Читать книгу Antennas - Yi Huang - Страница 17

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:

(1.5)