Quad Detector Detail

Assume that the rotating frame frequency is n_o and that the signal in the laboratory frame of reference is at n. The input signal to the quadrature detector, and hence the two DBMs, is

Cos(2π ν t).

The input reference to the quadrature detector is

Cos(2π ν_o t).

The inputs to the two DBMs within the quadrature detector become

Cos(2π ν_o t) and Sin(2π ν_o t)

since one of the DBMs has a 90^o phase shifter before it.

According to the trigonometric identities introduced in Chapter 2, the outputs from the two DBMs in the quadratre detector are

Cos(2π ν t) Cos(2π ν_o t) = 1/2 Cos(2π ν t - 2π ν_o t) + 1/2 Cos(2π ν t + 2π ν_o t)

Sin(2π ν t) Cos(2π ν_o t) = 1/2 Sin(2π ν t + 2π ν_o t) + 1/2 Sin(2π ν t - 2π ν_o t)

The filters remove the high frequency components, leaving

RE = 1/2 Cos(2π ν t - 2π ν_o t)

IM = 1/2 Sin(2π ν t - 2π ν_o t)

Note that one output of the quadrature detector gives the cosine and the other the sine of the rotating frame frequencies.