Can anyone verify my equation on elliptical polarization

时间：03-26 整理：3721RD 点击：

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

$3SE_y=E_2(\sin{\omega}t\cos\delta\;+\;\cos\omega{t}\sin\delta)$

$3S\sin\omega{t}=\frac{E_x}{E_1}\;,\;\cos\omega{t}=\sqrt{1-(\frac{E_x}{E_1})^2}$

$3S\Rightarrow\;E_y=\frac{E_2E_x\cos\delta}{E_1}\;+\;E_2\sqrt{1-(\frac{E_x}{E_1})^2}\;\sin\delta$

$3S\Rightarrow\;\sin\delta\;=\;\frac{E_y}{E_2\sqrt{1-(\frac{E_x}{E_1})^2}}\;-\;\frac{E_x\cos\delta}{E_1\sqrt{1-(\frac{E_x}{E_1})^2}}$

$3S\Rightarrow\;\sin^2\delta\;=\;\frac{E^2_y}{E_2^2\;(1\;-\;(\frac{E_x}{E_1})^2)}\;-\;\frac{2E_y\;E_x\;\cos\delta}{E_1\;E_2\;(1\;-\;(\frac{E_x}{E_1})^2}\;+\;\frac{E_x^2\;\cos^2\;\delta}{E_1^2\;(1\;-\;(\frac{E_x}{E_1})^2)}$

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

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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