The input reference to the quadrature detector is
The inputs to the two DBMs within the quadrature detector become
since one of the DBMs has a 90o phase shifter before it.
According to the trigonometric identities introduced in Chapter 2, the outputs from the two DBMs in the quadratre detector are
Sin(2π ν t) Cos(2π νo t) = 1/2 Sin(2π ν t + 2π νo t) + 1/2 Sin(2π ν t - 2π νo t)
The filters remove the high frequency components, leaving
IM = 1/2 Sin(2π ν t - 2π νo t)
Note that one output of the quadrature detector gives the cosine and the other the sine of the rotating frame frequencies.
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