2 and 3, respectively. Thus, there is no need for the reader to study these subjects using other books.
In some applications where space is very limited (such as hand portables and aircrafts), it is desirable to integrate the antenna and its feed line. In other applications (such as the reception of TV broadcasting), the antenna is far away from the receiver and a long transmission line has to be used.
Unlike other devices in a radio system (such as filters and amplifiers), the antenna is a very special device; it deals with electrical signals (voltages and currents) as well as EM waves (electric fields and magnetic fields) that have made antenna design an interesting and difficult subject. For different applications, the requirements on the antenna could be very different, even for the same frequency band.
In conclusion, the subject of antennas is about how to design a suitable device that will be well matched with its feed line and radiate/receive the radio wave in an efficient and desired manner.
1.3 Necessary Mathematics
To thoroughly understand antenna theory requires a considerable amount of mathematics. However, the intention of this book is to provide the reader with a solid foundation of antenna theory and apply the theory to practical antenna design. Here, we are just going to introduce and review the essential and important mathematics required for this book. More in‐depth study materials can be obtained from other references [1, 2].
1.3.1 Complex Numbers
In mathematics, a complex number, Z, consists of real and imaginary parts, that is
(1.1)
where R is called the real part of the complex number Z, i.e. Re(Z), and X is defined as the imaginary part of Z, i.e. Im(Z). Both R and X are real numbers, and j (not the traditional notion i in mathematics to avoid confusion with a changing current in electrical engineering) is the imaginary unit and defined by
(1.2)
Thus,
(1.3)
Geometrically, a complex number can be presented in a two‐dimensional plane where the imaginary part is found on the vertical axis while the real part is presented by the horizontal axis as shown in Figure 1.6.
Figure 1.6 Complex plane
In this model, multiplication by −1 corresponds to a rotation of 180 degrees about the origin. Multiplication by j corresponds to a 90‐degree rotation anti‐clockwise, and the equation j2 = −1 is interpreted as saying that if we apply two 90‐degree rotations about the origin, the net result is a single 180‐degree rotation. Note that a 90‐degree rotation clockwise also satisfies this interpretation.
Another representation of a complex number Z is to use the amplitude and phase form:
(1.4)
where A is the amplitude and φ is the phase of the complex number Z, which are also shown in Figure 1.6. The two different representations are linked by the following equations:
1.3.2 Vectors and Vector Operation
A scalar is a one‐dimensional quantity that has magnitude only, whereas a complex number is a two‐dimensional quantity. A vector can be viewed as a three‐dimensional (3D) quantity, and a special one – it has both a magnitude and a direction. For example, force and velocity are vectors. A position in space is a 3D quantity, but it does not have a direction, thus it is not a vector. Figure 1.7 is an illustration of vector A in Cartesian coordinates. It has three orthogonal components (Ax, Ay, Az) along the x, y, and z directions, respectively. To distinguish vectors from scalars, the letter representing the vector is printed in bold, as A or a, and a unit vector is printed in bold with a hat over the letter as
Figure 1.7 Vector A in Cartesian coordinates
The magnitude of vector A is given by
(1.6)
Now let us consider two vectors A and B:
The addition and subtraction of vectors can be expressed as
(1.7)
Obviously, the addition obeys the commutative law, that is A + B = B + A.
Figure 1.8 shows what the addition and subtraction mean geometrically. A vector may be multiplied or divided by a scalar. The magnitude changes but its direction remains the same. However, the multiplication of two vectors is complicated. There are two types of multiplication: dot product and cross product.
Figure 1.8 Vector addition and subtraction
The dot product of two vectors is defined as
(1.8)
where θ is the angle between vector A and vector B and cos θ is also called the direction cosine. The dot • between A and B indicates the dot product that results in a scalar, thus it is also called a scalar product. If the angle θ is zero, A and B are in parallel – the dot product maximized, whereas for an angle of 90 degrees, i.e. when A and B are orthogonal, the dot product is zero.
It is worth noting that the dot product obeys the commutative law, that is, A • B = B