example 5.7

ZIP with MATLAB scripts and note:

example 5.7 notes:

3 section Chebyshev transformer (Chebyshev º equal ripple) matching ZL=100 ohm to Zg=50 ohm with pass band attenuation not higher than gamma_m=0.05 Cheby polynomials:

syms x

chebyshevT([1, 2, 3, 4, 5], x)

for n=[1, 2, 3, 4, 5]

figure(1);

hp1(n)=fplot(chebyshevT(n, x))

hold on

end

patch('XData',[-1 1 1 -1],'YData',[1 1 -1 -1],'LineStyle','--','FaceAlpha',0)

axis([-1.5, 1.5, -6, 6]);grid on;ylabel('T_n(x)')

hL1=legend(hp1([1:5]),'T_1(x)','T_2(x)','T_3(x)','T_4(x)','T_5(x)')

hL1.Location='southeast'

title('Chebyshev polynomials of the first kind')

ans =

[ x, 2*x^2 - 1, 4*x^3 - 3*x, 8*x^4 - 8*x^2 + 1, 16*x^5 - 20*x^3 + 5*x]

[POZAR] doesn't mention it but Chebyshev 2nd class polynomials are of no use to keep a flat pass band

chebyshevU([1, 2, 3, 4, 5], x)

for n=[1, 2, 3, 4, 5]

figure(2);

hp2(n)=fplot(chebyshevU(n, x))

hold on

end

patch('XData',[-1 1 1 -1],'YData',[1 1 -1 -1],'LineStyle','--','FaceAlpha',0)

axis([-1.5, 1.5, -6, 6]);grid on;ylabel('U_n(x)')

hL2=legend(hp2([1:5]),'U_1(x)', 'U_2(x)', 'U_3(x)', 'U_4(x)', 'U_5(x)')

hL2.Location='southeast'

title('Cheby polynomials 2nd kind don''t fit')

ans =

[ 2*x, 4*x^2 - 1, 8*x^3 - 4*x, 16*x^4 - 12*x^2 + 1, 32*x^5 - 32*x^3 + 6*x]

3 sections means N=3 reflection coefficient (filter input) is:

gamma(a)=2*exp(-1j*N*a)*Chebyshev(N,x)

gamma_m=(SWR-1)/(SWR+1)

One ca get theta_m by solving

T1=1/gamma_m*abs((ZL-Z0)/(ZL+Z0))

sec(theta_m)=cosh(1/N*acosh(T1))

N=3;ZL=100;Zg=50; gamma_m=0.05;

cosh(1/N*acosh(1/.05*abs((ZL-Zg)/(ZL+Zg))))

theta_m=asec(cosh(1/N*acosh(1/gamma_m*abs((ZL-Zg)/(ZL+Zg)))))*180/pi % degree

With MATLAB there is no need for POZAR approximation (5.63)

cosh(1/N*acosh(abs(log(ZL/Zg)/(2*.05))))

theta_m_approx= asec(cosh(1/N*acosh(abs(log(ZL/Zg)/(2*gamma_m)))))*180/pi

= 1.394648447495264

theta_m = 44.190468508828843

= 1.407530092552086

theta_m_approx = 44.727288553390935