# Basic Concepts

Basic antenna and EM concepts are given that provide a foundation for understanding the different solver methods in Feko.

## What is an Antenna?

An antenna is a conducting structure consisting of surfaces and/or wires designed to be of specific characteristic dimensions to radiate or receive an electromagnetic wave.

The primary purpose of an antenna is to make an impedance match between a signal (electrical current or wave) travelling in a coaxial cable (transmission line) or wave guide and waves travelling in free space. A secondary purpose is to send the signal in a specific direction. 12

## What is a Far Field?

When calculating fields at a specific point P in space over a great distance from the radiator, the following assumptions hold:
• Differences in the distances from P to the different points on the radiator have a negligible effect on the magnitude of the field.
• Differences in the distances from P to the different points on the radiator should be accounted for when calculating the phase, but certain assumptions could be made.
• All field components that decay faster than $\frac{1}{r}$ can be considered negligible compared to those that decay with $\frac{1}{r}$ .3

The far field of an antenna is the minimum distance from the antenna where the field components do not contain reactive components, or where these components can be considered negligibly small. This distance is generally written as:

(1) $r=\frac{2{D}^{2}}{\lambda }$

where D is the largest dimension of the antenna.

In the far field, the antenna is considered a point source. Equation 1 is derived under the assumption that the varying distances to the radiator do not contribute to phase errors larger than 22.5°.4

## What is a Near Field?

The near field of an antenna is the region in close proximity to the antenna where the electric and magnetic fields are not in phase. The fields are reactive and there is also a strong radial component. The radial component of the field has no $\frac{1}{r}$ dependency but does have $\frac{1}{{r}^{2}}$ , $\frac{1}{{r}^{3}}$ and even higher dependencies. Naturally, these field components vanish very quickly with increasing distance.5

## What is a Transmission Line?

A transmission line is a coaxial cable, microstrip, stripline, waveguide or some specialized structure designed to conduct a radio frequency signal. The frequency of the signal is high enough, such that the wave behaviour of the signal cannot be ignored. While there are several purposes, in radio frequency engineering, transmission lines are typically used to connect transmitters, receivers and antennas.

From a radio frequency engineering point of view, typical parameters of interest are the input reflection coefficient and the voltage standing wave ratio (VSWR).

## What are S-Parameters?

S-parameters characterize the relationship between the input and output ports of a system in terms of power waves. While the relationship could also be described in terms of other network parameters such as ABCD, Z and Y-parameters, calculating these parameters require the termination of the ports in open or short circuits. Achieving purely open or short circuits, especially over wide bands, are not feasible. In addition, some devices are not stable if they are open or short-circuited.

However, when calculating S-parameters, it only requires termination of the ports in the system impedance.6

## What is the Reflection Coefficient?

The reflection coefficient is a quantity or figure describing how much of an electromagnetic wave is reflected due to an impedance mismatch (discontinuity) in the transmission line or transmission medium. The reflection coefficient is calculated as the ratio of the magnitude of the reflected wave to the incident wave. It can be calculated from the characteristic impedance of the transmission medium (line), ${Z}_{0}$ , and the impedance of the discontinuity (often the load impedance at the end of a transmission line), ${Z}_{l}$ , as follows:

(2) $\Gamma =\frac{{Z}_{l}-{Z}_{0}}{{Z}_{l}+{Z}_{0}}$

For the special case of a one-port device, the reflection coefficient is the same as the S-parameter, S11. In the case of a device with two ports or more, the parameters, Snn (for example S11, S22), is the same as the reflection coefficient if all the ports are loaded with the port or system impedances.
Note: A Feko model with a single port does not require an S-parameter request. The input reflection coefficient is the same as the S-parameter request.

## What is a Smith Chart?

A Smith chart is a graphical representation of impedance, admittance, phase, wavelength and reflection coefficient. The Smith chart consists of a family of normalized resistance circles and reactance circles. The circles represent the value of the input impedance of some system (network, circuit or load) as measured a certain distance away from the system over a transmission line. 7 The Smith chart provides a transformation between reflection coefficient and impedance over a transmission line. It represents wave behaviour on a transmission line.8 The advantage of the Smith chart is that you can represent all complex values from positive to zero and negative infinity for the real and imaginary parts.

## What is CEM?

Computational electromagnetics (CEM) refers to a numerical solution or a computer-based approximation of the currents or the fields. In the numerical solution the currents or fields are firstly divided into many small parts. Subsequently the physical equations (typically Maxwell's equations such as Ampère's law, Faraday's law) that describe the relationships between fields, currents and charges are then used to obtain the magnitude and phase of each current or field element. Finally the summing (integration) of these current or field elements yield antenna parameters such as input impedance and far fields.

Consider the Yagi-Uda antenna shown in Figure 5. Only the current-carrying parts of the antenna are shown. More specifically, the antenna is represented as an assembly of subdivided sections where each section (or rather, each junction between sections) carries a small but uniform current.

It is initially assumed that on each junction between sections, a constant current is flowing, but the magnitude and phase of these currents are not known. Maxwell's equations are used to find the magnitude and phase of each small current element.

Once the magnitude and phase of each current element is known, all the currents are added together (integrated). A transformation of the currents then give the electric and magnetic near and far fields as well as other antenna parameters.

In post-processing, the fields as seen in Figure 6 can be visualised.
1 Fields and Waves in Communication Electronics, Third Edition, Ramo, Whinnery and Van Duzer, p. 584-586; 599
2 Electromagnetic Fields and Energy. Haus and Melcher, p. 547
3 Fields and Waves in Communication Electronics, Third Edition, Ramo, Whinnery and Van Duzer, p. 593
4 Antenna Engineering Handbook, Fourth Edition, John L. Volakis, p. 1-8
5 Advanced Engineering Electromagnetics, Second Edition, Constantine A. Balanis, p. 283
6 High-Frequency Circuit Design and Measurements, Peter C.L. Yip, p. 33-34
7 High-Frequency Circuit Design and Measurements. Peter C.L. Yip, p. 15
8 High-Frequency Amplifiers, Second Edition, Ralph S. Carson, p. 56