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Can anyone verify my equation on elliptical polarization

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Alan0354

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Attached is a page in Kraus Antenna book, I cannot verify the equation on the last line. Here is my work

\[E_y=E_2(\sin{\omega} t \cos \delta \;+\; \cos \omega {t} \sin \delta)\] , \[ \sin\omega {t} =\frac {E_x}{E_1}\;,\; \cos \omega {t} =\sqrt{1-(\frac{E_x}{E_1})^2}\]

\[\Rightarrow\; E_y=\frac {E_2 E_x\cos \delta}{E_1}\;+\;E_2\sqrt{1-(\frac {E_x}{E_1})^2} \;\sin\delta\]

\[\Rightarrow\; \sin \delta \;=\;\frac {E_y}{E_2\sqrt{1-(\frac{E_x}{E_1})^2}}\;-\; \frac{E_x\cos\delta}{E_1 \sqrt{1-(\frac{E_x}{E_1})^2}}\]

\[\Rightarrow\; \sin^2\delta\;=\;\frac{E^2_y}{E_2^2\;(1\;-\;(\frac{E_x}{E_1})^2)}\;-\;\frac{2E_y\;E_x\;\cos\delta}{E_1\;E_2\;(1\;-\;(\frac{E_x}{E_1})^2}\;+\;\frac {E_x^2\;\cos^2\;\delta}{E_1^2\;(1\;-\;(\frac{E_x}{E_1})^2)}\]

Compare to the last line in the book, I just cannot get the last equation of the book. Please take a look and see what I did wrong.

Thanks

Alan
 

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OK, first of all let's substitute sin(ωt) with Ey/E1 and cos(ωt) with sqrt(1-(Ex/E1)²) obtaining

Ey/E2=Ex/E1•cosδ+sqrt[1-(Ex/E1)²]•sinδ from which:

Ey/E2-Ex/E1•cosδ=sqrt[1-(Ex/E1)²]•sinδ then squaring:

[Ey/E2-Ex/E1•cosδ]²=[1-(Ex/E1)²]•sin²δ that is:

(Ey/E2)²+(Ex/E1)²•cos²δ-2•(Ex•Ey•cosδ)/(E1•E2)=sin²δ-(Ex/E1)²•sin²δ rearranging:

(Ey/E2)²+(Ex/E1)²•cos²δ+(Ex/E1)²•sin²δ-2•(Ex•Ey•cosδ)/(E1•E2)=sin²δ

since cos²δ+sin²δ=1 then

(Ey/E2)²+(Ex/E1)²-2•(Ex•Ey•cosδ)/(E1•E2)=sin²δ
 
Thank you. I've gone down the wrong track and missed the simplification.
 

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