I have some doubts about the realization and the operation of a phase detector for a PLL. My reference book is "The Design of CMOS Radio-Frequency Integrated Circuits" (Thomas H. Lee).
Its aim is that of generating a signal which has the same frequency and, more precisely, the same instantaneous phase, of a reference input signal.
The Phase Detector is defined as a component whose output is a signal whose amplitude is proportional to the difference of phases of its input signals. The simplest way to realize it is that of using an analog multiplier, i.e. a mixer:
where \$\ S_i (t)\$ and \$\ S_f (t)\$ are the input and output signal of the PLL, both put at the input terminals of the Phase Detector.
The output of a mixer contains a term proportional to \$ cos(\phi)\$ and a term which contains \$ \omega \$, and the last one is deleted through a filter (not shown in the scheme) in order to get only a signal proportional to the cosine of the phase difference.
Now, I have the following questions:
1) The book says that the best choice is that of realizing a PLL which perform the lock condition with a phase difference equal to \$ \phi = 90°\$ in order to maximize the phase detector gain, whose expression is:
But, how can I decide at which angle does the PLL stabilize and perform the lock? I think that in theory the previous PLL may generate an output signal with any value of phase difference with respect to the input signal and I do not understand how can we decide it: it seems to be a casual number. In fact, the following stage is a VCO, whose output signal's frequency is proportional to the voltage signal applied to it.
Therefore, it is sufficient for the output of the Phase Detector to be constant, in order to have a stable oscillation frequency of the VCO. So, it seems to me that the loop may stabilize with any casual value of the phase difference: the only important thing is that it must be constant in time.
2) My book proposes other types of Phase Detectors for the situation in which one of its input signal is a square wave (while the other one is a sine wave).
I do not understand if the square wave is an approximation of a sine wave (and in that case I do not understand why approximate only 1 of them in this wave) or if it is properly chosen to be a square wave (and in this case I do not understand in which sense the PLL may synchronize it with the other sine wave, since they are different waveforms).