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Poynting Vector and Radiation Power Density. In an electromagnetic wave, energy is stored in equal amounts in the electric and magnetic fields, which together constitute the wave. The power flow is found by making use of the Poynting vector, S, defined as

S=ExH

where E (V/m) and H (A/m) are the field vectors. Since the Poynting vector represents a surface power density (W/m2), the integral of its normal component over a closed surface always gives the total power through the surface. That is,

where P is the total power (W) flowing out of closed surface A, and dA = ndA, и being the unit vector normal to surface. The Poynting vector S and the power P in the above relations are instantaneous values.

Normally, it is the time-averaged Poynting vector Sav, which represents the average power density, that is of practical interest. It is given by

where Re stands for the real part of the complex number and the asterisk denotes the complex conjugate. Note that /? and H in Eq. (3) are the electric and magnetic fields written as complex numbers to include the change with time. That is, for a plane wave traveling in the positive г direction with electric and magnetic field components in the x and у directions, respectively, the electric field is E =xEx0e’coT while in Eq. (1) it is E =xEx0. The factor | appears because the fields represent peak values; it should be omitted for rms values. The average power Pav flowing outward through a closed surface can now be obtained by integrating Eq.

(3):

Consider the case that the electromagnetic wave is radiated by an antenna. If the closed surface is taken around the antenna within the far-field region, then this integration results in the average power radiated by the antenna. This is called radiation power, Prad, while Eq. (3) represents the radiation power density, Sav, of the antenna. The imaginary part of Eq. (3) represents the reactive power density stored in the near field of an antenna. Since the electromagnetic fields of an antenna in its far-field region are predominately real, Eq. (3) is enough for our purposes.

The average power density radiated by the antenna as a function of direction, taken on a large sphere of constant radius in the far-field region, results in the power pattern of the antenna.

РГ /*2тг /*7Т Pr&d = II Si’dA= I I [rSi(r)l • [fr2sinedBd<t>] = 4;7T2Sj JJa Jо Jo |

As an example, for an isotropic radiator, the total radiation power is given by

Here, because of symmetry, the Poynting vector Si =! Stir) is taken independent of the spherical coordinate angles в and ф, having only a radial component.

From Eq. (5) the power density can be found:

The above result can also be reached if we assume that the radiated power expands radially in all directions with the same velocity and is evenly distributed on the surface of a spherical wavefront of radius r.

As we will see later, an electromagnetic wave may have an electric field consisting of two orthogonal linear components of different amplitudes, Ex0 and Ey0, respectively, and a phase angle between them, S. Thus, the total electric field vector, called an elliptically polarized vector, becomes

which at г = 0 becomes

So i£is a complex vector (phasor-vector), which is resolvable into two components хЁх and >’ЁУ. The total H vector associated with/?, at г = 0, is then

where f is the phase lag of Hy with respect to Ёх. From Eq. (9) the complex conjugate magnetic field can be found changing only the signs of exponents.

Now the average Poynting vector can be calculated using the above fields:

It should be noted that Sav is independent of S, the phase angle between the electric field components.

In a lossless medium t;= 0, because the electric and magnetic fields are in time phase and Ex0/Hx0 = Exl,/Hxl,

j/£’2 /£2 jJJ2 , JJ2

= i], where i] is the intrinsic impedance of the medium, which is real. If E = V *() J’11 and H = V

are the amplitudes of the total E and H fields respectively, then |

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The above expressions are the most general form of radiation power density of an elliptically polarized wave or of an elliptically polarized antenna, respectively, and hold for all cases, including the linear and circular polarization cases, that we will introduce later on.

Radiation Intensity. Radiation intensity is a far-field parameter, in terms of which any antenna radiation power pattern can be determined. Thus, the antenna power pattern, as a function of angle, can be expressed in terms of its radiation intensity as (2,3)

ЩЄ, ф) = S^vr2

r2 л

= ~Е(гв, ф)2 2ц

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= ^[Е, Лг,9,ф)2 + Еф(г, Є,ф)*] ~ 2~ 0)|2 + Еф{$, 0)|2

where

U(Q, ф) = radiation intensity (W/unit solid angle)

Sav = radiation density, or radial component of Poynting vector (W/m2)

E(r,0,ф) = total transverse electric field (V/m)

H(r,6,ф) = total transverse magnetic field (A/m) r = distance from antenna to point of measurement (m)

П = intrinsic impedance of medium (^/square)

In Eq. (12) the electric and magnetic fields are expressed in spherical coordinates.

What makes radiation intensity important is that it is independent of distance. This is because in the far field the Poynting vector is entirely radial, which means the fields are entirely transverse and E and H vary as 1/r.

Since the radiation intensity is a function of angle, it can also be defined as the power radiated from an antenna per unit solid angle. The unit of solid angle is the steradian, defined as the solid angle with its vertex at the center of a sphere of radius r that is subtended by an area on the sphere equal to r2. But the area of a sphere of radius r is given by A = 4nr2, so in the whole sphere there are 4nr2/r2 = 4n sr. For a sphere of radius r, an infinitesimal area dA on the surface of it can be written as

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dA = r2 sin вdвdф (ш2)

Thus, the total power can be obtained by integrating the radiation intensity, as given by Eq. (12), over the entire solid angle of 4n as

and therefore the element of solid angle dQ of a sphere is given by |

dA dSl = —5- = sin в d в d ф (sr) rB |

As an example, for the isotropic radiator ideal antenna, the radiation intensity и(в, ф) will be independent of the angles в and ф, and the total radiated power will be

Prad = §CiU-ld^ = Ui f f sm6d6d^ = Ui$iidn = 4xUi (16) Jo Jo

or Ui = Prad/4n, which is the power density of Eq. (6) multiplied by r2.

Dividing и(в, ф) by its maximum value Umax(e, ф), we obtain the normalized antenna power pattern,

A term associated with the normalized power pattern is the beam solid angle QA defined as the solid angle through which all the power from a radiating antenna would flow if the power per unit solid angle were constant over that solid angle and equal to its maximum value (Fig. 8). This means that, for typical patterns, the beam solid angle is approximately equal to the half-power beamwidth (HPBW), that is,

If the integration is done over the main lobe, the main-lobe solid angle, QM, results, and the difference of QA — QM gives the minor-lobe solid angle. These definitions hold for patterns with clearly defined lobes. The beam efficiency (BE) of an antenna is defined as the ratio QM/QA and is a measure of the amount of power in the major lobe compared to the total power. A high beam efficiency means that most of the power is concentrated in the major lobe and that minor lobes are minimized.

Directivity and Gain. A very important antenna parameter, which indicates how well an antenna concentrates power into a limited solid angle, is its directivity D, defined as the ratio of the maximum radiation intensity to the radiation intensity averaged over all directions. The average radiation intensity is calculated

Fig. 8. Power pattern and beam solid angle of an antenna. |

by dividing the total power radiated by 4n sr. Hence,

since from Eq. (16), Prad/4n = Ui. So, alternatively, the directivity of an antenna can be defined as the ratio of its radiation intensity in a given direction (which usually is taken to be the direction of maximum radiation intensity) to the radiation intensity of an isotropic source with the same total radiation intensity. Equation

(19) can also be written

Thus, the directivity of an antenna is equal to the solid angle of a sphere, which is 4n sr, divided by the antenna beam solid angle QA. We can say that by this relation the value of directivity is derived from the antenna pattern. It is obvious from this relation that the smaller the beam solid angle, the larger the directivity, or, stated in a different way, an antenna that concentrates its power in a narrow main lobe has a large directivity.

Obviously, the directivity of an isotropic antenna is unity. By definition an isotropic source radiates equally in all directions. If we use Eq. (20), QA = 4n, since Un(e, ф) = 1. This is the smallest directivity value that one can attain. However, if we consider the directivity in a specified direction, for example D(e, ф), its value can be smaller than unity.

As an example let us calculate the directivity of the very short dipole. We can calculate its normalized radiated power using the electric or the magnetic field components, given in Table 1. Using the electric field Ee for the far-field region, from Eq. (12) we have

and

Alternatively, we can work using power densities instead of power intensities. The power flowing in a particular direction can be calculated using Eq. (3) and the electric and magnetic far-field components given in Table 1:

By integrating over all angles the total power flowing outwards is seen to be |

The directivity is the ratio of the maximum power density to the average power density. For the very short dipole antenna, the maximum power density is in the в = 90° direction (Fig. 2), and the average power density is found by averaging the total power PT from Eq. (24) over a sphere of surface area 4n r2. So

Thus, the directivity of a very short dipole is 1.5, which means that the maximum radiation intensity is 1.5 times the power of the isotropic radiator. This is often expressed in decibels:

dB |

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D = 10 log10d dB = 10 log10(1.5) = 1.76

Here, we use a lowercase letter for the absolute value and a capital letter for the logarithmic value of the directivity, as is a common in the field of antennas and propagation.

The gain of an antenna is another basic property for its characterization. Gain is closely associated with directivity, which is dependent upon the radiation patterns of an antenna. The gain is commonly defined as the ratio of the maximum radiation intensity in a given direction to the maximum radiation intensity produced in the same direction from a reference antenna with the same power input. Any convenient type of antenna may be taken as the reference. Many times the type of the reference antenna is dictated by the application area, but the most commonly used one is the isotropic radiator, the hypothetical lossless antenna with uniform radiation intensity in all directions. So

where the radiation intensity of the reference antenna (isotropic radiator) is equal to the power in the input, Pin, of the antenna divided by 4n.

Real antennas are not lossless, which means that if they accept an input a power Pin, the radiated power Prad generally will be less than Pin. The antenna efficiency k is defined as the ratio of these two powers:

where Rr is the radiation resistance of the antenna. Rr is defined as an equivalent resistance in which the same current as that flowing at the antenna terminals would produce power equal to that produced by the antenna. Rioss is the loss resistance, which allows for any heat loss due to the finite conductivity of the materials used to construct the antenna or due to the dielectric structure of the antenna. So, for a real antenna with losses, its radiation intensity at a given direction U(e, ф) will be

where U0(e, ф) is the radiation intensity of the same antenna with no losses.

Using Eq. (29) in Eq. (27) yields the expression for the gain in terms of the antenna directivity:

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(6’Ф) _ Щnax(£. Ф) _ kD

Vi |

Thus, the gain of an antenna over a lossless isotropic radiator equals its directivity if the antenna efficiency is k = 1, and it is less than the directivity if k < 1.

The values of gain range between zero and infinity, while those of directivity range between unity and infinity. However, while the directivity can be computed from either theoretical considerations or measured radiation patterns, the gain of an antenna is almost always determined by a direct comparison of measurement against a reference, usually a standard-gain antenna.

Gain is expressed also in decibels:

where, as in Eq. (26), lowercase and capital letters mean absolute and logarithmic values, respectively. The reference antenna used is sometimes declared in a subscript; for example, dBi means decibels over isotropic.