Acoustic impedance characteristics of linear compressors *

Transcription

1 9 Gn t l / J Zhjing Univ-Sci A (Appl Phys & Eng) (7):9-5 Journl of Zhjing Univrsity-SCIENCE A (Applid Physics & Enginring) ISSN X (Print); ISSN (Onlin) wwwzjuducn/jzus; wwwspringrlinkco E-il: jzus@zjuducn Acoustic ipdnc chrctristics of linr coprssors * Zhi-hu GAN, Long-yi ANG, Shng-ying ZHAO, Yu-jing SONG, i-wi ANG, Yi-nong U ( Institut of Cryognics nd Rfrigrtion, Zhjing Univrsity, Hngzhou 7, Chin) ( School of Infortion nd Elctricl Enginring, Zhjing Univrsity City Collg, Hngzhou 5, Chin) ( Shnghi Institut of Tchnicl Physics, Chins Acdy of Scincs, Shnghi 8, Chin) E-il: zhosy@zuccducn Rcivd Spt, ; Rvision ccptd Jun 7, ; Crosschckd Jun, Abstrct: Th coustic fild of linr coprssor srvs to dlivr th coprssion work to th lod, such s th connctd cold hd of cryocoolr; it plys n quivlntly iportnt rol s th lctricl nd chnicl prts, spcilly in th ipdnc tch issu This ppr studis th coustic ipdnc chrctristics of linr coprssor Th prtrs including th currnt, th piston displcnt, th prssur plitud, th lctricl powr dissiption, th powr fctor, th prssur-volutric (PV) powr dlivrd, nd th fficincy r thorticlly nd xprintlly invstigtd Diffrnt fro prvious thorticl studis, optiiztion for th oprtions wy fro th rsonnc is lso includd Mor gnrl optiiztion rsults iply rlvnc btwn throcoustic ngins nd linr coprssors Th prdictd rsults r vlidtd by th xprints prford on linr coprssor with n djustbl rsistiv-cpcitiv (RC) coustic lod Th coprisons btwn th clcultions nd th surnts r prsntd nd nlyzd Th rsults provid dpr insight into th chnis of th linr coprssor nd th ipdnc tch in cryocoolr syst Ky words: Linr coprssor, Acoustic ipdnc, Rsistiv-cpcitiv (RC) lod doi:6/jzusa Docunt cod: A CLC nubr: TB65 Introduction Th linr coprssor posssss th dvntgs of oil-fr, high rlibility, low vibrtion, nd long-lif oprtion Sinc bing introducd for th first ti in 98 (Dvy, 98), thy r oftn usd to driv cryocoolrs to nsur long-lif nd high-rlibility (Nst t l, 6; Rb nd Twrd, ) Th oprtion of th linr coprssor copriss coplx trnsition procsss ong th lctricl, chnicl, nd coustic filds Prvious studis rportd th dynic chrctristics of linr coprssors Howvr, th coustic ipdnc on th piston ws odld using sipl Corrsponding uthor * Projct supportd by th Ntionl Nturl Scinc Foundtion of Chin (No 57665), nd th Opn Projct Progr of th Ky Lbortory of Infrrd Iging Mtrils nd Dtctors (No IIMDKFJJ--7), Chin Zhjing Univrsity nd Springr-Vrlg Brlin Hidlbrg quivlnt dping cofficint nd gs spring stiffnss (Koh t l, ; Prk t l, ; Chn t l, 7; Chn nd Zhu, ) For th ppliction in cryognic filds, th coustic fild is quivlntly iportnt bcus it dlivrs th coprssion powr to th lod, which is usully connctd cold hd (Rdbugh t l, ) Th ipdnc tch btwn th is ky to optiizing th syst prfornc Thrfor, quntittiv clcultions of th coustic chrctristics nd thir coupling with th lctric nd chnicl filds r fundntl klnd () usd n quivlnt circuit odl to nlyz th prfornc nd optiizd th coustic rsistnc R undr th rsonnt condition H found tht th optiu fficincy of linr coprssor cn b obtind whn R =σr /A, whr σ is coprssor constnt, R is th chnicl rsistnc, nd A is th piston r Swift () lso drivd th fficincy using coustic ipdnc nlyss, nd optiizd th piston r A t rsonnc

2 Gn t l / J Zhjing Univ-Sci A (Appl Phys & Eng) (7): Rdbugh t l () discussd th coustic ipdnc tch bsd on givn corcil linr coprssor Di t l () nlyzd th ipdnc tching principl btwn th linr coprssor nd th cold hd fro th prspctiv of n nrgy blnc Howvr, ost of th work ntiond bov wr focusd on th rsonnt oprtion, which in prctic is difficult to chiv For xpl, th ipdnc of cold hd is oftn optilly dsignd such tht th volu flow rt lds th prssur wv by tns of dgrs t th hot nd, tht is t th piston (Gn t l, 8), which dos not lwys fit th bst output chrctristics of givn coprssor, hnc non-rsonnc usully occurs Also, th ffcts of coustic ipdnc on th prtrs, xcpt for prssur-volutric (PV) powr nd fficincy, rin unnswrd Bo t l (6) studid th ffct of rsistiv-cpcitiv (RC) lod on th prfornc of throcoustic syst nd invstigtd wid rng of coustic ipdnc It ws xprintlly found tht xiu PV powr or fficincy cn b obtind whn th coustic rsistnc quls th coustic rctnc Howvr, this is diffrnt fro th conclusion of klnd () nd not lwys vlid for linr coprssor (Gn t l, ; ng t l, ) This ppr focuss on dvloping gnrl xprssion for th dpndnc of prfornc on vrity of rltd prtrs, including th currnt, th piston displcnt, th prssur plitud, th lctricl powr dissiption, nd th powr fctor Th PV powr dlivrd nd th fficincy of linr coprssor r optiizd without th ssuption tht th coprssor nd th lod r in rsonnt sttus Th ffcts of coustic ipdnc on th output chrctristics of th coprssor r discussd Exprints r crrid out using th RC lod pproch to vlidt th clcultions Vc ks Vc I p car j M, A () A whr U is th coplx voltg cross th lctric trinls, th bold font rprsnting th coplx vctors, R is th lctric rsistnc, I is th coplx lctric currnt, L is th lctric inductnc, α is th trnsduction cofficint which is th product of th gntic fild nd th lngth of th wir in th fild, V c is th volu flow rt t th piston fc, p c is th prssur plitud insid th coprssion volu, j quls, M is th oving ss, ω=πf is th ngulr frquncy, nd k s is th chnicl spring stiffnss Ths two qutions xprss Oh s lw nd Nwton s lw, rspctivly Hr, th lctricl ipdnc Z, th chnicl ipdnc Z, nd th coustic ipdnc on th piston Z r dfind s Z R jl R j X, () Z R j( M ks / ) R j X, () pc Z R j X, V (5) c whr R is th coustic rsistnc, X, X, nd X xprss th lctricl, chnicl nd coustic rctnc, rspctivly By solving Eqs () (5), w driv th xprssions of I, piston displcnt x, nd p c s follows: U I, Z A Z Z Vc U x ja j A ZZ ZZ, (6) (7) Thorticl nlysis Th linr coprssor shown in Fig is kind of lctrodynic dvic, which cn b dscribd by th following linr hronic-pproxition qutions (Swift, ): c U RI j LI V, () A U, I, R, X ks α R x p c, V c, R, X Elctricl Mchnicl Acoustic Fig Physicl odl of linr coprssor M

3 96 Gn t l / J Zhjing Univ-Sci A (Appl Phys & Eng) (7):9-5 pc A Z AU ( Z Z ) Z (8) Thn th input lctricl powr cn b obtind: R R A R () Substituting Eqs () nd () into Eqs (9) nd (), rspctivly, lds to sipl nd ccssibl fors: R U UI ( AR R ) R ( AR R ) ( AX X ) A ZZ ZZ (9) Th PV powr on th piston is xprssd s U ( A) R R pv A ZZ ZZ PV c c () U R + R ( R X ) A ZZ ZZ =, PV ( A) R R R R X = ( ) A totl ipdnc is dfind s U Z Z I A Z Z R X R X j R X R X () (5) (6) Hnc, th rtio of Eqs (9) nd () givs th fficincy of th trnsfortion of lctric to PV powr on th piston: = PV ( A) R ( AR R ) R[( AR R ) ( AX X ) ] () Eq () is quivlnt to tht in (klnd, ; Swift, ) but is drivd in strightforwrd wy It is found tht X is bsnt in Eq (), which ns tht th lctricl inductnc hs no influnc on η lthough it ffcts th nd PV trs An quivlnt chnicl rctnc X cobining th chnicl nd coustic ipdncs is dfind s (klnd, ) X X A X () Not tht X hr hs th ssntil ning bcus whn X=, th syst is known to b t rsonnc In siilr wy, n quivlnt chnicl rsistnc R cobining th chnicl nd coustic ipdncs is dfind s Th first tr of Eq (6), in othr words th rl prt of Z, rprsnts th quivlnt lctricl rsistnc, whil th scond tr, in othr words th iginry prt of Z, rprsnts th quivlnt lctricl rctnc Sinc th powr cn b only dissiptd in th rl prt of th ipdnc, th PV powr fficincy cn b lso obtind fro th frctionl contribution of th coustic rsistnc to th totl quivlnt rsistnc fro Eq (6), hnc, th s rsult s Eq (5) cn b obtind Th powr fctor cn b xprssd s cos U-I (7) ( R X ) X X ( R X ) R R Eq (7) indicts tht th powr fctor is not only rltd to th lctricl prt, but lso to th chnicl s wll s th coustic doin Th highst powr fctor dos not ccopny th highst fficincy (X=) In othr words, whn th powr fctor quls, it dos not n tht th syst is t rsonnc A cpcitor in sris with th coil is ncssry to shift th lctricl rsonnc (cosθ U-I =) to th chnicl rsonnc (X=) which llows X = Diffrntiting Eq () with rspct to R nd

4 Gn t l / J Zhjing Univ-Sci A (Appl Phys & Eng) (7): stting th rsult qul to zro givs th xiu PV powr: R j A Z X Z Z, _PV A Z ja Z X Z Z U PV(x) Z A ZZ ZZ (8) (9) Diffrntiting Eq () with rspct to R without th ssuption of X=, nd quting th rsult to zro, w obtin th optiu of R s R R R X R X, () _ A R A whr /( RR )+ dtrins th highst possibl fficincy of linr coprssor (klnd, ) For this vlu of R, th fficincy hs its xiu vlu of x = R X R X R R R X R X () Eqs () nd () r obtind for th first ti by th prsnt uthors without th rsonnc ssuption hn th coprssor is t rsonnc (X=), Eqs () nd () r siplifid to Exprintl stup To vrify th nlyss bov nd to invstigt th ffcts of coustic ipdnc on th prfornc of th linr coprssor ovr wid rng of R, xprintl studis r crrid out Th xprintl stup consists of linr coprssor with swpt volu of 6 c nd n RC lod Th surd prtrs of th linr coprssor r listd in Tbl Tbl Prtrs of th linr coprssor R (Ω) L (H) R (kg/s) M (kg) k s (N/) α (N/A) A ( ) Th RC lod copriss ndl vlv nd rsrvoir, s shown in Fig Hliu gs ws ployd hr Two pizolctric-typ prssur trnsducrs p c nd p r r ountd bfor nd ftr th vlv in th xprints, rspctivly Th currnt, th powr fctor, nd th lctricl powr r obtind fro th powr tr A linr vribl displcnt trnsducr (LVDT) is ountd on on piston nd of th coprssor to sur th displcnt Fig shows th surd prssur nd displcnt wvs, which r ll sinusoidl Th dlivrd PV powr cn b clcultd s PV R π sin, c c fa c c pv p x p x () R R _, x = () A + p c p r Eq () is th s s drivd by klnd () nd Di t l () For throcoustic syst, thr r no chnicl coponnts, thrfor R =X =, nd Eq () siplifis to R _ X, which is th s s found by Bo t l (6) Although throcoustic ngins r fr or coplictd thn tht dscribd in Eqs () nd (), th forgoing nlyss without rsonnc ssuption iply so rlvnc btwn throcoustic ngins nd linr coprssors x Fig Schtic of n RC lod drivn by linr coprssor R C

5 98 Gn t l / J Zhjing Univ-Sci A (Appl Phys & Eng) (7):9-5 p (kp) 5 5 whr p c x is th phs diffrnc btwn p c nd x hn linr coprssor is oprting, th PV powr dlivrd to th RC lod is dissiptd in th vlv (R) Th coprssion volu is sll nough to b ignord, nd th rsrvoir is d of stinlss stl which cn b considrd s n dibtic rsrvoir; hnc, R nd X r qul to th rsistnc of th vlv (R) nd th cpcitnc of th rsrvoir (C), rspctivly, which r givn by (Bo t l, 6) whr -5 - R X c r M c pc pr Vc πfv pr c r p c p r -5 x t (s) Fig Msurd wvs of prssurs nd displcnt p p pm, fv p p sin, () p p is th phs diffrnc btwn p c nd p r, γ is th spcific ht rtio of th working gs, is th n chrging prssur, nd V is th volu of th rsrvoir During th xprints, R is chngd by rgulting th vlv opnings, whil X is chngd by using vrious rsrvoir volus nd oprting frquncis Anlyss nd discussion Volu influnc R =8 8 P s/ = MP, V= c, T= K, f= Hz Figs copr th thorticl prdictions of th root n squr (RMS) currnt I RMS, piston displcnt plitud x, prssur plitud p c, lctric powr, powr fctor, PV powr, nd th fficincy with th corrsponding xprintl rsults - x () with diffrnt volus of th rsrvoir In th clcultions nd xprints, th input RMS voltg U RMS of th coprssor is V, th chrging prssur in th syst is MP, nd f is Hz A R is usd s th bsciss instd of R bcus it hs th s unit of R Th clcultions r crrid out in wid rng of A R btwn 5 P s, whil th xprints r prford within th rng of A R btwn P s Hr, diffrnt volus will only rsult in diffrnt vlus of X On th whol, th odl nd xprintl rsults gr wll with ch othr Th dvitions btwn surnts nd clcultions r du to svrl spcts tht r nglctd in our clcultions, for xpl th ddy currnt loss nd th hystrsis loss insid th yoks (Rd t l, 6), th blow-by through th clrnc sl (Rd t l, 6), th irrvrsibl ht trnsfr insid th coprssion volu (Rd t l, ), nd th prssur plitud in th bck volu So th surd PV is lss thn th clcultions (Fig 9), thus th fficincy is lowr (Fig ) Morovr, th sttic lctricl inductnc ployd hr is diffrnt fro th dynic lctricl inductnc, which rsults in th dvition btwn surnts nd clcultions in Fig 8 It is shown in Figs tht ll ths prtrs (I RMS, x, p c,, powr fctor, PV, nd η) r ffctd by A R in wid rng ( 5 P s), whn R is uch sllr thn X, thn X, or in othr words V, is th dcisiv fctor for th prfornc hn R is coprbl with X, both R nd X hv n quivlnt ipct on th prfornc hn R is uch lrgr thn X, it dtrins th prfornc of th linr coprssor I RMS (A) 6 c, clcultion 5 c, clcultion 5 5 c, surnt 5 c, clcultion 5 c, surnt f= Hz, p M = MP A R Fig RMS currnt vs coustic rsistnc undr diffrnt volus

6 Gn t l / J Zhjing Univ-Sci A (Appl Phys & Eng) (7): x () 5 f= Hz, = MP c, clcultion 5 c, clcultion 5 c, surnt 5 c, clcultion 5 c, surnt A R p c (kp) c, clcultion 5 c, clcultion 5 c, surnt 5 c, clcultion 5 c, surnt f= Hz, = MP A R Fig 5 Piston displcnt plitud vs coustic rsistnc undr diffrnt volus Fig 6 Prssur plitud t lod inlt vs coustic rsistnc undr diffrnt volus () 8 6 c, clcultion 5 c, clcultion 5 c, surnt 5 c, clcultion 5 c, surnt f= Hz, = MP A R Powr fctor 8 6 f= Hz, = MP c, clcultion 5 c, clcultion 5 c, surnt 5 c, clcultion 5 c, surnt A R Fig 7 Elctricl input powr vs coustic rsistnc undr diffrnt volus Fig 8 Powr fctor vs coustic rsistnc undr diffrnt volus PV () 5 c, clcultion 5 c, clcultion 5 c, surnt 5 c, clcultion 5 c, surnt f= Hz, = MP 8 6 c, clcultion 5 c, clcultion 5 c, surnt 5 c, clcultion 5 c, surnt f= Hz, = MP A R Fig 9 PV powr dlivrd vs coustic rsistnc undr diffrnt volus A R Fig Efficincy vs coustic rsistnc undr diffrnt volus

7 5 Gn t l / J Zhjing Univ-Sci A (Appl Phys & Eng) (7):9-5 Along with th incrs of R, th gnitud Z of th totl quivlnt ipdnc Z, shown in Eq (6), dcrss; thrfor I RMS nd incrs for givn voltg Th quivlnt dping cofficint R incrss, hnc, x dcrss Thus, s indictd by Eq (), or otor forc is pplid to th gs insid th coprssion volu, s rsult, p c incrss In th rng whr x dcrss (Fig 5) nd p c incrss (Fig 6), PV xhibits pk s shown in Fig 9 As for th powr fctor, it rchs pk vlu of t th condition of ( R X ) X X, nd it will finlly rch th vlu R / R X whn R bcos infinit Th vrition of η cn b xplind fro th tndncis of both PV nd, or by xining th vrition of th frctionl contribution of th coustic rsistnc to th totl quivlnt rsistnc s givn by Eq (6) As R gos up, this contribution first incrss, pks, nd thn dcrss As th rsrvoir volu, V incrss, x incrss nd p c dcrss, I RMS nd dcrs t first nd thn incrs, whil th powr fctor nd η incrs t first nd thn dcrs ith this linr coprssor, for th givn conditions of Hz nd MP, w hv X = kg/s, nd A X = 56, 7, nd 5 P s for V of c, 5 c nd 5 c, rspctivly Hr, th volu of 5 c provids th pproxit rsonnt oprtion (X=), nd th fficincy η lso rchs its xiu It is lso intrsting to not tht th curvs in Fig r sytric bout R on th logrithic scl A brif thticl xplntion of this ftur is providd in th Appndix Frquncy influnc Th ffcts of frquncy on th coprssor prfornc hv lso bn invstigtd Figs 7 copr th thorticl prdictions with th corrsponding xprintl rsults t diffrnt frquncis Hr, in th clcultions nd xprints, th input voltg U RMS of th coprssor is V, th chrging prssur in th syst is MP, nd V is 5 c Unlik th ffct of V, diffrnt vlus of f will rsult in not only diffrnt vlus of X, but lso diffrnt vlus of X nd X Also, it is found tht th tndncis in th thorticl rsults dscrib th xprintl dt I RMS (A) x () Hz, clcultion Hz, clcultion Hz, surnt 6 Hz, clcultion 6 Hz, surnt V=5 c, = MP Hz, clcultion Hz, clcultion Hz, surnt 6 Hz, clcultion 6 Hz, surnt V=5 c, = MP A R Fig RMS currnt vs coustic rsistnc undr diffrnt frquncis A R Fig Piston displcnt plitud vs coustic rsistnc undr diffrnt frquncis p c (kp) 6 5 Hz, clcultion Hz, clcultion Hz, surnt 6 Hz, clcultion 6 Hz, surnt V=5 c, = MP A R Fig Prssur plitud t lod inlt vs coustic rsistnc undr diffrnt frquncis

8 Gn t l / J Zhjing Univ-Sci A (Appl Phys & Eng) (7):9-5 5 () 5 Hz, clcultion Hz, clcultion Hz, surnt 6 Hz, clcultion 6 Hz, surnt V=5 c, = MP A R Fig Elctricl input powr vs coustic rsistnc undr diffrnt frquncis Powr fctor 8 6 V=5 c, = MP Hz, clcultion Hz, clcultion Hz, surnt 6 Hz, clcultion 6 Hz, surnt A R Fig 5 Powr fctor vs coustic rsistnc undr diffrnt frquncis PV () 8 6 Hz, clcultion Hz, clcultion Hz, surnt 6 Hz, clcultion 6 Hz, surnt V=5 c, = MP 8 6 Hz, clcultion Hz, clcultion Hz, surnt 6 Hz, clcultion 6 Hz, surnt V=5 c, = MP A R Fig 6 PV powr dlivrd vs coustic rsistnc undr diffrnt frquncis A R Fig 7 Efficincy vs coustic rsistnc undr diffrnt frquncis Th ffcts of coustic ipdnc t diffrnt frquncis f r siilr to thos t diffrnt rsrvoir volus V An xcption occurs whn R is uch lrgr thn X ; in this rng of R th prfornc is inly dtrind not only by R but lso by f (bcus f ffcts X nd X ) hn R is uch sllr thn X, n incrs in f rsults in dcrs in both x nd p c, whrs I RMS nd dcrs t first nd thn incrs, nd η incrss t first nd thn dcrss Also, so losss, for xpl th ddy currnt loss nd th hystrsis loss insid th yoks, incrs with frquncy incrss, tht is why th PV powr of 6 Hz in Fig 6 is sllr thn tht of Hz t thir pk vlus Eqs (6), (8) nd (9) indict tht whn R bcos infinit, I, p c, nd will rch thir xiu: I p x U Z, U cx A Z U R x R X, ( + ) (5) Thus, in th rng whr R is uch lrgr thn X, n incrs in f will cus I RMS, p c, nd ll to dcrs Likwis, th curvs of fficincy on th logrithic scl r sytric Th fficincy η rchs its xiu t Hz, which is th so-clld rsonnt frquncy

9 5 Gn t l / J Zhjing Univ-Sci A (Appl Phys & Eng) (7):9-5 5 Conclusions A dtild study focusing on th ffcts of coustic ipdnc in linr coprssor hs bn prford Th prtrs including th currnt, th piston displcnt, th prssur plitud, th lctricl powr dissiption, th powr fctor, th PV powr dlivrd, nd th fficincy hv bn thorticlly nd xprintlly invstigtd An optiizd coustic rsistnc hs bn dvlopd without th ssuption of rsonnc, thrby providing or gnrl xprssions thn in prvious studis nd corrlting chrctristics btwn linr coprssor nd throcoustic ngin Both clcultions nd xprints hv bn crrid out which show tht n pproprit coustic ipdnc is crucil in ordr to chiv th bst prfornc Th rsrch provids bttr undrstnding of th oprtion chnis of linr coprssor nd th ipdnc tch in cryocoolr syst Rfrncs Bo, R, Chn, GB, Tng, K, Ji, ZZ, Co, H, 6 Effct of RC lod on prfornc of throcoustic ngin Cryognics, 6(9): [doi:6/jcryognics6] Chn, N, Tng, YJ, u, YN, Chn, X, Xu, L, 7 Study on sttic nd dynic chrctristics of oving gnt linr coprssors Cryognics, 7(9-):57-67 [doi:6/jcryognics7] Chn, X, Zhu, ZQ, Modling nd vlution of linr oscillting ctutors Journl of Intrntionl Confrnc on Elctricl Mchins nd Systs, ():57-5 Di,, Luo, EC, ng, XT, u, ZH, Ipdnc tch for Stirling typ cryocoolrs Cryognics, 5(): 68-7 [doi:6/jcryognics] Dvy, G, 98 Th Oxford Univrsity Minitur Cryognic Rfrigrtor Intrntionl Confrnc on Advncd Infrrd Dtctors nd Systs, London, p9 Gn, ZH, Liu, GJ, u, YZ, Co, Q, Qiu, LM, Chn, GB, Pfotnhur, JM, 8 Study on 5 /8 K singl stg Stirling typ puls tub cryocoolr Journl of Zhjing Univrsity-SCIENCE A, 9(9):77-8 [doi: 6/jzusA8] Gn, ZH, ng, LY, Liu, DL, Zhng, XJ, u, YN, Prfornc tsting with RC lod pproch in linr coprssors Journl of Enginring Throphysics, (9):75-78 (in Chins) Koh, DY, Hong, YJ, Prk, SJ, Ki, HB, L, KS, A study on th linr coprssor chrctristics of th Stirling cryocoolr Cryognics, (6-7):7- [doi: 6/S-75()6-] Nst, T, Olson, J, Chpgn, P, Evtiov, B, Frnk, D, Roth, E, Rnn, T, 6 Ovrviw of Lockhd Mrtin cryocoolrs Cryognics, 6(-):6-68 [doi:6/j cryognics56] Prk, SJ, Hong, YJ, Ki, HB, Koh, DY, Ki, JH, Yu, BK, L, KB, Th ffct of oprting prtrs in th Stirling cryocoolr Cryognics, (6-7):9-5 [doi:6/s-75()6-] Rb, J, Twrd, E, Northrop Grun rospc systs cryocoolr ovrviw Cryognics, 5(9):57-58 [doi:6/jcryognics9] Rdbugh, R, Grwy, I, Vprik, AM, Dvlopnt of Minitur, High Frquncy Puls Tub Cryocoolrs Procdings of SPIE, 766:766J-- [doi: 7/85766] Rd, J, Bily, PB, Ddd, M, Dvis, T, 6 Motor nd throdynic losss in linr cryocoolr coprssors Advncs in Cryognic Enginring, 5:6-68 [doi: 6/6] Rd, JS, Dvy, G, Ddd, M, Bily, PB, Coprssion losss in cryocoolrs Cryocoolrs, :9- [doi:7/ _] Swift, G, Throcoustics: A Unifying Prspctiv for So Engins nd Rfrigrtors Acousticl Socity of Aric Publictions, Nw York, Aric, p-8 klnd, RS, Us of lctrodynic drivrs in throcoustic rfrigrtors Journl of th Acousticl Socity of Aric, 7():87-8 [doi:/ 865] ng, LY, Zhou, J, Gn, ZH, Prfornc tsting of linr coprssors with RC pproch Advncs in Cryognic Enginring, 57:6-6 [doi:6/ 779] Appndix: Explntion of th ftur of η whr Eq () cn b writtn s C C CR C R C ( A), C A R, C R ( R X ) R, C A R A Hr, R is writtn in n xponntil for so tht R = i, nd w dfin th function f(r ) s f R C ( ) CR, R,

10 Gn t l / J Zhjing Univ-Sci A (Appl Phys & Eng) (7):9-5 5 nd th function g(i) bout i s whr gi f R in ( in) () ( ), C C n C, lg C It is found tht g(i) is sytric bout i on th linr coordints, hnc, f(r ) nd η r sytric bout R on th logrithic coordints, s shown in Fig