Question

Problem 3: Explain how the PLL loop shown below works. Show mathematically that for FM input given by u(t) Accos(2 fe+ (t) where (t) = 2TK, m(t)at, the output can be designed so that к m(t). vkv v(t) (r) Loop filter 8(1) d(t) 21

Answer 1

phase lok cd loop (PLL) ^o uoed in thackina the phane the and Kcauency eb he Cosia Cemponet ot an inComung FM Somal . PLL n aho obul fo demodulaluig fm Signalo n plesence of lasge noise aud ow Sioynal powev ft phase -lod Loro? (PLL)i negatve teadlok banically Sytcwo oghe Contiolled Osclalt (Vco)onet ed Voltage negatwe loTp the ot fmWawe elt) Sct) bCt) VCO 9iven tha u t= Ac Cos attcttg Ct Ct) Tt Kf P CAw e Ct) 9ct) SiCt) uakin opalm- ther the operaluon ob a PLL in Sumla o an

adbock Stem. An auy feedbok Sytem , the teadback Sigmal teuda to foUow the ipud Sigmal . Tt-Hhe Siamal teodbouk io not equaJ io the nput Si omal the ea SiqmaJ uu chauge the Walve of the fealbak Signal Junl J cqual I the mpit Sigmal. the citaence siqmal blwen Sit) and b+) i^ Caled uanli ded Similas punaple eacept te fot that te bak i nt the amplitud, bul a geneaalised Phase t) the esad Signal d dittaence Sigmal c(t) io ulil'secd to adjut the. Vco equency in Sucha Way that the intautaneusphae augle omea cae' to the augle of the inConim (t) t thia pot the Juo Siqmab S(t) aud blt ) Symchiniom and the. PLL j lok ed to the inloming Sigmal s(t) Math cmali Cal Stcps Cewnd e that the adjwi ed VCO when the Conhol voltages Gomeso initially So that luo Consitus are Satidfe 3ero, the olloulung the Set at O the equen y of the VCo i^ pieubcly unmodulot d Casic keaueuay aud the Vco opu hao a 90° phaoe huft w.K.ts unm odusoJed Cawa ubve

la e fM inpud mC d Let the vco bput be detn ed ky bct)- Av Co (wet + P,(t)) pCt)aKu the un Comang Siqmal FM o slt) and the vco output mutip lie appli ed to a the Output ob the multipli a hao and (a wet t o (t) È 42 (t)) Km AAv KmA Au Sm O ct) -P, Ct) Km o multiplic gam the ih hequenly Compenet Can be eluminat ed y ig the ebbeclve unpt lo the lou pan fu aturer km A A 19 G e Ct ect)KmAAv Svo (Ø,Ct)-i(t)) e Ct)Ct)- ,Ct). e t) t) the losp hurca opaates Signal t flduCe. an eaad the autput 19 (t) ..Ct)- ecy hCt-) dY

e()ht-)dtey e Ct) t)- a TIKm K» AA » - O A C e (t)- (t)-an o PP C-44 (Ca)7ug ditactiaing om both Sides dpelt) (P t Sm CPe (Y) h (t-7)d T an Ko CO et) KmA Av Gvm e (t). aud (t)- ( elT)h (t-Y)d A non n cau k eauwaleut model o- PLL in &TI KO ct) e (t) htt) uct) (}' when phase e de (t) s 3o 4hen PLLin Said to be uken the phoane phane Loked ene e t) at all times s Emall Cepaned to | ^adtan then the qo- dibbueence ¢e(h(t-7) dy d¢(1) dt

ain the ie thauw t ko HCf) oren ortaare fmdo L (f)= Ko t CF) ltL C Ko HC e f K 19 Aating douie thaufom j 2P(-)C)a JLCVe ) L e ko 2 Ko Ko () UJ) Jt L)eC)= J LC) K19 [tLC C 1tL) 9K19 d t ОПВКУ d t 1 (}1 ØP 2TIK 10 dt 2Пку 9Ct) Kf ank m t) mct) dt dt 6 the Out put Da Ct) of PLL in apploximadely Same txlept \a mlulalwig Stating dat Si'gmal met. ao the арn] P