High-Sensitivity Inductive Pressure Sensor

2960 IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 60, NO. 8, AUGUST 2011

High-Sensitivity Inductive Pressure SensorEzzat G. Bakhoum, Senior Member, IEEE, and Marvin H. M. Cheng, Member, IEEE

Abstract—A new type of pressure sensors with extremely highsensitivity is introduced. Unlike piezoresistive, capacitive, andlinear-variable-differential-transformer-based pressure sensors,the new sensor is based on a technique for substantially changingthe inductance of a coil. The prototype device has demonstrated achange in inductance of approximately 34.5 mH over a pressurerange of 10 kPa. The sensor offers a number of desirable features,including linearity, low temperature, and pressure hysteresis, inaddition to small size.

Index Terms—Inductance-to-frequency converter, pressure hys-teresis, pressure sensor, relative permeability, temperature hys-teresis, variable inductance.

I. INTRODUCTION

P RESSURE sensors exist mainly in two varieties: piezore-sistive and capacitive. A third type of pressure sensors

that is essentially a derivative of a displacement transduceris the linear variable differential transformer (LVDT) [1], [2].Piezoresistive pressure sensors are characterized by good lin-earity and acceptable sensitivity, but temperature hysteresis inthose sensors is usually quite large [1]–[4]. Capacitive pres-sure sensors can achieve much higher sensitivity and lowertemperature hysteresis, but they are usually nonlinear [5]–[15].LVDT-based pressure sensors, on the other hand, offer the bestlinearity, highest sensitivity, and lowest temperature hysteresis,but they are usually very bulky by comparison with othersensors [16]–[18]. This paper introduces a new type of pressuresensors that is characterized by: 1) miniature size (the sensor fitsinside an integrated circuit (IC) package); 2) excellent linearityover an arbitrarily chosen pressure range; 3) substantially highsensitivity; and 4) substantially low-temperature hysteresis. Theprinciple behind the new sensor is shown in Fig. 1.

The principle of the new device, as shown in Fig. 1, is to cre-ate a highly variable inductance mechanism. In the mechanismshown in Fig. 1, the change in inductance is equal to the relativemagnetic permeability of the core material and is typically4000-fold or higher. By comparison with the LVDT, this sensoroffers substantially higher sensitivity and does not requirealternating-current (ac) excitation, and since it is composed ofonly one coil, it is substantially more compact than a typical

Manuscript received June 15, 2010; revised September 13, 2010; acceptedOctober 11, 2010. Date of publication March 14, 2011; date of current versionJuly 13, 2011. The Associate Editor coordinating the review process for thispaper was Dr. Salvatore Baglio.

E. G. Bakhoum is with the Department of Electrical and Computer En-gineering, University of West Florida, Pensacola, FL 32514 USA (e-mail:[email protected]).

M. H. M. Cheng is with the Department of Mechanical and AerospaceEngineering, West Virginia University, Morgantown, WV 26506 USA (e-mail:[email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TIM.2011.2118910

Fig. 1. (a) Movable iron core is positioned inside the core of a verticalinductor (coil). (b) As pressure acts on the iron core in the vertical direction,the core can be totally displaced outside the coil. The change in inductancebetween the two configurations is equal to the relative magnetic permeabilityof the core material and is typically 4000-fold or higher.

LVDT [16]–[18]. The prototype device that was built and testedby the authors has demonstrated a change in inductance ofapproximately 34.5 mH over a pressure range of 0.3 to 10 kPa.Its sensitivity is therefore 3.56 mH/kPa, which is substantiallyhigh. The detailed structure of the new sensor is described inSection II. The theory of operation of the new sensor, which isalso substantially different from the LVDT principle, is given inSection III. Testing results are presented in Section IV.

Table I lists the four most important parameters of the newsensor: sensitivity, linearity, pressure hysteresis, and temper-ature hysteresis, as compared with the other known types ofpressure sensors.

In addition to the parameters shown in Table I, other impor-tant aspects of comparison include cost and size. Piezoresistiveand capacitive pressure sensors are generally miniature sensors,with a cost that ranges from a fraction of a dollar up to a fewdollars. LVDT-based pressure sensors, on the other hand, arelarge bulky sensors [16]–[18], with a cost that generally exceeds$100. The new sensor introduced here is of the miniature low-cost variety.

II. STRUCTURE OF THE NEW SENSOR

The structure of the new sensor is shown schematically inFig. 2. Fig. 3 is a photograph of the actual components of thesensor.

As shown in Figs. 2 and 3, a vertical coil of a height of4 mm and a diameter of 12 mm is totally embedded insidean open-cavity 24-pin dual-in-line (DIP) IC package. A smallcylindrical iron core of a height of 4 mm and a diameter of6 mm is positioned inside the coil, surrounded by a smooth

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BAKHOUM AND CHENG: HIGH-SENSITIVITY INDUCTIVE PRESSURE SENSOR 2961

TABLE ICOMPARISON OF THE NEW SENSOR TO THE OTHER KNOWN TYPES OF PRESSURE SENSORS (SEE [1] AND [2]. NOTE THAT UNCERTAINTY IN THE

SENSITIVITY DATA IS TYPICALLY NOT PROVIDED BY MANUFACTURERS)

Fig. 2. Mechanical structure of the sensor (scale: 3 : 1).

Fig. 3. Components of the sensor. The coil is totally mounted inside a standard24-pin DIP IC package (with dimensions of 30 × 14 mm). The iron core andthe plastic sleeve are positioned inside the core of the coil, and the plastic domeshown in the figure is mounted on top of the coil and attached permanently withan adhesive.

Teflon sleeve. In the device shown, the pressure acts on the ironcore in the upward direction. However, this is only one possibleconfiguration, and it is equally possible to configure the sensorsuch that the pressure acts downwardly. In addition, as shownin the figures, a semispherical plastic dome is positioned on topof the coil in order to contain the iron core as it gets displaced.In the internal cavity of the dome (see Fig. 2), a spring with aknown spring constant is mounted. As the displaced iron coreexerts force on the spring, the displacement will be proportionalto the force (and, hence, pressure) that is acting on the iron core.The displacement of the iron core, in addition, will be related tothe observed inductance of the coil. These two characteristics

Fig. 4. As a result of the displacement of the iron core, the coil is effectivelysplit into two inductors connected in series.

allow the pressure acting on the sensor to be calculated as afunction of the observed inductance.

III. THEORY OF OPERATION

A. Inductance as a Function of the Position of the Iron Core

Inductance L of any inductor is given by the following well-known equation [19]:

L =µ0µrN

2A

l(1)

where µ0 is the magnetic permeability of free space, µr is therelative permeability of the material present in the core, N isthe number of turns in the coil, A is the cross-sectional areaof the coil, and l is its length. Fig. 4 shows the creation oftwo inductors in series as the iron core is displaced by thesmall distance x from its original position. The first inductorcontains the iron core, and its inductance is labeled L1. Thesecond inductor contains only air in its core, and its inductanceis labeled L2. As the figure shows, the total length of the coil inthe device is l.

Based on the aforementioned formula, inductances L1 andL2 will be now given in terms of displacement x as follows:

L1 =µ0µrN

21 A

(l − x)

L2 =µ0N

22 A

x(2)

where N1 is the number of turns in the first inductor, and N2 isthe number of turns in the second inductor. The following two


Fig. 5. Total inductance (in millihenry) as a function of the displacement x,according to (7).

relationships now must hold:

N1 + N2 =N (3)

N2

N=

x

l. (4)

These two relationships can be alternatively written as follows:

N1 = N(1 − x

l

)

N2 = N(x

l

). (5)

By substitution for N1 and N2 from the aforementioned twoidentities into the equations in (2), we obtain, after simplifica-tion of the results, the following:

L1 =µ0µrN

2A

l

(1 − x

l

)

L2 =µ0N

2Ax

l2. (6)

From circuit theory, the total inductance of a series combi-nation of two inductors is equal to the sum of the individualinductances. By taking the sum L1 + L2 and simplifying theresulting expression, we obtain the following:

L =L1 + L2 =µ0N

2A

l

[µr − x

l(µr − 1)

]

=L0

[µr − x

l(µr − 1)

](7)

where L is the total observed inductance, and L0 is the induc-tance of the coil with an air core (minimum inductance). Usingthe parameters of the present device (see Section IV), Fig. 5shows a plot of L as a function of displacement x.

As expected, the observed inductance varies linearly fromthe maximum value Lmax (the inductance of the coil with theiron core) to the minimum value L0 (which has a value of11.5 µH in the present prototype and therefore appears negli-gible by comparison with Lmax). As can be seen in (7), theratio Lmax/L0 = µr, which is approximately equal to 4000

for iron. The plot in Fig. 5 also shows the experimentallymeasured values of L. As can be seen, the plot actually deviatesfrom the theoretically expected linear behavior when L is veryclose to L0. This is due to the simplifying assumptions ofthe theory of inductance (specifically, the core of the coil isnot actually purely air when the iron cylinder is placed inthe immediate vicinity of the coil). Nevertheless, the linearresponse is obtained for most of the dynamic range of thesensor. In the present prototype, the region of operation is onlythe linear region (where 0 mm ≤ x ≤ 3 mm).

B. Position of the Iron Core as a Functionof the Applied Pressure

In the present prototype, the pressure acts on the iron core inthe upward direction. Accordingly, the minimum force Fmin =mg must be applied, where m is the mass of the iron core, and gis the acceleration of gravity. The minimum pressure Pmin thatmust be applied to the sensor is therefore equal to

Pmin =Fmin

A. (8)

The sensor cannot respond to any pressure less than Pmin,which is approximately equal to 0.3 kPa in the present proto-type. Pressures larger than Pmin will result in a force that willcompress the vertical spring (see Fig. 2). In this case, the netforce acting to compress the spring will be given by

F = kx = (P − Pmin)A (9)

where k is the spring constant, and P is the pressure acting onthe sensor. Hence, the displacement x of the iron core will begiven as a function of pressure by

x =(P − Pmin)A

k. (10)

C. Applied Pressure as a Function of theObserved Inductance L

By substituting (10) into (7), we obtain a relationship be-tween L and the applied pressure as follows:

L = L0

[µr − (µr − 1)

(P − Pmin)Akl

]. (11)

Solving for P , we obtain the following:

P = Pmin +kl

A

(µr − L/L0

µr − 1

). (12)

It should be pointed out that the sensor is essentially a differ-ential pressure sensor (since ambient air exists on both sidesof the moving iron core). The pressure calculated in (12)is, thus, the pressure that exceeds atmospheric pressure. Toobtain a measurement of absolute pressure, the atmosphericpressure must simply be added to the pressure calculatedin (12).


Fig. 6. Pressure P as a function of the measured inductance L.

IV. EXPERIMENTAL RESULTS

A. Basic Results

For the prototype device described here, the following are thedimensions and the physical constants.

1) The spring constant is k = 91.33 N/m.2) The relative permeability of the iron core (i.e., electrical

steel) is µr = 3978.3) The diameter of the core/iron is core = 6 mm.4) The height of the iron core and the height of the coil are

l = 4 mm.5) The mass of the iron core is m = 0.89 g.6) The total number of turns in the coil is N = 36.

Based on the aforementioned data, the minimum inductanceL0 = 11.5 µH and the maximum inductance Lmax is found tobe equal to 46 mH [see (1)]. From (8), Pmin = 0.3 kPa, andin (10), Pmax is found to be approximately equal to 10 kPa(at displacement x = 3 mm). The range of 0.3–10 kPa istherefore the dynamic range of the present prototype (of course,the dynamic range can be changed by varying the parameters,particularly the spring constant). Fig. 6 shows a plot of pressureP predicted by (12) as a function of the measured inductanceL, along with the actual pressure values that were measured ina pressure chamber.

The measurements were performed in a commercial-qualitypressure chamber1 that is pressurized with compressed air. Thechamber is equipped with two different types of commerciallyavailable precalibrated pressure sensors,2 to eliminate any pos-sibility for errors in the measurements. The inductance wasmeasured directly with an LRC meter that was connected tothe sensor. As Fig. 6 shows, the agreement between the exper-imental and the theoretically calculated values of the pressureis excellent (it is to be pointed out that the lowest inductanceshown in the plot is approximately 11.5 mH, to avoid thenonlinear region near L0; see Fig. 5). Within the linear region

1Chamber type 175-10000 from Allied High Tech products, Inc.2a 61CP Series ceramic capacitive pressure sensor from Sensata Technolo-

gies, Inc; and MAX1450 evaluation kit, with a GE NovaSensor pressure sensor,from Maxim Integrated Circuits, Inc.

Fig. 7. Pressure hysteresis curves at −10 ◦C and +80 ◦C, in the vicinityof the full-scale pressure (10 kPa). The maximum hysteresis error is less than±0.05% FSO.

shown in the figure, it can be clearly seen that the dynamicrange of the present sensor is indeed 0.3–10 kPa. It can alsobe seen that the sensitivity (or change in inductance per unitpressure) is about 3.56 mH/kPa.

B. Pressure Hysteresis

The hysteresis in the values of the calculated pressure wasdetermined by cycling the pressure applied to the sensor atfixed temperature and plotting the measured pressure versus theactual applied pressure. The results are shown in Fig. 7, for twodifferent extreme values of temperature. As the plots in Fig. 7show, the maximum hysteresis error was found to be ±0.05%,which is essentially negligible (it is to be pointed out that thesensor was subjected to 100 pressure cycles in each test, andthe results did not vary significantly). The pressure hysteresis


Fig. 8. Measured pressure versus temperature for a fixed applied pressure of 10 kPa (a) and (b) and for a fixed applied pressure of 0.3 kPa (c) and (d). Eachfigure shows one complete cycle, starting at 25 ◦C.

in this sensor is almost totally due to the “memory effect” inthe spring.

C. Temperature Hysteresis

Fig. 8 shows temperature hysteresis curves that were ob-tained at a fixed pressure of 10 kPa (a) and (b) and at a fixedpressure of 0.3 kPa (c) and (d). As the figures clearly show,the maximum temperature hysteresis error is about ±0.1% full-scale output (FSO). The temperature hysteresis is believed to betotally due to variations in the relative permeability of the ironcore as temperature is cycled.

D. Susceptibility to Mechanical Shocks

If the sensor is shocked in the vertical direction, the ironcore will be momentarily displaced, and it was observed thata “recovery time” is needed for the iron core to return toits original position (and, hence, for the momentary error todisappear). The sensor was tested on a standard electrodynamicshaker3 that provides shock pulses of a magnitude of 34 g

3Model DSS-M100 from Dynamic Solutions, Inc.

and a duration of 10 ms [20], [21]. In order to determine therecovery time that is needed for the error to disappear afterthe shock, the frequency of an interface circuit that uses a 555oscillator (see Section V) was monitored with a digital storageoscilloscope. The nominal frequency of the interface circuitwas noted, and the time duration that lapsed from the onsetof the shock until the frequency returned to its nominal valuewas observed on the scope. The results of that test are shownin Fig. 9. It should be mentioned that while the sensor wasmounted on the electrodynamic shaker, it was pressurized witha small mechanical fixture that uses a screw for exerting forceon the iron core. By measuring the inductance of the coil, thepressure in each phase of the test was determined precisely.

V. SENSOR INTERFACE CIRCUIT

A word is now in order concerning the interface circuitthat is used with the sensor. An interface circuit that uses a555 timer chip working in an oscillator mode was integratedonto the printed-circuit board on which the sensor is mounted.The oscillator circuit essentially converts the unknown induc-tance of the coil to frequency. Such a circuit is very wellknown in the literature and is described in references such


Fig. 9. Aftershock recovery time as a function of the applied pressure.

Fig. 10. Block diagram of the interface circuit used to measure inductance L.

as in [22]. The block diagram of the circuit is shown inFig. 10.

The equation that characterize the circuit shown in Fig. 10 is[22] as follows:

f = 0.092R

L(13)

where f is the frequency of the resulting square wave inhertz. Given the known resistance R, the value of the unknowninductance L can be easily calculated from that equation bysimply observing the frequency. This is how plots such as theplots in Figs. 5 and 6 were obtained. It is to be pointed outthat the interface circuit does not amplify or compensate for thecharacteristics of the sensor (i.e., the sensor is uncompensated,as shown in Table I).

VI. CONCLUSION

The new inductive pressure sensor introduced in this paperhas sensitivity that is substantially high by comparison withother pressure sensors. The prototype device that was fabri-cated and tested has demonstrated a change in inductance ofapproximately 34.5 mH over a pressure range of 10 kPa. Highersensitivity and different pressure ranges can be achieved byvarying the mechanical parameters of the sensor. The sensoroffers a number of desirable features, including linearity, lowpressure and temperature hysteresis, in addition to small size.The equations that relate the applied pressure to the measuredinductance of the device are fairly simple, and testing has shownvery good agreement between theory and experiment.

APPENDIX

ERROR ANALYSIS

It is possible that the measurements of inductance L and therelative permeability µr may be in error by increments ΔL andΔµr. We shall now examine the effect of such possible errorterms on the pressure P measured by the sensor.

Effect of an Error in µr: By replacing µr by µr + Δµr in(12) and assuming that quantity µr − 1 ≈ µr, we obtain thefollowing:

P ≈ Pmin +kl

A

(1 − 1

µr + Δµr

L

L0

). (14)

By taking difference P − P0, where P0 is the error-free expres-sion given by (12), and simplifying the result, we can verifythat

P − P0

P0=

ΔP

P0≈ Δµr

µ2r

. (15)

Given that Δµr is usually ±10% of µr and that µr is a largenumber (≈4000 in the present sensor), it can be seen that theerror predicted by the aforementioned equation is negligible,more specifically, on the order of 10−6.

Effect of Error in L: By replacing L by L + ΔL in (12) andconducting a similar analysis, it is straightforward to verify that

ΔP

P0≈ ΔL

L. (16)

Depending on the magnitude of error ΔL, the error percentagegiven by the above expression can be substantial. Accordingly,it is clear from the above analysis that any error in the relativepermeability µr will lead to a negligible error in the calculatedpressure, whereas errors in inductance L can have a substantialeffect.

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Ezzat G. Bakhoum (SM’08) received the B.S. de-gree in electrical engineering from Ain Shams Uni-versity, Cairo, Egypt, in 1986 and the M.S. and Ph.D.degrees in electrical engineering from Duke Univer-sity, Durham, NC, in 1989 and 1994, respectively.

From 1994 to 1996, he was a Senior Engineerand a Managing Partner with ESD Research, Inc.,Durnham. From 1996 to 2000, he was a Senior En-gineer with Lockheed Martin/L3 Communications,Inc., Camden, NJ. From 2000 to 2005, he served as aLecturer with the Department of Electrical Engineer-

ing, New Jersey Institute of Technology, Newark. Currently, he is an AssistantProfessor with the University of West Florida, Pensacola.

Marvin H. M. Cheng (M’04) received the B.S. andM.S. degrees in mechanical engineering from Na-tional Sun Yat-Sen University, Kaohsiung, Taiwan,in 1994 and 1996 and the Ph.D. degree in mechanicalengineering from Purdue University, West Lafayette,IN, on December 2005.

From 1997 to 1999, he was a Research andDevelopment Engineer with National SynchrotronRadiation Research Center, Hsinchu, Taiwan, wherehe developed a real-time monitoring system forhigh-energy emission systems. From 2006 to 2010,

he was an Assistant Professor of mechanical engineering technology withGeorgia Southern University, Statesboro. He is currently an Assistant Professorof mechanical engineering with West Virginia University, Morgantown. Hiscurrent research interests include mechatronics, controller synthesis with theconsideration of finite word length, precision, motion control, fast imaging ofatomic force microscope, and embedded controllers.

High-Sensitivity Inductive Pressure Sensor

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