EE 4990/6990 Antennas Fall 2002
Page Lecture Material from Balanis Problems
1 Ch. 1, Introduction, antenna types 2 Radiation, Ch. 2, Antenna patterns 2.2 3 Average power, radiation intensity 2.4, 2.7 4 Directivity, numerical evaluation of directivity 2.4, 2.7 5 Antenna gain 2.11, 2.13, 6 Antenna efficiency and impedance 2.17(a), 2.21 7 Loss resistance, transmission lines 2.27, 2.39 8 Transmit/receive systems, Polarization 2.41, 2.46 9 Equivalent areas, effective aperture 2.29, 2.48 10 Friis transmission equation 2.53, 2.56, 2.58 11 Radar systems, radar cross section 2.62, 2.66 12 Problem Session 13 Quiz #1 [Ch. 1,2] 14 Ch. 3, Radiated fields 15 Use of potential functions 16 Far fields, duality, reciprocity 4.1 85 Ch. 4, Wire antennas, infinitesimal dipole 4.3 18 Infinitesimal dipole 4.5 19 Poynting’s theorem, total power 4.11, 4.15 20 Radiation resistance, Short dipole 4.18(b), 4.21 21 Center-fed dipole 4.31 123 Half-wave dipole 4.25, 4.26 23 Dipole characteristics 4.27, 4.33 24 Image theory, antennas over ground 4.37 25 Monopole 4.41, 4.44 26 Ground Effects on Antennas 27 Quiz #2 [Ch. 3,4] 28 Ch. 5, Small loop antenna 5.4 29 Dual sources 5.17 30 Loop characteristics 5.21 162 Ch. 6, Antenna arrays 6.3 32 Broadside arrays 6.6 33 Endfire arrays 6.16 34 Hansen-Woodyard array, Binomial arrays 6.24, 6.28 35 Dolph-Chebyshev array, 6.41 191 Ch. 9, folded dipole 9.8, 9.10, 9.12 37 Ch. 10, Traveling wave antennas 10.4, 10.6 38 Terminations, vee antenna, 10.28 39 rhombic antenna, Yagi-Uda arrays 10.28 40 Ch. 11, Log-periodic antenna 11.8 41 Problem Session 42 Quiz #3 [Ch. 5,6,9,10,11] 43 Ch. 12, Aperture antennas 44 Ch. 13, Horn antennas 13.7, 13.12 45 Course review Antennas
Antenna - a device used to efficiently transmit and/or receive electromagnetic waves.
Example Antenna Applications Wireless communications Personal Communications Systems (PCS) Global Positioning Satellite (GPS) Systems Wireless Local Area Networks (WLAN) Direct Broadcast Satellite (DBS) Television Mobile Communications Telephone Microwave/Satellite Links Broadcast Television and Radio, etc.
Remote Sensing Radar [active remote sensing - radiate and receive] Military applications (target search and tracking) Weather radar, Air traffic control Automobile speed detection Traffic control (magnetometer) Ground penetrating radar (GPR) Agricultural applications Radiometry [passive remote sensing - receive emissions] Military applications (threat avoidance, signal interception)
Antenna Types Wire antennas (monopoles, dipoles, loops, etc.) Aperture antennas (sectoral horn, pyramidal horn, slots, etc.) Reflector antennas (parabolic dish, corner reflector, etc.) Lens antennas Microstrip antennas Antenna arrays Antenna Performance Parameters
Radiation pattern - angular plot of the radiation. Omnidirectional pattern - uniform radiation in one plane Directive patterns - narrow beam(s) of high radiation
Directivity - ratio of antenna power density at a distant point relative to that of an isotropic radiator [isotropic radiator - an antenna that radiates uniformly in all directions (point source radiator)].
Gain - directivity reduced by losses.
Polarization - trace of the radiated electric field vector (linear, circular, elliptical).
Impedance - antenna input impedance at its terminals.
Bandwidth - range of frequencies over which performance is acceptable (resonant antennas, broadband antennas).
Beam scanning - movement in the direction of maximum radiation by mechanical or electrical means.
Other system design constraints - size, weight, cost, power handling, radar cross section, etc. Fundamentals of Antenna Radiation
An antenna may be thought of as a matching network between a wave-guiding device (transmission line, waveguide) and the surrounding medium. Transmitting antenna guided wave input 6 antenna 6 unguided wave output
Receiving antenna unguided wave input 6 antenna 6 guided wave output
Antenna as the termination of a transmission line
The open-circuited transmission line does not radiate effectively because the transmission line currents are equal and opposite (and very close together). The radiated fields of these currents tend to cancel one another. The current on the arms of the dipole antenna are aligned in the same direction so that these radiated fields tend to add together making the dipole and efficient radiator. Antenna as the termination of a waveguide
The open-ended waveguide will radiate, but not as effectively as the waveguide terminated by the horn antenna. The wave impedance inside the waveguide does not match that of the surrounding medium creating a mismatch at the open end of the waveguide. Thus, a portion of the outgoing wave is reflected back into the waveguide. The horn antenna acts as a matching network, with a gradual transition in the wave impedance from that of the waveguide to that of the surrounding medium. With a matched termination, the reflected wave is minimized and the radiated field is maximized. Antenna Patterns (Radiation Patterns)
Antenna Pattern - a graphical representation of the antenna radiation properties as a function of position (spherical coordinates).
Common Types of Antenna Patterns Power Pattern - normalized power vs. spherical coordinate position. Field Pattern - normalized *E* or *H* vs. spherical coordinate position.
Antenna Field Types Reactive field - the portion of the antenna field characterized by standing (stationary) waves which represent stored energy. Radiation field - the portion of the antenna field characterized by radiating (propagating) waves which represent transmitted energy.
Antenna Field Regions Reactive Near Field Region - the region immediately surrounding the antenna where the reactive field (stored energy - standing waves) is dominant. Near-Field (Fresnel) Region - the region between the reactive near- field and the far-field where the radiation fields are dominant and the field distribution is dependent on the distance from the antenna. Far-Field (Fraunhofer) Region - the region farthest away from the antenna where the field distribution is essentially independent of the distance from the antenna (propagating waves). Antenna Field Regions Antenna Pattern Definitions Isotropic Pattern - an antenna pattern defined by uniform radiation in all directions, produced by an isotropic radiator (point source, a non-physical antenna which is the only nondirectional antenna). Directional Pattern - a pattern characterized by more efficient radiation in one direction than another (all physically realizable antennas are directional antennas). Omnidirectional Pattern - a pattern which is uniform in a given plane. Principal Plane Patterns - the E-plane and H-plane patterns of a linearly polarized antenna. E-plane - the plane containing the electric field vector and the direction of maximum radiation. H-plane - the plane containing the magnetic field vector and the direction of maximum radiation.
Antenna Pattern Parameters Radiation Lobe - a clear peak in the radiation intensity surrounded by regions of weaker radiation intensity. Main Lobe (major lobe, main beam) - radiation lobe in the direction of maximum radiation. Minor Lobe - any radiation lobe other than the main lobe. Side Lobe - a radiation lobe in any direction other than the direction(s) of intended radiation. Back Lobe - the radiation lobe opposite to the main lobe. Half-Power Beamwidth (HPBW) - the angular width of the main beam at the half-power points. First Null Beamwidth (FNBW) - angular width between the first nulls on either side of the main beam.
Antenna Pattern Parameters (Normalized Power Pattern) Maxwell’s Equations (Instantaneous and Phasor Forms)
Maxwell’s Equations (instantaneous form) % (
' + Ã-
'
%
(Ã+Ã'Ã%Ã- - instantaneous vectors [( =( (x,y,z,t), etc.] D t - instantaneous scalar
Maxwell’s Equations (phasor form, time-harmonic form)
E, H, D, B, J - phasor vectors [E=E(x,y,z), etc.] D - phasor scalar
Relation of instantaneous quantities to phasor quantities ... ( (x,y,z,t) = Re{E(x,y,z)ejTt}, etc. Average Power Radiated by an Antenna
To determine the average power radiated by an antenna, we start with the instantaneous Poynting vector 6 (vector power density) defined by
6à Ã(ÃðÃ+ÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃÃ(V/m × A/m = W/m2)
Assume the antenna is enclosed by some surface S.
= S =s
ds
3 The total instantaneous radiated power rad leaving the surface S is found by integrating the instantaneous Poynting vector over the surface. = 3 à ÃçÃ6ÃÃ@ çà (ÃðÃ+à Ã@ rad ds = ( ) ds ds = s ds SS
= ds = differential surface s = unit vector normal to ds For time-harmonic fields, the time average instantaneous Poynting vector (time average vector power density) is found by integrating the instantaneous Poynting vector over one period (T) and dividing by the period. 1 à ÃÃçà (ÃðÃ+à Pavg = ( ) dt T T ( = Re{Ee jTt}
+ = Re{He jTt}
The instantaneous magnetic field may be rewritten as
+ = Re{½ [ He jTt + H*e!jTt ]} which gives an instantaneous Poynting vector of
(ÃðÃ+ÃÃà Ãý Re {[E ð H]ej2Tt + [E ð H*]} ~~~~~~~~~~~~~~~ ~~~~~~~ time-harmonic independent of time (integrates to zero over T ) and the time-average vector power density becomes 1 ð * ç Pavg = Re [E H ] dt 2T T = ½ Re [E ð H*]
The total time-average power radiated by the antenna (Prad) is found by integrating the time-average power density over S.
à Ãçà @ ç ð * Ã@ Prad Pavg ds = ½ Re [E H ] ds S S Radiation Intensity
Radiation Intensity - radiated power per solid angle (radiated power normalized to a unit sphere).
à Ãçà @ Prad Pavg ds S In the far field, the radiation electric and magnetic fields vary as 1/r and the direction of the vector power density (Pavg) is radially outward. If we assume that the surface S is a sphere of radius r, then the integral for the total time-average radiated power becomes
2 2 N If we defined Pavg r = U( , ) as the radiation intensity, then
where dS = sin2d2dN defines the differential solid angle. The units on the radiation intensity are defined as watts per unit solid angle. The average radiation intensity is found by dividing the radiation intensity by the area of the unit sphere (4B) which gives
The average radiation intensity for a given antenna represents the radiation intensity of a point source producing the same amount of radiated power as the antenna. Radian
2S radians in full circle arc length of circle r T
Fig. 2.10(a) Geometrical arrangements for defining a radian
19
Steradian
one steradian subtends an area of A r 2
4ʌ steradians in entire sphere
dA r 2 sin T dT dI
dA d: sin T dT dI r 2
Fig. 2.10(b) Geometrical arrangements for defining a steradian.
20 Radiation power density
Instantaneous Time average Poynting vector Poynting vector G G G G G G W E H 1 u [ W/m ² ] Wavg Re >Eu H @ [ W/m ² ] 2 [2-8] [2-3]
Total instantaneous Average radiated Power Power G G G G G [ W ] P W s Prad Wavg x d s x d [ W ] ³³ [2-9] ³³ s s [2-4]
21
Radiation intensity
“Power radiated per unit solid angle”
2 U r Wavg [W/unit solid angle]
2 G r 2 U (T,I) E(r,T,M) 2K 2 r 2 2 > ET (r,T,I) EI(r,T,I) @ 2K [2-12a]
far zone fields without 1/ r factor
22 Directivity
Directivity (D) - the ratio of the radiation intensity in a given direction from the antenna to the radiation intensity averaged over all directions.
The directivity of an isotropic radiator is D(2,N) = 1.
2 N The maximum directivity is defined as [D( , )]max = Do. # 2 N # The directivity range for any antenna is 0 D( , ) Do.
Directivity in dB
Directivity in terms of Beam Solid Angle
We may define the radiation intensity as
2 N where Bo is a constant and F( , ) is the radiation intensity pattern function. The directivity then becomes
and the radiated power is Inserting the expression for Prad into the directivity expression yields
The maximum directivity is
S where the term A in the previous equation is defined as the beam solid angle and is defined by
Beam Solid Angle - the solid angle through which all of the antenna 2 N power would flow if the radiation intensity were [U( , )]max for all S angles in A. Example (Directivity/Beam Solid Angle/Maximum Directivity) 2 N S Determine the directivity [D( , )], the beam solid angle A and the 2 N maximum directivity [Do] of an antenna defined by F( , ) = sin22cos22. 2 N In order to find [F( , )]max, we must solve MATLAB m-file for plotting this directivity function for i=1:100 theta(i)=pi*(i-1)/99; d(i)=7.5*((cos(theta(i)))^2)*((sin(theta(i)))^2); end polar(theta,d)
90 2 120 60
1.5
150 1 30
0.5
180 0
210 330
240 300
270 Directivity/Beam Solid Angle Approximations
Given an antenna with one narrow major lobe and negligible radiation in its minor lobes, the beam solid angle may be approximated by
2 2 where 1 and 2 are the half-power beamwidths (in radians) which are perpendicular to each other. The maximum directivity, in this case, is approximated by
If the beamwidths are measured in degrees, we have
Example (Approximate Directivity)
A horn antenna with low side lobes has half-power beamwidths of 29o in both principal planes (E-plane and H-plane). Determine the approximate directivity (dB) of the horn antenna. Numerical Evaluation of Directivity
The maximum directivity of a given antenna may be written as
2N 2 N where U( ) = Bo F( , ). The integrals related to the radiated power in the denominators of the terms above may not be analytically integrable. In this case, the integrals must be evaluated using numerical techniques. If we assume that the dependence of the radiation intensity on 2 and N is separable, then we may write
The radiated power integral then becomes Note that the assumption of a separable radiation intensity pattern function results in the product of two separate integrals for the radiated power. We may employ a variety of numerical integration techniques to evaluate the integrals. The most straightforward of these techniques is the rectangular rule (others include the trapezoidal rule, Gaussian quadrature, etc.) If we first consider the 2-dependent integral, the range of 2 is first subdivided into N equal intervals of length
The known function f (2) is then evaluated at the center of each subinterval. The center of each subinterval is defined by
The area of each rectangular sub-region is given by The overall integral is then approximated by
Using the same technique on the N-dependent integral yields
Combining the 2 and N dependent integration results gives the approximate radiated power.
The approximate radiated power for antennas that are omnidirectional with respect to N [g(N) = 1] reduces to The approximate radiated power for antennas that are omnidirectional with respect to 2 [ f(2) = 1] reduces to
For antennas which have a radiation intensity which is not separable in 2 and N, the a two-dimensional numerical integration must be performed which yields
Example (Numerical evaluation of directivity)
Determine the directivity of a half-wave dipole given the radiation intensity of The maximum value of the radiation intensity for a half-wave dipole occurs at 2 = B/2 so that
MATLAB m-file sum=0.0; N=input(’Enter the number of segments in the theta direction’) for i=1:N thetai=(pi/N)*(i-0.5); sum=sum+(cos((pi/2)*cos(thetai)))^2/sin(thetai); end D=(2*N)/(pi*sum)
NDo 5 1.6428 10 1.6410 15 1.6409 20 1.6409 Antenna Efficiency
When an antenna is driven by a voltage source (generator), the total power radiated by the antenna will not be the total power available from the generator. The loss factors which affect the antenna efficiency can be identified by considering the common example of a generator connected to a transmitting antenna via a transmission line as shown below.
Zg - source impedance
ZA - antenna impedance
Zo - transmission line characteristic impedance
Pin - total power delivered to the antenna terminals
2 Pohmic - antenna ohmic (I R) losses [conduction loss + dielectric loss]
Prad - total power radiated by the antenna
The total power delivered to the antenna terminals is less than that available from the generator given the effects of mismatch at the source/t- line connection, losses in the t-line, and mismatch at the t-line/antenna connection. The total power delivered to the antenna terminals must equal that lost to I2R (ohmic) losses plus that radiated by the antenna. We may define the antenna radiation efficiency (ecd) as
which gives a measure of how efficient the antenna is at radiating the power delivered to its terminals. The antenna radiation efficiency may be written as a product of the conduction efficiency (ec) and the dielectric efficiency (ed).
ec - conduction efficiency (conduction losses only)
ed - dielectric efficiency (dielectric losses only)
However, these individual efficiency terms are difficult to compute so that they are typically determined by experimental measurement. This antenna measurement yields the total antenna radiation efficiency such that the individual terms cannot be separated. Note that the antenna radiation efficiency does not include the mismatch (reflection) losses at the t-line/antenna connection. This loss factor is not included in the antenna radiation efficiency because it is not inherent to the antenna alone. The reflection loss factor depends on the t- line connected to the antenna. We can define the total antenna efficiency
(eo), which includes the losses due to mismatch as
eo - total antenna efficiency (all losses)
er - reflection efficiency (mismatch losses)
The reflection efficiency represents the ratio of power delivered to the antenna terminals to the total power incident on the t-line/antenna connection. The reflection efficiency is easily found from transmission line theory in terms of the reflection coefficient (').
The total antenna efficiency then becomes
The definition of antenna efficiency (specifically, the antenna radiation efficiency) plays an important role in the definition of antenna gain.
Antenna Gain
The definitions of antenna directivity and antenna gain are essentially the same except for the power terms used in the definitions.
Directivity [D(2,N)] - ratio of the antenna radiated power density at a
distant point to the total antenna radiated power (Prad) radiated isotropically.
Gain [G(2,N)] - ratio of the antenna radiated power density at a distant
point to the total antenna input power (Pin) radiated isotropically.
Thus, the antenna gain, being dependent on the total power delivered to the antenna input terminals, accounts for the ohmic losses in the antenna while the antenna directivity, being dependent on the total radiated power, does not include the effect of ohmic losses. The equations for directivity and gain are
The relationship between the directivity and gain of an antenna may be found using the definition of the radiation efficiency of the antenna.
Gain in dB Antenna Impedance
The complex antenna impedance is defined in terms of resistive (real) and reactive (imaginary) components.
RA - Antenna resistance [(dissipation ) ohmic losses + radiation]
XA - Antenna reactance [(energy storage) antenna near field]
We may define the antenna resistance as the sum of two resistances which separately represent the ohmic losses and the radiation.
Rr - Antenna radiation resistance (radiation)
RL - Antenna loss resistance (ohmic loss)
The typical transmitting system can be defined by a generator, transmission line and transmitting antenna as shown below.
The generator is modeled by a complex source voltage Vg and a complex source impedance Zg. In some cases, the generator may be connected directly to the antenna.
Inserting the complete source and antenna impedances yields
The complex power associated with any element in the equivalent circuit is given by
where the * denotes the complex conjugate. We will assume peak values for all voltages and currents in expressing the radiated power, the power associated with ohmic losses, and the reactive power in terms of specific components of the antenna impedance. The peak current for the simple series circuit shown above is The power radiated by the antenna (Pr) may be written as
The power dissipated as heat (PL) may be written
The reactive power (imaginary component of the complex power) stored in the antenna near field (PX) is From the equivalent circuit for the generator/antenna system, we see that maximum power transfer occurs when
The circuit current in this case is
The power radiated by the antenna is
The power dissipated in heat is
The power available from the generator source is The power dissipated in the generator resistance is
Transmitting antenna system summary (maximum power transfer)
Power dissipated in the generator [P/2]
Power available from the generator [P] Power dissipated by the ! antenna [(1 ecd)(P/2)]
Power delivered to the antenna [P/2]
Power radiated by the
antenna [ecd (P/2)]
With an ideal transmitting antenna (ecd = 1) given maximum power transfer, one-half of the power available from the generator is radiated by the antenna. The typical receiving system can be defined by a generator (receiving antenna), transmission line and load (receiver) as shown below.
Assuming the receiving antenna is connected directly to the receiver
For the receiving system, maximum power transfer occurs when The circuit current in this case is
The power captured by the receiving antenna is
Some of the power captured by the receiving antenna is re-radiated
(scattered). The power scattered by the antenna (Pscat) is
The power dissipated by the receiving antenna in the form of heat is
The power delivered to the receiver is Receiving antenna system summary (maximum power transfer)
Power delivered to the receiver [P/2]
Power captured by Power dissipated by the ! the antenna [P] antenna [(1 ecd)(P/2)]
Power delivered to the antenna [P/2]
Power scattered by the
antenna [ecd (P/2)]
With an ideal receiving antenna (ecd = 1) given maximum power transfer, one-half of the power captured by the antenna is re-radiated (scattered) by the antenna. Antenna Radiation Efficiency
The radiation efficiency (ecd) of a given antenna has previously been defined in terms of the total power radiated by the antenna (Prad) and the total power dissipated by the antenna in the form of ohmic losses (Pohmic).
The total radiated power and the total ohmic losses were determined for the general case of a transmitting antenna using the equivalent circuit. The total radiated power is that “dissipated” in the antenna radiation resistance (Rr).
The total ohmic losses for the antenna are those dissipated in the antenna loss resistance (RL).
Inserting the equivalent circuit results for Prad and Pohmic into the equation for the antenna radiation efficiency yields
Thus, the antenna radiation efficiency may be found directly from the antenna equivalent circuit parameters. Antenna Loss Resistance
The antenna loss resistance (conductor and dielectric losses) for many antennas is typically difficult to calculate. In these cases, the loss resistance is normally measured experimentally. However, the loss resistance of wire antennas can be calculated easily and accurately. Assuming a conductor of length l and cross-sectional area A which carries a uniform current density, the DC resistance is
where F is the conductivity of the conductor. At high frequencies, the current tends to crowd toward the outer surface of the conductor (skin effect). The HF resistance can be defined in terms of the skin depth *.
where : is the permeability of the material and f is the frequency in Hz.
F 7 ® : : B !7 The skin depth for copper ( = 5.8×10 /m, = o = 4 ×10 H/m) may be written as If we define the perimeter distance of the conductor as dp, then the HF resistance of the conductor can be written as
where Rs is defined as the surface resistance of the material.
For the RHF equation to be accurate, the skin depth should be a small fraction of the conductor maximum cross-sectional dimension. In the case . B of a cylindrical conductor (dp 2 a), the HF resistance is
f * R 4 S 0 RDC = 0.818 m 1 kHz 2.09 mm ~ S 10 kHz 0.661 mm RHF = 1.60 m S 100 kHz 0.209 mm RHF = 5.07 m S 1 MHz 0.0661 RHF = 16.0 m mm
Resistance of 1 m of #10 AWG (a = 2.59 mm) copper wire. The high frequency resistance formula assumes that the current through the conductor is sinusoidal in time and independent of position along the T conductor [Iz(z,t) = Io cos( t)]. On most antennas, the current is not necessarily independent of position. However, given the actual current distribution on the antenna, an equivalent RL can be calculated.
Example (Problem 2.44) [Loss resistance calculation] A dipole antenna consists of a circular wire of length l. Assuming the current distribution on the wire is cosinusoidal, i.e.,
Equivalent circuit equation
(uniform current, Io - peak)
Integration of incremental power along the antenna Thus, the loss resistance of a dipole antenna of length l is one-half that of a the same conductor carrying a uniform current. Lossless Transmission Line Fundamentals
Transmission line equations (voltage and current)
~~~~~~~ ~~~~~~~ +z directed !z directed waves waves
Transmitting/Receiving Systems with Transmission Lines
Using transmission line theory, the impedance seen looking into the input terminals of the transmission line (Zin) is
The resulting equivalent circuit is shown below.
The current and voltage at the transmission line input terminals are The power available from the generator is
The power delivered to the transmission line input terminals is
The power associated with the generator impedance is
Given the current and the voltage at the input to the transmission line, the values at any point on the line can be found using the transmission line equations.
+ The unknown coefficient Vo may be determined from either V(0) or I(0) which were found in the input equivalent circuit. Using V(0) gives where
+ Given the coefficient Vo , the current and voltage at the load, from the transmission line equations are
The power delivered to the load is then
The complexity of the previous equations leads to solutions which are typically determined by computer or Smith chart. MATLAB m-file (generator/t-line/load)
Vg=input(’Enter the complex generator voltage ’); Zg=input(’Enter the complex generator impedance ’); Zo=input(’Enter the lossless t-line characteristic impedance ’); l=input(’Enter the lossless t-line length in wavelengths ’); Zl=input(’Enter the complex load impedance ’); j=0+1j; betal=2*pi*l; Zin=Zo*(Zl+j*Zo*tan(betal))/(Zo+j*Zl*tan(betal)); gammal=(Zl-Zo)/(Zl+Zo); gamma0=gammal*exp(-j*2*betal); Ig=Vg/(Zg+Zin); Pg=0.5*Vg*conj(Ig); V0=Ig*Zin; P0=0.5*V0*conj(Ig); Vcoeff=V0/(1+gamma0); Vl=Vcoeff*exp(-j*betal)*(1+gammal); Il=Vcoeff*exp(-j*betal)*(1-gammal)/Zo; Pl=0.5*Vl*conj(Il); s=(1+abs(gammal))/(1-abs(gammal)); format compact Generator_voltage=Vg Generator_current=Ig Generator_power=Pg Generator_impedance_voltage=Vg-V0 Generator_impedance_current=Ig Generator_impedance_power=Pg-P0 T_line_input_voltage=V0 T_line_input_current=Ig T_line_input_power=P0 T_line_input_impedance=Zin T_line_input_reflection_coeff=gamma0 T_line_standing_wave_ratio=s Load_voltage=Vl Load_current=Il Load_power=Pl Load_reflection_coeff=gammal
S 8 Given Vg = (10+j0) V, Zg = (100+j0) and l = 5.125 , the following results are found.
*' * *' * Zo ZL Zin (0) = (l) Pg s P(l) 100 75 96+j28 0.1429 0.25 1.3333 0.1224 100 100 100 0 0.25 1 0.125 100 125 98!j22 0.1111 0.25 1.25 0.1235 75 100 72!j21 0.1429 0.2864 1.3333 0.1199 100 100 100 0 0.25 1 0.125 125 100 122+j27 0.1111 0.2219 1.25 0.1219 Antenna Polarization
The polarization of an plane wave is defined by the figure traced by the instantaneous electric field at a fixed observation point. The following are the most commonly encountered polarizations assuming the wave is approaching. The polarization of the antenna in a given direction is defined as the polarization of the wave radiated in that direction by the antenna. Note that any of the previous polarization figures may be rotated by some arbitrary angle.
Polarization loss factor
Incident wave polarization
Antenna polarization
Polarization loss factor (PLF)
PLF in dB General Polarization Ellipse
The vector electric field associated with a +z-directed plane wave can be written in general phasor form as
where Ex and Ey are complex phasors which may be defined in terms of magnitude and phase. The instantaneous components of the electric field are found by multiplying the phasor components by e jT t and taking the real part.
( x (z,t)
( y (z,t)
The relative positions of the instantaneous electric field components on the general polarization ellipse defines the polarization of the plane wave.
Linear Polarization
If we define the phase shift between the two electric field components as
we find that a phase shift of
defines a linearly polarized wave. ( x (z,t)
( y (z,t)
Examples of linear polarization:
Y J If Eyo = 0 Linear polarization in the x-direction ( = 0) Y J o If Exo = 0 Linear polarization in the y-direction ( = 90 ) Y J o If Exo = Eyo and n is even Linear polarization ( = 45 ) Y J o If Exo = Eyo and n is odd Linear polarization ( = 135 ) Circular Polarization
If Exo = Eyo and
then ( x (z,t)
( y (z,t)
This is left-hand circular polarization.
If Exo = Eyo and
then ( x (z,t)
( y (z,t)
This is right-hand circular polarization.
Elliptical Polarization
Elliptical polarization follows definitions as circular polarization ú except that Exo Eyo. ú )N B Y Exo Eyo, = (2n+½) left-hand elliptical polarization ú )N ! B Y Exo Eyo, = (2n+½) right-hand elliptical polarization Antenna Equivalent Areas
Antenna Effective Aperture (Area)
Given a receiving antenna oriented for maximum response, polarization matched to the incident wave, and impedance matched to its load, the resulting power delivered to the receiver (Prec) may be defined in terms of the antenna effective aperture (Ae) as
where S is the power density of the incident wave (magnitude of the Poynting vector) defined by
According to the equivalent circuit under matched conditions,
We may solve for the antenna effective aperture which gives Antenna Scattering Area
The total power scattered by the receiving antenna is defined as the product of the incident power density and the antenna scattering area (As).
From the equivalent circuit, the total scattered power is
which gives
Antenna Loss Area
The total power dissipated as heat by the receiving antenna is defined as the product of the incident power density and the antenna loss area
(AL).
From the equivalent circuit, the total dissipated power is
which gives Antenna Capture Area
The total power captured by the receiving antenna (power delivered to the load + power scattered by the antenna + power dissipated in the form of heat) is defined as the product of the incident power density and the antenna capture area (Ac).
The total power captured by the antenna is
which gives
Note that Ac = Ae + As + AL. Maximum Directivity and Effective Aperture
Assume the transmitting and receiving antennas are lossless and oriented for maximum response.
Aet, Dot - transmit antenna effective aperture and maximum directivity Aer, Dor - receive antenna effective aperture and maximum directivity
If we assume that the total power transmitted by the transmit antenna is Pt, the power density at the receive antenna (Wr) is
The total power received by the receive antenna (Pr) is
which gives
If we interchange the transmit and receive antennas, the previous equation still holds true by interchanging the respective transmit and receive quantities (assuming a linear, isotropic medium), which gives These two equations yield or
If the transmit antenna is an isotropic radiator, we will later show that
which gives
Therefore, the equivalent aperture of a lossless antenna may be defined in terms of the maximum directivity as
The overall antenna efficiency (eo) may be included to account for the ohmic losses and mismatch losses in an antenna with losses.
The effect of polarization loss can also be included to yield Effective Area and Gain______Hon Tat Hui
λ2 λ2 Proof of A ()θ,φ = D()θ,φ = g()θ,φ e 4π 4π
Extracted from the book:
Kai Fong Lee, Principles of Antenna Theory, John Wiley & Sons, 1984, pp. 74-76.
1 Effective Area and Gain______Hon Tat Hui
2 Friis Transmission Equation
The Friis transmission equation defines the relationship between transmitted power and received power in an arbitrary transmit/receive antenna system. Given arbitrarily oriented transmitting and receiving antennas, the power density at the receiving antenna (Wr) is
where Pt is the input power at the terminals of the transmit antenna and where the transmit antenna gain and directivity for the system performance are related by the overall efficiency
' where ecdt is the radiation efficiency of the transmit antenna and t is the reflection coefficient at the transmit antenna terminals. Notice that this definition of the transmit antenna gain includes the mismatch losses for the transmit system in addition to the conduction and dielectric losses. A manufacturer’s specification for the antenna gain will not include the mismatch losses. The total received power delivered to the terminals of the receiving antenna (Pr) is
where the effective aperture of the receiving antenna (Aer) must take into account the orientation of the antenna. We may extend our previous definition of the antenna effective aperture (obtained using the maximum directivity) to a general effective aperture for any antenna orientation.
The total received power is then
such that the ratio of received power to transmitted power is
Including the polarization losses yields
For antennas aligned for maximum response, reflection-matched and polarization matched, the Friis transmission equation reduces to Radar Range Equation and Radar Cross Section
The Friis transmission formula can be used to determine the radar range equation. In order to determine the maximum range at which a given target can be detected by radar, the type of radar system (monostatic or bistatic) and the scattering properties of the target (radar cross section) must be known.
Monostatic radar system - transmit and receive antennas at the same location.
Bistatic radar system- transmit and receive antennas at separate locations. Radar cross section (RCS) - a measure of the ability of a target to reflect (scatter) electromagnetic energy (units = m2). The area which intercepts that amount of total power which, when scattered isotropically, produces the same power density at the receiver as the actual target.
If we define F = radar cross section (m2) 2 Wi = incident power density at the target (W/m ) Pc = equivalent power captured by the target (W) 2 Ws = scattered power density at the receiver (W/m )
According to the definition of the target RCS, the relationship between the incident power density at the target and the scattered power density at the receive antenna is
The limit is usually included since we must be in the far-field of the target for the radar cross section to yield an accurate result. The radar cross section may be written as
where (Ei, Hi) are the incident electric and magnetic fields at the target and
(Es, Hs) are the scattered electric and magnetic fields at the receiver. The incident power density at the target generated by the transmitting antenna ' (Pt, Gt, Dt, eot, t, at ) is given by
The total power captured by the target (Pc) is
The power captured by the target is scattered isotropically so that the scattered power density at the receiver is
The power delivered to the receiving antenna load is Showing the conduction losses, mismatch losses and polarization losses explicitly, the ratio of the received power to transmitted power becomes
where
aw - polarization unit vector for the scattered waves
ar - polarization unit vector for the receive antenna
Given matched antennas aligned for maximum response and polarization matched, the general radar range equation reduces to Example
Problem 2.65 A radar antenna, used for both transmitting and receiving, has a gain of 150 at its operating frequency of 5 GHz. It transmits 100 kW, and is aligned for maximum directional radiation and reception to a target 1 km away having a cross section of 3 m2. The received signal matches the polarization of the transmitted signal. Find the received power. Determination of Antenna Radiation Fields Using Potential Functions
Sources of Antenna 6 J - vector electric current density (A/m2) Radiation Fields M - vector magnetic current density (V/m2)
Some problems involving electric currents can be cast in equivalent forms involving magnetic currents (the use of magnetic currents is simply a mathematical tool, they have never been proven to exist).
A - magnetic vector potential (due to J) F - electric vector potential (due to M)
In order to account for both electric current and/or magnetic current sources, the symmetric form of Maxwell’s equations must be utilized to determine the resulting radiation fields. The symmetric form of Maxwell’s equations include additional radiation sources (electric charge density - D D and magnetic charge density m). However, these charges can always be related directly to the current via conservation of charge equations. Maxwell’s equations (symmetric, time-harmonic form)
The use of potentials in the solution of radiation fields employs the concept of superposition of fields.
Electric current Y Magnetic vector Y Radiation fields D source (J, ) potential (A) (EA, H A)
Magnetic current Y Electric vector Y Radiation fields D source (M, m) potential (F) (EF, H F)
The total radiation fields (E, H) are the sum of the fields due to electric currents (EA, H A) and the fields due to the magnetic currents (EF, H F).
Maxwell’s Equations (electric sources only Y F = 0) Maxwell’s Equations (magnetic sources only Y A = 0)
Based on the vector identity, any vector with zero divergence (rotational or solenoidal field) can be expressed as the curl of some other vector. From Maxwell’s equations with electric or magnetic sources only [Equations (1d) and (2c)], we find
so that we may define these vectors as
where A and F are the magnetic and electric vector potentials, respectively. The flux density definitions in Equations (3a) and (3b) lead to the following field definitions:
Inserting (3a) into (1a) and (3b) into (2b) yields Equations (5a) and (5b) can be rewritten as
Based on the vector identity the bracketed terms in (6a) and (6b) represent non-solenoidal (lamellar or irrotational fields) and may each be written as the gradient of some scalar
N N where e is the electric scalar potential and m is the magnetic scalar potential. Solving equations (7a) and (7b) for the electric and magnetic fields yields
Equations (4a) and (8a) give the fields (EA, HA) due to electric sources while Equations (4b) and (8b) give the fields (EF, HF) due to magnetic sources. Note that these radiated fields are obtained by differentiating the respective vector and scalar potentials. The integrals which define the vector and scalar potential can be found by first taking the curl of both sides of Equations (4a) and (4b):
According to the vector identity
and Equations (1b) and (2a), we find Inserting Equations (7a) and (7b) into (10a) and (10b), respectively gives
We have defined the rotational (curl) properties of the magnetic and electric vector potentials [Equations (3a) and (3b)] but have not yet defined the irrotational (divergence) properties. If we choose
Then, Equations (11a) and (11b) reduce to
The relationship chosen for the vector and scalar potentials defined in Equations (12a) and (12b) is defined as the Lorentz gauge [other choices for these relationships are possible]. Equations (13a) and (13b) are defined as inhomogenous Helmholtz vector wave equations which have solutions of the form where r locates the field point (where the field is measured) and rN locates the source point (where the current is located). Similar inhomogeneous Helmholtz scalar wave equations can be found for the electric and magnetic scalar potentials.
The solutions to the scalar potential equations are Determination of Radiation Fields Using Potentials - Summary Notice in the previous set equations for the radiated fields in terms of potentials that the equations for EA and HF both contain a complex differentiation involving the gradient and divergence operators. In order to avoid this complex differentiation, we may alternatively determine EA and HF directly from Maxwell’s equations once EF and HA have been determined using potentials. From Maxwell’s equations for electric currents and magnetic currents, we have (1) (2) In antenna problems, the regions where we want to determine the radiated fields are away from the sources. Thus, we may set J = 0 in Equation (1) to solve for EA and set M = 0 in Equation (2) to solve for HF. This yields
The total fields by superposition are which gives Antenna Far Fields in Terms of Potentials
As shown previously, the magnetic vector potential and electric vector potentials are defined as integrals of the (antenna) electric or magnetic current density.
If we are interested in determining the antenna far fields, then we must determine the potentials in the far field. We will find that the integrals defining the potentials simplify in the far field. In the far field, the vectors r and r !rN becomes nearly parallel.
(1) Using the approximation in (1) in the appropriate terms of the potential integrals yields
(2)
N If we assume that r >> (r )max, then the denominator of (2) may be simplified to give
(3)
Note that the rN term in the numerator complex exponential term in (3) cannot be neglected since it represents a phase shift term that may still be significant even in the far field. The r-dependent terms can be brought outside the integral since the potential integrals are integrated over the source (primed) coordinates. Thus, the far field integrals defining the potentials become
(4)
(5)
The potentials have the form of spherical waves as we would expect in the far field of the antenna. Also note that the complete r-dependence of the potentials is given outside the integrals. The rN term in the potential integrands can be expressed in terms of whatever coordinate system best fits the geometry of the source current. Spherical coordinates should always be used for the field coordinates in the far field based on the spherical symmetry of the far fields. Rectangular coordinate source
Cylindrical coordinate source
Spherical coordinate source
The results of the far field potential integrations in Equations (4) and (5) may be written as The electric field due to an electric current source (EA) and the magnetic field due to a magnetic current source (HF) are defined by
(6)
(7)
If we expand the differential operators in Equations (6) and (7) in spherical coordinates, given the known r-dependence, we find that the ar-dependent terms cancel and all of the other terms produced by this differentiation are of dependence r!2 or lower. These field contributions are much smaller in the far field than the contributions from the first terms in Equations (6) and !1 (7) which vary as r . Thus, in the far field, EA and HF may be approximated as (8) (9)
The corresponding components of the fields (HA and EF) can be found using the basic plane wave relationship between the electric and magnetic field in the far field of the antenna. Since the radiated far field must behave like a outward propagating spherical wave which looks essentially 64 like a plane wave as r , the far field components of HA and EF are related to the far field components of EA and HF by Solving the previous equations for the individual components of HA and EF yields
Thus, once the far field potential integral is evaluated, the corresponding far field can be found using the simple algebraic formulas above (the differentiation has already been performed). Duality
Duality - If the equations governing two different phenomena are identical in mathematical form, then the solutions also take on the same mathematical form (dual quantities).
Dual Equations
Electric Sources Magnetic Sources Dual Quantities
Electric Sources Magnetic Sources Reciprocity
Consider two sets of sources defined by (Ja , Ma) within the volume
Va and (Jb , Mb) within the volume Vb radiating at the same frequency. The sources (Ja , Ma) radiate the fields (Ea , Ha) while the sources (Jb , Mb) radiate the fields (Eb , Hb). The sources are assumed to be of finite extent and the region between the antennas is assumed to be isotropic and linear. We may write two separate sets of Maxwell’s equations for the two sets of sources.
If we dot (1a) with Eb and dot (2b) with Ha, we find
Adding Equations (3a) and (3b) yields The previous equation may be rewritten using the following vector identity. which gives
If we dot (1b) with Ea and dot (2a) with Hb, and perform the same operations, then we find
Subtracting (4a) from (4b) gives
If we integrate both sides of Equation (5) throughout all space and apply the divergence theorem to the left hand side, then
The surface on the left hand side of Equation (6) is a sphere of infinite radius on which the radiated fields approach zero. The volume V includes all space. Therefore, we may write Note that the left hand side of the previous integral depends on the “b” set of sources while the right hand side depends on the “a” set of sources.
Since we have limited the sources to the volumes Va and Vb, we may limit the volume integrals in (7) to the respective source volumes so that
Equation (8) represents the general form of the reciprocity theorem. We may use the reciprocity theorem to analyze a transmitting- receiving antenna system. Consider the antenna system shown below. For mathematical simplicity, let’s assume that the antennas are perfectly- conducting, electrically short dipole antennas.
The source integrals in the general 3-D reciprocity theorem of Equation (8) simplify to line integrals for the case of wire antennas.
Furthermore, the electric field along the perfectly conducting wire is zero so that the integration can be reduced to the antenna terminals (gaps). If we further assume that the antenna current is uniform over the electrically short dipole antennas, then
The line integral of the electric field transmitted by the opposite antenna over the antenna terminal gives the resulting induced open circuit voltage.
If we write the two port equations for the antenna system, we find
Note that the impedances Zab and Zba have been shown to be equal from the reciprocity theorem. Therefore, if we place a current source on antenna a and measure the response at antenna b, then switch the current source to antenna b and measure the response at antenna a, we find the same response (magnitude and phase). Also, since the transfer impedances (Zab and Zba) are identical, the transmit and receive patterns of a given antenna are identical. Thus, we may measure the pattern of a given antenna in either the transmitting mode or receiving mode, whichever is more convenient. Wire Antennas
Electrical Size of an Antenna - the physical dimensions of the antenna defined relative to wavelength.
Electrically small antenna - the dimensions of the antenna are small relative to wavelength.
Electrically large antenna - the dimensions of the antenna are large relative to wavelength.
Example Consider a dipole antenna of length L = 1m. Determine the electrical length of the dipole at f = 3 MHz and f = 30 GHz.
f = 3 MHz f = 30 GHz (8 = 100m) (8 = 0.01m) Electrically small Electrically large Infinitesimal Dipole ()l . 8/50, a << 8)
We assume that the axial current along the infinitesimal dipole is uniform. With a << 8, we may assume that any circumferential currents are negligible and treat the dipole as a current filament.
The infinitesimal dipole with a constant current along its length is a non- physical antenna. However, the infinitesimal dipole approximates several physically realizable antennas. Capacitor-plate antenna (top-hat-loaded antenna)
The “capacitor plates” can be actual conductors or simply the wire equivalent. The fields radiated by the radial currents tend to cancel each other in the far field so that the far fields of the capacitor plate antenna can be approximated by the infinitesimal dipole.
Transmission line loaded antenna
If we assume that L . 8/4, then the current along the antenna resembles that of a half-wave dipole. Inverted-L antenna
Using image theory, the inverted-L antenna is equivalent to the transmission line loaded antenna.
Based on the current distributions on these antennas, the far fields of the capacitor plate antenna, the transmission line loaded antenna and the inverted-L antenna can all be approximated by the far fields of the infinitesimal dipole. To determine the fields radiated by the infinitesimal dipole, we first determine the magnetic vector potential A due to the given electric current source J (M = 0, F = 0).
The infinitesimal dipole magnetic vector potential given in the previous equation is a rectangular coordinate vector with the magnitude defined in terms of spherical coordinates. The rectangular coordinate vector can be transformed into spherical coordinates using the standard coordinate transformation. The total magnetic vector potential may then be written in vector form as
Because of the true point source nature of the infinitesimal dipole ()l . 8/50), the equation above for the magnetic vector potential of the infinitesimal dipole is valid everywhere. We may use this expression for A to determine both near fields and far fields.
The radiated fields of the infinitesimal dipole are found by differentiating the magnetic vector potential.
The electric field is found using either potential theory or Maxwell’s equations.
Potential Theory
Maxwell’s Equations (J = 0 away from the source)
Note that electric field expression in terms of potentials requires two levels of differentiation while the Maxwell’s equations equation requires only one level of differentiation. Thus, using Maxwell’s equations, we find fields radiated by an infinitesimal dipole Field Regions of the Infinitesimal Dipole
We may separate the fields of the infinitesimal dipole into the three standard regions:
³ Reactive near field kr << 1 ´ Radiating near field kr > 1 µ Far field kr >> 1
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0 0 2 4 6 8 10 12
Considering the bracketed terms [ ] in the radiated field expressions for the infinitesimal dipole ...
³ Reactive near field (kr << 1) (kr)-2 terms dominate ´ Radiating near field (kr > 1) constant terms dominate if present otherwise, (kr)-1 terms dominate µ Far field (kr >> 1) constant terms dominate Reactive near field [ kr << 1 or r << 8/2B ]
When kr << 1, the terms which vary inversely with the highest power of kr are dominant. Thus, the near field of the infinitesimal dipole is given by
Infinitesimal dipole near fields
Note the 90o phase difference between the electric field components and the magnetic field component (these components are in phase quadrature) which indicates reactive power (stored energy, not radiation). If we investigate the Poynting vector of the dominant near field terms, we find
The Poynting vector (complex vector power density) for the infinitesimal dipole near field is purely imaginary. An imaginary Poynting vector corresponds to standing waves or stored energy (reactive power). The vector form of the near electric field is the same as that for an electrostatic dipole (charges +q and !q separated by a distance )l).
0 If we replace the term (Io /k) by in the near electric field terms by its charge equivalent expression, we find
The electric field expression above is identical to that of the electrostatic dipole except for the complex exponential term (the infinitesimal dipole electric field oscillates). This result is related to the assumption of a uniform current over the length of the infinitesimal dipole. The only way for the current to be uniform, even at the ends of the wire, is for charge to build up and decay at the ends of the dipole as the current oscillates. The near magnetic field of the infinitesimal dipole can be shown to be mathematically equivalent to that of a short DC current segment multiplied by the same complex exponential term. Radiating near field [ kr ù 1 or r ù 8/2B ]
The dominant terms for the radiating near field of the infinitesimal dipole are the terms which are constant with respect to kr for E2 and HN -1 and the term proportional to (kr) for Er.
Infinitesimal dipole radiating near field
Note that E2 and HN are now in phase which yields a Poynting vector for these two components which is purely real (radiation). The direction of this component of the Poynting vector is outward radially denoting the outward radiating real power.
Far field [ kr >> 1 or r >> 8/2B ]
The dominant terms for the far field of the infinitesimal dipole are the terms which are constant with respect to kr.
Infinitesimal dipole far field Note that the far field components of E and H are the same two components which produced the radially-directed real-valued Poynting vector (radiated power) for the radiating near field. Also note that there is no radial component of E or H so that the propagating wave is a transverse electromagnetic (TEM) wave. For very large values of r, this TEM wave approaches a plane wave. The ratio of the far electric field to the far magnetic field for the infinitesimal dipole yields the intrinsic impedance of the medium. Far Field of an Arbitrarily Oriented Infinitesimal Dipole
Given the equations for the far field of an infinitesimal dipole oriented along the z-axis, we may generalize these equations for an infinitesimal dipole antenna oriented in any direction. The far fields of infinitesimal dipole oriented along the z-axis are
If we rotate the antenna by some arbitrary angle " and define the new direction of the current flow by the unit vector a", the resulting far fields are simply a rotated version of the original equations above. In the rotated coordinate system, we must define new angles (",$) that correspond to the spherical coordinate angles (2,N) in the original coordinate system. The angle $ is shown below referenced to the x-axis (as N is defined) but can be referenced to any convenient axis that could represent a rotation in the N-direction. Note that the infinitesimal far fields in the original coordinate system depend on the spherical coordinates r and 2. The value of r is identical in the two coordinates systems since it represents the distance from the coordinate origin. However, we must determine the transformation from 2 to ". The transformations of the far fields in the original coordinate system to those in the rotated coordinate system can be written as
Specifically, we need the definition of sin". According to the trigonometric identity we may write
Based on the definition of the dot product, the cos" term may be written as so that
Inserting our result for the sin" term yields Example
Determine the far fields of an infinitesimal dipole oriented along the y-axis. Poynting’s Theorem (Conservation of Power)
Poynting’s theorem defines the basic principle of conservation of power which may be applied to radiating antennas. The derivation of the time-harmonic form of Poynting’s vector begins with the following vector identity
If we insert the Poynting vector (S = E × H*) in the left hand side of the above identity, we find
From Maxwell’s equations, the curl of E and H are
such that
Integrating both sides of this equation over any volume V and applying the divergence theorem to the left hand side gives
The current density in the equation above consists of two components: the impressed (source) current (Ji) and the conduction current (Jc). Inserting the current expression and dividing both sides of the equation by 2 yields Poynting’s theorem.
The individual terms in the above equation may be identified as
Poynting’s theorem may then be written as Total Power and Radiation Resistance
To determine the total complex power (radiated plus reactive) produced by the infinitesimal dipole, we integrate the Poynting vector over a spherical surface enclosing the antenna. We must use the complete field expressions to determine both the radiated and reactive power. The time- average complex Poynting vector is
The total complex power passing through the spherical surface of radius r is found by integrating the normal component of the Poynting vector over the surface. N N The terms We and Wm represent the radial electric and magnetic energy flow through the spherical surface S.
The total power through the sphere is The real and imaginary parts of the complex power are
The radiation resistance for the infinitesimal dipole is found according to
Infinitesimal dipole radiation resistance Infinitesimal Dipole Radiation Intensity and Directivity
The radiation intensity of the infinitesimal dipole may be found by using the previously determined total fields.
Infinitesimal dipole directivity function Infinitesimal dipole Maximum directivity Infinitesimal Dipole Effective Aperture and Solid Beam Angle
The effective aperture of the infinitesimal dipole is found from the maximum directivity:
Infinitesimal dipole effective aperture
The beam solid angle for the infinitesimal dipole can be found from the maximum directivity,
or can be determined directly from the radiation intensity function.
Infinitesimal dipole beam solid angle Short Dipole (8/50 # l # 8/10, a <<8) Note that the magnetic vector potential of the short dipole (length = l, peak current = Io) is one half that of the equivalent infinitesimal dipole (length ) l = l, current = Io). The average current on the short dipole is one half that of the equivalent infinitesimal dipole. Therefore, the fields produced by the short dipole are exactly one half those produced by the equivalent infinitesimal dipole.
Short dipole radiated fields
Short dipole near fields
Short dipole radiating near field Short dipole far field
Since the fields produced by the short dipole are one half those of the equivalent infinitesimal dipole, the real power radiated by the short dipole is one fourth that of the infinitesimal dipole. Thus, Prad for the short dipole is
and the associated radiation resistance is
Short dipole radiation resistance
The directivity function, the maximum directivity, effective area and beam solid angle of the short dipole are all identical to the corresponding value for the infinitesimal dipole. Center-Fed Dipole Antenna (a << 8)
If we assume that the dipole antenna is driven at its center, we may assume that the current distribution is symmetrical along the antenna.
We use the previously defined approximations for the far field magnetic vector potential to determine the far fields of the center-fed dipole. field coordinates (spherical)
Source coordinates (rectangular)
For the center-fed dipole lying along the z-axis, xN = yN = 0, so that
Transforming the z-directed vector potential to spherical coordinates gives
(Center-fed dipole far field magnetic vector potential )
The far fields of the center-fed dipole in terms of the magnetic vector potential are
(Center-fed dipole far field electric field)
(Center-fed dipole far field magnetic field) The time-average complex Poynting vector in the far field of the center-fed dipole is
The radiation intensity function for the center-fed dipole is given by
(Center-fed dipole radiation intensity function)
2 2 We may plot the normalized radiation intensity function [U( ) = BoF( )] to determine the effect of the antenna length on its radiation pattern. l = 8/10 l = 8/2
l = 8 l = 38/2
In general, we see that the directivity of the antenna increases as the length goes from a short dipole (a fraction of a wavelength) to a full wavelength. As the length increases above a wavelength, more lobes are introduced into the radiation pattern. l = 8/10 l = 8/2
1 1
0.9 0.9
0.8 0.8
0.7 0.7
0.6 0.6 o o 0.5 0.5 I(z) / I I(z) I(z) / I I(z) 0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 0 -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 z/λ z/λ
l = 8 l = 38/2
1 1
0.9 0.8
0.8 0.6
0.7 0.4
0.6 0.2 o o 0.5 0 I(z) / I I(z) / I I(z) 0.4 -0.2
0.3 -0.4
0.2 -0.6
0.1 -0.8
0 -1 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 z/λ z/λ The total real power radiated by the center-fed dipole is
The 2-dependent integral in the radiated power expression cannot be integrated analytically. However, the integral may be manipulated, using several transformations of variables, into a form containing some commonly encountered special functions (integrals) known as the sine integral and cosine integral.
The radiated power of the center-fed dipole becomes The radiated power is related to the radiation resistance of the antenna by
which gives
(Center-fed dipole radiation resistance)
The directivity function of the center-fed dipole is given by Center-fed dipole directivity function
The maximum directivity is
Center-fed dipole maximum directivity
The effective aperture is
Center-fed dipole effective aperture
Center-fed dipole Solid beam angle Half-Wave Dipole
Center-fed half-wave dipole far fields
Center-fed half-wave dipole radiation intensity function Center-fed half-wave dipole radiation resistance (in air)
Center-fed half-wave dipole directivity function
Center-fed half-wave dipole maximum directivity
Center-fed half-wave dipole effective aperture Dipole Input Impedance
The input impedance of the dipole is defined as the ratio of voltage to current at the antenna feed point.
The real and reactive time-average power delivered to the terminals of the antenna may be written as
If we assume that the antenna is lossless (RL = 0), then the real power delivered to the input terminals equals that radiated by the antenna. Thus, and the antenna input resistance is related to the antenna radiation resistance by
In a similar fashion, we may equate the reactive power delivered to the antenna input terminals to that stored in the near field of the antenna.
or
The general dipole current is defined by
N The current Iin is the current at the feed point of the dipole (z = 0) so that
The input resistance and reactance of the antenna are then related to the equivalent circuit values of radiation resistance and the antenna reactance by The dipole reactance may be determined in closed form using a technique known as the induced EMF method (Chapter 8) but requires that the radius of the wire (a) be included. The resulting dipole reactance is
(Center-fed dipole reactance)
The input resistance and reactance are plotted in Figure 8.16 (p.411) for a dipole of radius a = 10-58. If the dipole is 0.58 in length, the input impedance is found to be approximately (73 + j42.5) S. The first dipole resonance (Xin = 0) occurs when the dipole length is slightly less than one- half wavelength. The exact resonant length depends on the wire radius, but for wires that are electrically very thin, the resonant length of the dipole is approximately 0.488. As the wire radius increases, the resonant length decreases slightly [see Figure 8.17 (p.412)]. Antenna and Scatterers
All of the antennas considered thus far have been assumed to be radiating in a homogeneous medium of infinite extent. When an antenna radiates in the presence of a conductor(inhomogeneous medium), currents are induced on the conductor which re-radiate (scatter) additional fields. The total fields produced by an antenna in the presence of a scatterer are the superposition of the original radiated fields (incident fields, [E inc,H inc] those produced by the antenna in the absence of the scatterer) plus the fields produced by the currents induced on the scatterer (scattered fields, [E scat,H scat]).
To evaluate the total fields, we must first determine the scattered fields which depend on the currents flowing on the scatterer. The determination of the scatterer currents typically requires a numerical scheme (integral equation in terms of the scatterer currents or a differential equation in the form of a boundary value problem). However, for simple scatterer shapes, we may use image theory to simplify the problem. Image Theory
Given an antenna radiating over a perfect conducting ground plane, [perfect electric conductor (PEC), perfect magnetic conductor (PMC)] we may use image theory to formulate the total fields without ever having to determine the surface currents induced on the ground plane. Image theory is based on the electric or magnetic field boundary condition on the surface of the perfect conductor (the tangential electric field is zero on the surface of a PEC, the tangential magnetic field is zero on the surface of a PMC). Using image theory, the ground plane can be replaced by the equivalent image current located an equal distance below the ground plane. The original current and its image radiate in a homogeneous medium of infinite extent and we may use the corresponding homogeneous medium equations.
Example (vertical electric dipole) Currents over a PEC
Currents over a PMC Vertical Infinitesimal Dipole Over Ground
Give a vertical infinitesimal electric dipole (z-directed) located a distance h over a PEC ground plane, we may use image theory to determine the overall radiated fields.
The individual contributions to the electric field by the original dipole and its image are
In the far field, the lines defining r, r1 and r2 become almost parallel so that The previous expressions for r1 and r2 are necessary for the phase terms in the dipole electric field expressions. But, for amplitude terms, we may . . assume that r1 r2 r. The total field becomes
The normalized power pattern for the vertical infinitesimal dipole over a PEC ground is
h = 0.18 h = 0.258 h = 0.58 h = 8
h = 28 h = 108 Since the radiated fields of the infinitesimal dipole over ground are different from those of the isolated antenna, the basic parameters of the antenna are also different. The far fields of the infinitesimal dipole are
The time-average Poynting vector is
The corresponding radiation intensity function is
The maximum value of the radiation intensity function is found at 2 = B/2.
The radiated power is found by integrating the radiation intensity function. (Infinitesimal dipole over ground radiation resistance)
The directivity function of the infinitesimal dipole over ground is
so that the maximum directivity (at 2 = B/2) is given by
(Infinitesimal dipole over ground maximum directivity) Given an infinitesimal dipole of length )l = 8/50, we may plot the radiation resistance and maximum directivity as a function of the antenna height to see the effect of the ground plane.
0.8 8
0.7 7
0.6 6
0.5 5 ) Ω o
( 0.4 r 4 D R
0.3 3
0.2 2
0.1 1
0 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 λ h/ h/λ
For an isolated infinitesimal dipole of length )l = 8/50, the radiation resistance is
and the maximum directivity (independent of antenna length) is Do = 1.5. Note that Rr of the infinitesimal dipole over ground approaches twice that 6 of Rr for an isolated dipole as h 0 (see the relationship between a monopole antenna and its equivalent dipole antenna in the next section). As the height is increased, the radiation resistance of the infinitesimal dipole over ground approaches that of an isolated dipole. The directivity of the infinitesimal dipole over ground approaches a value twice that of the isolated dipole as h 6 0 and four times that of the isolated dipole as h grows large. This follows from our definition of the total radiated power and maximum directivity for the isolated antenna and the antenna over ground. First, we note the relationship between Umax for the isolated dipole and the dipole over ground.
Note that Umax for the antenna over ground is independent of the height of the antenna over ground. h 6 0
h 6 large Monopole
Using image theory, the monopole antenna over a PEC ground plane may be shown to be equivalent to a dipole antenna in a homogeneous region. The equivalent dipole is twice the length of the monopole and is driven with twice the antenna source voltage. These equivalent antennas generate the same fields in the region above the ground plane. The input impedance of the equivalent antennas is given by
The input impedance of the monopole is exactly one-half that of the equivalent dipole. Therefore, we may determine the monopole radiation resistance for monopoles of different lengths according to the results of the equivalent dipole.
Infinitesimal dipole [length = )l < 8/50]
Infinitesimal monopole [length = )l < 8/100]
Short dipole [length = l, (8/50 # l # 8/10)]
Short monopole [length = l, (8/100 # l # 8/20)]
Lossless half-wave dipole [length = l = 8/2]
Lossless quarter-wave monopole [length = l = 8/4] The total power radiated by the monopole is one-half that of the equivalent dipole. But, the monopole radiates into one-half the volume of the dipole yielding equivalent fields and power densities in the upper half space.
The directivities of the two equivalent antennas are related by
Infinitesimal dipole [length = )l < 8/50]
Infinitesimal monopole [length = )l < 8/100]
Lossless half-wave dipole [length = l = 8/2]
Lossless quarter-wave monopole [length = l = 8/4] Ground Effects on Antennas
At most frequencies, the conductivity of the earth is such that the ground may be accurately approximated by a PEC. Given an antenna located over a PEC ground plane, the radiated fields of the antenna over ground can be determined easily using image theory. The fields radiated by the antenna over a PEC ground excite currents on the surface of the ground plane which re-radiate (scatter) the incident waves from the antenna. We may also view the PEC ground plane as a perfect reflector of the incident EM waves. The direct wave/reflected wave interpretation of the image theory results for the infinitesimal dipole over a PEC ground is shown below.
~~~~~~~~~ ~~~~~~~~~~ direct wave reflected wave At lower frequencies (approximately 100 MHz and below), the electric fields associated with the incident wave may penetrate into the lossy ground, exciting currents in the ground which produce ohmic losses. These losses reduce the radiation efficiency of the antenna. They also effect the radiation pattern of the antenna since the incident waves are not perfectly reflected by the ground plane. Image theory can still be used for the lossy ground case, although the magnitude of the reflected wave must be reduced from that found in the PEC ground case. The strength of the image antenna in the lossy ground case can be found by multiplying the strength of the image antenna in the PEC ground case by the appropriate ' plane wave reflection coefficient for the proper polarization ( V). If we plot the radiation pattern of the vertical dipole over ground for cases of a PEC ground and a lossy ground, we find that the elevation plane pattern for the lossy ground case is tilted upward such that the radiation maximum does not occur on the ground plane but at some angle tilted upward from the ground plane (see Figure 4.28, p. 183). This alignment of the radiation maximum may or may not cause a problem depending on the application. However, if both the transmit and receive antennas are located close to a lossy ground, then a very inefficient system will result. The antenna over lossy ground can be made to behave more like an antenna over perfect ground by constructing a ground plane beneath the antenna. At low frequencies, a solid conducting sheet is impractical because of its size. However, a system of wires known as a radial ground system can significantly enhance the performance of the antenna over lossy ground.
Monopole with a radial ground system
The radial wires provide a return path for the currents produced within the lossy ground. Broadcast AM transmitting antennas typically use a radial ground system with 120 quarter wavelength radial wires (3o spacing). The reflection coefficient scheme can also be applied to horizontal antennas above a lossy ground plane. The proper reflection coefficient must be used based on the orientation of the electric field (parallel or perpendicular polarization).
The Effect of Earth Curvature
Antennas on spacecraft and aircraft in flight see the same effect that antennas located close to the ground experience except that the height of the antenna over the conducting ground means that the shape of the ground (curvature of the earth) can have a significant effect on the scattered field. In cases like these, the curvature of the reflecting ground must be accounted for to yield accurate values for the reflected waves.
Antennas in Wireless Communications
Wire antennas such as dipoles and monopoles are used extensively in wireless communications applications. The base stations in wireless communications are most often arrays (Ch. 6) of dipoles. Hand-held units such as cell phones typically use monopoles. Monopoles are simple, small, cheap, efficient, easy to match, omnidirectional (according to their orientation) and relatively broadband antennas. The equations for the performance of a monopole antenna presented in this chapter have assumed that the antenna is located over an infinite ground plane. The monopole on the hand-held unit is not driven relative to the earth ground but rather (a.) the conducting case of the unit or (b.) the circuit board of the unit. The resonant frequency and input impedance of the hand-held monopole are not greatly different than that of the monopole over a infinite ground plane. The pattern of the hand-held unit monopole is different than that of the monopole over an infinite ground plane due to the different distribution of currents. Other antennas used on hand-held units are loops (Ch. 5), microstrip (patch) antennas (Ch. 14) and the planar inverted F antenna (PIFA). In wireless applications, the antenna can be designed to perform in a typical scenario, but we cannot account for all scatterer geometries which we may encounter (power lines, buildings, etc.). Thus, the scattered signals from nearby conductors can have an adverse effect on the system performance. The detrimental effect of these unwanted scattered signals is commonly referred to as multipath. Loop Antennas
Loop antennas have the same desirable characteristics as dipoles and monopoles in that they are inexpensive and simple to construct. Loop antennas come in a variety of shapes (circular, rectangular, elliptical, etc.) but the fundamental characteristics of the loop antenna radiation pattern (far field) are largely independent of the loop shape. Just as the electrical length of the dipoles and monopoles effect the efficiency of these antennas, the electrical size of the loop (circumference) determines the efficiency of the loop antenna. Loop antennas are usually classified as either electrically small or electrically large based on the circumference of the loop.
electrically small loop Y circumference î 8/10
electrically large loop Y circumference . 8
The electrically small loop antenna is the dual antenna to the electrically short dipole antenna when oriented as shown below. That is, the far-field electric field of a small loop antenna is identical to the far-field magnetic field of the short dipole antenna and the far-field magnetic field of a small loop antenna is identical to the far-field electric field of the short dipole antenna. Given that the radiated fields of the short dipole and small loop antennas are dual quantities, the radiated power for both antennas is the same and therefore, the radiation patterns are the same. This means that the plane of maximum radiation for the loop is in the plane of the loop. When operated as a receiving antenna, we know that the short dipole must be oriented such that the electric field is parallel to the wire for maximum response. Using the concept of duality, we find that the small loop must be oriented such that the magnetic field is perpendicular to the loop for maximum response. The radiation resistance of the small loop is much smaller than that of the short dipole. The loss resistance of the small loop antenna is frequently much larger than the radiation resistance. Therefore, the small loop antenna is rarely used as a transmit antenna due to its extremely small radiation efficiency. However, the small loop antenna is acceptable as a receive antenna since signal-to-noise ratio is the driving factor, not antenna efficiency. The fact that a significant portion of the received signal is lost to heat is not of consequence as long as the antenna provides a large enough signal-to-noise ratio for the given receiver. Small loop antennas are frequently used for receiving applications such as pagers, low- frequency portable radios, and direction finding. Small loops can also be used at higher frequencies as field probes providing a voltage at the loop terminals which is proportional to the field passing through the loop. Electrically Small Loop Antenna
The far fields of an electrically small loop antenna are dependent on the loop area but are independent of the loop shape. Since the magnetic vector potential integrations required for a circular loop are more complex than those for a square loop, the square loop is considered in the derivation of the far fields of an electrically small loop antenna. The square loop, located in the x-y plane and centered at the coordinate origin, is assumed ) 2 to have an area of l and carry a uniform current Io.
The square loop may be viewed as four segments which each represent an infinitesimal dipole carrying current in a different direction. In the far field, the distance vectors from the centers of the four segments become almost parallel. As always in far field expressions, the above approximations are used in the phase terms of the magnetic vector potential, but we may assume that . . . . R1 R2 R3 R4 r for the magnitude terms. The far field magnetic vector potential of a z-directed infinitesimal dipole centered at the origin is
The individual far field magnetic vector potential contributions due to the four segments of the current loop are
Combining the x-directed and y-directed terms yields For an electrically small loop ()l << 8), the arguments of the sine functions above are very small and may be approximated according to which gives
The overall vector potential becomes
where )S = )l2 = loop area. The bracketed term above is the spherical coordinate unit vector aN.
Electrically small current loop far field magnetic vector potential The corresponding far fields are
Electrically small current loop far fields
The fields radiated by an electrically small loop antenna can be increased by adding multiple turns. For the far fields, the added height of multiple turns is immaterial and the resulting far fields for a multiple turn loop antenna can be found by simply multiplying the single turn loop antenna fields by the number of turns N.
Electrically small multiple turn current loop far fields Dual and Equivalent Sources (Electric and Magnetic Dipoles and Loops)
If we compare the far fields of the infinitesimal dipole and the electrically small current loop with electric and magnetic currents, we find pairs of equivalent sources and dual sources.
Infinitesimal electric dipole Small electric current loop
Using duality, we may determine the far fields of the corresponding magnetic geometries.
Electric source Magnetic source
Infinitesimal magnetic dipole Small magnetic current loop If, for the small electric current loop and the infinitesimal magnetic dipole, we choose then the far fields radiated by these two sources are identical (the small electric current loop and the infinitesimal magnetic dipole are equivalent sources).
Similarly, for the small magnetic current loop and the infinitesimal electric dipole, if we choose then the far fields radiated by these two sources are identical (the small magnetic current loop and the infinitesimal electric dipole are equivalent sources). The infinitesimal electric and magnetic dipoles are defined as dual sources since the magnetic field of one is identical to the electric field of the other when the currents and dimensions are chosen appropriately. Likewise, the small electric and magnetic current loops are dual sources. We also find from this discussion of dual and equivalent sources that the polarization of the far fields for the dual sources are orthogonal. In the plane of maximum radiation (x-y plane), the four sources have the following far field polarizations infinitesimal electric dipole Y vertical polarization infinitesimal magnetic dipole Y horizontal polarization small electric current loop Y horizontal polarization small magnetic current loop Y vertical polarization Loop Antenna Characteristics
The time-average Poynting vector in the far field of the multiple-turn electrically small loop is
The radiation intensity function is
Loop antenna radiation intensity function
The maximum value of the radiation intensity function is
The radiated power is
Loop antenna radiated power The radiation resistance of the loop antenna is found from the radiated power.
Loop antenna in air radiation resistance
The directivity of the loop antenna is defined by
Loop antenna directivity function
Given the same directivity function as the infinitesimal dipole, the loop antenna has the same maximum directivity, effective aperture and beam solid angle as the infinitesimal dipole.
Loop antenna maximum directivity, effective aperture, and beam solid angle If we compare the radiation resistances of the electrically short dipole and the electrically small loop (both antennas in air), we find that the radiation resistance of the small loop decreases much faster than that of the short dipole with decreasing frequency since
8!2 Rr (short dipole) ~
8!4 Rr (small loop) ~
The radiation resistance of the small loop can be increased significantly by 2 adding multiple turns (Rr ~ N ). However, the addition of more conductor length also increases the antenna loss resistance which reduces the overall antenna efficiency. To increase the radiation resistance without significantly reducing the antenna efficiency, the number of turns can be decreased when a ferrite material is used as the core of the winding. The general radiation resistance formula for a small loop with any material as its core is
A multiturn loop which is wound on a linear ferrite core is commonly referred to as a loop-stick antenna. The loop-stick antenna is commonly used as a low-frequency receiving antenna.
Loop-stick antenna Impedance of Electrically Small Antennas
The current density was assumed to be uniform on the electrically small current loop for our far field calculations. For a circular loop, the assumption of uniform current is accurate up to a loop circumference of about 0.28.
b = loop radius
a = wire radius
The restriction on the size of the constant current loop in terms of the loop radius is
The electrically small current loop was found to be a dual source to the infinitesimal dipole. If we investigate the reactance of these dual electrically small antennas, we find that the dipole is capacitive while the loop is inductive. The exact reactance of the current loop is dependent on the shape of the loop. Approximate formulas for the reactance are given below for a short dipole and an electrically small circular current loop.
Infinitesimal Dipole (length = )l, wire radius = a) Electrically Small Circular Current Loop (loop radius = b, wire radius = a)
Example (Impedances of electrically small antennas)
Determine the total impedance and radiation efficiency of the following electrically small antennas operating at 1, 10 and 100 MHz. Both antennas are constructed using #10 AWG copper wire (a = 2.59 mm, F = 5.8 × 107 ®/m). Infinitesimal Dipole
) f (MHz) lRr RL ecd jXA 1 0.00028 31.6 :S 0.962 mS 3.18 % !204 kS 10 0.0028 3.16 mS 3.04 mS 51.0 % !20.4 kS 100 0.028 0.316 S 9.62 mS 97.0 % !2.04 kS
Small Loop
f (MHz) bRr RL ecd jXA 1 0.000328 3.09 nS 9.57 mS 3.2×10-5 % 2.76 S 10 0.00328 30.9 :S 30.3 mS 0.102 % 27.6 S 100 0.0328 0.309 S 95.7 mS 76.4% 276 S Antenna Arrays
Antennas with a given radiation pattern may be arranged in a pattern (line, circle, plane, etc.) to yield a different radiation pattern.
Antenna array - a configuration of multiple antennas (elements) arranged to achieve a given radiation pattern.
Linear array - antenna elements arranged along a straight line.
Circular array - antenna elements arranged around a circular ring.
Planar array - antenna elements arranged over some planar surface (example - rectangular array).
Conformal array - antenna elements arranged to conform to some non-planar surface (such as an aircraft skin).
There are several array design variables which can be changed to achieve the overall array pattern design.
Array Design Variables
1. General array shape (linear, circular, planar, etc.). 2. Element spacing. 3. Element excitation amplitude. 4. Element excitation phase. 5. Patterns of array elements.
Phased array - an array of identical elements which achieves a given pattern through the control of the element excitation phasing. Phased arrays can be used to steer the main beam of the antenna without physically moving the antenna. Given an antenna array of identical elements, the radiation pattern of the antenna array may be found according to the pattern multiplication theorem.
Pattern multiplication theorem
Array element pattern - the pattern of the individual array element. Array factor - a function dependent only on the geometry of the array and the excitation (amplitude, phase) of the elements.
Example (Pattern multiplication - infinitesimal dipole over ground)
The far field of this two element array was found using image theory to be
«®®®®®®®®®®®®®®®®®®®®®®¬ «®®®®®®®®®®®®®®®¬ element pattern array factor N-Element Linear Array
The array factor AF is independent of the antenna type assuming all of the elements are identical. Thus, isotropic radiators may be utilized in the derivation of the array factor to simplify the algebra. The field of an isotropic radiator located at the origin may be written as (assuming 2- polarization)
We assume that the elements of the array are uniformly-spaced with a separation distance d.
In the far field of the array
The current magnitudes the array elements are assumed to be equal and the current on the array element located at the origin is used as the phase reference (zero phase). The far fields of the individual array elements are
The overall array far field is found using superposition.
(Array factor for a uniformly-spaced N-element linear array) Uniform N-Element Linear Array (uniform spacing, uniform amplitude, linear phase progression)
A uniform array is defined by uniformly-spaced identical elements of equal magnitude with a linearly progressive phase from element to element.
Inserting this linear phase progression into the formula for the general N- element array gives
The function R is defined as the array phase function and is a function of the element spacing, phase shift, frequency and elevation angle. If the array factor is multiplied by e jR, the result is
Subtracting the array factor from the equation above gives
The complex exponential term in the last expression of the above equation represents the phase shift of the array phase center relative to the origin. If the position of the array is shifted so that the center of the array is located at the origin, this phase term goes away. The array factor then becomes
Below are plots of the array factor AF vs. the array phase function R as the number of elements in the array is increased. Note that these are not plots of AF vs. the elevation angle 2.
Some general characteristics of the array factor AF with respect to R: R (1) [AF ]max = N at = 0 (main lobe). (2) Total number of lobes = N!1 (one main lobe, N!2 sidelobes). (3) Main lobe width = 4B/N, minor lobe width = 2B/N The array factor may be normalized so that the maximum value for any value of N is unity. The normalized array factor is
The nulls of the array function are found by determining the zeros of the numerator term where the denominator is not simultaneously zero.
The peaks of the array function are found by determining the zeros of the numerator term where the denominator is simultaneously zero.
The m = 0 term,
represents the angle which makes R = 0 (main lobe). Broadside and End-fire Arrays
The phasing of the uniform linear array elements may be chosen such that the main lobe of the array pattern lies along the array axis (end-fire array) or normal to the array axis (broadside array).
End-fire array main lobe at 2 = 0o or 2 = 180o Broadside array main lobe at 2 = 90o
The maximum of the array factor occurs when the array phase function is zero.
For a broadside array, in order for the above equation to be satisfied with 2 = 90o, the phase angle " must be zero. In other words, all elements of the array must be driven with the same phase. With " = 0o, the normalized array factor reduces to
Normalized array function Broadside array, " = 0o
Consider a 5-element broadside array (" = 0o) as the element spacing is varied. In general, as the element spacing is increased, the main lobe beamwidth is decreased. However, grating lobes (maxima in directions other than the main lobe direction) are introduced when the element spacing is greater than or equal to one wavelength. If the array pattern design requires that no grating lobes be present, then the array element spacing should be chosen to be less than one wavelength.
If we consider the broadside array factor as a function of the number of array elements, we find that, in general, the main beam is sharpened as the number of elements increases. Below are plots of AF for a broadside array (" = 0o) with elements separated by d = 0.258 for N = 2, 5, 10 and 20. Using the pattern multiplication theorem, the overall array pattern is obtained by multiplying the element pattern by the array factor. As an example, consider an broadside array (" = 0o) of seven short vertical dipoles spaced 0.58 apart along the z-axis.
The normalized element field pattern for the infinitesimal dipole is
The array factor for the seven element array is
The overall normalized array pattern is 90 1 90 1 120 60 120 60 0.8
0.6 150 30 150 0.5 30 0.4
0.2
180 0 180 0
element pattern array factor
210 330 210 330
240 300 240 300 270 270 If we consider the same array with horizontal (x-directed) short dipoles, the resulting normalized element field pattern is
Since the element pattern depends on the angle N, we must choose a value of N to plot the pattern. If we choose N = 0o, the element pattern becomes
and the array pattern is given by
If we plot the array pattern for N = 90o, we find that the element pattern is unity and the array pattern is the same as the array factor. Thus, the main beam of the array of x-directed short dipoles lies along the y-axis. The nulls of the array element pattern along the x-axis prevent the array from radiating efficiently in that broadside direction. End-fire arrays may be designed to focus the main beam of the array factor along the array axis in either the 2=0o or 2=180o directions. Given that the maximum of the array factor occurs when in order for the above equation to be satisfied with 2 = 0o, the phase angle " must be
For 2 = 180o, the phase angle " must be which gives
The normalized array factor for an end-fire array reduces to
Normalized array function
Consider a 5-element end-fire array (2 = 0o) as the element spacing is varied. Note that the phase angle " must change as the spacing changes in order to keep the main beam of the array function in the same direction. If the corresponding positive phase angles are chosen, the array factor plots are mirror images of the above plots (about 2 = 90o ). Note that the end- fire array grating lobes are introduced for element spacings of d $ 0.58. 7-element array end-fire array, vertical short dipoles (d = 0.258, " = !90o)
The normalized array factor for the 7-element end-fire array is
The overall array field pattern is 90 90 1 1 120 60 120 60 0.8 0.8
0.6 0.6 150 30 150 30 0.4 0.4
0.2 0.2
180 0 180 0
element pattern array factor
210 330 210 330
240 300 240 300 270 270
90 0.5 120 60 0.4
0.3 150 30 0.2
0.1
180 0
array pattern
210 330
240 300 270
7-element end-fire array, x-directed horizontal short dipoles (d = 0.258, " = !90o)
The overall array pattern in the N = 0o plane is 90 90 1 1 120 60 120 60 0.8 0.8
0.6 0.6 150 30 150 30 0.4 0.4
0.2 0.2
180 0 180 0
element pattern array factor
210 330 210 330
240 300 240 300 270 270 90 1 120 60 0.8
0.6 150 30 0.4
0.2
180 0
array pattern
210 330
240 300 270 Hansen-Woodyard End-fire Array
The Hansen-Woodyard end-fire array is a special array designed for maximum directivity.
Ordinary end-fire array Y " = ±kd
Hansen-Woodyard end-fire array Y " = ± (kd + *)
In order to increase the directivity in a closely-spaced electrically long end- fire array, Hansen and Woodyard analyzed the patterns and found that a additional phase shift of
increased the directivity of the array over that of the ordinary end-fire array given an element spacing of
For very long arrays (N - large), the element spacing in the Hansen- Woodyard end-fire array approaches one-quarter wavelength. The Hansen- Woodyard design shown here does not necessarily produce the maximum directivity for a given linear array but does produce a directivity larger than that of the ordinary end-fire array [by a factor of approximately 1.79 (2.5 dB)]. The Hansen-Woodyard end-fire array design can be summarized as
where the upper sign produces a maximum in the 2 = 0o direction and the lower sign produces a maximum in the 2 = 180o direction. The Hansen- Woodyard end-fire design increases the directivity of the array at the expense of higher sidelobe levels. Non-Uniformly Excited, Equally-Spaced Arrays
Given a two element array with equal current amplitudes and spacing, the array factor is
For a broadside array (" = 0o) with element spacing d less than one-half wavelength, the array factor has no sidelobes. An array formed by taking the product of two arrays of this type gives
This array factor, being the square of an array factor with no sidelobes, also has no sidelobes. Mathematically, the array factor above represents a 3- element equally-spaced array driven by current amplitudes with ratios of 1:2:1. In a similar fashion, equivalent arrays with more elements may be formed.
The current coefficients of the resulting N-element array take the form of a binomial series. The array is known as a binomial array.
Binomial array The excitation coefficients for the binomial array are given by Pascal’s triangle.
The binomial array has the special property that the array factor has no sidelobes for element spacings of 8/2 or less. Sidelobes are introduced for element spacings larger than 8/2.
N = 5, d = 0.58 N = 10, d = 0.58 Array Factor - Uniform Spacing, Nonuniform Amplitude
Consider an array of isotropic elements positioned symmetrically along the z-axis (total number of elements = P). The array factor for this array will be determined assuming that all elements are excited with the same current phase (N = 0o for simplicity) but nonuniform current amplitudes. The amplitude distribution assumed to be symmetric about the origin.
P = 2M + 1 (Odd) P = 2M (Even) P = 2M + 1 (Odd) P = 2M (Even) P = 2M + 1 (Odd)
where
P = 2M (Even)
Note that the array factors are coefficients multiplied by cosines with arguments that are integer multiples of u. Using trigonometric identities, these cosine functions can be written as powers of u. Through the transformation of x = cos u, the terms may be written as a set of polynomials [Chebyshev polynomials - Tn(x)].
Using properties of the Chebyshev polynomials, we may design arrays with specific sidelobe characteristics. Namely, we may design arrays with all sidelobes at some prescribed level. Chebyshev Polynomials
Properties of Chebyshev Polynomials
1. Even ordered Chebyshev polynomials are even functions. 2. Odd ordered Chebyshev polynomials are odd functions. 3. The magnitude of any Chebyshev polynomial is unity or less in the range of !1 # x #1.
4. Tn (1) = 1 for all Chebyshev polynomials. 5. All zeros (roots) of the Chebshev polynomials lie within the range of !1 # x #1.
Using the properties of Chebyshev polynomials, we may design arrays with all sidelobes at a prescribed level below the main beam (Dolph-Chebyshev array). The order of the Chebyshev polynomial should be one less than the total number of elements in the array (P!1). Dolph-Chebyshev Array Design Procedure
(1.) Select the appropriate AF for the total number of elements (P).
(2.) Replace each cos(mu) term in the array factor by its expansion in terms of powers of cos(u).
(3.) For the required main lobe to side lobe ratio (Ro), find xo such that
(4.) Substitute cos(u) = x/xo into the array factor of step 2. This substitution normalizes the array factor sidelobes to a peak of unity.
(5.) Equate the array factor of step 4 to TP-1(x) and determine the array coefficients.
Example
Design a 5-element Dolph-Chebyshev array with d = 0.58 and sidelobes which are 20 dB below the main beam.
(1.) P = 5, M = 2 (2.)
(3.)
(4.)
(5.) Equate coefficients and solve for a1, a2, and a3. Folded Dipole
A folded dipole is formed by connecting two parallel dipoles of radius a and length l at the ends to form a narrow loop. The center-to- center separation of the parallel wires is s. The separation distance s is always assumed to be small relative to wavelength.
The input impedance of the folded dipole is defined (as is any other antenna) by the ratio of voltage to current at the antenna feed point.
The folded dipole operates as an unbalanced transmission line. The current on the folded dipole can be decomposed into two distinct modes: an antenna mode (currents flowing in the same direction yielding significant radiation) and a transmission line mode (currents flowing in opposite directions yielding little radiation). Transmission line mode Antenna mode
Note that the superposition of the two modes yields the folded dipole input voltage V on the left wire and zero on the right wire. The transmission line current It in both antenna conductors must be the same in order to satisfy Kirchoff’s current law at the ends of the antenna. The total antenna current
Ia must be split equally between the two antenna conductors to yield the proper results for the radiated fields (the folded dipole radiates like two closely spaced dipoles). The total folded dipole input current can then be defined as the sum of the transmission line and antenna currents such that
so that the folded dipole input impedance may be written as
The folded dipole impedance is determined by relating the transmission line and antenna mode currents to the corresponding input voltage. We may insert an equivalent set of voltage sources into the transmission line mode problem in order to view the folded dipole as a set of two shorted transmission lines of length l/2. Note that both of the shorted transmission lines are driven with a source voltage of V/2 across its input terminals. The voltage and current for the transmission lines are related by
where Zt is the input impedance of a shorted two-wire line of length l/2 with wire of radii a with a center-to-center spacing of s. The general equation for the input impedance of a transmission line of characteristic impedance Zo and length l terminated with an load impedance
ZL is
For the shorted line, ZL = 0 and the length is l/2 so that
The characteristic impedance of the two wire line transmission line is The folded dipole antenna current can be related to an equivalent dipole (treating the parallel currents as coincident for far field purposes) by
where Zd is the input impedance of a dipole of length l and equivalent radius ae. The equivalent radius is necessary because of the close proximity of the two wires (capacitance) which alters the current distribution from that seen on an isolated dipole. The equivalent radius is given by
The impedance Zd is given by
Given the relationships between the transmission line and antenna mode currents and voltages, the input impedance of the folded dipole can be written as
For the special case of a folded dipole of length l = 8/2, the input impedance of the equivalent transmission line is that of a shorted quarter- wavelength transmission line (open-circuit).
The impedance of the half-wave folded dipole becomes
The half-wave folded dipole can be made resonant with an impedance of approximately 300 S which matches a common transmission line impedance (twin-lead). Thus, the half-wave folded dipole can be connected directly to a twin-lead line without any matching network necessary. In general, the folded dipole has a larger bandwidth than a dipole of the same size. Traveling Wave Antennas
Antennas with open-ended wires where the current must go to zero (dipoles, monopoles, etc.) can be characterized as standing wave antennas or resonant antennas. The current on these antennas can be written as a sum of waves traveling in opposite directions (waves which travel toward the end of the wire and are reflected in the opposite direction). For example, the current on a dipole of length l is given by
The current on the upper arm of the dipole can be written as
«¬ «¬ +z directed !z directed wave wave
Traveling wave antennas are characterized by matched terminations (not open circuits) so that the current is defined in terms of waves traveling in only one direction (a complex exponential as opposed to a sine or cosine). A traveling wave antenna can be formed by a single wire transmission line (single wire over ground) which is terminated with a matched load (no reflection). Typically, the length of the transmission line is several wavelengths.
The antenna shown above is commonly called a Beverage or wave antenna. This antenna can be analyzed as a rectangular loop, according to image theory. However, the effects of an imperfect ground may be significant and can be included using the reflection coefficient approach. The contribution to the far fields due to the vertical conductors is typically neglected since it is small if l >> h. Note that the antenna does not radiate efficiently if the height h is small relative to wavelength. In an alternative technique of analyzing this antenna, the far field produced by a long isolated wire of length l can be determined and the overall far field found using the 2 element array factor. Traveling wave antennas are commonly formed using wire segments with different geometries. Therefore, the antenna far field can be obtained by superposition using the far fields of the individual segments. Thus, the radiation characteristics of a long straight segment of wire carrying a traveling wave type of current are necessary to analyze the typical traveling wave antenna. Consider a segment of a traveling wave antenna (an electrically long wire of length l lying along the z-axis) as shown below. A traveling wave current flows in the z-direction.
" - attenuation constant
$ - phase constant
If the losses for the antenna are negligible (ohmic loss in the conductors, loss due to imperfect ground, etc.), then the current can be written as
The far field vector potential is If we let , then
The far fields in terms of the far field vector potential are
(Far-field of a traveling wave segment) We know that the phase constant of a transmission line wave (guided wave) can be very different than that of an unbounded medium (unguided wave). However, for a traveling wave antenna, the electrical height of the conductor above ground is typically large and the phase constant approaches that of an unbounded medium (k). If we assume that the phase constant of the traveling wave antenna is the same as an unbounded medium ($ = k), then
Given the far field of the traveling wave segment, we may determine the time-average radiated power density according to the definition of the Poynting vector such that The total power radiated by the traveling wave segment is found by integrating the Poynting vector.
and the radiation resistance is
The radiation resistance of the ideal traveling wave antenna (VSWR = 1) is purely real just as the input impedance of a matched transmission line is purely real. Below is a plot of the radiation resistance of the traveling wave segment as a function of segment length.
The radiation resistance of the traveling wave antenna is much more uniform than that seen in resonant antennas. Thus, the traveling wave antenna is classified as a broadband antenna. The pattern function of the traveling wave antenna segment is given by
The normalized pattern function can be written as
The normalized pattern function of the traveling wave segment is shown below for segment lengths of 58, 108, 158 and 208.
l = 58 l = 108 l = 158 l = 208
As the electrical length of the traveling wave segment increases, the main beam becomes slightly sharper while the angle of the main beam moves slightly toward the axis of the antenna. Note that the pattern function of the traveling wave segment always has a null at 2 = 0o. Also note that with l >> 8, the sine function in the normalized pattern function varies much more rapidly (more peaks and nulls) than the cotangent function. The approximate angle of the main lobe for the traveling wave segment is found by determining the first peak of the sine function in the normalized pattern function. o#2 # o The values of m which yield 0 m 180 (visible region) are negative 2 values of m. The smallest value of m in the visible region defines the location of main beam (m = !1)
If we also account for the cotangent function in the determination of the main beam angle, we find The directivity of the traveling wave segment is
The maximum directivity can be approximated by
where the sine term in the numerator of the directivity function is assumed to be unity at the main beam. Traveling Wave Antenna Terminations
Given a traveling wave antenna segment located horizontally above a ground plane, the termination RL required to match the uniform transmission line formed by the cylindrical conductor over ground (radius = a, height over ground = s/2) is the characteristic impedance of the corresponding one-wire transmission line. If the conductor height above the ground plane varies with position, the conductor and the ground plane form a non-uniform transmission line. The characteristic impedance of a non-uniform transmission line is a function of position. In either case, image theory may be employed to determine the overall performance characteristics of the traveling wave antenna.
Two-wire transmission line
If s >> a, then
In air, One-wire transmission line
If s >> a, then
In air, Vee Traveling Wave Antenna
The main beam of a single electrically long wire guiding waves in one direction (traveling wave segment) was found to be inclined at an angle relative to the axis of the wire. Traveling wave antennas are typically formed by multiple traveling wave segments. These traveling wave segments can be oriented such that the main beams of the component wires combine to enhance the directivity of the overall antenna. A vee traveling wave antenna is formed by connecting two matched traveling wave 2 segments to the end of a transmission line feed at an angle of 2 o relative to each other.
The beam angle of a traveling wave segment relative to the axis of the wire 2 ( max) has been shown to be dependent on the length of the wire. Given the 2 length of the wires in the vee traveling wave antenna, the angle 2 o may be chosen such that the main beams of the two tilted wires combine to form an antenna with increased directivity over that of a single wire. A complete analysis which takes into account the spatial separation effects of the antenna arms (the two wires are not co-located) reveals that by 2 . 2 choosing o 0.8 max, the total directivity of the vee traveling wave antenna is approximately twice that of a single conductor. Note that the overall pattern of the vee antenna is essentially unidirectional given matched conductors. If, on the other hand, the conductors of the vee traveling wave antenna are resonant conductors (vee dipole antenna), there are reflected waves which produce significant beams in the opposite direction. Thus, traveling wave antennas, in general, have the advantage of essentially unidirectional patterns when compared to the patterns of most resonant antennas. Rhombic Antenna
A rhombic antenna is formed by connecting two vee traveling wave antennas at their open ends. The antenna feed is located at one end of the rhombus and a matched termination is located at the opposite end. As with all traveling wave antennas, we assume that the reflections from the load are negligible. Typically, all four conductors of the rhombic antenna are assumed to be the same length. Note that the rhombic antenna is an example of a non-uniform transmission line.
A rhombic antenna can also be constructed using an inverted vee antenna over a ground plane. The termination resistance is one-half that required for the isolated rhombic antenna. To produce an single antenna main lobe along the axis of the rhombic antenna, the individual conductors of the rhombic antenna should be aligned such that the components lobes numbered 2, 3, 5 and 8 are aligned (accounting for spatial separation effects). Beam pairs (1, 7) and (4,6) combine to form significant sidelobes but at a level smaller than the main lobe. Yagi-Uda Array
In the previous examples of array design, all of the elements in the array were assumed to be driven with some source. A Yagi-Uda array is an example of a parasitic array. Any element in an array which is not connected to the source (in the case of a transmitting antenna) or the receiver (in the case of a receiving antenna) is defined as a parasitic element. A parasitic array is any array which employs parasitic elements. The general form of the N-element Yagi-Uda array is shown below.
Driven element - usually a resonant dipole or folded dipole.
Reflector - slightly longer than the driven element so that it is inductive (its current lags that of the driven element).
Director - slightly shorter than the driven element so that it is capacitive (its current leads that of the driven element). Yagi-Uda Array Advantages
! Lightweight ! Low cost ! Simple construction ! Unidirectional beam (front-to-back ratio) ! Increased directivity over other simple wire antennas ! Practical for use at HF (3-30 MHz), VHF (30-300 MHz), and UHF (300 MHz - 3 GHz)
Typical Yagi-Uda Array Parameters
Driven element ! half-wave resonant dipole or folded dipole, (Length = 0.458 to 0.498, dependent on radius), folded dipoles are employed as driven elements to increase the array input impedance.
Director ! Length = 0.48 to 0.458 (approximately 10 to 20 % shorter than the driven element), not necessarily uniform.
Reflector ! Length . 0.58 (approximately 5 to 10 % longer than the driven element).
Director spacing ! approximately 0.2 to 0.48, not necessarily uniform.
Reflector spacing ! 0.1 to 0.258 Example (Yagi-Uda Array)
Given a simple 3-element Yagi-Uda array (one reflector - length = 0.58, one director - length = 0.458, driven element - length = 0.4758) 8 where all the elements are the same radius (a = 0.005 ). For sR = sD = 0.18, 0.28 and 0.38, determine the E-plane and H-plane patterns, the 3dB beamwidths in the E- and H-planes, the front-to-back ratios (dB) in the E- and H-planes, and the maximum directivity (dB). Also, plot the currents along the elements in each case. Use the FORTRAN program provided with the textbook (yagi-uda.for). Use 8 modes per element in the method of moments solution.
The individual element currents given as outputs of the FORTRAN code are all normalized to the current at the feed point of the antenna. 8 sR = sD = 0.1
8 sR = sD = 0.2
8 sR = sD = 0.3
8 sR = sD = 0.1
3-dB beamwidth E-Plane = 62.71o 3-dB beamwidth H-Plane = 86.15o Front-to-back ratio E-Plane = 15.8606 dB Front-to-back-ratio H-Plane = 15.8558 dB Maximum directivity = 7.784 dB
8 sR = sD = 0.2
3-dB beamwidth E-Plane = 55.84o 3-dB beamwidth H-Plane = 69.50o Front-to-back ratio E-Plane = 9.2044 dB Front-to-back-ratio H-Plane = 9.1993 dB Maximum directivity = 9.094 dB
8 sR = sD = 0.3
3-dB beamwidth E-Plane = 51.89o 3-dB beamwidth H-Plane = 61.71o Front-to-back ratio E-Plane = 5.4930 dB Front-to-back-ratio H-Plane = 5.4883 dB Maximum directivity = 8.973 dB Example 15-element Yagi-Uda Array (13 directors, 1 reflector, 1 driven element) reflector length = 0.58 reflector spacing = 0.258 director lengths = 0.4068 director spacing = 0.348 driven element length = 0.478 conductor radii = 0.0038 3-dB beamwidth E-Plane = 26.79o 3-dB beamwidth H-Plane = 27.74o Front-to-back ratio E-Plane = 36.4422 dB Front-to-back-ratio H-Plane = 36.3741 dB Maximum directivity = 14.700 dB Log-Periodic Antenna
A log-periodic antenna is classified as a frequency-independent antenna. No antenna is truly frequency-independent but antennas capable of bandwidth ratios of 10:1 ( fmax : fmin ) or more are normally classified as frequency-independent.
The elements of the log periodic dipole are bounded by a wedge of angle 2". The element spacing is defined in terms of a scale factor J such that
(1) where J < 1. Using similar triangles, the angle " is related to the element lengths and positions according to
(2) or
(3)
Combining equations (1) and (3), we find that the ratio of adjacent element lengths and the ratio of adjacent element positions are both equal to the scale factor.
(4)
The spacing factor F of the log periodic dipole is defined by
where dn is the distance from element n to element n+1 .
(5)
From (2), we may write
(6)
Inserting (6) into (5) yields (7)
Combining equation (3) with equation (7) gives
(8) or
(9)
According to equation (8), the ratio of element spacing to element length remains constant for all of the elements in the array.
(10)
Combining equations (3) and (10) shows that z-coordinates, the element lengths, and the element separation distances all follow the same ratio.
(11)
Log Periodic Dipole Design
We may solve equation (9) for the array angle " to obtain an equation for " in terms of the scale factor J and the spacing factor F.
Figure 11.13 (p. 561) gives the spacing factor as a function of the scale factor for a given maximum directivity Do. The designed bandwidth Bs is given by the following empirical equation.
The overall length of the array from the shortest element to the longest element (L) is given by
where
The total number of elements in the array is given by
Operation of the Log Periodic Dipole Antenna
The log periodic dipole antenna basically behaves like a Yagi-Uda array over a wide frequency range. As the frequency varies, the active set of elements for the log periodic antenna (those elements which carry the significant current) moves from the long-element end at low frequency to the short-element end at high frequency. The director element current in the Yagi array lags that of the driven element while the reflector element current leads that of the driven element. This current distribution in the Yagi array points the main beam in the direction of the director. In order to obtain the same phasing in the log periodic antenna with all of the elements in parallel, the source would have to be located on the long-element end of the array. However, at frequencies where the smallest elements are resonant at 8/2, there may be longer elements which are also resonant at lengths of n8/2. Thus, as the power flows from the long- element end of the array, it would be radiated by these long resonant elements before it arrives at the short end of the antenna. For this reason, the log periodic dipole array must be driven from the short element end. But this arrangement gives the exact opposite phasing required to point the beam in the direction of the shorter elements. It can be shown that by alternating the connections from element to element, the phasing of the log periodic dipole elements points the beam in the proper direction.
Sometimes, the log periodic antenna is terminated on the long- element end of the antenna with a transmission line and load. This is done to prevent any energy that reaches the long-element end of the antenna from being reflected back toward the short-element end. For the ideal log periodic array, not only should the element lengths and positions follow the scale factor J, but the element feed gaps and radii should also follow the scale factor. In practice, the feed gaps are typically kept constant at a constant spacing. If different radii elements are used, two or three different radii are used over portions of the antenna. Example
Design a log periodic dipole antenna to cover the complete VHF TV band from 54 to 216 MHz with a directivity of 8 dB. Assume that the input impedance is 50 S and the length to diameter ratio of the elements is 145.
From Figure 11.13, with Do = 8 dB, the optimum value for the spacing factor F is 0.157 while the corresponding scale factor J is 0.865. The angle of the array is
The computer program “log-perd.for” performs an analysis of the log periodic dipole based on the previously defined design equations. Please see Log-Perd.DOC for information about these parameters 1 Design Title 2 Upper Design Frequency (MHz) 236.20000 MHz 3 Lower Design Frequency (MHz) 33.70000 MHz 4 Tau, Sigma and Directivity Choices... Directivity: 8.00000 dBi 5 Length to Diameter Ratio 145.00000 6 Source Resistance .00000 Ohms 7 Length of Source Transmission Line .00000 m 8 Impedance of Source Transmission Line 50.00000 +j0 Ohms 9 Boom Spacing Choices... Boom Diameter : 1.90000 cm Desired Input Impedance : 45.00000 Ohms 10 Length of Termination Transmission Line .00000 m 11 Termination Impedance 50.00000 +j0 Ohms 12 Tube Quantization Choices... 13 Design Summary and Analysis Choices... Design Summary : E- and H-plane Patterns : Custom Plane Patterns : Swept Frequency Analysis : 14 Begin Design and Analysis : Please enter a line number or enter 15 to save and exit. DIPOLE ARRAY DESIGN
Ele. Z L D (m) (m) (cm) Term. .8861 ******* ******* 1 .8861 .3780 .26066 2 1.0243 .4369 .30134 3 1.1842 .5051 .34837 4 1.3690 .5840 .40273 5 1.5827 .6751 .46559 6 1.8297 .7805 .53825 7 2.1153 .9023 .62226 8 2.4454 1.0431 .71937 9 2.8271 1.2059 .83164 10 3.2683 1.3941 .96144 11 3.7784 1.6117 1.11149 12 4.3680 1.8632 1.28496 13 5.0498 2.1540 1.48550 14 5.8379 2.4901 1.71734 15 6.7490 2.8788 1.98537 16 7.8023 3.3281 2.29522 17 9.0200 3.8475 2.65344 18 10.4277 4.4480 3.06756 Source 10.4277 ******* *******
Design Parameters Upper Design Frequency (MHz) : 236.20000 Lower Design Frequency (MHz) : 33.70000 Tau : .86500 Sigma : .15825 Alpha (deg) : 12.03942 Desired Directivity : 8.00000
Source and Source Transmission Line Source Resistance (Ohms) : .00000 Transmission Line Length (m) : .00000 Characteristic Impedance (Ohms) : 50.00000 + j .00000
Antenna and Antenna Transmission Line Length-to-Diameter Ratio : 145.00000 Boom Diameter (cm) : 1.90000 Boom Spacing (cm) : 2.07954 Characteristic impedance (Ohms) : 51.76521 + j .00000 Desired Input Impedance (Ohms) : 45.00000
Termination and Termination Transmission Line Termination impedance (Ohms) : 50.00000 + j .00000 Transmission Line Length (m) : .00000 Characteristic impedance (Ohms) : 51.76521 + j .00000
DIPOLE ARRAY DESIGN
Ele. Z L D (m) (m) (cm) Term. .9877 ******* ******* 1 .9877 .4213 .29056 2 1.1419 .4871 .33591 3 1.3201 .5631 .38833 4 1.5261 .6510 .44894 5 1.7643 .7526 .51901 6 2.0396 .8700 .60001 7 2.3580 1.0058 .69365 8 2.7260 1.1628 .80191 9 3.1514 1.3442 .92706 10 3.6433 1.5540 1.07175 11 4.2119 1.7966 1.23902 12 4.8692 2.0770 1.43239 13 5.6291 2.4011 1.65594 14 6.5077 2.7759 1.91438 Source 6.5077 ******* *******
Design Parameters Upper Design Frequency (MHz) : 216.00000 Lower Design Frequency (MHz) : 54.00000 Tau : .86500 Sigma : .15825 Alpha (deg) : 12.03942 Desired Directivity : 8.00000 Source and Source Transmission Line Source Resistance (Ohms) : .00000 Transmission Line Length (m) : .00000 Characteristic Impedance (Ohms) : 50.00000 + j .00000
Antenna and Antenna Transmission Line Length-to-Diameter Ratio : 145.00000 Boom Diameter (cm) : 1.90000 Boom Spacing (cm) : 2.07954 Characteristic impedance (Ohms) : 51.76521 + j .00000 Desired Input Impedance (Ohms) : 45.00000
Termination and Termination Transmission Line Termination impedance (Ohms) : 50.00000 + j .00000 Transmission Line Length (m) : .00000 Characteristic impedance (Ohms) : 51.76521 + j .00000
90 5 120 60 4
3 150 30 2
1
180 0
210 E-Plane, f = 54 MHz 330
240 300 270
90 5 120 60 4
3 150 30 2
1
180 0
210 330 H-Plane, f = 54 MHz
240 300 270 90 8 120 60 6
150 4 30
2
180 0
210 E-Plane, f = 216 MHz 330
240 300 270
90 8 120 60 6
150 4 30
2
180 0
210 H-Plane, f = 216 MHz 330
240 300 270
10
9
8
7
6
5
Gain (dB) 4
3
2
1
0 0 50 100 150 200 250 Frequency (MHz) Aperture Antennas
An aperture antenna contains some sort of opening through which electromagnetic waves are transmitted or received. Examples of aperture antennas include slots, waveguides, horns, reflectors and lenses. The analysis of aperture antennas is typically quite different than the analysis of wire antennas. Rather than using the antenna current distribution to determine the radiated fields, the fields within the aperture are used to determine the antenna radiation patterns. Aperture antennas are commonly used in aircraft or spacecraft applications. The aperture can be mounted flush with the surface of the vehicle, and the opening can be covered with a dielectric which allows electromagnetic energy to pass through.
Open Ended Rectangular Waveguide
Consider an open-ended rectangular waveguide which connects to a conducting ground plane which covers the x-y plane. If we assume that the waveguide carries only the dominant TE10 mode, the field distribution in the aperture of the waveguide is where
The resulting radiated far fields are The fields in the E-plane (N = 90o) and H-plane (N = 0o) reduce to
a = 38, b = 28
a = 98, b = 68 Horn Antennas
The horn antenna represents a transition or matching section from the guided mode inside the waveguide to the unguided (free-space) mode outside the waveguide. The horn antenna, as a matching section, reduces reflections and leads to a lower standing wave ratio. There are three basic types of horn antennas: (a.) the E-plane sectoral horn (flared in the direction of the E-plane only), (b.) the H-plane sectoral horn (flared in the direction of the H-plane only), and (c.) the pyramidal horn antenna (flared in both the E-plane and H-plane). The flare of the horns considered here is assumed to be linear although some horn antennas are formed by other flare types such as an exponential flare. The horn antenna is mounted on a waveguide that is almost always excited in single-mode operation. That is, the waveguide is operated at a frequency above the cutoff frequency of the TE10 mode but below the cutoff frequency of the next highest mode.
E-Plane Sectoral Horn E-plane Sectoral Horn E-plane Far Field (N = B/2) E-plane Sectoral Horn H-plane Far Field (N = B/2)
The directivity of the E-plane sectoral horn (DE) is given by
A plot of the E-plane and H-plane patterns for the E-plane horn shows that the H-plane pattern is much broader than the E-plane pattern. Thus, the E-plane sectoral horn tends to focus the beam of the antenna in the E-plane (see Figures 13.3 and 13.4). Design curves for the E-plane sectoral horn are given in Figure 13.8. Example (E-plane sectoral horn design, Problem 13.6)
An E-plane horn is fed by a WR 90 (X-band) rectangular waveguide (a = 2.286 cm, b = 1.016 cm). Design the horn so that its maximum directivity at 11 GHz is 30 (14.77 dB). H-Plane Sectoral Horn H-plane Sectoral Horn E-plane Far Field (N = B/2)
H-plane Sectoral Horn H-plane Far Field (N = B/2)
The directivity of the H-plane sectoral horn (DH) is given by A plot of the E-plane and H-plane patterns for the H-plane horn shows that the E-plane pattern is much broader than the H-plane pattern. Thus, the H-plane sectoral horn tends to focus the beam of the antenna in the H-plane (see Figures 13.11 and 13.12). Design curves for the H-plane sectoral horn are given in Figure 13.16.
Pyramidal Horn
Based on the pattern characteristics of the E-plane and H-plane sectoral horns, the pyramidal horn should focus the beam patterns in both the E-plane and the H-plane. In fact, the E-plane and H-plane patterns of the pyramidal horn are identical to the E-plane pattern of the E-plane sectoral horn and the H-plane pattern of the H-plane sectoral horn, respectively (See Figure 13.19).
The directivity of the pyramidal horn (DP) can be written in terms of the directivities of the E-plane and H-plane sectoral horns: Reflector Antennas
A reflector antenna utilizes some sort of reflecting (conducting) surface to increase the gain of the antenna. A typical reflector antenna couples a small feed antenna with a reflecting surface that is large relative to wavelength. Reflector antennas can achieve very high gains and are commonly used in such applications as long distance communications, radioastronomy and high-resolution radar.
Corner Reflector
The corner reflector antenna shown below utilizes a reflector formed by two plates (each plate area = l × h) connected at an included angle ". The feed antenna, located within the included angle, can be one of many antennas although simple dipoles are the most commonly used. The most commonly used included angle " for corner reflectors is o 90 . The electrical size of the aperture (Da) for the corner reflector antenna is typically between one and two wavelengths. Given a linear dipole as the feed element of a 90o corner reflector antenna, the far field of this antenna can be approximated using image theory. If the two plates of the reflector are electrically large, they can be approximated by infinite plates. This allows the use of image theory in the determination of the antenna far field.
For analysis purposes, the current on the feed element is assumed to be z-directed. The image element #2 represents the image of the feed element (#1) to plate #1. Together, elements #1 and #2 satisfy the electric field boundary condition on plate #1. Similarly, image element #3 represents the image of the feed element to plate #2. In order for image element #2 to satisfy the electric field boundary condition on plate #2, an additional image element (#4) is required. Note that the inclusion of image element #4 also allows image element #3 to satisfy the electric field boundary condition on plate #1. The system of four elements (four- element array) yields the overall field within the included angle of the reflector antenna (!45o # N # 45o). The four- element array can be treated as 2 two-element arrays (a two-element array along the x-axis and the two- element array along the y-axis). Given a two-element array aligned along the z-axis with equal amplitude, equal phase elements which are separated by a distance 2s, the resulting array factor was found to be
If we rotate this 2-element array so that it lies along the x-axis, we must transform the array factor according to which yields
Similarly, rotating the two-element array so that it lies along the y-axis, and noting that the current is opposite to that of the array along the x-axis, we find
The overall array factor for the 90o corner reflector becomes
In the azimuth plane (2 = B/2), the 90o corner reflector array factor is
The corner reflector array factor can be shown to be quite sensitive to the placement of the feed element, as would be expected. The following plots show the azimuth plane array factor for various feed spacings. s = 0.18 s = 0.78
s = 0.88 s = 1.08 A variety of reflecting surface shapes are utilized in reflector antennas. Some reflector antennas employ a parabolic cylinder as the reflecting surface while a more common reflecting surface shape is the paraboloid (parabolic dish antenna). The so-called Cassegrain antenna uses dual reflecting surfaces (the main reflector is a paraboloid, the sub- reflector is a hyperboloid).