handbook of modern sensors

6.2 Microwave Motion Detectors 231
and since c 0 /v ≫ 1, the following holds after substituting Eq. (6.1):
f ≈ v λ 0
. (6.6)
Therefore, the signal frequency at the output of the mixer is linearly proportional
to the velocity of a moving target. For instance, if a person walks toward the detectors
with a velocity of 0.6 m/s, a Doppler frequency for the X-band detector is
f = 0.6/0.03 = 20 Hz.
Equation (6.6) holds true only for movements in the normal direction. When the
target moves at angles with respect to the detector, the Doppler frequency is
f ≈ v λ 0
cos . (6.7)
This implies that Doppler detectors theoretically become insensitive when a target
moves at angles approaching 90 ◦ . In the velocity meters, to determine the velocity of
a target, it is required is to measure the Doppler frequency and the phase to determine
the direction of the movement (Fig. 6.1A). This method is used in police radars. For
supermarket door openers and security alarms, instead of measuring the frequency, a
threshold comparator is used to indicate the presence of a moving target (Fig. 6.1B).
It should be noted that even if Eq. (6.7) predicts that the Doppler frequency is near
zero for targets moving at angles = 90 ◦ , the entering of a target into the protected
area at any angle results in an abrupt change in the received signal amplitude, and
the output voltage from the mixer changes accordingly. Usually, this is sufficient to
trigger the response of a threshold detector.
The signal from the mixer is in the range from microvolts to millivolts, so the
amplification is needed for signal processing. Because the Doppler frequency is in
the audio range, the amplifier is relatively simple; however, it generally must be
accompanied by so-called notch filters, which reject a power line frequency and the
main harmonic from full-wave rectifiers and fluorescent light fixtures: 60 and 120 Hz
(or 50 and 100 Hz). For the normal operation, the received power must be sufficiently
high. It depends on several factors, including the antenna aperture area A, target area
a, and distance to the target r:
P r = ρ P 0A 2 a
4πλ 2 r 4 , (6.8)
where P 0 is the transmitted power. For effective operation, the target’s cross-sectional
area a must be relatively large, because for λ 2 ≤ a, the received signal is drastically
reduced. Further, the reflectivity ρ of a target in the operating wavelength is also very
important for the magnitude of the received signal. Generally, conductive materials
and objects with high dielectric constants are good reflectors of electromagnetic radiation,
whereas many dielectrics absorb energy and reflect very little. Plastics and
ceramics are quite transmissive and can be used as windows in the microwave detectors.
The best target for a microwave detector is a smooth, flat conductive plate
positioned normally toward the detector. A flat conductive surface makes a very good