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Sensors and Actuators A 147 (2008) 210–215 Contents lists available at ScienceDirect Sensors and Actuators A: Physical journal homepage: www.elsevier.com/locate/sna Direct interface circuit to linearise resistive sensor bridges Ernesto Sifuentes a , b , ∗ , Oscar Casas a , Ferran Reverter a , Ramon Pall`as-Areny a a Instrumentation, Sensors and Interfaces (ISI) Group, Castelldefels School of Technology (EPSC), Universitat Polit`ecnica de Catalunya (UPC), Avda. del Canal Ol´ımpic 15, Edifici C4,08860 Castelldefels, Barcelona, Spain b Computer and Electrical Engineering Department, Autonomous University of Ciudad Juarez (UACJ), Avda. del Charro 450 nte., Juarez, Chihuahua, Mexico a r t i c l e i n f o Article history: Received 16 November 2007Received in revised form 10 March 2008Accepted 4 May 2008Available online 23 May 2008 Keywords: MicrocontrollerResistive sensor bridgeSensor electronic interfaceMagnetoresistive sensor a b s t r a c t Thispaperproposesthedirectconnectionofdifferentconfigurationsofresistivesensorbridgestoamicro-controllerwithoutanyintermediateactivecomponent.Suchadirectinterfacecircuitreliesonmeasuringthedischargingtimeofa RC networkthatincludestheresistancesofthesensorbridge.Forquarter-,half-,and full-bridge circuits, we combine the discharging times to estimate the fractional resistance change x of the bridge arms. Experimental results for half- and full-bridge circuits emulated by resistors yield anonlinearity error below 0.3%FSR (full-scale range) for x between 0 and 0.1 and an effective resolutionof 11 bit. Measurements on two commercial magnetoresistive sensors yield higher nonlinearity errors:1.8%FSRforanAMR(AnisotropicMagnetoresistive)sensorand5.8%FSRforaGMR(GiantMagnetoresistive)sensor, which are mainly due to the nonlinearity of the sensors themselves. Therefore, the nonlinearityof the measurement is limited by the sensors, not by the proposed interface circuit and linearisationalgorithm.© 2008 Published by Elsevier B.V. 1. Introduction Resistive sensors are often connected in a Wheatstone bridgebecause placing several sensors (modelled by resistances R 1 , R 2 , R 3 , and R 4 in Fig. 1) in different bridge arms increases sensitiv- ity and compensates for interference such as temperature [1]. This is the case, for example, for pressure and magnetoresistive sen-sors. The common approach to interface a resistive sensor bridgeto a microcontroller (MCU) is by supplying a constant voltage orcurrenttothebridge.Thentheoutputvoltageisamplifiedanddigi-tised with an analog-to-digital converter (ADC), either external orembedded into the MCU. Table 1 shows some typical configura-tions of the Wheatstone bridge circuit in Fig. 1 and the resultingoutput voltage ( V o ) when a constant voltage ( V r ) is applied [1]; R 0 is the nominal resistance of the sensors and x is the fractionalresistance change ( x = R / R 0 ). A quarter-bridge circuit (row 1 inTable 1) has a single active arm and the output voltage is nonlin-ear with respect to x . A half-bridge circuit (rows 2–4 in Table 1)has two active arms, but the transfer characteristic is linear onlyfor case 4. A full-bridge circuit (row 5 in Table 1) has four active arms and a linear transfer characteristic [2]. The fractional resis- tancechange x canbecertainlyestimatedfromanyoftheequationsinTable1.However,thecalculationsarequiteinvolvedforthecon- ∗ Corresponding author. Tel.: +34 934137089; fax: +34 934137007. E-mail addresses:
[email protected],
[email protected] (E. Sifuentes). figuration in row 3, and include divisions for those in rows 1 and 2[1].Several circuits have been proposed to obtain a linear responsefrombridgeswhoseoutputvoltageisnonlinearwith x .Onesolutionis to include the bridge into a relaxation oscillator whose outputfrequency is proportional to the resistance unbalance in one of the four bridge arms; the nonlinearity error achieved was smallerthan0.5%FSR(full-scalerange)for − 0.005< x <0.005[3].TheCMOS interface circuit for resistive sensor bridges proposed in Ref. [4]compensates for offset and gain effects in the analog domain andprovidesalineardigitaloutputbyaprogrammableamplifierandan8bitADC.ThecircuitproposedinRef.[5]usesafeedbackloopwith a transadmittance amplifier to reduce the nonlinearity to about ± 0.4% for − 0.2< x <0.2; however, minimal offset and nonlinearityerrors require matched bridge components, amplifiers, and gainsettings.Resistive sensors can be directly connected to a MCU by mea-suring the time needed to discharge a capacitor through them [6].This approach is simple, compact, and low-cost because only thesensor and a MCU are required. This method has been applied to(full-bridge) piezoresistive pressure sensors [7] by considering the bridge as a circuit with one input terminal and three output ter-minals, instead of the usual two input terminals and two outputterminals considered in amplitude-based interface circuits. How-ever, there is not any known proposal to apply this method toquarter- and half-bridge resistive sensors; neither is there any sys-tematicanalysisaimedtoestimatethefractionalresistancechange 0924-4247/$ – see front matter © 2008 Published by Elsevier B.V.doi:10.1016/j.sna.2008.05.023 E. Sifuentes et al. / Sensors and Actuators A 147 (2008) 210–215 211 Fig. 1. Wheatstone bridge circuit built from four resistive sensors supplied by aconstant voltage. Table 1 Output voltage for quarter-, half-, and full-bridge circuits (Fig. 1) when supplied by a constant voltage V r [1] R 1 R 2 R 3 R 4 Output voltage ( V o )(1) R 0 R 0 R 0 R 0 (1+ x ) V r x 2(2 + x )(2) R 0 (1+ x ) R 0 R 0 R 0 (1+ x ) V r x 2 + x (3) R 0 (1 − x ) R 0 R 0 R 0 (1+ x ) V r − x 2 4 − x 2 (4) R 0 R 0 (1 − x ) R 0 R 0 (1+ x ) V r x 2(5) R 0 (1+ x ) R 0 (1 − x ) R 0 (1 − x ) R 0 (1+ x ) V r x x whatever the number and position of active arms. This workextends the method described in Ref. [7] and proposes a general equationtoestimate x ineitherquarter-,half-,andfull-bridgeresis-tive sensors. The proposed method is then applied to commercialmagnetoresistive (MR) sensors based on bridge circuits. 2. Measurement method Fig. 2 shows the proposed direct interface circuit for a resistivesensor bridge, which is considered to have one input terminal andthree output terminals. This circuit relies on measuring the dis-charging time of a RC network [6] that includes the resistances of thesensorbridge.Themeasurementinvolvestwostages:(a)charg-ingand(b)dischargingandtimemeasurement.Duringthechargingstage,pinA4issetasanoutputprovidingadigital“1”,whereaspinsA1, A2, and A3 are set as an input, offering high impedance. There-fore, capacitor C is charged towards V DD (positive supply voltage Fig. 2. Direct connection between a Wheatstone bridge circuit and a microcon-troller. Fig. 3. Voltage waveform across C in Fig. 2 during the measurement sequence. forI/Opins)throughtheinternalresistanceoftheMCUpinandtheresistor R p for a time longer than 5 R p C ; R p improves the rejectionofpowersupplyinterference[8].Duringthesecondstage,pinA4is set as a high-impedance input, pin A1 is set as an output providinga digital “0”, and pins A2 and A3 do not change their state. At thisinstant, C is discharged towards V SS (ground reference for I/O pins)throughtheequivalentresistance( R eq1 )betweennodeAandpinA1,andtheembeddedtimerstartsthemeasurementofthedischargingtime. When the voltage across C reaches the low-threshold voltage( V TL )oftheSchmitt-trigger(ST)bufferembeddedintoA4,thetimerstops. Fig. 3 shows the waveform of the voltage across C during themeasurementsequence[6].Afterwards, C ischargedagainandthendischarged through the equivalent resistance ( R eq2 ) between nodeA and pin A2. Finally, C is charged once more and then dischargedthrough the equivalent resistance ( R eq3 ) between node A and pinA3.Table2showstheresultingvaluesofthethreeequivalentresis- tances ( R eq1 , R eq2 and R eq3 ) for different bridge configurations. If the internal resistance ( R in i ) of the MCU pins is assumed to bemuch smaller than those equivalent resistances (i.e., R in i << R eq i ),the discharging time equals t i = R eq i C ln V DD − V SS V TL − V SS . (1)If V DD , V TL , V SS and C remain constant during each measurement,then t i is proportional to R eq i .The goal is to have an interface circuit able to estimate the frac-tional resistance change x of the bridge. Because none of the times t i are proportional to x , we propose to use a linearisation algo-rithm that combines the three discharging times measured andthatshouldbesimpleenoughtobeappliedbyanycommonmicro-controller. To compensate some errors, we can calculate a ratio (tocancelmultiplicativeerrors)ofadifferencebetweenmeasureddis-chargingtimes(tocanceladditiveerrors).Hence,ageneralequationto estimate x for any bridge configuration is x ∗ = at 1 + bt 2 + ct 3 dt 1 + et 2 + ft 3 = aR eq1 + bR eq2 + cR eq3 dR eq1 + eR eq2 + fR eq3 , (2)where t 1 , t 2 , t 3 are the respective discharging times through R eq1 , R eq2 , R eq3 , and a , b , c , d , e , and f are the coefficients of the time-basedequation.Because R eq i isaparallelandseriescombinationof R 0 , R 0 (1+ x ), and R 0 (1 − x ), its general value can be written as R eq i = i + i x + i x 2 Den( x ) , (3)where ˛ i , ˇ i , and ı i are known coefficients, and the denomina-tor Den( x ) is common to the three equivalent resistances for eachbridge configuration, as shown in Table 2. Replacing (3) in (2) and 212 E. Sifuentes et al. / Sensors and Actuators A 147 (2008) 210–215 Table 2 Equivalent bridge resistances and equations to estimate the fractional resistance change for quarter-, half- and full-bridge circuits directly connected to a MCU (Fig. 2)Bridge configuration (Fig. 2) R eq1 R eq2 R eq3 Time-based equation(1) R 1 = R 2 = R 3 = R 0 R 0 (3 + 3 x )4 + xR 0 (4 + 2 x )4 + xR 0 (3 + x )4 + x x ∗ = 2( t 1 − t 3 ) t 2 + t 3 − t 1 R 4 = R 0 =(1+ x )(2) R 1 = R 4 = R 0 (1+ x ) R 0 (3 + x )(1 + x )4 + 2 xR 0 (2 + x ) 2 4 + 2 xR 0 (3 + 2 x )4 + 2 x x ∗ = 2( t 1 − t 3 ) t 2 + t 3 − t 1 R 2 = R 3 = R 0 (3) R 1 = R 0 (1 − x ) R 0 (3 − x )(1 + x )4 R 0 (4 − x 2 )43 R 0 4 x ∗ = 2( t 1 − t 3 ) t 2 + t 3 − t 1 R 4 = R 0 (1+ x ) R 2 = R 3 = R 0 (4) R 1 = R 3 = R 0 R 0 (3 − x )(1 + x )4 R 0 (4 − x 2 )4 R 0 (3 + x )(1 − x )4 x ∗ = t 3 − t 1 2( t 1 + t 3 − 2 t 2 ) R 4 = R 0 (1+ x ) R 2 = R 0 (1 − x )(5) R 1 = R 4 = R 0 (1+ x ) R 0 (3 − x )(1 + x )44 R 0 4 R 0 (3 + x )(1 − x )4 x ∗ = t 1 − t 3 t 2 R 2 = R 3 = R 0 (1 − x ) rearranging yields x ∗ = A + Bx + Cx 2 D + Ex + Fx 2 , (4)where A = a 1 + b 2 + c 3 ,B = a 1 + b 2 + c 3 ,C = a 1 + b 2 + c 3 ,D = d 1 + e 2 + f 3 ,E = d 1 + e 2 + f 3 ,F = d 1 + e 2 + f 3 . (5)To achieve x * = x , we first need A = F =0 in (4) to have x ∗ = Bx + Cx 2 D + Ex = x ( B + Cx ) D + Ex . (6)Then, from (6), we also need B = kD and C = kE . For k =1, we have a 1 + b 2 + c 3 = 0 ,d 1 + e 2 + f 3 = 0 ,a 1 + b 2 + c 3 = d 1 + e 2 + f 3 ,a 1 + b 2 + c 3 = d 1 + e 2 + f 3 . (7)Because we have six unknowns but only four equations, there areseveral possible solutions to solve (7) f or a , b , c , d , e , and f foreach bridge configuration. Therefore, different equations based on(2) can estimate x in each bridge circuit. Table 2 shows the sim-plest equations to estimate x for quarter-, half-, and full-bridgecircuits. The operations involved in such equations are the samerequired to estimate x from the voltage outputs in rows 1 and2 in Table 1, which in addition need voltage amplification and digitisation. 3. Materials and method The circuit in Fig. 2 was implemented using a MSP430F123microcontroller (Texas Instruments) running at 4MHz (quartzoscillator clock). The embedded 16 bit timer counted the discharg-ing time by incrementing its value every 250ns. Pins P3.3, P3.6,P3.7 and P1.2 implemented the function of pins A1, A2, A3 and A4in Fig. 2, respectively. To reduce the effect of the internal trigger noise, the MCU was set in LPM0 mode (disabled CPU but enabledtimer and interruptions) when the discharging times were beingmeasured [6]. Pin P1.2 (external interrupt pin with an internal ST buffer) was set to interrupt the MCU on the falling edge. The valueof C was selected to obtain a suitable time constant ( = R eq i C ) forthe discharge circuit. The optimal time constant in terms of speed-resolution trade-off for our circuit when measuring 1k resistorswas experimentally determined to be about 2–3ms. Hence weselected C =2.2 F ( ± 5% tolerance and 100 × 10 − 6 / ◦ C temperaturecoefficient). The resistor R p was 120 .The experimental work to assess the validity of the proposedmeasurement method was divided in two stages: (a) measure-ments on half- and full-bridge circuits emulated by resistors, and(b)measurementsontwocommercialMRsensorsbasedonbridgecircuits. For each value of x applied, the discharging times t 1 , t 2 and t 3 were measured 100 times and were sent to a personal com-puter (PC) via RS-232. Then, x * was calculated by the equation of row 5 in Table 2 f or full-bridge circuits, and by the equation of row 2 for half-bridge circuits. We selected these two bridge configura-tions because they are the same as those of the MR sensors laterused.To allow for a flexible use of computing and memory resources,andgraphicalrepresentation,theequationstoestimate x inTable2were computed using a PC. Nevertheless, for the MSP430 it is pos-sible to define floating-point variables, which results in a precisionof around seven decimal digits for the basic arithmetic operations( − ,+,/,and*)whenusingC-libraryroutines.Consequently,wealsocalculated x * using the MCU to asses whether or not the proposedmethod requires any capabilities beyond those of a common MCU,or an excessive computation time. 3.1. Bridge circuits emulated by resistors We built a full-bridge circuit (row 5 in Table 2) by using resis- tors R 0 =1k (1%). Changes in each bridge arm were introducedby placing resistors in parallel with R 2 and R 3 to produce R 0 (1 − x ),and resistors in series with R 1 and R 4 to produce R 0 (1+ x ). We alsobuiltahalf-bridgecircuit(row2inTable2)withthesameresistors, butonly R 1 and R 4 werechangedtoproduce R 0 (1+ x ).Inbothbridgecircuits,theaddedresistorswereselectedtoobtainfractionalresis-tance variations ( x ) from 0.5% to 10%. The actual value of x appliedwas determined in voltage mode using the equations in Table 1. 3.2. AMR and GMR sensors ThetwoMRsensorstestedwere:HMC1052(Honeywell),whichis an AMR (Anisotropic Magnetoresistive) sensor with full-bridgetopology and a typical resistance R 0 =1k , and AAH002 (NVE),which is a GMR (Giant Magnetoresistive) sensor with half-bridgetopology and R 0 =2k . To create a known magnetic field to beapplied to the sensors, we build a 55cm long ( L ) solenoid with2cminradius( R )and330turns( N )ofAWG20enamelcopperwire(Fig.4).Themagneticfluxdensity( B )alongthe z -axisisuniformin E. Sifuentes et al. / Sensors and Actuators A 147 (2008) 210–215 213 Fig. 4. Solenoid to create a known magnetic field. the center of the solenoid ( z =0) and its value is [9]B = o IN/L 1 + 4 R 2 /L 2 , (8)where o is the permeability of the air and I is the applied current.Inaninfinitelylongsolenoid, B wouldbeconstantalongitscentralaxis, but for the L and R values in our case, B values deviated byless than ± 0.3% from that constant value. The sensors were placedabout the center of the solenoid with their sensitivity axis paralleltothemagneticfield(Fig.4).Bycontrollingthecurrentapplied,we set B whosevalueswereselectedfrom75 Tto600 TfortheAMR sensorandfrom75 Tto290 TfortheGMRsensor,whicharetheirrespective linear measurement ranges. The actual response of theAMR and GMR sensors available was determined in voltage mode:a 3V supply voltage was applied to each sensor and its differentialoutput voltage was measured by a multimeter (Agilent 34401A). 4. Experimental results and discussion 4.1. Bridge circuits emulated by resistors Figs. 5 and 6 show the nonlinearity error (calculated from thebest-fit straight line) of the estimated x * versus actual x for thefull- and half-bridge circuits, respectively. The maximal nonlin-earity error was 0.2%FSR (Fig. 5) for the full-bridge circuit and 0.3%FSR(Fig.6)f orthehalf-bridgecircuit,whichareprobablydueto the quantization error in the discharging-time measurement. Theresults were similar when x was estimated by the MCU, and thecomputation time was smaller than 1ms. High R in i values couldyield offset and gain errors but they do would not increase thenonlinearity [10]. The interface circuit herein proposed has non- linearity errors similar to those in Ref. [3,5] but it is more compact, economic and simple to design.The standard deviation of x * remained constant over the mea-surement range for both bridge configurations. It was equal to7.4 × 10 − 5 for the full-bridge circuit and 1.3 × 10 − 4 for the half-bridge circuit. This variability is mainly due to power supply noiseand noise resulting from CPU activity [6]. If we define the resolu- tionastheamplitudeoftheuncertaintyintervalanduseacoveragefactor of two, then the resolution equals four times the standard Fig. 5. Nonlinearity error of the estimated x * vs. x for a full-bridge circuit. deviation. Accordingly, the measurement of x has an effective res-olution of 8 bit for a measuring time of about 10ms. If we usethe mean of one hundred values of x * as an estimate of the mea-surement, then the resolution increases to 11 bit, but of course themeasuring time is 100 times longer. 4.2. AMR and GMR sensors Figs. 7 and 8 show the nonlinearity error (calculated from thebest-fit straight line) of the estimated x * versus B for the AMR and GMR commercial sensors, respectively. The maximal nonlin-earityerrorequals1.8%FSR(Fig.7)fortheAMRsensorand5.8%FSR (Fig. 8) for the GMR sensor. These errors and their dependence on B , hence on x , are very similar to those obtained when the sen-sors were measured in voltage mode, which are also shown inFigs. 7 and 8. Therefore, the nonlinearity error of the completesystemcanbeattributedtothenonlinearcharacteristicofthesen-sor,ratherthantotheinterfacecircuitandlinearisationalgorithmsproposed. This error can be reduced by dividing the measurementrange in subranges. For example, for the GMR sensor, limiting themeasurementrangeto75 T< B <195 T,yieldsamaximalnonlin-earity of 2.2%FSR.The standard deviation of x * remained constant over the mea-surement range for both sensors. It was equal to 7.7 × 10 − 5 for the Fig. 6. Nonlinearity error of the estimated x * vs. x for a half-bridge circuit. 214 E. Sifuentes et al. / Sensors and Actuators A 147 (2008) 210–215 Fig.7. Nonlinearityerrorof(a)theestimated x * vs. B ,and(b)thedifferentialvoltagevs. B for the HMC1052 AMR (full-bridge) sensor. Fig.8. Nonlinearityerrorof(a)theestimated x * vs. B ,and(b)thedifferentialvoltagevs. B for the AAH002 GMR (half-bridge) sensor. AMR sensor and 1.4 × 10 − 4 for the GMR sensor, which are closeto the values obtained when the bridge circuit was emulated byresistors. If x was estimated by averaging one hundred values of x * , then the effective resolution would be 7 bit for the AMR sensorfor 0.001< x <0.006 (which correspond to 75 T< B <600 T) and10 bit for the GMR sensor for 0.02< x <0.06 (which correspond to75 T< B <290 T). 5. Conclusions Quarter-, half-, and full-bridge sensor circuits can be directlyconnected to a microcontroller without using any analogue inte-gratedcircuitinthesignalpath,thusresultinginacompact,simpleand low-cost design solution. The fractional resistance change of the resistive sensor bridge caneasilybe estimatedbyatime-basedequation(Table2)thatcombinesthreedischarging-timemeasure- ments. These equations in Table 2 involve operations that can be performed by a microcontroller. The maximal nonlinearity errorof the proposed interface circuit is 0.3%FSR for half-bridge circuitsand 0.2%FSR for full-bridge circuits for 0< x <0.1 and an effectiveresolutionof11bit,whichareacceptableformanyindustrialappli-cations. Acknowledgments This work has been funded in part by the Spanish Min-istry of Education and Science, projects TEC2004–05520 andDPI2006–04017;andbytheEuropeanRegionalDevelopmentFund.Ernesto Sifuentes has been funded by PROMEP and UACJ Mex-ico. The authors also acknowledge the technical support of FrancisL ´opez. References [1] R.Pall`as-Areny,J.G.Webster,SensorsandSignalConditioning,seconded.,JohnWiley & Sons, New York, 2001.[2] Wheatstone Bridge Nonlinearity, Technical Note 507, Vishay MeasurementsGroup, 1999.[3] V. Ferrari, C. Ghidini, D. Marioli, A. Taroni, Oscillator-based signal conditioningwith improved linearity for resistive sensors, IEEE Trans. Instrum. Meas. 47 (1)(1998) 293–298.[4] A.J. Lopez-Martin, J.I. Osa, M. Zuza, A. Carlosena, A CMOS interface for resistivebridge transducers, in: IEEE International Symposium on Circuits and SystemsISCAS, vol. 2, 2002, pp. 153–156.[5] G. De Graaf, R.F. Wolffenbuttel, Systematic approach for the linearization andreadoutofnonsymmetricimpedancebridges,IEEETrans.Instrum.Meas.55(5)(2006) 1566–1572.[6] F. Reverter, J. Jordana, M. Gasulla, R. Pall`as-Areny, Accuracy and resolutionof direct resistive sensor-to-microcontroller interfaces, Sens. Actuators A 121(2005) 78–87.[7] J. Jordana, R. Pall`as-Areny, A simple efficient interface circuit for piezoresistivepressure sensors, Sens. Actuators A 127 (1) (2006) 69–73.[8] F. Reverter, M. Gasulla, R. Pall`as-Areny, Analysis of power-supply interfer- enceeffectsondirectsensor-to-microcontrollerinterfaces,IEEETrans.Instrum.Meas. 56 (1) (2007) 171–177.[9] P. Lorrain, D.R. Corson, Electromagnetism: Principles and Applications, W.H.Freeman and Company, New York, 1990.[10] E. Sifuentes, O. Casas, F. Reverter, R. Pall`as-Areny, Improved direct interface circuit for resistive full- and half-bridge sensors, in: Fourth International Con-ference on Electrical and Electronics Engineering ICEEE, Mexico, September,2007, pp. 197–200. Biographies Ernesto Sifuentes received his BSc degree in electronic engineering from Techno-logical Institute of Durango (ITD), Durango, Mexico in January 1999; he receivedhisMScdegreeinelectronicengineeringfromTechnologicalInstituteofChihuahua(ITCH), Chihuahua, Mexico in May 2002. Since August 2002, he has been as a titu-lar professor (PTA) in the UACJ (Autonomous University of Ciudad Juarez), Mexico.SinceSeptember2005,hehasbeenintheCastelldefelsSchoolofTechnology,whichbelongs to the Technical University of Catalonia (UPC) Barcelona, Spain, where heis actually working in his PhD thesis research funded in part by PROMEP and UACJ,Mexico. His research interests are electronic instrumentation, embedded systems,direct sensor-to-microcontroller interfaces, and sensor networks. Oscar Casas was born in Barcelona, Spain, on April 15, 1970. He received the Inge-nierodeTelecomunicaci´onandDoctorIngenierodeTelecomunicaci´ondegreesfromthe Universitat Polit`ecnica de Catalunya (UPC), Barcelona, Spain, in 1994 and 1998, respectively. He is an associate professor of Electronic Engineering at the UPCand teaches courses in several areas of electronic instrumentation. His researchincludes sensor interfaces, autonomous sensors, electronic instrumentation, non-invasive physiological measurements and sensors based on electrical impedancemeasurements. Ferran Reverter was born in Llagostera, Spain, on January 4, 1976. He receivedthe BSc degree in Industrial Electronic Engineering from the University of Girona,Girona, Spain, in 1998, the MSc degree in Electronic Engineering from the Uni-versity of Barcelona, Barcelona, Spain, in 2001, and the PhD degree in ElectronicEngineering from the Technical University of Catalonia (UPC), Barcelona, in 2004.Since 2001, he has been an assistant professor in Analogue Electronics and Digi-tal Systems with the UPC. From 2005 to 2007, he was a Visiting Postdoctoral FellowwiththeDelftUniversityofTechnology,Delft,TheNetherlands.Heiscoauthor(withRamonPall`as-Areny)ofthebookDirectsensor-to-microcontrollerinterfacecircuits(Barcelona: Marcombo, 2005). His research interests are in the field of electronicinstrumentation, in particular, the design and characterisation of interface circuitsfor sensors.
