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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 2, FEBRUARY 2008
937
A Low-Cost Integrated Approach for Balancing
an Array of Piezoresistive Sensors for
Mass Production Applications
Hing Kai Chan,
Senior Member, IEEE
Abstract
—Offset error is inherently present in piezoresistive
sensors. This is caused by a heterogeneous excitation response and
static resistance of different sensing elements. Such an error makes
applications with multiple sensors difficult because a proper cali-
bration or normalization scheme is desired. A low-cost integrated
approach which can provide a closed form mathematical solution
for normalizing an array of piezoresistive sensors is presented
in this paper. The proposed solution just needs a few external
resistors and hence is most suitable for mass commercial applica-
tions. Necessary and sufficient conditions to derive the solution are
considered. This technique could be used offline so that the sensors
array could be used directly after going through the normalization
procedures.
Index Terms
—Normalization, piezoresistive, sensors array.
applications. This research thus presents a low-cost approach
which can provide a closed form mathematical solution for
normalizing output of a set of sensor arrays to eliminate errors
due to the heterogeneous nature of different sensors. Only a
few external passive components are required and hence this
is suitable for mass commercial applications. Necessary and
sufficient conditions are also addressed. This technique could
also be implemented offline so that sensors can be normalized
prior to actual usage, and hence further lower the production
cost for such applications.
The rest of this paper is organized as follows. Section II
presents the first step of the proposed solution to balance a sin-
gle half-bridge piezoresistive sensor. Before we can normalize
a sensor array, we have to know how to balance a single sensor
first. Section III then formulates the mathematical solution for
normalizing a set of sensor arrays. The objective is to transform
a set of sensors to the form discussed in Section II so that
they could be normalized in the same way. Section IV is the
concluding section.
I. I
NTRODUCTION
P
IEZORESISTIVE sensors (e.g., strain gauges) are perhaps
the most widely used transducers for force, pressure, or re-
lated measurements [1]. Various applications have been devel-
oped by employing such technology (to quote a few, [2]–[5]).
However, one problem of typical piezoresistive sensors is that
there is an intrinsic offset error [6]. For example, the behavior
of two sensing elements in a half-bridge transducer is not
the same, which implies that output of each element subject
to excitation is not the same as well. The temperature effect
of different sensors could also contribute to this problem [7].
However, this problem is not that significant if only one sensor
is presented because compensation for such error is rather easy,
or not even a problem at all with a proper circuit design.
Recently, multiple sensor applications have become popu-
lar in real-life situations, e.g., large-scale weighing systems
for trucks, which involves making measurements at different
locations from a monitoring station [8]. In such applications,
it is impractical to employ a sole sensor to achieve accu-
rate measurements. Due to the offset error mentioned above,
homogenous sensors can rarely be obtained. Consequently,
problems due to the heterogeneity of sensors are not uncom-
mon in such applications. The problem becomes complicated
when normalizing a set of sensor arrays, instead of balancing
the output of a single sensor only, is needed. Some specific
methodologies have been developed to solve similar problems
(e.g., [9]), but they are not suitable for low-cost mass production
II. B
ALANCING A
S
INGLE
H
ALF
-B
RIDGE
P
IEZORESISTIVE
S
ENSOR
A half-bridge piezoresistive sensor is normally unbalanced,
i.e., variations of the two sensing elements (i.e., ∆
R
1
and ∆
R
2
as shown in Fig. 1) are not subject to the same loading. The
following procedures set out to balance a half-bridge piezo-
resistive sensor by simply adding two resistors,
R
series
and
R
para
as shown in Fig. 1.
When the sensor is not subject to excitation (i.e., no load),
∆
R
1
and ∆
R
2
(hence, ∆
R
eff
) are equal to zero and
R
eff
is
given by
R
eff
=
R
series
+
R
para
//R
2
R
para
R
2
R
series
+
R
2
.
=
R
series
+
(1)
On the other hand, a more general equation could be formulated
when the sensor is excited (i.e., loaded)
R
eff
−
∆
R
eff
=
R
series
+
R
para
//
(
R
2
−
∆
R
2
)
Manuscript received August 17, 2005; revised October 16, 2007.
The author is with Norwich Business School, University of East Anglia,
NR4 7TJ Norwich, U.K. (e-mail: h.chan@uea.ac.uk).
Digital Object Identifier 10.1109/TIE.2007.896409
∆
R
2
)
R
para
+(
R
2
−
R
para
(
R
2
−
=
R
series
+
∆
R
2
)
.
(2)
0278-0046/$25.00 © 2008 IEEE
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 938
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 2, FEBRUARY 2008
Therefore, (8) can be further simplified as the following
quadratic equation:
∆
R
2
)(
R
para
)
2
+∆
R
1
(2
R
2
−
(∆
R
1
−
∆
R
2
)
R
para
+∆
R
1
R
2
(
R
2
−
∆
R
2
)=0
.
(9)
To guarantee (9) has real solution(s) for
R
para
,thefollowing
condition should hold:
∆
R
1
(2
R
2
−
∆
R
2
)
2
−
4(∆
R
1
−
∆
R
2
)∆
R
1
R
2
(
R
2
−
∆
R
2
)
≥
0
.
(10)
Since
R
2
∆
R
2
,theterm(
R
2
−
∆
R
2
) must be positive.
if ∆
R
2
>
∆
R
1
, i.e.,
Therefore,
the
inequality
(10)
holds
(∆
R
1
−
∆
R
2
)
<
0, since other terms are all positive.
To satisfy this condition, we have to measure ∆
R
1
and ∆
R
2
prior to selecting the proper values of
R
para
and
R
series
for
balancing the sensor. The solution could then be guaranteed
by configuring the sensor as shown in Fig. 1 such that ∆
R
2
>
∆
R
1
. In such case,
R
para
can be found by solving the quadratic
equation (9), and
R
series
can be found by (6) directly when
R
para
is known. There is only one exception when the sensor
could not be configured this way – when ∆
R
2
=∆
R
1
,in
which case we do not need to balance the sensor at all by just
setting
R
para
to open the circuit and calculate
R
series
, which is
equal to
R
1
−
Fig. 1. Equivalent circuit to balance a half-bridge piezoresistive sensor.
Assumptions: (a) ∆
R
1
and ∆
R
2
=0 at no load; (b) ∆
R
1
and ∆
R
2
are
positive; and (c) it is easy to verify that ∆
R
eff
also satisfies conditions
(a) and (b).
R
2
. The physical meaning of the negative value
of
R
series
is that the corresponding resistor should be added in
between the supply voltage and
R
1
.
To find
R
para
, rewrite the quadratic equation (9) as follows:
Combining (1) and (2) gives an expression for ∆
R
eff
as
R
para
R
2
R
para
+
R
2
−
∆
R
2
)
R
para
+(
R
2
−
R
para
(
R
2
−
A
(
R
para
)
2
+
BR
para
+
C
=0
∆
R
eff
=
∆
R
2
)
.
(3)
where
To balance the bridge, we need the following relationships:
A
=(∆
R
1
−
∆
R
2
)
B
=∆
R
1
(2
R
2
−
R
1
=
R
eff
(4)
∆
R
2
)
C
=∆
R
1
R
2
(
R
2
−
∆
R
1
=∆
R
eff
.
(5)
∆
R
2
)
.
Combining (1) and (4), we can express
R
series
as
Then,
R
para
can be found by
−
√
B
2
R
para
R
2
R
para
+
R
2
.
R
para
=
−
B
−
4
AC
R
series
=
R
1
−
(6)
.
(11)
2
A
Remark:
A negative solution is rejected.
It can be observed from the above procedures that
R
para
is
mainly used to balance the variation of the sensor (i.e., to equate
∆
R
1
and ∆
R
eff
only).
R
series
is used to ensure the output of the
sensor is biased at half supply voltage. Therefore, by extending
the same principle,
R
series
can be selected to offset the output
of the sensor to a certain level so that the static (i.e., no load)
output is not at half supply voltage of the sensor for particular
applications.
Theoretically, the solution is free from error from a mathe-
matical perspective, unless the stated assumptions are invalid.
However, this technique suffers from the following limitations
in real-life applications:
1) accuracy of measuring ∆
R
1
and ∆
R
2
, including the
precision limit of measuring instruments, and the fixture
for adding load, etc.;
On the other hand, by combining (3) and (5), we can express
∆
R
1
as
R
para
R
2
R
para
+
R
2
−
∆
R
2
)
R
para
+(
R
2
−
R
para
(
R
2
−
∆
R
1
=
∆
R
2
)
.
(7)
Equation (7) could then be rearranged as
∆
R
1
(
R
para
+
R
2
)[
R
para
+(
R
2
−
∆
R
2
)]
=(
R
para
R
2
)[
R
para
+(
R
2
−
∆
R
2
)]
−
R
para
(
R
2
−
∆
R
2
)(
R
para
+
R
2
)
.
(8)
L.H.S. of (8) =∆
R
1
[(
R
para
)
2
+(2
R
2
−
∆
R
2
)
R
para
+
R
2
(
R
2
−
∆
R
2
)].R.H.S. of (8)=
R
para
[
R
2
R
para
+
R
2
−
R
2
∆
R
2
−
R
2
R
para
−
R
2
+∆
R
2
R
para
+
R
2
∆
R
2
]=∆
R
2
R
para
.
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CHAN: LOW COST INTEGRATED APPROACH FOR BALANCING AN ARRAY OF PIEZORESISTIVE SENSORS
939
a particular sensor. From Fig. 2, we can formulate ∆
R
1eff
as in
Section II and set this value to
k
according to
∆
R
1eff
=
R
norm
(
R
1
+∆
R
1
)
R
norm
R
1
R
norm
+
R
1
=
k.
R
norm
+(
R
1
+∆
R
1
)
−
(13)
Following the same line of thought to derive the quadratic
equation (9), (13) can be expressed by the quadratic equation
k
)(
R
norm
)
2
(∆
R
1
−
−
k
(2
R
1
+∆
R
1
)
R
norm
−
kR
1
(
R
1
+∆
R
1
)=0
.
(14)
From (12), we know that ∆
R
1
>k
(or equivalently, ∆
R
1
i
>k
for all
i
, except where ∆
R
1
i
=
k
). Therefore, we can confirm
the following inequality is true:
k
2
(2
R
1
+∆
R
1
)
2
+4(∆
R
1
−
k
)[
kR
1
(
R
1
+∆
R
1
)]
>
0
.
(15)
In other words, we are able to solve for
R
norm
by solving the
quadratic equation (14). After finding
R
1eff
and ∆
R
1eff
,the
equivalent circuit of a particular piezoresistive sensor becomes
Fig. 1, with
R
1
=
R
1eff
and ∆
R
1
=∆
R
1eff
.
R
para
and
R
series
can then be solved by following the procedures as discussed in
Section II.
To verify the assumption that ∆
R
1eff
should be less than
∆
R
2
to derive a valid solution as described in Section II, we
get the following inequality from (13):
Fig. 2. Equivalent circuit to balance an array of half-bridge piezoresistive
sensors. Assumptions: (a) ∆
R
1
and ∆
R
2
=0 at no load; (b) ∆
R
1
and
∆
R
2
are positive; (c) it is easy to verify that ∆
R
1eff
and ∆
R
eff
also satisfy
conditions (a) and (b); and (d) ∆
R
1
<
∆
R
2
=
>
∆
R
1eff
<
∆
R
2
.
R
norm
(
R
1
+∆
R
1
)
R
norm
+(
R
1
+∆
R
1
)
−
R
norm
R
1
R
norm
+
R
1
∆
R
1eff
=
2) temperature effect, which can also affect the accuracy of
measured ∆
R
1
and ∆
R
2
;
3) error in
R
para
and
R
series
due to standard resistor values.
<
R
norm
(
R
1
+∆
R
1
)
R
norm
+
R
1
R
norm
R
1
R
norm
+
R
1
−
=
R
norm
∆
R
1
R
norm
+
R
1
.
III. N
ORMALIZING AN
A
RRAY OF
P
IEZORESISTIVE
S
ENSORS
The similar concept as derived in Section II can be extended
to normalize an array of half-bridge piezoresistive sensors. The
procedures would have to be more complicated. If the sensors
are normalized, their outputs are all balanced and biased at the
same level. Fig. 2 depicts the equivalent circuit diagram of a
particular sensor of the array, which is going to be normalized.
One more resistor,
R
norm
, is required to complete the task.
Assume the array of
I
sensors are indexed by
i
, (
i
=1
,
2
,...,I
). From Section II we know that if we want to balance
a half-bridge sensor, we need to configure the sensor as shown
in Fig. 1 such that ∆
R
2
>
∆
R
1
. Therefore, this condition is
assumed implicitly in the following discussion for any sensors.
We will validate this assumption later. Before we can select
R
para
and
R
series
as in Section II, the first step is to find the
following value when the sensors are all subject to the same
excitation:
We can further deduce that the right-hand side of the above
inequality has an upper bound of ∆
R
1
R
norm
+
R
1
<
(
R
norm
+
R
1
)∆
R
1
R
norm
∆
R
1
=∆
R
1
.
R
norm
+
R
1
Therefore, ∆
R
2
>
∆
R
1
, which implies ∆
R
2
>
∆
R
1eff
.In
other words, the prerequisite for solving
R
series
and
R
para
as
discussed in Section II is satisfied.
By finding
R
norm
for all sensors (except the one with the
minimum ∆
R
1
i
), the array of sensors can be reduced as an
array of sensors similar to Fig. 1. By balancing each sensor
independently, the array of sensors can then be normalized.
IV. C
ONCLUSION
A closed form mathematical solution is derived in this paper
to normalize an array of piezoresistive sensors. The proposed
procedures could be done offline so that compensation is not
required in the final product. Although the solution is free from
mathematical errors, the availability of standard resistors to
match the calculated values is a main source of error affecting
accuracy of the proposed solution. A capacitive effect may
k
= min(∆
R
1
i
)
∀
i.
(12)
The sensor which has ∆
R
1
i
=
k
does not need to add
R
norm
.
∆
R
1
i
refers to ∆
R
1
of sensor
i
. In the following discussion,
the subscript
i
is discarded for simplicity since the focus is on
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940
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 55, NO. 2, FEBRUARY 2008
also reduce accuracy of the proposed normalization procedures
(e.g., [10], [11]). However, if the excitation source is dc, this
problem could be minimized by waiting for a sufficient settling
time, say one to two seconds, until a stable reading is obtained.
On the other hand, if the source is ac, this is not a problem at
all because above discussions concerning resistances could be
replaced by impedances directly.
One may argue why only half-bridge sensors are limited in
the proposed normalization procedures. In fact, a full bridge
sensor is equivalent to two half-bridge sensors in the above
discussion. Therefore, normalizing
k
full bridge sensors is
equivalent to normalizing 2
k
half bridge sensors. In other
words, the proposed method is also applicable to full bridge
piezoresistive sensors.
Although it has not been stated explicitly in this paper, the
heterogeneous nature of piezoresistive sensors discussed in this
paper was not restricted to the intrinsic resistance values of the
piezoresistive sensors, i.e., mismatch of no load and loaded
resistances of individual sensors. It is only one of the many
possible sources to cause heterogeneity of the sensor array.
For example, heterogeneity may originate from adhesion of the
sensors to, say load beams, the deformation of which may cause
heterogeneity of the array.
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Hing Kai Chan
(M’98–SM’04) received the B.Eng.
degree in electrical and electronic engineering, the
M.Sc. degree in industrial engineering and industrial
management, and the Ph.D. degree from the Univer-
sity of Hong Kong, Hong Kong.
Prior to reading his Ph.D., he was a Design and
Project Engineer, with focus on instrumentation and
measurement, in the electronic manufacturing sector.
He is currently a Lecturer in the University of East
Anglia, Norwich, U.K. His current research interests
include industrial informatics and applications of
soft computing on intelligent industrial systems and supply chains.
Dr. Chan is a recipient of the IEEE Industrial Electronics Society Student
Travel Grant in 2005. He is an Organizing Committee Member of the IEEE
Region Ten Annual Technical Conference (TENCON) 2006, and is an Inter-
national Technical Programme Committee Member of the IEEE International
Conference on Industrial Informatics 2006 and 2007. He is a member of the
Institution of the Electrical Engineers, and the Chartered Institute of Marketing,
U.K. He is a Chartered Engineer and a Chartered Marketer, U.K.
R
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