Root Causes of Quartz Sensor Drift - Paroscientific
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Paroscientific, Inc. Technical Note
Precision Pressure Instrumentation Doc. No. G8101 Rev. A
Root Causes of Quartz Sensor Drift
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Paroscientific, Inc. 1Paroscientific, Inc. Technical Note
Precision Pressure Instrumentation Doc. No. G8101 Rev. A
Root Causes of Quartz Sensor Drift
Jerome M. Paros and Taro Kobayashi
Abstract:
The root causes of Quartz Sensor drift are analyzed. Models are developed for drift mechanisms that
occur when the Quartz Sensors are unloaded (“outgassing”) and loaded (“viscoelastic creep”).
Background:
Quartz Nano-Resolution Pressure Sensors, Accelerometers, and Tiltmeters are used in disaster
warning systems to detect earthquakes, tsunamis, and severe weather. Reference 1 describes in-situ
calibration methods that can provide the measurements needed for geodesy and long-term forecasting.
Drift is similar on all of the Quartz Sensors despite completely different mechanisms that apply loads to
the quartz crystals. For example, the pressure sensors use bellows, Bourdon tubes, or diaphragms as
pressure-to-force converters whereas the accelerometers generate forces due to accelerations acting
on inertially suspended masses. The common elements are the crystals and their attachments to the
force-producing structures.
Call the drift at zero load "outgassing" and the drift at load (e.g. full-scale) "creep". Both effects diminish
exponentially with time but drift in opposite directions. Outgassing (aging) of quartz crystals can
produce increasing frequencies with apparently higher force outputs. These frequency changes must
be converted to equivalent error forces through the conformance (linearization) equation. Creep is in
the direction of lower outputs and is due to attachment or mechanism “creep” deflections that work
against the spring rate of the mechanism to generate viscoelastic error forces. The combined drift
effects are dependent on the deployment history as illustrated in Figure 1.
The fits to 7 years of typical drift data at 0 were extrapolated and subtracted from 4 months of drift data
held mostly at pressure A = 100 MPa. The resulting curves illustrate Drift @ 0 (outgassing), Drift @ A
(creep), and Drift @ A combined. Data were provided by Dr. Hiroaki Kajikawa of the National Metrology
Institute of Japan--(AIST).
Drift @ 0, Drift @ A-Creep, Drift @ A-Combined
260
240
220
Drift relative to July 2007 [ppm]
200
180
160
140
120
100
80
60
40
20
0
7/13/2007 5/18/2009 3/24/2011 1/27/2013 12/3/2014 10/8/2016 8/14/2018 6/19/2020 4/25/2022
Time (Date)
Figure 1
Paroscientific, Inc. 2Paroscientific, Inc. Technical Note
Precision Pressure Instrumentation Doc. No. G8101 Rev. A
As shown in Figure 2, the combined drift can look quite different depending on the pressure profile.
Drift due to Outgassing, Creep, and Combined
for Different Start Times of Pressurization
250
200
R elative D rift [ppm ]
150
100
50
0
0 1 2 3 4 5 6 7 8 9
-50
-100
Sensor Age (years)
Figure 2
Outgassing:
The Quartz Crystal Resonator Force Sensors have a 10% change in frequency with applied full-scale
load. A 1 part-per-million (ppm) drift in frequency represents 10 ppm drift of full-scale output. Figure 3 is
a slide from Dr. John Vig’s presentation at: http://tf.nist.gov/sim/2010_Seminar/vig3.ppt. The aging
curves of standard (unloaded) quartz crystal resonators due to outgassing are in the direction of higher
frequencies and look very similar to the pressure sensor drift curves at zero load.
Aging and Short-Term Stability
Short-term instability
(Noise)
30
25
Δf/f (ppm)
20
15
10
5 10 15 20 25 Time (days)
4-5
Figure 3
Paroscientific, Inc. 3Paroscientific, Inc. Technical Note
Precision Pressure Instrumentation Doc. No. G8101 Rev. A
In Reference 2, Quartz Sensor stability was mathematically modeled using data from Paroscientific
pressure sensors and Quartz Seismic Sensors accelerometers. Qualitatively, we looked for models that
related to physical reality and quantitatively we looked for the best fits with the fewest free parameters
and the best predictive behavior. Stability data were fit with various models and the residuals between
the data and each fit were compared. As shown in Figure 4, the drift curves for the unloaded Quartz
Sensors are similar in shape to the aging curve shown above for standard crystals used in counter-
timer applications and the mathematical fits are quite good. In the quartz sensors, the frequency
changes due to crystal outgassing must be converted to equivalent error forces through the
conformance (linearization) equation.
SN123148
Comparison of Full Fits (Fit over entire 200 Days)
3 120
2
100
1
0 80
Residuals (ppm)
Power+Log
Drift (ppm)
-1
Log
60
Exp+Linear
-2 Drift
-3 40
-4
20
-5
-6 0
03-Nov-11 23-Dec-11 11-Feb-12 01-Apr-12 21-May-12 10-Jul-12
Figure 4
Creep:
Loads applied to the attachment joints produce viscoelastic creep (deflections) that act against the
spring rates of the mechanisms to generate error forces. The quartz resonator cannot distinguish the
error forces due to creep from the sensed input forces.
The Kelvin-Voigt viscoelastic model (Appendix I) predicts that drift due to creep is proportional to the
reactive spring rate of the mechanism and the applied load. Creep is inversely proportional to the
modulus. The time dependence is an exponential function with a time constant equal to the modulus
divided by the viscosity. When the load is removed, a viscoelastic “recovery” occurs.
Dr. Hiroaki Kajikawa and his colleagues at the National Metrology Institute of Japan--(AIST) followed
seven years of testing with quartz pressure sensors mostly at zero pressure with four months at full-
scale pressure at 100 MPa and then returned to zero pressure to monitor the viscoelastic recovery as
illustrated below. The fit to the creep model was reversed in sign to model the recovery and the
outgassing was added to predict the combined recovery curve. As shown in Figure 5, the predicted
recovery compares well to the data plots excerpted from Reference 3.
Paroscientific, Inc. 4Paroscientific, Inc. Technical Note
Precision Pressure Instrumentation Doc. No. G8101 Rev. A
Drift @ 0, Drift @ A-Combined, and Recovery
370
350
330
310
290
270
Deviation from standard [ppm]
250
230
210
190
170
150
130
110
90
70
50
30
10
7/13/2007 11/24/2008 4/8/2010 8/21/2011 1/2/2013 5/17/2014 9/29/2015
Time [Date]
Predicted Recovery Based on Kelvin-Voigt M odel +
Outgassing
60
58
56
54
DI100 [kPa]
52
50
48
46
44
0 20 40 60 80 100 120 140 160 180 200 220 240 260 280
Tim e [day]
Actual Creep & Recovery Predicted Creep & Recovery
Figure 5
Paroscientific, Inc. 5Paroscientific, Inc. Technical Note
Precision Pressure Instrumentation Doc. No. G8101 Rev. A
Conclusions:
There are two independent root causes for drift--"outgassing" and "creep". Both effects diminish
exponentially with time. At zero load, outgassing generates increasing frequencies with apparently
higher outputs. When the crystals are loaded, creep is in the opposite direction to outgassing and
generates lower outputs. The combined drift effects are dependent on the deployment history.
References:
(1) J.M. Paros and T. Kobayashi, “Calibration Methods to Eliminate Sensor Drift”, G8097,
Paroscientific, Inc., Technical Note
(2) J.M. Paros and T. Kobayashi, “Mathematical Models of Quartz Sensor Stability”, G8095
Paroscientific, Inc., Technical Note.
(3) H. Kajikawa and T. Kobata. “Long-term drift of hydraulic pressure transducers constantly subjected
to high pressure.” Proceedings of the 32nd Sensing Forum, Osaka, Japan, 10-11 September 2015.
pp.261-266.
Paroscientific, Inc. 6Paroscientific, Inc. Technical Note
Precision Pressure Instrumentation Doc. No. G8101 Rev. A
Appendix I Kelvin–Voigt Model of Creep Drift
The Kelvin–Voigt model represents loads applied to a purely viscous damper, D, and purely elastic
spring, S, connected in parallel as shown in Figure 1.
Load Load
Figure 1 Kelvin–Voigt Model
Since the spring and damper are arranged in parallel, the strains in each component are equal:
Strain = εTotal = εD = ε S
The total stress will be the sum of the stress in each component:
Stress = σTotal = σD + σS
The stress in the spring equals the modulus of elasticity, E, times the strain in the spring. The stress in
the damper equals the viscosity, η, times the rate of change of strain in the damper. Thus in a Kelvin–
Voigt material, stress σ, strain ε, and their rates of change with respect to time t are given by:
dε (t )
σ (t ) = Eε (t ) + η
dt
The above equation may be used for both shear stress and/or normal stress load applications.
Response to a Stress Step Function
If a constant stress, σ0, is suddenly applied to a Kelvin–Voigt material, then the strain approaches the
σ0
strain of the pure elastic material, , as a decaying exponential:
E
σ
Strain ( t ) = 0
(1 − e − λ t ) , where t is time and λ is the relaxation rate, λ =
E
E η
Paroscientific, Inc. 7Paroscientific, Inc. Technical Note
Precision Pressure Instrumentation Doc. No. G8101 Rev. A
If the stress is suddenly removed at time, t1, then the deformation is retarded in the return to zero
deformation:
Strain (t > t1 ) = ε (t1 )(1 − e − λ (t −t1 ) )
Figure 2 shows the deformation versus time when constant stress on the material is applied suddenly
at time, t = 0, and suddenly released at the later time, t1.
t > t1
Strain
t1 Time
Figure 2: Deformation versus time for sudden application and release of constant stress
The deformation, ∆L, over the length of the attachment, L, is:
⎛ Lσ 0 ⎞ − λt
ΔL = ⎜ ⎟(1 − e )
⎝ E ⎠
The reactive spring rate of the mechanism, K, acts against the deformation, ∆L, to generate a creep
force, ∆F.
⎛ KLσ 0 ⎞ −λt
ΔF = ⎜ ⎟(1− e )
⎝ E ⎠
Drift due to the creep force, ∆F, can be expressed as a fraction of the full-scale force, FFS:
ΔF ⎛ KLσ 0 ⎞
= ⎜⎜ ⎟⎟(1 − e −λt )
FFS ⎝ FFS E ⎠
The drift due to creep is proportional to the reactive spring rate of the mechanism and the
applied load. Creep is inversely proportional to the modulus. The time dependence is an
exponential function with a time constant equal to the modulus divided by the viscosity.
References:
Meyers, Marc A., and Krishan Kumar Chawla. "Creep and Superplasticity." Mechanical
Behavior of Materials. 2nd ed. Cambridge: Cambridge UP, 2009. 653-688. Print.
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