Complex Numbers

Contents

rectangular/polar conversions

% example 1
z = complex(3,4);
rho = abs(z);
theta = angle(z)*180/pi;
fprintf('given: z = complex(3,4)\n');
fprintf('rho %g theta %g\n',rho,theta);

% example 2
x = 5;
y = 12;
fprintf('given: x %g y %g\n',x,y);
[theta,rho] = cart2pol(x,y);
fprintf('rho %g theta %g\n',rho,theta*180/pi);

% example 3
rho = 2.0;
theta = 45;
[x,y] = pol2cart(theta*pi/180,rho);
fprintf('given: rho %g theta %g\n',rho,theta);
fprintf('x %g y %g\n',x,y);

given: z = complex(3,4)
rho 5 theta 53.1301
given: x 5 y 12
rho 13 theta 67.3801
given: rho 2 theta 45
x 1.41421 y 1.41421

angle calculations

x = -1;
y =  sqrt(3);

theta1 = atan2(y,x)*180/pi;
theta2 = angle(complex(x,y))*180/pi;
if (theta1==theta2)
    fprintf('calculations match, theta = %g degrees\n',theta1);
end

calculations match, theta = 120 degrees

Using phasors to add sinuosoids

Suppose that

 v1 = 20 cos(wt-45 deg)
 v2 = 10 sin(wt+60 deg)
 Reduce the sum vs(t) = v1(t) + v2(t) to a single sum.
   Reference: Hambley, Example 5.3

% first phasor is
v1 = phasor2complex(20,-45);
fprintf(['v1 ' ctext(v1) '\n' ]);
% subtract 90 degree from second phasor because v2 is a sine function
v2 = phasor2complex(10,-30);
fprintf('v2 %s\n', ctext(v2));
vx = v1+v2;
[rho,theta] = complex2phasor(vx);
fprintf('result: rho %g theta %g\n',rho,theta);

v1 14.1421 - 14.1421j 
v2 8.66025 - 5j 
result: rho 29.772 theta -40.0128

Lagging/Leading

close all
t = linspace(-1.5,2.5,201);
v1 = 3*cos(2*pi*t+40*pi/180);
v2 = 4*cos(2*pi*t-20*pi/180);
z1 = phasor2complex(3,40);
z2 = phasor2complex(4,-20);
plot([0 real(z1)],[0 imag(z1)],'b',[0 real(z2)],[0 imag(z2)],'k');
w = 5;
axis([-w w -w w]);
axis square;
%[arrowx arrowy] = dsxy2figxy(gca,[0 real(z1)],[0 imag(z1)]);
%annotation('arrow',arrowx,arrowy);
%[arrowx arrowy] = dsxy2figxy(gca,[0 real(z2)],[0 imag(z2)]);
%annotation('arrow',arrowx,arrowy);
figure;
plot(t,v1,'b',t,v2,'k','LineWidth',2);
axis([-1.5 2.5 -4.5 4.5]);
grid

Published with MATLAB® 7.6