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## Question on characterization of elliptical polarization of EM wave.

I am using "Advanced Engineering Electromagnetics" 2nd edition by Balanis AND "Antenna Theory" 3rd edition also by Balanis. I found an inconsistency in how to characterize RHC (CW) and LHC ( CCW) elliptical polarization.

1) In Advanced EE Page 159, for
$$\vec E(0,t)=Re[\hat x (E_R+E_L)e^{j\omega t}+\hat y (E_R-E_L)e^{j(\omega t+\Delta \phi)}]$$
$$\hbox { Where}\;\Delta\phi=\phi_x-\phi_y≠\frac{n\pi}{2}\;\hbox {where }\;n=0,2,4,6.....$$
If $\Delta \phi ≥ 0$, then, it is CW if $E_R>E_L$, CCW if $E_R<E_L$
If $\Delta \phi ≤ 0$, then, it is CCW if $E_R>E_L$, CW if $E_R<E_L$

2) In Antenna Theory Page 74,
$$\Delta\phi=\phi_y-\phi_x≠^+_-\frac{n\pi}{2}\;\hbox {where }\;n=0,1,2,3.....$$
If $\Delta \phi ≥ 0$, then, it is CW.
If $\Delta \phi ≤ 0$, then, it is CCW.

To avoid confusion, just use one example where $\Delta\phi=\frac {\pi}{4}$, you can see using Advanced EE, there are two condition that can give you CW or CCW. But in Antenna, there is only one condition which is CW.

How do you explain the inconsistency? Yes, there are confusion as the definition of $\Delta\phi$ is opposite between the two. But if you look pass the difference, you can still see the inconsistency. Am I missing something?

Thanks

Alan
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