How to solve Microwave Engineering Pozar chapter 05 exercise 02 with MATLAB

Published by John_2016 in the blog How to solve Microwave Engineering Pozar examples and exercises with MATLAB. Views: 100


Question

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the literature source is available here:
https://www.amazon.co.uk/Microwave-Engineering-David-M-Pozar/dp/0470631554

There's also a solutions manual available here:
https://www.scribd.com/doc/176505749/Microwave-engineering-pozar-4th-Ed-solutions-manual



Answer

1.-

What kind of loads can be matched with a single positive reactance?

Any load that already has real(ZL)=Z0. Then the match is achieved with X=-imag(ZL)

So the only impedances that series jX may match are located along the red arch on the top Smith chart.

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2.-
What kind of loads can be matched with a single negative reactance?

This is the complementary case to (1) where jB in parallel can only match loads on the upper red arch along circle R=Z0.

[​IMG]


This single reactance approach doesn't match resistive loads.


3.-
Example ZL=2*Z0 imag(ZL)=0 inside circle constant R=Z0.

To match this impedance, Load to Generator, Red marker to Green marker, add a parallel resistance same value as ZL.

clc;clear all;close all
Z0=50
ZL=2*Z0

hf1=figure(1);sm1=smithchart; ax1=hf1.CurrentAxes;
hold(ax1,'on');

gamma_ZL=(ZL-Z0)/(ZL+Z0);

plot(ax1,real(gamma_ZL),imag(gamma_ZL),'o','Color',[1 0 0],'MarkerFaceColor',[1 0 0]) % ZL

gamma_Z0=(Z0-Z0)/(Z0+Z0);

plot(ax1,real(gamma_Z0),imag(gamma_Z0),'o','Color',[0 1 0],'MarkerFaceColor',[0 1 0]) % Z0
legend('ZL','Z0')
title(' ZL=2*Z0')


[​IMG]


4.-

Example ZL=1/2*Z0 imag(ZL)=0ZL outside circle constant R=Z0.
To match impedance, Load to Generator, Red marker to Green marker, add a series resistor same value as ZL.

ZL=.5*Z0

hf2=figure(2);sm2=smithchart; ax2=hf2.CurrentAxes;
hold(ax2,'on');

gamma_ZL=(ZL-Z0)/(ZL+Z0);

plot(ax2,real(gamma_ZL),imag(gamma_ZL),'o','Color',[1 0 0],'MarkerFaceColor',[1 0 0]) % ZL

gamma_Z0=(Z0-Z0)/(Z0+Z0);

plot(ax2,real(gamma_Z0),imag(gamma_Z0),'o','Color',[0 1 0],'MarkerFaceColor',[0 1 0]) % Z0
legend('ZL','Z0')
title(' ZL=Z0/2')


[​IMG]

Following, another 2 examples where Z0' is no longer the centre of the Smith chart:

5.-
Z Smith Chart: Randomly selecting ZL and Z0, but both on same constant R circle.
Load to Generator: Red marker to Green marker.

SC_ref=50;

N=10

real_Z0=.01*randi([round([SC_ref/N SC_ref*N*100],4)],1,1)
Z0=real_Z0+1j*.01*randi([round([SC_ref/N*1000 SC_ref*N*100],4)],1,1)
ZL=real_Z0+1j*.01*randi([round([SC_ref/N*100 SC_ref*N*100],4)],1,1)

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

gamma_ZL=(ZL-SC_ref)/(ZL+SC_ref);

plot(ax3,real(gamma_ZL),imag(gamma_ZL),'o','Color',[1 0 0],'MarkerFaceColor',[1 0 0]) % ZL

gamma_Z0=(Z0-SC_ref)/(Z0+SC_ref);

plot(ax3,real(gamma_Z0),imag(gamma_Z0),'o','Color',[0 1 0],'MarkerFaceColor',[0 1 0])
% Z0

Smith_plotRcircle(ax3,real(Z0),SC_ref,[.8 .2 .2])
legend('ZL','Z0','circle constant R')


[​IMG]

If Z Load on right hand side of Z0', both on same circle constant R,
add series capacitance to rotate ZL Counter Clock Wise (CCW) along circle constant R.


[​IMG]

If Z Load on left hand side of Z0', both on same circle constant R,
add series inductance to rotate ZL Clock Wise (CW) along circle constant R.



6.-
Using Y Smith Chart constant G circles on Z Smith Chart:
Randomly selecting YL and Y0, but both on same constant G circle.
Load to Generator: Red marker to Green marker.

SC_ref=10;
N=10

real_Y0=(.01*randi([round([SC_ref/N SC_ref*N],4)],1,1))^-1
Y0=real_Y0+1j*.01*randi([round([SC_ref/N SC_ref*N*100],4)],1,1)
YL=real_Y0+1j*.01*randi([round([SC_ref/N SC_ref*N*100],4)],1,1)

Z0=1/Y0
ZL=1/YL

hf4=figure(4);sm4=smithchart; ax4=hf4.CurrentAxes;


To use Y Smith chart field Type of the Smith Chart handle has to be changed:

sm4.Type='y'

hold(ax4,'on');

gamma_YL=(SC_ref-YL)/(SC_ref+YL);

plot(ax4,real(gamma_YL),imag(gamma_YL),'o','Color',[1 0 0],'MarkerFaceColor',[1 0 0]) % YL

gamma_Y0=(SC_ref-Y0)/(SC_ref+Y0);

plot(ax4,real(gamma_Y0),imag(gamma_Y0),'o','Color',[0 1 0],'MarkerFaceColor',[0 1 0]) % Y0

Smith_plotGcircle(ax4,1/real(Y0),1/SC_ref,[.8 .2 .2])
legend('YL','Y0','circle constant G')



[​IMG]


But as mentioned, it's more practical to use Y Smith Chart constant G circles on the Z Smith Chart instead.

hf5=figure(5);sm5=smithchart; ax5=hf5.CurrentAxes;
sm5.Type='z' % Z is the default Smith Chart type

hold(ax5,'on');

gamma_ZL=-gamma_YL
% =(SC_ref-YL)/(SC_ref+YL);
plot(ax5,real(gamma_ZL),imag(gamma_ZL),'o','Color',[1 0 0],'MarkerFaceColor',[1 0 0]) % ZL

gamma_Z0=-gamma_Y0
% =(SC_ref-Y0)/(SC_ref+Y0);
plot(ax5,real(gamma_Z0),imag(gamma_Z0),'o','Color',[0 1 0],'MarkerFaceColor',[0 1 0]) % Z0

Smith_plotRcircle(ax5,real(Y0),SC_ref,[.8 .2 .2])
legend('ZL','Z0','circle constant G','Location','northeastoutside')


[​IMG]

So, for the readers who are not that familiar with the Smith Chart, this
Smith Chart plot is a Z Smith Chart, showing ZL and Z0 with red and
green markers respectively, yet the red circle is a constant G circle
belonging to the reversed Y Smith Chart, all on the same chart.

If Y Load on right hand side of Y0', both being on same constant G circle,
add series capacitance to rotate YL Counter Clock Wise (CCW) along circle constant G.

If Y Load on left hand side of Y0', both being on same constant G circle,
add series inductance to rotate YL Clock Wise (CW) along circle constant G.



Summary:

When real(ZL)=real(Z0) and imag(ZL)!=0 there is a single series or parallel reactance that can match this type of loads.



And in general:

When adding series capacitance: Impedance rotates CCW along Z Smith Chart constant resistance circle.

When adding series inductance: Impedance rotates CW along Z Smith Chart constant resistance circle.

When adding parallel capacitance: Admittance rotates CW along Y Smith Chart constant G (conductance) circle.

When adding parallel inductance: Admittance rotates CCW along Y Smith Chart constant G circle.
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