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POLYNOMIAL REPRESENTATION
Curve 10 can be expressed by a polynomial equation based on the Chebychev polynomials. Four separate ranges are
required to accurately describe the curve. Table 1 lists the parameters for these ranges. The polynomials represent
Curve 10 on the preceding page with RMS deviations of 10 mK. The Chebychev equation is:
where T(x) = temperature in kelvin, t
a normalized variable given by:
where V = voltage and VL & VU = lower and upper limit of the voltage over the fit range. The Chebychev polynomials can
be generated from the recursion relation:
Alternately, these polynomials are given by:
The use of Chebychev polynomials is no more complicated than the use of the regular power series and they offer
significant advantages in the actual fitting process. The first step is to transform the measured voltage into the normalized
variable using Equation 2. Equation 1 is then used in combination with equations 3 and 4 to calculate the temperature.
Programs 1 and 2 provide sample BASIC subroutines which will take the voltage and return the temperature T calculated
from Chebychev fits. The subroutines assume the values VL and VU have been input along with the degree of the fit. The
Chebychev coefficients are also assumed to be in any array A(0), A(1),..., A(i
An interesting property of the Chebychev fits is evident in the form of the Chebychev polynomial given in Equation 4. No
term in Equation 1 will be greater than the absolute value of the coefficient. This property makes it easy to determine the
contribution of each term to the temperature calculation and where to truncate the series if full accuracy is not required.
FUNCTION Chebychev (Z as double)as double
REM Evaluation of Chebychev series
X=((Z-ZL)-(ZU-Z))/(ZU-ZL)
Tc(0)=1
Tc(1)=X
T=A(0)+A(1)*X
FOR I=2 to Ubound(A())
Tc(I)=2*X*Tc(I-1)-Tc(I-2)
T=T+A(I)*Tc(I)
NEXT I
Chebychev=T
END FUNCTION
Program 1. BASIC subroutine for evaluating the
temperature T from the Chebychev series using
Equations (1) and (3). An array T
dimensioned. See text for details.
==
2.0 K to 12.0 K
VL = 1.32412
VU = 1.69812
A(0) = 7.556358
A(1) = -5.917261
A(2) = 0.237238
A(3) = -0.334636
A(4) = -0.058642
A(5) = -0.019929
A(6) = -0.020715
A(7) = -0.014814
A(8) = -0.008789
A(9) = -0.008554
A(10) = 0.039255
Application Notes
Lake Shore Cryotronics, Inc.
∑
( ) =
T x
=
i
(x) = a Chebychev polynomial, and a
i
−
(
V VL
=
x
( ) =
t
x
+
i
1
( ) =
t x
1,
0
( ) =
t x
cos
i
(i
) should be
c
degree
Table 1. Chebychev Fit Coefficients
12.0 K to 24.5 K
VL = 1.32412
VU = 1.69812
A(0) = 17.304227
A(1) = -7.894688
A(2) = 0.453442
A(3) = 0.002243
A(4) = 0.158036
A(5) = -0.193093
A(6) = 0.155717
A(7) = -0.085185
A(8) = 0.078550
A(9) = -0.018312
A(10) = -0.116823
n
( )
a t x
i i
0
= the Chebychev coefficient. The parameter x is
i
) −
−
(
)
VU V
−
(
)
VU VL
( ) −
( )
2
xt x
t
x
−
i
i
1
( ) =
t x
x
1
×
( )
i
arccos
x
).
degree
FUNCTION Chebychev (Z as double)as double
REM Evaluation of Chebychev series
X=((Z-ZL)-(ZU-Z))/(ZU-ZL)
T=0
FOR I=0 to Ubound(A())
T=T+A(I)*COS(I*ARCCOS(X))
NEXT I
Chebychev=T
END FUNCTION
( ) =
NOTE:
arccos
X
Program 2. BASIC subroutine for evaluating the
temperature T from the Chebychev series using
Equations (1) and (4). Double precision calculations are
recommended.
24.5 K to 100.0 K
VL = 1.32412
VU = 1.69812
A(0) = 71.818025
A(1) = -53.799888
A(2) = 1.669931
A(3) = 2.314228
A(4) = 1.566635
A(5) = 0.723026
A(6) = -0.149503
A(7) = 0.046876
A(8) = -0.388555
A(9) = 0.056889
A(10) = 0.015619
A(11) = 0.058580
(1)
(2)
(3)
(4)
π
X
−
arctan
−
2
2
1
X
100 K to 475 K
VL = 1.32412
VU = 1.69812
A(0) = 287.756797
A(1) = -194.144823
A(2) = -3.837903
A(3) = -1.318325
A(4) = -0.109120
A(5) = -0.393265
A(6) = 0.146911
A(7) = -0.111192
A(8) = 0.028877
A(9) = -0.029286
9
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