exercise 5.2

ZIP with MATLAB scripts and note:

exercise 5.2 notes:

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.

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.

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')

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')

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')

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.

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')

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')

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.

exercise 5.2

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.