Ea=e−jkhEr=e−jk(R+h)
R fr(ψ, χ)
But for the parabola, we haveR+h=2F. Thus, the aperture field is given by:
Ea=e−2jkF
R fa(ψ, χ) (aperture field) (18.9.11) where we definedfa=fr, so that:
fa= −fi+2ˆn(ˆnãfi) (18.9.12) Because|fa| = |fr| = |fi| =
2ηUfeed, it follows that Eq. (18.9.11) is consistent with Eq. (18.8.2). As plane waves propagating in thez-direction, the reflected and aperture fields are Huygens sources. Therefore, the corresponding magnetic fields will be:
Hr= 1
ηˆz×Er, Ha= 1 ηˆz×Ea
The surface currents induced on the reflector are obtained by noting that the total fields areEi+Er=2ˆn(nˆãEi)andHi+Hr=2Hi−2ˆn(nˆãHi). Thus, we have:
Js=ˆn×(Hi+Hr)=2 ˆn×Hi= 2 η
e−jkR R ˆR×fi
Jms= −nˆ×(Ei+Er)=0
18.10 Radiation Patterns of Reflector Antennas
The radiation patterns of the reflector antenna are obtained either from the aperture fieldsEa,Haintegrated over the effective aperture using Eq. (17.4.2), or from the cur- rentsJsandJms=0 integrated over the curved reflector surface using Eq. (17.4.1).
We discuss in detail only the aperture-field case. The radiation fields at some large distancerin the direction defined by the polar anglesθ, φare given by Eq. (17.5.3). The unit vector ˆrin the direction ofθ, φis shown in Fig. 18.7.2. We have:
Eθ=jke−jkr 2πr
1+cosθ 2
fxcosφ+fysinφ
Eφ=jke−jkr 2πr
1+cosθ 2
fycosφ−fxsinφ (18.10.1)
where the vectorf=ˆxfx+ˆyfyis the Fourier transform over the aperture:
f(θ, φ)=
a 0
2π 0
Ea(ρ, χ) ejkãrρdρdχ (18.10.2)
758 18. Aperture Antennas The vectorrlies on the aperture plane and is given in cylindrical coordinates by r=ρρρρˆ=ρ(ˆxcosχ+ˆysinχ). Thus,
kãr=kρ(ˆxcosφsinθ+ˆysinφsinθ+ˆzcosθ)ã(ˆxcosχ+ˆysinχ)
=kρsinθ(cosφcosχ+sinφsinχ)=kρsinθcos(φ−χ) It follows that:
f(θ, φ)= a
0
2π 0
Ea(ρ, χ) ejkρsinθcos(φ−χ)ρ dρ dχ (18.10.3) We may convert this into an integral over the feed anglesψ, χby using Eq. (18.9.11) anddρ=R dψ,ρ=2Ftan(ψ/2), andρ dρ=2FRtan(ψ/2) dψ. Then, the 1/Rfactor inEais canceled, resulting in:
f(θ, φ)=2Fe−2jkF ψ0
0
2π
0
fa(ψ, χ)e2jkFtanψ2sinθcos(φ−χ)tanψ
2 dψ dχ (18.10.4) Given a feed patternfi(ψ, χ), the aperture patternfa(ψ, χ)is determined from Eq. (18.9.12) and the integrations in (18.10.4) are done numerically.
Because of the condition ˆRãfi=0, the vectorfiwill have components only along the ˆψψψand ˆχχχdirections. We assume thatfihas the following more specific form:
fi=ψψψ Fˆ 1sinχ+χχχ Fˆ 2cosχ (y-polarized feeds) (18.10.5) whereF1, F2are functions ofψ, χ, but often assumed to be functions only ofψ, repre- senting the patterns along the principal planesχ=90oandχ=0o.
Such feeds are referred to as “y-polarized” and include y-directed dipoles, and waveguides and horns in which the electric field on the horn aperture is polarized along theydirection (thex-polarized case is obtained by a rotation, replacingχbyχ+90o.) Using Eqs. (18.9.1) and (18.9.10), the corresponding patternfacan be worked out:
fa= −ˆy
F1sin2χ+F2cos2χ
−ˆx
(F1−F2)cosχsinχ
(18.10.6) IfF1 =F2, we havefa= −ˆyF1. But ifF1 =F2, the aperture fieldEadevelops a
“cross-polarized” component along thexdirection. Various definitions of cross polar- ization have been discussed by Ludwig [1206].
As examples, we consider the cases of ay-directed Hertzian dipole feed, and waveg- uide and horn feeds. Adapting their radiation patterns given in Sections 16.2, 18.1, and 18.3, to theR, ψ, χcoordinate system, we obtain the following feed patterns, which are special cases of (18.10.5):
fi(ψ, χ)=Fdˆ ψ
ψψcosψsinχ+χχχˆcosχ
(dipole feed) fi(ψ, χ)=Fw(ψ, χ)ˆ
ψ ψ
ψsinχ+χχˆχcosχ
(waveguide feed) fi(ψ, χ)=Fh(ψ, χ)ˆ
ψ
ψψsinχ+χχχˆcosχ
(horn feed)
(18.10.7)
whereFdis the constantFd= −jη(Il)/2λ, andFw, Fhare given by:
18.10. Radiation Patterns of Reflector Antennas 759
Fw(ψ, χ)= −jabE0
πλ (1+cosψ)cos(πvx) 1−4vx2
sin(πvy) πvy Fh(ψ, χ)= −jABE0
8λ (1+cosψ)F1(vx, σa) F0(vy, σb)
(18.10.8)
whereI, lare the current and length of the Hertzian dipole,a, bandA, Bare the di- mensions of the waveguide and horn apertures, and vx = (a/λ)sinψcosχ, vy = (b/λ)sinψsinχfor the waveguide, andvx=(A/λ)sinψcosχ,vy=(B/λ)sinψsinχ, for the horn, andF1, F0are the horn pattern functions defined in Sec. 18.3. The corre- sponding aperture patternsfaare in the three cases:
fa(ψ, χ)= −yˆFd
cosψsin2χ+cos2χ
−ˆxFd
(cosψ−1)sinχcosχ fa(ψ, χ)= −yˆFw(ψ, χ)
fa(ψ, χ)= −yˆFh(ψ, χ)
(18.10.9)
In the general case, a more convenient form of Eq. (18.10.6) is obtained by writing it in terms of the sum and difference patterns:
A=F1+F2
2 , B=F1−F2
2 F1=A+B , F2=A−B (18.10.10) Using some trigonometric identities, we may write (18.10.6) in the form:
fa= −ˆy
A−Bcos 2χ
−ˆx
Bsin 2χ
(18.10.11) In general,A, Bwill be functions ofψ, χ(as in the waveguide and horn cases.) If we assume that they are functions only ofψ, then theχ-integration in the radiation pattern integral (18.10.4) can be done explicitly leaving an integral overψonly. Using (18.10.11) and the Bessel-function identities,
2π 0
ejucos(φ−χ)
cosnχ sinnχ
dχ=2πjn
cosnφ sinnφ
Jn(u) (18.10.12) we obtain:
f(θ, φ)= −ˆy
fA(θ)−fB(θ)cos 2φ
−ˆx
fB(θ)sin 2φ
(18.10.13) where the functionsfA(θ)andfB(θ)are defined by:
fA(θ)= 4πFe−2jkF ψ0
0
A(ψ) J0
4πF λ tan
ψ 2 sinθ
tanψ
2dψ fB(θ)= −4πFe−2jkF
ψ0
0
B(ψ) J2
4πF λ tan
ψ 2sinθ
tanψ
2 dψ
(18.10.14)
Using Eq. (18.10.13) and some trigonometric identities, we obtain:
fxcosφ+fysinφ= −(fA+fB)sinφ fycosφ−fxsinφ= −(fA−fB)cosφ
760 18. Aperture Antennas It follows that the radiation fields (18.10.1) are given by:
Eθ= −je−jkr λr
1+cosθ 2
fA(θ)+fB(θ) sinφ Eφ= −je−jkr
λr
1+cosθ 2
fA(θ)−fB(θ) cosφ
(18.10.15)
Example 18.10.1: Parabolic Reflector with Hertzian Dipole Feed.We compute numerically the gain patterns for ay-directed Hertzian dipole feed. We takeF=10λandD=40λ, so that F/D=0.25 andψ0=90o. These choices are similar to those in [1204].
Ignoring the constantFdin (18.10.7), we haveF1(ψ)=cosψandF2(ψ)=1. Thus, the sum and difference patters areA(ψ)=(cosψ+1)/2 andB(ψ)=(cosψ−1)/2. Up to some overall constants, the required gain integrals will have the form:
fA(θ)=
ψ0 0
FA(ψ, θ) dψ , fB(θ)=
ψ0 0
FB(ψ, θ) dψ (18.10.16) where
FA(ψ, θ)=(1+cosψ) J0
4πF λ tan
ψ 2sinθ
tanψ
2 FB(ψ, θ)=(1−cosψ) J2
4πF λ tan
ψ 2sinθ
tanψ
2
(18.10.17)
The integrals are evaluated numerically using Gauss-Legendre quadrature integration, which approximates an integral as a weighted sum [1298]:
fA(θ)=
N i=1
wiFA(ψi, θ)=wTFA
wherewi, ψiare the Gauss-Legendre weights and evaluation points within the integration interval[0, ψ0], whereFAis the column vector withith componentFA(ψi, θ).
For higher accuracy, this interval may be subdivided into a number of subintervals, the quantitieswi, ψiare then determined on each subinterval, and the total integral is evalu- ated as the sum of the integrals over all the subintervals.
We have written a MATLAB function,quadrs, that determines the quantitieswi, ψiover all the subintervals. It is built on the functionquadr, which determines the weights over a single interval.
The following MATLAB code evaluates and plots in Fig. 18.10.1 theE- andH-plane patterns (18.10.15) over the polar angles 0≤θ≤5o.
F = 10; D = 40; psi0 = 2*acot(4*F/D); %F/D=0.25,ψ0=90o ab = linspace(0, psi0, 5); % 4 integration subintervals in[0, ψ0] [w,psi] = quadrs(ab); % quadrature weights and evaluation points
% uses 16 weights per subinterval c = cos(psi); t = tan(psi/2); % cosψ,tan(ψ/2)at quadrature points th = linspace(0, 5, 251); % angleθin degrees over 0≤θ≤5o for i=1:length(th),
u = 4*pi*F*sin(th(i)*pi/180); %u=2kFsinθ
18.10. Radiation Patterns of Reflector Antennas 761
−50 −4 −3 −2 −1 0 1 2 3 4 5
0.2 0.4 0.6 0.8 1
θ (degrees)
field strength
Paraboloid with Dipole Feed, D = 40λ
E− plane H− plane
−50 −4 −3 −2 −1 0 1 2 3 4 5
0.2 0.4 0.6 0.8 1
θ (degrees)
field strength
Paraboloid with Dipole Feed, D = 80λ
E− plane H− plane
Fig. 18.10.1 Parabolic reflector patterns with dipole feed.
FA = (1+c) .* besselj(0, u*t) .* t; % integrand offA(θ) fA(i) = w’ * FA; % integral evaluated atθ FB = (1-c) .* besselj(2, u*t) .* t; % integrand offB(θ) fB(i) = w’ * FB;
end
gh = abs((1+cos(th*pi/180)).*(fA-fB)); gh = gh/max(gh); % gain patterns ge = abs((1+cos(th*pi/180)).*(fA+fB)); ge = ge/max(ge);
plot(-th,ge,’-’, th,ge, ’-’, -th,gh,’--’,th,gh,’--’);
The graph on the right hasψ0=90oandD=80λ, resulting in a narrower main beam.
Example 18.10.2: Parabolic Reflector with Waveguide Feed.We calculate the reflector radiation patterns for a waveguide feed radiating in the TE10mode with ay-directed electric field.
The feed pattern was given in Eq. (18.10.7). Ignoring some overall constants, we have with vx=(a/λ)sinψcosχandvy=(b/λ)sinψsinχ:
fi=(1+cosψ)cos(πvx) 1−4v2x
sin(πvy) πvy
(ψψψˆsinχ+χχχˆcosχ) (18.10.18) To avoid the double integration in theψandχvariables, we follow Jones’ procedure [1204] of choosing thea, bsuch that theE- andH-plane illuminations of the paraboloid are essentially identical. This is accomplished whenais approximatelya=1.37b. Then, the above feed pattern may be simplified by replacing it by itsE-plane pattern:
fi=(1+cosψ)sin(πvy) πvy
(ψψψˆsinχ+χχχˆcosχ) (18.10.19) wherevy=(b/λ)sinψ. Thus,F1=F2and
A(ψ)=(1+cosψ)sin(πbsinψ/λ)
πbsinψ/λ and B(ψ)=0 (18.10.20) The radiated field is given by Eq. (18.10.15) with a normalized gain:
g(θ)=
1+cosθ 2
fA(θ) fA(0)
2 (18.10.21)
762 18. Aperture Antennas
wherefA(θ)is defined up to a constant by Eq. (18.10.14):
fA(θ)=
ψ0
0
A(ψ) J0
4πF λ tan
ψ 2 sinθ
tanψ
2 dψ (18.10.22)
We choose a parabolic antenna with diameterD=40λand subtended angle ofψ0=60o, so thatF=Dcot(ψ0/2)/4=17.3205λ. The lengthbof the waveguide is chosen such as to achieve an edge illumination of−11 dB on the paraboloid. This gives the condition on b, where the extra factor of(1+cosψ)arises from the space attenuation factor 1/R:
|Ei(ψ0)|
|Ei(0)| =
1+cosψ0
2 2
sin(πbsinψ0/λ) πbsinψ0/λ
=10−11/20=0.2818 (18.10.23)
It has solutionb=0.6958λand therefore,a=1.37b=0.9533λ. The illumination effi- ciency given in Eq. (18.8.12) may be taken to be a measure of the overall aperture efficiency of the reflector. Because 2ηUfeed= |fi|2= |fa|2= |A(ψ)|2, the integrals in (18.8.12) may be calculated numerically, givingea=0.71 and a gain of 40.5 dB.
The pattern functionfA(θ)may be calculated numerically as in the previous example. The left graph in Fig. 18.10.2 shows theE- andH-plane illumination patterns versusψof the actual feed given by (18.10.18), that is, the normalized gains:
gE(ψ)=
(1+cosψ)2 4
sin(πbsinψ/λ) πbsinψ/λ
2 gH(ψ)=
(1+cosψ)2 4
cos(πasinψ/λ) 1−4(πasinψ/λ)2
2
They are essentially identical provideda =1.37b(the graph actually plots the square roots of these quantities.) The right graph shows the calculated radiation patterng(θ) (or, rather its square root) of the paraboloid.
−600 −40 −20 0 20 40 60
0.2 0.4 0.6 0.8 1
ψ (degrees)
field strength
Feed Illumination Patterns
−11 dB
E− plane H− plane
−80 −6 −4 −2 0 2 4 6 8
0.2 0.4 0.6 0.8 1
3− dB width
θ (degrees)
field strength
Paraboloid Reflector Pattern
Fig. 18.10.2 Feed illumination and reflector radiation patterns.
The following MATLAB code solves (18.10.23) forb, and then calculates the illumination pattern and the reflector pattern:
18.10. Radiation Patterns of Reflector Antennas 763
F = 17.3205; D = 40; psi0 = 2*acot(4*F/D); %ψ0=60o f = inline(’(1+cos(x)).^2/4 * abs(sinc(b*sin(x))) - A’,’b’,’x’,’A’);
Aedge = 11;
b = fzero(f,0.8,optimset(’display’,’off’), psi0, 10^(-Aedge/20));
a = 1.37 * b;
psi = linspace(-psi0, psi0, 201); ps = psi * 180/pi;
gE = abs((1+cos(psi)).^2/4 .* sinc(b*sin(psi)));
gH = abs((1+cos(psi)).^2/4 .* dsinc(a*sin(psi)));
figure; plot(ps,gE,’-’, ps,gH,’--’);
[w,psi] = quadrs(linspace(0, psi0, 5)); % quadrature weights and points s = sin(psi); c = cos(psi); t = tan(psi/2);
A = (1+c) .* sinc(b*s); % the patternA(ψ) thd = linspace(0, 5, 251); th = thd*pi/180;
for i=1:length(th), u = 4*pi*F*sin(th(i));
FA = A .* besselj(0, u*t) .* t;
fA(i) = w’ * FA;
end
g = abs((1+cos(th)) .* fA); g = g/max(g);
figure; plot(-thd,g,’-’, thd,g);
The 3-dB width was calculated from Eq. (18.8.18) and is placed on the graph. The angle factor was 1.05Aedge+55.95=67.5, so thatΔθ3dB=67.5oλ/D=67.5/40=1.69o. The gain-beamwidth product isp=G(Δθ3dB)2=1040.5/10(1.69o)2=32 046 deg2. Example 18.10.3: Parabolic Reflector with Horn Feed.Fig. 18.10.3 shows the illumination and reflector patterns if a rectangular horn antenna feed is used instead of a waveguide. The design requirements were again that the edge illumination be -11 dB and thatD=40λ andψ0=60o. The illumination pattern is (up to a scale factor):
fi=(1+cosψ)F1(vx, σa) F0(vy, σb) (ψψψˆsinχ+χχˆχcosχ)
TheE- andH-plane illumination patterns are virtually identical over the angular range 0 ≤ ψ ≤ψ0, provided one chooses the horn sides such thatA = 1.48B. Then, the illumination field may be simplified by replacing it by theE-plane pattern and the lengthB is determined by requiring that the edge illumination be -11 dB. Therefore, we work with:
fi=(1+cosψ)F0(vy, σb) (ψψψˆsinχ+χχχˆcosχ) , vy=B λsinψ
Then,A(ψ)=(1+cosψ)F0(vy, σb)andB(ψ)=0 for the sum and difference patterns.
The edge illumination condition reads now:
1+cosψ0
2 2
F0(Bsinψ0/λ, σb) F0(0, σb)
=10−11/20
764 18. Aperture Antennas
Its solution isB=0.7806λ, and henceA=1.48B=1.1553λ. The left graph in Fig. 18.10.3 shows theE- andH-plane illumination gain patterns of the actual horn feed:
gE(ψ)=
(1+cosψ)2 4
F0(Bsinψ/λ, σb) F0(0, σb)
2 gH(ψ)=
(1+cosψ)2 4
F1(Asinψ/λ, σa) F1(0, σa)
2
They are seen to be almost identical. The right graph shows the reflector radiation pattern computed numerically as in the previous example. The following MATLAB code illustrates this computation:
[w,psi] = quadrs(linspace(0, psi0, 5)); % 4 subintervals in[0, ψ0] s = sin(psi); c = cos(psi); t = tan(psi/2); % evaluate at quadrature points Apsi = (1+c) .* (diffint(B*s, sb, 0)); % the patternA(ψ) thd = linspace(0, 8, 251); th = thd*pi/180;
for i=1:length(th), u = 4*pi*F*sin(th(i));
FA = Apsi .* besselj(0, u*t) .* t;
fA(i) = w’ * FA;
end
g = abs((1+cos(th)) .* fA); g = g/max(g);
figure; plot(-thd,g,’-’, thd,g);
The horn’sσ-parameters were chosen to have the usual optimum values ofσa=1.2593 andσb=1.0246. The 3-dB width is the same as in the previous example, that is, 1.69o and is shown on the graph. The computed antenna efficiency is nowea=0.67 and the corresponding gain 40.24 dB, so thatp=G(Δθ3dB)2=1040.24/10(1.69o)2=30 184 deg2
for the gain-beamwidth product.
Example 18.10.4: Here, we compare the approximate symmetrized patterns of the previous two examples with the exact patterns obtained by performing the double-integration over the aperture variablesψ, χ.
Both the waveguide and horn examples have ay-directed two-dimensional Fourier trans- form pattern of the form:
fA(θ, φ)=fy(θ, φ)=
ψ0
0
2π
0
FA(ψ, χ, θ, φ) dψ dχ (18.10.24) where the integrand depends on the feed patternA(ψ, χ):
FA(ψ, χ, θ, φ)=A(ψ, χ) ej2kFtan(ψ/2)sinθcos(φ−χ)tanψ
2 (18.10.25)
and, up to constant factors, the functionA(ψ, χ)is given in the two cases by:
A(ψ, χ)=(1+cosψ)cos(πvx) 1−4v2x
sin(πvy) πvy
A(ψ, χ)=(1+cosψ) F1(vx, σa) F0(vy, σb)
(18.10.26)
18.10. Radiation Patterns of Reflector Antennas 765
−600 −40 −20 0 20 40 60
0.2 0.4 0.6 0.8 1
ψ (degrees)
field strength
Feed Illumination Patterns
−11 dB
E− plane H− plane
−80 −6 −4 −2 0 2 4 6 8
0.2 0.4 0.6 0.8 1
3− dB width
θ (degrees)
field strength
Paraboloid Reflector Pattern
Fig. 18.10.3 Feed and reflector radiation patterns.
wherevx = (a/λ)sinψcosχandvy = (b/λ)sinψsinχfor the waveguide case, and vx=(A/λ)sinψcosχandvy=(B/λ)sinψsinχfor the horn.
Once,fA(θ, φ)is computed, we obtain the (un-normalized)H- andE-plane radiation pat- terns for the reflector by settingφ=0oand 90o, that is,
gH(θ)= (1+cosθ) fA(θ,0o) 2, gE(θ)= (1+cosθ) fA(θ,90o) 2 (18.10.27) The numerical evaluation of Eq. (18.10.24) can be done with two-dimensional Gauss-Legendre quadratures, approximating the integral by the double sum:
fA(θ, φ)=
N1
i=1 N2
j=1
w1iFA(ψi, χj)w2j=wT1FAw2 (18.10.28)
where{w1i, ψi}and{w2j, χj}are the quadrature weights and evaluation points over the intervals[0, ψ0]and[0,2π], andFAis the matrixFA(ψi, χj). The functionquadrs, called on these two intervals, will generate these weights.
Fig. 18.10.4 shows the patterns (18.10.27) of the horn and waveguide cases evaluated nu- merically and plotted together with the approximate symmetrized patterns of the previous two examples. The symmetrized patterns agree very well with the exact patterns and fall between them. The following MATLAB code illustrates this computation for the horn case:
[w1, psi] = quadrs(linspace(0, psi0, Ni)); % quadrature over[0, ψ0],Ni=5 [w2, chi] = quadrs(linspace(0, 2*pi, Ni)); % quadrature over[0,2π],Ni=5 sinpsi = sin(psi); cospsi = cos(psi); tanpsi = tan(psi/2);
sinchi = sin(chi); coschi = cos(chi);
for i = 1:length(chi), % build matrixA(ψi, χj)columnwise Apsi(:,i) = diffint(A*sinpsi*coschi(i), sa, 1) ...
.* diffint(B*sinpsi*sinchi(i), sb, 0);
end
Apsi = repmat(tanpsi.*(1+cospsi), 1, length(psi)) .* Apsi;
th = linspace(0, 8, 401) * pi/180;
766 18. Aperture Antennas
−8 −6 −4 −2 0 2 4 6 8
−50
−40
−30
−20
−10 0
θ (degrees)
gains in dB
Reflector Pattern with Horn Feed symmetrized H− plane E− plane
−8 −6 −4 −2 0 2 4 6 8
−50
−40
−30
−20
−10 0
θ (degrees)
gains in dB
Reflector Pattern with Waveguide Feed symmetrized H− plane E− plane
Fig. 18.10.4 Exact and approximate reflector radiation patterns.
for i=1:length(th),
u = 4*pi*F*sin(th(i)); %u=2kFsinθ FH = Apsi .* exp(j*u*tanpsi*coschi’); %H-plane,φ=0o FE = Apsi .* exp(j*u*tanpsi*sinchi’); %E-plane,φ=90o fH(i) = w1’ * FH * w2; % evaluate double integral fE(i) = w1’ * FE * w2;
end
gH = abs((1+cos(th)).*fH); gH = gH/max(gH); % radiation patterns gE = abs((1+cos(th)).*fE); gE = gE/max(gE);
The patterns are plotted in dB, which accentuates the differences among the curves and also shows the sidelobe levels. In the waveguide case the resulting curves are almost
indistinguishable to be seen as separate.