Combined Differential and Relative Pressure Sensor based on a by yte37472


									       Combined Differential and Relative Pressure
       Sensor based on a Double-Bridged Structure.
                  C. Pedersen∗ , S. T. Jespersen† , J. P. Krog∗ , C. Christensen∗ and E. V. Thomsen†
                         ∗ Grundfos   A/S, Poul Due Jensens Vej 7, DK-8850 Bjerringbro, Denmark,

                     † Mikroelektronik   Centret, Build. 345 East, DTU, DK-2800 Kgs. Lyngby, Denmark

   Abstract— A combined differential and relative MEMS pres-                                         Pressure
sure sensor based on a double piezoresistive Wheatstone bridge
structure is presented. The developed sensor has a conventional
(inner) bridge on a micro machined membrane and a sec-
ondary (outer) bridge on the chip substrate. The double-bridge
structure has previously been used e.g. in the compensation of                     Patm                                  Patm
temperature-induced errors in MEMS pressure sensors [1], [2],                                         Pa
[3]. A novel approach is demonstrated for a combined measure-
ment of sensor output from inner and outer bridge, leading to                                        Pressure
the deduction of both differential and relative media pressure
(with respect to atm. pressure), and a significant improvement
                                                                   Fig. 1. A cross section sketch of an O-ring clamped double-bridged MEMS
in differential pressure sensor accuracy. Output from both         pressure sensor.
bridges depends linearly on both differential and absolute media
pressure. Furthermore, the sensor stress distributions involved
are studied and supported by extensive 3D FEM stress analysis.                            O-ring contact area

                     I. I NTRODUCTION
   O-ring clamping, as sketched in Fig. 1, has just recently
shown great promise in the field of MEMS pressure sensor
packaging [4]. However, recent studies have shown that the
output signal from the conventional inner bridge has a small
dependency on the absolute media pressure, Pa , relative
to atmospheric pressure, Patm [5]. The relative pressure is
defined as Prel = Pa − Patm . Based on this observation we
have developed a MEMS pressure sensor with a secondary
outer piezoresistive measurement bridge placed on the chip                                     Piezoresistors
substrate; see also Fig. 1. The conventional inner bridge
                                                                   Fig. 2.   A top view sketch of the double-bridge sensor layout used in this
mainly depends on differential pressure, Pd = Pa,high − Pa .       work.
However, the outer bridge has sensitivities to changes in
differential and absolute pressures, which are of the same
order of magnitude. By combining the outputs from inner            process. The sensors were all based on piezoresistive read-
and outer bridge using a microprocessor, we are able to            out from piezoresistors interconnected in Wheatstone bridge
deduce both differential and relative media pressure from          structures. Furthermore, all sensor dies were encapsulated
the sensor. Using this approach, a compensation for any            using O-ring clamping as sketched in Fig. 1. Fig. 2 shows
media pressure dependency can be achieved, leading to more         a sketch of the double-bridged die layout used in this work.
precise differential pressure sensors in combination with a        Notice that the contact surface from the O-ring clamping on
relative pressure sensor.                                          the sensor die is indicated by a light gray circle in Fig. 2.
                                                                   Also notice that the two edges of the KOH-etched hole on the
                    II. S ENSOR DESIGN
                                                                   backside of the sensor die are shown as two dashed squares
  All sensors used in this work were manufactured using            and that the bonding pads to the outer Wheatstone bridge
a conventional bulk silicon micro machining fabrication            structure are divided in each corner. A collection of design
             Parameter                  Dimensions                                  Diff. Pressure            Abs. Pressure
             Die                        4000×4000 µm                                    Charact.                 Charact.
             Membrane                   1000×1000 µm
             Membrane thickness         25 µm                                          Vi                       Vi
             Inner piezores. distance                                   Inner                    Sid(Pa)
             from die center            380-490 µm                      Bridge                           Oia(Pd)          Sia(Pd)
                                                                                            1                        1
             Outer piezores. distance                                          Oid(Pa)
             from die center            990-1040 µm                                    0             Pd         0             Pa
                            TABLE I
                 A LIST OF DESIGN PARAMETERS .                                       Vo                               Vo
                                                                        Outer                                                   Soa(Pd)
                                                                        Bridge O (P )                Sod(Pa) Ooa(Pd)
                                                                                od a             1                          1
                                                                                         0               Pd            0                  Pa
parameters are listed in Table I.
                                                                   Fig. 3. A sketch defining the sensitivities and offsets for a double bridged
                          III. T HEORY                             MEMS pressure sensor with indicated indices.

A. Output voltage versus differential pressure
   In the following we focus on the inner conventional             bridge structure are defined in a similar way replacing the
Wheatstone bridge structure of the pressure sensor shown           index i with an o. All above definitions are illustrated in
in Fig. 2. When supplying the sensor bridge with a constant        Fig. 3. Notice how a pressure dependency of the sensitivities
current Ib and subjecting the sensor to a differential pressure    and offsets are also indicated in this sketch.
Pd , the bridge output voltage V is normally written as:              Using the notation in Fig. 3, we write the output from the
                         V = S Pd + O                       (1)    inner and outer bridge as:
                                                                                 Vi (Pd , Pa )= Sid (Pa )Pd + Oid (Pa )                         (3)
where we have introduced the differential pressure sensitivity
S and the output offset O. The differential pressure sensitiv-                                = Sia (Pd )Pa + Oia (Pd )                         (4)
ity is defined as:                                                               Vo (Pd , Pa ) = Sod (Pa )Pd + Ood (Pa )                         (5)
                         S=                                (2)                                   =    Soa (Pd )Pa + Ooa (Pd )                   (6)
The temperature dependency of the sensitivity S and offset         in which Eq. (3) and (5) corresponds to differential pressure
O is characterized by the well known parameters TCS and            characteristics on the two bridges and Eq. (4) and (6)
TCO, respectively. For the sake of simplicity, this work does      to absolute pressure characteristics. From the experimental
not treat the temperature dependency of the used sensors.          data later presented in Sec. IV, a close coupling between
However, earlier unpublished studies have shown only a             differential and absolute pressure characteristics are found
moderate dependency on temperature.                                both on inner and outer sensor bridge. This coupled Pd -Pa
                                                                   dependency can be written as:
B. Notation for double-bridged sensor
                                                                                 Vi    =     α i Pd + β i Pa + γ i Pd Pa + δ i                  (7)
   In the following we expand the definitions reviewed in
Sec. III-A. We use the indices i and o to indicate whether                       Vo    =     α o Pd + β o Pa + γ o Pd Pa + δ o                  (8)
a parameter is measured or deduced on the inner or outer           where the γx -term (x = i,o) in both equations is a coupling
bridge, respectively. Furthermore we use the indices d and a       term between the dependency on differential and absolute
to indicate whether a parameter is deduced from a differential     media pressure. Similarly the coefficients αx and βx can be
or an absolute pressure characteristic. By a differential and an   interpreted as the differential and absolute pressure sensitiv-
absolute pressure characteristic we mean the output voltage        ity if ignoring the coupling term. Finally the parameters δx
measured as function of differential and absolute pressure,        corresponds to the conventional sensor offset at Pd = 0 bar
respectively. In the rest of the text, the differential and        and Pa = 0 bar. Using the proposed output dependency in
absolute pressures are written as Pd and Pa .                      Eq. (7) and (8), the sensitivities introduced in Eq. (3-6) can
   Using the above indices, the output voltage from the inner      be written as:
and outer bridge will be written as Vi and Vo respectively.
Similarly, the sensitivities to changes in differential and                Sid (Pa ) = γi Pa + αi , Oid (Pa ) = βi Pa + δi                      (9)
absolute pressure on the inner bridge will be written as Sid               Sia (Pd ) = γi Pd + βi , Oia (Pd ) = αi Pd + δi                     (10)
and Sia respectively. The notation for output offsets follows           Sod (Pa ) = γo Pa + αo , Ood (Pa ) = βo Pa + δo                        (11)
the same principle, Oid and Oia being the output offset on the
                                                                        Soa (Pd ) = γo Pd + βo , Ooa (Pd ) = αo Pd + δo                        (12)
inner bridge, found from differential and absolute pressure
characteristics respectively. The sensitivities Sod and Soa and    These equations are later used to deduce the coefficients αi ,
output offsets Ood and Ooa measured or deduced on the outer        βi , γi , δi , αo , βo , γo and δo on each sensor bridge.
                                 Parameter                  Data                                    Parameter        Fitted value   Unit
                                 Temperature                25◦ C                                   αi                65.9 ± 0.3    mV/bar
                                 Constant current           1.0 mA                                  βi               0.52 ± 0.03    mV/bar
                                 Pd interval                0-2 bar                                 γi              -0.17 ± 0.06    mV/bar2
                                 Pa,high interval           2-9 bar                                 δi                17.9 ± 0.5    mV
                                 Pa interval                2-7 bar
                                                                                                            TABLE III
                                              TABLE II                           PARAMETERS IN E Q . (9) AND (10) OBTAINED ON THE INNER BRIDGE .
                        A LIST OF EXPERIMENTAL SETTINGS .

                                86                                                          −22        P = 2.0 bar
                 200                 P = 1.0bar                                                          d
                       V (mV)

                                85                                                                              1.5 bar
                 150            84

                                                                                      Vo (mV)
                                83                                                                              1.0 bar
       Vi (mV)

                                 0        2         4   6      8
                 100                          P (bar)                                                           0.5 bar
                                                                                                                0.0 bar
                                                              Pa = 2.0bar
                   0                                                                            0               2            4          6     8
                          0               0.5           1             1.5   2                                          Pa (bar)
                                                    Pd (bar)
                                                                                Fig. 5. Output versus absolute media pressure measured on outer bridge
Fig. 4. Output versus differential and absolute media pressure on the inner     at different differential pressures.
bridge at Pa = 2 bar and Pd = 1 bar, respectively.

                                                                                   The inner bridge coefficients αi , βi , γi and δi defined in
                        IV. E XPERIMENTAL R ESULTS
                                                                                Eq. (7) were found from an analysis of Sid (Pa ), Sia (Pd ),
  In the following sections, data measured on inner and outer                   Oid (Pa ) and Oia (Pd ) defined in Sec. III-B. From differential
sensor bridge are presented separately.                                         and absolute pressure characteristics as in Fig. 4 an analysis
                                                                                of the inner bridge sensitivities and offsets showed approxi-
A. Experiments                                                                  mate linear relations as the proposed first order approxima-
   A compressor supplied compressed air to each side of the                     tion in Eq. (9) and (10). The inner bridge coefficients αi , βi ,
mounted sensor in Fig. 1. The absolute pressure, Pa,high and                    γi and δi finally obtained are listed in Table III.
Pa , of the compressed air on each side of the sensor were
measured by two high precision reference absolute pressure
                                                                                C. Outer bridge results
sensors. Similarly, the differential pressure Pd between the
two pressure lines were measured by a high precision refer-                        In Fig. 5 is shown a Vo -Pa -plot measured on the outer
ence differential pressure sensor. During measurements, both                    bridge in Fig. 2. During measurement the sensor bridge was
inner and outer sensor bridge in Fig. 2 were supplied with                      closed by externally connected pads. As on the inner sensor
a constant current Ib , while a high precision multimeter was                   bridge in Fig. 4, the sensor output depends linearly on Pa
used to measure sensor outputs. A selection of experimental                     but with a sensitivity Soa of the opposite sign at all applied
settings are listed in Table II.                                                differential pressures. However, the sensor output also has
                                                                                a dependency on Pd of the same order of magnitude and
B. Inner bridge results                                                         with a sensitivity Sod of the same sign as on the inner sensor
   Fig. 4 shows a Vi -Pd and a Vi -Pa plot measured on the                      bridge; see Fig. 4.
inner bridge in Fig. 2 at Pa = 2 bar and Pd = 1 bar,                               As on the inner bridge, the outer bridge coefficients αo ,
respectively. As expected, the output depends linearly on                       βo , γo and δo defined in Eq. (8) were found from an
the primary pressure parameter, Pd , but also has a small                       analysis of Sod (Pa ), Soa (Pd ), Ood (Pa ) and Ooa (Pd ). From
(max. 1.5% of Full Span, FS) linear dependency on Pa                            differential and absolute pressure characteristics as in Fig. 5
(inserted plot). This dependency on absolute media pressure                     an analysis of the outer bridge sensitivities and offsets also
has been presented and discussed elsewhere [5]. Similar                         showed approximate linear relations as the proposed first
linear relations as in Fig. 4 were found at other Pd and Pa                     order approximation in Eq. (11) and (12). The resulting outer
values.                                                                         bridge coefficients αo , βo , γo and δo are listed in Table IV.
                Parameter        Fitted value   Unit
                αo               3.42 ± 0.01    mV/bar                                                                                 35 Inner bridge
                             -0.226 ± 0.005     mV/bar

                βo                                                                                    1.0                                       Pd = 1.0bar

                                                                                   Stress σxx (MPa)
                γo            0.005 ± 0.002     mV/bar2
                δo            -28.93 ± 0.05     mV

                            TABLE IV
                                                                                                      0.9                              34
                                                                                                                                        0           2         4       6     8
                                                                                                                                                        P (bar)

            y (mm
                 )                                                                                    0.8       Outer bridge
                                                                                                                P = 1.0bar
                                                                                                           0               2                    4                 6          8
                                                         )                                                                             P (bar)
                                                    x (mm                                                                                   a

                                                                            Fig. 7. Longitudinal stress at the piezoresistor position of inner (inserted)
                                                                            and outer bridge piezoresistors.

                            500                                                                       2
                                                                                                               P =7bar
Fig. 6. Sketch of FE model used to simulate the media pressure dependency                                                                           Compressive
                                                                                                 1.5 Pa=4bar

of O-ring clamped MEMS pressure sensors.
                                                                                                               P =1bar

                    V. S IMULATION RESULTS                                                            1
                                                                                   Stress σ
   A series of FEM simulations were carried out to support
the experimental findings in Sec. IV.                                                             0.5

A. 3D FE model                                                                                                          Tensile
   In Fig. 6 is shown a picture of the developed finite element                                         0            200     400         600         800           1000    1200
model (FEM) used to investigate the stress distribution in O-                                                              Position x (µm)
ring clamped sensors, resulting from variations in absolute
                                                                            Fig. 8. Stress versus position for a selection of absolute media pressures
media pressure. The model was developed using the com-                      at Pd = 0 bar.
mercial FE program ANSYSTM and used roughly 120.000
elements in a mapped mesh configuration. An anisotropic
material model of silicon [8] was used in the calculations,                 Pd and Pa on both inner and outer piezoresistor positions.
showing a difference of approximately 9% compared with                      Similar linear relations were found at other Pd and Pa values
calculations normally based on isotropic material models.                   on both inner and outer piezoresistor positions.
As illustrated in Fig. 6, calculations were performed only
on one quarter of the die using symmetry constraints around                 C. Stress profiles
the symmetry planes of the die. The model was finally                          Fig. 8 shows calculated stress profiles along the x-axis
clamped in the z-direction of the bottom O-ring mounting                    in Fig. 6 at Pa = 1, 4 and 7 bar, all calculated at Pd =
and symmetry was added to the symmetry-planes locking the                   0 bar. From the profile calculated at Pa = 1 bar and
model in the x- and y-directions. These boundary conditions                 Pd = 0 bar (atmospheric pressure all around the sensor) we
proved to be sufficient for an analysis of the media pressure                notice that the O-ring clamping induces a tensile stress in the
dependency investigated in this work.                                       whole region inside the O-ring contact area. Furthermore, the
                                                                            induced stress close to the O-ring contact area around x =
B. Stress versus pressure                                                   1200 µm seems to drop exponentially towards the membrane.
   The stress distribution in the top surface of the die was                However, because of the much smaller dimensions, the stress
calculated for a number of differential and absolute pressures              concentration in the membrane region is relatively high
for comparison with the experimental data in Fig. 4 and 5.                  compared to the stress concentration around x = 800 µm.
Fig. 7 shows calculated longitudinal stress, σxx , as function
of Pa and Pd , respectively. Results from both inner (inserted                Fig. 8 further shows, how an increase in absolute media
plot) and outer piezoresistor positions at Pd = 1 bar and                   pressure Pa induces an additional tensile stress in the region
Pa = 1 bar are shown. As seen from the output data in Fig. 4                x = 0−650 µm. This corresponds to the increase in stress as
and 5, the stress σxx follows a similar linear dependency on                plotted for the inner piezoresistor position in the inserted plot
                                                                                  Pa :         Com-                       Com-
                                                                                              pressive      Tensile      pressive
                                                                                  Pd :            Tensile pressive Tensile
                    Compressive Stress (sxx<0)

                                                                         Fig. 10. The distribution in induced stress in the top surface of the sensor
Fig. 9. A FEM contour plot of the stress component σxx induced by a      die, resulting from an increase in differential and absolute media pressure.
high absolute media pressure of Pa = 7 bar (zoom of membrane region in
Fig. 6).

                                                                         B. Pressure Characteristics
in Fig. 7. Fig. 8 further illustrates how an increase in absolute           The experimental data presented in Fig. 4 and 5 showed
media pressure results in an additional compressive stress in            a close coupling between differential and absolute pressure
the region x > 650 µm. This corresponds to the decrease in               characteristics on both inner and outer sensor bridge. As
stress plotted for the outer piezoresistor positions in Fig. 7.          proposed in Sec. III-B, this coupling between differential
                                                                         and absolute media pressure is well approximated by the
                        VI. D ISCUSSION                                  following expressions:
   In the following section we take a deeper look into
                                                                                         Vi   =    α i Pd + β i Pa + γ i Pd Pa + δ i           (13)
the mechanisms behind the observed effects of changes in
absolute media pressure and furthermore discuss promising                                Vo   =    α o Pd + β o Pa + γ o Pd Pa + δ o           (14)
applications of the double-bridged pressure sensor.                      From the experimentally obtained coefficients in Table III
                                                                         and IV, we notice that the ratio αi /βi between differential
A. The stress distribution
                                                                         and absolute pressure sensitivity the inner bridge (ignoring
   The main contributor to the observed absolute media                   the coupling term with coefficient γi ) is of the order 100. In
pressure dependency in Fig. 4 results from the changes in                the full span of absolute media pressures this means, that the
the absolute media pressure supported by the KOH etched                  absolute media pressure contributes with less than 1.5%FS
sidewalls of the sensor die [5]. In Fig. 9 is shown a FEM                to the output on the inner sensor bridge. However, on the
contour plot of the stress component σxx for a relatively high           outer sensor bridge the ratio αo /βo is of the order 10 and the
absolute media pressure Pa = 7 bar (axis direction corre-                coupling coefficient γo almost negligible. Also visible from
spond to Fig. 6). We notice the relatively high compressive              the data in Fig. 5 this means, that in the full span of absolute
stress amplitude (σxx < 0) in the region around the KOH                  media pressures, the output contribution from changes in
etched sidewalls resulting from the effect mentioned above.              differential and absolute media pressures are comparable in
As illustrated in Fig. 8 the media pressure on the sidewalls             size on the outer bridge structure.
induces an additional tensile and compressive stress at the
position of the inner and outer piezoresistors, respectively.            C. Algorithm for combined pressure measurement
This distribution in induced stress is sketched in Fig. 10                  We now propose a simple but efficient approach on how to
both for an increase in absolute and differential pressure. The          deduce both differential and relative media pressure (relative
sketched effect fully explains the absolute media pressure               to atm. pressure) from the sensor based on the double-bridge
dependency observed on both the inner and outer sensor                   pressure characteristics in Eq. (13) and (14) above.
bridge in Fig. 4 and 5, respectively.                                       We assume having characterized the double-bridged pres-
   The differential pressure dependency of the outer sensor              sure sensor both as function of Pd and Pa including having
bridge is a simple consequence of the moment M acting                    found the parameters αi , βi , γi , δi , αo , βo , γo and δo defined
on the die substrate for Pd > 0 bar; see Fig. 10. A                      in Eq. (13) and (14). For outputs Vi and Vo measured on
differential pressure Pd > 0 bar slightly bends the die                  the inner and outer sensor bridge, respectively, the goal is
substrate downwards and induces an additional tensile stress             to convert these into the unknown differential and absolute
in the top surface of the die including at the outer bridge              pressure values, Pd and Pa . Since Pd and Pa are the only
position. This explains the increase in outer bridge output              unknowns in Eq. (13) and (14), we are left with two non-
for an increase in differential pressure; see Fig. 5. A sketch           linear equations with two unknown parameters - Pd and Pa .
of the overall stress distributions resulting from an increase           These can easily be found by an iterative loop as proposed
in absolute and differential pressure is included in Fig. 10.            below:
   1) From Eq. (13) we use the approximation Vi (Pd , Pa )               relative pressure could be deduced with an accuracy of order
       Vi (Pd ) = αi Pd +δi . Using Vi (Pd , Pa ), αi , δi as input in   ±10% of full relative media pressure span. Finally, the media
       this equation, we find Pd (approximate value for Pd ).
                                                                         pressure induced stress distribution in the sensor was studied
   2) Now using Vo (Pd , Pa ), αo , βo , γo , δo and Pd as input
                                                                         and fully confirmed by 3D FEM simulations.
       in Eq. (14), we find Pa∗ (approximate value for Pa ).
                                                                                                      R EFERENCES
   3) Next we use the full version of Eq. (13). By using
       Vi (Pd , Pa ), αi , βi , γi , δi and Pa∗ as input, we find Pd ∗∗    [1] M. Akbar and M. A. Shanblatt, Temperature compensation of piezore-
                                                                              sistive pressure sensors, Sensors and Actuators A, 33 (1992), p. 155
       (new approximate value for Pd ).                                   [2] J. J.Dziuban, A. Grecka-Drzazga, U. Lipowicz and W. Indyka, Self-
   4) Either the loop in step 2)+3) proceeds a given period or                compensating piezoresistive pressure sensor, Sensors and Actuators A,
       until the step-by-step change in Pd and Pa are smaller                 41-42 (1994), p. 368
                                                                          [3] Young-Tae Lee and Hee-Don Seo, Compensation method of offset
       than a preset value.                                                   and its temperature drift in silicon piezoresistive pressure sensor using
In a double-bridged pressure application, a microprocessor                    double Wheatstone-bridge configuration, Proc. of Transducers‘95,
                                                                              Vol.2, 1995, p. 570
could be programmed with the above algorithm, making it                   [4] Grundfos. Pressure sensor or differential pressure sensor. Patent
possible to deduce both the the differential and absolute                     EP00801293B1.
media pressure based on measured outputs Vi and Vo . From                 [5] C. Pedersen, S. T. Jespersen, J. P. Krog, K. W. Jacobsen, C. Christensen
                                                                              and E. V. Thomsen, Characterization of MEMS Pressure Sensor
these results, the relative pressure is found by Prel = Pa −                  Packaging Concept using O-rings as Hermetic Sealing, Eurosensors
Patm .                                                                        XVII, Portugal, 2003
                                                                          [6] C. Christensen, R. de Reus and S. Bouwstra, Tantalum oxide thin films
D. Sensor calibration and applications                                        as protective coating for sensors, MEMS’99, USA, 1999, p. 267
                                                                          [7] R. de Reus, C. Christensen, S. Weichel, S. Bouwstra, J. Janting,
   As noted in Sec. I, use of the O-ring packaging concept                    G. Friis Eriksen, K. Dyrbye, T. Romedahl Brown, J. P. Krog, O. Snder-
in Fig. 1 results in a small dependency on absolute media                     grd Jensen and P. Graversen, Reliability of industrial packaging for
                                                                              microsystems, Microelectronics Reliability, 38 (1998) p. 1251
pressure; see Fig. 4. However, using the double-bridge output             [8] ”Properties of silicon”, Inspec., ISBN: 0852964757, 1988, p. 3
model established in Sec. VI-B and VI-C a compensation
technique for this small media pressure dependency is also
achieved. By implementing this technique, the error induced
by the absolute media pressure dependency can be reduced
by more than a factor of 10. For future applications, this will
allow a fabrication of more precise piezoresistive differential
pressure sensors based on the O-ring packaging principle
sketched in Fig. 1. Furthermore, since the inclusion of an
outer bridge structure on the chip substrate does not involve
additional processing steps, the double-bridge concept also
has certain economical advantageous.
   From the sensor parameters listed in Table IV we note,
that the sensitivity to changes in absolute media pressure are
somewhat limited on the outer sensor bridge in the current
design. However, in certain applications there are no need
for a high precision absolute media pressure measurement
but only an estimate on the overall pressure level. In such
applications, a combined differential and absolute pressure
measurement (Prel = Pa − Patm ) based on the double-bridged
pressure sensor is an interesting sensor candidate. Further-
more, new improved designs of double bridged MEMS
devices are already under development.

                       VII. C ONCLUSION
   A combined differential and relative MEMS pressure
sensor was developed, based on novel use of a double-
bridged structure. By utilizing linear pressure dependencies
on the two bridges, both Pd and Prel = Pa − Patm could
be deduced by an iterative process. By use of this process,
small measurement errors induced by variations in absolute
media pressure could be reduced by more than a factor
of 10. Furthermore, using the double-bridge technique the

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