**Fundamentals of Antennas and Radiating systems:**
**Introduction: **
In wireless communication systems, signals are radiated in space as an electromagnetic wave by using a receiving transmitting antenna and a fraction of this radiated power is intercepted by using a receiving antenna. Thus, an antenna is a device used for radiating or receiver radio waves. An antenna can also be thought of as a transitional structure between free space and a guiding device (such as transmission line or waveguide). Usually antennas are metallic structures, but dielectric antennas are also used now a day. In our discussion we shall consider only metallic antennas. Here we shall restrict our discussion to some very commonly used antenna structures. Some of the most commonly used antenna structures are shown.
**Fundamental of Radiation: **
Radiation is the process of emitting energy from a source. Electromagnetic radiation can be at all frequencies except zero (DC), but radiation at various frequencies may take different forms. At relatively lower frequencies it is in the form of electromagnetic waves, in the visible domain the emission is in the form of light and at still higher frequencies it may be in the form of ultra violet or X-ray radiation. The energy associated with the radiation depends on the frequency.
Time varying currents radiate electromagnetic waves. A time varying current generates time varying electric and magnetic fields. When such fields exist, power is generated and propagated. Although, theoretically any structure carrying time varying current can radiate electromagnetic waves, all structures are not equally efficient in doing that. While in many applications we try to reduce the radiation, when radiation is intended, launching of waves into space is accomplished with the aid of specially designed structures called antennas. If the time varying current density established on an antenna structure is unknown, the radiated fields can be calculated without great difficulty. A more difficult problem is determination of current density *J* on an antenna such that the resultant field will satisfy the required boundary conditions on the antenna surface.
In many practical antenna structures it is often possible to estimate the current distribution with sufficient accuracy to obtain good approximation of radiated fields. However, in order to calculate the impedance properties of an antenna, the current distribution is required to be known with higher accuracy.
As we have mentioned that if the time varying current density *J* on the antenna is known, the radiated *E* and *H* fields can be determined. However, it is often advantageous to compute the radiated fields in an indirect manner, by introducing potential functions. We illustrate the procedure below.
In a region not containing tree charges, for time harmonic case we can write Maxwell's equations as:
................................ (7.1a)
..................................... (7.1b)
................................................ (7.1c)
............................................... (7.1d)
From equations (7.1a) and (7.1b)
where
Using we can write
………………………… (7.2)
By solving equation (7.2) electric field *E* can be determined for a specified current density *J* and *H* can be computed from *E* by using (7.1b). However, as mentioned such direct computations are often difficult. Simplification can be obtained by introducing potential functions.
Since, we can write
…………………………………... (7.3)
Because and A is the vector potential.
From (7.1) and (7.3)
or,
A curl tree vector function can be expressed as the gradient of a scalar function. Therefore we assume,
……………………….. (7.4)
From equation (7.3) and (7.1a),
Or,
By substituting from (7.4)
Or, ....................... (7.5)
So far we have defined and free to specify divergence of **A**. If we choose
………………………………. (7.6)
which is known as Lorentz condition, we can simplify equation (7.5) as
Equation (7.&) can be solved to determine **A** and when **A** is known we can find,
………………………. (7.8)
From equation (7.4) and (7.6) we can write
………………. (7.9)
From equations (7.7) - (7.9) we find that both *E* and *H* can be computed when the vector potential *A* is known.
**Radiated field of a Herzian dipole:** In the previous section we have outlined the procedure for computing the electric magnetic field distribution of a known current density J. In this section we consider the radiation from a shaft current filament. We consider an ideal short linear element (the length of the element d*l *<< operating wavelength) with current considered uniform over its length. More complex antennas can be considered to be made up of a large number of such differential antennas with proper magnitude and phase of their current. For current element under consideration, by continuity, equal and opposite time varying charges must exist on both ends of dl/2 so that such elements are also called a Herzian dipole.
**Fig 7.2**
As shown in the Fig 7.2, the current element is located at the origin and oriented in the *z*-direction. We consider the time harmonic case where the current varies sinusoid ally with time and *I* represents the current phasor. For a *z*-directed current density located in free space, from equation (7.7) we can write
………………………… (7.10)
In the source free region the wave equation reduces to
............................ (7.11)
Since the current element is infinitesimally small, it can be regarded as a point source so that away from the source *A*_{z} will be a function of *r* only. Therefore equ.(7.11) can be written as:
.............. (7.12)
If we substitute , then
or,
Therefore, Eqn.(7.12) can be written as
Or, …………………(7.13)
Equation (7.13) has solution of the form and where *C*_{1} and *C*_{2} are constants. Out of these two solutions, represents a wave solution which represents an outward traveling wave. Therefore we consider this solution.
Hence, ………………………. (7.14)
In the static case, and and Eqn(7.14) simplifies to
. ............. .................................................(7.15)
For *k* = 0, eqn.(7.10) simplifies to
...... ................................................. (7.16)
(7.16) is recognized to be as the Poisson's equation and therefore and therefore the solution can be written as,
................................................. (7.17)
Both (7.15) and (7.17) represents the solutions of the equation (7.10) for *k* = 0.
From (7.14) and (7.15) we observe that time varying soln (7.14) is obtained by multiplying static case solution (7.15) by multiplying the factor . In an analysis manner we can write the time varying solution from (7.17) as,
....................................... (7.18)
If the current densities were in *x* or *y* direction similar expression could be obtained. Therefore, in general we can write:
........................................ (7.19)
If the source is placed at a position (*x'*, *y'*, *z'*) instead of origin, the vector potential at a point (*x*, *y*, *z*) can be written as
......... (7.20)
Where .
Returning back to our problem of computation of radiated field for the current element *I* d*l*, we observe that the current *I* is assumed to be constant over the length of the dipole and can be replaced with . Further, dl being very small, and hence, we can write
……………………….. (7.22)
From (7.22) using (7.8) and (7.9) we obtain
..................................... (7.23a)
and ………….(7.23b)
when the distance *r* is very large compared to , amplitude variations corresponding to 1/*r* are important and 1/*r*^{n }terms can be neglected. Retaining the terms containing 1/*r* variations, which constitute the radiated field or the field in the far zone, the electric and magnetic field components can be written as:
………………….. (7.24a)
…………………….. (7.24b)
We find that the radiated field has transverse components only and they satisfy the relation
..................................................... (7.25a)
....................................................... (7.25b)
The Poynting vector of the radiation field is directed radially outward.
The terms varying as 1/*r*^{2} and 1/*r*^{3} in reactive field for. These fields do not contribute to the radiated power; rather they represent stored electric and magnetic energy in space in the vicinity of the antenna and account for the reactive past of the impedance seen looking into the antenna terminals. Therefore, in antenna impedance calculation, the near fields are to be taken into account.
**Basic Antenna Parameters:**
An antenna does not radiate uniformly in all directions. For the sake of a reference, we consider a hypothetical antenna called an isotropic radiator having equal radiation in all directions. A directional antenna is one which can radiate or receive electromagnetic waves more effectively in some directions than in others. The relative distribution of radiated power as a function of direction in space (i.e., as function of and ) is called the radiation pattern of the antenna. Instead of 3D surface, it is common practice to show planar cross section radiation pattern. E-plane and H-plane patterns give two most important views. The E-plane pattern is a view obtained from a section containing maximum value of the radiated field and electric field lies in the plane of the section. Similarly when such a section is taken such that the plane of the section contains H field and the direction of maximum radiation.
A typical radiation patter plot is shown in fig (7.3).
Fig 7.3(a) shows a typical radiation pattern plot in polar coordinates and Fig 7.3(b) shows the same in rectangle coordinates.
Fig 7.3 (a): Typical radiation pattern in Polar
Coordinates
Fig 7.3 (b): Typical radiation pattern in rectangular
Coordinates
The main lobe contains the direction of maximum radiation. However in some antennas, more than one major lobe may exist. Lobe other than major lobe are called minor lobes. Minor lobes can be further represent radiation in the considered direction and require to be minimized.
HPBW or half power beam width refers to the angular width between the points at which the radiated power per unit area is one half of the maximum.
Similarly FNBW (First null beam width) refers to the angular width between the first two nulls as shown in Fig 7.3. By the term beam width we usually refer to 3 dB beam width or HPBW.
**Directivity and gain:**
We have already mentioned that an antenna does not radiate uniformly in all directions. Directivity function describes the variation of the radiation intensity. The directivity function is defined by
=
If P_{r} is the radiated power, the gives the amount of power radiated per unit solid angle. Had this power beam uniformly radiated in all directions then average power radiated per unit solid angle is .
............................. (7.27)
The maximum of directivity function is called the directivity.
In defining directivity function total radiated power is taken as the reference. Another parameter called the gain of an antenna is defined in the similar manner which takes into account the total input power rather than the total radiated power is used as the reference. The amount of power given as input to the antenna is not fully radiated.
…………………………………… (7.28)
where is the radiation efficiency of the antenna.
The gain of the antenna is defined as
The maximum gain function is termed as gain of the antenna.
Another parameter which incorporates the gain is effective isotropic radiated power or EIRP which is defined as the product of the input power and maximum gain or simply the gain. An antenna with a gain of 100 and input power of 1 W is equally effective as an antenna having a gain of 50 and input power 2 W.
**Radiation resistance:**
The radiation resistance of an antenna is defined as the equivalent resistance that would dissipate the same amount power as is radiated by the antenna. For the elementary current element we have discussed so far. From equation (7.26) we find that radiated power density
Radiated power
................................... (7.29)
Further,
……………… (7.30)
From (7.29) and (7.30)
Directivity which occurs at .
If *R*_{r} is the radiation resistance of the elementary dipole antenna, then
Substituting P_{r} from (7.29) we get
.
Substituting
………………….. (7.32)
For such an elementary dipole antenna the principal *E* and *H* plane pattern are shown in Fig 7.4(a) and (b).
Fig 7.4 (a) Principal E plane pattern of an elementary
Dipole.
Fig 7.4 (b) Principal H plane pattern of an elementary
Dipole.
The bandwidth (3 dB beam width) can be found to be 90^{0} in the *E* plane.
**Effective Area of an Antenna: **
An antenna operating as a receiving antenna extracts power from an incident electromagnetic wave. The incident wave on a receiving antenna may be assumed to be a uniform plane wave being intercepted by the antenna. This is illustrated in Fig 7.5. The incident electric field sets up currents in the antenna and delivers power to any load connected to the antenna. The induced current also re-radiates fields known as scattered field. The total electric field outside the antenna will be sum of the incident and scattered fields and for perfectly conducing antenna the total tangential electric field component must vanish on the antenna surface.
Fig 7.5: Plane wave intercepted by an antenna
Let *P*_{inc} represents the power density of the incident wave at the location of the receiving antenna and *P*_{L} represents the maximum average power delivered to the load under matched conditions with the receiving antenna properly oriented with respect to the polarization of the incident wave.
We can write,
................................ (7.33)
where and the term *A*_{em} is called the maximum effective aperture of the antenna. *A*_{em } is related to the directivity of the antenna *D* as,
If the antenna is lossy then some amount of the power intercepted by the antenna will be dissipated in the antenna.
From eqn. (7.28) we find that
Therefore, from (7.34),
....................................................(7.35)
is called the effective aperture of the antenna ( in m^{2}).
So effective area or aperture *A*_{e} of an antenna is defined as that equivalent area which when intercepted by the incident power density *P*_{in} gives the same amount of received power *P*_{R} which is available at the antenna output terminals.
If the antenna has a physical aperture *A* then aperture efficiency
**Effective length/height of the antenna: **
When a receiving antenna intercepts incident electromagnetic waves, a voltage is induced across the antenna terminals. The effective length he of a receiving antenna is defined as the ratio of the open circuit terminal voltage to the incident electric field strength in the direction of antennas polarization.
……………………………….. (7.36)
where *V*_{oc} = open circuit voltage
*E* = electric field strength
Effective length he is also referred to as effective height.
**Antenna Equivalent Circuit:**
To a generator feeding a transmitting antenna, the antenna appears as a lead. In the same manner, the receiver circuitry connected to a receiving antenna's output terminal will appear as load impedance. Both transmitting and receiving antennas can be represented by equivalent circuits as shown by figure 7.6(a) and figure 7.6(b).
Fig 7.6 (a): Equivalent circuit of a *T*_{x} antenna
*V*_{g} = open circuit voltage of the generator
*Z*_{g} = antenna impedance
*Z*_{0} = Characteristics impedance of the transmission line connecting generator to the antenna.
*P*_{inc} = Incident power to the antenna terminal
*P*_{refl}_{ }= Power reflected from the antenna terminal.
*P*_{in} = Input power to the antenna
*X*_{A} = Antenna reactance
*R*_{l} = Loss resistance of the antenna
R_{r} = Radiation resistance
antenna impedance.
Fig 7.6 (b): Equivalent circuit of receiving antenna
.
*h*_{e} = effective length
*E* = incident field strength
*V*_{oc} = h0 E open circuit voltage
*Z*_{load} = Input impedance of the receiver.
*R*_{e}, *R*_{r} and *X*_{A} as defined earlier.
From equation (7.7) to (7.9) we have seen that solution for *E* and *H* can be obtained provided solution of A is unknown for a given J. Further while computation of radiated fields for a Herzian dipole, in equation (7.23a) and (7.23b) we have neglected the higher order terms of and retained only those terms having variation. In fact, once *A* is known the radiation field components can be completed for the far field region as:
The relationship stated above equation (7.37a) - (7.37f) may be verified for a Herzian dipole using equations (7.22), (7.24a) and (7.24b).
**Half Wave Dipole Antenna: **
Let us consider linear antennas of finite length and having negligible diameter. For such antennas, when fed at the center, a reasonably good approximation of the current is given by,
Fig 7.7: Current distribution on a center fed dipole antenna
This distribution assumes that the current vanishes at the two end points i.e., . The plots of current distribution are shown in the figure 7.7 for different '*l*'.
This distribution assumes that the current vanishes at the two end points i.e. . The plots of current distribution are shown in the figure 7.7 for different '*s*'.
For a half wave dipole, i.e., , the current distribution expressed as
………………………….. (7.39)
Fig 7.8(a): Half wave dipole
Fig 7.8(b): Far field approximation for half wave dipole
From equation (7.21) we can write
……………………… (7.40)
From Fig 7.8(b), for the far field calculation, for the phase variation and for amplitude term.
..................... (7.41)
Substituting from (7.39) to (7.41) we get
............................ (7.42)
Therefore the vector potential for the half wave dipole can be written as:
From (7.37b),
…………………. (7.44)
Similarly from (7.37c)
........................................................................ (7.45)
and from (7.37e) and (7.37f)
................................ (7.46)
and ………………………………………….(7.47)
The radiated power can be computed as
……………………………………. (7.48)
Therefore the radiation resistance of the half wave dipole antenna is =
Further, using Eqn(7.27) the directivity function for the dipole antenna can be written as
…………………. (7.49)
Thus directivity of such dipole antenna is 1.04 as compared to 1.5 for an elementary dipole. The half power beam width in the E-plane can be found to be 780 as compared to 900 for a horizon dipole.
**Quarter Wave Monopole Antenna:**
A quarter wave monopole antenna is half of a dipole antenna placed over a grounded plane. The geometry of such antennas is shown in Fig 7.9(a) and equivalent half wave dipole is shown in fig 7.9(b).
Fig 7.9 (a): Quarter wave monopole (b) Equivalent Half wave dipole
If the ground plane is perfectly conducting, the monopole antenna shown in Fig 7.9(a) will be equivalent to a half wave dipole shown in Fig 7.9(b) taking image into account.
The radiation pattern above the grounded plane ( in the upper hemisphere) will be same as that of a half wave dipole, however, the total radiated power will be half of that of a dipole since the field will be radiated only in the upper hemisphere.
An ideal quarter wave antenna mounted over a perfectly conducting ground plane has radiation resistance 36.56, half that of a dipole antenna, radiating in free space. The directivity of such antennas become double of that of dipole antennas.
Quarter wave monopole antennas are often used as vehicle mounted antennas, the evhicle providing required ground plane for the antenna. For quarter-wave antennas mounted above earth, the poor conductivity of the soil results in excessive power loss from the induced amount in the soil.
The effect of poor ground conductivity is taken care of by installing a ground screen consisting of radial wires extending outward from the antenna base for a distance of **....****.** Such arrangement is shown in Fig 7.10.
radial wires of length buried below grounded surface
Fig 7.10: Grounded screen for improving performance of monopole antennas operating near earth surface.
**Small Loop Antennas: **
Loop antennas may take many different forms such as circle, square, rectangle etc. Loop antennas are generally classified into two categories viz, electrically small and electrically large antennas. Electrically small antennas are those whose overall length is less than one tenth is number of terms in the loop times the circumference of the loop. Here we shall keep our discussion confined to small loop antennas only.
Fig 11: Small Current loop
Small loops are usually not used as transmitting antennas as they have radiation resistance smaller compared to ..... dipoles. However many unintentional sources of radiation such as transformers, inductor, printed circuit boards etc essentially behave as small loop antennas. A small loop of current is also called a magnetic dipole and its magnetic dipole moment is equal to the product of the area with the current it carries. Thus for these types of small current loops, the shape of the loop is not important. For a given current, it is the area of the loop that determines the magnitude of the radiated fields.
Fig. 11 shows a small current loop of radius peaced on the xy plane with its axis oriented in the z direction. The loop carries a current I0.
For , the loop may be treated as a point source. As shown in Fig11, the elementary current element placed at has a vector orientation . For this current element the vector potential by equation (7.21)
…………………….. (7.50)
Where,
Since in the far field
[ By Bionomial expansion after neglecting w.r.to 1]
For amplitude variation, we assume and for phase variation
{By approximating since is very small as )
…………………………. (7.51)
Using (7.37c) and (7.37e), the radiated field components can be written as
.........................................(7.52a)
.........................................(7.52b)
where is the dipole moment of the loop.
The field radiated by a small loop antenna is dual of that small dipole antenna, i.e., a short current filament, the role of electric and magnetic fields are interchanged.
From (7.52),
Therefore the radiated power
The radiation resistance of a loop antenna can be found be
If the antenna consists of *N* number of turns; the radiation resistance increases by a factor of *N*^{2} .
Small loop antennas are often used as receiving antennas.
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**Introduction to Antenna Arrays:**
An antenna array is an assembly of radiating elements. Radiation pattern of a single element is relatively wide, each element provides low values of directivity. However, in many applications we require antennas with very high directive characteristics. The directive characteristics of the antennas can be improved by increasing the electrical size of the antenna. One way to increase the dimension of the antenna without necessarily increasing the size of the individual elements is to form an array of antenna elements. The total field of the array is determined by the vector addition of the fields radiated by the individual elements. The elements of the array need not be identical, but it is often convenient and simpler to design such arrays when the individual elements are considered to be identical. Therefore, here we will consider an array with identical elements. In designing arrays we have several controls such as geometrical configuration of the overall array, distance between the elements, excitation (amplitude and phase) and pattern of individual elements.
**Array of identical elements:**
In this section we establish the basic methodology for analyzing an array of identical elements.
Fig. 7.12: A general N – element array.
As shown in fig 7.12, let us consider an array of N identical elements. The position vector of the *i*^{th} element is given by. The excitation of *i*^{th} element is given where and are respectively the relative amplitudes and phases.
Let the electric field radiated by an element, when placed at the origin and with an unity excitation is given by
……………………..(7.54)
The distance from the *i*^{th }element to the far field point of interest is for phase variation and for amplitude variation.
The total electric field at the point *P* is given by
……………………….(7.55)
As can be seen from (7.56), the total radiation field is given by the product of the radiation field of the reference element and the term .
The term ………………………....(7.56) is called the array factor of the antenna array.
The directivity of the array. Thus we find that the radiation pattern of an array is the product of the ……… function of the individual element with the array pattern function. This termed as principle of pattern multiplication.
If we consider isotropic elements then ; hence the radiation pattern of the array depends only on the array factor . Further, it is worth mentioning here that while discussing the properties of array we are neglecting the effect of radiation of one element on the source distribution of the other, i.e., we assume that mutual coupling effect among the elements of the array are neglected. Such effects are included when very accurate characterisation of arrays is required.
**Two element array:**
In equation (7.57) we derived the expression for the array factor for an N- element array. To simplify our discussion, let us consider a two-element array. Further, we consider the elements are to be isotropic point sources. The array configuration under consideration is shown in Fig. 7.13.
Fig 7.13: Two element array of isotropic point sources.
For this array, from (7.57) the array factor is given by
We now consider some specific cases.
Case -1:
Point sources have same amplitude and phase.
For this case we consider
&
Let us plot the array pattern on *xy* plane i.e., . Fig 7.14 (a) – Fig 7.14(d) show the nature of variation of the array factor as a function of .
It can be seen that for , the maximum radiation take place in a direction perpendicular to array axis( broad side direction) and no radiation along the axis of the array (endfix) for the radiation increases along the array axis.
Case 2:
Point sources have equal amplitude and opposite phase.
For this case let
And
Once again we plot array pattern on the *xy* plane, i.e., . The same is shown in Fig 7.15(a) to Fig 7.15(d).
It can be seen from Fig 7.14(b) and Fig 7.15(b), that for spacing, broadside pattern is obtained for elements having same phase while end side pattern is obtained when the elements are excited in the opposite phase.
(a) (b)
** (c) ** (d)
Fig 7.14: Plot of for different values of , the elements excited in the same phase.
(a) (b)
(c) _{ } (d) _{ }
Fig 7.15: Plot of _{ } for different values of _{ }, the elements excited in different phase.
Uniform One dimensional array:
So far, we have considered the behavior of arrays having only two elements. Let us now consider a uniform array having N +1 point sources. Each antenna element is assumed to have same amplitude _{ } and a progressive phase shift of _{ }between two elements where‘d’ is the separation between the elements. Thus, with reference to the Fig 7.1b, the ith element has a phase _{ }.
Fig7.16: Uniform linear array
_{ }
Where _{ }
Using the relation
_{ }…………………………………….(7.59)
For a G.P. , from (7.61) we can write
_{ }…………………….. (7.60)
If we define _{ }
and _{ }, then from (7.63) we can write array field pattern _{ } to be _{ }………………………..(7.61)
The function defined by equation (7.64) is a periodic function whose peak value occurs at _{ } and when ever _{ } is an integer. The peak value is _{ }.
Since _{ } lies in the range _{ }, the corresponding range of u, _{ } is the physical space or visible region. The plot of array factors _{ } as a function of u is shown in Fig 7.17.
……………..
As we can see from Fig 7.17, along with the major lobe, in the visible space there are several smaller maxima. These smaller maxima corresponds to ride lobes.
**Broad side Case**:
If _{ }, i.e., all the elements are in the same phase, then the maximum occurs at u = 0 i.e., _{ }.
i.e., _{ }. Thus the maximum radiation occurs broad side to array axis. If we consider the pattern in the y plane for which _{ }. Then _{ }. i.e., maximum radiation is along y-axis.
End fire Case:
If _{ } is chosen to be _{ }, then the beam maximum is formed along _{ }, i.e., _{ } maximum of the array pattern is formed along the array axis.
**Array pattern synthesis:**
So far we have discussed the nature of the pattern produced by the arrays in which the excitations (amplitude and phase) of the elements are specified. Alternatively, if we have an array pattern specified a priori, the same can be approximately realized (synthesis) by proper choice of element spacing, amplitudes and phases of the individual elements. This process of realizing a specified pattern is known as array pattern synthesis or simply array synthesis.
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