Table of Contents

# Introduction

Antennas act as converters between conducted waves and electromagnetic waves propagating freely in space (see Figure 1). Their name is borrowed from zoology, in which the Latin word antennae is used to describe the long, thin feelers possessed by many insects.

The oldest existing antennas, such as those used by Heinrich Hertz in 1888 during his first experiments to prove the existence of electromagnetic waves, were in theory and in practice not so very different from an RF generator.

An antenna can be derived from a parallel circuit which consists of an inductor and a capacitor. If the plates of the capacitor are bent open, and the inductor is reduced to the inductance of the wire itself, one ends up with a dipole antenna as shown at the very right position of Figure 2.

In fact, resonant circuits are still frequently used even today as a means of explaining the individual properties of antennas. It was not until around 1900 or even later, when transmitting and receiving stations were being built, that a clear distinction was made and antennas were classified as separate components of radio systems.

In Figure 3 it can be seen that the antenna is an important element in any radio system because it acts like a link of a chain. So the overall performance is significantly influenced by the performance of transmit and receive antennas.

At first glance, modern antennas may still look very similar to the ancient model. However they are nowadays optimized at great expense for their intended application. Communications antenna technology primarily strives to transform one wave type into another with as little loss as possible.

This requirement is less important in the case of test antennas, which are intended to provide a precise measurement of the field strength at the installation site to a connected test receiver; instead, their physical properties need to be known with high accuracy.

The explanation of the physical parameters by which the behavior of each antenna can be both described and evaluated is probably of wider general use; however the following chapters can describe only a few of the many forms of antenna that are in use today.

# Fundamentals of Wave Propagation

## Maxwell’s Equations

The equations postulated by the Scottish physicist James Clerk Maxwell in 1864 in his article A Dynamic Theory of the Electromagnetic Field are the foundations of classical electrodynamics, classical optics and electric circuits. This set of partial differential equations describes how electric and magnetic fields are generated and altered by each other and by the influence of charges or currents.

Equation (1) is Ampere’s law. It basically states that any change of the electric field over time causes a magnetic field. Equation (2) is Faraday’s law of induction, which describes that any change of the magnetic field over time causes an electric field. The other two equations relate to Gauss’s law. (3) states that any magnetic field is solenoid and (4) defines that the displacement current through a surface is equal to the encapsulated charge.

From Maxwell’s equations and the so called **material equations**

it is possible to derive a second order differential equation known as the **telegraph equation**:

? is a placeholder which stands for either ? ⃗or ?⃗.

If one assumes that the conductivity of the medium in which a wave propagates is very small (? → 0) and if one limits all signals to sinusoidal signals with an angular frequency ω, the so called **wave equation** can be derived:

The simplest solution to this equation is known as a plane wave propagating in loss free homogenous space. For this wave, the following condition applies:

The vectors of the electric and magnetic field strength are perpendicular to each other and mutually also to the direction of propagation (see Figure).

Consequently the electric and magnetic field strengths are connected to each other via the so called impedance of free space:

## Plane wave description

There are two preconditions for the existence of plane waves:

- Far field conditions must be reached.
- Free space conditions are present.

Only if both of them are met, the assumption that the electric field strength drops with the factor 1/r over the distance r, can be made.

## Wavelength

In linear media, any wave pattern can be described in terms of the independent propagation of sinusoidal components. The wavelength of a sinusoidal waveform travelling at constant speed is given by:

The wavelength is a measure of the distance between repetitions of a shape feature such as peaks, troughs or zero-crossings.

## Far Field Conditions

The distance from an antenna, where far field conditions are met, depends on the dimensions of the antenna in respect to the wave length. For smaller antennas (e.g. a half-wave dipole) the wave fronts radiated from the antenna become almost parallel at much closer distance compared to electrically large antennas. A good approximation for small antennas is that far field conditions are reached at:

For larger antennas (i.e. reflector antennas or array antennas) where the dimensions of the antenna (L) are significantly larger compared to the wave length (L >> λ), the following approximation for the far field distance applies:

Free space conditions require a direct line of sight between the two antennas involved. Consequently no obstacles must reach into the path between them. Furthermore in order to avoid the majority of effects caused by superposition of direct and reflected signals, it is necessary that the first Fresnel ellipsoid (see Figure 5) is completely free of obstacles:

The first Fresnel ellipsoid is defined as a rotational ellipsoid with the two antennas at its focal points. Within this ellipsoid the phase difference between two potential paths is less than half a wavelength.

The radius (b) at the center of the ellipsoid can be calculated based on the formula: