These findings thus lend support for the hypothesis that the decreased values (in true active S-parameters) is perhaps the result of total radiation caused by the antenna array elements. This leads us to assume that perhaps four antenna elements is not enough to characterize fully the effect of total radiation. Most importantly we need to test practically bigger array (more than 16 elements at least) to consolidate the proposed hypothesis. Nevertheless, it has been shown that simultaneously excited antenna array exhibits different values as compared to standalone values of antenna elements in presence of practically negligible coupling.

**5.10 Conclusion**

In the first part of this chapter, the impact of mutual coupling in microstrip antenna arrays parameters is studied and discussed. A theoretical background about antenna mutual coupling was presented where several useful concepts were revised. A literature review about the different decoupling techniques was cited precisely, along the ones that were simulated and analyzed.

A two element patch antenna array oriented along E-plane with novel DGS (μ-shaped) design was simulated. This technique has the advantage of simple execution and improves the isolation between the microstrip antenna elements upto -33 dB. A parametric analysis of coupling with respect to width (W1) of the DGS was carried out. It was found that a variation in the width of the DGS produces a shift in the cutoff frequency for this particular design. As a limitation, we could find a narrow bandwidth of the stop band filter.

In the second part of this chapter, a pin loaded patch antenna has been shown to outperform a conventional patch antenna in terms of suppressing surface waves and radiation in horizontal directions resulting in better directivity. Moreover, when these pin loaded patch antennas are set in arrays, at a separation distance of 0.7λ₀ between their feed points, they share -5 dB and -4 dB less coupled power (simulated value) as compared to conventional patch antenna array arranged along E-plane and H-plane orientations, respectively. For further reduction of coupling, pin loaded patch antennas arrays are complemented with metallic walls erected in between the antenna elements to reduce the effect of free space waves coupling. When a combination of pins and metallic wall is utilized, a reduction of -19 dB and -15 dB in E-plane and H-plane, respectively, has been observed (measured value) at the resonant frequency. It has also been shown that apart from suppression of mutual coupling, this technique presents a wide band rejection characteristic of transmission coefficient along both E and H-plane orientations.

The third part of this chapter further highlights the importance of simultaneous excitation of antenna elements when there is negligible mutual coupling. It has been observed that -37 dB (measured value) of coupled power presents same values as that of passive S-parameters and active S-parameters (based on passive scattering). However, in addition to that it has been observed that when all antennas are excited simultaneously according to coherent mode of excitation of VNA, the true active S-parameters present decreased values as compared to so called active S-parameters based on passive scattering. Even if the model of pin loaded patch antenna has not been develop yet, the results obtained tend to prove that the passive scattering approach to evaluate the active one is not
accurate enough as a shift is always observed when comparing the true active S matrix and
its approximate value through equation 4.3 using measured values.
<figure><img src="image_83.png" /></figure>

# Conclusion & Perspectives

## 6.1 Conclusion

The objective of this thesis is to propose and clarify input impedance evaluation from the effect of total radiation.

In chapter 1, an introduction about antenna and antenna arrays has been presented. In addition to that a background study about different antenna analysis methods has been presented along with motivation, thesis contribution and thesis structure.

In chapter 2, we have developed the basic concepts related to antenna radiating field regions and antenna analysis parameters. These basic concepts are used frequently to define and characterize antenna and antenna array performance.

A model to evaluate input impedance of individual antenna elements in an array based on total radiation has been proposed in chapter 3. In this regard, we have built a theoretical model to evaluate input impedance of different antennas (short dipole, resonant half wave length dipole, patch antenna) in different combinations (linear and planar). It has been demonstrated that by arraying the antennas the input impedance of each element in the array changes compared to that of single element which is not arrayed. In other words input impedance changes versus antenna number and position. In this approach, we do not take into account mutual coupling. However, the results suggest that even though there is no mutual coupling, the input impedance of individual antenna elements varies in an array inferring the cause being the total radiation. In our approach, we have considered free space combination as a probable contributor so we tried to investigate this part by isolating this contributor considering the others as null. Nevertheless, it will be interesting to have a deeper approach of all other possible contributors which in terms could explain the impedance shift on antenna array. It's also highly likely that some aspects are not taken into account in the array calculations.

Further, in order to consolidate the theoretical aspect of the effect of total radiation, an array of patch antennas has been manufactured in chapter 4. In this part, the problem to address is to excite all antennas simultaneously. This is done with the help of existing four ports VNA. The used VNA (R&S ZVA24) is capable of providing two coherent signals out of four ports. Thus, a novel method of exciting any number of antenna elements is introduced with the help of power splitter. This work further proposes an important aspect of the evaluation of active S-parameters in antenna arrays. It has been observed that the active S-parameters obtained by using the passive scattering matrix do not correspond to the actual values obtained during a simultaneous excitation of all the elements.

Since the theoretical code does not include the effect of mutual coupling, hence to verify the effect of total radiation practically, we need to decouple an antenna array to a
level where mutual coupling presents no change of active S-parameters as compared to passive S-parameters. In this context, in the first section of chapter 5, a novel DGS (μ shaped) has been introduced. This work reduced coupling upto -33 dB in two E-plane oriented patch antenna arrays with limitation of narrow band rejection characteristics of transmission coefficient (S12).

To have even better rejection in coupled power and wide band rejection characteristics, a method of loaded pins with metallic walls has been introduced in the second section of chapter 5. A method of loaded patch antennas with shorting pins and erected walls in between patch antenna arrays is introduced to reduce surface waves and free space wave coupling in both E and H-plane, respectively. This simple technique works equally well in both orientations by reducing coupling upto -39 dB and -44 dB (measured value) in E-plane and H-plane, respectively, as compared to conventional patch antenna array. Apart from mutual coupling reduction, this technique presents wide band rejection characteristics of transmission coefficient (S12). This section further highlights that coupling less than -34.88 dB is fully negligible if we consider active scattering through passive matrix as the impedance does not change much.

Further, a four element patch antenna array in E-plane with -37 dB of coupled power, in between two inner antennas, has been manufactured and put under test. It has been shown that despite having negligible mutual coupling, the active S-parameters evaluated through passive scatterings do not correspond to same values when all antennas radiate simultaneously according to coherent mode of VNA. Finally, input impedance of measured four element antenna array, evaluated through the true active S-parameters is analyzed. These findings lend support for the hypothesis that the decreased values (in true active S-parameters) are perhaps the result of total radiation caused by the antenna array elements.

## 6.2 Perspectives

The work in this thesis correlates the theory of the effect of total radiation on the impedance value of an array. Apart from this, it has been observed that active S-parameters evaluated through passive scattering matrix do not correspond to actual values obtained through simultaneously excited antenna array (through coherent mode of excitation of VNA). In light of this, the research has also opened a horizon for carrying out further analysis starting from the observations carried out in this work. Even though these techniques do not result in the desired end-state of implementation in operational arrays, still they can be useful in laboratory and simulation set-ups to facilitate and enhance greater understanding of total radiation from an array and its corresponding effects on different antenna types and array arrangements.

Thus, further research can be focused on;

* To address the aspect of the physical explanation behind the change in input impedance of individual antenna elements due to the effect of total radiation.

* To test and verify bigger arrays through measurements (more than 16 elements) to verify and consolidate further the effect of total radiation.
* To develop cavity like model for shorted pin loaded patch antenna to evaluate its input impedance. The model should be consistent with that of the measured one when coupling is observed negligible.

* To verify the effect of total radiation in other types of antenna elements (wide band and non-resonant antennas).

* The research can also be focused on this issue of mismatch which has not been apprehended so for. Development of equations or physical solutions may still be taken into account to propose so that the effect of mismatch may properly be understood and overcome.
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Appendix A
Matlab Code

clear all;
clc;
tic;
warning('off','all')
format shortG
freq = 5*10^9;
c = 3*10^8;
lambda = c./freq;
k = (2.*pi)/lambda;
eta = 120.*pi;
d = lambda;
A=[1 4 16 64 ];
B=[1 2 4 8 16 64 ];

% ResonantWireAntenna

for i=A

Etot1=@(theta,phi) 0.5.*sin theta).*eta.*cos((pi./2).*cos theta)).*
cos((pi./2).*cos theta))./((2.*pi).* (2.*pi).*sin theta).*sin theta
)).*((sin(((sqrt(i)).*k.*d./2).*sin theta).*cos phi)).*sin(((sqrt(
i)).*k.*d./2).*sin theta).*sin phi))./sin((k.*d./2).*sin theta).*c
os phi))./sin((k.*d./2).*sin theta).*sin phi))).^2);
P1 = integral2(Etot1,0,pi,0,2*pi);
Zres(i) = P1.*2./((i));

end

for i=B

Etotx=@(theta,phi) 0.5.*sin theta).*eta.*cos((pi./2).*cos theta)).*
cos((pi./2).*cos theta))./((2.*pi).* (2.*pi).*sin theta).*sin theta
)).*((sin((i.*k.*d./2).*sin theta).*cos phi)).*sin((k.*d./2).*sin(
theta).*sin phi))./sin((k.*d./2).*sin theta).*cos phi))./sin((k.*d
./2).*sin theta).*sin phi))).^2);
Etoty=@(theta,phi) 0.5.*sin theta).*eta.*cos((pi./2).*cos theta)).*
cos((pi./2).*cos theta))./((2.*pi).* (2.*pi).*sin theta).*sin theta
)).*((sin((k.*d./2).*sin theta).*cos phi)).*sin((i.*k.*d./2).*sin(
theta).*sin phi))./sin((k.*d./2).*sin theta).*cos phi))./sin((k.*d
./2).*sin theta).*sin phi))).^2);
Px = integral2(Etotx,0,pi,0,2*pi);
Py = integral2(Etoty,0,pi,0,2*pi);
Zresx(i) = Px.*2./i;
Zresy(i) = Py.*2./i;

end

Zres = Zres.find(Zres>0);
Zresx = Zresx.find(Zresx>0);
Zresy = Zresy.find(Zresy>0);
resonantwire_planar = [A;Zres]';
resonantwire_XY = [B;Zresx;Zresy]';

% Short Dipole Antenna

dl = lambda./10;
for i=A
    Etot1=@(theta,phi)((k.*dl.*sin theta)./(4.*pi)).^2.*eta.*sin(the
    ta)/2).*(sin((sqrt(i)).*k.*d./2).*sin theta).*cos phi).*sin((
    sqrt(i)).*k.*d./2).*sin theta).*sin phi)/sin((k.*d./2).*sin(the
    t a).*cos phi)/sin((k.*d./2).*sin theta).*sin phi)).^2);
    P1 = integral2(Etot1,0,pi,0,2*pi);
    Zdip(i) = P1.*2./((i));
end

end

for i=B
    Etotx=@(theta,phi)((k.*dl.*sin theta)./(4.*pi)).^2.*eta.*sin(the
    ta)/2).*(sin((i.*k.*d./2).*sin theta).*cos phi).*sin((k.*d./2).
    *sin theta).*sin phi)/sin((k.*d./2).*sin theta).*cos phi)/sin(
    (k.*d./2).*sin theta).*sin phi)).^2);
    Etoty=@(theta,phi)((k.*dl.*sin theta)./(4.*pi)).^2.*eta.*sin(the
    ta)/2).*(sin((k.*d./2).*sin theta).*cos phi).*sin((i.*k.*d./2).
    *sin theta).*sin phi)/sin((k.*d./2).*sin theta).*cos phi)/sin(
    (k.*d./2).*sin theta).*sin phi)).^2);
    Px = integral2(Etotx,0,pi,0,2*pi);
    Zdipx(i) = Px.*2./i;
    Py = integral2(Etoty,0,pi,0,2*pi);
    Zdipy(i) = Py.*2./i;
end

end

Zdip = Zdip.find(Zdip>0);
Zdipx = Zdipx.find(Zdipx>0);
Zdipy = Zdipy.find(Zdipy>0);
shortdipole_planar = [A;Zdip]';
shortdipole_XY = [B;Zdipx;Zdipy]';

% Patch Antenna

h = 1.6*(10^-3);
k = (2.*pi)/(lambda);
w = (lambda./2).*((er+1)/2).^-0.5);
eff = ((er+1)/2)+((er-1)/2).*((1+(10*h/w)).^(-0.5));
delta = 0.412*h*(eff+0.3)*((w/h)+0.262)./((eff-
0.258)*((w/h)+0.813));
l = (lambda/(2*sqrt(eff)))-(2*delta);
d = (2*l)+(2*delta);
d2 = l+delta;

% Testing with single antenna aperture

G2 = (pi.*w.* (1-(((k.*h)^2)/24)))./(eta.*lambda);
Zin1_with_single_aperture_test = 1/(G2);
E2=@(theta,phi) (2.*sin theta).*sin phi.*sin phi.*sin ((pi.*w./lam
bda).*sin theta).*sin phi).*sin((pi.*w./lambda).*sin theta).*sin(
phi)).*cos(pi.*cos phi).*sin theta./ (2.*sqrt(eff))).*cos(pi.*cos(
phi).*sin theta./ (2.*sqrt(eff)))./(pi.*pi.*sin theta).*sin phi.
*sin theta).*sin phi));
P2 = integral2(E2,0,pi/2,0,2*pi)/eta;
Zin2_with_single_aperture_test = 1/(P2.*2);
patch_with_single_aperture_compare=[Zin1_with_single_aperture_test
;Zin2_with_single_aperture_test] '

% Arrays with single aperture

for i=A
    Etot2=@(theta,phi) (2.*sin theta).*sin phi.*sin phi.*sin ((pi.*w./
lambda).*sin theta).*sin phi)).*sin((pi.*w./lambda).*sin theta).*s
in phi)).*cos pi.*cos phi).*sin theta)./(2.*sqrt(eff))).*cos pi.*c
os phi).*sin theta)./(2.*sqrt(eff)))./(pi.*pi.*sin theta).*sin(ph
i).*sin theta).*sin phi)).*((sin((sqrt(i)).*k.*d./2).*sin theta).
*cos phi)).*sin((sqrt(i)).*k.*d./2).*sin theta).*sin phi))./sin((
k.*d./2).*sin theta).*cos phi))./sin((k.*d./2).*sin theta).*sin(ph
i))).^2);
P2 = integral2(Etot2,0,pi/2,0,2*pi)/eta;
Zpatch2(i) = 1./((P2.*2./((i))));

end

for i=B
    Etotx2=@(theta,phi) (2.*sin theta).*sin phi.*sin phi.*sin ((pi.*w./
lambda).*sin theta).*sin phi)).*sin((pi.*w./lambda).*sin theta).*
sin phi)).*cos pi.*cos phi).*sin theta)./(2.*sqrt(eff))).*cos pi.*
cos phi).*sin theta)./(2.*sqrt(eff)))./(pi.*pi.*sin theta).*sin(p
hi).*sin theta).*sin phi)).*((sin((i.*k.*d./2).*sin theta).*cos ph
i)).*sin((k.*d./2).*sin theta).*sin phi))./sin((k.*d./2).*sin thet
a).*cos phi))./sin((k.*d./2).*sin theta).*sin phi))).^2);
    Etoty2=@(theta,phi) (2.*sin theta).*sin phi.*sin phi.*sin ((pi.*w./
lambda).*sin theta).*sin phi)).*sin((pi.*w./lambda).*sin theta).*
sin phi)).*cos pi.*cos phi).*sin theta)./(2.*sqrt(eff))).*cos pi.*
cos phi).*sin theta)./(2.*sqrt(eff)))./(pi.*pi.*sin theta).*sin(p
hi).*sin theta).*sin phi)).*((sin((k.*d./2).*sin theta).*cos phi))
.*sin((i.*k.*d./2).*sin theta).*sin phi))./sin((k.*d./2).*sin thet
a).*cos phi))./sin((k.*d./2).*sin theta).*sin phi))).^2);
Px2 = integral2(Etotx2,0,pi/2,0,2*pi)/eta;
Py2 = integral2(Etoty2,0,pi/2,0,2*pi)/eta;
Zpatchx2(i) = 1./ (Px2.*2./ (i));
Zpatchy2(i) = 1./ (Py2.*2./ (i));

end

Zpatch2 = Zpatch2 (find(Zpatch2>0));
ZpatchE2 = Zpatchx2 (find(Zpatchx2>0));
ZpatchH2 = Zpatchy2 (find(Zpatchy2>0));
patch_planar_with_single_aperture = [A;Zpatch2]';
patch_EH_with_single_aperture = [B;ZpatchE2;ZpatchH2]';

% Testing with one antenna element with two apertures

G = (pi.*w.* (1-(((k.*h)^2)/24)))./ (eta.*lambda);
Zin1_with_aperture_test = 1/(2.*G);
E=@(theta,phi) (2.*sin theta).*sin phi).*sin phi).*sin ((pi.*w./lamb
da).*sin theta).*sin phi)).*sin ((pi.*w./lambda).*sin theta).*sin(p
hi)).*cos(pi.*cos phi).*sin theta)./(2.*sqrt(eff))).*cos(pi.*cos(p
hi).*sin theta)./(2.*sqrt(eff)))./(pi.*pi.*sin theta).*sin phi).*
sin theta).*sin phi));
P = integral2(E,0,pi,0,2*pi)/eta;
Zin2_with_aperture_test = 1/(P.*2);
patch_with_aperture_compare =
[Zin1_with_aperture_test;Zin2_with_aperture_test]';

% Now for arrays

for i=A
    Etot1=@(theta,phi) (2.*sin theta).*sin((pi.*w./lambda).*sin theta).*sin phi)).*sin((pi.*w./lambda).*sin theta).*sin phi)).*cos(pi.*cos s phi).*sin theta)./(2.*sqrt(eff))).*cos(pi.*cos phi).*sin theta)./(2.*sqrt(eff))).*sin phi).*sin phi))./(pi.*pi.*sin theta).*sin ph i).*sin theta).*sin phi)).*((sin((sqrt(i)).*k.*d./2).*sin theta).*cos phi)).*sin((sqrt(i)).*k.*d./2).*sin theta).*sin phi))./sin(( k.*d./2).*sin theta).*cos phi))./sin((k.*d./2).*sin theta).*sin ph i))).^2);
P1 = integral2(Etot1,0,pi,0,2*pi)/eta;
Zpatch(i) = 1./ (P1.*2./ ((i)));
end

end

for i=B
    Etotx=@(theta,phi) (2.*sin theta).*sin phi).*sin phi).*sin((pi.*w./
lambda).*sin theta).*sin phi)).*sin((pi.*w./lambda).*sin theta).*s
in phi)).*cos(pi.*cos phi).*sin theta)./(2.*sqrt(eff))).*cos(pi.*c
os phi).*sin theta)./(2.*sqrt(eff)))./(pi.*pi.*sin theta).*sin ph
i).*sin theta).*sin phi)).*((sin((i.*k.*d./2).*sin theta).*cos phi
```matlab
)).*sin((k.*d./2).*sin theta).*sin phi))./sin((k.*d./2).*sin theta
).*cos phi))./sin((k.*d./2).*sin theta).*sin phi))).^2);
Etoty=@(theta,phi) (2.*sin theta).*sin phi).*sin phi).*sin((pi.*w./
lambda).*sin theta).*sin phi)).*sin((pi.*w./lambda).*sin theta).*s
in (phi)).*cos(pi.*cos phi).*sin theta)./(2.*sqrt(eff))).*cos(pi.*c
os phi).*sin theta)./(2.*sqrt(eff)))./(pi.*pi.*sin theta).*sin(ph
i).*sin theta).*sin phi)).*((sin((k.*d./2).*sin theta).*cos phi)).
*sin((i.*k.*d./2).*sin theta).*sin phi))./sin((k.*d./2).*sin theta
).*cos phi))./sin((k.*d./2).*sin theta).*sin phi))).^2);
Px = integral2(Etotx,0,pi,0,2*pi)/eta;
Py = integral2(Etoty,0,pi,0,2*pi)/eta;
Zpatchx(i) = 1./((Px.*2./i));
Zpatchy(i) = 1./((Py.*2./i));
end
```

```matlab
Zpatch = Zpatch find(Zpatch>0));
ZpatchE = Zpatchx find(Zpatchx>0));
ZpatchH = Zpatchy find(Zpatchy>0));
patch_planar_with_aperture = [A;Zpatch]';
patch_EH_with_aperture = [B;ZpatchE;ZpatchH]';
```

## % Plotting

```matlab
semilogx(A,Zres)
xlabel('No.of Antennas')
ylabel('Impedence')
title('PlanarArray')
grid on
hold on
semilogx(A,Zdip,'k')
hold on
semilogx(A,Zpatch,'g')
hold on
semilogx(A,Zpatch2,'b')
hold on
legend('ResonantWireAntennaplanar','Short DipoleAntennaplanar','Patch Antennaplanarwithaperture','PatchAntennaplanarwithout aperture');
plot(A,Zres,'r*')
hold on
plot(A,Zdip,'r*')
hold on
plot(A,Zpatch,'r*')
hold on
plot(A,Zpatch2,'r*')
hold on
figure
semilogx(B,Zresx,'--')
xlabel('No.of Antennas')
ylabel('Impedence')
title('patch antenna-Eplane')
```
grid on
hold on
semilogx (B, Zresy, ':')
hold on
semilogx (B, Zdipx, 'r')
hold on
semilogx (B, Zdipy, 'c')
hold on
semilogx (B, ZpatchE, 'b')
hold on
semilogx (B, ZpatchH, 'b')
hold on
plot (B, ZpatchE, 'r*')
hold on
plot (B, ZpatchH, 'r*');
<figure><img src="image_84.png" /></figure>
**Titre :** Etude de l'adaptation d'impédance dans le réseau d'antennes.

**Mots clés :** Réseau d'antennes, rayonnement total, impédance d'entrée, couplage mutuel, paramètres S actifs.

**Résumé :** La désadaptation d'impédance dans les réseaux d'antennes est un problème important, en particulier lorsqu'un grand nombre d'éléments rayonnants sont placés côte à côte. Cette désadaptation est généralement due au couplage mutuel entre les éléments d'antenne. Le phénomène de couplage mutuel peut cependant provenir des ondes de surface, du rayonnement de champ proche ou du rayonnement de champ lointain. Le phénomène de désadaptation se révèle néanmoins important lorsque l'ensemble du réseau est réglé pour rayonner même temps même si le couplage est faible. Dans cette thèse, nous concentrons nos travaux sur l'effet du rayonnement total en champ lointain. En mettant les antennes en réseau, l'impédance d'entrée de chaque élément du réseau d'antennes change par rapport à celle de l'élément unique lorsqu'il n'est pas en réseau.

En d'autres termes, la valeur de l'impédance d'entrée des éléments rayonnants individuels change en fonction du nombre d'éléments et de la position de l'antenne.

Ce travail propose en outre un aspect important de l'évaluation des paramètres S actifs dans les réseaux d'antennes. Il a été montré que les paramètres S actifs obtenus en utilisant la matrice de distribution passive ne correspondent pas aux valeurs réelles obtenues lors d'une excitation simultanée de tous les éléments.

Enfin, cette étude a fait un pas de plus vers deux nouvelles méthodes de découplage afin de valider la théorie du rayonnement total et l'évaluation des paramètres S actifs.

Les résultats théoriques sont complétés par des simulations 3D et des mesures.

**Title :** Study of impedance matching in antenna array.

**Keywords:** Antenna array, total radiation, input impedance, mutual coupling, active S-parameters.

**Abstract:** Impedance mismatching in antenna arrays is an important issue especially when a huge number of antenna elements are placed side by side. This mismatch is generally blamed to exist because of mutual coupling between antenna elements. The phenomenon of mutual coupling, however, may result due to surface waves, near field radiation or far field radiation. The phenomenon of mismatch, nevertheless, reveals itself strongly when the whole array is set to radiate even in presence of less significant coupling. In this thesis, we focus our work to the effect of total radiation due to far field radiation only. It says that by arraying the antennas the input impedance of each element of the antenna array changes as compared to that of the single element which is not arrayed.

In other words, the value of the input impedance of individual antenna elements changes versus antenna number and position. This work further proposes an important aspect of the evaluation of active S-parameters in antenna arrays. It has been shown that the active S-parameters obtained by using the passive scattering matrix do not correspond to the actual values obtained during a simultaneous excitation of all the elements.

Finally, this study has taken a step further towards two novel decoupling methods in order to validate the theory of total radiation and evaluation of active S-parameters.

Theoretical results are complemented with 3D simulations and measurements.