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exercise 5.10

ZIP with MATLAB scripts and note:

Small tag OK.jpg
pozar_05_exercise_10_question.jpg

 exercise 5.10 notes:

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double SC stub TL 3 lambda 8th.jpg

Analytical solution with |s11| surface

Z0=50;ZL=Z0/(.4+1j*1.2);

D3=3/8;

Dstep=.0002;drange=[0:Dstep:.5];

D1=drange;D2=D1;

 

Zin4=zeros(numel(drange),numel(drange));

s11=zeros(numel(drange),numel(drange));

 

for k=1:1:numel(D1)

    for s=1:1:numel(D2)

        Zin_stub1=Z0*1j*tan(2*pi*D1(k));

        Zin2=(Zin_stub1.^-1+ZL.^-1).^-1;

        Zin3=Z0*(Zin2+1j*Z0*tan(2*pi*D3))./(Z0+1j*Zin2*tan(2*pi*D3));

        Zin_stub2=Z0*1j*tan(2*pi*D2(s));

        Zin4(k,s)=(Zin_stub2.^-1+Zin3.^-1).^-1;

        s11(k,s)=(Zin4(k,s)-Z0)/(Zin4(k,s)+Z0);

     end

end

 

hf1=figure(1);hs1=surf(D1,D2,abs(s11))

ax1=hf1.CurrentAxes;

hs1.LineStyle='none'

xlabel('D2');ylabel('D1')

 

 

 

 

 

 

 

campos(ax1,10*[0   0   100]);

camroll(ax1,-45)

 

ax1.PlotBoxAspectRatio=[1 1 1]  

 

hold(ax1,'on')

n0=find(drange==.5)               % plotting corner to box D1<.5 D2<.5

plot3(ax1,[drange(n0) drange(n0) 0],[0 drange(n0) drange(n0)],…

[5 5 5],'Color',[1 0 0],'LineWidth',3)

001-1.jpg
001-2.jpg

Conditioning surface to catch |s11| nulls

% V=1-(abs(s11));           

V=10^3*del2(abs(s11));      

 

hf2=figure(2);ax2=gca;surf(V,'Lines','none');xlabel('D1');ylabel('D2');

ax2.XTickLabelMode='manual';

ax2.YTickLabelMode='manual';

ax2.XTickLabel=[0:0.1:.5];ax2.YTickLabel=[0:0.1:.5];

 

[pks,locs]=findpeaks(V(:),'MinPeakHeight',.99,'MinPeakDistance',200); 

 

[nd1,nd2]=ind2sub(size(V),locs);

hold all;plot3(nd2,nd1,V(nd2,nd1)+2,'ro');  % verifying peaks locations

 

plot3(ax2,[drange(n0) drange(n0) 0],[0 drange(n0) drange(n0)],…

[5 5 5],'Color',[1 0 0],'LineWidth',3)

 

Dstep=.0002;drange=[0:Dstep:.5];

D1=drange;D2=D1;

 

 

 

 

 

 

 

 

D1=unique(sort(D1(nd1)))

 

 

 

 

% same as

D1=drange;D2=D1;

D1=D1(unique(sort(nd1)))

 

 

 

 

 

 

 

D2=unique(sort(D2(nd2)))

002-1.jpg

D1 =

  Columns 1 through 3

    0.050600000000000   0.050800000000000   0.051000000000000

   Columns 4 through 6

    0.051200000000000   0.051400000000000   0.051600000000000

   Columns 7 through 9

    0.051800000000000   0.098600000000000   0.098800000000000

 

 D1 =

   Columns 1 through 3

    0.050600000000000   0.050800000000000   0.051000000000000

   Columns 4 through 6

    0.051200000000000   0.051400000000000   0.051600000000000

   Columns 7 through 9

    0.051800000000000   0.098600000000000   0.098800000000000

 

 D2 =

   Columns 1 through 3

    0.048600000000000   0.048800000000000   0.049000000000000

   Columns 4 through 6

    0.049200000000000   0.049400000000000   0.049600000000000

..

    0.374400000000000   0.374600000000000   0.374800000000000

   Columns 31 through 33

    0.375000000000000   0.375200000000000   0.375400000000000

   Columns 34 through 35

    0.375600000000000   0.375800000000000

Now the quite different shape of the peaks returns a different amount of peak locations despite from both previous graphs one can tell that there are 2 clear |s11| nulls. Here is where function kmeans comes quite handy

On the right hand side, the closest that the above surface and chosen conditioning gets us to peaks locations, so far.

 

Since from |s11| it's highly likely that |s11| has 2 nulls within [0 .5] x [0 .5] wavelength fraction D, L stub ength, L=D*l , let's use kmeans with k=2.

locs1

=

      608003

      610504

      613005

      615506

      618006

      620507

      623008

      625509

      628009

      630510

      633011

      635512

      638012

      640513

      643014

      645514

      648015

      650516

      653017

      655517

      658018

      660519

      663020

      665520

      668021

      670522

locs2

=

     4679866

     4682367

     4684868

     4687369

     4689870

     4692370

     4694871

     4697372

     4699873

locs3

=

   1.037676749975155

   1.127321441557838

   1.219759148078254

   1.307902495524382

   1.436938848741699

   1.626818258880166

   1.828290158620465

   2.008170018700442

   2.327551443905220

   2.920029348588793

   3.609215877084772

   4.005216276204536

   4.917328558982023

  10.188490651197503

   5.714683330692673

   3.641781956334297

   3.487384399634491

   2.877672242184274

   2.296011113485124

   1.878303169964850

   1.711972650510920

   1.518129552003200

   1.332318001956141

   1.185153344920222

   1.096700695677261

   1.005028697436138

locs4

=

   1.022294122904170

   1.288112420134564

   1.616498278070448

   1.813192017180561

   1.479922650697164

   1.467160456281476

   1.526533687583176

   1.328279542622017

   1.096739121053541

[idx,C]=kmeans(locs,2)              % grouping peaks locations

nlocs=[1:1:numel(locs)]      % finging linear indices of peaks locations in locs

nlocs1=nlocs(idx==1)        % linear indices of 1st peak locations in locs

nlocs2=nlocs(idx==2)

locs1=locs(nlocs1)            % locations in surface of 1st group of peaks

locs2=locs(nlocs2)

pks1=pks(nlocs1)                          %   values of 1st group of peaks

pks2=pks(nlocs2)

 

[max_pks1,n_max_pks1]=max(pks1)         % what value has each peak

[max_pks2,n_max_pks2]=max(pks2)

pks1(n_max_pks1)

pks2(n_max_pks2)

 

nlocs1(n_max_pks1) % peaks locs in string locator returned by findpeaks

nlocs2(n_max_pks2)

 

 

loc_peak1=locs(nlocs1(n_max_pks1)) % peaks locations in V and |s11|

loc_peak2=locs(nlocs2(n_max_pks2))

 

[n1_pk1,n2_pk1]=ind2sub(size(V),loc_peak1)           

 

640513 is pointer equivalent to [257 257]

 

 

 

[n1_pk2,n2_pk2]=ind2sub(size(V),loc_peak2)           

 

 4687369 is pointer equivalent to [495 1875]

 

 

check on V surface

 

hold all;plot3(n2_pk1,n1_pk1,V(n2_pk1,n1_pk1)+2,'go',...

                                                                        'MarkerEdgeColor','g',…

                                                                        'MarkerFaceColor','g',…

                                                                        'MarkerSize',10)

 

hold all;plot3(n2_pk2,n1_pk2,V(n2_pk2,n1_pk2)+2,'ro',...

                                                                        'MarkerEdgeColor','g',…

                                                                        'MarkerFaceColor','g',…

                                                                        'MarkerSize',10)

 

 

 

 

 

 

 

=   10.188490651197503

 =   1.813192017180561

 

 =    14

 =     30

 

 

  =       640513

 =      4687369

 

 

 n1_pk1 =

    257

 

n2_pk1 =

    257

 

n1_pk2 =

    495

​

 n2_pk2 =

         1875

002-2.jpg

campos(1.0e+04*[-0.8847438 -2.2297478  0.0012768])

                                                                   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

D1=drange;D2=D1;

D11=D1(n1_pk1)

D12=D1(n1_pk2)

 

D21=D2(n2_pk1)

D22=D2(n2_pk2)

 

putting all stub lengths in same matrix

 

D=[D11 D12;D21 D22]                                           

002-3.jpg

D11 =    0.051200000000000

D12 =    0.098800000000000

 

D21 =    0.051200000000000

D22 =    0.374800000000000

 

 

 

D =

   0.051200000000000   0.098800000000000

   0.051200000000000   0.374800000000000

Frequency response directly connecting component impedances

c0=2.998e8;

 

stubs physical lengths                    

 

L=D*c0/f0                                       

 L3=D3*c0/f0

 

f0=2e9;

df=1e6;

 

defining frequency band to sweep with f1 f2 start stop frequencies

 

f1=1e9;f2=3e9; f=[f1:df:f2];

frel=linspace(f1/f0,f2/f0,numel(f));

 

connecting impedances

 

Zin_stub1=Z0*1j*tan(2*pi*frel*f0/c0*L(1,1))

Zin2=(Zin_stub1.^-1+ZL.^-1).^-1;

Zin3=Z0*(Zin2+1j*Z0*tan(2*pi*frel*f0/c0*L3))./…

(Z0+1j*Zin2.*tan(2*pi*frel*f0/c0*L3)) ;

Zin_stub2=Z0*1j*tan(2*pi*frel*f0/c0*L(2,1));

Zin4=(Zin_stub2.^-1+Zin3.^-1).^-1

s11_1=(Zin4-Z0)./(Zin4+Z0)

 

 Zin_stub1=Z0*1j*tan(2*pi*frel*f0/c0*L(1,2))

Zin2=(Zin_stub1.^-1+ZL.^-1).^-1;

Zin3=Z0*(Zin2+1j*Z0*tan(2*pi*frel*f0/c0*L3))./…

(Z0+1j*Zin2.*tan(2*pi*frel*f0/c0*L3)) ;

Zin_stub2=Z0*1j*tan(2*pi*frel*f0/c0*L(2,2));

Zin4=(Zin_stub2.^-1+Zin3.^-1).^-1

s11_2=(Zin4-Z0)./(Zin4+Z0)

 

hf3=figure(3); %ax3=hf3.CurrentAxes

plot(frel,abs(s11_1),frel,abs(s11_2));grid on;

xlabel('f/f0')

title('|s11(f)|')

legend({'D(1,1) D(2,1)','D(1,2) D(2,2)'},… 'location','northeastoutside')

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