Antennas. Yi Huang

cross product of two vectors is defined as

(1.9)

where is a unit vector normal to the plane containing A and B. The cross × between A and B indicates the cross product that results in a vector C, thus, it is also called a vector product. The vector C is orthogonal to both A and B, and the direction of C follows a so‐called right‐hand rule as shown in Figure 1.9. If the angle θ is zero or 180 degrees, that is, A and B are in parallel, the cross product is zero, whereas for an angle of 90 degrees, i.e. A and B are orthogonal, the cross product of these two vectors reaches a maximum. Unlike the dot product, the cross product does not obey the commutative law.

Figure 1.9 The cross product of vectors A and B

      The cross product may be expressed in determinant form as follows, which is the same as Equation (1.9) but it may be easier for some people to memorize:

      (1.10)

      Another important thing about vectors is that any vector can be decomposed into three orthogonal components (such as x, y, and z components) in 3D or two orthogonal components in a 2D plane.

      Example 1.1 Vector operation

      Vectors and . Find:

      Solution

      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 A_r, A_θ, A_φ 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

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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