handbook of modern sensors

230 6 Occupancy and Motion Detectors
An antenna transmits the frequency f 0 which is defined by the wavelength λ 0 as
f 0 = c 0
λ 0
, (6.1)
where c 0 is the speed of light. When the target moves toward or away from the transmitting
antenna, the frequency of the reflected radiation will change. Thus, if the
target is moving away with velocity v, the reflected frequency will decrease and it
will increase for the approaching targets. This is called the Doppler effect, after the
Austrian scientist Christian Johann Doppler (1803–1853). 3 Although the effect first
was discovered for sound, it is applicable to electromagnetic radiation as well. However,
in contrast to sound waves that may propagate with velocities dependent on the
movement of the source of the sound, electromagnetic waves propagate with the speed
of light, which is an absolute constant. The frequency of reflected electromagnetic
waves can be predicted by the theory of relativity as
f r = f 0
√
1 − (v/c0 ) 2
1 + v/c 0
. (6.2)
For practical purposes, however, the quantity (v/c 0 ) 2 is very small as compared with
unity; hence, it can be ignored. Then, the equation for the frequency of the reflected
waves becomes identical to that for the acoustic waves:
1
f r = f 0 . (6.3)
1 + v/c 0
Due to a Doppler effect, the reflected waves have a different frequency f r . The mixing
diode combines the radiated (reference) and reflected frequencies and, being a nonlinear
device, produces a signal which contains multiple harmonics of both frequencies.
The electric current through the diode may be represented by a polynomial:
n∑
i = i 0 + a k (U 1 cos 2πf 0 t + U 2 cos 2πf r t) k , (6.4)
k=1
where i 0 is a dc component, a k are harmonic coefficients which depend on a diode
operating point, U 1 and U 2 are amplitudes of the reference and received signals,
respectively, and t is time. The current through a diode contains an infinite number
of harmonics, among which there is an harmonic of a differential frequency:
a 2 U 1 U 2 cos 2π(f 0 − f r )t, which is called a Doppler frequency f .
The Doppler frequency in the mixer can be found from Eq. (6.3):
f = f 0 − f r = f 0
1
c 0 /v + 1 ; (6.5)
3 One hundred fifty years ago acoustical instruments for precision measurements were not
available; yet, to prove his theory, Doppler placed trumpeters on a railroad flatcar and musicians
with a sense of absolute pitch near the tracks. A locomotive engine pulled the flatcar
back and forth at different speeds for two days. The musicians on the ground “recorded”
the trumpet notes as the train approached and receded. The equations held up.