ZIP with MATLAB scripts and note:
exercise 6.2
exercise 6.2 notes:
The following list of approximations are used in the solutions manual:
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for L= lambda/2 and L=l the behaviour of a short-circuited transmission line input impedance is quite the same on the observation port.
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thus, the general expression of Zin on the left becomes:
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that in turn, with trigonometric basic relations between tanh and tan is:
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f~f0, more precisely f=f0+df or
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alpha constant
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alpha*L<<1 implies
All the above reduces Zin to the following simplification:
c0=2.998792586
er=1
f1=4e9
f2=6e9
f0=5e9
Nf=1e6 % amount frequency points between f1 f2
df=abs(f1-f2)/(Nf+1) % frequency resolution
f=[f1:df:f2];
delta_f=f-f0;
lambda1=c0/f1
lambda2=c0/f2
lambda0=c0/f0
dlambda=abs(lambda1-lambda2)/(Nf+1)
lambda=[lambda2:dlambda:lambda1];
L=lambda0
RLoad=.1;Rgen=50;
ZL=RLoad; % load assumed constant resistance over all band
Z0=Rgen;
alpha=.1
beta=2*pi./lambda;
D=1
Zin_1=...
Z0*(ZL+Z0*tanh(alpha*D*c0./f+1j*2*pi*f/c0*L))./...
(Z0+ZL*tanh(alpha*D*c0./f+1j*2*pi*f/c0*L));
% lossy TL general expression
[min_absZin nf0]=min(abs(Zin_1))
f(nf0)
Zin_3=Z0*(alpha*L+1j*2*pi*delta_f*2*pi/(2*pi*f0));
% 2nd approximation
Zin_1_lossless=Z0*1j*tan(2*pi*f/c0*L); % lossless TL
figure(1); % lossy transmission line
subplot(2,1,1);plot(f,abs(Zin_1));
title('exact |Zin1|');grid on
subplot(2,1,2);plot(f,angle(Zin_1));
title('exact phase(Zin1)');grid on
figure(2); % lossy transmission line
subplot(2,1,1);
plot(f,real(Zin_1));title('exact real(Zin1)');grid on
subplot(2,1,2);plot(f,imag(Zin_1));
title('exact Im(Zin1)');grid on
figure(3); % lossless transmission line
plot(f,abs(Zin_1_lossless));title('ideal TL |Zin1|');grid on
figure(5); % 2nd approximation
subplot(2,1,1);hold all;
plot(f,real(Zin_3));
title('2nd approximation real(Zin3)');grid on
subplot(2,1,2);hold all;plot(f,imag(Zin_3));
title('2nd approximation Im(Zin3)');grid on
R=real(Zin_1);
X=imag(Zin_1./(delta_f));
figure(6);plot(f,X);grid on;title('L with exact Zin')
R=real(Zin_3);
X=imag(Zin_3./(2*delta_f));
figure(7);plot(f,X);grid on;title('L with approximate Zin')
L=X(nf0)
R=R(nf0)
Q=2*pi*f0*L/R
% or Q=pi*Z0/(Z0*alpha*length_TL)
pi/(alpha*lambda0)
% or Q=beta/(2*alpha)
beta(nf0)/(2*alpha)
L =
3.141592653589793e-08
R =
2.998792586000000e-09
Q =
3.291192744428560e+11
=
5.238095939439863e+10
=
5.028571096149054e+10