THAI JOURNAL OF PHYSICS, SERIES 9, (2013)
Interpolation Equation for Calibration Results of Thermo-Hygrometers
T. Sinhaneti*, P. Phuauntharo, and T. Keawprasert
National Institute of Metrology Thailand, 3/4-5 Moo 3 Klonghar Klong Luang,
Pathum Thani, 12120 Thailand
*Corresponding author. E-mail: thasorn@nimt.or.th
Abstract
Most of commercial thermo-hygrometers applied the linear equation as its calibration equation for both relative humidity
and temperature, so three calibration points for each parameter are required for the thermo-hygrometer calibration by their
manufacturer. Meanwhile calibration laboratories calibrate thermo-hygrometers in terms of correction at points. As a
result, it is rather difficult for users to apply any calibration results for a whole range since the instrument’s behavior is
non-linear and sensitive to temperature. In this paper, two selected types of high accuracy thermo-hygrometer were
intensively calibrated for their relative humidity corrections throughout the relative humidity range of 30 % to 80 % at
different temperatures of 15 °C to 30 °C. For each type, the resulting corrections were evaluated by fitting to different
polynomial equations to determine its best interpolation equation. It was found that the quadratic polynomial equation of
two variables for the relative humidity with the linear function of temperature is accurate enough to use as an interpolation
equation for these two types of thermo-hygrometers. Moreover, for one type of them, the quadratic polynomial equation of
two variables both for relative humidity and temperature is very accurate, yielding the residual error of lower than 0.1% for
relative humidity. From this result, this type of thermo-hygrometer and its best appropriate interpolation equation will be
implemented for the next inter-laboratory comparison in humidity measurement among calibration laboratories in
Thailand.
Keywords: Thermo-hygrometer, Calibration, Interpolation equation
Introduction
Problem
Presently thermo-hygrometers are widely used in
many applications especially for environment control and
industrial process control. For traditional application, a
narrow operating temperature range is enough, while a
wide range is required for industry. Normally thermohygrometer manufacturers require sensors which their
response is linear for relative humidity measurement so that
a linear equation can be used as their calibration equation.
Moreover, relative humidity at a given temperature is
defined as the ratio between the actual partial vapor
pressure to the saturation vapor pressure of the air
temperature [1]. That is the air temperature is a main factor
affecting to the accuracy of thermo-hygrometers, including
non-linearity, thermal radiation, hysteresis, contamination
and condensation.
Fig. 1 show measurement results of a thermohygrometer in terms of relative humidity correction for the
range of 40% to 80% at different temperatures. It can be
clearly seen that the correction values at any relative
humidity are changed by the air temperature. Moreover the
non-linearity can be found at the results of 40%, 50% and
60% since the manufacture use a linear equation as its
calibration equation.
A main task of this work is to study the relationship
between the thermo-hygrometer output and the actual
relative humidity at any temperatures. The aim is to find
out an appropriate interpolate equation which can describe
the behavior of thermo-hygrometer. It is expected that the
resulting equation will be used to determine correction
values of relative humidity at any temperatures accurately.
Fig.1 Corrections at difference temperatures
For properly use, thermo-hygrometers must be used at
same temperature on certificate of calibration to avoid such
a problem. Moreover 3 calibration points for each
parameter (temperature and relative humidity) are not
enough for a whole range calibration.
* Corresponding author. Tel: 02-577-5100
Fax: 02-577-5092
; E-mail: thasorn@nimt.or.th
© 2013 Thai Physics Society
T. Sinhaneti
Methodology
In this work, two types of high accuracy thermohygrometer were selected in this work. Firstly these
thermo-hygrometers were intensively calibrated for their
correction covering their measuring range as much as
possible. A dew-point hygrometer and a platinum
resistance thermometer were used as reference
thermometers to calibrate the test thermo-hygrometers by
comparison method in a 4.5 ft2 climatic chamber. An
accuracy of the used dew-point hygrometer and the PRT is
traceable to NIMT national standard in humidity and
temperature respectively. The climatic chamber used to
generate humid air was characterized for its stability and
uniformity for a whole operation range.
The calibration for each item was started from the
lowest temperature to the highest temperature. For each
temperature, at least 5 calibration points in the relative
humidity range from 30% to 80% are required. Measuring
relative humidity of thermo-hygrometer, and the relative
humidity and the air temperature measured by the dewpoint hygrometer and the PRT were the main parameters
which must be recorded in this measurement.
Next the correction values at the calibration points
H(R,t) can be calculated by subtracting the actual relative
humidity value obtaining from the reference thermometers
(RSTD) with the reading value of the test thermo-hygrometer
(RUUC.) as follows
H ( R, t )  RSTD (t )  RUUC (t )
Fig 2. Measured correction and estimate correction
calculated from the interpolate equation for sensor A
From the measurement result of the sensor A, the nonlinearity behavior can be clearly observed at low
temperatures.
(1).
After that, the resulting corrections were fitted to
different polynomial function both for the relative humidity
and the air temperature using the regression method.
Finally the resulting interpolation equation was used to
estimate correction values at any relative humidity and
temperature in order to compare to the measuring
correction values.
Figure 3 shows residual errors of within ±0.6% from
the measured value for the whole relative humidity range. It
seems the residual errors increases with increasing relative
humidity.
Results and Discussion
Measurement results of one sensor for each type are
shown and analyzed. Here the first type and the other one
are defined as sensor A and sensor B respectively. From the
method mentioned above, it was found that the polynomial
function of the two variables in Equation (2) to be
sufficiently accurate for both of them.
H ( R, t )  b0  b1 R  b2 R 2  b3t  b4 R t  b5 R 2t
 b6t 2  b7 R t 2
(2),
where H(R,t) is resulting correction of the test thermohygrometer at the relative humidity R and the air
temperature t measured by the dew-point thermometer and
the PRT. The bi parameter is fitting coefficient calculated
from the measurements results of R and t.
Figure 2 shows the measured correction values of the
sensor A in comparison to estimate correction values
calculated from the interpolate equation above.
Fig 3. Residual error from the measured value for the
thermo-hygrometer sensor A
For the sensor A, a poor fitting result was found with the r2
of lower than 0.9. Even increasing the order of the
polynomial function, the measurement results are still
unable to fit accurately.
T. Sinhaneti
237
each type must be investigated to ensure their behavior.
Therefore an ultimate goal of the subsequent work is to
identify the characteristic of frequently used types of
thermo-hygrometer.
Conclusions
Two types of high accurate thermo-hygrometer were
investigated to find out the sufficiently general
interpolation equation for each type. With the quadratic
polynomial equation of two variables for relative humidity
and temperature, a good agreement between the measured
values and the estimate values was found for the calibration
result of one sensor. As a result, this type of thermohygrometer and its appropriate interpolation equation are
already implemented for the in progress inter-laboratory
comparison in humidity measurement of Thailand.
1.
Fig 4. Measured correction and estimate correction
calculated from the interpolate equation for sensor B
Figure 4 shows the measured correction values of the
sensor B and its estimate correction values calculated from
the interpolate equation in Equation 2. For the sensor B, a
good fitting result was found for its measurement results
with r2 of greater than 0.99. Furthermore, this sensor seems
to response to relative humidity linearly for all test
temperatures.
For the sensor B, the associate residual errors from the
measuring value are not larger than +0.1% for the whole
range of 30% to 80%. The advantage of applying this
interpolate equation for the calibration results is that the
calibrated thermo-hygrometer can be used at any
temperatures cover the whole calibration range with much
lower error. At this stage, the maximum of residual error is
considered as an uncertainty for the whole range
calibration.
Fig 5. Residual error from the measured value for the
thermo-hygrometer sensor B
Even a good agreement was found for this type of thermohygrometer, it should be noted that this is the results from
one sample of this type. In fact, more than two sensors for
REFERENCES
J. W. Lovell-Smith and H. Pearson, “On the concept
of relative humidity”, Metrologia 43 (2006), 129-134.