AN ELECTROMECHANICAL SIGMA-DELTA MODULATOR FOR MEMS GYROSCOPE. Byung Su Chang*, Jang Gyu Lee*, Taesam Kang **, and Woon-Tahk Sung*

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1 AN ELECTROECHANICAL SIGA-DELTA ODULATOR FOR ES GYROSCOPE Bung Su Chang*, Jang Gu Lee*, Taeam Kang **, and Woon-Tahk Sung* * School of Electrical Engineering and Computer Science, Seoul National Univerit, Seoul, 5-744, Korea ** Department of Aeropace Engineering, Konkuk Univerit, 43-7, Seoul, Korea Abtract: Thi paper preent a deign and anali of electromechanical igma-delta for ES grocope, which enable u to control the proof ma and to obtain an eact digital output without additional A/D converion. The tem tructure and the circuit realization of the igma-delta modulation are impler than thoe of the analog ening and feedback circuit. Baed on the electrical igma-delta theor, a compenator i deigned to improve the cloed loop performance of the enor. With the deigned compenator, a cloed loop performance of the grocope i enhanced to have wider bandwidth, larger dnamic range and direct digital output compared with thoe of an analog open loop tem. Copright 5 IFAC Keword: Grocope, ES, Sigma-delta, Inertial enor, INS. INTRODUCTION The inertial enor are baic element that compoe Inertial Navigation Stem (INS). The grocope i baic inertial enor, which can meaure eternal angular rate. The ES grocope i an inertial angular rate enor fabricated uing ES technolog. It ha man good propertie: it i mall, light, and low power conuming. oreover, it can be ma-produced inepenivel. Therefore, it ha man application in variou field. The grocope i a high-q tem, which reult in mall bandwidth and ha theoreticall nonlinear characteritic. When eternal angular rate i applied to the ES grocope, the proof ma vibrating at reonant frequenc i forced to vibrate in orthogonal direction due to the Corioli force. The angular rate can be etimated b meauring the amplitude of the orthogonal ocillation. However, in thi open loop cae, the dnamic range of grocope i limited. Furthermore, the nonlinearit become larger a the amplitude of the orthogonal ocillation become larger (Sung, et al., 4). To overcome thee diadvantage, a cloed loop controller named rebalance loop can be ued. A rebalance loop i a kind of feedback controller that make the orthogonal ocillation be mall. The magnitude of the feedback control i proportional to the Corioli force due to the angular rate input. Therefore, the control input i ued for the meaurement of eternal angular rate input. Furthermore, the bandwidth of the grocope can be made large b uing uitable compenator. ot of feedback control technique i analog control. Analog feedback ha good performance and high enitivit. However, the implementation of the high performance A/D converter, which i necear for the navigation tem, i ver difficult. Therefore, igma-delta modulation technique, which ha been widel ued in high-reolution A/D and D/A converter (Norworth, et al., 997), ha been tudied to realize high-performance ES enor tem becaue it give precie digital output without A/D converter. It i epeciall attractive in enor application where the ignal are relativel narrow-band and a large over-ampling ratio can be eail achieved. Electromechanical igma-delta feedback, however, i

2 different from electrical igma-delta. Electrical igma-delta conit of pure electric ignal, but electromechanical igma-delta feedback i a ver complicated mied ignal feedback tem. It ha building block in multiple domain, uch a mechanical and electrical, continuou-time and dicrete-time, analog and digital, a well a linear and nonlinear. apping uch an electromechanical igma-delta feedback into pure electrical igma-delta will be helpful in undertanding and deigning. After tem mapping, the cloe loop tem i well deigned uing the etablihed theor of conventional igma-delta. In thi paper, model equation of ES grocope are obtained and an electromechanical igma-delta i deigned via electrical theor. A imulation uing ATLAB Simulink i performed to verif the performance of the cloed loop rebalance loop with deigned. The reult how the deigned rebalance loop uing improve bandwidth a well a it give digital output ignal, directl. function of the ening mode can be epreed a (4) and (5), repectivel. + + = t C t K t F t G corioli ene ( ) ( ) Y Fcorioli + + Fig.. Simple model of ES vibrating grocope C K (4) (5). ODEL EQUATION OF ES GYROSCOPE ES grocope i generall compoed of a ma and beam that i deigned to ocillate. The ma readil ocillate in two orthogonal direction. A imple model of a vibrating grocope i hown in Fig.. The governing equation of the vibrator grocope can be epreed uing equation ()-(6). To ue Corioli force, the proof ma i driven at it reonance frequenc. When the proof ma i driven b the electrotatic force or F driving (t) along the -ai (driving ai), the equation of the dnamic motion of the ma can be impl epreed a a econd order ma-damper-pring tem. + + = t C t K t F t driving () Where i the ma of the moving plate, C, the da mping coefficient of the air-damper and K i the pri ng contant along the driving ai. Then the tranfer f unction from the driving force to the diplacement of the plate along -ai i given b G drive ( ) X F C + + K () If the angular rate input Ω z (t) i applied along z-ai (input ai), Corioli force i induced, which i given b (3). () = () Ω () F t t t corioli z (3) Then, the equation of the dnamic motion along - ai (ening ai) i obtained b the ame manner in -ai cae. The dnamic equation and the tranfer 3. SIGA-DELTA ODULATOR 3. Electrical igma-delta theor Sigma-delta modulation technique wa firt introduced b Inoe and Yauda (Inoe and Yauda, 963) and it became more popular for uing a a high reolution A/D and D/A converter. Since thi tem contain a delta and an integrator, the are named igma-delta, where igma denoted the ummation performed b integrator. The igma-delta conit of a coare (low reolution) quantizer preceded b a filter, both embedded into a negative feedback loop. It ue overampling and quantization error haping to achieve high accurac. In other word, it trade peed for reolution, and analog circuit accurac for imple digital circuit. To rigoroul analze a igma-delta in the frequenc domain i a difficult tak due to the preence of the nonlinear quantizer. If quantization error i uncorrelated with the input ignal and i uniforml ditributed in quantizer output band, and i an independent identicall ditributed (iid.) equence, then thi anali can be implified a the input independent additive white noie approimation for the quantization error (Norworth, et al., 997; Ki, ). Therefore, the determinitic nonlinear tem i replaced b a linear tochatic tem. The linearized model of the igma-delta i preented in Fig.. Then, tranfer function from input U(z) and quantization error E(z) to output V(z) are epreed a (6) = ( ) + H( z) = + + H( z) + H( z) V z U z V z H z E z V z U z E z (6)

3 Fig.. Linearized model of igma-delta The block diagram of the igma-delta feedback loop i hown in Fig. 4. The mechanical tranducer i embedded in the loop. In addition, a compenator for improving performance i added in the loop. Thi i repreented in multiple domain, different from the electrical cae. Poition ening part convert the diplacement to the voltage ignal, and actuator in feedback loop convert the voltage ignal to the feedback force. From (6), it reult that the igma-delta proce i independentl eparated into the ignal and the noie component. Therefore, it ignal tranfer function, STF(z) and noie tranfer function, NTF(z) can be defined a follow. STF z NTF z H ( z) H ( z) V z U z + E z = H ( z) V z E z U z = + (7) Alo, one can write the output ignal V(z) a the combination of the input ignal, U(z) and the quantization noie ignal, Q(z). Each of them i filtered b the correponding tranfer function, V ( z) = STF( z) U ( z) + NTF( z) Q( z). (8) If one chooe a low-pa loop filter H(z), which ha large magnitude over the low frequencie of interet ignal band (alo called ignal band or in band or baeband ), and mall magnitude (large attenuation) over high frequencie (alo called out of band ), then the magnitude of the ignal tranfer function STF(z) will be approimated into unit over the ignal band, hence it will not ditort the ignal. However, the magnitude of the noie tranfer function, NTF(z) will be approimated into zero over the ame band, hence the quantization noie will be reduced ignificantl. B doing o, the ignalband pectral compoition of the analog input and digital output ignal will be linearl related, but outide the ignal band, the pectral compoition of them will differ ubtantiall. An eample of frequenc repone of igma-delta i hown Fig. 3. Fig. 3. STF and NTF of conventional igma-delta 3. Electromechanical force feedback b igma-delta Fig. 4. Block diagram of electromechanical igmadelta For ea anali, mapping of electromechanical igma-delta into a pure electrical tem will be helpful in undertanding and deigning cloed loop grocope tem. The mechanical tranducer, which i embedded in the loop, ha nd order tranfer function. That i repreented a (5). Since igmadelta i dicrete tem, it i necear to convert continuou tranfer function into dicrete one. Aume that ampling period i T and ampling model i zero-order-hold model (Oppenheim and Schafer, 999), then the dicrete tranfer function of mechanical tranducer i repreented a (9). z G( z) = Z G( ) z Az + B = at at K z e ( cobt ) + e C K a =, b = a at a at A = e cobt e inbt b at a at at B = e + e in bt e cobt b (9) Becaue bandwidth of poition ening part and actuator are much higher than mechanical tructure, the poition ening part and the actuator are modelled a a DC gain block, K and K f repectivel. Then, a loop filter of electromechanical igma-delta i epreed a (). G z K C z () The linearized model of the electromechanical igma-delta i preented in Fig. 5. Therefore, the NTF and STF are calculated a (), (). NTF z = + KKGzC f ( z) ()

4 STF z KG z C z = + KKG f zc z () Fig. 5. Linearized model of electromechanical igmadelta 4. DESIGN OF ELECTROECHANICAL SIGA- DELTA ODULATOR Now, a compenator block i deigned to improve the reolution of the cloed loop tem. Firt of all, an intereted ignal band hould be determined. In ES grocope tem, driving ignal i inuoidal wave whoe frequenc i a reonance frequenc of the ma-damper-pring tem. Then the eternal angular rate ignal i modulated b the driving ignal. Since the bandwidth of conventional grocope i about Hz, the intereted ignal band i around the reonance frequenc with additional ±Hz. 4. Bandpa igma-delta theor In the cae of ordinar igma-delta, the intereted ignal band i a low frequenc region. Therefore, the loop filter of the ha a frequenc repone characteritic like a low pa filter. In thi cae, the intereted ignal band i around the reonance frequenc. Therefore, the loop filter of thi cae ha a bandpa propert. Bandpa igma-delta operate in much the ame manner a conventional (low pa). The deign of bandpa converter ha much in common with low pa deign (The magnitude of the ignal tranfer function STF(z) will be approimated into unit over the ignal band, the magnitude of the noie tranfer function NTF(z) will be approimated into zero over the ame band). An eample of frequenc repone of bandpa igma-delta i hown in Fig Deign of compenator Firt, NTF ntheize proce i accomplihed. To reduce quantization noie around the reonance frequenc, NTF i deigned a band reject filter via a generalized filter approimator/optimizer (Jantzi, et al., 994). The optimizer adjut the pole and zero of NTF uch that it magnitude of frequenc repone cloel matche the meaure of ideal value. In thi paper, the 6 th order NTF i deigned. Fig. 7 repreent Bode plot of deigned NTF. It how that the deigned NTF reject the quantization noie in ignal band. Then, compenator tranfer function C(z) i calculated b (). Table how the parameter of the ES grocope. The Bode plot of deigned compenator i hown in Fig. 8. After obtaining the compenator, STF i calculated b (). The bode plot of cloed loop tem and open loop tem. From Fig. 9, the diplacement of the ma along ening ai i regulated well. oreover, Fig. how that the cloed loop tem ha wider bandwidth and lower phae ditortion then the open loop cae. Table Parameter of ES grocope Parameter Value Qualit factor 5 Reonant frequenc rad/ec a Kg Spring contant K N/m Damping coef. C N/m/ec K.5 8 V/m.5-9 N/V K f ignal band bandwidth Hz Fig. 7. Bode plot of deigned NTF Fig. 6. STF and NTF of bandpa igma-delta

5 5 5. DECIATION FILTER AND DEODULATOR Fig. 8. Bode plot of deigned compenator Phae (deg) SD cloe loop open loop SD cloe loop open loop Fig. 9. Bode plot of open loop and cloed loop tem 5. Decimation filter The deigned igma-delta i onl half of the whole grocope tem. Becaue the output ignal i not eternal angular rate ignal but overampled -bit digital pule ignal, the ignal i digitall converted to the deired form through a digital filtering and reampling proce called decimation. To recontruct the digitized ignal with high SNR, the out-of-band quantization noie mut be rejected. If deign proce i accomplihed well, the quantization noie in the ignal band i minimized. It i then left to decimator to filter out the unwanted noie in out-of-band region o that it i not aliaed into the ignal band b the decimation proce. Let u then dicu traditional decimator deign method uch a inc K filter. Thi method i ver attractive for implementation becaue it doen t require multiplier, but adder and regiter onl (Hoegenauer, 98; Norworth, et al., 997). The tranfer function for inc K decimation filter ha the general form a (3), H deci R z = z K (3) where R i decimation factor which repreent reampling rate to remove all ignal energ above π / R frequenc range. K i the order of the inc K filter. In order to prevent aliaing, K hould be at leat plu the order of the The mot efficient implementation i formulated b cacading K tage of accumulator operation at the high ampling rate, followed b K tage of cacaded differentiator operating at the low ampling rate (Dijktra, et al., 988). Fig. how a block diagram of the 3 rd order inc K filter. -45 Phae (deg) (a) Open loop Fig.. Block diagram of the 3 rd decimation filter order inc K Phae (deg) (b) Cloed loop Fig.. Bode plot of open loop and cloed loop tem: magnified view around the ignal band 5. De After the decimation proce, we obtain ening ai diplacement ignal, which i converted to digital form. Since the diplacement along ening ai due to the Corioli acceleration i modulated b the driving ignal, the demodulation proce i eentiall needed to etract the eternal angular rate ignal. Becaue the modulated ignal i converted to digital form, de can be implemented in oftware. In addition, analog electronic part uch a analog multiplier and phae hifter are not needed.

6 6. SIULATION RESULT Uing the deigned tem and parameter of the grocope, a imulation i performed b ATLAB Simulink. Fig. how diplacement output of ma when the grocope operate in open loop. That i the form, whoe input angular rate are modulated inuoidal driving ignal. Fig. 3 repreent diplacement output when the grocope operate with electromechanical igmadelta feedback. Firt plot how output with compenator and econd plot how output without compenator. From the reult, it i confirmed the compenator much enhance the performance of the igma-delta feedback loop. Fig. 4 how that the output of whole grocope tem when deg/ec tep input i applied. From thi reult, decimation filter and de operate a deired. 7. CONCLUSION In thi tud, an electromechanical igma-delta for ES grocope i propoed, which enable u to control the proof ma and to obtain a digital output directl. Baed on electrical igmadelta theor, deigned compenator improve cloed loop performance uch a larger bandwidth and digital output comparing with the reult of analog open-loop tem. oreover the dnamic range of grocope i improved, ince diplacement along ening ai i well regulated Fig.. Diplacement of the ma in open loop tem with compenator without compenator ample Fig. 3. Diplacement of the ma when the igmadelta cloed loop grocope operate with and without compenator eternal angular rate (deg/ec) time (dec) Fig. 4. Output of whole grocope tem when deg/ec tep input i applied at. ec ACKNOWLEDGEENTS Thi work ha been upported b the BK SNU-KU Reearch Diviion for Information Technolog, Seoul National Univerit, Automation and Stem Reearch Intitute (ASRI) in Seoul National Univerit, the Korea Reearch Foundation Grant (KRF B47), and Aeropace Technolog Development Project. REFERENCES Dijktra, E., N, O., Piguet, C. and Degrauwe,. (998). On the ue of modulo arithmetic comb Filter in igma delta, IEEE proc. ICASSP 88, -4. Hoegenauer, E. G. (98). An economical cla of digital filter for decimation and interpolation, IEEE Tranaction on Acoutic, Speech and Signal Proceing, vol. ASSP-9, Inoe, H. and Yauda, Y. (963). A unit bit coding method b negative feedback, Proc. IEEE, vol. 5, Jantzi, S., Ouli, C. and Sedra, A.S. (994). The deign of tranfer function for delta-igma converter, Proc. 994 IEEE Int. Smp. Circuit St., vol.5, Jiang, X. and Bhave, S. A. (). Σ Capacitive interface for a verticall-driven X&Y-ai rate grocope, in 8th European Solid-State Circuit Conference, Ki, P. () Deign guide of high-order deltaigma - an empirical tud, Technical memorandum # , Agere Stem, New Jere: urra Hill. Norworth, S., Scherier, R. and Teme, G. C. (997). Delta-Sigma Data Converter theor, deign, and imulation, IEEE pre, New York. Oppenheim, A. V. and Schafer, R. W. (999). Dicrete-time ignal proceing, Prentice Hall, New Jere. Sung, W.T., Lee, J.G., Song, J.W. and Kang, T. (4). H Controller deign of ES grocope and it performance tet. Proc. IEEE PLANS 4,