1.3.3 Coordinates
In addition to the well‐known Cartesian coordinates, the spherical coordinates (r, θ, ϕ), as shown in Figure 1.10, will also be used frequently throughout this book. These two coordinate systems have the following relations:
(1.11)
and
(1.12)
Figure 1.10 Cartesian and spherical coordinates
The dot products of unit vectors in these two coordinator systems are
(1.13)
Thus, we can express a quantity in one coordinate system using the known parameters in the other coordinate system. For example, if Ar, Aθ, Aφ are known, we can find
1.4 Basics of EMs
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 |
| ||
