case, the next intersection point of the current ray will be calculated. The step of the intersection 17

23 Chapter 3: Total power-flow method point calculation is coloured in order to point out that such process is very timeconsuming. Determination of emission angles Ray initialization and emission Calculation of the intersection point Receiver Ray saving max. attenuation exceed? no Obstacle Interaction (i.e. Reflection) yes next ray Fig a: Program flow chart of the ray tracing For example, Figure b illustrates a tunnel section, where four cars, a lorry and ten single spherical receivers have been inserted. The computer must test at every surface whether an intersection point exists. That is to say, there are 44 possible surfaces of interaction and 44 tests of intersection must be made for each interaction and for each ray. After checking that, the computer finds 7 possible intersection points. However, only one point is right one: the one closest to the last intersection point. Therefore, the computer must calculate the point of minimum distance (see number 1 in Fig b) To sum up, many test of intersection, that do not make sense, are carried out. 18

24 Chapter 3: Total power-flow method CAR ❼ CAR CAR Transmitter Single spherical receivers CAR ➈ LORRY Fig b: Complexity of ray tracing The space grids technique consists in dividing the tunnel longitudinally into slices of the same size (see Fig c). Now, the computer must exclusively find a point of intersection among the obstacles that are situated inside each grid. Concerning the last example, only 12 tests of intersection must be made. Thus, the program can be accelerated. Grids CAR CAR CAR Transmitter Single spherical receivers CAR LORRY Fig c: The space grid technique Space grids help to avoid unnecessary tests of interaction. However, space grids imply a management that spends processing time too. The program is written in C++ and the maintenance of the space grids resides in a list that contains the obstacles belonging to each grid. Additionally, the use of a string of transparent walls between 19

25 Chapter 3: Total power-flow method grids is necessary in order to know when a ray goes from a grid to another (see the red broken lines in Fig c). To conclude, space grid realizes a profit but has a disadvantage. An optimum should be found as a means to obtain faster calculations. This task will be discussed in the next section Influence of the space grid on the computing time Paying attention to the size of the space grid, it can not be selected too small because the computing time increases due to the management of the space grids. Otherwise, the space grid can not be too large because the amount of tests of intersection rises. In order to study the influence of the space grids on the computing time, two situations have been considered: a straight tunnel 100 m long with rectangular crosssection (8x6 m), and a straight tunnel 1000 m long with rectangular cross-section with vehicular traffic. In the first case, the power-flow was calculated by means of the total power-flow method. Therefore, 81 plane receivers were inserted. Figure a shows a minimum at seven surfaces per grid. In other words, the optimum lies on three plane receivers plus the four walls of the tunnel. 20

26 Chapter 3: Total power-flow method Normalized calculating time 4 3,5 3 2,5 2 1, Tests per grid Fig a: Influence of the space grid on the computing time in a straight tunnel 100 m long with rectangular cross-section (8x6 m) In the second simulation, 20 plane receivers were inserted in a tunnel 1000 m long with rectangular cross-section. Moreover, there were 20 cars in the tunnel, that were allocated in positions from 25 m to 975 m. The minimum is registered at 11 surfaces of interaction per grid (see Fig b). Taking into account that a car consists of 6 walls, the optimum corresponds to a car plus a plane receiver plus the four walls of the tunnel. In addition, figure b shows that the computing time is very similar, when the space grids are not applied and when too many grids are used. That reflects a tradeoff between the space grid and its costs of maintenance. Summarizing, two optimum values of space grid have been found depending on if there are vehicles in the tunnel or if not. These results can be considered in order to optimize the space grid by creating an intelligent and adaptive space grid. This can be a task to work on in the future. 21

27 Chapter 3: Total power-flow method Normalized calculating time 2,4 2,2 2 1,8 1,6 1,4 1,2 1 too many grids Number of vehicles without grids Tests per grid Fig b: Influence of the space grid on the computing time in a straight tunnel 1000 m long with rectangular cross-section (8x6 m). There are 20 cars, that are allocated in arbitrary positions from 25 m to 975 m 3.3 Comparison between the power-trace method and the new total power-flow method The new total power-flow method is based on the power-trace method. It should be clarified, that the power-trace method allows to calculate the power-flow, too. In the power-trace method, each plane receiver is divided into a grid of pixels in order to know exactly where the ray intersects the plane receiver and then, the correction of phase is performed. Therefore, the power-trace method is able to obtain the powerflow by inserting plane receivers with the rudest resolution. However, it takes a quite long time. When the new total power-flow method is used, the phases of the electric field strength are not required, because an incoherent addition is made. Thus, neither resolution of plane receivers nor correction of phase are necessary in the new total power-flow method. 22

28 Chapter 3: Total power-flow method To compare both methods in relation to computing time, six simulations with different number of rays were carried out. The environment of simulation consisted of a straight tunnel 1000 m long with rectangular cross-section (6x6 m), and 100 plane receivers were inserted. The figure 3.3-a displays the results that manifest a great difference between both methods regarding the calculating time. Overall, the new total power-flow method saves 80% of the computing time. Normalized computing-time Number of rays (million) Incoherent power-trace method using plane receivers with the rudest resolution New total power flow method Fig. 3.3-a: Comparison between the power-trace method and the new total power-flow method 3.4 Dependence of the results on the number of rays The new total power-flow method allows to obtain a quick characterization of the behaviour of the tunnel. To reach a greater gain in the computing time, the minimum number of emitted rays that provides accurate results must be estimated. Thus, further simulations at a frequency of 1 GHz were made in order to study the dependence of the results on the number of transmitted rays. A curved tunnel 1500 m long with elliptic cross-section was employed. The radius of curvature was 1500 m long. The transmitter was located at the tunnel entry and the plane of analysis at the tunnel exit. Additionally, a car and a lorry were inserted at 1000 m and 180 m 23

29 Chapter 3: Total power-flow method from tunnel entrance respectively. It can be seen in Figure 3.4-a that there are few fluctuations starting from rays, approximately 0.2 db, and in any case, the fluctuations are always less than 0.5 db for a number of rays equal to If the power-trace method with a coherent addition was used, between 10 and 50 millions rays must be emitted. Now, only rays are necessary. That means an enormous gain of calculating time, but a high loss in accuracy. Normalized power-flow (db) Number of rays (in thousand) Fig. 3.4-a: Dependence of the results on the number of transmitted rays The next sections are focussed on the study of the tunnel parameters and their influence on the wave propagation. The effect of the cross-section area is simulated first. Secondly, the influence of the shape of the tunnel cross-section on the propagation slope is studied, where the tunnel cross-section shape is varied but the area always remains the same. Thirdly, several simulations are carried out in order to see the influence of the permittivity on the wave propagation. Finally, a verification of the power-flow method is made with the aid of measurements coming from Spain. 24

30 Chapter 3: Total power-flow method 3.5 Influence of the cross-section area on the wave propagation The area of the tunnel cross-section may have an influence on the wave propagation. A series of simulations were made in order to characterize this effect. The power-flow method with stochastic sending algorithm was used and the tunnel was defined by means of the following parameters: Frequency: 1 GHz. Straight section. Length: 1500 m Relative permittivity: 5 Transmitter: - at the tunnel entrance, in the centre of the cross-section - omnidirectional pattern - vertical polarization Sending range: 180 in θ and Φ (Ω=2π) Sending algorithm: stochastic Number of rays: Plane receivers: 150 plane receivers from 1 m to 1491 m. A simulation explains the behaviour of an elliptic cross-section and another one that of a rectangular one. The different areas that were employed are introduced in the following table: 25

31 Chapter 3: Total power-flow method Horizontal ellipse a x b (m) Area (m 2 ) 3x x x x Vertical ellipse a x b (m) Area (m 2 ) 2x x x x Horizontal rectangle width x height (m) Area (m 2 ) 3x2 6 6x4 24 6x x Vertical rectangle width x height (m) Area (m 2 ) 2x3 6 4x6 24 6x x As said, several different cross-sections were simulated with the rectangular and the arched shapes of the tunnel. The following figures show the results for the different tunnels: 26

32 Chapter 3: Total power-flow method Normalized Power-flow (db) Vertical 2x3 m Vertical 4x6 m Vertical 6x10 m Vertical 12x15 m Distance (m) Fig. 3.5-a: Normalized power-flow in vertical rectangular tunnels of different cross-section area Normalized Power-flow (db) Horizontal 3x2 m Horizontal 6x4 m Horizontal 10x6 m Horizontal 15x12 m Distance (m) Fig. 3.5-b: Normalized power-flow in horizontal rectangular tunnels of different cross-section area 27

33 Chapter 3: Total power-flow method Normalized Power-flow (db) Vertical 2x3 m Vertical 4x6 m Vertical 6x10 m Vertical 12x15 m Distance (m) Fig. 3.5-c: Normalized power-flow in vertical elliptic tunnels of different cross-section area Normalized Power-flow (db) Horizontal 3x2 m Horizontal 6x4 m Horizontal 10x6 m Horizontal 15x12 m Distance (m) Fig. 3.5-d: Normalized power-flow in horizontal elliptic tunnels of different crosssection area 28

34 Chapter 3: Total power-flow method A same tendency is noticeable in the graphics above. The smaller the area of the cross-section, the greater the attenuation of the signal. This behaviour is in accordance with [Lam98] and it can be explained with the aid of Figure 3.5-e. Tx Plane receiver Tx Plane receiver Fig. 3.5-e: The smaller the area of the cross-section, the greater the attenuation of the signal Figure 3.5-e shows two tunnel sections each with a transmitter. It can be seen that there are more reflections in the small tunnel (left) than in the big one (right). In other words, the larger the area of the tunnel cross-section, the higher is the powerflow level. Ideally, if the area of the tunnel cross-section was infinite, there would not be any reflection and the power-flow would be maximal. In this way, a comparison between tunnels of different cross-section areas is not interesting. From now on, we make use of the power-flow per cross-section area in order to compare tunnels of different cross-section areas. To determine the effect of the cross-section area on the power-flow, a series of simulations were made by considering a straight tunnel 1500 m long with elliptic cross-section. The cross-section area was progressively increased, starting from an ellipse with horizontal semiaxis a=1 m and vertical semiaxis b=1.5 m and ending with an ellipse (a=150 m, b=500 m). Another parameters employed in the simulations are: 29

35 Chapter 3: Total power-flow method Frequency: 1 GHz Relative permittivity: 5 Transmitter: - at the tunnel entrance, in the centre of the cross-section - omnidirectional pattern - vertical polarization Sending range: 180 in θ and Φ (Ω=2π) Sending algorithm: stochastic Number of rays: Plane receivers: 1 plane receiver at the end of the tunnel. The results can be seen in Figure 3.5-f. This figure shows a logarithmical proportion between the cross-section area and the power-flow. This relation becomes a straight line when the areas are represented logarithmically (see Fig. 3.5-g). Power-Flow/Cross section area (db) , , Cross section area (m 2 ) Fig. 3.5-f: Power flow per cross-section area. It can be seen from Figure 3.5-g that the slope is -5 db/dec. The confined spaces of tunnels considerably constitute special radio propagation. Propagation losses become even lower than in free space because of the waveguide effect. Thus, if the radio propagation would be considered as a free space propagation, the slope would be -10 db/dec. 30

36 Chapter 3: Total power-flow method Power-Flow/Cross section area (db) Qualitative free space propagation Cross section area (m 2 ) Fig. 3.5-g: Power flow per cross-section area and comparison with the free space propagation. 3.6 Influence of the tunnel shape on the wave propagation A series of graphics are presented in order to explain how the tunnel shape can influence the characteristics of the wave propagation. The simulations were carried out in a tunnel with following values: Straight section. Length: 1500 m Frequency: 1 GHz Relative permittivity: 5 Transmitter: - at the tunnel entrance, in the centre of the cross-section - omnidirectional pattern - vertical polarization Sending range: 180 in θ and Φ (Ω=2π) Sending algorithm: stochastic Number of rays: Plane receivers: 150 plane receivers from 1 m to 1491 m. 31

37 Chapter 3: Total power-flow method Firstly, a rectangular geometry was used. Starting from a square (see Fig. 3.6-a), the shape of the tunnel cross-section was progressively modified, in order to obtain horizontal and vertical rectangles, always with the same area (see Fig. 3.6-b). Tx h w w=h= m Fig 3.6-a: Square cross-section w Tx h w w= m h=8.86 m w=9.24 m h=8.5 m w=11.22 m h=7 m w=15.7 m h=5 m w=31.42 m h=2.5 m w=78.54 m h=1 m w=157 m h=0.5 m Tx h w=8.86 m h= m w=8.5 m h=9.24 m w=7 m h=11.22 m w=5 m h=15.7 m w=2.5 m h=31.42 m w=1 m h=78.54 m w=0.5 m h=157 m Fig 3.6-b: Horizontal and vertical rectangular cross-sections After having explained how the simulations were carried out, the results are presented: 32

38 Chapter 3: Total power-flow method Normalized Power-flow (db) 0-5 V-polarization x m x8.86 m 8.86x m Distance (m) Fig 3.6-c: Normalized power-flow in vertical and horizontal rectangular cross-sections (width x height) in comparison with a square cross-section Normalized Power-flow (db) 0-5 V-polarization x m 31.42x2.5 m 2.5x31.42 m Distance (m) Fig 3.6-d: Normalized power-flow in vertical and horizontal rectangular cross-sections (width x height) in comparison with a square cross-section 33

39 Chapter 3: Total power-flow method Normalized Power-flow (db) 0-5 V-polarization x m 78.54x1 m 1x78.54 m Distance (m) Fig 3.6-e: Normalized power-flow in vertical and horizontal rectangular cross-sections (width x height) in comparison with a square cross-section Normalized Power-flow (db) 0-5 V-polarization x m 157x0.5 m 0.5x157 m Distance (m) Fig 3.6 -f: Normalized power-flow in vertical and horizontal rectangular cross-sections (width x height) in comparison with a square cross-section 34

40 Chapter 3: Total power-flow method The results for the square tunnel appear in all the graphics. In this way, it is always possible to compare the behaviour of the rectangular cross-section with a square cross-section. The conclusion of this comparison is that there are few changes in the received power-flow at the plane receivers. This variation is important only when the rectangular cross-section of the tunnel differs considerably from a square (see Fig 3.6-e and Fig. 3.6-f). Moreover, the behaviour of a vertical and a horizontal rectangular cross-section is different. The vertical rectangle has a greater deviation from the square case. The reason is, on the one hand, that a vertical polarization is considered. On the other hand, according to [Gen96], the reflection factors R and R are similar when the angle of incidence is lower than 20º or near to 90º. Consequently, when the power-flow is measured starting from 500 m (angle of incidence near to 90º) or near the tunnel entrance (angle of incidence lower than 20º), the measured power-flow is practically independent of the cross-section. However, when the distance is between 50 m and 500 m, the measured power-flow is different for the vertical and horizontal cross-section. Secondly, the elliptic geometry was considered. Starting from a circle (see Fig g), the shape of the tunnel section was progressively modified in order to obtain more and more eccentric ellipses (see Fig. 3.6-h), but again the area remains constant. Additionally, the area of the cross-section remains constant and it is equal to the rectangular cross-sections. 35

41 Chapter 3: Total power-flow method a b Tx a=b=5 m Fig 3.6-g: Circular cross-section a b Tx a b a=5.005 m b=4.995 m a=5.5 m b= m a=6 m b= m a=10 m b=2.5 m a=20 m b=1.25 m a=25 m b=1 m a=50 m b=0.5 m Tx a=4.995 m b=5.005 m a= m b=5.5 m a= m b=6 m a=2.5 m b=10 m a=1.25 m b=20 m a=1 m b=25 m a=0.5 m b=50 m Fig 3.6-h: Horizontal and vertical elliptic cross-sections 36

42 Chapter 3: Total power-flow method And the results are: Normalized Power-Flow (db) V-polarization 0-5 5x5 m 5.005x4.995 m 4.995x5.005 m Distance (m) Fig 3.6-i: Normalized power-flow in vertical and horizontal elliptic cross-sections (a x b) in comparison with a circular cross-section Normalized Power-Flow (db) V-polarization 0-5 5x5 m 5.5x m x5.5 m Distance (m) Fig 3.6-j: Normalized power-flow in vertical and horizontal elliptic cross-sections (a x b) in comparison with a circular cross-section 37

43 Chapter 3: Total power-flow method Normalized Power-Flow (db) V-polarization 0-5 5x5 m 6x m x6 m Distance (m) Fig 3.6-k: Normalized power-flow in vertical and horizontal elliptic cross-sections (a x b) in comparison with a circular cross-section Normalized Power-Flow (db) V-polarization 0-5 5x5 m 10x2.5 m 2.5x10 m Distance (m) Fig 3.6-l: Normalized power-flow in vertical and horizontal elliptic cross-sections (a x b) in comparison with a circular cross-section 38

44 Chapter 3: Total power-flow method Normalized Power-Flow (db) V-polarization 0-5 5x5 m 20x1.25 m 1.25x20 m Distance (m) Fig 3.6-m: Normalized power-flow in vertical and horizontal elliptic cross-sections (a x b) in comparison with a circular cross-section Normalized Power-Flow (db) V-polarization 0-5 5x5 m 25x1 m 1x25 m Distance (m) Fig 3.6-n: Normalized power-flow in vertical and horizontal elliptic cross-sections (a x b) in comparison with a circular cross-section 39

45 Chapter 3: Total power-flow method Normalized Power-Flow (db) V-polarization 0-5 5x5 m 50x0.5 m 0.5x50 m Distance (m) Fig 3.6-o: Normalized power-flow in vertical and horizontal elliptic cross-sections (a x b) in comparison with a circular cross-section The following trend can be observed from the graphics: the behaviour of the horizontal ellipse differs more from that of the circle. There is more attenuation. When the vertical ellipse is considered, the response is at first almost as if the crosssection was circular. Nevertheless, when the ellipse is quite eccentric, the deviation is already appreciable (see Fig. 3.6-m, Fig. 3.6-n and Fig. 3.6-o). To sum up, a last graphic with the behaviour of the circle, the square, and the extreme cases (rectangle with very great height and very eccentric vertical ellipse) are presented in Figure 3.6-p. 40

46 Chapter 3: Total power-flow method Normalized Power-flow/Cross section area (db) V-polarization Circle Square Ellipse Rectangle distance (m) Fig. 3.6-p: Comparison between circle, square, rectangle and ellipse The best tunnel shape is the circle, which has less attenuation. The square is the following shape with a good wave propagation. Finally, it can be observed that the ellipse shows a lower power-flow than the rectangle. Consequently, the ellipse is the worst tunnel shape for the power-flow. Additional simulations with the transmitter at an eccentric position were carried out. This eccentric position of the sender was calculated in order to keep a proportion. The procedure is as before: starting from a square (see Fig. 3.6-q) or circular crosssection (see Fig. 3.6-s), the dimensions of the tunnel geometry are changed in order to obtain narrower and narrower rectangles or more and more eccentric ellipses. Consequently, the position of the sender must vary with the geometry (see Fig. 3.6-r and Fig. 3.6-t). 41

47 Chapter 3: Total power-flow method dis_middle h Tx height w w=h= m height= m dis_middle= m Fig 3.6-q: Square cross-section with the transmitter at an eccentric position dis_middle h Tx height dis_middle Tx w w= m h=8.86 m dis_middle= m height= m w=9.24 m h=8.5 m dis_middle= m height= m w=11.22 m h=7 m dis_middle= m height= m w=15.7 m h=5 m dis_middle= m height= m w=31.42 m h=2.5 m dis_middle= m height= m w=78.54 m h=1 m dis_middle= m height= m w=157 m h=0.5 m dis_middle= m height= m h w h e i g h t w=8.86 m h= m dis_middle= m height= m w=8.5 m h=9.24 m dis_middle= m height= m w=7 m h=11.22 m dis_middle= m height= m w=5 m h=15.7 m dis_middle= m height= m w=2.5 m h=31.42 m dis_middle= m height= m w=1 m h=78.54 m dis_middle= m height= m w=0.5 m h=157 m dis_middle= m height= m Fig 3.6-r: Horizontal and vertical rectangular cross-sections with the transmitter at an eccentric position 42