Modeling Mutual Coupling in Large Arrays Using Embedded Element Pattern

Model an Array of Dipoles Using Isolated Element Pattern

In [4], it was discussed that the central element of a 5$$\lambda$$-by-5$$\lambda$$ array starts to behave like it is in an infinite array. Such an aperture would correspond to a 10-by-10 array of half-wavelength spaced radiators. You choose to slightly exceed this limit and consider a 11-by-11 array of $$\lambda /2$$ dipoles.

Nrow = 11;
Ncol = 11;
drow = 0.5*lambda;
dcol = 0.5*lambda;

Dipole as Antenna Element

Choose a dipole as the individual element. Choose its length to be slightly lower than $$\lambda /2$$ and radius approximately $$\lambda /150$$.

mydipole = dipole;
mydipole.Length = 0.47*lambda;
mydipole.Width = cylinder2strip(0.191e-3);

URA with an Isolated Dipole

Create a 11-by-11 URA and assign the isolated dipole as its element. Adjust the spacing to be half-wavelength at 10 GHz. The dipole tilt is now set to zero, so that its orientation matches the array geometry in the Y-Z plane.

myURA2 = phased.URA;
myURA2.Element = mydipole;
myURA2.Size = [Nrow Ncol];
myURA2.ElementSpacing = [drow dcol];

Create Full-Wave Model of the 11-by-11 Array

Use the Antenna Toolbox™ to create a full-wave model of the 11-by-11 array of resonant dipoles. Since the default orientation of the dipole element in the library is along z-axis, tilt it so that the array is initially formed in the X-Y plane and then tilt the array to match the array axis of the URA.

myFullWaveArray = rectangularArray;
myFullWaveArray.Element = mydipole;
myFullWaveArray.Element.Tilt = 90;
myFullWaveArray.Element.TiltAxis = [0 1 0];
myFullWaveArray.Size = [Nrow Ncol];
myFullWaveArray.RowSpacing = drow;
myFullWaveArray.ColumnSpacing = dcol;
myFullWaveArray.Tilt = 90;
myFullWaveArray.TiltAxis = "Y";
figure
show(myFullWaveArray)
title("Rectangular 11-by-11 Array of Dipole Antennas")

drawnow

Model Array of Dipoles Using Embedded Element Pattern

Calculate Embedded Element Pattern

Calculate the full 3D embedded element pattern in terms of the electric field magnitude. In [3], the scan resistance and scan reactance for an infinite array of resonant dipoles spaced $$\lambda /2$$ apart is provided. Choose the resistance at broadside as the termination for all elements. To calculate the embedded element pattern, use the pattern function and pass in additional input parameters of the element number(index of the center element) and termination resistance.

Zinf = 76 + 1i*31;
ElemCenter = (prod(myFullWaveArray.Size)-1)/2 + 1;
az = -180:2:180;
el = -90:2:90;
h = waitbar(0,"Calculating center element embedded pattern....");
embpattern = pattern(myFullWaveArray,freq,az,el,...
    ElementNumber=ElemCenter,Termination=real(Zinf),...
    Type="efield");
waitbar(1,h,"Pattern computation complete");
delete(h);

URA with Embedded Element Pattern

Import this embedded element pattern into the custom antenna element.

embpattern = 20*log10(embpattern); 
fmin = freq - 0.1*freq;
fmax = freq + 0.1*freq;
freqVector = [fmin fmax];
EmbAnt = phased.CustomAntennaElement(FrequencyVector=freqVector,...
    AzimuthAngles=az,ElevationAngles=el,...
    MagnitudePattern=embpattern,PhasePattern=zeros(size(embpattern)));

Create a uniform rectangular array(URA), with the custom antenna element, which has the embedded element pattern.

myURA1 = phased.URA;
myURA1.Element = EmbAnt;
myURA1.Size = [Nrow Ncol];
myURA1.ElementSpacing = [drow dcol];

Compare Array Pattern in Elevation and Azimuth Plane

Calculate the pattern in the elevation plane (specified by azimuth = 0 deg and also called the E-plane) and azimuth plane (specified by elevation = 0 deg and called the H-plane) for the three arrays based on the isolated element pattern, the embedded element pattern, and the full-wave model.

Eplane1 = pattern(myURA1,freq,0,el);
Eplane2 = pattern(myURA2,freq,0,el);
[Eplane3,~,el3e] = pattern(myFullWaveArray,freq,0,el);
figure
plot(el,Eplane2,el,Eplane1,el3e,Eplane3,LineWidth=1.5);
axis([min(el) max(el) -60 30])
grid on
xlabel("Elevation Angle (deg.)");
ylabel("Directivity (dBi)");
title("E-plane Array Directivity Comparison")
legend("With Isolated Pattern","With Embedded Pattern","Full Wave Solution")

drawnow

Hplane1 = pattern(myURA1,freq,az/2,0);
Hplane2 = pattern(myURA2,freq,az/2,0);
Hplane3 = pattern(myFullWaveArray,freq,az/2,0);
figure
plot(az/2,Hplane2,az/2,Hplane1,az/2,Hplane3,LineWidth=1.5);
axis([min(az/2) max(az/2) -60 30])
grid on
xlabel("Azimuth Angle (deg.)");
ylabel("Directivity (dBi)");
title("H-plane Array Directivity Comparison")
legend("With Isolated Pattern","With Embedded Pattern","Full Wave Solution")

drawnow

The array directivity is approximately 23 dBi. This result is close to the theoretical calculation for the peak directivity [5] after taking into account the absence of a reflector, D = 4 $$\pi$$ $$A$$ / $${\lambda }^2$$ $$NrowNcol$$, $$A=drow*dcol$$.

Normalize the directivity for the three arrays and plot it for comparison.

figure
Eplanenormlz1 = Eplane1 - max(Eplane1);
Eplanenormlz2 = Eplane2 - max(Eplane2);
Eplanenormlz3 = Eplane3 - max(Eplane3);
plot(el,Eplanenormlz2,el,Eplanenormlz1,el,Eplanenormlz3,LineWidth=1.5);
axis([min(el) max(el) -60 0])
grid on
xlabel("Elevation Angle (deg.)");
ylabel("Directivity (dB)");
title("Normalized E-plane Array Directivity Comparison")
legend("With Isolated Pattern","With Embedded Pattern","Full Wave Solution")

drawnow

figure
Hplanenormlz1 = Hplane1 - max(Hplane1);
Hplanenormlz2 = Hplane2 - max(Hplane2);
Hplanenormlz3 = Hplane3 - max(Hplane3);
plot(az/2,Hplanenormlz2,az/2,Hplanenormlz1,az/2,Hplanenormlz3,LineWidth=1.5);
axis([min(el) max(el) -60 0])
grid on
xlabel("Azimuth Angle (deg.)");
ylabel("Directivity (dB)");
title("Normalized H-plane Array Directivity Comparison")
legend("With Isolated Pattern","With Embedded Pattern","Full Wave Solution")

drawnow

The pattern comparison suggests that the main beam and the first side lobes are aligned for all three cases. Moving away from the main beam shows the increasing effect of coupling on the side lobe level. As expected,the embedded element pattern approach suggests a coupling level in between the full-wave simulation model and the isolated element pattern approach.

Comparison with 25-by-25 Array

The behavior of the array pattern is intimately linked to the embedded element pattern. To understand how the choice of a 11-by-11 array impacts the center element behavior, increase the array size to a 25-by-25 array (12.5$$\lambda$$-by-12.5$$\lambda$$ aperture size). Note that the triangular mesh size for the full wave Method of Moments (MoM) analysis with 625 elements increases to 25000 triangles (40 triangles per dipole) and the computation for the embedded element pattern takes approximately 12 minutes on a 2.4 GHz machine with 32 GB memory. This time can be reduced by lowering the mesh size per element by meshing manually using a maximum edge length of $$\lambda /20$$.

load dipolearray
embpattern = 20*log10(DipoleArrayPatData.ElemPat); 
EmbAnt2 = clone(EmbAnt);
EmbAnt2.AzimuthAngles = DipoleArrayPatData.AzAngles;
EmbAnt2.ElevationAngles = DipoleArrayPatData.ElAngles;
EmbAnt2.MagnitudePattern = embpattern;
Eplane1 = pattern(EmbAnt2,freq,0,el);
Eplane1 = Eplane1 - max(Eplane1);
Eplane2 = pattern(mydipole,freq,0,el);
Eplane2 = Eplane2 - max(Eplane2);
embpatE = pattern(EmbAnt,freq,0,el);
embpatE = embpatE - max(embpatE);
figure
plot(el,Eplane2,el,embpatE,el,Eplane1,LineWidth=1.5);
axis([min(el) max(el) -60 0])
grid on
xlabel("Elevation Angle (deg.)");
ylabel("Directivity (dB)");
title("Normalized E-plane Element Directivity Comparison")
legend("IsolatedPattern","Embedded Pattern: 11-by-11","Embedded Pattern: 25-by-25",Location="best")

drawnow

Hplane1 = pattern(EmbAnt2,freq,0,az/2);
Hplane1 = Hplane1 - max(Hplane1);
Hplane2 = pattern(mydipole,freq,0,az/2);
Hplane2 = Hplane2 - max(Hplane2);
embpatH = pattern(EmbAnt,freq,az/2,0);
embpatH = embpatH - max(embpatH);
figure
plot(az/2,Hplane2,az/2,embpatH,az/2,Hplane1,LineWidth=1.5);
axis([min(el) max(el) -60 0])
grid on
xlabel("Azimuth Angle (deg.)");
ylabel("Directivity (dB)");
title("Normalized H-plane Element Directivity Comparison")
legend("IsolatedPattern","Embedded Pattern: 11-by-11","Embedded Pattern: 25-by-25",Location="best")

drawnow

The plot above reveals that the difference between embedded element patterns of the 11-by-11 and the 25-by-25 array, respectively, is less than 0.5 dB, in the E-plane. However, the H-plane shows more variation for the 11-by-11 array as compared with the 25-by-25 array.

Scan Behavior and Embedded Element Pattern

Scan the array based on the embedded element pattern in the elevation plane defined by azimuth = 0 deg and plot the normalized directivity. Also, overlay the normalized embedded element pattern. Note the overall shape of the normalized array pattern approximately follows the normalized embedded element pattern. This is also predicted by the pattern multiplication principle.

eplane_indx = find(az==0);
scan_el1 = -30:10:30;
scan_az1 = zeros(1,numel(scan_el1));
scanEplane = [scan_az1;scan_el1];
hsv = phased.SteeringVector;
hsv.SensorArray = myURA1;
hsv.IncludeElementResponse = true;
weights = step(hsv,freq,scanEplane);
legend_string1 = cell(1,numel(scan_el1)+1);
legend_string1{end} = "Embedded element";
scanEPat = nan(numel(el),numel(scan_el1));
for i = 1:numel(scan_el1)
    scanEPat(:,i) = pattern(myURA1,freq,scan_az1(i),el,Weights=weights(:,i)); % -23.13;
    legend_string1{i} = strcat("scan = ",num2str(scan_el1(i)));
end
scanEPat = scanEPat - max(max(scanEPat));
figure
plot(el,scanEPat,LineWidth=1.5);
hold on
grid on
plot(el,embpatE,'-.',LineWidth=1.5);
axis([min(el) max(el) -50 0])
xlabel("Elevation (deg.)")
ylabel("Directivity (dB)")
title("E-plane Scan Comparison")
legend(legend_string1,Location="southeast")
hold off

drawnow

Scan Blindness

In large arrays, it is possible that the array directivity to reduce drastically at certain scan angles. At these scan angle, referred to as the blind angles, the array does not radiate the power supplied at its input terminals [3]. Two common mechanisms under which blindness conditions occur are

It is possible to detect scan blindness in large finite arrays by studying the embedded element pattern (also known as array element pattern in the infinite array analysis). The array being investigated in this example does not have a dielectric substrate/ground plane, and therefore the surface waves are eliminated. However the grating lobe excitation mechanism can be investigated. To do so, increase the spacing across rows and columns of the array to be 0.7 $$\lambda$$. Since this spacing is greater than the half-wavelength limit, expect grating lobes in the visible space beyond a specific scan angle. As pointed out in [3], to accurately predict the depth of grating lobe blind angles in the finite array of dipoles, you need to have an array of the size 41-by-41 or higher. You compare 3 array sizes 11-by-11, 25-by-25, and 41-by-41 and check if the existence of blind angles can at least be observed in the 11-by-11 array. As mentioned earlier, the results were precomputed in Antenna Toolbox™ and saved in a MAT file. To reduce the computational time, the elements were meshed with maximum edge length of $$\lambda /20$$.

load dipolearrayblindness.mat

The normalized E-plane embedded element pattern for arrays of three sizes

The normalized H-plane embedded element pattern for arrays of three sizes. Notice the blind angle around 24-26 deg.

Conclusion

The embedded element pattern approach is one possible way of performing the analysis of large finite arrays. They need to be so large that the edge effects can be ignored. The isolated element pattern is replaced with the embedded element pattern which includes the effect of mutual coupling.

Reference

[1] R. J. Mailloux, 'Phased Array Antenna Handbook', Artech House,2nd edition, 2005

[2] W. Stutzman, G. Thiele, 'Antenna Theory and Design', John Wiley & Sons Inc., 3rd Edition, 2013.

[3] R. C. Hansen, Phased Array Antennas, Chapter 7 and 8, John Wiley & Sons Inc.,2nd Edition, 1998.

[4] H. Holter, H. Steyskal, "On the size requirement for finite phased-array models," IEEE Transactions on Antennas and Propagation, vol.50, no.6, pp.836-840, Jun 2002.

[5] P. W. Hannan, "The Element-Gain Paradox for a Phased-Array Antenna," IEEE Transactions on Antennas Propagation, vol. 12, no. 4, July 1964, pp. 423-433.