A Theoretical Model of a Voltage Controlled Oscillator

Transcription

1 A Theoreical Model of a Volage Conrolled Ocillaor Yenming Chen Advior: Dr. Rober Scholz Communicaion Science Iniue Univeriy of Souhern California UWB Workhop, April 11-1, 6

2 Inroducion Moivaion The volage conrolled ocillaor of he PLL ha been modeled a a imple inegraor, i doe no capure he eence of dynamic of ocillaion Ocillaor inabiliy iue criical for communicaion yem Need a high peed clock for ulra-wideband (UWB) communicaion yem Synchronizaion analyi for UWB yem Problem aemen Need a mahemaical model of VCO ha fully incorporae a mean of non-lineariy for abilizing ocillaor 1

3 Schemaic of he propoed VCO wih noie y( Z( x x c( F( F(: inernal noie of he clock a b a b d d ( ) K ω v ( W ( -1 pole filer v ~ ( ) : he racking loop plu he conroller noie v(: he conrolled volage y(: he ocillaor waveform v ~ ( ) : inpu ignal from he racking loop W ( :inpu noie from he racking loop

4 Theoreical model The volage conrolled ocillaor can be decribed by he following ochaic differenial equaion which include he inernal noie, F(, of he clock and he racking loop plu he conroller noie,. K( v& n& ( ) && y y& = ω K ( v ) [ ω K ( v ) ] y c( F( where F( i aumed o be a whie Gauian noie, independen of. In addiion, C ([, )), y( i he ocillaor waveform, v( i he conrolled volage, c( i a caled facor, ω i he clock re frequency ( ω > ), and K i he VCO gain. The oupu afer he hard-limier i Z ( = g y( ) 3

5 Analyic oluion of he model Iniial condiion: y() =, v() =, y 1 () =, y () = aˆω, aˆ R When a if where a, C are conan, and i he ime of eady ae, unle v ( = C if > oherwie pecified. Analyic oluion Phae noie Saiical properie No noie preen Inernal ~ noie F ( preen ) Inernal noie and exernal noie preen y ( ) = 1 Brownian y1( = i ω K aτ dτ ) aˆ i ω y ( = aˆ in Φ 1 where Φ moion ) v( = a, ω = ω K ( = ω i ω c ( ) ω ( where c( ) i he caling facor for he noie F (, )) db in bˆ(, ( c( ) db ω K ( a )) bˆ(, = ω ( ) K a, and aτ dτ K B aτ dτ i a 1- dimeniona l K τ ) dτ, K τ ) dτ ϕ( = an ~ ϕ ( = an N/A 1 n1 ( = K coω n ( = K K = -1 n ( 1 n1 ( inω c ω db n~ ( 1 n~ 1 ( db, n~ = ~ 1 ( K(, ))co Φ ( ) db n~ = ~ ( K(, ))in Φ ( ) db ~ c K(, )) = ω K ( a )) ϕ ( n = K z) dz E c 3 4ω N/A [ ϕ )] [ co( ω ) 1] R ϕ ( (, ) R (, ) n c iω mi, ) = mi, ω ω ~ E[ ϕ ( ] and R~ ϕ (, ) are boh ime-varying and no cloed form oluion can be obained. 4 Rϕ (, ) K Rn ( τ, z) dzdτ n =

6 Reul o dae: imulaion reul inernal noie only = upward zero-croing Averaged upward zero-croing jier ideal zero-croing Simulaion done a VCO frequency of ~ khz and F ( ) i on. 5

7 Simulaion reul Hiogram of upward zero croing jier a differen ime inan, 1 run. Noe: Diffuion occur 6

8 Simulaion reul Cycle o cycle jier imulaed for 1 ec. Cycle-o-cycle jier Chi-quared goodne-of-fi e i ued o e he normaliy of he e aiic The parameric Pearon correlaion e i performed o ee ha he cycle o cycle jier are independen. 7

9 Simulaion reul for inernal exernal noie iming jier when boh F( and are preen Cycle number 8

10 Summary of reul A heoreical model of a volage conrolled ocillaor i propoed baed on phyical dynamic wih noie. Analyi of he reuling phae noie along wih he imulaion ugge ha he iming jier proce ha a random walk behavior (wih reoring force) and he upward zero croing jier i normal diribued. The incremen of he iming jier proce, cycle-o-cycle jier, i hown o behave a normal diribued and independen. Reul ugge ha he racking loop plu he conroller noie ~ caue he ocillaor phae o drif while he inernal noie F ( ) end o caue diffuion on he ocillaor phae. 9