Phase Noise in LC Oscillators: From Basic Concepts to Advanced Topologies. Carlo Samori. Politecnico di Milano Milano, Italy

Transcription

1 Phase Nise in LC Oscillatrs: Frm Basic Cncepts t Advanced Tplgies Carl Samri carl.samri@plimi.it Plitecnic di Milan Milan, Italy Carl Samri

2 Outline Basics f LC Vltage-Cntrlled Oscillatrs (VCOs) Phase Nise thery and mdels (simplified) Practical issues and trade-ffs Advanced LC scillatr s tplgies Carl Samri

3 LC Tank L C V(t) L C V(t) Ideal V(t) ω 0 = 1 LC Real V(t) R SL R SC t t L & C quality factrs Qs Q SL = ω L 0 R SL 1 Q SC = ω 0 CR SC Carl Samri

4 Tank Lsses and Q L C R SL R SC Equivalent at ω 0 L C R Tank quality factr: Q T 1 = Q T Q SC Q SL Q = ω 0 RC = R ω 0 L Frm: Q T = Q R Q f the equivalent tank derives frm lsses f the real ne Carl Samri

5 Crss-Cupled MOS LC VCO V DD L V tune L V tune V DD C C C C R -G m =-g m / R V + V - I T I T I T 0 t Start-up: g m / > 1/R V tune cntrls the frequency: VCO Steady state Operates in class-b Carl Samri

6 Output Amplitude i c.m. V + C L V tune R i diff. V DD C i c.m. V - i diff. 0 i c.m. 0 V + I T / t - I T / I T / t V DD V - = π I T R I T is the differential utput amplitude Carl Samri

7 Current vs. Vltage Limited / V + C L V tune R V DD C V - Optimum bias pint π I T R Current lim. V DD Vltage lim. - / 0 I T I T Maximum is V DD (ideal) Optimum bias at the transitin between the tw regimes Carl Samri

8 Crss-Cupled CMOS LC VCO V DD At start-up (g mp +g mn )/>1/R V tune All I T is differentially flwing in the tank N c.m. current in the tank R 0 I T t = 4 π I T R I T Als perates in class-b Carl Samri

9 Output Amplitude V DD V + / V tune V DD / V + V - R - / V - = 4 π I T R I T Same amplitude f single pair (in current lim.) with half I T Ideally the maximum differential amplitude is =V DD Carl Samri

10 A Large Signal Circuit In bth the VCOs, is much larger than input linear range I(t) -I s + I T The circuit behaves apprximately as an hard-limiter (utput is a current) Carl Samri

11 Example L = 1nH, C = 4pF è ω 0 /π =.5 GHz Q=15 è R=(Q/ω 0 C) = 39 Ω L V tune V DD V DD = 1. V, = V = I T R(/π) è I T = 13 ma / V + C R C V - P = V DD I T = 15.6 mw - / I T Carl Samri

12 Outline Basics f LC Vltage-Cntrlled Oscilladars (VCOs) Phase Nise thery and mdels (simplified) Practical issues and trade-ffs Advanced LC scillatr s tplgies Carl Samri

13 Phase Nise Time Frequency Ideal t ω 0 ω dbm Real Zer-crssing errrs (jitter) accumulate in time White nise prduces (1/ ) -shaped tails arund ω 0 : phase nise A srt f integratin is taking place, why? Carl Samri

14 Frequency and Phase Mdulatin ω ( t ) = ω 0 + Δω 0 cs ( t) V ( t ) = A cs # ω t + Δω 0 0 0! # # # "# Φ $ & sin( ω t & )& m & %& 0 S Φ # 1 Δω & 0 % ( $ ' ω Phase spectrum (SSB) if Δω 0 << 1 rad V ( t ) cs ( ω 0 t) + Δω 0 0 S V ω 0 - ω 0 ω 0 +! 1 Δω $ 0 # & " % cs! ω ω " 0 + m ω ( ) t ( ) t # $ Δω 0 cs! ω ω " 0 m Vltage spectrum (SSB) # $ Carl Samri

15 Frequency and Phase Mdulatin 0 S Φ # 1 Δω & 0 % ( $ ' ω 0 S V ω 0 - ω 0 ω 0 +! 1 Δω $ 0 # & " % ω Single sideband pwer Carrier pwer = " $ # Δω 0 % ' & 1 = 1 " 4 Δω % 0 $ ' # & = S φ What d we expect if the frequency is mdulated by white (plus 1/f) nise? Carl Samri

16 Phase Nise ω t S Φ [db] 1 3 Nisy ( ) = ω 0 + Δω n ( t )! Φ t $ # & # t & V 0 ( t ) = cs # ω 0 t + Δω n ( τ ) dτ& # 0 & # & "# %& ( ) S V [db] ( ) = Nise pwer in 1Hz band L Carrier pwer = S ω V ( ± ω 0 m) A 0 ω 0 = S ω φ ( m) ω Phase nise L is expressed as 10lg 10 (L) (measured in dbc/hz) Carl Samri

17 Phase Nise Spectrum 1/f 3 (- 30 db/dec) (flicker r clse-in p.n.) L! " dbc/hz# $ L (dbc/hz) E E E E+07 Frequency [Hz] 1/f (- 0 db/dec) ( white phase nise) Carl Samri

18 Lrentzian Spectrum D, ften < 100 Hz -3 db L ( ) = D D + φ ( t ) = Dt ω 0 ω Phase diffusin prcess with D as diffusin cnstant Derives frm Brwnian mtin thery ( white phase nise nly) D.Ham, A. Hajimiri, IEEE JSSC, March 003 A. Mirzaei, A. A. Abidi, IEEE TCAS-I, March 010 Carl Samri

19 Impact f Phase Nise Signal -99dBm 00kHz band Blcker -3 dbm RF LO IF ω 0 (ω 0 + ) ω 0 ω L (@ ) P Carrier = P Nise (@ )/1Hz ω 0 ω L (@3MHz) 3 dbm + 10 lg 10 ( 10 5 ) < 108 dbm L (@3MHz) < -138 dbc/hz Blcker at ( /π) = 3MHz frm signal Required SNR=9 db, i.e. the ttal nise < - (99+9) dbm E. Hegazi, A. A. Abidi, IEEE JSSC, May 003 Carl Samri

20 Leesn s Empirical Mdel F accunts fr nise frm transcnductr + S ni = 4kT R ( 1 + F ) L C R Ideal, lssless, tank impedance (magnitude squared) at (ω 0 +/- ) ( ) S 1 ω ni ( C m ) D.B. Leesn, Prc. IEEE, Feb Carl Samri

21 AM & PM Nise V n = Nise tne at (ω 0 + ) referred t the carrier frame rtating at ω 0 V n / AM PM - V n / ω 0 ω + ω 0 ω Carl Samri

22 Phase Nise (1/)S ni + Phase Nise: lssless tank ( ) = 1 S 1 ω ni ( C m ) L = 1 kt R 1 ( 1 + F ) ω m C A 0 Only half f the nise cntributes t phase nise It is a general results, nt nly valid fr an hard limiter Carl Samri

23 AM Nise vs. Phase Nise (1/)S ni + S V Phase nise E.g. AM nise fr the tank 1! 4kT $ # & " R R = ktr % AM Nise: lsses nt balanced ω 0 ω All f the physical surces cause AM nise It has always a negligible impact, but Can be cnverted again in t phase nise by a nn-linearity Carl Samri

24 Nise frm the Transcnductr (1/)S ni + S ni = 4kT R ( 1 + F ) F = L Transc. L Tank Evaluatin f F prvides the transcnductr s cntributin t L One f the main difficulties in VCOs design Carl Samri

25 Nise frm the Transcnductr When diff. utput V(t)=0 When diff. utput V(t)=V DD The nise transfer t the tank is switched The scillatr can be cnsidered Linear and Time-Variant Carl Samri

26 LTV Mdels Tw (almst) equivalent mdels capture LTV behavir Time dmain: A. Hajimiri, T. Lee, IEEE JSSC, Feb Frequency dmain: C. Samri, A. Lacaita, F. Zappa, F. Villa, IEEE TCAS_II, July 1998 D. Murphy, J. J. Rael, A. A. Abidi, IEEE TCAS-I, June 010 Key feature f a LTV system is nise flding, as fr mixer and samplers E.g.: nise at (nω 0 + ) is flded at bth at (ω 0 + ) and at (ω 0 - ) Carl Samri

27 LTV Mdel in Time Dmain i n (t) ff ff t i n V(t) t In time dmain the impulse respnse is cnsidered The impulse respnse depends n the particular nise generatr The effect n the utput wavefrm depends n the injectin instant A. Hajimiri, T. Lee, IEEE JSSC, Feb Carl Samri

28 LTV Mdel in Time Dmain: ISF i n Δq. δ(t-τ) i n Δq. δ(t-τ) V(t) τ τ t t V(t) τ τ t t Δφ = 0; fr t < τ Δφ = Δq q MAX Γ ω 0 τ ( ) ; fr t > τ Zer phase errr Max. phase errr Phase errr depends n the injectin instant τ Time-variant system, described by an impulse sensitivity functin Γ Γ is peridic and depends n the surce cnsidered Carl Samri

29 Frm Nise t Phase Nise Γ ( t ) = a 0 + a n n=1 cs ( nω 0 t + θ n ) S ni Fr a generic current nise i n : S ni0 Δφ ( t ) = 1 q max t Δq Γ ω 0 τ i n τ ( ) ( ) dτ Flding and Integratin 0 ω 0 ω 0 3ω a 0 0 S Φ a 1 a a 3 ω E.g. fr a white nise with PSD S ni0 : S Φ ( ) = S ni0 4q MAX a n = S ni0 q MAX Γ rms ω Carl Samri

30 Tank Nise: Evaluatin with ISF 4KT/R + V(t) sin(ω 0 t) L C R ( ) = S ni,r L q MAX Γ rms 1/ = 1 kt R 1 C Γ Tank (t) t cs(ω 0 t) t q MAX =C Γ fr the tank can be derived frm state-space representatin Carl Samri

31 Diff. Pair Nise: Evaluatin with ISF V(t) i n / i n / V v t i n T 0 t -1/ T W T W V v /ω 0 Nise is injected nly at zer crssing, every T 0 / sec. Injectin windw is T W What is the cntributin t Γ? (thus t F?) Carl Samri

32 Diff. Pair Nise: Evaluatin with ISF V(t) V v t i n / i n / i n Γ MOS T W T 0 1/ t -1/ F = Γ rms,mos 1 S ni,mos ( ) T T W 0 4kT R 1 4kTγg m 4kT R = γ Factr t accunt fr the tw MOSs The rigrus evaluatin f Γ is nt straightfrward (see P. Andreani, X.Wang, L.Vandi, A. Fard, IEEE JSSC, May 005 P. Andreani, A.Fard. IEEE JSSC, Dec. 006) Carl Samri

33 Is the CMOS Oscillatr Any Better? fr γ n = γ p = γ F T T 4kTγg ( W 0) m 1 $ & % 4kT R γ p + γ n ' ) ( = γ = 4 π I T R T W V OV ω 0 I T Assume tail nise can be eliminated in sme way The cntributin t F frm the 4 MOS is still γ (r the γ s averages) w.r.t. a single crss-cupled VCO with same tank, I T and V v : n Same g m, duble, T W is halved, but twice nise surces Carl Samri

34 A General Result If active devices d nt add lsses (i.e. d nt lad the tank) If active devices nise is is prprtinal t g m K + F = L Transc. L Tank = γ K K can be a niseless vltage gain (e.g. an ideal transfrmer) Bias current, device type and size d nt impact n minimum F This is the best case, the minimum nise achievable J. Bank, Ph.D. Dissertatin, Chalmers Univ. Tech., 006 A. Mazzanti, P. Andreani, IEEE JSSC, Dec. 008 Carl Samri

35 Outline Basics f LC Vltage-Cntrlled Oscillatrs (VCOs) Phase Nise thery and mdels (simplified) Practical issues and trade-ffs Advanced LC scillatr s tplgies Carl Samri

36 Phase nise vs. Pwer Trade-Off Frm the expressin f L and Q = ω 0 RC L ( ) = 1 kt R 1 ( 1 + F ) = kt ω m C A 0 A 0 R # 1 ω & 0 Q % ( $ ' # 1 + F & % ( $ ' P RF : RF pwer in the tank In gd designs is prprtinal bth t V DD and t bias current I T Therefre P RF is prprtinal t VCO pwer P DC =V DD I T Carl Samri

37 Figure f Merit (FM) P. Kinget, in Analg Circuit Design, Kluwer 1999 *, $ FM = 10 lg 10, L P DC & % +, ω 0 ' ) ( - / /. / P DC in mw, FM expressed in db Only fr 1/ω m ( white ) phase nise Des nt depend n and ω 0 By defining the pwer efficiency η = P RF /P DC : ( ) " kt 1 + F FM = 10 lg $ 10 $ # Q η % ' ' = lg! $ 10 "# Q %& & FOM MAX " 1 + F % 10 lg 10 $ ' # $ η &' Excess Nise Factr (ENF) M. Garampazzi et al., IEEE JSSC, March 014 Carl Samri

38 Cmparisn fr Ideal Cases Pwer efficiency η = P RF /P DC =[(I ω 0,rms /I DC )(V ω 0,rms /V DC )]= η I η V V DD F = γ F = γ V DD = π ( ) I T R = 4 π ( ) I T R η I = π η I = π,max = V DD,MAX = V DD η V = η V = 1 I T η = η I η V = π η = η I η V = π I T Same peak FM: fr γ=1 and η = /π the ENF shuld be 4.9dB In practical circuits ENF 10 db Carl Samri

39 Practical Issues: e.g. Tail Capacitr IDEAL REAL Tail parasitic capacitance can filter tail nise, but The diff. pair nise reaches the tank als when the pair is switched Wavefrm can be distrted: F increases Carl Samri

40 Example * -, $ ω ' FM = 10 lg 10 L P DC m /, & ω ) / % 0 ( +,. / " kt ( 1 + F % ) FM = 10 lg $ ' 10 $ # Q η ' = lg! $ 10 "# Q %& & FOM MAX " 1 + F % 10 lg 10 $ ' # $ η &' Excess Nise Factr (ENF) Q=15, L = 1nH, C = 4pF, ω 0 /π =.5 GHz, P = 15.6 mw Q=15, ENF = 4.9 db è FM = (ideal case) Assume real FM = 185, what is L(3MHz)? L(3MHz) = dbc/hz Carl Samri

41 Nise vs. Tuning Range Phase nise is nt nly traded with pwer (FM desn t tell all) A gd tuning range (e.g. Δω/ω 0 = 30%) affects the phase nise: n n n A large slpe f f 0 vs. V tune characteristic implies high sensitivity t tuning nise The varactr s Q usually reduces if its capacitance range increases Switched tuning features the same trade-ffs A mdified FM shuld include als the tuning range Carl Samri

42 Outline Basics f LC Vltage-Cntrlled Oscillatrs (VCOs) Phase Nise thery and mdels (simplified) Practical issues and trade-ffs Advanced LC scillatr s tplgies Carl Samri

43 Scaling Phase Nise and Pwer Hw t explit phase nise vs. pwer trade-ff? Example: save half f the pwer while increasing L by 3 dbs. Is it a gd idea t divide by tw the tail current in the VCO? I T R V DD ( ) = kt L R # 1 ω & 0 Q % ( $ ' # 1 + F & % ( $ ' 0 I T NO. If I T is halved the dissipatin drps by 3dB, but L increases by 6dB The ptimum bias is at ne particular tail current (transitin pint) Carl Samri

44 Change Tank t Save Current L C R Q = ω 0 RC = R ω 0 L ( ) = kt L # 1 ω & 0 R Q % ( $ ' I T 1 R C 1 same same ω 0, Q $ L & P 1 % DC same ω 0, Q & ' L The tank must be changed t keep the peak FM # 1 + F & % ( $ ' Is it pssible t switch the tank? (fr dynamical adjustment f nise at peak FM) Difficult: switching L degrades the Q. Better use tw scillatrs Carl Samri

45 Recnfiguring Transcnductr V DD V DD = V DD I T = V DD π R P DC = V DD π R = V DD I T = V DD R π 4 P DC = V DD R π 4 I T I T Same V DD and same tank Single pair features 4 X P DC (6 db larger pwer) Single pair feature als X (6 db better phase nise) Carl Samri

46 Recnfiguring Transcnductr V DD V DD Real bias is mre cmplex 55nm CMOS, 1.5 V n-nly: 4mA, L(MHz)= dbc/hz p-n: 6mA, L(MHz)= dbc/hz FM = 185 db A. Liscidini et al., IEEE JSSC, March 014 Carl Samri

47 Tail Filter / V DD - / Tail filter resnates at ω 0 ω 0 I T L T C T C L Large C L is practically a shrt at ω 0 E. Hegazi, H. Sjland, A. A. Abidi, IEEE JSSC, Dec. 001 Carl Samri

48 Tail Filter V DD V + V - V + V - V DD I T L T V S C T V S = π I T R C L Surces vltage ges belw grund η I is unchanged, but > V DD (imprves η V, ENF 4.5dB) Carl Samri

49 Tail Filter L V DD I T L T C T C L (E. Hegazi, H. Sjland, A. A. Abidi, IEEE JSSC, Dec. 001) L(3MHz) = -153 dbc/hz, 1 GHz - 1. GHz FM=195.4dB, 0.35 µm CMOS, Q=14 Carl Samri

50 Tail Filter V DD L Advantages n Large scillatin amplitude (imprves η V ) n MOS in tride withut lading the tank n Tail nise filtered by C L Drawbacks n T large scillatin amplitude (reliability) I T L T C T n n Tail filter either limit T.R. r must be tuned Additinal inductr is required C L Carl Samri

51 Class-C VCO L V DD V + V - V + V - t I(t) I(@ω 0 ) I T V B t I T C T Transistrs are perated in class-c, η I increases t 1/ FM ideally imprves by 3.9dB with respect t a class-b VCO Carl Samri A. Mazzanti, P. Andreani, IEEE JSSC, Dec. 008

52 Class-C VCO L V DD C T als filters tail nise Transistrs must avid tride regin: < V DD V B + V T V B can be nisy V B Additinal adaptive bias needed fr the start-up I T C T FM=193.5 db 196 db, 130nm CMOS, T.R. = 10.5 %, Q=17 Carl Samri

53 Wrap Up Basics f LC VCOs Linear time-variant analysis f 1/f phase nise Trade-ff phase nise vs. pwer (and tuning range) Trends f advanced tplgies: n n n Reach phase nise limit Imprve pwer efficiency Mdify tank tplgy Other imprtant issues nt discussed are the impact f 1/f nise, vltage-biased VCO (e.g. class D), quadrature VCOs, pulling frm supply, etc Carl Samri

54 Acknwledgment Thanks t A. Bnfanti fr suggestins and fr carefully checking this tutrial, t A. L. Lacaita, S. Levantin, P. Wambacq, A.Sheikhleslami fr their suggestins Carl Samri