Simon Haykin, Communication Systems, chapter 3, Costas Loop. The frequency of the local oscillator is assumed to be the same as the carrier frequency, the difference is only in the phase shift. The DC component is extracted from the output of the phase discriminator, and used to adjust the phase of the local oscillator until it reaches zero. For VCOs, I know that the DC voltage is used to control the frequency of the oscillator, can the DC voltage affect the phase of the oscillator not the frequency, as in this case? Also, I noticed that the demodulated output signal is the information signal multiplied by cos phi (constant value), thus the phase shift phi affects only the amplitude of the demodulated signal, which can easily be solved with no need to the loop 
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Since frequency is just the derivative of phase with respect to time, the phase is just the integral of frequency.
Can't change frequency without affecting the phase!
answered Nov 5 at 22:17
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$\begingroup$ I noticed that the phase shift affects only the amplitude of the output, not its phase nor its frequency, thus the spectral components are kept correctly. I think there is no problem with the change in amplitude, what is the benefit then of this circuit? $\endgroup$– NohaCommented Nov 9 at 21:59
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$\begingroup$ @Noha you will have to read up on Inphase and Quadrature components. All our anwers assume you've read that, because your figures mention them, and they appear in textbooks with such figures before and around these figures. $\endgroup$ Commented Nov 10 at 14:09