Oscillator design using two-port describing functions

Transcription

1 Oscillor sign ug wo-por scriing funcions G. Mészáros, J. Lvánszky n. Brcli, Fllow, EEE Asrc Our gol is o show h h scriing funcion concp is usful for oscillor sign. Dscriing funcions hv n x for h cs of wo-pors. om propris of wopor scriing funcions hv n prsn. wo-por minc scriing funcions of simplifi rnsisor mol hv n rmin. As n pplicion, Colpis oscillor hs n sign for h MHz rng. hr is goo coincinc wn circui nlysis n msurmns, n hs rsuls vi from scriing funcion mho. his inics molling prolm h will solv in our nx pulicion. nx rms Oscillor, nonlinriy, scriing funcion concp, comprison, AWR, msurmn D. NRODUCON: WO-POR DECRBNG FUNCON, PROPERE scriing funcion concp s ck s long s o 968 whn h fmous monogrph y Gl n vn r l ppr []. A scriing funcion is n pproxim chrcrizion of nonlinr opror whn usoil nswr for usoil xciion s funcion of signl innsiy is consir. h ci rfrnc gnrlizs h concp for mulipl inpus. n our rly work, so cll wo-por minc n scring scriing funcions hv n inrouc []. npu n oupu currns r scri s funcions of h mgniu of inpu n oupu volgs n h phs wn hm. Anohr gnrlizion is h scriing funcion mrix, h is, gnrlizion of h scriing funcion concp for mulipl hrmonics []. Whn oh mulipl pors n mulipl hrmonics r consir, h chrcrizion is cll s X-prmrs [4] ssuming h h nonlinriy is concnr on h firs por only. n his ppr, pplicion of h wo-por scriing funcions for h cs of oscillor sign hs n sui. wo-por minc scriing funcions for simplifi rnsisor mol hv n riv in cion. hn hs rsuls hv n us in h sign of low frquncy Colpis oscillor in cion. G. Mészáros is wih h Bups Univrsiy of chnology n Economics, Dprmn of Bron nfocommunicions n Elcromgnic hory, H Bups, Egry Józsf uc 8., Hungry (-mil:mszros@m.hv.hu). J. Lvánszky is wih h Ericsson lcom Hungry L., H7 Bups, rinyi J. u. 4-, Hungry (-mil: Jnos.Lvnszky@Ericsson.com). Now w fin wo-por scriing funcions n summriz hir propris. A nonlinr im-invrin, ynmic wo-por [5] cn chrcriz pproximly y usoil inpu wo-por scriing funcions in rms of h missil signl pirs ξ i, η i (i=, ) for h ih por. Amissil signl pirs [5] cn volg n currn, incin n rflc wv prmrs, or ny linr cominions of h volg n currn im funcions. Lr on, w us minc scriing funcions whr missil signl pir is currn n volg. n our ppr [], w show microwv msurmn xmpl whr missil signl pirs r rflc n incin wv prmrs. W ssum h im pnc of h missil signl pirs is usoil. his is consqunc of som un circuis inclu in h ming circui roun h wo-por. From his poin, w no y ξ n η, h complx ffciv vlus of h por quniis. hn wo-por scriing funcions r fin s follows: ξ = DF ( η, η, φ)η () ξ = DF ( η, η, φ)η () No h wo-por scriing funcions cn fin in mny iffrn wys. his finiion xprsss h fc h h ffc of our wo-por o h nvironmn is rflcion (DF ) n rnsmission (DF ). n h Eqs. (), (), φ is h phs iffrnc wn η n η. As shown in Eqs. () n (), w consir inpu n rnsfr s nonlinr funcions of η, η, φ. Nx w consir h rlion wn scriing funcions n circui prmrs if h wo-por is linr. A linr wo-por cn chrcriz s follows: ξ = CP η + CP η () ξ = CP η + CP η (4) whr CP ij is h circui prmr wih xciion n nswr por j n, rspcivly.. Brcli is wih h Bups Univrsiy of chnology n Economics, Dprmn of Bron nfocommunicions n Elcromgnic hory, H Bups, Egry Józsf uc 8., Hungry (-mil:brcli@m.hv.hu).

2 h comprison wn Eqs. (,) n (,4) yils DF = CP + CP η /η (5) DF = CP + CP η /η (6) Eq. (5) sys h if h frquncy is fix, DF is circl wih cnr n rius CP n CP, rspcivly, if h mplius of η n η r inicl n h phs wn hm is vri. iuion is vry similr for DF. Rciprociy implis CP = CP so in cs of h wo circls, h cnr of DF is rl o h rius of DF. ymmry implis rciprociy plus CP = CP so h cnr of DF is rl o h rius of DF. Losslssnss is scri iffrnly for iffrn choic of missil signl pirs, so losslssnss is sn iffrnly in rms of iffrn scriing funcions. n h following, w us minc scriing funcions for low frquncy sign, n scring scriing funcions for microwv sign. For minc scriing funcions, h missil signl pir is currn, volg n for scring scriing funcions, rflc n incin wv vrils. n oy s sign, mosly nonlinr wo-pors (rnsisors) r us s n civ lmn. hrfor, w fin possil gnrlizion of h scriing funcion concp for h cs of low frquncy wo-pors []: = Y (,, φ ) (7) = Y (,, φ ) (8) whr,, n r h complx mplius of h firs hrmonic currns n volgs h firs n scon pors wih h rfrnc ircions givn in Fig., φ is h phs wn h wo volg xciions. h inics n sn for inpu n rnsfr, rspcivly. Wih hs prliminry informion, w rmin h wo-por scriing funcions of nonlinr rnsisor mol in cion, n in cion w us hm in h sign of n oscillor. Avng of h scriing funcion pproch is h w cn pproxim wll h nonlinr hvior vn if i is srong. W xpc h ug his pproch will rsul in r coincinc wn our msur n nlyz prmrs hn for. W us h following inx convnions: - Lowrcs lr, upprcs inx: Gnrl im funcion - Upprcs lr, upprcs inx: DC componn - Lowrcs lr, lowrcs inx: Alrning componn - Upprcs lr, lowrcs inx: Complx ffciv vlu of h firs hrmonic Nx w riv h rlions wn minc n scring scriing funcions. L us sr wih h finiions. = v+z i Z (9) = v Z i Z () v = Z (+) () i = Z () usiuion of Eqs. (,) ino (7,8) n rrrnging, rsuls in h rnsformion from minc o scring scripion. R = Y Z +Y Z () An h invrs rnsformion is: = Y Z +Y Z (4) Y Z = R + R (5) Y Z = + (6) Nx w riv rnsformion wn Q n Q W : Figur. Rfrnc ircions h mns h nonlinr civ wo-por is chrcriz y wo scriing funcions, on for h inpu n h ohr for h rnsfr from h firs por o h scon. As on phs cn rirrily chosn, possiiliy is o choos s rl n hn φ is h phs of h complx volg mpliu h scon por. usiuing Eq. () ino (7): Q = (7) Q W = A A [] (8)

3 Q = A A +B A = +B A A +B + B = Q w+ (9) + R A. WO-POR ADMANCE DECRBNG FUNCON OF A MPLFED RANOR MODEL As h forhcoming clculion my lnghy for rl rnsisor mol, w hv o propos h simpls possil mol. Firs w in o ry low frquncy rlizion of h oscillor hus rnsisor prsiic lmns r nglc ll. Furhr simplificion is possil if w us such circui lmn vlus in our sign h supprss ffcivly h influnc of rl rnsisor inpu. n h plnn Colpis oscillor, h rnsisor oprs wih groun collcor, hus h inpu n oupu r h sis n h mir s shown in h following Figur. i v BE if vbe if v if v BE BE (6) is sy o show h circui consig wo-pors in prlll hs h following wo-por minc scriing funcions: Y (,, ) Y (,, ) Y (,, ) (7) Y (,, ) Y (,, ) Y (,, ) (8) L h firs wo-por compris h ios only n h scon compris h nonlinr conroll sourc. Circui quions of h firs -por r = E / B * ( / n BE ) (9) = - E / B * ( / n BE ) () whr is h moifi Bssl funcion of h firs orr, E n B r h DC mir currn n h DC currn gin, rspcivly. hrfor h scriing funcions of h firs wopor r: Figur. rnsisor mol (AC) W ssum io n nonlinr volg conroll currn sourc wih h following chrcrisic: Y () Y () Now w clcul h scriing funcions of h scon wopor. is ovious h Y =. Bs-mir volg is v BE ( ) () whr h mpliu is chosn s rl. W hv o clcul h mpliu of h firs hrmonic currn h scon por. As h nonlinriy is rsisiv, h oupu currn is in phs wih h inpu volg. For in xciion, hr will only in componn of h firs hrmonic. hr r hr css: Figur. Nonlinr rnsconucnc hn.. hn (4)

4 (5) (6) c. hn (7) (8) (9) Now w vlu h. n c. css.. 4 (4) (4) Y (4) c. 4 (4) (44) (45)

5 Y (46) Now susiuion of (,) n (4,46) ino (7,8) rsuls in h wo-por minc scriing funcions of h rnsisor mol: (,, ) = (,, ) = ( / ) E BE Y Y n B (47) Figur 4. Emir currn mpliu s funcion of h s volg mpliu. Ngiv currn is consqunc of h rfrnc ircions if if if Y ),, ( (48) whr w choos s rl. W show pr of h mir currn mpliu coming from h nonlinr rnsconucnc in h nx Figur, wih prmrs =9.mA, =.57 n =.99. Plo in Fig. 4 hs n gnr y h following Ml progrm (rnsfr_scr_fn.m): =9.-; =.57; =.99; min=; mx=5; N=; sp=(mx-min)/n; (:N)=; (:N)=min+(:N).*sp; for i=(:n) if (i)< (i)=; if <=(i) & (i)< (i)=-//pi*/(- )*(((i)*(/(i))-*sqr(- (/(i))^))); if <=(i) (i)=-//pi*(/(- )*((i)*((/(i))- (/(i)))-*sqr(- (/(i))^)-(-*)*sqr(- (/(i))^))+**sqr(- (/(i))^)); plo(,); il('pr of mir currn mpliu'); xll(' in ols'); yll(' in Amps'); gri on;. DEGN OF A LOW FREQUENCY COLP OCLLAOR UNG WO-POR DECRBNG FUNCON W solv h sign sk y cring smll Ml progrm h simuls h oscillor in frquncy omin ug h scriing funcions formul ov. Circui igrm of Colpis oscillor is shown in h nx Figur.

6 Figur 5. Circui igrm of Colpis oscillor. DC is lmns r no shown h Ml fil is h following (osc_sim4.m): =9.-; % [A] =.57; % [] =.99; % [] C=8-; % [F] C=44-; % [F] % L=-6; R=47; C=47-; % [F] L=-6; % [H] om=/sqr(l*/(/c+/c+/c)); R=.7; % [Ohm] nbe=.847; BE=4.75-; nbc=.99; BC=.67-; E=.5; B=6; P=; %P=.5-*E; B=nBE*log(E/B/BE+); N=; (:N+)=; (:N+)=; (:N+)=; (:N)=; (:N)=; Y(:,:)=; Z(:,:)=; (i)=; if <=(i) && (i)< (i)=-//pi*/(- )*(((i)*(/(i))-*sqr(- (/(i))^))); if <=(i) (i)=-//pi*(/(- )*((i)*((/(i))- (/(i)))-*sqr(- (/(i))^)-(-*)*sqr(- (/(i))^))+**sqr(- (/(i))^)); (i)=e/b*ssli(,((i)- (i))/nbe); (i)=(i)-e/b*ssli(,((i)- (i))/nbe); (i)=-(i); (i)=-(i); ; (i+)=z(,)*(i)+z(,)*(i); (i+)=z(,)*(i)+z(,)*(i); plo((:n),s(),'r.',(:n),s(),'+ ','LinWih',); il('oscillor volg mplius'); yll('(r), (lu) in ols'); xll('rion sps'); gri on; h only prolm is h i is vry ifficul o fin propr iniil vlus. hrfor w ci o implmn h picwis linr conroll sourc in AWR. Circui ils n rsuls r shown in h nx Figurs. Y(,)=/(j*om*L+R+/(j*om*C))+j* om*c; Y(,)=-j*om*C; Y(,)=Y(,); Y(,)=j*om*C+/R+j*om*C; Z=inv(Y); ()=.; ()=.; for i=(:n) (i)=s((i)-(i)); if (i)< Figur 6. Circui rlizion of h picwis linr volg conroll currn sourc wih chrcrisics shown in Fig.. Brkpoins r rliz y ios wih high surion currn n

7 vry low iliy fcor. C prvns DC volg h oupu (firs hrmonic scriing funcion), L ls DC currn o h groun. Figur. Circui igrm Figur 7. h oscillor circui in AWR Figur 8. Oscillor volgs for h scriing funcion mho. R:, lu: Figur. rnsisor mol Figur 9. h oupu spcrum Figur. Oscillor volgs. R:, lu: Now w nlyz h circui in AWR. h circui igrm is:

8 Figur. pcrum of h oupu volg. Pls compr his Figur o Fig. 8 Figur 6. Phoo of h mir volg msurmn l. Comprison of Dscriing funcion (pp).896 (Fig. 8) (pp).5 (Fig. 8) AWR.4 (Fig. ).75 (Fig. ) Msurmn.594 (Fig. 5).75 (Fig. 6). CONCLUON n his ppr w sui oscillor sign s on scriing funcions. A qusion my why his procur is r hn fully compuriz on whr oscillor prmrs (frquncy, powr) r mnully jus. n our opinion, h only righ nswr is h h mho prsn hr givs p insigh ino circui oprion. Firs w inrouc wo-por scriing funcions n hir propris. hn h rnsformion wn hm hs n riv. W us minc scriing funcions for low frquncy sign. hr is goo coincinc wn circui nlysis n msurmns, n hs rsuls vi from scriing funcion mho, s pls l. his inics molling prolm h will solv in our nx pulicion. Our pln is sris comprig hr pprs, his is h firs on wih inroucion o h prolm. n h scon on w solv h ccurcy prolm. n h hir on w prsn microwv rlizion. Figur 4. Phoo of h oscillor rlizion. ACKNOWLEDGMEN his work is in s our mos commmorion of h h nnivrsry of Prof. K. imonyi s irh. Circui uiling n msurmns hv n on h Opicl Lorory of our Univrsiy whil pprwork hs n on Ericsson lcom Hungry. Accoringly, Mr.. Bski n Mr. J. Bnkő Ericsson lcom Hungry r cknowlg for h xclln rsrch coniions. Also, w r grful for Mr.. zili n Mr. K. M. Osáh for hir prcis inrnl rviws. Rfrncs [] A. Gl, W. vn r l: Mulipl-npu Dscriing Funcions n Nonlinr ysm Dsign, McGrw Hill, Nw York, 968 [] A. Brnyi, J. Lvánszky, G. Kolumán: Accur Lrg-ignl Chrcrizion of Microwv rnsisors, Procings of h 7h Colloquium on Microwv Communicions, Bups, Hungry, 6- pmr, 98, pp Figur 5. Phoo of h s volg msurmn

9 [] A. Ms: h scriing funcion mrix, MA Journl of Appli Mhmics, 97, olum, ssu, pp [4] Dvi E. Roo, Jn rspch, Json Horn, Mihi Mrcu: X- Prmrs: Chrcrizion, Moling, n Dsign of Nonlinr RF n Microwv Componns, Pr of h Cmrig RF n Microwv Enginring ris, [5] L. O. Chu: hory of Nonlinr Circuis, uoril lcur, ECCD 976, Lusnn, wizrln Grgly Mészáros rciv his B.c. gr in lcricl nginring from Bups Univrsiy of chnology n Economics in follow y M.c. gr from h sm univrsiy in. H is currnly pursuing h Ph.D. gr in lcricl nginring from h Dprmn of Bron nfocommunicions n Elcromgnic hory h Bups Univrsiy of chnology n Economics. His rsrch inrss covr h phs nois of h oscillors n microwv vics. projcs. H pricip in h Europn NEFER n Nwork of Excllnc n svrl CO co-oprions. Prof. Brcli is h uhor of 86 pprs n 6 ooks pulish in English. H lso hs 6 pns. H ws visiing profssor Polychnic nsiu of Brooklyn in 964, Univrsiy Collg Lonon in 986, Drxl Univrsiy (Phillphi) in , chnicl Univrsiy of Hmurg- Hrurg in 99, Osk Univrsiy in 99, chnicl Univrsiy of Grnol in 994, Hlki Univrsiy of chnology in n h yny Univrsiy in 4. János Lvánszky rciv his Mc n PhD grs in lcricl nginring from h Bups Univrsiy of chnology n h Hungrin Acmy of cincs in 978 n 988, rspcivly. h il of his PhD hsis is Prolms in Nonlinr, Microwv Circui Dsign: Powr Mching, Error Corrcion for Microwv -Prmr Msurmn. During his firs mploymn h Rsrch nsiu for lcommunicions, Bups, Hungry, h lrn h founions of circui hory wih microwv pplicions, n microwv phoonics, from 976 o. hn h h n for n yrs wih Ausri Mikro ysm AG, Grz, Ausri, whr h lrn sysm hory n sysm lvl hinking. inc h hs n wih Ericsson lcom Hungry s sysm nginr, working on prolms in nois rucion, MMO, wvgui iplxr sign, signl ingriy n prsnly rio ovr fir. H ws mmr of h lcommunicion ysms Commi of h Hungrin Acmy of cincs (985-97). H is n uhor or co-uhor of mor hn on hunr n fify pulicions, incluing svn pns. H ws gus rsrchr in Hlki (99), Bologn (994), Gohnurg () n ockholm (, 4 n 5). Prof. ior Brcli m significn conriuions in h fil of microwv n opicl chnologis. His civiy is concnr on h nonlinr procsss. Prof. Brcli pricip in mny inrnionl rsrch projcs. oghr wih his group h conriu o h Europn ACCORD, Coprnicus, MOK, FRAN, LABEL n GANDALF