Phasor examples

56. Phasor examples#

Recall the unit circle from trig. The sine and cosine are generated by rotating the unit circle counter-clockwise and reading the cosine off of the x axis and the sine off of the y axis.

Note that the sine and the cosine are out of phase by an angle of 90 degrees, with the sine leading the cosine. That is, at any point on the circle

\[ \sin(\theta) = \cos(\theta + 90) \]
import matplotlib.pyplot as plt
from numpy import pi
import numpy as np

56.1. Phase shift of \(\phi=0\)#

  • sin(0) =0

  • cos(0) = 1

fig1, axis1 =plt.subplots(1,1, subplot_kw=dict(polar=True))
theta=np.array([0.,0.])*pi/180.;
print(f"{np.cos(theta[0])=:.3},{np.sin(theta[0])=:.3}")
rho=np.array([0.,1.]);
line1=axis1.plot(theta,rho,'b-',linewidth=3);
point1=axis1.plot(0,1,'bo');
axis1.set_rticks([]);

np.cos(theta[0])=1.0,np.sin(theta[0])=0.0
../../_images/cfda21d981a012afbb02dda41d3fcdba42ad8cb27584fd8221719c041d90bfd0.png

56.2. Phase shift of \(\phi=30\ deg\)#

  • sin(0) =0

  • cos(0) = 1

theta=np.array([30.,30.])*pi/180.;
rho=np.array([0,1]);
line1=axis1.plot(theta,rho,'c-',lw=4);
point1=axis1.plot(theta[0],1,'co')
print(f"{np.cos(theta[0])=:.3},{np.sin(theta[0])=:.3}")
display(fig1)

np.cos(theta[0])=0.866,np.sin(theta[0])=0.5
../../_images/938540a90e4bd3c78bb5c3ea04a46aaf1db30f9a13e8fca156416cb7b637130f.png

56.3. Phase shift of \(\phi=60\ deg\)#

theta=np.array([60.,60.])*pi/180.;
print(f"{np.cos(theta[0])=:.3},{np.sin(theta[0])=:.3}")
rho=np.array([0,1]);
line1=axis1.plot(theta,rho,'g-',lw=4);
point1=axis1.plot(theta[0],1,'go')
display(fig1)

np.cos(theta[0])=0.5,np.sin(theta[0])=0.866
../../_images/8cc56d77a8db4e1f3d53e193e90b46018a773c49c35e7120907ba4d20df9a1b3.png

56.4. Phase shift of \(\phi=90\ deg\)#

theta=np.array([90.,90.])*pi/180.;
print(f"{np.cos(theta[0])=:.3},{np.sin(theta[0])=:.3}")
rho=np.array([0,1]);
line1=axis1.plot(theta,rho,'k-',lw=4);
point1=axis1.plot(theta[0],1,'ko');
axis1.set_title('phase shifts of 0, 30, 60, 90 degrees')
display(fig1)

np.cos(theta[0])=6.12e-17,np.sin(theta[0])=1.0
../../_images/336705b9e62f395895a69f60e2dd21561d21738a87d9fb474510e1dd467c9de7.png
Contents