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Lake Shore Model 330 Autotuning Temperature Controller User's Manual
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
is a normalized variable given by:
where V = voltage and VL and 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 V"U 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.
Program 1. BASIC Subroutine to evaluate temperature
T from the Chebychev series using Equations 1 and 3.
An array Tc (Ndegree) should be dimensioned.
100
REM Evaluation of Chebychev series
110
X = ((V-VL) -(VU-V)) / (VU-VL)
120
Tc(0) = 1
130
Tc(1) = x
140
T = A(0) + A(1) * X
150
FOR I = 2 to Ndegree
160
Tc(I) = 2 * X * Tc(I-1) - Tc(I-2)
170
T = T + A(I) * Tc(I)
180
NEXT I
190
RETURN
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
Application Notes
n
a f
∑
=
T x
=
i
0
(x) = a Chebychev polynomial, and a
i
−
(
V VL
=
x
(
VU VL
t
(x) = 2xt
i+1
t
(x) = 1, t
(x) = x
0
1
t
(x) = cos[i x arccos (x)]
i
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)= 0.039255
a f
a t x
i i
= the Chebychev coefficient. The parameter x
i
−
−
) (
VU V
)
−
)
(x) - t
(x)
i
i-1
degree
Program 2. BASIC Subroutine to evaluate temperature
T from the Chebychev series using Equations 1 and 4.
ACS is used to represent the arccosine function.
100
REM Evaluation of Chebychev series
110
X = ((V-VL) -(VU-V)) / (VU-VL)
120
T = 0
130
FOR I = 0 to Ndegree
140
T = T + A(I) * COS(I * ACS(X))
150
NEXT I
160
RETURN
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)=-0.116823
A(11)= 0.058580
(1)
(2)
(3)
(4)
).
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
A(10)=
0.015619
B-9
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