exercise 5.9

ZIP with MATLAB scripts and note:

exercise 5.9 notes:

Analytic solution with a surface

Z0=50;ZL=Z0/(.4+1j*1.2);D3=1/8;

Dstep=.001;drange=[0:Dstep:1];

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

abs(s11);

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

ax1=hf1.CurrentAxes;ax1_backup=ax1

hs1.LineStyle='none'

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

campos([.5 .5 9.1602])

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)

Conditioning surface to capture peaks, it's useful that the peaks are sharper than just 1./abs(s11):

V=1e3*del2(abs(s11));

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

ax2.XTickLabelMode='manual';

ax2.XTickLabel=[0:0.1:1];ax2.YTickLabel=[0:0.1:1];

ax2.XTick=[0:100:1000];ax2.YTick=[0:100:1000];

[pks,locs]=findpeaks(V(:),'MinPeakHeight',9,'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[0 drange(n0) drange(n0)],[5 5 5],'Color',[1 0 0],'LineWidth',3)

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

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

abs(s11(sub2ind(size(V),nd1,nd2))) % |s11| linear values

D_surf=[D1;D2]

taking only stub lengths below lambda/2

D_surf=D_surf([1 2],[1 2])

D11*360

D12*360

findpeaks options minpeakheigh and mineapkdist manually trimmed to avoid getting too many peaks

numel(nd1)

V(sub2ind(size(V),nd1,nd2))

the |s11| linear values onmatch points:

abs(s11(sub2ind(size(V),nd1,nd2)))

1st solution, stubs with lengths below 0.5:

D11=0.086

D12=0.375

2nd solution

D21=0.199

D22=0.375

Not mentioned in the solutions manual, but choosing the shortest stubs doesn't mean stubs longer than lambda/2 are not perfectly valid.

One can manually change the view point with the rotate button on the figure window toolbar:

=

Columns 1 through 3

0.086000000000000 0.375000000000000 0.586000000000000

Column 4

0.875000000000000

=

Columns 1 through 3

0.199000000000000 0.375000000000000 0.699000000000000

Column 4

0.875000000000000

D_surf =

0.0860000 0.3750000 0.5860000 0.8750000

0.1990000 0.3750000 0.6990000 0.8750000

D_surf =

0.086000000000000 0.375000000000000

0.199000000000000 0.375000000000000

= 1.350019397711374e+02

= 30.957734133059738

= 8

=

15.367723269270918

15.367723269272101

10.995122045537965

10.995122045537279

15.367723269270671

15.367723269271853

10.995122045537727

10.995122045537039

=

0.006499310969800

0.006499310969799

0.000000000000001

0.006499310969801

0.006499310969800

0.000000000000001

for loops are faster than compact but slower expressions

The direct assault, without double for loops, with my current platform, imposes a drastic reduction in the D1 D2 resolution steps, both up to 0.01, and it takes about 10 fold more time to produce result than with the double for.

Dstep=.01;drange=[0:Dstep:1];

[D1,D2]=meshgrid(drange,drange);

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

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));

[Zin_stub2,Zin3]=meshgrid(zin_stub2,zin3);

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

s11=(Zin4-Z0)./(Zin4+Z0);

When attempting even a modest d=0.001 without double for loop a serious RAM shortage takes place:

and d=0.0001

Apparently 'smart' compact nice looking expressions sometimes require more processing capacity than equivalent vectorised expressions.

'Looking smart' goes after 'being smart' not the other way round.

Error using repmat

Requested 1002001x10001 (149.3GB) array exceeds maximum array size

preference. Creation of arrays greater than this limit may take a

long time and cause MATLAB to become unresponsive. See array size

limit or preference panel for more information.

Error in meshgrid (line 58)

xx = repmat(xrow,size(ycol));

Error using repmat

Requested 100020001x10001 (14905.6GB) array exceeds maximum array size preference …

With Smith Chart

1st OC stub, next to load, 1st shortest length:

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

hf3=figure(3);sm3=smithchart; ax3=hf3.CurrentAxes;hold(ax3,'on')

gammaZL=(ZL-Z0)/(ZL+Z0); % mark ZL

[gammaZL_angle,gammaZL_mod]=cart2pol(real(gammaZL),imag(gammaZL))

hold all;plot(real(gammaZL),imag(gammaZL),'ro','LineWidth',1.5);

YL=1/ZL;

gammaYL=(YL-1/Z0)/(YL+1/Z0); % mark YL, no need to switch to Y Smith chart

[gammaYL_angle,gammaYL_mod]=cart2pol(real(gammaYL),imag(gammaYL))

plot(real(gammaYL),imag(gammaYL),'go','LineWidth',1.5);

D3=1/8; % stretch of TL between stubs

1st shunt stub circle

[xc1,yc1]=Smith_plotStubCircle(ax,1/Z0,D3,[0 1 1]);

[x_r1,y_r1]=Smith_plotRcircle(ax,Z0^2/ZL,Z0,[1 0.5 .5]);

[px1,py1]=intersections(xc1,yc1,x_r1,y_r1); % constant R circle for YL

plot(px1,py1,'bo');

Left hand side of YL 1st intersection:

gammaY11=px1(1)+1j*py1(1)

reverse reflection coeff to admittance

gammaZ11=-gammaY11

z11=(1-gammaZ11)/(1+gammaZ11)

y11=1/z11

Y11=y11/Z0

[x_Barc_stub11_1,y_Barc_stub11_1]=Smith_plotBarc(ax3,YL,1/Z0,[1 0 0])

[x_Barc_stub11_2,y_Barc_stub11_2]=Smith_plotBarc(ax3,1/Y11,Z0,[1 0 0])

[dX11,arcSC_stub11]=Smith_arcStub(ax3,gammaYL,gammaY11,Z0,1,[0 0 1])

y11_stub1=-1j*dX11/Z0

D11=atan(y11_stub1/1j)/(2*pi)

if D11<0 D11=D11+.5; end

D11

D11*360

gammaY11 = -0.399969149078284 + 0.200061529810982i

gammaY12 = 0.461478371609742 + 0.692294433963756i

D11 = 0.375005388253159

=

1.350019397711374e+02

1st OC stub, 2nd shortest length:

hf4=figure(4);sm4=smithchart;

ax4=hf4.CurrentAxes

hold(ax4,'on')

plot(ax4,real(gammaZL),imag(gammaZL),'ro','LineWidth',1.5);

plot(ax4,real(gammaYL),imag(gammaYL),'go','LineWidth',1.5);

[xc1,yc1]=Smith_plotStubCircle(ax4,1/Z0,D3,[0 1 1]);

plot constant r=1 circle shifted l/8 towards generator on Y Smith chart 180deg shifted

[x_r1,y_r1]=Smith_plotRcircle(ax4,Z0^2/ZL,Z0,[1 0.5 5]);

plot(ax4,px1,py1,'bo');

1st intersection 2nd point, on Smith chart right hand side to YL, shunt stub has to shift impedance YL to Y12 CW, d=1

gammaY12=px1(2)+1j*py1(2)

gammaZ12=-gammaY12

from reflection coeff to admittance

z12=(1-gammaZ12)/(1+gammaZ12)

y12=1/z12;Y12=y12/Z0

[x_Barc_stub12_1,y_Barc_stub12_1]=Smith_plotBarc(ax4,YL,1/Z0,[1 0 0])

[x_Barc_stub12_2,y_Barc_stub12_2]=Smith_plotBarc(ax4,1/Y12,Z0,[1 0 0])

[dX12,arcSC_stub12]=Smith_arcStub(ax4,gammaYL,gammaY12,Z0,-1,[0 0 1])

y12_stub1=1j*dX12/Z0

D12=atan(y12_stub1/1j)/(2*pi)

if D12<0 D12=D12+.5; end

D12

ADS by default supplies stub length results in degree

D12*360

D12 = 0.08599370592516

= 30.957734133059738

2nd OC stub, 1st shortest length:

hf5=figure(5);sm5=smithchart;

ax5=hf5.CurrentAxes

hold(ax5,'on')

plot(ax5,real(gammaZL),imag(gammaZL),'ro','LineWidth',1.5);

plot(ax5,real(gammaYL),imag(gammaYL),'go','LineWidth',1.5);

plot(ax5,px1,py1,'o','Color',[.6 .6 1]); % intersection points for 1st stub

Smith_plotStubCircle(ax5,1/Z0,D3,[.7 1 1]);

% r=1 circle rotated lambda/8 away from generator Y Smith chart

D3_TLshift=pi/4 % TL shift lambda/8 in rad

a1=-2*D3_TLshift % D3 shift on Smith chart, towards generator on Y Smith chart

rotation matrix

G=[cos(a1) -sin(a1);sin(a1) cos(a1)];

Y11=y11/Z0;Y12=y12/Z0;

[x_GCz21,y_GCz21]=Smith_plotGammaCircle(ax5,1/Y11,Z0,[1 .8 .8]);

[x_GCz22,y_GCz22]=Smith_plotGammaCircle(ax5,1/Y12,Z0,[1 .8 .8]);

constant resistance circle g=re(YL)

[x_r1,y_r1]=Smith_plotRcircle(ax5,Z0^2/ZL,Z0,[1 0.5 .5]);

Pc2=G*[xc1;yc1] % rotating stub circle

xc2=Pc2(1,:);yc2=Pc2(2,:);

plot(ax5,xc2,yc2,'c') % circle r=1

P2=G*[px1';py1'] % rotating intersection points

px2=P2(1,:);py2=P2(2,:);

plot(ax5,px2,py2,'bo') % intersection points on r=1 circle

gammaY21=px2(1)+1j*py2(1)

gammaZ21=-gammaY21 % from reflection coeff to admittance

z21=(1-gammaZ21)/(1+gammaZ21)

Z21=z21*Z0

y21=1/z21

Y21=y21/Z0

[x_Barc_stub21,y_Barc_stub21]=Smith_plotBarc(ax5,1/Y21,Z0,[1 0 0])

[x_Barc_stub22,y_Barc_stub22]=Smith_plotBarc(ax5,0,Z0,[1 0 0])

[dX21,arcSC_stub21]=Smith_arcStub(ax5,-1,gammaY21,Z0,-1,[0 0 1])

y21_stub2=-1j*dX21/Z0 % -1 quick fix, pending more robust fix

D21=atan(y21_stub2/1j)/(2*pi)

if D21<0 D21=D21+.5; end; D21

D21 = 0.374993889549368

hf6=figure(6);sm6=smithchart;

ax6=hf6.CurrentAxes

hold(ax6,'on')

plot(ax6,real(gammaZL),imag(gammaZL),'ro','LineWidth',1.5);

plot(ax6,real(gammaYL),imag(gammaYL),'go','LineWidth',1.5);

plot(ax6,px1,py1,'o','Color',[.6 .6 1]); Smith_plotStubCircle(ax6,1/Z0,D3,[.7 1 1]);

[x_GCz21,y_GCz21]=Smith_plotGammaCircle(ax6,1/Y11,Z0,[1 .8 .8]);

[x_GCz22,y_GCz22]=Smith_plotGammaCircle(ax6,1/Y12,Z0,[1 .8 .8]);

[x_r1,y_r1]=Smith_plotRcircle(ax6,Z0^2/ZL,Z0,[1 0.5 .5]);

% constant resistance circle g=re(YL)

plot(ax6,xc2,yc2,'c') % r=1 circle

plot(ax6,px2,py2,'bo') % intersection points on r=1 circle

gammaY22=px2(2)+1j*py2(2)

gammaZ22=-gammaY22

z22=(1-gammaZ22)/(1+gammaZ22)

Z22=z22*Z0

y22=1/z22

Y22=y22/Z0

[x_Barc_stub12_2,y_Barc_stub12_2]=Smith_plotBarc(ax6,1/Y22,Z0,[1 0 0])

[x_Barc_stub12_2,y_Barc_stub12_2]=Smith_plotBarc(ax6,Z0,Z0,[1 0 0])

[dX22,arcSC_stub22]=Smith_arcStub(ax6,-1,gammaY22,Z0,1,[0 0 1])

y22_stub2=1j*dX22/Z0

D22=atan(y22_stub2/1j)/(2*pi)

if D22<0 D22=D22+.5; end

D22

D22*360

D22 = 0.198792933210378

= 71.565455955736013

I'm a paragraph. Click here to add your own text and edit me. It's easy.

Frequency Response

c0=2.998e8;f0=2e9;

df=1e6;f1=1e9;f2=3e9;f=[f1:df:f2];D_11=drange;

Putting all stub lengths together in same matrix

D=[D11 D12;D21 D22]

a1=atan(D11/D21)

a1*180/pi

close(hf1)

D1=drange;D2=D1;

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

ax1=hf1.CurrentAxes;ax1_backup=ax1

hs1.LineStyle='none'

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

ax1.PlotBoxAspectRatio=[1 1 1]

hold(ax1,'on')

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

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

check what way cross sections should go.

plot3(ax1,[0 3*D21],[0 3*D11],[1 1],'-r','LineWidth',1.5)

plot3(ax1,[0 5*D22],[0 5*D12],[1 1],'-r','LineWidth',1.5)

plot3 pushes camera position away, but only for certain data, just to avoid the resulting view point being left in the middle of the data. The following 2 lines restore cam pos to bird eye position used at the beginning of this script.

campos(ax1,10*[0 0 100]);

camroll(ax1,-45) % input angle in degree

D = 0.375005388253159 0.085993705925166

0.374993889549368 0.198792933210378

= 0.785413495017266

= 45.000878437108625

a1=atan(D11/D21)

a1*180/pi

if a1>pi/4

p=[1 floor(numel(drange)*D21/D11);1 numel(drange)]; end

if a1<pi/4

p=[1 numel(drange);1 floor(numel(drange)*D11/D21)]; end

if a1==pi/4 p=ones(2); end

p=p';

ny1=floor(linspace(p(1,1),p(2,1),max(abs(p(1,1)-p(2,1))+1,…

abs(p(1,2)-p(2,2))+1)));

nx1=floor(linspace(p(1,2),p(2,2),max(abs(p(1,1)-p(2,1))+1,…

abs(p(1,2)-p(2,2))+1)));

abs_S11=abs(s11);

abs_S11_xsection1=[]

for k=1:1:numel(nx1)

abs_S11_xsection1=[abs_S11_xsection1 abs_S11(nx1(k),ny1(k))];

end

hf201=figure(201);

subplot(2,1,1);plot(D_11(ny1),abs_S11_xsection1);grid on;xlabel('D2')

subplot(2,1,2);plot(D_11(nx1),abs_S11_xsection1);grid on;xlabel('D1')

a2=atan(D12/D22)

a2*180/pi

if a2>pi/4 p=[1 floor(numel(drange)*D22/D12);1 numel(drange)]; end

if a2<pi/4 p=[1 numel(drange);1 floor(numel(drange)*D12/D22)]; end

if a2==pi/4 p=ones(2); end

p=p';

ny2=floor(linspace(p(1,1),p(2,1),max(abs(p(1,1)-…

p(2,1))+1,abs(p(1,2)-p(2,2))+1)));

nx2=floor(linspace(p(1,2),p(2,2),max(abs(p(1,1)-p(2,1))+1,abs(p(1,2)-p(2,2))+1)));

abs_S11_xsection2=[]

for k=1:1:numel(nx1)

abs_S11_xsection2=[abs_S11_xsection2 abs_S11(nx2(k),ny2(k))];

end

hf202=figure(202);

subplot(2,1,1);plot(D_11(ny2),abs_S11_xsection2);grid on;xlabel('D2')

subplot(2,1,2);plot(D_11(nx2),abs_S11_xsection2);grid on;xlabel('D1')

= 0.785413495017266

= 45.000878437108625

= 0.408272819072597

= 23.392309422768086

Experiment, sweep Transmission line length between both stubs

In the last part of the script, not reproduced here because there's still an error pending correction, with a basic for loop that sweeps D3 between 1.1*l/8 and 0.75*l/8.

Resulting OC stub lengths stored in D_11 D_12 D_21 and D_22