ELECTROMAGNETIC WAVE PROPAGATION EC 442 Prof. Darwish Abdel Aziz
CHAPTER 6 LINEAR WIRE ANTENNAS INFINITESIMAL DIPOLE
INTRODUCTION Wire antennas, linear or curved, are some of the oldest, simplest, cheapest, and in many cases the most versatile for many applications. 3
1 - INTRODUCTION An infinitesimal linear wire is positioned symmetrically at the origin of the coordinate system as shown in Figure (6-1). Figure 6-1 Infinitesimal dipole 4
Although infinitesimal dipoles are not very practical, they are used to represent capacitor plate ( also referred to as tophat-loaded) antennas. In addition, they are utilized as building blocks of more complex geometries. The end plates are used to provide capacitive loading in order to maintain the current on the dipole nearly uniform. Since the end plates are assumed to be small, their radiation is usually negligible. 5
2 CURRENT DISTRIBUTION The wire, in addition to being very small, is very thin. The spatial variation of the current is assumed to be constant and it current element is given by Where,. The remaining two equations are unchanged from 6
3 RADIATION EQUATIONS Since So,, and Where, and and 7
3 RADIATION EQUATIONS So, and, 8
4 AUXILIARY VECTOR POTENTIAL FUNCTION So the electric vector potential components are: While the magnetic vector potential components are:, and 9
5 THE RADIATED FIELD COMPONENTS The Magnetic Field Components can be found as follows: 10
and INFINITESIMAL DIPOLE 5 THE RADIATED FIELD COMPONENTS The Electric Field Components can be found as follows: 11
5 THE RADIATED FIELD COMPONENTS So, 12
and INFINITESIMAL DIPOLE 5 THE RADIATED FIELD COMPONENTS and 13
6 THE RADIAL AND TRANSVERSE POWER DENSITY For the infinitesimal dipole, the complex Poynting vector can be written using (6-6a) - (6-6b) and (6-8a) - (6-8c) as Whose radial and transverse components are given, respectively, by 14
7 THE RADIAL POWER The complex power moving in the radial direction is obtained by integrating (6-9) (6-10b) over a closed sphere of radius. Thus it can be written as which reduces to 15
8 THE REACTIVE POWER The transverse component of the power density does not contribute to the integral. Thus (6-12) does not represent the total complex power radiated by the antenna. Since, as given by (6-11b), is purely imaginary, it will not contribute to any real radiated power. However, it does contribute to the imaginary (reactive) power which along with the second term of (6-12) can be used to determine the total reactive power of the antenna. 16
9 THE TOTAL OUTWARDLY RADIAL POWER The reactive power density, which is most dominant for small values of components., has both radial and transverse Equation (6-11b), which gives the real and imaginary power that is moving outwardly, can also be written as 17
9 THE TOTAL OUTWARDLY RADIAL POWER Where From (6-12) 18
9 THE TOTAL OUTWARDLY RADIAL POWER It is clear from (6-15) that the radial electric energy must be larger than the radial magnetic energy. For large values of, the reactive power diminishes and vanishes when. 19
10 RADIAN DISTANCE AND RADIAN SPHERE for infinitesimal dipole, as represented by (6-6a) - (6-6c) and (6-8a) - (6-8b), are valid everywhere (except on the source itself). An inspection of these equations reveals the following: At a distance, which is referred to as the radian distance, the magnitude of the first and second terms within the brackets of (6-6c) and (6-8a) is the same. 20
10 RADIAN DISTANCE AND RADIAN SPHERE Also at the radian distance the magnitude of all three terms within the bracket of (6 8b) is identical; the only term that contributes to the total field is the second, because the first and third terms cancel each other. At distances less than the radian distance, o the magnitude of the second term within the brackets of (6-6c) and (6 8a) is greater than the first term and begins to dominate as. 21
10 RADIAN DISTANCE AND RADIAN SPHERE For (6-8b) and, the magnitude of the third term within the brackets is greater than the magnitude of the first and second terms while the magnitude of the second term is greater than that of the first one; each of these terms begins to dominate as. The near-field region, is defined as the region region is basically imaginary (stored)., and the energy in that 22
10 RADIAN DISTANCE AND RADIAN SPHERE At distances greater than the radian distance, The first term within the brackets of (6-6c) and (6-8a) is greater than the magnitude of the second term and begins to dominate as. For (6-8b) and, the first term within the brackets is greater than the magnitude of the second and third terms while the magnitude of the second term is greater than that of the third; each of these terms begins to dominate as. 23
10 RADIAN DISTANCE AND RADIAN SPHERE The intermediate - field region is defined as the region The far- field region is defined as the region, and the energy in that region is basically real (radiated). The radian sphere is defined as the sphere with radius equal to the radian distance. 24
10 RADIAN DISTANCE AND RADIAN SPHERE The radian sphere defines the region within which the reactive power density is greater than the radiated power density. For an antenna, the radian sphere represents the volume occupied mainly by the stored energy of the antenna s electric and magnetic fields. Outside the radian sphere the radiated power density is greater than the reactive power density and begins to dominate as. 25
10 RADIAN DISTANCE AND RADIAN SPHERE 26
10 RADIAN DISTANCE AND RADIAN SPHERE The radian sphere can be used as a reference, and it defines the transition between stored energy pulsating primarily in the direction [represented by (6-10b)] and energy radiating in the radial direction [represented by (6-10a); the second term represents stored energy pulsating inwardly and outwardly in the radial direction]. Similar behavior, where the power density near the antenna is primarily reactive and far away is primarily real, is exhibited by all antennas, although not exactly at. 27
11 NEAR FIELD REGION An inspection of (6-6a)- (6-6b) and (6-8a)- (6-8c) reveals that for or they can be reduced in much simpler form and can be approximated by 28
11 NEAR FIELD REGION The components, are in time- phase but they are time- phase quadrature with the component ; therefore there is no time-average power flow associated with them. This is demonstrated by forming the time- average power density as which by using (6-16a)- (6-16d) reduces to 29
12 INTERMEDIATE FIELD REGION As the values of begin to increase and become greater than unity, the terms that were dominant for smaller and eventually vanish. For moderate values of the become components lose their in-phase condition and approach time-phase quadrature. Since their magnitude is not the same, in general, they form a rotating vector whose extremity traces an ellipse. 30
12 INTERMEDIATE FIELD REGION At these intermediate values of, components approach time-phase, which is an indication of the formation time-average power flow in the outward (radial) direction (radiation phenomenon). As the values of become moderate, the field expression can be approximated again but in a different form. In contrast to the region where, the first term within the brackets in (6-6b) and (6-8a) becomes more dominant and the second term can be neglected. 31
12 INTERMEDIATE FIELD REGION The same is true for (6-8b) where the second and third terms become less dominant than the first. Thus we can write for 32
13 FAR - FIELD REGION Since (6-19a) - (6-19d) are valid only for values of, then will be smaller than because is inversely proportional to where is inversely proportional to. In a region where, (6-19a) - (6-19d) can be simplified and approximated by 33
13 FAR - FIELD REGION The ratio of to is equal to where The components are perpendicular to each other, transverse to the radial direction of propagation, and the variations are separable from of variations. 34
13 FAR - FIELD REGION The shape of the pattern is not a function of the radial distance, and the fields form a Transverse ElectroMagnetic (TEM) wave whose wave impedance is equal to the intrinsic impedance of the medium. As it will become even more evident, this relationship is applicable in the far-field region of all antennas of finite dimensions. 35
14 FAR FIELD RADIATED COMPONENTS The far field components of (6-20a) - (6-20c) can also be derived using the procedure outlined and relationships developed in chapter-5 of auxiliary vector potential functions. The far field radiated components using the radiation equations written as: can be 36
14 FAR FIELD RADIATED COMPONENTS 37
15 POWERE DENSITY AND RADIATION RESISTANCE The input impedance of an antenna, which consists of real and imaginary parts as discussed in Chapter- 4 (Fundamental Parameters of Antenna). For a lossless antenna, the real of the input impedance was designated as radiation resistance, through which the radiated power is transferred from the guided wave to the free space wave. To find the input resistance for a lossless antenna, it is required to find the time average poynting vector as 38
15 POWERE DENSITY AND RADIATION RESISTANCE The total radiated power in the radial direction is obtained by integrating (6-23e) over a closed sphere of radius. Thus it can be written as: 39
15 POWERE DENSITY AND RADIATION RESISTANCE 40
15 POWERE DENSITY AND RADIATION RESISTANCE Since the antenna radiates its real power through the radiation resistance, for the infinitesimal dipole it can be written that For free space medium,, where is the intrinsic impedance, so 41
16 DIRECTIVITY As was shown before, the average power density of the infinitesimal dipole is given by (6-23e) as As was discussed in Chapter- 4 (Fundamental Parameters of Antenna), the radiation intensity can be obtained from 42
16 DIRECTIVITY The maximum value of the radiation intensity occurs at and it is equal to The real power radiated by the infinitesimal dipole is given by (6-24e) as 43
16 DIRECTIVITY As was discussed in Chapter- 4 (Fundamental Parameters of Antenna), the directivity is given as As was discussed in Chapter- 4 (Fundamental Parameters of Antenna), for lossless antenna, the relation between the directivity and the maximum effective aperture area is given as 44