Optimized calibration of cryogenic silicon thermodiodes
Shown in this paper is the efficiency of smoothing cubic spline approximation aimed at optimization of calibration of cryogenic silicon diode thermometers (SDTs). The proposed algorithm allows to significantly reduce time and material costs associated with calibration of SDTs.
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2011
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| Назва видання: | Semiconductor Physics Quantum Electronics & Optoelectronics |
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| Цитувати: | Optimized calibration of cryogenic silicon thermodiodes / O.M. Ivashchenko, Yu.M. Shwarts, M.M. Shwarts, D.P. Kopko, M.I. Sypko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 4. — С. 399-402. — Бібліогр.: 4 назв. — англ. |
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irk-123456789-1177842017-05-27T03:05:19Z Optimized calibration of cryogenic silicon thermodiodes Ivashchenko, O.M. Shwarts, Yu.M. Shwarts, M.M. Kopko, D.P. Sypko, M.I. Shown in this paper is the efficiency of smoothing cubic spline approximation aimed at optimization of calibration of cryogenic silicon diode thermometers (SDTs). The proposed algorithm allows to significantly reduce time and material costs associated with calibration of SDTs. 2011 Article Optimized calibration of cryogenic silicon thermodiodes / O.M. Ivashchenko, Yu.M. Shwarts, M.M. Shwarts, D.P. Kopko, M.I. Sypko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 4. — С. 399-402. — Бібліогр.: 4 назв. — англ. 1560-8034 PACS 06.20.Fb; 02.60.Ed; 02.60.Gf; 07.20.Dt http://dspace.nbuv.gov.ua/handle/123456789/117784 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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English |
| description |
Shown in this paper is the efficiency of smoothing cubic spline approximation
aimed at optimization of calibration of cryogenic silicon diode thermometers (SDTs).
The proposed algorithm allows to significantly reduce time and material costs associated
with calibration of SDTs. |
| format |
Article |
| author |
Ivashchenko, O.M. Shwarts, Yu.M. Shwarts, M.M. Kopko, D.P. Sypko, M.I. |
| spellingShingle |
Ivashchenko, O.M. Shwarts, Yu.M. Shwarts, M.M. Kopko, D.P. Sypko, M.I. Optimized calibration of cryogenic silicon thermodiodes Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Ivashchenko, O.M. Shwarts, Yu.M. Shwarts, M.M. Kopko, D.P. Sypko, M.I. |
| author_sort |
Ivashchenko, O.M. |
| title |
Optimized calibration of cryogenic silicon thermodiodes |
| title_short |
Optimized calibration of cryogenic silicon thermodiodes |
| title_full |
Optimized calibration of cryogenic silicon thermodiodes |
| title_fullStr |
Optimized calibration of cryogenic silicon thermodiodes |
| title_full_unstemmed |
Optimized calibration of cryogenic silicon thermodiodes |
| title_sort |
optimized calibration of cryogenic silicon thermodiodes |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2011 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117784 |
| citation_txt |
Optimized calibration of cryogenic silicon thermodiodes / O.M. Ivashchenko, Yu.M. Shwarts, M.M. Shwarts, D.P. Kopko, M.I. Sypko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 4. — С. 399-402. — Бібліогр.: 4 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
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2025-07-08T12:47:42Z |
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| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 3. P. 399-402.
PACS 06.20.Fb; 02.60.Ed; 02.60.Gf; 07.20.Dt
Optimized calibration of cryogenic silicon thermodiodes
O.M. Ivashchenko1, Yu.M. Shwarts2, M.M. Shwarts3, D.P. Kopko4, M.I. Sypko
V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine,
41, prosp. Nauki, 03028 Kyiv, Ukraine
Phone/ Fax: +38/044/525-7463
1E-mail: iva@isp.kiev.ua
2E-mail: shwarts@isp.kiev.ua
3E-mail: mshwarts@isp.kiev.ua
4E-mail: kdp2007@ukr.net
Abstract. Shown in this paper is the efficiency of smoothing cubic spline approximation
aimed at optimization of calibration of cryogenic silicon diode thermometers (SDTs).
The proposed algorithm allows to significantly reduce time and material costs associated
with calibration of SDTs.
Keywords: spline, free-knot, silicon diode, temperature sensor, calibration.
Manuscript received 21.12.10; revised manuscript received 10.08.11; accepted for
publication 14.09.11; published online 30.11.11.
1. Introduction
The procedure aimed at calibration of temperature
sensors includes stabilization and measurement of a
selected temperature, maintaining a stable operation
current, measurement of electrical signals (voltages,
currents) and resistances. To achieve high precision in
measurements, it is necessary to use highly accurate
instruments based on modern microcontrollers as well as
automated systems to control and keep temperature.
Thus, calibration is a complex technical task that
requires significant time and material costs, which
substantially increases the cost of thermal sensors.
To solve this task, proposed in the paper [1] is the
method that simplifies calibration of interchangeable
cryogenic thermistors of the Cernox type within several
Kelvins by reducing the number of measurements from
50 down to 30 within the temperature range 1.6 – 300 K.
The method is based on the heuristic analysis of large
array experimental data. Algorithm proposed by the
authors using the model of a cubic spline with free knots
allows finding the position and the minimum number of
experimental points within the operation temperature
range for thermistors necessary for calculation of their
thermometric characteristics (TCh). At the same time,
the algorithm requires a number of specific criteria, for
example, the fourth derivative of the sensor TCh should
not be equal to zero.
Previously, we reported on the development of
cryogenic silicon diode thermometers (SDTs) [2]. As
shown in [3], their individual TCh are well described by
a cubic spline with free knots, when using a large
amount of experimental data. At the same time, the
method [1] can not be used to optimize the thermal diode
calibration due to violations of the criterion for its
applicability: a weak temperature dependence of the
sensitivity for these SDTs within the range 100 to 373
K does not allow determining its derivatives with a high
precision.
The purpose of this article is to develop an
optimization algorithm aimed at calibration of
interchangeable silicon diode thermometers by
minimizing the number of measurements necessary to
calculate TCh with the required accuracy by using the
model of smoothing cubic spline with free knots.
2. Samples and experimental technique
To develop a method for optimizing the calibration of
the developed SDTs, we made 300 samples. Before
grading the SDTs, we exposed them up to 30 thermal
shocks (room temperature - liquid nitrogen and room
temperature - liquid helium), which was followed by
controlling the stability of sensor readings at
temperatures of liquid nitrogen and/or helium.
Calibration of the SDTs for three values of the operation
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
399
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 3. P. 399-402.
current I was carried out using the automated
metrological stand UGT-A with a flow cryostat [2].
According to our estimates, the total measurement error
did not exceed ±40 mK, the standard deviation (SD) of
the random component of the experimental error in the
temperature equivalent was less than 10 mK.
3. Results
The detailed calibration of the sensor volume sample C
was carried out using a large number of experimental
points (for I=1 μA, the number of experimental points N
= 169 within the range 4.2 to 373 K, and for I = 10 and
100 μA N = 106 within the range 30 to 373 K).
According to the criteria selected by us in [3], using a
cubic smoothing spline found were the approximating
functions for experimental dependences Ui(Т) as well as
temperature sensitivity of sensors dTdFS ii /= .
Moreover, as shown in Table 1, to calculate F (T)i for
similar SDTs we used splines with the same number and
distribution of knots over the temperature axis.
Table 1. Parameters of the approximating spline
F (T)i
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
TT for SDTs
Amounts of non-repeated knots
in temperature subranges, K
Cur-
rent,
μA
SDT
num-
ber 4.2…20 20…50 50…150 150…373
Total
amount
of
knots
1 12 6 8 6 32
2 12 6 7 5 30 1
3 12 6 8 5 31
1 5 7 5 17
2 5 10 5 20 10
3 5 7 5 17
1 6 8 5 19
2 6 8 5 19 100
3 7 7 5 19
To determine the level of interchangeability for the
developed SDTs, first we found the standard
characteristics ( )TF for each operation current:
( )
( )
C
TF
TF
C
i
i∑
== 1 , (1)
where C = 26 for I = 1 μA and C = 11 for currents 10
and 100 μA, and then the temperature deviations of
individual calculated TChs from the standard
( ) ( ) ( )
( ) dTTFd
TFTFT i
Fi
−
=Δ . (2)
The performed analysis has shown that the
developed sensors can be characterized by five tolerance
bands (Table 2):
Table 2. Tolerance bands for interchangeability of the
developed SDTs
Ope-
ration
cur-
rent,
μA
Designa-
tion of
tolerance
bands
Deviation from
the standard
TCh
(in temperature
equivalents), K
Temperature
range, K
А ±0.25 4.2 - 373
B ±0.35
±0.25
4.2 - 50
50 - 373
C ±0.60 4.2 - 373
D ±0.50
±1
4.2 - 100
100 - 373
1
E ±2.5 4.2 - 373
А ±0.35 30 - 373
B ±0.50 30 - 373
C ±1 30 - 373
D ±1.5 30 - 373
10
E ±3 30 - 373
А ±0.35 30-373
B ±0.50 30 - 373
C ±0.50
±1
30 - 78
78 - 373
D ±1.50 78 - 373
100
E ±3 30 - 373
The deviation ( )T
iFΔ of the SDT group from the
band C for the current 1 μA is illustrated in Fig 1. One
can see that the maximum deviation value over the
whole operation range for these SDTs does not exceed
±0.6 K.
As seen from the data adduced in Table 2, the
sensors of the A band are the closest ones to the standard
over the whole operation temperature range.
In general, ( )T
iFΔ for all operation currents does
not exceed ±3 K (band E). The data obtained on the
interchangeability of the SDTs and the ability to describe
the experimental TChs by splines with the same
parameters show that the algorithm of optimized
calibration can be applied to all the developed series of
SDTs.
Taking ( )TF as a model function, we find the
minimum number of experimental points M < N and
their location within the operation temperature range [T1,
TM] for chosing such a function ( )T F ′ for which the
condition
[ ]
( )( ) exp,
1.0max
1
Δ⋅≤Δ
∈
T
MTTT
, (3)
where
( ) ( ) ( )
/dTFd
T FT FT −′
=Δ (4)
is fulfilled.
400
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 3. P. 399-402.
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Fig. 1. Deviations of individual calculated TChs for groups of
the SDTs from the standard curve. Operation current is 1 μA.
The data are presented for samples from the band C.
From the mathematical viewpoint, the desired
function of the spline can be represented as:
( ) ( )TBTF j
n
j
j∑
=
=′
1
¦Б (5)
where j¦Б are the coefficients of the spline, ( )TB j - β -
spline of the third degree computed for the j-th part
between n-knots.
To define ( )T F ′ , let us take the vector of knot
approximation as }...1;¦У{¦У njj == and the data table
{Ti, Ui; i=1…M}. The voltage values corresponding to
the temperature Ti are found from the model
function ( )T F , i.e. Ui = ( )TF i . At the first stage of
searching, we find the knots allowing for description of
all the features inherent to the function ( )T F . At the
same time, to limit the search range we should use the
temperature points 4−= nMint (the case of
interpolation) coinciding with the knots (Fig. 2).
As a result of interpolation, we have obtained the
temperature intervals with their corresponding errors.
From this series of errors σ, we find the k-th and l-th
intervals with the highest σk and lowest σl errors,
respectively. According to [4], reduction of the mesh
size for interpolation significantly reduces the errors of
cubic spline interpolation. Let us make a decrease in the
k-th and the increase in l-th intervals by the value δТ up
to the moment when for a certain set of knots in all the
mesh intervals the errors become practically the same,
that is, until the condition
121 ... −σ≈≈σ≈σ M (6)
will be fulfilled.
This criterion allows us to find the optimal mesh of
knots τopt, using which we significantly reduce the
maximum interpolation error. However, the use of knots
τopt to approximate the actual experimental data does not
provide the condition (3). Therefore, it is necessary to
apply a smoothing spline constructed by the least
squares method, which requires additional experimental
points located between the knotes in the temperature
scale (Fig. 3).
As result of simulating the function ( )T F by the
function ( )T F ′
iS
with account of the condition (3), were
found the optimum temperature meshes Тopt and knots of
approximation τopt for calibration at three operation
currents. Their numbers were for the operation currents
as follows: for 1 μA Тopt = 95 points, τopt = 42 knots; for
10 μA Тopt = 31 points, τopt = 18 knots; for 100 μA Тopt =
36 points, τopt = 23 knots.
The found in this way optimal meshes Тopt and τopt
were used to simulate F (T)i and (Т) for the
investigated SDTs under various operation currents. The
obtained results have been illustrated in Figs 4 to 6.
Regardless of SDT affiliation to any tolerance band,
using the same meshes Тopt and τopt allows to
approximate F (T)i with the accuracy not worse than
4 mK.
Fig. 2. Arrangement of temperature points Ti and knots of the
interpolation spline τj on the temperature axis [Т1,ТMint]. The
operation current of the sensor is 1 μA.
Fig. 3. Arrangement of experimental points Ti and knots of the
approximating spline τj on the temperature axis [Т1, ТMapp]. The
operation current of the sensor is 1 μA.
Fig. 4. Approximation error for the function F (T)i describing
the sensor NKGPI20 when using the optimal meshes Тopt and
τopt. The operation current of the sensor is 1 μA.
401
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 3. P. 399-402.
Fig. 5. Approximation error for the function F (T)i describing
the sensor 220 when using the optimal meshes Тopt and τopt. The
operation current of the sensor is 10 μA.
Fig. 5. Approximation error for the function F (T)i describing
the sensor 233 when using the optimal meshes Тopt and τopt. The
operation current of the sensor is 100 μA.
The account of the error for setting the temperature
in the course of a real experiment made it possible to
determine the necessary and sufficient accuracy for
location of the mesh points Тopt with mandatory
condition (3). Our analysis of the obtained data has
shown that the accuracy of setting the temperature
should be: for 1 μA ±0.3 K at temperatures below 30 K,
±3 К at temperatures above 30 K, for 10 μA - ±1.5 К
over the entire operation range; for 100 μA - ±0.3 K
within the range of 30 to 80 K, ± 1.5 К at temperatures
above 80 K. These precision are quite feasible for the
installation of UGT-A.
4. Conclusions
1. It is shown that the developed SDTs are characterized
by the five tolerance bands. The dispersion of sensor
characteristics within a zone range from ± 0.25 up to ±
3.0 К.
2. Developed is the algorithm for optimizing the
calibration of interchangeable SDTs, which enables us to
minimize the number of measurements needed to
calculate thermometric characteristics with a given
accuracy using the model of smoothing cubic splines
with free knots. The proposed method makes it possible
to significantly reduce time and material expenses
associated with calibration of these sensors.
References
1. 1. P. Ciarlini, D. Ichim. Free-knot cubic spline
modelling in cryogenic thermometer calibration //
Measurement, 39(9), p. 815 – 820 (2006).
2. Yu. M. Shwarts, O. M. Ivashchenko, M. M.
Shwarts. Metrological providing diode
thermometry // Pribory, 8(86), p. 5 – 11 (2007).
3. O. M. Ivashchenko, Yu. M. Shwarts, M. M.
Shwarts, D. P. Kopko, N.I. Sypko. Smoothing
cubic spline approximation of thermometric
characteristics for silicon diode temperature sensors
// Semiconductor Physics, Quantum Electronics &
Optoelectronics, 14(4), p. … (2010).
4. V. P. Denisyuk, B. G. Marchenko. Splines and
their application to the tasks of modeling and
processing the measuring signals. Kyiv, Published
in Kyiv Polytechnical Institute, 1995. 246 p.
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
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