Solution 10.8.9.3 The MATLAB program w=logspace(-2,3,200);; s=j*w;; K=100 z1 = 1 p1 = 0 p2 = 4 p3 = 4 p4 = 20 mag = 20*log10( abs((K.*( +z1)) ./((s+p1).*(s+p2).*(s+p3).*(s+p4))));; phase=(angle(s+z1)-angle(s+p1)-angle(s+p2) -angle(s+p3)-angle(s+p4))*180/pi;; phi = -170 phi = phi*(pi/180);; N=tan(phi);; c=-0.5 + j* (1/(2*N));; r =0.5* sqrt( 1 + (1/N^2) );; theta = linspace(0,2*pi,1000);; s=c+r.*sin(theta) + j*r.*cos(theta);; gamma = angle(s)*180/pi;; b=zeros(1,1000);; d=gamma > b;; gamma = gamma -360*d;; rho = 20*log10(abs(s));; G= [gamma'];; R=[rho'];; % % phi = -190 phi = phi*(pi/180);; N=tan(phi);; c=-0.5 + j* (1/(2*N));; r =0.5* sqrt( 1 + (1/N^2) );; theta = linspace(0,2*pi,1000);; s=c+r.*sin(theta) + j*r.*cos(theta);; gamma = angle(s)*180/pi;; b=zeros(1,1000);; d=gamma > b;; gamma = gamma -360*d;; rho = 20*log10(abs(s));; G= [G gamma'];; 1 R=[R rho'];; % % phi = -210 phi = phi*(pi/180);; N=tan(phi);; c=-0.5 + j* (1/(2*N));; r =0.5* sqrt( 1 + (1/N^2) );; theta = linspace(0,2*pi,1000);; s=c+r.*sin(theta) + j*r.*cos(theta);; gamma = angle(s)*180/pi;; b=zeros(1,1000);; d=gamma > b;; gamma = gamma -360*d;; rho = 20*log10(abs(s));; G= [G gamma'];; R=[R rho'];; % % phi = -230 phi = phi*(pi/180);; N=tan(phi);; c=-0.5 + j* (1/(2*N));; r =0.5* sqrt( 1 + (1/N^2) );; theta = linspace(0,2*pi,1000);; s=c+r.*sin(theta) + j*r.*cos(theta);; gamma = angle(s)*180/pi;; b=zeros(1,1000);; d=gamma > b;; gamma = gamma -360*d;; rho = 20*log10(abs(s));; G= [G gamma'];; R=[R rho'];; % % phi = -250 phi = phi*(pi/180);; N=tan(phi);; c=-0.5 + j* (1/(2*N));; r =0.5* sqrt( 1 + (1/N^2) );; theta = linspace(0,2*pi,1000);; s=c+r.*sin(theta) + j*r.*cos(theta);; gamma = angle(s)*180/pi;; b=zeros(1,1000);; d=gamma > b;; 2 gamma = gamma -360*d;; rho = 20*log10(abs(s));; G= [G gamma'];; R=[R rho'];; % % phi = -270 phi = phi*(pi/180);; N=tan(phi);; c=-0.5 + j* (1/(2*N));; r =0.5* sqrt( 1 + (1/N^2) );; theta = linspace(0,2*pi,1000);; s=c+r.*sin(theta) + j*r.*cos(theta);; gamma = angle(s)*180/pi;; b=zeros(1,1000);; d=gamma > b;; gamma = gamma -360*d;; rho = 20*log10(abs(s));; G= [G gamma'];; R=[R rho'];; % % phi = -290 phi = phi*(pi/180);; N=tan(phi);; c=-0.5 + j* (1/(2*N));; r =0.5* sqrt( 1 + (1/N^2) );; theta = linspace(0,2*pi,1000);; s=c+r.*sin(theta) + j*r.*cos(theta);; gamma = angle(s)*180/pi;; b=zeros(1,1000);; d=gamma > b;; gamma = gamma -360*d;; rho = 20*log10(abs(s));; G= [G gamma'];; R=[R rho'];; % % phi = -310 phi = phi*(pi/180);; N=tan(phi);; c=-0.5 + j* (1/(2*N));; r =0.5* sqrt( 1 + (1/N^2) );; theta = linspace(0,2*pi,1000);; s=c+r.*sin(theta) + j*r.*cos(theta);; 3 gamma = angle(s)*180/pi;; b=zeros(1,1000);; d=gamma > b;; gamma = gamma -360*d;; rho = 20*log10(abs(s));; G= [G gamma'];; R=[R rho'];; % % phi = -330 phi = phi*(pi/180);; N=tan(phi);; c=-0.5 + j* (1/(2*N));; r =0.5* sqrt( 1 + (1/N^2) );; theta = linspace(0,2*pi,1000);; s=c+r.*sin(theta) + j*r.*cos(theta);; gamma = angle(s)*180/pi;; b=zeros(1,1000);; d=gamma > b;; gamma = gamma -360*d;; rho = 20*log10(abs(s));; G= [G gamma'];; R=[R rho'];; % % phi = -350 phi = phi*(pi/180);; N=tan(phi);; c=-0.5 + j* (1/(2*N));; r =0.5* sqrt( 1 + (1/N^2) );; theta = linspace(0,2*pi,1000);; s=c+r.*sin(theta) + j*r.*cos(theta);; gamma = angle(s)*180/pi;; b=zeros(1,1000);; d=gamma > b;; gamma = gamma -360*d;; rho = 20*log10(abs(s));; G= [G gamma'];; R=[R rho'];; plot(phase,mag,'k-',G,R,'k-') grid on axis([-360 0 -30 30]) print -deps 10893.eps draws the log magnitude plot in Figure 1 4 -350 -300 -250 -200 -150 -100 -50 0 -30 -20 -10 0 10 20 30 Figure 1: Log magnitude with M-circle 1.2 5