Determination of polarization for combination of linearly polarized vectors

[Ethan0718]

Question Source : Elements of Engineering Electromagnetics 6th edition by Rao. Page 202 problem3.30

Problem:
Three sinusoidally time-varying polarized vector fields are given at a point by

F1 = 3^(1/2) * ax * cos(wt +30)
F2 = az * cos(wt+30)
F3 = [ 0.5ax + 3^(1/2)ay + 0.5*3^(1/2)az ] * cos(wt - 60)

So, what is the polarization of F1 + F2 + F3?

I don't know how to use The mathematical discription of elliptical polarization to solve this problem.

http://en.m.wikipedia.org/wiki/Elliptical_polarization#section_1

I've tried to arrange and combine them in x,y,z component respectively...

 

[Nate810]

F1x = 3√axcos(wt + 30°) F1y = 0F1z = 0F2x = 0F2y = 0F2z = azcos(wt + 30°) F3x = 0.5ax + 3^(1/2)aycos(wt - 60°) F3y = 0.5ax - 3^(1/2)aycos(wt - 60°) F3z = 0.5√azcos(wt - 60°) Fx = 3√axcos(wt + 30°)+ 0.5ax + 3^(1/2)aycos(wt - 60°) Fy = 0 - 0.5ax - 3^(1/2)aycos(wt - 60°) Fz = azcos(wt + 30°) + 0.5√azcos(wt - 60°) Answer:The polarization of F1 + F2 + F3 is elliptical. To determine the exact shape of the ellipse, we can use the mathematical description of elliptical polarization. This description states that the two components of the electric field, Ex and Ey, can be written as:Ex = E0cos(wt + φ) Ey = E1cos(wt + φ + θ) where E0, E1, φ, and θ are constants. In this case, we can calculate E0, E1, φ, and θ from the given information:E0 = 3√a E1 = √(a^2 + 9a^2/4 + 9a^2/4) = 3√a φ = 30° θ = 60° Therefore, the polarization of F1 + F2 + F3 is an elliptically polarized wave with major axis 3√a, minor axis 3√a, and angle of orientation 30°.