Although the whole spectrum is infinite, the useful spectrum is limited and some frequency bands, such as the UHF, are already very congested. Normally significant license fees have to be paid to use the spectrum, although there are some license‐free bands: the most well‐known ones are the industrial, science, and medical (ISM) bands. The 433 MHz and 2.45 GHz bands are just two examples. Cable operators do not need to pay the spectrum license fees, but they have to pay other fees for things such as digging out the roads to bury the cables.
The wave velocity v is linked to the frequency f and wavelength λ by this simple equation:
(1.14)
It is well known that the speed of light (an EM wave) is about 3 × 108 m/s in free space. The higher the frequency, the shorter the wavelength. An illustration of how the frequency is linked to the wavelength is given in Figure 1.11, where both the frequency and wavelength are plotted on a logarithmic scale. The advantage, by doing so, is that we can see clearly how the function is changed even over a very large scale.
Figure 1.11 Frequency vs wavelength
Radio waves, lights, and X‐ray (f = 1016 to 1019 Hz) are EM waves at different frequencies although they seem to be very different. One thing that all the forms of EM waves have in common is that they can travel through empty space (vacuum). This is not true for other kinds of waves; sound waves, for example, need some kind of material, such as air or water, in which to move. EM energy is carried by photons, the energy of a photon (also called quantum energy) is hf, where h is Planck’s constant = 6.63 × 10−34 Js, and f is frequency in Hz. The higher the frequency, the more the energy of a photon. X‐Ray has been used for imaging just because of its high frequency: it carries very high energy and can penetrate through most objects. Also due to this high energy, X‐ray can kill our cells and cause ionizing radiation that is not safe for our health. However, lights and radio waves operate at lower frequencies and do not have such a problem.
Logarithmic scales are widely used in RF (radio frequency) engineering and antennas community since the signals we are dealing with change significantly (over 1000 times in many cases) in terms of the magnitude. The signal power is normally expressed in dB (decibel), which is defined as
(1.15)
Thus, 100 W is 20 dBW, or just expressed as 20 dB in most cases; 1 W is 0 dB or 30 dBm; and 0.5 W is −3 dB or 27 dBm. Based on this definition, we can also express other parameters in dB. For example, since the power is linked to voltage V by P = V2/R (so P ∝ V2), the voltage can be converted to dBV by
(1.16)
Thus, 300 kVolts is 70 dBV, and 0.5 V is −6 dBV (not −3 dBV) or 54 dBmV.
1.4.1 Electric Field
The electric field (in V/m) is defined as the force (in Newtons) per unit charge (in Coulombs). From this definition and Coulomb's law, the electric field E created by a single point charge Q at a distance r is
(1.17)
where
F is the electric force given by Coulomb's law ;
is a unit vector along r direction which is also the direction of the electric field E.
ε is the electric permittivity of the material. Its SI unit is Farads/m. In free space, it is a constant:(1.18)
The product of permittivity and the electric field is called the electric flex density, D, which is a measure of how much electric flux passes through a unit area, i.e.
(1.19)
where εr = ε/ε0 is called the relative permittivity (also called dielectric constant, but it is normally a function of frequency, not really a constant, thus relative permittivity is preferred in this book). The relative permittivities of some common materials are listed in Table 1.2. Note that they are functions of frequency and temperature. Normally, the higher the frequency, the smaller the permittivity in the radio frequency band. It should also be pointed out that almost all conductors have a relative permittivity of one.
Table 1.2 Relative permittivity of some common materials at 100 MHz
| Material | Relative permittivity | Material | Relative permittivity |
|---|---|---|---|
| ABS (plastic) | 2.4–3.8 | Polypropylene | 2.2 |
| Air | 1 | Polyvinylchloride (PVC) | 3 |
| Alumina | 9.8 | Porcelain | 5.1–5.9 |
| Aluminum silicate | 5.3–5.5 | PTFE‐teflon | 2.1 |
| Balsa wood | 1.37 @ 1 MHz | PTFE‐ceramic |
