Note: Descriptions are shown in the official language in which they were submitted.
.j 2101aG8
- 1 - EH 212 WO/CA
Pressure-Measuring Arrangement with
High Linearity
FIELD OF THE IN~ENTION
The present invention relates to a pressure-measuring
arrangement with a pressure sensor structure having a
diaphragm whose pressure-dependent deflection is measur-
able capacitively by means of a measuring electrode
forming a pressure-dependent measuring capacitance, and
with an evaluating circuit which derives the pressure
by capacitance measurement and whose transfer funct;on
is proportional to the difference between the measuring
capacitance and a reference capacitance as well as in-
versely proportional to a further capacitance.
8ACKGROUND OF THE INUENTION
A prior art pressure-measuring arrangement of the above
kind is shown in Figs. 17 to 19. A pressure sensor
structure, generally designated by the reference
numeral 1 and shown in a top view in Fig. 17 and in a
schematic cross-sectional view in Fig. 19, comprises a
sensor body 2 which defines a reference-pressure space
3 that is covered by a diaphragm 4. The diaphragm 4
comprises an inner, circular electrode 5, which forms
a pressure-dependent measuring capacitance Cs, and an
Ke/Lo
16.02.93
21 01~68
outer, substant;ally annular electrode 6, which forms
an essent;alLy pressure-independent reference capaci-
tance Cr.
An evaluating circuit for the pr;or art pressure-measur-
ing arrangement is illu~trated in Fig. 18 and includes
a DC voitage source UG. The reference capacitance Gr is
connectable via a first switching e~ement S1 either to
the DC voltage source U~ or to the inverting input of an
operational amp~;fier, whose noninverting input is grounded
A capacitor is connected between the inverting
input and the output of the operational amplifier. One
electrode of the measuring capacitance Cs is connectable
either to the inverting input or to the output of the
operational amp~ifier OPv by means of a second switching
element S2, while the other electrode of the measuring
capacitance Cs is grounded. A summing point SP is supplied
with the voltage of the DC voltage source UG and with
the output voltage, negative in sign, of the operational
amplifier OPV. It is evident to those skilled in the art
that the output voltage of the operational amplifier is
proportional to the reference capacitance Cr and inversely
proportional to the measuring capacitance Cs. Since this
output voltage is applied to the summing point SP with
negative sign, the prior art circuit has the following
transfer function:
C - C
s r
F
The following derivation will show that the cur~ature of the dia-
phragm 4 of Fig 19, which supports the electrodes 5, 6,
results in a nonlinearity of the output signal wh;ch is
2lnl0~s
dependent on the pressure to be measured.
For the deflection w(r) of the diaphragm, the follow;ng
relation holds, assuming that the thickness of the dia-
phragm is much smaller than its diameter and greater than
the deflection w:
(1) w(r) = (R2 _ r2)2
64D
where r is the radius under consideration, R is the ra-
dius of the diaphragm, p is the pressure, and D ;s the
flexuraL strength. The latter is given by
(2) D = Eh
12(1-V )
where E is the modulus of elasticity, h is the thickness
of the diaphragm (see Fig. 19), and v is Poisson's ratio.
For the sensor capacitance, the following integral holds:
r~R
~ .2-~-r~dr~ r = r/R (see Fig. 19)
(3) Cs(p) = _~ _
o d-w(r~)
where r~ is the normalized radius, and F o iS the permit-
tivity of vacuum. Solving the integral yields the follow-
ing dependence of the sensor capacitance Cs:
P + r~2 ~ + r~2 -1
~ P ~I P
(4) Cs(p) = - CO~ ln -- -
2 p P - r~2 ~ + r~2 -1
P ~I P
2~01~6~
-- 4 --
Equation (4~ includes the basic capacitance CO and the
support pressure pO as newly introduced constants. For
these quantities, the following relations hold:
(S) Cs(O) = r~2-CO
1r R2
(6) CO = o -
d
16 d E h3
(7) Po
3 R4 (1-~2)
It is apparent from the transfer function F of the
evaluating circuit of Fig. 18 and from the pressure
dependence of the pressure capacitance Cs given in Equa-
tion (4) that the prior art pressure-measuring arrange~
ment exhibits a nonlinear relationship between output
voltage;and pressure.
Since, to a first approximation, the characteristic of
the sensor capacitance is hyperbolic, a certain linea-
rization can be produced by forming the reciprocal,
which is done in the prior art circuit of Fig. 18 by in-
serting the sensor capacitance Cs ;nto the feedback path
of the evaluating circuit. Such a prior art circuit is
about four to five times more linear than pressure-measur-
ing arrangements in which the measuring capacitance and
the reference capacitance are located at the input and
in the feedback path of an evaluating circuit, respectively.
21010~8
S;nce, however, the characteristic of the measuring capa-
c;tance or sensor capacitance Cs is not exactly hyper-
bolic, ;t is not possible to generate a zero of the error
function with a pressure-measur;ng arrangement as shown
in Figs. 17 to 19.
The publication U. Schoneberg et al., "A CMOS-Readout-
Amplifier For Instrumentation Applications", ..., Ed.
Frontieres 1990, pages 208 to 217, shows a pressure-
sensor arrangement with an evaluating circuit ha~ing a
transfer function which is proportional to the pressure-
dependent measuring capacitance less a pressure-
independent reference capacitance. The further capacitance
given in the denominator of the transfer function is a
constant quantity. This circuit serves to measure the
capacitance ~alue of capacitive sensors, and thus also
of capacitive pressure sensors. This pressure-sensor
arrangement with a capacitive pressure sensor and the
evaluating c;rcuit is designed for a single pressure-de-
pendent capacitance, designated there by the reference
characters CSEN1, CSEN2. All other capacitances of the
prior art evaluating circuit are constant, pressure-in-
dependent quantities.
As was explained above, this prior art pressure-measuring
arrangement has an output voltage which is nonlinear be-
cause ofthe nonlinear relation between pressure and sen-
sor capacitance. --
Based on this prior art, the invention has for its ob-
ject to provide a pressure-measuring arrangement which
exhibits increased linearity~
21010~8
SUMMARY OF THE INVENTION
In accordance w;th the present invention, a pressure-
measuring arrangement ;s provided comprising
- a pressure sensor structure comprising a diaphragm
whose pre.ssure-dependent deflection is
measurable capacitively by means of a measur;ng
electrode disposed on ~he diaphragm and forming a
pressure-dependent measuring capacitance,
- an evaluating circuit wh;ch derives the pressure
by capacitance measurement and has the following
transfer function:
Cs (P)-Cr
f P
where Cs is the measuring capacitance, Cr is a constant
reference capacitance, and Cf i.s a further capacitance,
and
- a further electrode forming the further capacitance
and implemented on the diaphragm in such a way
that the further capacitance is pressure-dependent,
and that the measuring eLectrode and the further
electrode have angular extensions changing oppositely
to each other as a function of the radius.
The invention is based on the fundamental concept that
improved linearity of a pressure-measuring arrangement
can be achieved if the pressure sensor structure includes
a further electrode forming a pressure-dependent,further
2101068
capacitance Cf, and if an evaluating circuit is used
whose transfer function is proportiona~ to the difference
between measuring capacitance Cs and reference capac;tance
Cr divided by the further capacitance Cf. The use of such
an additional pressure-dependent capacitance Cf ;n an
evaluating circu;t with the above transfer function
makes it poss;ble to produce a zero in the error function f.
In the transfer function F, the difference between
measuring capacitance Cs and reference capacitance Cr
is so weighted with the further capacitance, which may
be connected as a feedback capacitance Cf, that in the
error function f, a zero is produced at the center of
the desired pressure range. This eliminates the quadrat;c
linear;ty error of the pr;or art pressure sensor arrange-
ments.
To ;ncrease the l;nearity, the invention, based on th;s
fundamental concept, provides that the measuring elec-
trode and the further electrode, which are both d;s-
posed on the d;aphragm, have angular extens;ons changing
oppositely to each other as a function of the rad;us.
Preferred developments are defined in the subclaims.
GRIEF DESCRlPTION OF THE DRAWINGS
Preferred embodiments of the pressure-measuring arrange-
ment will now be explained ;n more deta;l with reference
to the accompanying drawings, in which:
Fig. 1 shows a first embodiment of a pressure
sensor structure of a pressure-measuring
arrangement in accordance with the funda-
mental concept;
'
.; ~ .
2~01068
Fig. 2 shows a first embodiment of an e~aluating cir-
cuit for the pressure sensor structure of
Fig. 1;
Figs. 3A and 3B show a second embodiment of a pres-
sure sensor structure for a pressure sensor
arrangement in accordance with the fundamental
concept;
Fig. 4 shows a second embodiment of an evaluating
circuit for the second embodiment of the
pressure sensor structure, shown in Figs. 3A
and 3B;
Fig. S shows a third embodiment of a pressure sen-
sor structure of a pressure-measuring arranae-
ment in accordance with the fundamenta-l con-
cept;
Fig. 6 shows a third embodiment of an evaluating cir-
cuit for the pressure sensor structure of
Fig. 5;
Fig. 7 shows waveforms of control signals within the
circuit of Fig. 6;
Figs. 8A to 8D are diagrams serving to explain the
normalization of the functions;
Fig. 9 is a diagram showing the linearity error
of prior art pressure-measuring arrangements
as a function of the measured pressure in
comparison with the linearity error of the
pressure-measuring arrangement in accordance
with the fundamenta~ concept;
2101068
_ 9 _
Fig. 10 is a diagram showing the radius-dependent
angular extensions of a measuring electrode
and a further electrode of a fourth embodi-
ment of the pressure sensor structure of a
pressure-measuring arrangement in accordance
with the invention;
Fig. 11 shows the an~le prof,ile of the electrode
structure of Fig~ 10 on a normalized radius;
Fig. 12 shows a comparison between the linearity
error of the pressure-measuring arrangement
in accordance with the fundamental concept
and that of the pressure-measuring arrange-
ment in accordance with the invention;
Figs. 13 to 16 show different electrode structures
in accordance with the invention;
Fig. 17 shows a pressure sensor structure of a prior
art pressure-measuring arrangement;
Fig. 18 shows an evaluating circuit of a prior art
pressure-measuring arrangement, and
Fig. 1C~ is a schematic representation of the pressure
sensor structure of Fig. 17.
DETAILED DESCRIPTION OF THE DRAWINGS
Referring to Fig. 1, the pressure sensor structure accord-
ing to the first embodiment of the fundamental concept,
generally designated by the reference numeral 10, com-
prises two essentially sectorial electrode areas 11, 12, which
2101068
- 10 -
form a f;rst and a second measuring capacitance Cs1, Cs2,
and two further, likewise essentialLy sectorial electrode
areas 13, 14, which form two further pressure-dependent
capacitances Cf1, Cf 2'
These electrode areas 11, 12, 13, 14 are surrounded by
two reference electrode areas 15, 16 uhich are spaced
from the electrode areas 11, 12, 13, 14. The reference
e~ectrode areas 15 and 16 each have the shape of an essen-
tially semicircular ring and form a first reference capa-
citance Cr1 and a second reference capacitance Cr2, re-
spectively. The areas and, hence, the capacitances Cf1,
Cf2 of the further electrode areas 13, 14 are smaller
than those of the essentially sectorial electrode areas
11, 12, which form the measuring capacitances Cs1, Cs2.
The essentially sectorial electrode areas 11, 12, which
formthe measuring capacitances C51~ Cs2, have a smaller
percentage share in the centralarea of the diaphragm
than the further essentially sectorial electrode areas
13, 14, which form the further capacitances Cf1, Cf2.
As a result, the percentage change in the further capa-
citance Cf1, Cf2 as a function of the pressure is greater
than the percentage change in the measuring capacitance
Cs1, Cs2 as a function of the pressure. This realization
of the pressure dependence of the measuring capacitance
Cs1, Cs2 and the further capacitance Cf1, Cf2 contri-
butes to a further ;mprovement in the linearity of the
pressure sensor structure. In the embodiment shown in
Fig. 1, the stronger pressure dependence of the further
capacitance Cf1, Cf2 referred to the pressure dependence
of the measur;ng capacitance Cs1, Cs2 is achieved by pro-
viding the essentially sectorial electrode areas 11, 12
with a circular recess in their inner radial region,
while the further essentially sectorial electrode areas
~. .
~101 0 G8
- 11 -
13, 14 form semicircular electrode area e~ements 17, 18
in the central region.
Fig~ 2 shows a first embodiment of an evaluating circuit.
A DC vo~tage source UG is connectable in alternating
polarity to a first and a second node 22, 23 by means
of a first and a second switch 20, 21. Connected between
the nodes 22, 23 are, on the one hand, the series combina-
tion of the first measuring capacitance Cs1 and the sec-
ond reference capacitance Cr2 and, on the other hand,
the series combination of the first reference capacitance
Cr1 and the second measuring capacitance Cs2. The node
of the measuring capacitance C51 and the reference capa-
citance Cr2 and the node of the reference capacitance
Cr1 and the measuring capacitance Csz are coupled to the
inputs of a differential-path ampl;fier 24. Connected be-
tween the inverting input and the noninverting output of
the differential-path amplifier 24 is the first further
pressure-dependent capacitance Cf1. Connected between the
noninverting input and the inverting output is the second
further pressure-dependent capacitance Cf2. These further
pressure-dependent capacitances Cf1 and Cf2 are shunted
by a third switch 25 and a fourth switch 26, respectiveLy,
which are activated simultaneous~y to discharge these
capacitances in synchronism with the operation of the
switches 20, 21,
~ .
In the following description of the second embodiments
of the pressure sensor structure of Figs. 3A, 3B and of the
evaluating circuit of Fig. 4, parts and circuit elements
corresponding to the embodiments of f;gs. 1 and 2 will
be designated by the same reference characters,so that
the following description can be restricted to the de-
vlations of the second ~bod;ments of the pressure sensor
21010~8
- 12 -
structure and the evaluating circuit from the respect;ve
first embodiments.
Fig. 3A shows the arrangement of the electrodes on the
diaphragm, while Fig. 38 illustrates the arrangement of
the opposite electrodes at the bottom of the sensor body 2.
The arrangement of Fig. 38 corresponds to the struc-
ture shown in Fig. 1. The corresponding opposite elec-
trodes on the diaphragm side, shown in Fig. 3A, are de-
signated by like reference characters and provided with
a prime. In the case of the bottom electrodes, the first
reference electrode area 16 is connected to the first
further sectorial electrode area 13 and the first sen-
sor electrode 11. Similarly, the second further sectorial
electrode area 14 is connected to the second reference
electrode area 15 and the second sensor electrode 12
(corresponds to connection between nodes K1 and K2
three capacitors interconnected).
On the diaphragm-electrode side, the first sectorial
electrode area 11' is linked with the second reference
electrode area 15'. Similarly, the second sectorial
electrode area 12' is linked with the first reference
electrode area 16'. The two reference electrode areas
15', 16' on the diaphragm-electrode side are provided
with a first and a second contact K1, K2. Similarly, the
two further electrode areas 13', 14' are provided with a
fifth and a sixth contact KS, K6.
On the bottom-electrode side, the two reference electrode
areas 15, 16 are provided with a third and a fourth con-
tact K3, K4.
In Fig. 4, the double-throw switches 20, 21 of Fig. 2
have been replaced by single-throw switches 20', 20'',
21010~8
- 13 -
21', 21''. As can be seen, the first node 22 is connect-
ed to the first contact K1, and the second node 23 to
the second contact K2. The third contact K3 is connect-
ed to the inverting input of the differential-path ampli-
fier 24. The fourth contact K4 is connected to the non-
inverting input of this ampLifier. Two further, parasitic
capacitances Cm1, Cm2 are connected between these inputs
and ground.
Connected in series with the further pressure-dependent
capacitances Cf1 and Cf2 are a fifth switch 27 and a
sixth switch Z8, respectively, which are operated with the
second clock signal T2, which also controls the switch-
es 21', 20''. The series combination of the further
pressure-dependent capacitance Cf1 and the switch 27 is
shunted by the series combination of a hold capacitance
CH1 and a seventh switch 29, and the ser;es combination
of the further pressure-dependent capacitance Cf2 and
the switch 28 is shunted by the series combination of a
ho(d capacitance Ch2 and an eighth switch 30~ The
seventh and eighth switches 2~, ~0 are cLosed with a
signaL T1', which is deLayed with respect to the control
signa~ T1, with which the switches 20', 21'' are closed~
The node of the hold capacitance Ch1 and the seventh
switch 29 as well as the node of the hold capacitance
Ch2 and the eighth switch 30 are connectable to associated
voltage sources UM by means of a ninth switch 31 and a tenth
switch 32, respectively, on the occurrence of the delayed
second contro~ signal T2', and the node of the further pres-
sure-dependent capacitance Cf1 and the fifth switch 27 as well
as the node of the further pressure-dependent capacitance
Cf2 and the sixth switch 28 are connectable to the
associated voltage sources UM by means of an eleventh
2 1 0 1 ~ 6!8
- 14 -
switch 33 and a twelfth switch 34, respect;vely, on the
occurrence of the first control signal T1.
F;g. S shows a third embodiment of the pressure sensor
structure. Those parts of the pressure sensor structure
of Fig. 5 which correspond to the pressure sensor struc-
tures of Fig. 1 and 38 are designated by like reference
characters and have been supplemented with the distinguish-
ing character (a)~ Consequently, like or sim;Lar parts need
not be described again.
The preCsure sensor structure 1a of Fig~ 5 differs from
the pressure sensor structures of Figs~ 1 and 38 essen-
tia~ly in that a single reference electrode area 15a is
provided wh;ch has essentially the shape of a near~y
completely closed r;ng. Th;s reference electrode area
forms a single reference capac;tance Cr.
In the area of the semicircular electrode area elements
17, 18 of the embodiments of Figs. 1 and 3B, the two
further essential(y sectorial electrode areas 13, 14
now form a single further electrode area 11a, which pro-
vides the further pressure-dependent capacitance Cf.
Similarly, in the embodiment of Fig. 5, the two essen-
tially sectorial electrode areas 11, 12 of the embodiment
of Fig. 1 are interconnested and form a single measuring
capacitance Cs. The measuring capacitance Cs, the further
pressure-dependent capacitance Cf~ and the reference
capacitance Cr are measurable at terminals N1, N2, and
N3~ respectively. The further terminal N4 of Fig. 5
serves to make contact to a counterelectrode (not shown)
covering ihe whole surface
The evaluating circuit of Fig. 6, generally designated
by the reference numeral 35, includes a DC voltage
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- 15 -
source UG which is connectable via four switches 36
through 39 to a first terminal N1 and a third terminal N3
in a first and a second polarity, respectively, as describ-
ed above. The fourth terminal N4 of the pressure sensor
structure la is connected to the inverting input of an
amplifier 40, whose noninverting input is grounded. The
second terminal N2, to which the reference electrode 15a is
connected, can be grounded by means of a fifth switch 41
and is connectable to the output of the amplifier 40 by
means of a sixth swith 42. A hold capacitanGe CH is con-
nected in parallel with this feedback path and in series
with a seventh switch 43. The node of the seventh switch 43
and the hold capacitance CH can be grounded through an
eighth switch 44. The switches 36, 39 are controlled by a
second control signal T2. The first and second switches
37, 38 are controlled by a first control signal Tl. The
waveforms of these control signals are shown in Fig.7. The
fifth switch 41 is controlled by the first clock signal T1,
while the sixth switch 42 is controlled ~y the second clock
signal T2. The seventh and the eighth switches 43 and 44
are controlled by the delayed first and second clock
signals Tl' and T2', respectively, which are delayed with
respect to the first and the second clock signals.
At the beginning, the pressure-dependent capaci.tance of
a capacitive pressure sensor of the diaphragm type was
derived using Equations (1) through (7). The following
will show that, compared with the prior art pressure sen-
sor structure, the pressure sensor structure according to
the fundamental-concePt just described has a considerably
reduced linearity error. The linearity error of this
pressure sensor structure can be optimized not only by
21010~8
- 16 -
means of simple experiments, but also mathematically,
which can be done by means of the relationshi~ derived
in the following.
The reference capacitance with the structure shown in
Fig. 1 ;s calculated as the capacitance of an annulus
from Equation (4~ as follows:
(8) Cr(p) = C (p, r rA) ~ C (P~ r~rl)
With no pressure applied,the basic capacitance value is
(9) Cr(O) = (r ZrA r 2rl) CO
The transfer function F of the evaluating circuit is
(Cs1+CsZ) ~ (Cr1~Cr2) C ~C
( 10) F
Cfl + Cfz Cf
As indicated in Fig. 3~, the angle ~a is the radian mea-
sure of the further sectorial electrode areas 13, 14.
r~ is the radius of the semicircular e~ectrode area
elements 17, 18. rs~ is the ins;de radius of the sec-
torial electrode areas. rs* is the outside radius of
these electrode areas. rr1~ is the inside radius of ~he
reference electrode area whose outs;de radius R serves
as a reference ~uantity for normalizing the first-men-
tioned radi;.
210~068
From Ec~uation (10), the transfer function is
~ ) [Cs(p/ r~52)-CS(p~ r~sl)]~s ~C~r
(11) F
~Cs(p~ r~S2)~ cs(p~ rl)~f
where 5 relates to the sensor electrode, ar to the
reference electrode, and af to the feedback electrode.
The normalization of the functions wiLl now be explained
with reference to f;gs. 8A through 8D. F;g. 8A shows
the pressure-dependent prof;le of the transfer function
F, with F1 denot;ng the maximum value, and Fo the mini-
mum value.
Fig. 8B shows the profile of F(p) minus Fo~
Fig. 8C shows the curve of Fig. 8B referred to the swing,
i.e.,
F(p) - Fo
F - Fo~
Hence the linearity error shown in Fig. 8C is
F(p) - F p
t12) f~F) = F - F P
1 0 max
Using a computer-aided optimization technique, such as
the Levenberg-Marquardt technique, the linearity error
f(F) is optimized by varying the following parameters:
a : angle of the sensor sectors 13, 14;
21 ~106~
- 18 -
rl : radius of the electrode area element 17, 18;
rr : inside radius of the reference electrode 15, 16;
rs : outside radius of the sensor electrode 11, 12.
The following objecti~e functions were specified:
FA = F(p = O) - ~ O;
FB = F(p = 0.5 Pmax)
FC = F(p = PmaX) -1 --~ 0; and
FD = r +d . - r --~ O
s mln r
where PmaX = maximum applied nominal pressure
d ; = minimum electrode spacing.
In Fig. 9, the linearity error according to ~quation (12)
of the normalized output voltage for the prior art pres-
sure sensor structure uith a capacitive pressure sensor
element in the input branch is denoted by f(Cs). The
linearity error of an improved prior art circuit is de-
noted in Fig. 9 by
Cr \
f
As can also be seen in Fig. 9, the linearity error of
the circuit, denoted by
2101~68
_ 19 -
is a funct;on which, compared w;th the best attainable
linearity error curve of pressure sensor structures with
only two electrodes, is improved approximately by a fac-
tor of 10.
If, ;nstead of the optimization with four parameters as
just described, further parameters are used for lineariza-
tion, a further improvement in linearity can be achieved.
According to the invention~ unlike in the sensor struc-
ture described by way of example with reference to
Fig. 1, the measuring electrode 11b and the further
electrode 13b have angular extensions changing oppositely
to earh other as a function of their radius. In this
embodiment, therefore, the pressure-dependent measuring
electrode 11b and the pressure-dependent further elec-
trode 13b are separated by a boundary line which has
a radius-dependent angle profile atr). The measuring
electrode and the further electrode are separated by an
essentially constant distance dm. This distance relates
tothe distance of the electrodes 11b, 13b perpendicular
tothe respective radius-dependent angle profile a(r).
Together, these two pressure-dependent measuring elec-
trodes 11b, 13b cover the region of the diaphragm 4b up
to a normalized radius of 0.74 with the exception of
the aforementioned region formed by the distance dm,
while the reference electrode 15b, in the embodiment
shown here, covers the radial outer annuLar region with-
in a normalized radius range from 0.78 to 0.97.
As particularly a comparison of Figs. 10, 13, 14, 15,
and 16 shows, centrosymmetr;c subelectrode pairs with
21010~8
- 20 -
arbitrary external interconnection may be provided which
each cover the electrode area shown in Fig. 10. It is
aLso possible, of course, to realize only a single
measuring electrode and a single further electrode as
well as a s;ngLe reference electrode on a single dia-
phragm. In that case, the second half portion of the
electrode may adjoin mirror-symmetrically the electrode
half shown in Fig. 10, so that both the measuring
electrode and the further electrode each have the double
angular extension. Lt is also possible, even though~
of no further advantage, to further subdi~ide the area
of the full circle, e.g., into quadrants or octants on
each of which are realized one measuring electrode and
one further electrode with a radius-dependent angle
profile of the mutual boundary. In that case, the angles
given in Fig. 11 must be referred to the respective sec-
torial electrode area rather than to the semicircle of 180.
As can be seen in Fig. 10, the angle profile given there,
a(r),corresponds essentially to the angular extension of
the further electrode 13b, which forms the further pres-
sure-dependent capacitance Cf1. Conversely, the angle
profile of the measuring electrode 11b corresponds to
the value 180 less the value of the radius-dependent
angle profile a(r).
In other words, in the embodiment of Fig. 10, the two
radial extensions of the two pressure-dependent elec-
trodes 11b, 13b, with the exception o~ the above-mentioned
distance dm,supplement each other to give 180. As men-
tioned, however, the sum of the angles may also be equal
to a full circle of 360 or an arbitrary portion there-
of. In the latter case, a correspondingly recurring
2101068
- 21 -
pattern is formed.
F;g. 11 shows the radius-dependent angLe profîle re-
ferred to the normalized radius. As mentioned, the refer-
ence electrode 15b lies on a normalized radius range
from about 0.78 to 0.97. Therefore, the measuring elec-
trode 116 and the further pressure-dependent electrode
13b extend only up to a normalized radius of 0.74. The
angle profile begins at the center at a value of 90,
rises continuously up to a value of approx;mately 115
at a norma~ized rad;us of 0.14, shows a first slight
minim~m of about 111 at a normalized radius of about
0.18, sho~s a further weak minimum of about 116 at a
normalized radius of about 0.24, and then fa~Ls contin-
ously to a minimum va(ue of 90 at a normalized radius
of about 0.375, which is followed by a continuous rise
up to a distinct maximum of about 118 at a normaLized
radius of 0.5. After that, the radius-dependent angle
decreases to a value of ~, which is reached at the
normalized radius of 0.74.
Figs. 13, 14 and Figs~ 15, 16 show electrode pairs
belonging together. The top and bottom eLectrodes of
Figs. 13 and 14 have the same shape, which corresponds
tothe shape described with reference to Fig. 10. The
e(ectrode pairs of Figs. 15 and 16 differ in that the
counterelectrode of Fig. 16 covers the who~e surface,
while the electrode structure of Fig 15 differs from
that of Fig. 10 in that the two further pressure-clepen-
dent electrodes Cf are interconnected by a link extend-
ing through the center, while the two measuring elec-
trodes Cs are interconnected by a radially extending
link.
2101068
- 22 -
The electrode structure (electrode and counterelectrode)
shown in Figs. 13 and 14 is suited to being operated
with the differential-path amplifier of Fig. 4. The
simplified electrode-counterelectrode structure of
Figs. 15 and 16 can only be operated with the simple
amplif;er circuit of Fig. 7.
In Fig. 12, the error function or linearity error of
the pressure sensor arrangement according to the funda-
mental concept described with reference to Figs. 1 to 9
is compared with the pressure-measuring arrangement
according to the invention. It can be seen that with
the measures of the invention, a very small maximum
linearity error of less than 0.01 percent is achieved
over an extended pressure range. The maximum pressure
range lies just below the so-called bearing pressure
pO, at which the opposite electrodes touch. Compared
with the operating-pressure range of the pressure sen-
sor arrangement according to the fundamental concept de-
scribed with reference to Figs. 1 to 9, the operating-
pressure range of the pressure-measuring arrangement
according to the invention can be more than doubled.
