Page 473 - The Mechatronics Handbook

0066_frame_C19 Page 95 Wednesday, January 9, 2002 5:32 PM

d = φl/4π = φc/4πf if the emitter and receiver are at the same location, or d = φl/2π = φc/2πf if
the receiver is attached to the target, where c is the speed of travel, φ is the measured phase, and f is the
modulation frequency.
The phase shift between outgoing and reflected sine waves can be measured by multiplying the two
signals together in an electronic mixer, then averaging the product over many modulation cycles
(Woodbury et al., 1993). This integrating process can be relatively time consuming, making it difficult to
achieve extremely rapid update rates. The result can be expressed mathematically as follows (Woodbury
et al., 1993):

1 T  2pc 4pd   2pc 
lim --- ∫ sin  ---------t + ---------- sin  --------- dt (19.70)
T→∞ T 0 l l  l 

which reduces to

 4pd 
Acos  ---------- (19.71)
l 

where t is the time, T is the averaging interval, and A is the amplitude factor from gain of integrating
amplifier.
From the earlier expression for φ, it can be seen that the quantity actually measured is in fact the cosine
of the phase shift and not the phase shift itself (Woodbury et al., 1993). This situation introduces a so-
called ambiguity interval for scenarios where the round-trip distance exceeds the modulation wavelength
λ (i.e., the phase measurement becomes ambiguous once φ exceeds 360°). Conrad and Sampson (1990)
define this ambiguity interval as the maximum range that allows the phase difference to go through one
complete cycle of 360°:

c
R a = ---- (19.72)
2f
where R a is the ambiguity range interval.
Referring to Eq. (19.73), it can be seen that the total round-trip distance 2d is equal to some integer
number of wavelengths nλ plus the fractional wavelength distance x associated with the phase shift. Since
the cosine relationship is not single-valued for all of φ, there will be more than one distance d corre-
sponding to any given phase-shift measurement (Woodbury et al., 1993):

(
--------- =
cos φ = cos   4pd  cos   2p x + nλ)   (19.73)
---------------------------
l 
λ
where
d = (x + nλ)/2 = true distance to target,
x = distance corresponding to differential phase φ,
n = number of complete modulation cycles.
Careful re-examination of Eq. (19.73), in fact, shows that the cosine function is not single-valued even
within a solitary wavelength interval of 360°. Accordingly, if only the cosine of the phase angle is measured,
the ambiguity interval must be further reduced to half the modulation wavelength, or 180° (Scott, 1990).
In addition, the slope of the curve is such that the rate of change of the nonlinear cosine function is not
constant over the range of 0 ≤ φ ≤ 180°, and is in fact zero at either extreme. The achievable accuracy
of the phase-shift measurement technique thus varies as a function of target distance, from best-case

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