Tag Archives: Maxwell

Ibn al-Haytham to Maxwell: A Long Road

Fundamentals, History of Science, , , ,

As the Chinese proverb says “The journey of a thousand miles begins with a single step”. The journey that started with Ibn al-Haytham experimenting with his Camera Obscura in the eleventh century was completed eight hundred years later by James Clerk Maxwell and Heinrich Hertz. While Maxwell laid down the mathematical framework that described the behavior of Electromagnetic waves, Hertz conclusively proved the existing of these invisible waves through his experiments. There were several scientists on the way that played a crucial part in development of this Electromagnetic theory such as Gauss, Faraday and Ampere. Then there were others such as Huygens, Fresnel and Young who worked on nature of light, which was not known to be an Electromagnetic wave at that time. Once the theory  of Electromagnetic wave propagation was in place there was rapid progress in many fields, particularly in wireless communications (wireless telegraph, radio, radar etc.).

Maxwell’s equations that were proposed in 1861 were initially quite circuitous and were not well accepted. But later on these equations were simplified into the form we now know by Oliver Heaviside. There are still two popular forms of the equations, the integral form and the differential form. We present the integral form of these equations in this article as it is more intuitive and is also easier to represent graphically. The differential form requires understanding of the concepts of divergence and curl and we skip it in this article. The main take away from these equations (presented below) is that a changing Electric field produces a Magnetic field and a changing Magnetic field produces an Electric field. Another important result is that magnetic monopoles do not exist (simply put a magnet, however small, always has a north and south pole).

Maxwell's Equations in Integral Form
Maxwell’s Equations in Integral Form

Note:

  1. The dot product with a line segment means that only that component of the field vector is effective that is along the line segment. On the other hand the dot product with a surface means that only that component is considered that is perpendicular to the surface (since the unit vector of a surface is perpendicular to the surface). It means that only those field components are considered that are going perpendicularly in or out of the surface.
  2. For more on history of Maxwell equations visit IEEE Spectrum  and for a detailed explanation of the various forms of the Maxwell’s equations visit this page.
  3. In modern Electromagnetic simulation software the differential form is preferred and the algorithm used is called Finite Difference Time Domain (FDTD). However, if the area of interest is quite large (with respect to the wavelength) then the FDTD method becomes prohibitively complex and another method known as Ray-Tracing is used. Please do check out the Ray-Tracing engine that we have developed. Ray-Tracing is becoming increasingly important in RF Planning of Telecom Networks.

Ray-Tracing for Network Planning-II

Network Planning, , , , ,

It’s very easy to get lost in the jargon when selecting a simulation tool for planning your wireless network. You will be faced with complex terminology which would not make much sense. At one end of the spectrum are solutions based on simple empirical models while at the other end are solutions based on ray-tracing techniques. Empirical models are based on measurement data and are your best bet if you want a quick and cheap solution whereas ray-tracing techniques are based on laws of physics and promise more accurate results. In principle ray-tracing techniques are quite simple: just transmit a bunch of rays in all directions and see how they behave. However when the number of rays and their interactions becomes large the simulation time may become prohibitively expensive. The simulation time for complex geometries may vary from a few hours to several days.

Following are some of the factors that you must consider when selecting a ray-tracing simulator.

1. Upper limit on the number of interactions

Ray-tracing simulators essentially generate a bunch of rays (image based techniques are an exception) and then follow them around as they reflect, refract, diffract and scatter. Each interaction decreases the strength of the rays. The strength of the rays also decays with distance. As a result the simulator needs to decide when to terminate a ray path. This is usually done based upon the number of interactions that a ray undergoes (typically 8-10 interactions are considered) or based upon its strength (once the strength of a ray falls below -110 dBm there is no point following it any further). Higher the number of interactions considered, greater the accuracy of the simulation but higher the computational complexity.

2. Granularity in field calculations

Field calculations cannot be performed at each and every point within the simulation space. The usual approach is to divide the region under study into a grid such that locations closer to a transmitter are covered more finely and the regions further away are covered in lesser detail. The rays are then combined within each block of the grid to get the resultant field strength. The level of granularity determines the computation load. It would be prohibitively expensive to have a very high level of granularity for a large network.

3. Accuracy in modeling the various propagation phenomenon

As mentioned previously an accurate modeling of all propagation phenomena is required including reflection, refraction, diffraction and scattering. Some ray-tracing simulators might model reflection and refraction only while ignoring the other phenomenon such as diffraction. Furthermore some ray-tracing simulators might consider all reflections to be specular (no scattering). This is a good approximation for large smooth surfaces but is not such a good assumption for irregular terrain.

4. Granularity of the terrain database

Most state of the art ray-tracing tools use some sort of terrain database to perform their calculations. These terrain databases are required for determining the paths of the rays as they travel in dense urban environments. These databases may contain simple elevation data or actual 3D building data. These databases may have accuracy of 10m or 30m or maybe more. The accuracy of the simulation is highly dependent on the granularity of the terrain database.

5. Accuracy in representation of building materials

The wireless signal propagation within cities is governed by complex phenomena such as reflection, refraction, diffraction and scattering. Let’s take the example of the phenomenon of reflection. The percentage of signal reflected back at a particular interface is dependent on permittivity and permeability of the object. Based on these properties only 10% of the signal maybe reflected or 50% of the signal may be reflected. So, for accurate simulation not only should we have a high level of granularity of the 3D building data, we also need an accurate description of the building materials.

6. Dynamic Channel Behavior

A wireless channel is continuously changing i.e. the channel is dynamic (as opposed to being static). However the ray-tracing techniques available in the literature do not capture this dynamic behavior. The dynamic behavior of the channel is mainly due to the motion of the transmitter or receiver as well as motion of the surroundings. While the position of the transmitter and receiver can be varied in the ray-tracing simulation the surroundings are always stationary. Hence a ray-tracing simulator is unable to capture the time-varying behavior of the channel.

The accuracy of ray-tracing simulators is bound to increase as the computational power of computers increases and as accurate 3D building databases become available throughout the world. Until that time we would have to fall back to approximate simulations or maybe measurement results.