exercise 5.22
ZIP with MATLAB scripts and note:
exercise 5.22 notes:
clear all;close all;clc
syms K A L Z0 ZL z
f(z)=A*sin(pi*z/L)
F(z)=int(f,z)
F2(z)=K*exp(-(A*L*cos((pi*z)/L))/pi)
eq1=K*F2(0)==Z0
eq2=K*F2(L)==ZL
A0=solve(K*F2(0)-Z0==K*F2(L)-ZL,A)
K0=solve(K*F2(0)-Z0==K*F2(L)-ZL,K)
eq1*eq2
eq3=K^2-ZL*Z0==0
K1=solve(eq3,K)
eq1/eq2
eq4=exp(-(2*A*L)/pi) -Z0/ZL
A1=solve(eq4,A)
f(z) =A*sin((pi*z)/L)
F(z) =-(A*L*cos((pi*z)/L))/pi
F2(z) =K*exp(-(A*L*cos((pi*z)/L))/pi)
eq1 =K*exp(-(A*L)/pi) == Z0
eq2 =K*exp((A*L)/pi) == ZL
A0 =
(pi*log(-(Z0 - ZL + (4*K^4 + Z0^2 - 2*Z0*ZL + ZL^2)^(1/2))/(2*K^2)))/L
(pi*log((ZL - Z0 + (4*K^4 + Z0^2 - 2*Z0*ZL + ZL^2)^(1/2))/(2*K^2)))/L
K0 =
(exp((A*L)/(2*pi))*(-(exp((A*L)/pi) - 1)*(exp((A*L)/pi) + 1)*(Z0 - ZL))^(1/2))/(exp((2*A*L)/pi) - 1)
-(exp((A*L)/(2*pi))*(-(exp((A*L)/pi) - 1)*(exp((A*L)/pi) + 1)*(Z0 - ZL))^(1/2))/(exp((2*A*L)/pi) - 1)
=K^2 == Z0*ZL
=exp(-(2*A*L)/pi) == Z0/ZL
K1 =
Z0^(1/2)*ZL^(1/2)
-Z0^(1/2)*ZL^(1/2)
= exp(-(2*A*L)/pi) == Z0/ZL
eq4 = exp(-(2*A*L)/pi) - Z0/ZL
A1 = -(pi*log(Z0/ZL))/(2*L)
syms ZL Z0 L z
K= ZL^(1/2)*Z0^(1/2)
A = (pi*log(ZL/Z0))/(2*L)
Zin(z)=K*exp((-A*L*cos((pi*z)/L))/pi)
subs(Zin,ZL/Z0,1.5)
MATLAB does not catch Z0^(1/2)*ZL^(1/2)*..
when told to substitute ZL/Z0 with numeric value.
G(L)=int(.5*exp(-1j*2*pi*z)*A*sin((pi*z)/L),z,0,L)
fplot(@(L) G(L))
Warning: Error updating FunctionLine.
The following error was reported evaluating
the function in FunctionLine update: Division by zero.
Gnum=- log(ZL/Z0)./(16*L.^2 - 4) - (exp(-pi*L*2i)*log(ZL/Z0))./(16*L.^2 - 4);
Gnum=subs(Gnum,ZL/Z0,1.5)
L=[0:.001:4];
Gnum =- log(3/2)./(16*L.^2 - 4) - (exp(-pi*L*2i)*log(3/2))./(16*L.^2 - 4);
plot(pi*L,abs(Gnum));grid on
Zin(z) = Z0^(1/2)*ZL^(1/2)*exp(-(cos((pi*z)/L)*log(ZL/Z0))/2)
Zin(z) =Z0^(1/2)*ZL^(1/2)*exp(-(cos((pi*z)/L)*log(3/2))/2)
G = - log(ZL/Z0)/(4*(4*L^2 - 1)) - (exp(-pi*L*2i)*log(ZL/Z0))/(4*(4*L^2 - 1))
G = - log(3/2)/(16*L^2 - 4) - (exp(-pi*L*2i)*log(3/2))/(16*L^2 - 4)
once more, it's easier to solve the problem numerically than symbolically, let alone the time to find out what symbolic expression and conditions the symbolic solver will decide to solve, or just return trivial solutions.
Correction Solutions Manual:
