Intrumentation and Meaurement Technology Conference IMTC 007 Waraw, Poland, May 1-3, 007 Thermal Σ- Modulator: Anemometer Performance Analyi Will R. M. Almeida 1, Georgina M. Freita 1, Lígia S. Palma 3, Sebatian Y.C. Catunda, Raimundo C. S. Freire 1, Francico F. Santo 1, Amauri Oliveira 3, Haan Abouhady 4 1 Univeridade Federal de Campina Grande, Unidade Acadêmica de Engenharia Elétrica, Campina Grande - PB, Brazil. Univeridade Federal do Maranhão, Departamento de Engenharia de Eletricidade, São Lui MA, Brazil. 3 Univeridade Federal da Bahia, Departamento de Engenharia Elétrica, Salvador BA, Brazil. 4 Univerity of Pari VI, Pierre & Marie Curie, LIP6/ASIM Laboratory, 755 Pari, France Email: {willalmeida, rcfreire, ffanto}@dee.ufcg.edu.br, geo_maciel@hotmail.com, catunda@dee.ufma.br, {ligia, amauri}@ufba.br, Haan.Abouhady@lip6.fr. Abtract In thi paper we propoe a feedback architecture with a thermoreitive enor, baed on the thermal igma-delta principle to realize digital meaurement of phyical quantitie that interact with the enor: temperature, thermal radiation, fluid peed. Thi architecture ue a 1-bit igma-delta modulator for which a coniderable part of the converion function i performed by a thermoreitive enor. The enor i modelled uing the electrical equivalence principle and the contant temperature meaurement method i applied. We preent an analyi of the ytem performance, in term of frequency repone and ytem meaurement reolution, of a 1-bit firt-order Σ- thermal modulator. It i hown that the ytem performance depend on the ytem overampling ratio (OSR) and on the ytem tranfer function pole and zero, which in turn, depend on the thermal and phyical enor characteritic and on the ytem operating condition. Thi ytem i propoed a anemometer. Kword Thermoreitive enor, Anemometer, Sigma delta modulation, Contant temperature architecture, Microenor. I. INTRODUCTION The claical architecture of hot-wire anemometer are baed on the equivalence principle for which the ening element i a thermoreitive enor, the etimated output i an analog ignal and the configuration ued i an Wheattone bridge with the enor placed in one of it branche in a contant temperature arrangement. Other Wheattone bridge configuration ue pule width modulation in the feedback loop. The implicity and robutne of igma-delta A/D converter make thi category of A/D converter an excellent candidate for mart enor application [1, ]. Realizing both functional and economical characteritic of integrating enor and ignal proceing function on a chip et a challenge of complexity and component tolerance on the integrated circuit deign. An eaier alternative i the ue of igma delta configuration with the enor a part of the feedback loop. A mono-bit firt-order thermal Σ- modulator, a hown in Figure 1, wa propoed a anemometer, which i baed on the electric equivalence principle with the enor operating at a contant temperature [3]. Thi fluid peed meaurement ytem directly tranform the phyical ignal into it equivalent digital form and may be integrated with a microenor. In the propoed mono-bit firt-order Σ- modulator architecture, the um and integration operation are performed by the enor. In thi paper, the propoed anemometer ytem performance i analyzed and dicued. II. BACKGROUND DEFINITIONS The dynamic heat equation for a thermoreitive enor can be expreed by [3-5]: dt P+αHS e = hs ( T T f ) + mc. (1) dt where, αsh i the incident thermal radiation aborbed by the enor, P e = I R i the electrical power delivered to the enor, h i the heat tranfer coefficient referred to the enor urface area S, T i the enor temperature, T f i the fluid temperature, m i the enor ma, c i the enor pecific heat. The enor temperature, T, can i given by: ( ( )) 1 f T = αhs + I R hs T T dt mc () Figure 1 how the block diagram of a firt-order igmadelta modulator. The umming and integrating block are in evidence, howing the imilarity with (). The idea of including the microenor into a 1-bit firtorder igma delta loop come from the mentioned imilarity and from the fact that the enor temperature repone curve lead to an almot exponential function in repone to a quared current tep for mall tep amplitude. Thu, Figure 1. Block diagram of a firt-order igma-delta modulator. 1-444-0589-0/07/$0.00 007 IEEE
conidering that the ampling frequency, f, i much greater than the enor linear tranfer function pole frequency, thi exponential can be approximated by an integration function, for which the gain i the exponential function initial lope. Conidering that the tep repone for an ideal integrator and for the exponential are almot coincident until 10% of the exponential time contant, ome tudie were carried out analyzing the mall ignal model for the igma delta converter, employed for the meaurement of thermal radiation and environment temperature [6]. Baed on thee tudie and auming that H i equal to zero, we propoe a tructure baed on igma delta modulation for etimation fluid peed (ϑ). Hence, T, can be expreed by: 1 T = I R hs ( T Tf ) dt mc (3) The thermoreitive microenor ued ha a poitive temperature coefficient, PTC. The thermal behavior mathematical model of the PTC i given by: R=R o 1+β ( T Tf ) (4) where R o i the enor reitance at 0 o C and β i the thermal coefficient, which i a function of the enor material. Rewritten (3), conidering the ubtitution: I for Y and h = a + bϑ n and auming the condition of tatic thermal balance, the enor quared current (Y ), can be expreed by: 1 n f R ( ϑ )( ) Y= S a+b T T (5) Thu, auming that the enor temperature i kept almot contant, ϑ can be evaluated from the knowledge Y a: 1 YR ϑ= a b S( T Tf ) 1/ n A. Continuou Current modulator model Figure how the block diagram the continuou current (CC) igma-delta modulator behavioural model with the thermoreitive microenor working a the umming and integrating component. The fluid peed ϑ(t) and the enor quared current, Y (t), are the input ignal wherea the enor temperature, T, i the output ignal. The enor ubtitute the original 1-bit firt-order Σ- modulator um and integration function. (6) A the enor i deigned to operate at contant temperature, a comparator i included into feedback loop to verify enor temperature. If enor temperature i greater than a reference value the 1-bit D/A (quantizer) generate a 1 bit at the modulator output. Thi bit, introduced in modulator feedback path, reduce the enor current reducing alo the enor temperature. If enor temperature i maller than the reference value the 1-bit D/A generate a -1 bit at the modulator output. Thi bit introduced in modulator feedback path increae the enor current increaing alo the enor temperature. B. Puled current modulator model. Figure 3 how the block diagram of the puled-current (PC) igma-delta behavioural model. The quared current Y, from the continuou-ignal model, i replaced by a puled current I (PWM), generated by the PWM block, which i proportional to Y. The PWM generate a puled current with only two pule width, one pule width for quantizer output 1 and another pule width for quantizer output -1. In the thermal equilibrium tate, the pule width ha a theoretical value of 50% of the PWM period (T PWM ). The fluid peed information i now in the pule width, which ha a nonlinear relationhip with the former. III. PERFORMANCE ANALYSIS A performance analyi for the continuou current ytem i carried out in thi ection, in term of the ytem frequency repone and meaurement reolution. A. Continuou Signal Sytem Tranfer Function. Figure 4 how the block diagram of a ampled verion in the -domain of the propoed architecture, baed on the continuou ignal model, which i ued to analyze the ytem performance. In thi block diagram it wa ued the enor mall ignal model. The quantizer wa linearized and modelled by a white noie ource, E(), that wa added to the enor temperature, T (), which i a function of the fluid peed, ϑ(). The unity gain block between enor output and Figure. Block diagram of the continuou ignal Σ modulator model. Figure 3. Block diagram of the puled current Σ modulator model.
the white noie ource i ued to tranform the enor temperature cale to the quantizer cale. The block A o () i a zero-order holder and Y o i the quared current gain of the D/A converter in the modulator feedback path. The enor mall ignal model i decribed by: 1 T= Kϑϑ( +K ) Y Y ( ) (7) p Figure 4. Block diagram ued to analyze the ytem performance. In which, p = ((a+bϑ n )S+βR o Y o )/mc i the invere of the enor time contant, k ϑ =bϑ n S(T f -T )/mc, and k y =R o /mc are the coefficient aociated to the fluid peed and to the enor quared current Y, repectively. A zero-order holder, A o (), tranform the modulator output ample to a continuou ignal, with T a the overampling period. A o () i given by: T Ao ( ) = ( 1 e ) (8) With the model of the meaurement ytem tranfer function in the z-domain, it i poible to analye the behaviour of the quantization noie frequency pectrum at the Σ- modulator output and conequently the quantization inband noie power in the Σ- converter output. Finally, we can determinate the ignal/noie relationhip (SNR) and the effective reolution of the propoed meaurement ytem. B. Continuou Current Sytem Tranfer Function in the z- domain The velocity tep repone for the mono-bit firt-order Σ- modulator with the thermoreitive enor can be expreed by it z-domain tranfer function a: z r z r F() z = E( z) + ( z) (9) z q z q where, r i the quantizer error tranfer function zero and q i the ytem tranfer function pole. The quantizer noie (error) tranfer function (NTF) ha a zero that depend on the enor mall ignal model pole p, for r=e pt with T being the overampling period. The ytem tranfer function pole q alo depend on thi parameter, and can be foud a k y Yo q= ( 1+r ) +r. p The NTF zero degrade the noie attenuation in the ignal band once there i a finite attenuation at DC frequency intead of an infinite attenuation a in the ideal firt-order igma-delta modulator [7]. The frequency pectral denity magnitude of the propoed meaurement ytem quantization noie can be expreed by: wt E( f ) ( 1 r ) + 4rin Ey ( f ) = (10) wt ( 1 q ) + 4qin The Σ- converter output noie power ignal band with a firt order enor, σ, in the frequency domain, i calculated from Σ- modulator output noie pectral denity, auming that the modulator output ignal ha being filtered by an ideal filter on ignal band frequency. The ignal band noie power depend on the OSR and on the quadratic relationhip involving the NTF zero and pole. Then: σ rm ( 1 r) σ = ( Ey ( f )( df = (11) OSR 1 q ( ) C. Continuou Current Sytem Meaurement Reolution To obtain the ytem theoretical reolution, the ytem noie power wa compared to an N-bit Nyquit PCM noie power and the reolution, in number of bit, i given by (1). Thi expreion how the reolution dependence with OSR -1 and the quadratic relationhip involving the NTF zero and pole. 1 1 ( 1 r ) N= log (1) OSR ( 1 q) A better SNR can be obtained by increaing the modulator quantizer number of bit, by increaing the modulator order, or by limiting the input ignal band under the enor pole frequency. To verify the theoretical reult, the puled current ytem wa imulated in the time domain for a ine wave peed input. The ytem reolution i improved by half bit every time that the overampling ratio i doubled. IV. SIMULATION RESULTS The enor characteritic ued in the theoretical analyi and in the ytem imulation were: β = 0.000784 o C -1, R 0 = 10 Ω, S = 4x10-9 m, mc = 9x10-1 J C -1. The enor temperature theoretical operation point wa defined at 80 o C and the fluid peed range wa defined from ϑ min =0 to 0 m/. The ignal band frequency wa choen to be near ytem tranfer function pole frequency, f B = 0.9f r. Simulation were made for the continuou current model uing the developed analye preented in the previou ection, and are hown in Figure 5 to 7. Simulation for the puled current model, hown in Figure 3, were made and the reult are compared
with continuou model reult, a preented in Figure 8 and in Table 1. Figure 5 how the theoretical ytem output noie magnitude for the overampling ratio equal to 56, zoomed into the ignal band frequency. At DC there i a finite attenuation (-38.9 db). Thi attenuation i 35.9 db and 3 db at the NTF zero and pole frequencie, repectively, which limit the ytem application in the ignal band frequency. Figure 6 how the ytem NTF zero and pole location in the z-plane. Figure 7 how theoretical ytem reolution, obtained from (1), a a function of the OSR. The ytem reolution i improved by half bit every doubling of the overampling ratio. To verify theoretical reult the puled current ytem wa imulated in the time domain for a inuoidal fluid peed, expreed by (13), covering the full meaurement range. The ine wave frequency wa elected to be maller than the enor mall model pole frequency. ϑ(t)=[10+10 in (πt/10 5 )] (m/) (13) The fluid peed wa etimated, through imulation, from the data at the Σ- modulator output, uing a digital filter a preented in [8]. Figure 8 how the etimated fluid peed abolute error for the puled current ytem, for the fluid peed full range of 0 m/, and diregarding the dynamic reolution lot around the poitive peak of the etimated fluid peed due to the fluid temperature mathematical compenation. Thee abolute error are hown for the OSR equal to 64, 18 and 56. The mean quared error for the puled current ytem wa obtained from etimated fluid peed ample at the end of the converter, after the tabilization, and wa calculated by Na 1 σ = [ ϑn () i ϑ() i ] (14) ϑmax ϑmax Na 1 where N a i the number of ample, ϑ n i the etimated fluid peed at the output of the converter and ϑ i the fluid peed at the input of the converter. The etimated fluid peed mean quared error wa Figure 5. Sytem noie tranfer function magnitude. Figure 7. Sytem reolution. Figure 6. Sytem NTF zero-pole located in z-plane. Figure 8. Puled current ytem: etimated abolute error.
compared with the N-bit Nyquit PCM noie power to obtain ytem reolution in time domain. The ytem reolution wa given by equation (11): Na 1 1 [ ] N = 0.5log ( ) ( ) n i i ( max min ) ϑ ϑ ϑ ϑ N (15) a 1 Table 1 how ytem reolution reult obtained for three overampling ratio value. The firt column refer to ytem reolution reult calculated from (1) and the two column refer to the puled current ytem time domain imulation reult that were calculated from (15). The puled current ytem reolution value are wort due to the uncompenated environment temperature, when comparing to theoretical ytem. Table 1. Theoretical and Puled current ytem imulated meaurement reolution reult. Reolution (number of bit) OSR Theoretical Sytem Puled current Sytem 64 9.7 6.4 18 10. 7.3 56 10.7 8. V. CONCLUSION The puled current fluid peed architecture preented here realized the expected A/D converion with a lower reolution when compared to an ideal 1-bit Σ- modulator. To obtain better meaurement reolution, the input ignal band frequency mut be limited under enor mall ignal tranfer function pole frequency and the modulator output ample mut be filtered at the enor mall ignal pole frequency. The puled current ytem SNR value are lower them for an ideal 1-bit Σ- modulator, which i 1.5 bit for every doubling of the overampling ratio. Thi puled-current fluid peed meaurement architecture doe not need a 1-bit D/A converter in the feedback loop becaue thi function i realized by PWM. Fluid peed meaurement ytem baed on thi ame principle, with compenation of the environment temperature, will be our future objective. VI. ACKNOWLEDGMENT The author wih to thank CAPES, CNPq, CAPES/COFECUB and FAPESQ/PRONEX for financial upport and the award of fellowhip during invetigation period. VII. REFERENCES [1] C. A. Leme, M. Chevroulet, H. Balte, A flexible architecture for CMOS enor interface, Proceeding IEEE International Sympoium on Circuit and Sytem (ISCAS 9), pp. 188-1831 May 199 [] E. R. Riedijk, J. H. Huijing, A mart balanced thermal pyranometer uing a igma-delta A-to-D converter for direct communication with microcontroller, Senor and Actuator, pp. 16-5, volume 37-38, 1993. [3] A. Oliveira, L. S. Palma, A. S. Cota, R. C. S. Freire, A. C. C. Lima, A Contant Temperature Operation Thermoreitive Sigma-Delta Solar Radiometer, 10th IMEKO TC7 International Sympoium on Advance of Meaurement Science, vol. 1, pp 199-04, June 004. [4] R. C. S. Freire, G.S. Deep, C.C. Faria, Electrical equivalence olar radiometer configuration, in: Proc. XI Congreo Braileiro de Automática, vol. 3, 1996, pp. 149 154. [5] L. S Palma,. A. Oliveira, A. S. Cota, R. C. S. Freire,A. C. C Lima,. A Contant Temperature Operation Thermoreitive Sigma-Delta Solar Radiometer. Proceeding 10th IMEKO, 004. v. 1. p. 199-04. [6] A. Oliveira, L. S. Palma, A. S. Cota, R. C. S. Freire, A. C. C. Lima. A Contant Temperature Operation Thermo-Reitive Sigma Delta Tranducer. In: Intrumentation and Meaurement Technology Conference 004, 004, Como. Intrumentation and Meaurement Technology Conference 004, 004. v. 3. p. 1175-1181. [7] J. C. Candy, G. C. Teme, Overampling Method for A/D and D/A Converion, cap: Introduction, Overampling Delta-Sigma Data Converter Theory, Deign and Simulation, IEEE Pre, pp. 1-5. [8] S. Park, Principle of Sigma-Delta Modulation for Analog-to-Digital Converter. Motorola DSP: Strategic Application and Digital Signal Proceor Operation, Motorola, 1998.