Improving the metrological characteristics of differential-capacity transducers for displacement measurements
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UDC 531.7.087.92
Differential-capacity transducers have found extensive use in the device of data-measuring technology for various nonelectrical quantities (length, level, force, vibration, etc.) [1-4]. However, the complexity of determining the optimal transducer parameters in many cases makes it impossible to obtain good metrological characteristics. The conversion function (CF) of a differential-capacity displacement transducer [4] made in the form of two parallel-plate capacitors is written as [3]
F1(x) ::A
(
I d--x
1 )= A d+x
2x
(i)
- ' d- '2----~x'
where A = c c o S , e is the relative dielectric constant of the medium between the plates, co is the dielectric constant of a vacuum, S is the overlapping area of the plates, d is the initial separation between the plates, and x is the displacement of the movable plate. It follows from Eq. (i) that as d is reduced, the sensitivity of the transducer and the nonlinearity of its CF increase. The value of the linearity error for the CF depends directly on the form of the polynomial that approximates the true transducer characteristic. In the simplest case the CF can be represented for a given range of measurements with a certain approximation by a straight line, the points of the approximation being placed conveniently at the beginning and end of the measuring range (see Fig. i). Since the value of the linearity error is stipulated for measuring devices, the largest deviation in the values of the CF from the values of the approximating function must be within specified limits. Thus the problem is reduced to a determination of the initial separation between the plates ~uch that for a given measuring range of displacements Xmax and a given linearity error the measuring device has the best sensitivity. The equation for the approximating straight line has the form F2(x) = kx, where k = tan~ is the tangent of the angle of inclination for the straight line. Inasmuch as FI (Xmax) = Fa(xmax) at the end of the measuring range, we will obtain for the approximating straight line with Eq. (i) taken into account 2x F~(x) = A~2 __ xmax-
The value of the absolute linearity error for the CF at any point of the given measuring range will be defined by
A(x) =F~(x)--FI(x)=2Ax
"d 2
2 -
-
d2__x~
'
Xrria x
where A(x) is the absolute linearity error of CF at the point x.
]
Fig. 1
Translated from Izmeritel'naya Tekhnika, No. 5, pp~ 17-18, May, 1978,
0543-1972/78/2105-0617507.50
9 1978 Plenum Publishing Corporation
617
The magnitude of the absolute linearity error must be within the specified limits, i.e.,
( 1 2Ax d~___-X~ax
1 d~ - x2
)r
2Axmax
n
,
(2)
d2 . _ X~ax
where n is the linearity error of the CF referred to the limiting value of the quantity being measured. At some point xx it takes on the maximum value; i,e,, the equality of Eq. (2) is fulfilled. Consequently, the derivative of the function A(x) at the point x~ will be I A'(x0
-- 2A
E
~ d
d-Tx~ a
--
~
1
(~Z.-~
- - Xma x
=0.
(3)
After the transformation of Eqs. (2) and (3) a system of no
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