ECEN60: Netwrk Thery Bradband Circuit Design Fall 01 Lecture 16: VCO Phase Nise Sam Palerm Analg & Mixed-Signal Center Texas A&M University
Agenda Phase Nise Definitin and Impact Ideal Oscillatr Phase Nise Leesn Mdel Hajimiri Mdel
Phase Nise Definitin An ideal scillatr has an impulse shape in the frequency dmain A real scillatr has phase nise skirts centered at the carrier frequency Phase nise is quantified as the nrmalized nise pwer in a 1Hz bandwidth at a frequency ffset frm the carrier L ( ) = 10lg P sideband ( +,1Hz) P carrier ( dbc/hz) 3
Phase Nise Impact in RF Cmmunicatin RX Reciprcal Mixing Strng Nisy TX Interfering with RX At the RX, a large interferer can degrade the SNR f the wanted signal due t reciprcal mixing caused by the LO phase nise Having large phase nise at the TX can degrade the perfrmance f a nearby RX 4
Jitter Impact in HS Links RX sample clck jitter reduces the timing margin f the system fr a given bit-errr-rate TX jitter als reduces timing margin, and can be amplified by lw-pass channels 5
Ideal Oscillatr Phase Nise The tank resistance will intrduce thermal nise in f = 4kT R The spectral density f the mean - squared nise vltageis v f n = in f Z tank 6
Tank Impedance Near Resnance 7 ( ) ( ) ( ) ( ) ( ) tank tank tank tank 1 Tank 1 1 Cnsider frequencies clse t resnance 1 Resnance Frequency: 1 1 = = = = + = + = = = = Q R Z Q R j Z Q R C C R Q C j LC L j LC LC LC L j Z LC LC L j L j C j Z
Ideal Oscillatr Phase Nise v f n in = f Z tank = 4kT R R Q = 4kTR Q The Equipartitin Therem[Lee JSSC 000]states that, in equilibrium, amplitude and phase - nise pwer are equal. Therefre, this nise pwer is split 1 evenly between amplitude and phase. L kt Psig Q { } = 10lg ( dbc/hz) Phase nise due t thermal nise will display a - 0dB/dec slpe away frm the carrier 8
Other Phase Nise Surces [Perrtt] Tank thermal nise is nly ne piece f the phase nise puzzle Oscillatr transistrs intrduce their wn thermal nise and als flicker (1/f) nise 9
Leesn Phase Nise Mdel L FkT P sig Q 3 1 f { } = 10lg 1+ 1+ ( dbc/hz) Leesn s mdel mdifies the previusly derived expressin t accunt fr the high frequency nise flr and 1/f nise upcnversin A empirical fitting parameter F is intrduced t accunt fr increased thermal nise Mdel predicts that the (1/ ) 3 regin bundary is equal t the 1/f crner f device nise and the scillatr nise flattens at half the resnatr bandwidth 10
A 3.5GHz LC tank VCO Phase Nise Measure Phase nise -30dB/decade -0dB/decade -105dBc 11 ELEN60-IC Design f Bradband Cmmunicatin circuit
VCO Output Spectrum Example -0dBm RBW=10K PN=-85dBm-(-0dBm)- 10lg 10 (10e3) =-105dBc dbc---in db with respect t carrier -85dBm 1 ELEN60-IC Design f Bradband Cmmunicatin circuit
Leesn Mdel Issues The empirical fitting parameter F is nt knwn in advance and can vary with different prcess technlgies and scillatr tplgies The actual transitin frequencies predicted by the Leesn mdel des nt always match measured data 13
Harjimiri s Mdel (T. H. Lee) Injectin at Peak (amplitude nise nly) Injectin at Zer Crssing (maximum phase nise) YCLO AMSC-TAMU 14
A time-varying Phase Nise mdel: Hajimiri-Lee mdel Impulse applied t the tank t measure its sensitivity functin i n f The impulse respnse fr the phase variatin can be represented as Γ is the impulse sensitivity functin ISF qmax, the maximum charge displacement acrss the capacitr, is a nrmalizing factr 15 ELEN60-IC Design f Bradband Cmmunicatin circuit
ISF Mdel The phase variatin due t injecting nise can be mdeled as: The functin, Γ(x), is the time-varying prprtinality factr and called the impulse sensitivity functin. The phase shift is assumed linear t injectin charge. ISF has the same scillatin perid T f the scillatr itself. The unity phase impulse respnse can be written as: YCLO AMSC-TAMU 16
Hw t btain the Impulse sensitivity functin fr a LC scillatr Γ(τ) can be btained using Cadence Cnsider the effect n phase nise f each nise surce 17 ELEN60-IC Design f Bradband Cmmunicatin circuit
Typical ISF Example The ISF can be estimated analytically r calculated frm simulatin. The ISF reaches peak during zer crssing and zer at peak. YCLO AMSC-TAMU 18
Phase Nise Cmputatin The impulse sensitivity functin is used t btain the phase nise impulse functin h φ ( t, τ ) = ( τ ) u ( t τ ) Γ q max The phase nise can then be cmputed by the superpsitin (cnvlutin) integral f the any arbitrary nise current with the phase nise impulse functin φ ( t) = h ( t, τ ) i( τ ) dτ = Γ( τ ) i( τ ) dτ φ 1 q max t 19
ISF Decmpsitin w/ Furier Series In rder t gain further insight, and because the ISF is peridic, it may be expressed as a Γ c 0 ( τ ) = + c cs( n τ + θ ) n= 1 n n Furier series where the cefficients c n are real andθ is the phase f n the nth ISF harmnic. cefficients are deteremined. φ The phase nise can then be cmputed by 1 q c 0 ( t) = i( τ ) dτ + c i( τ ) cs( n τ ) max t n n= 1 This allws the excess phase frm an arbitrary nise surce t be cmputed nce the ISF Furier Essentially, the current nise is and scaled accrding t the ISF cefficients. t dτ mixed dwn frm different frequency bands 0
Phase Nise Frequency Cnversin First cnsider a simple case where we have a sinusidal nise current whse frequency is integer multiple m f the scillatin frequency i ( t) = I cs[ ( m + ) t] When perfrming the phase nise cmputatin integral, there will be a negligible cntributin frm all terms ther than n = m ( t) I ( t) The resulting frequency spectrum will shw tw equal sidebands at v φ P ut SBC m m ( t) = cs[ t + φ( t) ], ( ) cm sin q max I 10lg 4q m max 1 Nte that this pwer is prprtinal t there will be tw equally weighted sidebands symmetricabut the carrier with pwer c m. near an ±. Assuming a sinusidal wavefrm 1
Phase Nise Due t White & 1/f Surces Extending the previus analysis t the general case f a white nise surce results in P SBC ( ) i n f 10lg 4q m= 0 max m Here nise cmpnents near integer multiples f the carrier frequency all fld near the carrier 1 itself and are weighted by c. Nise near dc gets upcnverted, weighted by cefficient c, s1 0 f nise becmes1 f 3 nise near the carrier. Nise near the carrier stays there. White nise near higher integer multiples f the carrier gets dwncnverted and weighted by 1 f.
Hw t Minimize Phase Nise? In rder t minimize phase nise, the ISF cefficients c n shuld be minimized. Using Parseval's therem m= 0 c m π 1 = Γ π 0 ( x) dx = Γ rms The spectrum in the1 f regin can be expressed as Thus, reducing Γ rms L ( ) i n Γrms 10lg f = qmax will reduce the phase nise at all frequencies. 3
1/f Crner Frequency Cnsider current nise which includes1 f cntent i n,1 f 1 f = in where 1 f is the1 f crner frequency Frm the previusslide L ( ) i n c0 10lg f = 8qmax 1 f Thus, the1 f 3 crner frequency is 1 3 f = 1 f c 4Γ 0 rms = 1 f Γ Γ dc rms This is generally lwer than the1 f device/circuit nisecrner. If Γ dc is minimized thrugh rise - and fall - time symmetry,then there is the ptential fr dramatic reductins in 1 f nise. 4
Cyclstatinary Nise Treatment Transistr drain current, and thus nise, can change dramatically ver an scillatr cycle. The LTV mdel can easily handle this by treating it as the prduct f statinary white nise Here i n0 nise surce, and α ( x) and a peridic functin. i n Γ ( t) = i ( t) α( t) is a statinary white nise surce whse peak valueis equal t that f eff n0 frmulate an effective ISF ( x) = Γ( x) α( x) 0 the cyclstatinary is a peridic unitless functin with a peak valuef unity. Using this, we can 5
Key Oscillatr Design Pints As the LTI mdel predicts, scillatr signal pwer and Q shuld be maximized Ideally, the energy returned t the tank shuld be delivered all at nce when the ISF is minimum Oscillatrs with symmetry prperties that have small Γ dc will prvide minimum 1/f nise upcnversin 6
Phasr-Based Phase Nise Analysis Mdels nise at sideband frequencies with mdulatin terms The α 1 and α terms sum c-linear with the carrier phasr and prduce amplitude mdulatin (AM) The φ 1 and φ terms sum rthgnal with the carrier phasr and prduce phase mdulatin (PM) 7
Phasr-Based LC Oscillatr Analysis This phasr-based apprach can be used t find clsedfrm expressins fr LC scillatr phase nise that prvide design insight In particular, an accurate expressin fr the Leesn mdel F parameter is btained L FkT Psig Q { } = 10lg ( dbc/hz) 8
LC Oscillatr F Parameter L FkT Psig Q { } = 10lg ( dbc/hz) 1st Term = Tank Resistance Nise nd Term = Crss-Cupled Pair Nise 3rd Term = Tail Current Surce Nise The abve expressin gives us insight n hw t ptimize the scillatr t reduce phase nise The tail current surce is ften a significant cntributr t ttal nise 9
Tail Current Nise The switching differential pair can be mdeled as a mixer fr nise in the current surce Only the nise lcated at even harmnics will prduce phase nise. YCLO AMSC-TAMU 30
Lading in Current-Biased Oscillatr The current surce plays rles It sets the scillatr bias current Prvides a high impedance in series with the switching transistrs t prevent resnatr lading YCLO AMSC-TAMU 31
Nise Filtering in Oscillatr Only thermal nise in the current surce transistr arund nd harmnic f the scillatin causes phase nise. In balanced circuits, dd harmnics circulate in a differential path, while even harmnics flw in a cmmn-mde path A high impedance at the tail is nly required at the nd harmnic t stp the differential pair FETs in tride frm lading the resnatr. YCLO AMSC-TAMU 3
Nise Filtering in Oscillatr Tail-biased VCO with nise filtering. YCLO AMSC-TAMU 33
Phase Nise w/ Tail Current Filtering Tail current nise filtering prvides near 7dB imprvement 34
Nise Filtering in Oscillatr A tp-biased VCO ften prvides imprved substrate nise rejectin and reduced flicker nise YCLO AMSC-TAMU 35
Next Time Divider Circuits 36
In general the impulse sensitivity functin is peridic with a fundamental frequency equal t the scillating frequency The input sensitivity functin can be characterized as a Furier Series: An the phase nise is then Therefre: 37 ELEN60-IC Design f Bradband Cmmunicatin circuit
If the input nise is represented as Due t the peridicity f the terms, the series cnverge t (n=m term nly): Phase nise Nise Γ (m-1) 0 m 0 m 0 38 ELEN60-IC Design f Bradband Cmmunicatin circuit
If the input nise is represented by its pwer distributin functin Fr small nise level V V V ut ut ut = V V V 0 0 0 cs cs cs ( t + φ( t) ) ( t) V0 sin( t) sin( φ( t) ) ( t) V ( sin( t) ) φ( t) 0 39 ELEN60-IC Design f Bradband Cmmunicatin circuit