A Simple Approach to Control the Time-constant of Microwave Integrators

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1 5 VOL., NO.3, MA, A Simple Approch to Control the Time-constnt of Microwve Integrtors Dhrmendr K. Updhyy* nd Rkesh K. Singh NSIT, Division of Electronics & Communiction Engineering New Delhi-78, In Tel: ; Fx: -599; E-mil: Abstrct- In this pper, firstly the time-constnt of n integrtor is described in continuous s well s in the screte-time domins. Further, simple reltion is expressed to find the time-constnt of n integrtor from continuous-time domin to screte-time domin or vice-vers. Therefter, the microwve integrtors of fferent time-constnts re obtined by optimiing the chrcteristic impednce of n open-circuited shunt stub using liner progrmming pproch in -domin. It is cler tht ll the microwve integrtors of fferent time-constnts re relied only by using single open-circuited shunt stub. Index Terms- Digitl integrtor, microwve integrtor, open-circuited shunt stub, optimition, time-constnt. I. INTRODUCTION The integrtors re very useful systems to estimte the time integrls of mesured signls. These systems hve lot of pplictions for wveform shping, ccumultor nlysis, coherent detection, control systems etc. It is noticed tht the integrtors re generlly designed for systems of low-frequency pplictions nd rrely for high-frequency or microwve pplictions. The frequency response of n idel integrtor is defined by K Hi ( jω) ( jω) () Where j, K is scling constnt nd ω is the ngulr frequency in rns per second. A series resistor-cpcitor (RC) circuit, in conjunction with n opertionl mplifier, hs been generlly used to design n integrtor. However, such type of integrtor is useful only for low-frequency pplictions. It is interesting to note tht lot of reserch work is vilble in the re of gitl integrtors using recursive systems. Initilly, the recursive gitl integrtors hve been designed by tking the Z- trnsform of existing integrtion rules. Further, these re obtined by performing simple liner interpoltion between the mgnitude responses of clssicl rectngulr, trpeoidl nd Simpson gitl integrtors [-3]. Therefter, liner progrmming optimition pproch hs been proposed to design recursive gitl integrtors []. In [5], Ngo hs proposed recursive widebnd gitl integrtor of third-order system bsed on Newton-Cotes integrtion rule. Further, Gupt-Jin-Kumr hve proposed recursive widebnd gitl integrtors for comprtively lower percentge reltive errors (PREs) in mgnitude responses over widebnd with the idel integrtor [6]. Lter, Al-Aloui hs lso proposed novel clss of -segment, optimied 3-segment nd optimied -segment recursive gitl integrtors [7]. Recently, Updhyy et l. hve designed the recursive widebnd gitl integrtors for lower reltive errors in mgnitude responses by using coefficient nd pole-ero optimitions [8-]. It is noted tht the gitl integrtors cnnot be relied prcticlly in the frequency rnge of GH or microwve rnge by using the dder, multiplier nd dely circuits. Therefore, the designing of microwve integrtors is the emerging field in current reserch environment. IJMOT IAMOT

2 6 VOL., NO.3, MA In [], Hsue-Tsi-Kn hve designed the firstorder trpeoidl rule bsed microwve integrtor. Further, Hsue-Tsi-Tsi hve proposed the time-constnt control of microwve integrtors by cscng the fferent number of trnsmission line sections []. The combintion of time-constnt nd mgnitude response describe the system behvior of n integrtor. Lter, Tsi-Fng hve designed the second-order microwve integrtors by using two section open-circuited shunt stubs nd the seril trnsmission line sections [3]. It is noticed tht the existing Hsue-Tsi-Tsi (HTT) pproch for controlling the time-constnt of microwve integrtors is not simple pproch []. This pproch requires uto regression moving verge (ARMA) process in -domin. Further, this pproch hs the need of six to ten trnsmission line sections to design microwve integrtors of required timeconstnts. The sies of microstrips Lyouts for microwve integrtors of fferent time-constnts re pproximtely from.5 cm to 8.5 cm []. Therefore, there is the need to develop simple pproch for controlling the time-constnt of microwve integrtors with the constrint of compct design formultion. In this pper, simple pproch is proposed to control the time-constnt of microwve integrtors. All the designed integrtors of fferent time-constnts require trnsmission line i.e. shunted with single open-circuited stub. Section II describes the time-constnt of n integrtor. This section lso provides reltion to relte the time-constnt of n integrtor in continuous-time nd screte-time domins. Further, nlysis of microwve integrtor nd time-constnt control re given in section III. Therefter, results re scussed in section IV. Finlly, conclusions re given in section V. II. TIME-CONSTANT OF AN INTEGRATOR Severl techniques hve been developed to design recursive gitl integrtors in the study of gitl signl processing (DSP). In [], Hsue- Tsi-Tsi (HTT) hve designed new recursive gitl integrtor by pplying the liner interpoltion over trpeoidl integrtion rule nd the inverse of wide-bnd Hsue-Tsi-Chen (HTC) fferentitor []. The trnsfer function of HTT integrtor is given below Hi ( ) () Fig. shows the mgnitude response of HTT gitl integrtor with the idel integrtor of time-constnt 3.8 over the full Nyquist bnd. Mgnitude X:.35 Y:.3 HTT Idel Normlied frequency Fig.. Mgnitude response of HTT integrtor with idel integrtor ( 3.8) In [], HTT hve lso defined the time-constnt ( i ) of n integrtor by/( H( jω) Ω ) in continuous-time domin, where H ( jω) is the trnsfer function of n integrtor in frequency domin nd Ω is signl ngulr frequency. Similrly, the time-constnt ( ) of n integrtor j cn lso be defined by/( He ( ω ) ω ) in j screte-time domin, where H ( e ω ) is the IJMOT IAMOT

3 7 VOL., NO.3, MA trnsfer function of gitl integrtor nd ω is the normlied frequency of screte-time signl. Now the time-constnt of HTT gitl integrtor is determined t prticulr normlied frequency using Fig.. The procedure to determine the time-constnt is given below. 3.8 j He ( ω ) ω.3(.35 π) (3) It is cler in Eq. 3 tht the time-constnt of HTT integrtor is 3.8 for normlied frequency ω.35π. The percentge reltive error ( PRE ) of ny gitl integrtor in mgnitude response with the idel integrtor of corresponng time-constnt is defined by jω H( e ) PRE () Fig. shows the PRE of existing HTT gitl integrtor in mgnitude response over the full Nyquist bnd with the idel integrtor of timeconstnt 3.8. From Fig., it is observed tht the HTT integrtor hs mximum PRE of 3.86% in mgnitude response over the full Nyquist bnd with the idel integrtor of time-constnt 3.8. Now simple reltion is expressed to convert the time-constnts of n integrtor from screte-time to continuous-time domin or vice-vers s in [5]. The generlied reltion is defined s i H( j π f ) π f Y ( mgnitude) X ( frequency) f [ π ] (5) Where f is the normliing or mximum operting frequency. If f GH, then the time-constnt of existing HTT gitl integrtor ( 3.8) will be.59 ns in continuous-time domin. III. ANALYSIS OF MICROWAVE INTEGRATOR AND TIME CONSTANT CONTROL In [6, 7], the chin scttering prmeters of trnsmission line i.e., shunted with n opencircuited stub is defined in the -domin. These prmeters re given below in mtrix form. Reltive error (% ) Normlied frequency Fig.. PRE of HTT gitl integrtor with the idel integrtor ( 3.8) T T ( + c') + ( c') c'( ) (6) T T + + c' c ' ( c') + ( + c') Therefore, the trnsfer function of trnsmission line i.e., shunted with n open-circuited stub cn be written s T oc + ( ) T ( ) ( + c') + ( c') (7) Where c ' Z /( Z ) ; Z is the chrcteristic impednce of open-circuited shunt stub, Z is the chrcteristic impednce of min trnsmission line. IJMOT IAMOT

4 8 VOL., NO.3, MA It is noticed tht the system function Toc () behves s low-pss filter. Therefore, the microwve integrtor of required time-constnt ( i ) in continuous-time domin or fi in screte-time domin my be designed by using trnsmission line i.e. shunted with n open-circuited stub. The PRE of generl opencircuited shunt stub in mgnitude response with the idel integrtor of required time-constnt f ) is defined by ( i jω Toc ( e ) PRE (8) Now, the microwve integrtor of required timeconstnt ( i ) is obtined by minimiing the mximum PRE over specific Nyquist bnd using the liner progrmming pproch []. In this minimition, c ' is used s the optimition prmeter. In prticulr, the time-constnt of.53 ns (.6, if f GH) is chieved for the vlue of optimied prmeter c '.3 over specific Nyquist bnd. The trnsfer function of proposed microwve integrtor for the time-constnt of.53 ns (.6), thus obtined is given in (9). + Ti ( ).3.3 (9) Notice tht Z is.6ω, if Z is 5Ω. This revels tht trnsmission line i.e., shunted with n open-circuited stub of Z.6Ω nd length λ L (where λ is the wvelength i.e. corresponng to the normliing or mximum operting frequency of GH) cn be employed to design microwve integrtor of time-constnt (.53 ns) over specific operting bnd. This pproch is lso pplied to design microwve integrtors of fferent timeconstnts. Tble shows the suitble vlues of optimied prmeters ( c ') or chrcteristic impednces ( Z ) of open-circuited shunt stubs for designed microwve integrtors of fferent time-constnts ( i ). Fig. 3 shows the design formultion of microwve integrtors for fferent time-constnts. Tble : Suitble vlues of chrcteristic impednces of open-circuited stubs for designed microwve integrtors of fferent time-constnts, Microwve integrtors i in ns d i f i f GH Optimum vlue of c ' Z in Ω for Z 5 Ω Z L Z Z Fig.3. Proposed design formultion of microwve integrtors Now, the trnsfer function S ( f ) of proposed design formultion of microwve integrtors is obtined in terms of the optimied prmeter ( c ') nd the physicl length of open-circuited stub ( L ) by using the trnsmission line prmeters concept [8-]. The trnsmission line mtrix for open-circuited shunt stub is defined s A B jtn( βl) C D Y Z () IJMOT IAMOT

5 9 VOL., NO.3, MA, The scttering prmeter S of ny system reltes with the trnsmission line prmeters s S () A + ( B/ Z ) + CZ + D Therefore, the system function S of proposed design formultion for microwve integrtor thus obtined is given below. Fig. shows the mgnitude responses of designed microwve integrtors, nd in -domin with the idel integrtors of required time-constnts ( ) s.6,.6 nd.59, respectively. Further, Fig. 5 shows the PREs of designed microwve integrtors in mgnitude responses with the idel integrtors of required time-constnts ( ) s.6,.6 nd.59, respectively. S ( f ) π f + jc 'tn c ε eff L () Where c ' Z /( Z ), c is the velocity of light, f is the operting frequency, L is the physicl length of open-circuited shunt stub nd ε eff is the effective electric constnt of microstrips structure. Let the physicl length of microstrips structure is L λ /( ε ) eff (where λ is the wvelength nd it is corresponng to the normliing frequency of GH). IV. RESULTS AND DISCUSSION Reltiv e error (% ) Normlied frequency Fig.5. PREs of designed microwve integrtors with the idel integrtors of required time-constnts ( ) Mgnitude Idel-.6 Idel-.59 Idel Normlied frequency Fig.. Mgnitude responses of designed microwve integrtors in -domin with the idel integrtors From Figs. nd 5, it is observed tht the designed integrtors, nd hve not more thn 6% reltive error in mgnitude responses over the specific Nyquist bnds. Further, it is observed tht the lrger timeconstnt integrtor design () hs more Nyquist bndwidth s compred to the smller time-constnt integrtor design (). It is lso interesting to note tht the lrger time-constnt integrtor design is more preferble in lower bnd s compred to the smller time-constnt integrtor design. Fig. 6 shows the mgnitude responses of S ( f ) for designed, nd microwve integrtors with the idel ones of corresponng time-constnts ( i ). Further Fig. 7 shows the IJMOT IAMOT

6 VOL., NO.3, MA PREs of these designed integrtors with the idel ones of corresponng time-constnts ( i ). M gnitude (S) Idel-.6 Idel-.59 Idel Frequency (GH) Fig.6. Mgnitude responses (S) of designed microwve integrtors with the idel integrtors of corresponng time-constnts ( i ) Reltive error (% ) Frequency (GH) Fig.7. PREs of designed microwve integrtors with the idel integrtors of time-constnts ( i ) From Figs. 6 nd 7, it is observed tht the designed, nd microwve integrtors hve not more thn 6% reltive error in mgnitude responses over the specific operting bnds. Further, it is observed tht the microwve integrtor hs the time-constnt of.53 X - second over the frequency rnge of 3. GH to 6.7 GH, the integrtor hs the time-constnt of.795 X - second over the frequency rnge of.6 GH to 6. GH nd the microwve integrtor hs the timeconstnt of.59 X - second over the frequency rnge of.5 GH to 5. GH. Therefore, it is cler tht the microwve integrtor of lrger time-constnt hs more operting bndwidth s compred to the microwve integrtor of smller time-constnt. It is interesting to note tht the lrger time-constnt integrtor design () is useful in lower bnd s compred to the smller time-constnt integrtor design. It is lso cler tht the microwve integrtor of required time-constnt for bnd limited pplictions cn be esily designed by using the single open-circuited shunt stub. V. CONCLUSION A simple pproch is presented to control the time-constnt of microwve integrtors. This pproch is much simpler in comprison with the existing pproch nd it requires single trnsmission line element i.e. shunted with n open-circuited stub to control the time-constnt. It is shown tht the microwve integrtor of lrger time-constnt hs more operting bndwidth s compred to the integrtor of smller time-constnt. The min drwbck of proposed pproch is the smller operting bndwidth for smller time-constnt. However, the proposed pproch is preferble for lrger time-constnt over lower-bnd. ACKNOWLEDGMENT The uthors re highly grteful to Prof. Rj Senni (Director, NSIT Dwrk, New Delhi, In) nd Prof. D. S. Chuhn (Vice-Chncellor, UTU Dehrdun, In) for encourging the reserch ctivities. IJMOT IAMOT

7 VOL., NO.3, MA REFERENCES [] M.A. Aloui, Novel gitl integrtor nd fferentitor, Electron. Lett., Vol. 9, No., pp , Feb [] J.L. Bihn, Novel clss of gitl integrtors nd fferentitors, Electron. Lett., Vol. 9, No., pp , My 993. [3] M.A. Aloui, Novel IIR fferentitor from the Simpson integrtion rule, IEEE Trns. Circuits Syst. I, Fundm. Theory Appl., Vol., pp , Feb. 99. [] N. Ppmrkos nd C. Chms, A new pproch for the design of gitl integrtors, IEEE Trns. Circuits Syst. I, Fundm. Theory Appl., Vol. 3, pp , Sept [5] N.Q. Ngo, A New Approch for the Design of Widebnd Digitl Integrtor nd Differentitor, IEEE Trnsctions on circuits nd systems II, Vol. 53, pp , Sept. 6. [6] M. Gupt, M. Jin nd B. Kumr, Novel clss of stble widebnd recursive gitl integrtors nd fferentitors, IET Signl Process., Vol., pp , Oct.. [7] M.A. Aloui, Clss of gitl integrtors nd fferentitors, IET Signl Processing, Vol. 5, pp. 5-6, Mr.. [8] D.K. Updhyy, Recursive widebnd gitl fferentitors, Electron. Lett., Vol. 6, pp , Dec.. [9] D.K. Updhyy nd R. K. Singh, Recursive widebnd gitl fferentitor nd integrtor, Electron. Lett., Vol. 7, pp , My. [] D.K. Updhyy, Clss of recursive widebnd gitl fferentitors nd integrtors, Ro Engineering Journl, Vol., pp. 9 9, Sept.. [] C.-W. Hsue, L.-C. Tsi nd S.T. Kn, Implementtion of trpeoidl-rule microwve integrtor, Microw. And Opt. Tech. Lett., Vol. 8, pp. 8-85, Apr. 6. [] C.-W. Hsue, L.-C. Tsi nd Y.-H. Tsi, Timeconstnt control of microwve integrtors using trnsmission lines, IEEE Trns. Microw. Theory Tech., Vol. 5, pp. 3-7, Mrch 6. [3] L.-C. Tsi nd H.-S. Fng, Design nd implementtion of second-order microwve integrtors, Microw. And Opt. Tech. Lett., Vol. 53, pp , Sept.. [] C.-W. Hsue, L.-C. Tsi nd K.-L. Chen, Implementtion of first-order nd second-order microwve fferentitors, IEEE Trns. Microw. Theory Tech., Vol. 5, pp. 3-7, My. [5] D.K. Updhyy nd R. K. Singh, A simple pproch to control the time-constnt of microwve fferentitors, IJMOT., Vol. 7, pp , Jn.. [6] D-Ching Chng nd C.-W. Hsue, Design nd implementtion of filters using trnsfer function in the -domin, IEEE Trns. Microw. Theory Tech., Vol. 9, pp , My. [7] T.-R. Cheng nd C.-W. Hsue, High-speed wveform shping using non-uniform lines nd Z trnsform technique, Microwves, Antenns nd Propgtion, IEE Proceengs, Vol. 5, pp , Apr. 3. [8] D.A. Frickey, Conversions between S, Z, Y, h, ABCD nd T prmeters which re vlid for complex source nd lod impednces, IEEE Trns. Microw. Theory Tech., Vol., pp. 5-, Feb. 99. [9] R.B. Mrks nd D.F. Willims, Comments on Conversions between S, Z, Y, h, ABCD nd T prmeters which re vlid for complex source nd lod impednces, IEEE Trns. Microw. Theory Tech., Vol. 3, pp. 9-95, Apr [] D.M. Por, Microwve Engineering, 3 rd ed., Wiley, Singpore, 5. IJMOT IAMOT

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