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Sec. 4–13 Detector Circuits 275
the low-pass filter needs to be much smaller than the carrier frequency f c and much larger than
the bandwidth of the (detected) modulation waveform B. That is,
B
1
2pRC f c
(4–75)
where RC is the time constant of the filter.
The envelope detector is typically used to detect the modulation on AM signals. In this
case, v in (t) has the complex envelope g(t) = A c [1 + m(t)], where A c 7 0 represents the strength
of the received AM signal and m(t) is the modulation. If |m(t)| 6 1, then
v out = KR(t) = K|g(t)| = KA c [1 + m(t)] = KA c + KA c m(t)
KA c is a DC voltage that is used to provide automatic gain control (AGC) for the AM
receiver. That is, for KA c relatively small (a weak AM signal received), the receiver gain is
increased and vice versa. KA c m(t) is the detected modulation. For the case of audio (not
video) modulation, typical values for the components of the envelope detector are R = 10 kΩ
and C = 0.001 µfd. This combination of values provides a low-pass filter cutoff frequency
(3 dB down) of f co = 1(2pRC) = 15.9 kHz, much less than f c and larger than the highest
audio frequency, B, used in typical AM applications.
Product Detector
A product detector (Fig. 4–14) is a mixer circuit that down-converts the input (bandpass
signal plus noise) to baseband. The output of the multiplier is
v 1 (t) = R(t) cos[v c t + u(t)]A 0 cos(v c t + u 0 )
= 1 2 A 0R(t) cos [u(t) - u 0 ] + 1 2 A 0R(t) cos[2v c t + u(t) + u 0 ]
where the frequency of the oscillator is f c and the phase is u 0 . The low-pass filter passes only
the down-conversion term, so that the output is
v out (t) = 1 2 A 0R(t) cos[u(t) - u 0 ] = 1 2 A 0Re{g(t)e -ju 0
}
(4–76)
v in (t)=R(t) cos[ c t +¨(t)]
or
v in (t)=Re[g(t) e j c t ]
where g(t)=R(t) e j¨(t)
v 1 (t)
Low-pass
filter
v 0 (t)=A 0 cos[ c t +¨0]
1
v out (t)=–– A 0 Re[g(t) e
2
j¨0]
Oscillator
Figure 4–14
Product detector.