# Ibn al-Haytham to Maxwell: A Long Road

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).

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.

# How to Calculate the Surface Area Required by Solar Panels

You have estimated the size of the solar system that you need and are ready to get the equipment from the market to install it. But wait, are you sure you have enough space in your garden or your backyard or your rooftop to install the solar panels? How can you do a rough estimate of the area required by the solar panels? Here is a quick and easy way to go about it.

Lets assume that you want to install 10 solar panels rated at 100 Watts each and having a conversion efficiency of 18%. The total power output of the solar system can be calculated as:

Total Power Output = Total Area x Solar Irradiance x Conversion Efficiency

We know the required Total Output Power is 1000 Watts (10 panels x 100 Watts), the Solar Irradiance for a surface perpendicular to the sun’s rays at sea level on a clear day is about 1000 Watt/m2 and the Conversion Efficiency is 18%.  Plugging these number in the above equation we get:

1000 Watts = Total Area x 1000 Watts/m2 x 0.18

or

Total Area = 1000/180 = 5.56 m2

I you are going to install all the panels in one line you would need a space of approximately 1 m x 5.56 m (each panel having a size of 1 m x 0.556 m) on your rooftop. There you go. You have a rough estimate of the space required by the solar panels of your system.

Note:

1. Do remember that solar panels are usually installed at an angle to the earth’s surface and this may change the results somewhat. For an example of detailed calculation see the following post.

2. Appliances typically operate on AC voltage, whereas, solar panel produces DC voltage and battery also operates on DC. Therefore an inverter is needed to convert DC to AC and there can be substantial losses in conversion.

3. Imagine a solar panel has a conversion efficiency of 100% i.e. it converts all the solar energy into electrical energy then all you would need is a 1 m2 solar panel to produce 1000 Watts of electrical energy :).

# E and H Field of a Patch

The Electric and Magnetic Field variations within a patch are sometimes a bit confusing and difficult to visualize. The figure below shows the E and H Field variations within a rectangular patch of length L and width W.

E and H Field of a Patch

As can be seen the E-field varies along the length of the patch with minimum at the centre and maximum at the edges (maximum positive and maximum negative). The H-field also varies along the length is in a direction perpendicular to the E-field. The H-field is maximum at the center and minimum at the edges. Thus the impedance is zero at the center of the patch (using Z=V/I).

Another point to note is that the E-field does not completely terminate at the edges along the length of the antenna rather it extends at the outer periphery. These field extensions are known as fringing fields and cause the patch to radiate [1].

# Patch Antenna Design using Transmission Line Model

A microstrip antenna can be designed using either the transmission line model or the cavity model (more complex models also exist that suit a particular design). We here demonstrate the transmission line model since it is fairly simple to implement and results in antenna designs with reasonably good performance in terms of return loss and efficiency.

The design starts with selecting the operating frequency, selecting a substrate with the required permittivity, and defining the width of the substrate. Thick substrates with low permittivity result in antenna designs with high efficiency and large bandwidths. Thin substrates with high permittivity lead to a smaller antenna size but with a lower bandwidth and a high-radiation loss. The tradeoffs between substrate thickness and permittivity and antenna bandwidth and efficiency have been discussed in the literature.

According to the transmission line model, the length L and width W of the patch are calculated as

Although the design of the patch is quite simple, the design of the feeding mechanism is not that straightforward. There are four possible methods that can be used:

(1) Microstrip-line feed

(2) Probe feed

(3) Aperture-coupled feed

(4) Proximity-coupled feed

Reference

[1] Yasir Ahmed, Yang Hao, and Clive Parini, “A 31.5 GHz Patch Antenna Design for Medical Implants,” International Journal of Antennas and Propagation, vol. 2008, Article ID 167980, 6 pages, 2008.

# E-field of a Patch Antenna

A Microstrip Patch Antenna or simply a Patch Antenna is a very common antenna type used in cell phones and many other electronic devices. It basically consists of two metallic plates separated by a dielectric layer. The metallic plates are usually made of copper or some other highly conductive material. Another important feature of this antenna is the feeding mechanism, which is also made of a highly conductive material. A Microstrip Patch Antenna fed by a 50 ohm transmission line and a quarterwave transformer is shown below.

The E-field and H-field generated by the Patch Antenna can be calculated by using a simulation tool such as CST Microwave Studio. It is dependent on the length ‘L’ and width ‘W’ of the patch as well as on the wavelength ‘λ’ and the feeding mechanism. Other important parameters are the width of the substrate and the permittivity of the substrate. The E-field can be approximated as.

Part of the above equations is a sinc function which can be calculated by the MATLAB sinc function.

References:

# Solar Analogy

All electromagnetic energy travels in the form of rays. The most obvious example is solar energy that is radiated by the sun in all directions. The further away a body is from the sun the lower the energy that it receives. Objects in the path of these rays cause shadows but not complete darkness as rays reflect from other objects and also diffract around the edges. These rays also have a phase and frequency that determines their behaviour when interacting with objects. The amount of rays that can be collected by a receiver depends upon its size and orientation. Solar energy can be harmful when a body is exposed to it for longer periods.

All these concepts are extendable to wireless communications. Wireless signals decay with distance, suffer from shadowing, reflect, refract, diffract, scatter, have phase and frequency, can be collected by appropriately designed antennas and can be harmful as well. The major difference being that modern transmitters are not isotropic radiators. Practical transmitters are like a sun that radiates solar energy to the earth in a narrow beam while ignoring the other planets.