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Now let us use basic mathematics to deal with antennas, or precisely, EM problems in this section.

EM waves cover the whole spectrum; radio waves and optical waves are just two examples of EM waves. We can see light but cannot see radio waves. The whole spectrum is divided into many frequency bands. Some EM bands and their applications are listed in Table 1.1. There are other letter band designations from organizations such as NATO. Here, we have used the IEEE standard.

Table 1.1 EM frequency bands and applications

Frequency	Band	Wavelength	Applications
3–30 kHz	VLF	100–10 km	Navigation, sonar, fax
30–300 kHz	LF	10–1 km	Navigation
0.3–3 MHz	MF	1–0.1 km	AM broadcasting
3–30 MHz	HF	100–10 m	Tel, Fax, CB, ship communications
30–300 MHz	VHF	10–1 m	TV, FM broadcasting
0.3–3 GHz	UHF	1–0.1 m	TV, mobile, radar
3–30 GHz	SHF	100–10 mm	Radar, satellite, mobile, microwave links
30–300 GHz	EHF	10–1 mm	Radar, wireless communications
0.3–3 THz	THz	1–0.1 mm	THz imaging
3–430 THz	Infrared	0.1 mm–700 nm	Heating, communications, camera
430–770 THz	Light	700–400 nm	Lighting, camera
Radar frequency bands according to IEEE standard
1–2 GHz	L	0.3–0.15 m	Long wave, mobile radio
2–4 GHz	S	0.15–0.075 m	Short wave, mobile radio
4–8 GHz	C	7.5–3.75 cm	Compromise between S and X, radar
8–12 GHz	X	3.75–2.5 cm	Radar, satellite
12–18 GHz	Ku	2.5–1.7 cm	Satellite and radar
18–27 GHz	K	1.7–1.1 cm	Satellite and radar
27–40 GHz	Ka	11–7.5 mm	Communications and radar
40–75 GHz	V	7.5–4.0 mm	Communications and radar
75–110 GHz	W	4.0–2.7 mm	Communications and radar
Frequencies of some popular wireless systems
535–1605 kHz	AM radio broadcast band
3–30 MHz	Short‐wave radio broadcast band
13.56 MHz	NFC
88–108 MHz	FM radio broadcast band
175–240 MHz	DAB radio broadcast band
470–890 MHz	UHF TV (14‐83)
698–960 MHz	Cellular mobile radio (2/4G)
1710–2690 MHz	Cellular mobile radio (2/3/4G)
3.3–3.8 GHz	Cellular mobile radio (5G)
1.227 GHz	GPS L2 band
1.575 GHz	GPS L1 band
2.45 GHz	Microwave, Bluetooth, Wi‐Fi
3.1–10.6 GHz	UWB band
5.180–5.825	Wi‐Fi bands

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.