Chapter 4 Antenna.

Chapter 4 Antenna

Chapter Outlines Chapter 4 Antenna Fundamental Parameters of Antennas
Electrically Short Antennas Dipole Antennas Monopole Antennas Antenna Arrays Helical Antennas Yagi Uda Array Antennas Antennas for Wireless Communications

Introduction Wires passing an alternating current emit or radiate EM energy. The shape and size of the current carrying structure determine how much energy is radiated, direction of propagation and how well the radiation is captured. Henceforth, the structure to efficiently radiate in a preferred direction is called transmitting antenna, while the other side which is to capture radiation from preferable direction is called receiving antenna. In most cases, the efficiency and directional nature for an antenna is the same whether its receiving or transmitting.

Introduction (Cont’d..)
Common types of antennas:

Introduction (Cont’d..)
Generic antenna network. The antenna acts as a transducer between guided waves on the T-line and waves propagating in space.

4.1 Fundamental Parameters of Antennas
To describe the performance of an antenna, definitions of various parameters are discussed. The radiated power, beam pattern, directivity, and efficiency are all important parameters in characterizing antenna. RADIATED POWER Suppose transmitting antenna located at the origin of spherical coordinate. From this coordinate system, there are three components of radiated field, in r, θ and φ. But, the intensities of these components vary with radial distance as 1/r, 1/r2 and 1/r3.

Fundamental Parameters of Antennas (Cont’d..)
For almost all practical applications, a receiving antenna located far enough away from the transmitter (as a point source of radiation)  far field region. A distance r from the origin is generally accepted as being in the far field region if : L is the length of the largest dimension on the antenna element, and assumed L>λ. For smaller L, r should at least as large as λ

In the far field, the radiated waves resemble plane waves propagating in ar direction where time harmonic fields related by : The time averaged power density vector of the wave is by Poynting Theorem, Where in the far field,

So, the total power radiated by the antenna, Prad is found by integrating it over a close spherical surface :

Example 1 In free space, suppose a wave propagating radially away from an antenna at the origin has: Where the driving current phasor Find: ES P(r,φ,θ) Prad and Rrad

Solution to Example 1 To find ES, from time harmonic fields relation:
Then, to find P(r,φ,θ)

Solution to Example 1 (Cont’d..)
Then, Solving:

RADIATION PATTERNS From Balanis book:

Radiation patterns usually indicate either electric field, E intensity or power intensity. Magnetic field intensity, H has the same radiation pattern as E related by η0. The polarization or orientation of the E field vector is an important consideration in an E field plot. A transmit receive antenna pair must share same polarization for the most efficient communication.

Coordinate System

Since the actual field intensity is not only depends on radial distance, but also on how much power delivered to antenna, we use and plot normalized function  divide the field or power component with its maximum value. E.g. the normalized power function or normalized radiation intensity :

If the antenna radiates EM waves equally in all directions, it is termed as isotropic antenna, where the normalized power function is equal to 1. So, In contrast with isotropic antenna, a directional antenna radiates and receives preferentially in some direction.

The normalized radiation patterns for a generic antenna, called polar plot. A 3D plot of radiation pattern can be difficult to generate and work with, so take slices of the pattern and generate 2D plots (rectangular plots) for all θ at φ=π/2 and φ=3π/2 Rectangular plot (in dB) Polar plot

The polar plot also can be in terms of dB. Where normalized E field pattern, This will be identical to the power pattern in decibels if: whereas

The are some zeros and nulls in radiation pattern, indicating no radiations. These lobes shows the direction of radiation, where main or major lobe lies in the direction of maximum radiation. The other lobes divert power away from the main beam, so that good antenna design will seek to minimize the side and back lobes. Beam’s directional nature is beamwidth, or half power beamwidth or 3 dB beamwidth. It will shows the angular width of the beam measured at the half power or -3 dB points.

For linearly polarized antenna, performance is often described in terms of its principal E and H plane patterns. E plane : the plane containing the E field vector & the direction of max radiation H plane : the plane containing the H field vector & the direction of max radiation For next figure, the x-z plane (elevation plane, φ=0) is the principal E-plane the x-y plane (azimuthal plane; θ=π/2) is the principal H-plane.

For this figure, Infinite number of principal E-planes (elevation plane, φ=φc). One principal H-plane (azimuthal plane; θ=π/2).

DIRECTIVITY A measure of how well an antenna radiate most of the power fed into the main lobe. Before defining directivity, describe first the antenna’s pattern solid angle or beam solid angle. An arc with length equal to a circle’s radius defines a radian.

An area equal to the square of a sphere’s radius defines a steradian (sr). A differential solid angle dΩ in sr is: For sphere, the solid angle is found by integrating dΩ :

An antenna’s solid angle Ωp: To find the normalized power’s average value taken over the entire spherical solid angle :

The directive gain D(θ,φ) of an antenna is the ratio of the normalized power in particular direction to the average normalized power : The directivity Dmax is the maximum directive gain, So,

Directivity in decibels as: Useful relation: Total radiated power as: Or :

For (a), power gets radiated to the side and back lobes, so the pattern solid angle is large and the directivity is small. For (b), almost all the power gets radiated to the main beam, so pattern solid angle is small and directivity is high.

Example 2 For this normalized radiation intensity,
Find the beamwidth, pattern solid angle and the directivity.

Solution to Example 2 The beam is pointing in the +y direction.
WHY?!?! Due to the beam having function in θ and φ, so that: From previous equation, so that a 3dB beamwidth is at half of total power = 0.5.

To find BWθ, we fix φ = π/2 to get: and then set sin2θ equal to ½. Then, To find BWφ, we fix θ = π/2 to get:

and then set sin3θ equal to ½. Then, So, The pattern solid angle is:

Where each integral is solved as follows: Please continue on your own!! So, Finally, and the directivity,

IMPEDANCE & EFFICIENCY (a) A T-line terminated in a dipole antenna can be modeled with an antenna impedance, Zant (b) consisting of resistive and reactive components (c).

The antenna resistance consists of radiation resistance Rrad and a dissipative resistance Rdiss that arises from ohmic losses in the metal conductor. For antenna driven by phasor current, and also For maximum radiated power, Rrad need to be as large as possible but without being too large, for easily match with the feedline.

So then the antenna efficiency, e The power gain G(θ,φ) is likely its directive gain plus efficiency, where: And the max power gain is when the directivity is max. It’s been always expressed in dBi , indicating dB with respect to an isotropic antenna.

4.2 Electrically Short Antennas
If the current distribution of a radiating element is known, it’s possible to calculate the radiated fields by a direct integration but, the integrals can be very complex. For time harmonic fields, integration is performed to find a phasor called retarded vector magnetic potential, which then followed by simple differentiation to find the H field. We will begin with a derivation of the retarded vector magnetic potential, then find the radiated fields for Hertzian dipole (infinitesimally short element with uniform current along its length). From that, we can find the fields from longer structures via integration e.g. small loop antenna.

Electrically Short Antennas (Cont’d..)
Vector Magnetic Potential In working with electric fields, and analogous term for H field is the vector magnetic potential A, often used for antenna calculations. Where in the point form of Gauss Law for magnetic fields, With vector identity that states the div of the curl of any vector A is zero. So, Then we now seek a relation between vector A and a current source.

From Biot Savart law, Remember?!? The vector magnetic potential at the observation point (o) results from a current density distributed about the volume vd.

Due to long derivation and by using vector identity, we could get: Refer to text book! Then, More properly, Where the vector potential at point o is a function of the position of current element

We have yet to consider time dependence, so: Where: Where Jds is the retarded phasor quantity: k = β = 2π/λ

Using this current density, the phasor form is: This is what we called as the time harmonic equation for the retarded vector magnetic potential. In phasor notation, the vector magnetic potential is related to the magnetic flux density by :

From there, we could find H in free space : Because the radiation is propagating radially away from the source, it is then a simple matter to find E, where: Finally, the time averaged power radiated is:

Hertzian Dipole Suppose that a short line of current, It’s placed along the z axis as shown.

Here, the phasor current is: To maintain constant current over its entire length, imagine a pair of plates at the ends of the line that can store charge. The stored charge at the ends resembles an electric dipole, and the short line of oscillating current is then ‘Hertzian Dipole’. For Is in the +az direction through a cross sectional S, the current density at the source seen by the observation point:

Geometrical arrangement of an infinitesimal dipole and its associated electric field components on a spherical surface.

A differential volume of this element, So, Therefore, With assumption that the Hertzian dipole is very short, integrate to get A0S :

The unit vector az can be converted to its equivalent spherical coordinates, so that: It is now a relatively straightforward matter to find B0S and then, Regroup to get:

The second terms drops off with increasing radius much faster than the first term, where for far field condition: Therefore, Meanwhile, the far field value of E field :

The time averaged power density at observation point is: The term in brackets is the max power density. The sin2θ term is the normalized radiation intensity Pn(θ) plotted as :

The normalized radiation intensity can be used to find the pattern solid angle, The directivity is then: The total power radiated by a Hertzian dipole:

Also Rrad as: For Hertzian dipole, where l<<λ, Rrad will be very small and the antenna will not efficiently radiate power. Larger dipole antennas, have much higher Rrad and thus more efficient.

Example 3 Suppose a Hertzian dipole antenna is 1 cm long and is excited by a 10 mA amplitude current source at 100 MHz. What is the maximum power density radiated by this antenna at a 1 km distance? What is the antenna’s radiation resistance?

Solution to Example 3 We should calculate the wavelength,
The max. power radiated is: The antenna radiation resistance :

Small Loop Antenna A small loop of current located in the xy plane centered at the origin  small loop antenna or magnetic dipole.

Substitute, Giving us, Assume that a<<λ and A0s is in far field. It’s discovered that, Where, and thus,

The power density vector : Where, Since the normalized power function is the same as Hertzian dipole, Ωp=8π/3, Dmax = 1.5

Try this!! Calculation of Prad and Rrad yields The fields for the small loop antenna is similar to Hertzian dipole. It’s the dual of Hertzian (electric) dipole  magnetic dipole. The equations also valid for multi turn loop, as long as the loop small compared to wavelength. For N circular loop, S=Nπa2 and for square coil N loops, with each side length b, S=Nb2

4.3 Dipole Antennas Dipole Antennas
A drawback to Hertzian dipole as a practical antenna is its small radiation resistance. A longer will have higher radiation resistance, becomes more efficient. It as an L long conductor conveniently placed along the z axis with current distribution i(z,t).

Dipole Antennas (Cont’d..)
Assume sinusoidal current distribution on each arm, where the antenna is center-fed and the current vanishes at the end points. The distribution: Where, Division of the L long dipole into a series of infinitesimal Hertzian dipoles !!

For simplicity, assume phase term =0, and make use of current distribution term with magnetic field equation for a Hertzian dipole to get :

For far field, the vectors r and R appear to be parallel, so that θ’=θ and R=r, where: After pulling the components that don’t change with z, use table of integrals and with application to Euler’s identity,

The vector E0S is then easily found from : The time averaged power radiated is: Where, the pattern function is given by:

It’s not generally equivalent to the normalized power function since F(θ) can be greater than one. Therefore, the normalized power function is: And the max time averaged power density is then :

3D and 2D amplitude patterns for a thin dipole of l=1.25λ

Half Wave Dipole Antennas Because of its convenient radiation resistance, and because it’s a smallest resonant dipole antenna, the half wavelength dipole antenna merits special attention. With kL/2 = π/2, With the F(θ) is 1, the maximum power density is:

Therefore, the normalized power density is: The current distribution and normalized radiation pattern for a half wave dipole antenna.

For half wave dipole… and with L = λ/2 Try this!! We can then find the directivity as : Which is slightly higher than the directivity of Hertzian dipole, the radiation resistance is given by: Leads to:

This radiation resistance much higher than of Hertzian dipole, where it radiates more efficiently  easier to construct an impedance matching network for this antenna impedance. This antenna impedance also contains a reactive components, Xant, where for a λ/2 dipole antenna it is equal to 42.5Ω . Therefore, total impedance by neglecting Rdiss. For impedance matching, need to make reactance zero (in resonant condition). So, it can be achieved by making the antenna slightly shorter (reduced in length until reactance vanishes).

Example 4 Find the efficiency and maximum power gain of a λ/2 dipole antenna constructed with AWG#20 (0.406 mm radius) copper wire operating at 1.0 GHz. Compare your result with a 3 mm length dipole antenna (Hertzian Dipole) if the center of this antenna is driven with a 1.0 GHz sinusoidal current.

Solution to Example 4 We first find the skin depth of copper at 1.0 GHz, This is much smaller than the wire radius, so the wire area over which current is conducted by: At 1 GHz, the wavelength is 0.3m and the λ/2 is 0.15m long. The ohmic resistance is then:

Since the radiation resistance for half wave dipole is 73.2Ω, we have : A gain of: Meanwhile for Hertzian Dipole, the ohmic resistance of the small dipole is :

To find the radiation resistance, with value of wavelength is 0.3m, thus : Therefore, its efficiency and gain: Thus, the half wave dipole is clearly more efficient with a higher gain than the short dipole.

4.4 Monopole Antennas Consider the construction of half wave dipole for an AM radio station broadcasting at 1 MHz. At this f, the wavelength is 300m long and the half wave dipole antenna must be 150m tall. We can cut this in half, by employing image theory to build a quarter wave monopole antenna that is only 75m tall!!

Monopole Antennas (Cont’d..)
Consider pair of charges, +Q and –Q (as electric dipole), where the dashed line shows the location of zero potential surface. If we slide a conductive plane over the zero potential surface, the field lines in the upper half plane are unchanged.

Note that the charge can be in any distribution (point charge, line charge, surface or volume charge) and the image charge is a mirror image of opposite polarity. A monopole antenna is excited by a current source at its base. By image theory, the current in the image will be the same with the current in actual monopole. The pair of monopole resembles a dipole antenna. A monopole antenna placed over a conductive plane and half the length of a corresponding dipole antenna will have identical field patterns in the upper half plane.

For the upper half plane (00< θ<900), the time averaged power, max power density and normalized power density for the quarter wave monopole is the same with half wave dipole. But the pattern solid angle is different.

Since the normalized power density is zero for (900< θ<1800), the pattern solid angle: Integrate over all space will be half value of Ωp for half wave dipole. So, for quarter wave monopole, See that the radiation resistance is halved, and the antenna impedance :

Summary of Key Antenna Parameters

4.5 Antenna Arrays A Ka-Band Array Antenna
The antennas we have studied so far have all been omnidirectional – no variation in φ. A properly spaced collection of antennas, can have significant variation in φ leading to dramatic improvements in directivity. A Ka-Band Array Antenna

Antenna Arrays (Cont’d..)
An antenna array can be designed to give a particular shape of radiating pattern. Control of the phase and current driving each array element along with spacing of array elements can provide beam steering capability. For simplification: All antenna elements are identical The current amplitude is the same feeding each element. The radiation pattern lies only in xy plane, θ=π/2 The radiation pattern then can be controlled by: controlling the spacing between elements or controlling the phase of current driving for each element

For simple example, consider a pair of dipole antennas driven in phase current source and separated by λ/2 on the x axis. Assume each antenna radiates independently, at far field point P, the fields from 2 antennas will be 180 out-of-phase, owing to extra λ/2 distance travel by the wave from the farthest antenna  fields cancel in this direction. At point Q, the fields in phase and adds. The E field is then twice from single dipole, fourfold increase in power  broadside array  max radiation is directed broadside to axis of elements.

Modify with driving the pair of dipoles with current sources 180 out of phase. Then along x axis will be in phase and along y axis will be out of phase, as shown by the resulting beam pattern  endfire array  max radiation is directed at the ends of axis containing array elements.

Pair of Hertzian Dipoles Recall that the far field value of E field from Hertzian dipole at origin, But confining our discussion to the xy plane where θ = π/2,

Consider a pair of z oriented Hertzian dipole, with distance d, where the total field is the vector sum of the fields for both dipoles and the magnitude of currents the same but a phase shift  between them. Where,

Assumption, And,

Where geometrically we could get, Thus, the total E field becomes: With Euler’s identity, the total E field at far field observation point from two element Hertzian dipole array becomes :

To find radiated power, It can be written as:

Unit factor, Funit is the max time averaged power density for an individual antenna element at θ=π/2 An array factor, Farray is Where, This is the pattern function resulting from an array of two isotropic radiators.

Example 5 The λ/2 long antennas are driven in phase and are λ/2 apart. Find: the far field radiation pattern for a pair of half wave dipole shown. the maximum power density 1 km away from the array if each antenna is driven by a 1mA amplitude current source at 100 MHz.

Solution to Example 5 At 100 MHz, λ = 3m, so that 1 km away is definitely in far field. For a half wave dipole, we have : The unit factor can be found by evaluating above at θ = π/2 ,

The array factor, with d = λ/2 and =0 (due to antennas are driven at the same phase) : For the array, we now have : The normalized power function is:

This can be plotted as : But how to plot?!?!?! Use MATLAB!! The maximum radiated power density at 1000m is :

N - Element Linear Arrays The procedure of two-element array can be extended for an arbitrary number of array elements, by simplifying assumptions : The array is linear  antenna elements are evenly spaced, d along a line. The array is uniform  each antenna element driven by same magnitude current source, constant phase difference,  between adjacent elements.

The far field electric field intensity : Where, Manipulate this series to get: With the max value as :

So then the normalized power pattern for these elements is :

Example 6 Five antenna elements spaced λ/4 apart with progressive phase steps 300. The antennas are assumed to be linear array of z oriented dipoles on the x axis. Find: the normalized radiation pattern in xy plane the plot of the radiation pattern.

Solution to Example 6 To find the array factor, first need to find psi, Ψ: Inserting this ratio to array factor,

The normalized radiation pattern is : The plot is : But how to plot?!?!?! Use MATLAB!!

Parasitic Arrays Not all the elements in array need be directly driven by current source. A parasitic array typically one driven element and several parasitic elements, the best known parasitic array is Yagi-Uda antenna.

On one side of the driven element is reflector  length and spacing are chosen to cancel most of the radiation in that direction, as well as to enhance the direction to forward or main beam direction. Several directors (four to six)  focus the main beam in the forward direction  high gain and easy to construct. Parasitic elements tend to pull down the Rrad of the driven element. E.g. Rrad of dipole would drop from 73Ω to 20Ω when used as the driven element in Yagi-Uda antenna. But higher Rrad is more efficient, so use half wavelength folded dipole antenna (four times Rrad of half-wave dipole!)

4.6 Helical Antennas Diameter of ground plane at least 3λ/4

Helical Antennas (Cont’d..)

Important Parameters

For this special case, The radiated field is circularly polarized in all directions other than θ = 00

Parameters for End Fire mode

Feed Design for Helical Antennas

4.7 Yagi Uda Array Antennas

Yagi Uda Array Antennas (Cont’d..)

Example 7 With above parameters, design a Yagi Uda Array antenna by finding the element spacing, lengths and total array length.

Solution to Example 7 from given directivity = 9.2 dB

Solution to Example 7(Cont’d..)

Therefore, Total array length: 0.8 λ The spacing between directors: 0.2 λ The reflector spacing: 0.2 λ The actual elements length: L3 = L5 : 0.447λ L4 : 0.443λ L1 : 0.490λ

Fig 10.25

Fig 10.26

Yagi Uda Array Antennas (Cont’d..)

4.8 Antennas for Wireless Communications
Parabolic Reflectors Parabolic dish antenna Parabolic reflector antenna

Antennas for Wireless Communications (Cont’d..)
All of these parabolic reflectors operates based on the geometric optics principle that a point source of radiation placed at the focal point of parabolic reflector will radiate the energy incident on the dish in a narrow and collimated beam. Cassegrain reflector antenna For high efficient, the dish must be significantly larger than the radiation wavelength, and has a directive feed.

Patch Antennas Other shapes such as circles, triangles and annular rings also been used. It can be excited by an edge or probe fed, where its location is chosen for impedance match between cable and antenna.

Folded Dipole Antennas A pair of half-wavelength dipole elements are joined at the ends and fed from the center of one of the pair. If the two sections are close together (d on the order of λ/64), the impedance will be four times greater than the regular λ/2 dipole antenna. The directivity is the same but the bandwidth is significantly broader.

Antenna End

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