MATRUSRI ENGINEERING COLLEGE�DEPARTMENT OF ELECTRONICS & COMMUNICATION ENGINEERING

SUBJECT NAME: ANTENNA & WAVE PROPAGATION (PC504EC)

FACULTY NAME: Dr. Pallavi Khare

MATRUSRI

ENGINEERING COLLEGE

Current Distributions, Radiation from Infinitesimal Dipole, Half wave Dipole and Quarter wave Monopole, Loop Antennas - Introduction, Small Loop, Far field pattern of circular loop with uniform current, Comparison of far fields of small loop and short dipole, Slot Antennas, Helical Antennas-Helical Geometry, Helix modes, Practical Design considerations for Mono filar Helical Antenna in Axial and Normal Modes, wideband characteristics, radiation efficiency.

UNIT-2

OBJECTIVE

Familiarize with the design of different types of antennas for various frequency ranges and latest developments in the practical antennas.

OUTCOME

Design and analyze different types of antennas for various frequency ranges and different regions and get updated with latest developments in the practical antennas

Outline

• Introduction

• Infinitesimal dipole

• Small dipole

• Region separation

• Finite length dipole

• Half-wavelength dipole

Introduction

• Wire antennas, linear or curved, are some of the oldest, simplest, cheapest, and in many cases the most versatile for many applications.

• We begin our analysis of antennas by considering some of the oldest, simplest, and most basic configurations.

• Initially we will try to minimize the complexity of the antenna structure and geometry to keep the mathematical details to a minimum.

1. Infinitesimal dipole

• An infinitesimal linear wire (l << λ) is positioned symmetrically at the origin of the coordinate system and oriented along the z axis, as shown in Figure

• 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.

• The wire, in addition to being very small (l << λ), is very thin (a << λ). The spatial variation of the current is assumed to be constant and given by

where I₀ = constant

(2.1)

Radiated Fields

• Since the source only carries an electric current I_e, I_m and the potential function F are zero.

(x, y, z ) : the observation point coordinates

(x’ , y’ , z’ ) represent the coordinates of the source

R: the distance from any point on the source to the observation point

path C is along the length of the source

(2.4)

(2.3 d)

(2.3 c)

(2.3 b)

(2.3 a)

(2.2)

Radiated Fields

• The next step of the procedure is to find H_A using (a) and then E_A using (b) or (c) with J = 0.

• It is often much simpler to transform

from rectangular to spherical components and then use (a) and (b) or (c) in spherical coordinates to find H and E.

• The transformation between rectangular and spherical components in matrix form:

• For this problem, A_x = A_y = 0

(a)

(b)

(c)

(2.5)

(2.6 a)

(2.6 b)

(2.6 c)

Radiated Fields

• Using the symmetry of the problem (no φ variations), (a) can be expanded in spherical coordinates and written in simplified form as

• Substituting (2-6a)–(2-6c) into (2-7) reduces it to

• The electric field E can now be found using (c) or (b) with J = 0.

• Substituting (2-6a)–(2-6c) or (2-8a)–(2-8b) into (2-9) reduces it to

(a)

(c)

(b)

(2.8 b)

(2.8 a)

(2.6 c)

(2.6 b)

(2.6 a)

(2.7)

(2.10 a)

(2.10 b)

(2.10 c)

(2.9)

Power Density and Radiation Resistance

• For the infinitesimal dipole, the complex Poynting vector can be written using (2-8a)–(2-8b) and (2-10a)–(2-10c) as

• whose radial W_r and transverse W_θ components are given, respectively, by

(2-12 b)

(2-12 a)

(2-11)

(2-10c)

(2-10b)

(2-10a)

(2-8b)

(2-8a)

Power Density and Radiation Resistance

• The complex power moving in the radial direction can be written as

• Equation (4-13), which gives the real and imaginary power that is moving outwardly, can also be written as

(2-15)

(2-14)

(2-13)

Power Density and Radiation Resistance

• Since the antenna radiates its real power through the radiation resistance, for the infinitesimal dipole:

• Radiation resistance R:

• It should be pointed out that the radiation resistance of (2-19) represents the total radiation resistance since (2-12b) does not contribute to it.

(2-12 b)

(2-19)

(2-18)

(2-17)

(2-16)

Radian Distance and Radian Sphere

• The E- and H-fields for the infinitesimal dipole, as represented by

• Radian distance

r = λ/2π (kr =1)

• Near-field region

r < λ/2π (kr <1)

• Far-field region

r >> λ/2π (kr >>1)

• Radian sphere

Sphere radius = r = λ/2π (kr =1)

• Intermediate-field region

r > λ/2π (kr >1)

(2-10 c)

(2-10 b)

(2-10 a)

(2-8b)

(2-8a)

Near-Field (kr << 1) Region

• The E-field components, E_r and E_θ , are in time-phase but they are in time-phase quadrature with the H-field component H_φ; therefore there is no time-average power flow associated with them.

• Time-average power density

which by using (4-20a)–(4-20d) reduces to

• Equations (2-20a) and (2-20b) are similar to those of a static electric dipole and (2-20d) to that of a static current element.

• Thus we usually refer to (2-20a)–(2-20d) as the quasistationary fields.

(2-22 )

(2-21)

(2-20 d)

(2-20 c)

(2-20 a)

(2-20 b)

Intermediate-Field (kr > 1) Region

• The total electric field

whose magnitude can be written as

(2-25)

(2-24 )

(2-23 c)

(2-23 d)

(2-23 b)

(2-23 a)

Far-Field (kr >> 1) Region

(2-27)

(2-26 c)

(2-26 b)

(2-26 a)

• The ratio of Eθ to Hφ is equal to

• The E- and H-field components are perpendicular to each other, transverse to the radial direction of propagation, and the r variations are separable from those of θ and φ.

• The fields form a Transverse ElectroMagnetic (TEM) wave whose wave impedance is equal to the intrinsic impedance of the medium.

Directivity

• Average power density

• Radiation intensity U

• The maximum value occurs at θ = π/2

• Directivity

• Maximum effective aperture

(2-28)

(2-32)

(2-31)

(2-30)

(2-29)

2. Small dipole

• A better approximation of the current distribution of wire antennas, whose lengths are usually λ/50 < l ≤ λ/10, is the triangular variation of Figure (a). The sinusoidal variations of Figures(b)–(c) are more accurate representations of the current distribution of any length wire antenna.

• The radiation properties of an infinitesimal dipole, which is usually taken to have a length l ≤ λ/50, were discussed in the previous section. Its current distribution was assumed to be constant.

Small dipole

• The current distribution of a small dipole (λ/50 < l ≤ λ/10):

where I₀ = constant

(2-33)

Small dipole

• The vector potential:

• Because the overall length of the dipole is very small (usually l ≤ λ/10), the values of R for different values of z along the length of the wire (−l/2 ≤ z ≤ l/2) are not much different from r. Thus R can be approximated by R ≈ r through out the integration path.

• Performing the integration, (2-34) reduces to

which is one-half of that obtained in the previous section for the infinitesimal dipole and given by (2-4).

(2-4)

(2-35)

(2-34)

Small dipole

• Since the potential function for the triangular distribution is one-half of the corresponding one for the constant (uniform) current distribution, the corresponding fields of the former are one-half of the latter.

• The E and H-fields radiated by a small dipole as

• The radiation resistance of the antenna is strongly dependent upon the current distribution.

(2-37)

(2-36 c)

(2-36 a)

(2-36 b)

Region separation

• A very thin dipole of finite length l is symmetrically positioned about the origin with its length directed along the z-axis, as shown in Figure

(2-38 a)

(2-38)

Region separation

• The wire is assumed to be very thin (x’ = y’ =0)

where

• Using the binomial expansion, we can write (2-40) in a series as

(2-41)

(2-40 b)

(2-40 a)

(2-40)

(2-39)

Region separation

where D is the largest dimension of the antenna (D = l for a wire antenna).

3. Finite length dipole

• For a very thin dipole (ideally zero diameter), the current distribution can be written, to a good approximation, as

• Electric and Magnetic field components (far-field)

Finite length dipole

• Using the far-field approximations given by below , equ. can be written as

• Summing the contributions from all the infinitesimal elements

• The total field of the antenna is equal to the product of the element and space factors

• For the current distribution of Ie, Eθ can be written as

Finite length dipole

• Each one of the integrals in equation can be integrated using

• After some mathematical manipulations, Eθ takes the form of

• In a similar manner, the total Hφ component can be written as

Power Density, Radiation Intensity, and Radiation Resistance

• For the dipole, the average Poynting vector can be written as

• The radiation intensity

The normalized (to 0 dB) elevation power patterns, as given for l = λ/4, λ/2, 3λ/4, and λ are shown plotted in Figure

• As the length of the antenna increases, the beam becomes narrower.

Power Density, Radiation Intensity, and Radiation Resistance

• As the length of the dipole increases beyond one wavelength (l > λ), the number of lobes begin to increase.

• The normalized power pattern for a dipole with l = 1.25λ is shown in Figures

Power Density, Radiation Intensity, and Radiation Resistance

• The current distribution for the dipoles with l = λ/4, λ/2, λ, 3λ/2, and 2λ, as given by Below equ. of Ie is shown in Figure.

Power Density, Radiation Intensity, and Radiation Resistance

• To find the total power radiated, the average Poynting vector of Wav is integrated over a sphere of radius r.

• Using equation of Wav, we can write Prad as

Power Density, Radiation Intensity, and Radiation Resistance

• After some extensive mathematical manipulations, it can be shown that Prad reduces to

where C = 0.5772 (Euler’s constant) and Ci(x) and Si(x) are the cosine and sine integrals

• Ci(x) is related to Cin(x) by

where

Power Density, Radiation Intensity, and Radiation Resistance

• The radiation resistance can be obtained as

• Figure is a plot of R_r as a function of l (in wave lengths) when the antenna is radiating into free-space (η = 120π).

Power Density, Radiation Intensity, and Radiation Resistance

• The imaginary part of the impedance, relative to the current maximum, is given by

• To examine the effect the wire radius has on the values of the reactance, its values, as given above, are plotted in Figure for a = 10⁻⁵λ, 10⁻⁴λ, 10⁻³λ, and 10⁻²λ.

Directivity

• The directivity was defined mathematically by

• The radiation intensity U

• the dipole antenna of length l has

• Because the pattern is not a function of φ, reduces to

• The corresponding values of the maximum effective aperture are related to the directivity by

Input Resistance

• To refer the radiation resistance to the input terminals of the antenna, the antenna itself is first assumed to be lossless (RL = 0). Then the power at the input terminals is equated to the power at the current maximum.

• Referring to Figure , we can write

• For a dipole of length l, the current at the input terminals (I_in ) is related to the current maximum (I₀) referring to Figure, by

• Thus the input radiation resistance can be written as

Finite Feed Gap

• To analytically account for a nonzero current at the feed point for antennas with a finite gap at the terminals, Schelkunoff and Friis have changed the current of Ie by including a quadrature term in the distribution.

• The additional term is inserted to take into account the effects of radiation on the antenna current distribution.

• This reaction is included by modifying Ie to

• where p is a coefficient that is dependent upon the overall length of the antenna and the gap spacing at the terminals. The values of p become smaller as the radius of the wire and the gap decrease.

4. Half-wavelength dipole

• The electric and magnetic field components of a half-wavelength dipole can be obtained from Eθ and Hϕ by letting l = λ/2.

• The time-average power density

• Radiation intensity

Half-wavelength dipole

• The total power radiated can be obtained as

• which when integrated reduces, to

Half-wavelength dipole

• By the definition of C_in(x), as given , C_in (2π) is equal to

• The maximum directivity of the half-wavelength dipole reduces to

• The corresponding maximum effective area is equal to

Half-wavelength dipole

• The radiation resistance, for a free-space medium (η = 120π), is given by

• The total input impedance for l = λ/2 is equal to

• Depending on the radius of the wire, the length of the dipole for first resonance is about l = 0.47λ to 0.48λ; the thinner the wire, the closer the length is to 0.48λ. Thus, for thicker wires, a larger segment of the wire has to be removed from λ/2 to achieve resonance.