exercise 5.13

ZIP with MATLAB scripts and note:

exercise 5.13 notes:

1.- calculation of l/4 transformer characteristic impedance Z1

ZL=350;Z0=100;f0=4e9;

Z1=(ZL*Z0)^.5

SWR=2;g_mod=(SWR-1)/(SWR+1)

2.- Fractional Bandwidth Df/f0, as defined [POZAR] 248pg

df_over_f0=2-4/pi*acos((g_mod)/(1-g_mod^2)^.5*(2*(Z0*ZL)^.5)/abs(ZL-Z0))

This expression of fractional bandwidth is only valid when

approximating l/4 |G| transformer general expression with (sec(q))^2>>1, q=b*L
TEM line only; q=b*L=2*p*f/vp*vp/(4*f0)= p*f/(2*f0)

Checking frequency response

c0=2.998*10^8;

df=1e6;

f0=2e9;f1=0;f2=2*f0;

dfrel=df/f0;

frel=[f1/f0:dfrel:f2/f0];

D1=.25;L1=D1*c0/f0

Zin=Z1*(ZL+1j*Z1*tan(2*pi*frel.*f0/c0.*L1))./...

(Z1+1j*ZL*tan(2*pi*frel.*f0/c0.*L1))

gamma1=(Zin-Z0)./(Zin+Z0)

hf1=figure(1);ax1=gca

plot(ax1,frel,abs(gamma1));grid on

title(['|gamma| ZL/Z0=' num2str(ZL/Z0)])

xlabel('f/f0')

axis([0 2 0 1])

Z0=100;ZL1=10*Z0;Z1_1=(ZL1*Z0)^.5

Zin1=Z1_1*(ZL1+1j*Z1_1*tan(2*pi*frel.*f0/c0.*L1))./...

(Z1_1+1j*ZL1*tan(2*pi*frel.*f0/c0.*L1));

gamma1_mod=abs((Zin1-Z0)./(Zin1+Z0));

hf2=figure(2);ax2=gca

plot(ax2,frel,gamma1_mod);grid on

ZL2=4*Z0;Z1_2=(ZL2*Z0)^.5

Zin2=Z1_2*(ZL2+1j*Z1_2*tan(2*pi*frel.*f0/c0.*L1))./...

(Z1_2+1j*ZL2*tan(2*pi*frel.*f0/c0.*L1));

gamma2_mod=abs((Zin2-Z0)./(Zin2+Z0));

hold(ax2,'on')

plot(ax2,frel,gamma2_mod);grid on

ZL3=2*Z0;Z1_3=(ZL3*Z0)^.5

Zin3=Z1_3*(ZL3+1j*Z1_3*tan(2*pi*frel.*f0/c0.*L1))./...

(Z1_3+1j*ZL3*tan(2*pi*frel.*f0/c0.*L1));

gamma3_mod=abs((Zin3-Z0)./(Zin3+Z0));

plot(ax2,frel,gamma3_mod);grid on

str1=['ZL/Z0=' num2str(ZL1/Z0)]

str2=['ZL/Z0=' num2str(ZL2/Z0)]

str3=['ZL/Z0=' num2str(ZL3/Z0)]

egend(str1,str2,str3)

axis(ax2,[0 2 0 1]);xlabel('f/f0');ylabel('|\Gamma|')

3.- µ-strip dimensioning

A first approximation, sometimes valid, would be manual iteration suggested in the solutions manual:

Let be Wd for W/d:

for the microstrip already with Z0=100 generator side:

guess that W/d<2 for instance W/d=.5

d=1.59e-3;Wd0=.5;er=2.2

ee0=(er+1)/2+(er-1)/2*1/(1+12*1/Wd0)^.5

for the microstrip on the generator side

Z0=100;

A100=Z0/60*((er+1)/2)^.5+(er-1)/(er+1)*(.23+.11/er)

B100=377*pi/(2*Z0*er^.5)

then

Wd100=8*exp(A100)/(exp(2*A100)-2)

W1=Wd100*d

check

60/ee0^.5*log(8/Wd100+.24*Wd100)

quite close to Z0=100

for the l/4 transformer microstrip, also guess W/d<2

Z1=187;

A187=Z1/60*((er+1)/2)^.5+(er-1)/(er+1)*(.23+.11/er)

B187=377*pi/(2*Z1*er^.5)

Wd187=8*exp(A187)/(exp(2*A187)-2)

W187=Wd187*d

check

60/ee0^.5*log(8/Wd187+.24*Wd187)

again close enough to Z1=187

Then the length [metre] of the transformer would be

c0=2.998e8;f0=4e9;

lambda0=c0/f0

lambdag=lambda0/ee0^.5

L_trf=lambdag/4

er = 2.200000000000000

ee0 = 1.720000000000000

A100 = 2.213185106778920

B100 = 3.992545616410477

Wd100 = 0.896248722160686

W100 = 0.001425035468235

= 1.012342362687005e+02

A187 = 4.047306149676580

B187 = 2.135051131770309

Wd187 = 0.139840322091229

W187 = 2.223461121250534e-04

= 1.851614198712142e+02

L_trf = 0.014287209808036

The guessing whether W/d<2 is a try-and-error manual process that can be somehow simplified sweeping all possible values, in same way when seeking function roots.

Copy of the support function iter_Wovd appended to this note. Start values Wd=1.5 (W/d) and the step d_Wd=0.01

d=1.59e-3 %mm thick

er=2.2

% 1st round

Wd0=1.5

d_Wd=.01

% dZ1=.01*Z1

ee0=(er+1)/2+(er-1)/2*1/(1+12*1/Wd0)^.5

if Wd0>=1

Z1_t2=120*pi/(ee0^.5*(Wd0+1.393+.667*log(Wd0+1.444))); %W/d>=1

else

Z1_t2=60/ee0^.5*log(8/Wd0+.24*Wd0); % W/d<1

end

Z1_v=Z1_t2

errZ1_v=abs(Z1_t2-Z1) % .5*Z1 % Z1 error vector

Wd_v=Wd0 % W/d vector

k=1

while errZ1_v(end)>.01 && k<2e3 % && k<2e3 safety switch or the loop doesn't stop

[Z1_2,Wd1]=iter_Z1_Wovd(Z1_v(end),Wd_v(end),er)

[Z1_2_dZup,Wd1_dZup]=iter_Z1_Wovd(Z1_v(end),Wd_v(end)+d_Wd,er)

[Z1_2_dZdown,Wd1_dZdown]=iter_Z1_Wovd(Z1_v(end),Wd_v(end)-d_Wd,er)

if abs(Z1-Z1_2_dZup)<abs(Z1-Z1_2_dZdown)

errZ1_v=[errZ1_v abs(Z1-Z1_2_dZup)];

Wd_v=[Wd_v Wd1_dZup];

Z1_v=[Z1_v Z1_2_dZup];

end

if abs(Z1-Z1_2_dZup)>abs(Z1-Z1_2_dZdown)

errZ1_v=[errZ1_v abs(Z1-Z1_2_dZdown)];

Wd_v=[Wd_v Wd1_dZdown];

Z1_v=[Z1_v Z1_2_dZdown];

end

k=k+1;

end

figure(3);stem(errZ1_v);grid on;title('Z1 error vector')

figure(4);stem(Z1_v);grid on;title('Z1')

figure(5);stem(Wd_v);grid on;title('W/d')

When W/d step is too small the convergence of Z1 happens with offset dZ1=85 ohm.

85 ohm away from the expected Z1 is an excessive deviation.

Let's increase step: d_Wd=.1

When starting with larger W/d step Z1 is eventually reached

% 13.2 Microstrip

d=1.59e-3 %mm thick

er=2.2

% 1st round

Wd0=1.5

d_Wd=.1

% dZ1=.01*Z1

ee0=(er+1)/2+(er-1)/2*1/(1+12*1/Wd0)^.5 % eff rel diel permittivity

if Wd0>=1

Z1_t2=120*pi/(ee0^.5*(Wd0+1.393+.667*log(Wd0+1.444)));

%W/d>=1

else

Z1_t2=60/ee0^.5*log(8/Wd0+.24*Wd0); % W/d<1

end

Z1_v=Z1_t2

errZ1_v=abs(Z1_t2-Z1) % .5*Z1 % Z1 error vector

Wd_v=Wd0 % W/d vector

k=1

while errZ1_v(end)>.01

[Z1_2,Wd1]=iter_Z1_Wovd(Z1_v(end),Wd_v(end),er)

[Z1_2_dZup,Wd1_dZup]=…

iter_Z1_Wovd(Z1_v(end),Wd_v(end)+d_Wd,er)

[Z1_2_dZdown,Wd1_dZdown]=…

iter_Z1_Wovd(Z1_v(end),Wd_v(end)-d_Wd,er)

if abs(Z1-Z1_2_dZup)<abs(Z1-Z1_2_dZdown)

errZ1_v=[errZ1_v abs(Z1-Z1_2_dZup)];

Wd_v=[Wd_v Wd1_dZup];

Z1_v=[Z1_v Z1_2_dZup];

end

if abs(Z1-Z1_2_dZup)>abs(Z1-Z1_2_dZdown)

errZ1_v=[errZ1_v abs(Z1-Z1_2_dZdown)];

Wd_v=[Wd_v Wd1_dZdown];

Z1_v=[Z1_v Z1_2_dZdown];

end

% if abs(Z1-Z1_2_dZup)==abs(Z1-Z1_2_dZdown)

% return;

% end

figure(6);hold all;stem(errZ1_v);grid on;title('Z1 error vector')

figure(7);hold all;stem(Z1_v);grid on;title('Z1')

figure(8);hold all;stem(Wd_v);grid on;title('W/d')

k=k+1;

end

Same result if W/d>2 W/d=2.5 Wd0=2.5 and d_Wd=0.1

The error threshold to stop is too small compared to the step the span is a bit too wide, yet when attempting to reduce both, again convergence is too far from Z1.

The point that this basic solver loop doesn't stop for some W/d and error threshold values or the span of the correct convergence can be narrowed, are points that can be improved is about diverting time on the algorithm, for instance adding variable steps.

Yet since Z1 values eventually hang around the expected Z1 and the error close to zero, we can focus on the smallest errors and from there get W/d:

the following ends up within a few seconds and converges to Z1

find(errZ1_v<.2)

Wd_v(find(errZ1_v<.2))

so the result is

Wd=mean(Wd_v(find(errZ1_v<.2)))

d

W=Wd*d

er

ee1=(er+1)/2+(er-1)/2*1/(1+12*1/Wd)^.5

lambda0=c/f0;lambda_eff=lambda0/(ee1)^.5

and with c=3e8

L_transformer=lambda_eff/4 % m

= 497 628 957 1683 2400

2612 2882 2980 3119 3375

3422 3831 4083 4235 5663

= 0.140344773157983 0.140416696611814 0.140182123411192

0.140392081104524 0.140193473954178 0.139486015946577

0.139983812519741 0.140263297911968 0.139605857141912

0.139484812310120 0.140322941984743 0.139283038977218

0.140218003471145 0.139906790481617 0.139859977987934

Wd = 0.139996246464844

d = 0.001590000000000

W = 2.225940318791026e-04

er = 2.200000000000000

ee1 = 1.664431786024021

lambda_eff = 0.058133739859052

L_transformer = 0.014533434964763

L_transformer=14.55 mm
W transformer track width 0.22mm

Again, single shot approach, compact, for instance W/d=.5 Wd=.5

d=1.59e-3;Wd=.5,Z1=187;er=2.2

A1=Z1/60*((er+1)/2)^.5+(er-1)/(er+1)*(.23+.11/er)

Wd_check=8*exp(A1)/(exp(2*A1)-2) % <1, ok

% since Wd is also <1

ee=(er+1)/2+(er-1)/2*1/(1+12*Wd_check)^.5

Z1=60/ee^.5*log(8/Wd_check+.24*Wd_check)

W_Z1=Wd*d

Z1=173 ohm, rather close to 187 ohm, the solutions manual leaves this as good enough

d=1.59e-3;Z0=100;Wd=.5,Z1=Z0;er=2.2

A0=Z1/60*((er+1)/2)^.5+(er-1)/(er+1)*(.23+.11/er)

Wd_check=8*exp(A0)/(exp(2*A0)-2) % <1, ok

% since Wd is also <1

ee=(er+1)/2+(er-1)/2*1/(1+12*Wd_check)^.5

Z0_check=60/ee^.5*log(8/Wd_check+.24*Wd_check)

W_Z0=Wd_check*d

A1 = 4.047306149676580

Wd_check = 0.139840322091229

ee = 1.966639425647109

Z1 = 1.731619051045303e+02

W_Z1 = 7.950000000000000e-04

A0 = 2.213185106778920

Wd_check = 0.896248722160686

ee = 1.775000873695489

Z0_check = 99.653452321549082

W_Z0 = 0.001425035468235

In any case it's highly recommended to validate results with a real prototype or with reliable microstrip calculators like LineCalc.

LineCalc is ADS microstrip utility, also used in this solution.

Another microstrip application that may turn useful is the free online microstrip calculator Chemandy:

http://chemandy.com/calculators/microstrip-transmission-line-calculator.htm.

The online microstrip calculator Chemandy uses Z0(W/d) slightly different a bit more compact expressions than POZAR.

Now calculating ee(er) and ee(W/d)

first ee(W/d)

Wd=[.5:.5:3];er=[1:.1:15];

ee=zeros(numel(Wd),numel(er))

hold all

for s=1:1:numel(Wd)

ee(s,:)=(er+1)/2+(er-1)/2*1/(1+12*Wd(s))^.5;

semilogy(er,ee(s,:));

end

grid on;

str1=['W/d= ' num2str(Wd(1))];str2=['W/d= ' num2str(Wd(2))];

str3=['W/d= ' num2str(Wd(3))];str4=['W/d= ' num2str(Wd(4))];

str5=['W/d= ' num2str(Wd(5))];str6=['W/d= ' num2str(Wd(6))];

legend(str1,str2,str3,str4,str5,str6)

and now ee(er)

er=[1:1:6];Wd=[.5:.01:3];

ee=zeros(numel(er),numel(Wd));hold all

for s=1:1:numel(er)

ee(s,:)=(er(s)+1)/2+(er(s)-1)/2*1./(1+12*Wd).^.5;

semilogy(Wd,ee(s,:));

end

grid on;

str1=['er= ' num2str(er(1))];str2=['er= ' num2str(er(2))];

str3=['er= ' num2str(er(3))];str4=['er= ' num2str(er(4))];

str5=['er= ' num2str(er(5))];str6=['er= ' num2str(er(6))];

evalin('base','legend(str1, str2 , str3 , str4 , str5)')

legend(str1,str2,str3,str4,str5,str6)

evalin is a powerful function that enables code generation.

Support function iter_Z1_Wovd

function [Z1_t2,Wd_t2]=iter_Z1_Wovd(Z1_t1,Wd_t1,er)

ee=(er+1)/2+(er-1)/2*1/(1+12*Wd_t1)^.5 % effective relative dielectric permittivity

if Wd_t1>=1

Z1_t2=120*pi/(ee^.5*(Wd_t1+1.393+.667*log(Wd_t1+1.444))); %W/d>=1

else

Z1_t2=60/ee^.5*log(8/Wd_t1+.24*Wd_t1); % W/d<1

end

A=Z1_t2/60*((er+1)/2)^.5+(er-1)/(er+1)*(.23+.11/er)

B=377*pi/(2*Z1_t2*er^.5)

if (Wd_t1>2)

Wd_t2=2/pi*(B-1-log(2*B-1)+(er-1)/(2*er)*(log(B-1)+.39-.61/er));

else

Wd_t2=8*exp(A)/(exp(2*A)-2);

end

% err_Z1=abs(Z1-Z1_t2)

end