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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 45, NO. 2, APRIL 1996
53 1
A Very Accurate Measurement System
for Multielectrode Capacitive Sensors
Ferry N. Toth, Gerard C. M. Meijer, and IIarry M. M. Kerkvliet
Abstruct A very accurate capacitancemeasurement system
consisting of a discrete capacitancedependent oscillator and a
microcontroller has been developed. It can measure multielec
trode capacitors with capacitances up to 2 pF, with an accuracy
of 100 ppm with respect to a reference capacitor. The resolution
amounts to 50 aF with a total measurement time of 300 ms.
I. INTRODUCTION
HE capacitancemeasurement system presented here is
T based on earlier work [I]. The previous design involved a
linear capacitancecontrolled oscillator (the socalled modified
Martin oscillator) that allowed several capacitors and an offset
capacitance to be measured in exactly the same way. By ap
plying continuous autocalibration, gain and offset errors were
reduced to an insignificant level. However, the previous system
had two major disadvantages when applied in capacitive sensor
systems.
First, it only measured capacitances with a common elec
trode. More advanced multielectrode structures, which are
often required in capacitive position sensors [2][4], call for
multiplexing on both sides of the capacitances. Unfortunately,
adding an additional multiplexer at the input of the system can
disturb the continuous autocalibration scheme, since the par
asitic capacitances must not change during the measurement
cycle.
Second, it could only measure accurately over a 50 f F range.
With a 2 pF range the accuracy degraded to 0.4%. Applications
in accurate weighing equipment, angular encoders [5] and
position sensors require a much higher accuracy over a 1 pF
range.
The new circuit design implements doublesided multiplex
ing and also improves the linearity by more than one order of
magnitude. The concepts presented in [1] and [6] are used to
reduce the offset, the gain and the effect of the capacitance
of the connecting cables to an insignificant level. The circuit
can be integrated on a chip or realized using readily available
components, which allows an economical production of both
small and large series.
Manuscript received April 24, 1995; revised December 26, 1995. This work
was sponsored by Enraf BV and STW, the Duth Technology Foundation.
The authors are with the Faculty of Electrical EngineeringDIMES,
University of Technology, Mekelweg 4, 2628 AG Delft, The Netherlands.
Delft
Publisher Item Identifier S 00189456(96)035140.
h
Ccable,i
1
Fig. 1. Elirnination of parasitic capacitances.
11. BASIC PRINCIPLES
The system is based on a capacitancecontrolled oscillator.
Because a microcontroller is used to measure the period
and to control the multiplexer, several capacitances can be
measured sequentially. A number of measures have been taken
to minimize the effect of the main nonidealities. Each of these
will be diiscussed in turn below.
A. Shielding
Several parasitic capacitances can be seen in Fig. 1. As will
be shown below, the effects of cable capacitances Ccable, a and
Ccable,
since the parasitic capacitor C,,,
Cz,a it cannot be eliminated electronically. Still it can be
sufficiently reduced by shielding the terminals A and B.
zn will be reduced to an insignificant level. However,
is parallel to the measurand
B. TwoPort Measurement
Although shielding reduces C,,, , it greatly increases cable
Capacitances Ccable, i and (?,able, in. It is useful to look at the
capacitance as a twoport (Fig. 1). In that case, the effect of
Ccable,i can be eliminated by connecting an ideal voltage
source to terminal A, and Ccable,in can be eliminated by
connecting an ideal current meter to terminal B. In both cases,
the cable capacitances are effectively shorted.
A practical circuit is shown in Fig. 2. Here the NAND
gates operate as lowimpedance voltage sources with on/off
switches. The op amp operates together with Cf as a low
impedance charge amplifier. The capacitance CO, results from
00189456/96$05.00 0 1996 IEEE
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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 45, NO. 2, APRIL 1996
I
Fig. 2. The input stage of the capacitancecontrolled oscillator.
C , l f S , t
I
II I I
It
L X , Z
J
Fig. 3. The modified Martin oscillator
an imperfect shielding between the input of the charge am
plifier and the voltage source. This capacitor does not depend
on the selected capacitor and will be eliminated, as we will
show later on.
C. The CapacitanceControlled Oscillator
Martin [7] proposed a switchedcapacitor relaxation oscilla
tor with a linear relationship between the capacitance and the
period. In our system the switched capacitor has been replaced
by a resistor [SI (Fig. 3). The period of the oscillator amounts
to c11
T =4R(GAffSet + Cz,i),
with
(i = 1; 2)
(1)
D. The ThreeSignal Approach
To eliminate the unknown values CAffset and R in (l),
a continuous autocalibration process called the threesignal
approach [6) has been applied. The idea is to perform three
successive measurements (Czl, C z 2 ,
the same way. When a linear relationship is assumed be
tween the measured Deriods
and CAffset) in exactly
M3. M,,ff,,+) and the
.................
...........................
j
I multiplexer I
.................
output
j
Input
j
...........................
multiplexer
I
Fig. 4. Adding multiplexers to the input.
TABLE I
MULTIPLEXER
SETTINGS DURING 3SIGNAL MEASUREMENT
~ ~~ ~ ~ ~
Measurand
Outputmulnplerer Inputmulnplexer Period Result (eq 3)
CCjJW
off
c x I
Tmez
M+,
measurands, the capacitance ratio between Czl and Cza can
be calculated from
E. DoubleSided Multiplexing
Adding multiplexers to the input does not necessarily reduce
the circuit performance, since the ON resistances of the
switches are negligible and the parasitic capacitances only
add to the (much larger) cable capacitances. However, the
threesignal approach requires all three measurements to be
performed in exactly the same way. This means that the total
parasitic capacitance as seen from the input of the charge
amplifier must not change during the measurements.
This problem can be resolved as shown in Fig. 4. An extra
capacitor C,,S, which is not multiplexed, has been added. The
three measurements are then performed as shown in Table I.
Note that in all three measurements the inputmultiplexer is
in the same position.
111. NONIDEALITIES
In the threesignal approach all additive and multiplicative
errors that do not change during the three measurements will
be eliminated and along with them many errors caused by
nonidealities in the electronic circuit. However, errors that
cause nonlinearity will not be eliminated.
A. Additive and Multiplicative Errors
Nonidealities in the circuit that lead to additive and multi
plicative errors include the following.
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TOTH et al.: ACCURATE MEASUREMENT SYSTEM FOR MULTIELECTRODE CAPACITIVE SENSORS
~
533
1) Input Bias Current o f the OP Amp: The ratio between
period T’ with a bias current and T without a bias current is
=1+1
(4)
where I,,,,
Therefore I,,,,
period. Although the duty cycle will deviate from 50%, it will
not affect the measurement.
2) The Input Offset o f the OP Amp: It can easily be shown
that the influence of the total inputoffset voltage leads to a
multiplicative error, similar to that in the previous paragraph.
3) Delay Times o f the CMOS Inverters: These lead to an
additive error which increases the oscillator period T‘ to
is the bias current and I, the integration current.
merely causes a multiplicative error in the
T’ = T f 4Td
(5)
where T d is the delay time.
B. Nonlinearity
Nonidealities in the circuit that lead to nonlinearity include
the following.
I ) Output Resistance o f the CMOS Inverters: The invert
ers in the output multiplexer form an RCcircuit with a time
constant of Tcable, i = R,,(C,, i + Ccable, ;). The resulting
error is
For an error smaller than 10 ppm, the maximum Ccable, i can
be calculated from
(7)
For instance, with Ron = 100 R and T = 100 p u s , Ccable, ,
must be smaller than 41 nF.
2) Input Cable Capacitance and the Bandwidth of the OP
Amp: The parasitic capacitance
divider with Cf, which reduces the effective bandwidth of
the op amp. This causes an HF time constant
in forms a voltage
that must satisfy (7), with Tcable,, substituted by THF. For
instance, with fT = 2 MHz and Cf = 20 pF, Ccable,
be smaller than 1 nF.
3) DC Gain of the OP Amp: The DC gain (A) in combi
nation with Cf and R causes an LF time constant at ‘rLF =
ARCf. For a 10 ppm error this must satisfy
must
4) Slewrate: With bipolar op amps or when a switched re
sistor is used as current source, the equivalent input resistance
in combination with the slew rate of the op amp causes a
relative error
where V,, is the supply voltage and SR the slew rate. Adding
series resistances to C, and Coffset prevents slewing and
minimizes this error.
5) Comparator: The comparator delay is often modeled as
a constant. However, measurements have shown this delay
to be dependent on the oscillator period, which can lead to
nonlinearity .
C. Noise
Noise can originate either from the oscillator (thermal noise,
etc.) or from using a microcontroller to measure the period
(quantization noise).
I ) Noise Originating From the Oscillator: Van der Goes
[9] shows that the jitter equals
(11)
in which f~ is the bandwidth of the op amp, vpq the equivalent
input noise (V/Hzl/’) of the op amp (assumed to be white
noise), T,,,,
the measurement time and Ccable,
input capacitance.
2) Quantization Noise Caused by the Microcontroller:
During AT,,,
periods of the oscillator N,c periods of the
microcontroller are counted. The relative quantization noise
can then be calculated from [IO]
zn the total
En, FTzn ‘.fTCf
V,2,Tmeas
where Tcrk is the period of the clock.
IV. EXPERIMENTAL
RESULTS
The coimplete system is shown in Fig. 5. Both the integrator
and the comparator have been implemented with an LT1013,
which has a bandwidth of 500 kHz and a slew rate of 0.5 VIps.
The oscilllator is connected to the programmable counter array
of the INTEL 87C5 1 FA microcontroller, which has a 3 MHz
clock.
The measured integrator output voltage with Cref selected
is shown in Fig. 6. Several measurements have been per
formed on the system to determine the offset, nonlinearity
and resolution.
The offset can be calculated by not connecting any of
the capacitors Cz,z to the system and then performing the
threesignal measurement. The measured offset was 40 aF.
This offset appears to originate from the imperfect shielding
between points A and B in Fig. 5.
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534
IEEE TRANSACnONS ON INSTRUMENTATION AND
Select Cx,i
MEASUREMENT, VOL. 45, NO. 2, APRIL 1996
I I
U
Fig. 5. The complete circuit.
1000
500
E
Y
Q)
a
>o
c
c 
no
500
1 000 L
t
0
600 400 200
200
400 600
Time [us]
Fig. 6. Integrator output voltage.
1000 T
E
4
l
i

10
100
Measurement time [ms]
1000 10000
Fig. 7.
measurement time.
Standard deviation of the capacitance ratio as a function of the
The nonlinearity measured over a 2 pF range was found to
be less than 100 ppm of the full scale.
The resolution was determined by connecting a capacitance
of approximately 1 pF to the system, using two 1 m coaxial
cables of 1 mm in diameter (AXON), which has a capacitance
of 50 pF/m. The capacitance was then measured 100 times
by the system. Fig. 7 shows the standard deviation of the
capacitance ratio at various measurement times. At T,,,,
100 ms, the standard deviation corresponds to 50 aF.
Because of the continuous autocalibration, the absolute
accuracy of the measurement system nearly equals the nonlin
earity, provided that a good reference capacitor is available.
A lowcost highly stable reference capacitor has been given
in [ll]. For absolute accuracy this reference capacitor is
calibrated against a standard.
Y
=
V. CONCLUSION
In this paper a very lowcost and accurate capacitance
measurement system has been presented. Since the system
can be integrated on a chip or built from readily available
components, the economical production of both small and
large quantities is possible. Not only does the measurement
system allow for the connection of multiterminal capacitors,
but the accuracy has been improved compared to an earlier
design. Several have been built and tested, with excellent
results for
offset: 40 aF,
linearity: 100 ppm over a 2 pF range,
0 accuracy: less than 100 ppm over a 2 pF range when an
accurate reference capacitor is used, and
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TOTH et al.: ACCURATE MEASUREMENT SYSTEM FOR MULTIELECTRODE CAPACITIVE SENSORS
535
resolution: 50 aF with a total measurement time of
300 ms per capacitor, including the offset and reference
measurement.
ACKNOWLEDGMENT
The authors wish to thank Marcus Bonse of the Faculty
of Mechanical Engineering for the valuable discussions and
his suggestion to use doublesided multiplexing as shown in
Section 11.
REFERENCES
[1] F. N. Toth, G. C. M. Meijer, “A lowcost smart capacitive position
sensor,” IEEE Trans. Instrum. Meas., vol. 41, no. 6, pp. 10411044,
Dec. 1992.
[2] M. R. Wolffenbuttel, “Surface micromachined capacitive tactile image
sensor,” Ph.D. Dissertation, Delft University of Technology, 1994.
[3] M. H. W. Bonse, J. W. Spronck, and F. Zhu, “A new twodimensional
capacitive position transducer,” Sensors and Actuators A, vol. 4142,
pp. 2932, 1994.
[4] W. C. Heerens, “Basic principles in designing highly reliable multi
terminal capacitor sensors and performance of some laboratory test
models,” Sensors and Actuators A, vol. 3, pp. 137148, 1982/1983.
[5] G. W. de Jong, A. M. M. Aalsma, A. J. M. Bertels, K. van der Lingen, G.
C. M. Meijer, and J. W. Spronck, “A smart capacitive absolute angular
position sensor,” Sensors and Actuators A, vol. 4142, pp. 212216,
1994.
[6] G. C. M. Meijer, J. van Drecht, P. C. de Jong, and H. Nenteboom,
“New concepts for smart signal processors and their application to PSD
displacement transducers,” Sensors and Actuators A, vol. 35, pp. 2330,
1992.
[7] K. Martin, “A voltagecontrolled switchedcapacitor relaxation oscil
lator,” IEEE J. Solidstate Circuits, vol. SSC16, no. 4, pp. 412413,
Aug. 1981.
[8] J. van Drecht, “Relaxatie oscillator,” Pat. Appl. 91.01076, The Nether
lands, 1991.
[9] F. M. L. van der Goes, “Low cost sensor interfacing,” Ph.D. thesis,
Delft University of Technology, to be published, 1996.
[lo] A. B. Carlson, Communication Systems. New York: McGrawHill,
1986. ch. 12, p. 437.
1111 F. N. Toth, A. J. M. Bertels, and G. C. M. Meijer, “A lowcost highly
stable reference capacitor for capacitive sensor systems,” in IMTC’95,
Proc., Waltham, MA, Apr. 2426, 1995, pp. 412415.
Ferry N. Toth, for a photograph and biography, see this issue, p. 530.
Gerard C. M. Meijer, for a photograph and biography, see this issue, p. 530.
Harry M. M. Kerkvliet was born in Voorburg,
The Netherlands, on March 18, 1945. He graduated
in electronic engineering at the Royal Polytechnic
Institute PBNA, Arnhem, The Netherlands, in 1974
He joined the Department of Electrical Engineering
of the Delft University of Technology in 1968, and
was involved in areas of television signal processing
techniques and electronic system design. His current
interests include signal processing