example 5.1

ZIP with MATLAB scripts and note:

example 5.1 notes:

How to decide whether first next-to-load circuit element has to be shunt or series, when single frequency matching with 2-element L-shape networks.

Since near short-circuit impedances are prone to increased noise and may damage power supplies, [POZAR] suggests the criterion displayed on the right

The matching network orientation, whether leftward or rightward, is assigned to avoid short circuit (SC), open circuit (OC), or unnecessarily long impedance trajectories along the Smith chart. If the load on the Smith chart is located inside the 1+jx circle, then, in order to reduce impedance, start with shunt. If load is outside circle 1+jx, increase impedance with series component.

Comment: the term 'series' is here used as the jX component connected in both circuits above.

But in some literature sources like [MEDLY] a distinction is made between 'cascade' that would the jX component connection above, and 'series' would be as shown on the right hand side:

[MEDLY] Microwave and RF Circuits: Analysis, Synthesis and Design. Author Max W Medley Jr.

The following puts ZL on Smith Chart, tells whether ZL is inside or outside 1+jx, and calculates match values X and B for a single frequency. Ceq value has been previously calculated, it can be done with pen and paper, MATLAB or ADS LEMtch Assistant:

clc;clear all;format long;

ZL=200-1j*100;Z0=100;f0=5e8; % Hz

Ceq=1/(2*pi*f0*mag(ZL)); % 3.183pF at 500MHz for ADS LEMtch assistant

sm1=smithchart;hold all;

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

if imag(ZL)<0 sign1='-'; % show ZL on Smith chart adding text showing values

else sign1='+';

end

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

str1=['ZL =' num2str(real(ZL)) sign1 'j' num2str(abs(imag(ZL))) ' \rightarrow'];

text(real(gamma_L),imag(gamma_L)+.01, str1,...

'Color','blue',...

'FontSize',20,...

'HorizontalAlignment','right',...

'VerticalAlignment','middle');

RL=real(ZL);XL=imag(ZL);

if abs(real(ZL/Z0))>=1 % ZL inside 1+jx

disp(' ZL inside 1+jx');

B1=1/(RL^2+XL^2)*(XL+(RL/Z0)^.5*(RL^2+XL^2-Z0*RL)^.5)

B2=1/(RL^2+XL^2)*(XL-(RL/Z0)^.5*(RL^2+XL^2-Z0*RL)^.5)

X1=1/B1+XL*Z0/RL-Z0/(B1*RL) ; X2=1/B2+XL*Z0/RL-Z0/(B2*RL)

elseif abs(real(ZL/Z0))<=1 % ZL outside 1+jx

disp(' ZL outside 1+jx');X1=(RL*(Z0-RL))^.5-XL; X2=-(RL*(Z0-RL))^.5-XL

B1=1/Z0*((Z0-RL)/RL)^.5; B2=-1/Z0*((Z0-RL)/RL)^.5

end

[C1 ctype]=B2LC(B1,f0) % 1st solution

[L1 ltype]=X2LC(X1,f0)

[L2 ctype]=B2LC(B2,f0) % 2nd solution

[C2 ltype]=X2LC(X2,f0)

ZL inside 1+jx

B1 =

0.002898979485566

B2 =

-0.006898979485566

X1 =

1.224744871391589e+02

X2 =

-1.224744871391589e+02

C1 = 9.227738300997709e-13

ctype =C [Farad]

L1 = 3.898484006168380e-08

ltype =L [Henry]

L2 = 2.167378326912196e+07

ctype =L [Henry]

C2 = 3.847649490485592e+11

ltype =C [Farad]

To simplify reactance translation into capacitance or inductance the following functions B2LC.m and X2LC.m can be used:

function [LC char1]=B2LC(B_,f1)

% B2LC translates shunt reactance to capacitive or inductive

% impedance depending upon the sign of the input reactance

if B_>=0

LC=B_/(2*pi*f1); char1='C [Farad]';

end

if B_<0

LC=1/(abs(B_)*(2*pi*f1)); char1='L [Henry]';

end;

end

function [LC char1]=X2LC(X_,f1)

% B2LC translates shunt reactance to capacitive or inductive

% impedance depending upon the sign of the input reactance

if X_>=0

LC=X_/(2*pi*f1); char1='L [Henry]';

end

if X_<0

LC=1/(abs(X_)*2*pi*f1); char1='C [Farad]';

end;

end

Zin1=1j*2*pi*f*L1+1./((1j*2*pi*C1*f)+1/ZL);

gamma_in1=(Zin1-Z0)./(Zin1+Z0);

Zin2=1/(1j*2*pi*f*C2)+1./(1./(1j*2*pi*L2*f)+1/ZL);

gamma_in1=(Zin1-Z0)./(Zin1+Z0);

figure(2)

plot(f,abs(gamma_in1),'LineWidth',2)

hold all

plot(f,abs(gamma_in2),'--', 'LineWidth',2)

grid on

title('L-shape 2-element single f0 impedance match');xlabel(f);