DESIGN METHODS,TRANSISTOR MODELING, AND NUMERICAL SIMULATION OF LOW PHASE NOISE OSCILLATORS

Transcription

1 DESIGN METHODS,TRANSISTOR MODELING, AND NUMERICAL SIMULATION OF LOW PHASE NOISE OSCILLATORS JC NALLATAMBY* - M PRIGENT* -M CAMIADE** - J OBREGÓN* * IRCOM-CNRS - Université de Limges - 13 Av A Thmas Limges - France ** UMS - Rute Départementale Orsay Cedex - France 1

2 Must the design methds f lw phase nise scillatrs be revisited? - New design methds allwing t clarify design prcess, are regularly prpsed -New deep insights n phase nise generatin in free running scillatrs, are regularly presented -New accurate mdels f transistrs and their assciated nise surces, are cntinually develped

3 OUTLINE - Intrductin Design bjectives fr lw-nise peratin - Mdern linear design apprach A simplified transistr mdel, valid fr FETs and HBTs Open-lp gain cncept: calculatin f the starting scillatin frequency Fast analytic calculatin f phase nise: Leesn frmula revisited 3

4 - Nnlinear mdels and design tls Semi-cnductr device mdeling: - Large signal mdels f FETs and HBTs Nise surce mdeling riented t CAD f nnlinear circuits: - Lw frequency nise surces - RF nise surces Simulatin tls: - Steady state analysis - Lcal stability - Phase nise simulatin 4

5 - Practical examples Breadbard example MMIC-based scillatrs fr high vlume prductin - Cnclusin 5

6 S(ω) IRCOM UMR CNRS n University f Limges Design bjectives fr lw-nise peratin 0 Ω ω -Ω ω ω +Ω Fig1 : Nise cnversin in scillatrs ω A well-designed scillatr requires the ptimizatin f the circuit behavir: - n the ne hand at the scillatin-frequency (large signal) - n the ther hand at the intermdulatin frequencies resulting frm the interactin between (small signal) nise surces and (large signal) steady-state 6

7 Optimizatin bjectives fr lw phase nise generatin - At the scillatin frequency : ω Maximizatin f the transistr added pwer - Arund the scillatin frequency : Maximizatin f the phase slpe f the scillatr pen lp gain - At lw-frequencies near DC: The impedance presented by the scillatr circuit and the bias netwrk must be lw in rder t reduce the influence f the transistr lw-frequency nise surces n the phase nise 7

8 A simplified transistr mdel: IRCOM UMR CNRS n University f Limges Mdern linear design apprach Frm a circuit-design pint f view, a transistr is fundamentally a nn-linear vltage-cntrlled current surce The cntrlling vltage is taken at a dide prt : frward biased in HBTs and reverse biased in HEMTs Due t the highly nnlinear equivalent capacitance f this input dide, the phase nise generatin by up cnversin takes place principally at this cntrlling vltage prt Parasitics Gi n Ci Iut=f Vc NL (Vc,Vut Vut n ) Cntrlled current-surce 8

9 A linearizatin f the nnlinear mdel gives the linear simplified mdel: V ~ ( ω ) in G in C in G m V ( ω ) in G ut This mdel can be used fr a first linear analysis f an scillatr circuit: V ~ ( ω ) in G in C in G ut G m V ( ω ) in Feedback Path Includes the utput lad 9

10 The pen lp gain cncept: E ~ ( ω ) ext V ~ ( ω ) in G in C in G ut G m E ( ω ) ext Feedback Path ~ V ~ ( ω ) G = pen lp gain = OL ~ in E ( ) ext ω 10

11 - In a first small signal apprximatin, the scillatin frequency is given fr the cnditins ~ G OL > 1 ~ G ϕ OL OLG ( ω ) > 1 ~ ( ω ) = G ( ω ) OL 0 : During the transient G ~ OLreduces t a large signal perating pint where ~ = OL ( ω ) G ( ω ) 0 ~ G LS ω ( ) = 1 : During the transient ϕ OLG ω shift in frequency t the perating pint where ( ) = 0 and : ϕ ( ) OLG ω large signal ω small signal ϕ LS ω 11

12 Phase (degree) Gain(dB) Phase (degree) Gain(dB) 0 0 f f f spurius = 800MHz f = 10GHz scil Simulatin results f linear pen-lp gain fr an MMIC HEMT scillatr Simulatin results f linear pen-lp gain fr an MMIC HEMT scillatr with circuit stabilizatin 1

13 Phase (degree) Gain(dB) 0 f f Simulatin results f linear-lp gain fr a transistr scillatr Questin : Are the previus cnditins sufficient t find a stable scillatin? Answer : N, the lcal stability f the large signal perating pint must be analyzed See sectin Nn linear simulatin tl 13

14 A first respnse can be given by the linear analysis If : ~ G ϕ OL OL ( ω ) dϕ OL dω > 1 ~ ( ω ) = G ( ω ) ω < 0 OL = 0 Then the final steady-state scillatin shuld be stable 14

15 Linear evaluatin f phase nise in free running scillatrs: - A useful tl: The mdified Leesn frmula In the field f linear feedback-systems frmalism, the Leesn frmula is a useful tl fr the determinatin f phase nise in feedback scillatrs A direct applicatin f the Leesn mdel withut care, can lead t errneus results because the frmula cntains hidden parameters which are generally unwittingly ill-evaluated, r neglected A detailed analysis enables us t enlighten the hidden parameters leading t a mdified Leesn frmula which is valid fr all scillatr circuits 15

16 Amplifying device Feedback path This general representatin f a feedback scillatr includes - the amplifying device: the transistr - the feedback path, which includes the lad cnductance white nise surce I n V 1 Passive tw-prt G M0 V 1 A linear representatin (after nnlinear equivalent linearizatin) f the feedback scillatr in the frequency dmain allws t highlight: The amplifying (active) functin f the transistr :the vltage cntrlled current surce f the transistr is islated G M V 1 frm the passive elements f the transistr mdel, which are nw included in the passive, reciprcal feedback path Fr phase nise calculatin purpses a carrier vltage f peak valuev 1 at the scillatin frequency ω 0 is implied at the cntrlling input prt f the transistr 16

17 I n I 1 V 1 G M V 1 V A C B D I Feedback scillatr with a chain matrix descriptin f the passive circuit One nise surce alne is nt always sufficient t characterize a nisy transistr Hwever in the mst general case tw crrelated surces : - In1 input and In input are sufficient In that case In becmes In = In1 input + In input + R In1 * input In input Fr phase nise calculatin, this input nise surce must be carefully evaluated accrding t the lcalizatin f the physical nise surces in the transistr 17

18 Determinatin f S ϕin In the field f linear feedback system frmalism, the equivalent clsed lp representatin f the figure belw is deduced In this figure, we have: G M0 V 1 = I ut, V = V ut, V n represents the equivalent input nise vltage surce due t I n In rder t determine S ϕin, we first calculate V n arund the scillatin frequency ω I n A C V n + V 1 I ut + G M0 1 C 1 B G M 0 AG M 0 V ut Let ω n = ω + ω We btain V n = I n A 0 C 0 A and C are the chain-matrix cefficients taken at ω 18

19 Determinatin f S ϕin (cntinued) A cnventinal treatment shws that by additin t a pure carrier signal f peak value V 1 at the frequency ω, tw uncrrelated cmpnents at ω + ω and ω ω give rise t a mdulated carrier with a phase-nise spectral density S ϕin At an ffset frequency ω frm the carrier, we btain V n S = ϕ ω in V 01 Finally we btain a white phase-nise spectral density in a 1-Hz bandwidth I n S S A 0 I A = n 0 ϕ C = in C V 0 V

20 Determinatin f the scillatin frequency In rder t find the scillatin frequency, by setting I n =0 and pening the feedback lp, the figure belw is btained V ext G M0 I ut 1 BG M 0 AG M 0 V ut V 1 1 C We have: I G V and ut V 0 ext = I M ut = C 1 The pen-lp gain can be written as: ~ V G = 1 = G V M 0 ext C Where G ~ dentes the cmplex gain in the frequency dmain 0

21 Determinatin f the scillatin frequency (cntinued) Oscillatin cnditins are fulfilled fr V 1 = V ext It fllws that: ~ G G G ( ω ) = 1= - M 0 = C C M 0 Nte that frm the abve expressin C must be real and negative Let C = C R +jc I, The scillatin frequency is determined by : C I (ω 0 ) = 0 and then C R (ω 0 ) = C 0 =-G M0 1

22 Determinatin f the laded Q factr f the scillatr The lp gain is written as : ~ G G = Ge j ϕ = - M 0 C At the scillatin frequency ω ~ j G C G = 1 ( ω ) 0 ϕ( ω ) = 0 0 dϕ dω Frm G = Ge ϕ = - M 0, the slpe f the lp gain can be btained at the scillatin frequency as:

23 Determinatin f the laded Q factr f the scillatr (cntinued) dϕ C' d C I dc = ω with C ' = ω 0 0 I dω I ω 0 We dente the scillatr laded Q factr Q Lscill as Q Lscill ω ω ' d C = ϕ ω = dω C I ω 0 - Nte as a general rule that the scillatr laded Q factr Q Lscill des nt cincide with the laded Q factr f the passive circuit - It cincides nly fr sme elemental feedback netwrks The relevant cefficient fr the calculatin f the utput phase nise:q Lscill is prprtinal t the grup delay f the feedback path It is directly related t the phase-frequency relatinship f the scillatr lp gain 3

24 4 IRCOM UMR CNRS n University f Limges + + ϕ ut 1 + ω 0 ' 1 1 C i C j ϕ in h ut =1 ϕ ut Determinatin f the utput phase nise The fllwing nrmalized representatin f the feedback scillatr can be deduced We have successively:, ' = ω ϕ ϕ C I C j ut in, ' 0 1- = ω ϕ ϕ I C C j in ut + = 1 ' 0 1 ω ϕ ϕ I C C in S ut S + = 1 ' ω ϕ I C C C A V I n ut S Explicitly, we btain

25 Determinatin f the utput phase nise (cntinued) The near carrier frequency, S ϕut becmes I n A S = 1 ϕ C' 0 ut ω V I 01 Τhese equatins must nw be cmpared t the crrespnding Leesn frmulas recalled belw: ω S = S 1+ 0 ϕut ϕ in 4Q ω Leesn and near carrier frequency ω S = S 0 ϕ ut ϕ in 4Q ω Leesn By cmparisn with the previus equatins we have ω C = d ω = = Q C I ' Q ϕ Leesnω d Lscill ω ω ω 0 The hidden parameters included in the Leesn frmula, may result in a Q Lscill different frm the laded Q factr Q L f the feedback tank by mre than ne rder f magnitude 5

26 In rder t accunt fr such a pssible difference, let us lk at the fllwing example Dielectric resnatr λ, 4 Z C θ R, Z C Nnlinear nisy transistr and its bias circuits Output matching circuit G Lad Feedback tw-prt The circuit includes a dielectric resnatr cupled by means f a lssless transmissin line f characteristic impedance, and electrical length θ R at the scillatin frequency The purpse f intrducing the line is t allw variatin in the laded Q factr f the scillatr (withut simultaneusly varying the resnant frequency r, at the first rder, the laded Q f the passive circuit) by the varying length θ R 6

27 L C A simplified linear schematic λ, 4 Z C C C G C θ R, Z C G in I n V in G V M in G ut G L 1 : N with the electrical equivalent circuit I n V in G M V in I in 1 : N I ut V in G in Y f G L G ut V ut 7

28 A clsed lp representatin can be deduced I n 1 Y T e n + + V in G M N G M V ut N Y T with Y = Y + G in + N T f G + ut G L Y f is the admittance f the whle resnatr circuit brught back by the cupling line t the transistr input prt Arund the scillatin frequency, a straightfrward calculatin gives π G C c c ω Y = G + j ( θ ) ω f G c 1 cs c The term: C c is due t the lumped resnatr, and is due t the pen circuit stub π G c ω 8

29 Phase nise calculatin One can nw represent the scillatr arund the scillatin frequency ω by the nrmalized clsed lp representatin f the figure belw S ϕ in + + ~ =1 G S φ ut H ( ω ) The nrmalized transfer functin arund the scillatin frequency is written as: H ( ω ) = 1 1 = j dϕ ω dω 1+ j 1 πg C c c ω G tt cs ( θ ) ω 9

30 The utput phase nise is calculated as IRCOM UMR CNRS n University f Limges Frm this equatin, it can be shwn that A numerical simulatin cnfirms this result G S = S 1+ tt 1 ϕ ut ϕ in πg cs ( ) C c θ ω c ω S δϕut is peridic with θ ο θ ο, G R G R C R L R Gate R D Drain L G R G δi 1/f Intrinsic nnlinear Mdel L D C pg C pd G Lad R S L S Surce HEMT Phase nise has been successively simulated with tw nise surces : a white nise surce and 1/f nise surce 30

31 As expected, a phase nise rll-ff, respectively f 0dB/dec and 30 db/dec versus frequency ffset frm carrier is btained Besides, fr a fixed frequency ffset, bth nise surces give similar curves as a functin f the electrical length: θ ο This fig shws the results btained at 100 khz frm carrier, with the white nise surce: curve (a) is the analytical result and curve (b) is the simulatin result UNSTABLE REGION curve (b) curve (a) Withut transmissin line Applicatin f the Leesn frmula gives a phase nise f -11 dbc practically cnstant as a functin f θ ο Nte that the unladed Q factr f the resnatr used was :Q =

32 Nte: - The Q factr f the scillatr circuit defined as : Q Lscill ω ω = dϕ dω ω Is nt, as a general rule, equal t the Q factr f the circuit defined as: Q circuit ω ε =ω P 3

33 As an example, let us cnsider the fllwing simple scillatr circuit: I n 1 : 1 V C G in G M0 V C G ut G Lad G R C R L R Zc 1 = ; θ G R R 33

34 The energy stred is : - In the resnatr cupling line - In the resnatr : - Pwer dissipated : Then, P = 1 ε = line 1 θ V Z c cω ( ω ) ε 1 resnatr = C R V c G + G + G in L ut Q L ω ε = ω tt P V c = ω ( ω ) ( ) ω = 1 ( ω ) G tt V c θ C + R Z ω c G tt 34

35 The lp gain phase slpe arund ω is dϕ C = G R cs θ dω ω tt and Q scill = ω dϕ C = ω d G R cs θ ω ω tt Q scill is different frm Q L, Hwever fr θ = 0 Q = Q scill L 35

36 Large signal mdels Nise surces IRCOM UMR CNRS n University f Limges Mdeling f Semi-cnductr devices Large signal electrical mdels must be physically based in rder t accurately take int accunt the interactins between the nise surces and the nnlinear elements in the semi-cnductr devices The lw frequency nise surces up cnverted near the scillatin frequency must be accurately lcalized in the semi-cnductr devices 36

37 The cunter example L chke I n1 Niseless accurate NL transistr mdel I n L chke - Let us suppse I n1 and I n are tw lw frequency nise surces representing the LF nise surces extracted frm the transistr If this mdel was physically accurate, it wuld be sufficient t bias the transistr thrugh tw chke inductances in rder t eliminate all pssible up-cnversins 37

38 Unfrtunately in nnlinear design, the nise surces must be lcalized exactly where they are generated An example f nnlinear distributed mdel f HEMT Gate Lg Cpg RgGi Intrinsic transistr Si Rs Di Rd Cds Ld Cpd Drain Gi Cb gs Ls Surce Cg(Vg 1 ) Cg(Vg k ) Cg(Vg N ) Vg k VgN Cb gd Si Id(Vg 1, Vc 1 ) Id(Vg k, Vc k ) Id(Vg N, Vc N ) Vc 1 Vc k Vc N Vc k Di 38

39 In field-effect transistrs, accurate mdeling f the interactin between steady-state large signal and lw-frequency nise surces distributed alng the surce-drain channel, requires a distributed mdel Fr this purpse, a nn-linear, nn-unifrm, distributed mdel f FETs has been develped Gate Vg 1 Vg K Vg N V gd Surce Drain δic 1 δic K δic N Vc 1 Vc k VcN Nnlinear HEMT Mdel including the LF nise surces f the channel 39

40 - The channel is cnsidered as a nn-unifrm, nn-linear, active transmissin line with N unit cells The number f cells depends n the channel length and the perating frequency A gd cmprmise fr micrwave devices is 10 N 0 - Every unit cell includes : A nn-linear gate-channel capacitance: C=fc(Vg k ) in parallel with a Schttky dide: I = fg(vg k ) These elements are a functin f their wn prt vltage Vg k A nn-linear channel current-cntrlled surce: Ic = fi(vg k, Vc k ) which depends n the tw cell-vltages: Vc k and Vg k 40

41 - The mdel has been extracted fr many HEMTs frm different prcesses Figure 1 shws a cmparisn between the mdel and measured characteristics fr a 4*05*50 µm PHEMT mdeled with 10 unit-cells - It must be pinted ut that : The mdel is fully extracted frm measurements The functins fc, fg, fi are identical fr all unit cells The mdel reprduces the nn-unifrm nn-quasi-static behavir f the transistr in a natural way 41

42 Ids (A) MEASURED AND MODELED I-V CHARACTERISTIC PH5-4 x 50 µm Vgs= -08 V, Vds=51 V IRCOM UMR CNRS n University f Limges Measures Mdel Vgs= +10 V +08 V +06 V +0 V +00 V -0 V -04 V -06 V Vds (V) Fig 1 : PH 5 Transistr distributed mdel DISTRIBUTED MODEL : Vc k AND Vg k Cell number : k Gate - Channel ptentiel Vg k (V) Cell number : k Nte that such a distributed mdel including distributed nise surces has been recently prpsed fr MOSFETs 4

43 Sme biplar transistr mdels including a distributed circuit (fr the base regin), have been prpsed fr linear applicatins (Te-Winkel, Pritchard, Van der Ziel, ) Hwever t ur knwledge, nnlinear distributed mdels f HBTs suitable fr CAD f nnlinear circuits have nt been prpsed In Rb 1/f Rc Lc Lb Cpb Rbb Cpc In CE Sht In BE Sht In BE G-R In BE 1/f Re In Re 1/f Le This figure shws an example f HBT nnlinear mdel including the main nise surces used fr CAD f scillatr circuits Nte that the circuit shws pssible lcalizatin f nise surces Frtunately many f them can be neglected, accrding t the technlgy 43

44 Nise surce mdeling Physical nise surces: G-R Nise Lw-frequency nise surces Fundamental 1/f nise Linear resistances RF nise surces: diffusin nise FET-Channel Sht nise in SC junctins 44

45 Tw physical surces f lw-frequency nise must be cnsidered : - δn(t) : Randm fluctuatins f the carrier number : N - δµ(t) : Randm fluctuatins f the lw-field mbility : µ The first ne is due t traps present in the semi-cnductr The secnd is named the fundamental 1/f nise, and applies als t the lw-frequency fluctuatins f the diffusin cefficient δd(t) One physical surce f RF nise : The diffusin nise gives rise t white nise surces: - Lw electric field : thermal nise - High electric field: diffusin nise - Nise f minrity carriers in semi-cnductr junctins: sht nise 45

46 Circuit-CAD riented mdeling f lw-frequency nise surces in semi-cnductr devices LOCAL CURRENT- NOISE SOURCES IN HOMOGENEOUS LINEAR SAMPLES x Current Applied vltage : V(t) Slice f hmgeneus linear sample - Let us cnsider a slice : x with a crss-sectin : A, and a carrier cncentratin : n - the velcity f carriers due t the deterministic lcal applied vltage : V(t) writes : V N x c (t) = µ - A deterministic current I(t) fllws : q x V(t) x I (t) = q V c(t) = V(t) µ N with N = n A x 46

47 - If lw-frequency randm fluctuatins : δn(t) and δµ(t) are present in the slice : A x Lcal Nrtn nise surces are given by : δi(t) = DC+RF Deterministic applied vltage q V(t) δµ x [ µ δn(t) + N (t)] Lw-frequency randm fluctuatins Lw-frequency randm fluctuatins Cnclusin: The initial lw-frequency randm fluctuatins : δn(t) and δµ(t) are prbed by the DC+RF deterministic applied vltage, which results in Nrtn nise surces with the initial lw-frequency spectrum transpsed arund the Furier cmpnents f the deterministic applied vltage giving rise t cyclstatinary nise surces 47

48 RF nise: Since Dragne, we have knwn that the diffusin nise with a δ crrelatin functin is mdulated by a large-signal peridic applied vltage - A lcal nise surce described fr a DC applied by its spectral density: I ( x) = q Dn ( ) x x f A x δ 4 δ ' - becmes cyclstatinary when a peridic large signal is applied: δi k * = 4 DN( k j) δ ' j A ( x) I ( x) q δ x x f 48

49 Where N( k j) is the ( k j) th harmnic f n(x,t) In ther wrds: In the presence f a large signal peridic electric field: - the lw frequency current nise surces - and the RF current nise surces Are mdulated and create upper and lwer side-bands arund every harmnic cmpnent f the large signal peridic electric field 49

50 An external DC Applied Vltage gives rise t: - DC lcal current density int the SC device: J ( x) = q N ( x) v( x) DC LF randm fluctuatins f traps ccupancy : - LF randm fluctuatins f free carriers : δn x, t N T x, t - LF randm fluctuatins f velcity recmbinatin : δv x, t It results tw main generatins: N T x, t δn x, t LF δv x, t LF δn v LF DC δv LF N DC gives: bulk LF bulk nise current surce LF surface recmbinatin nise current surce surface 50

51 An external Large Signal Applied Vltage gives rise t: - Large signal lcal current density int the SC device: J x, t = q N x, t v x, t DC and RF cmpnents LF randm fluctuatins f traps ccupancy : - LF randm fluctuatins f free carriers : δn x, t N T x, t - LF randm fluctuatins f velcity recmbinatin : δv x, t It results tw main generatins: N T x, t δn x, t LF δv x, t LF δn v LF DC+RF δv LF N DC+RF gives: bulk Cyclstatinary bulk nise current surce surface Cyclstatinary surface recmbinatin nise current surce 51

52 Simulatin tls A cherent set f rbust and reliable simulatin-tls is required in rder t : Find the steady-state scillatin-frequency and variables at every nde f the circuit Perfrm a linear/nn-linear stability analysis and find pssible spurius frequencies Calculate AM and PM nise spectrum Steady-state analysis is generally perfrmed using the well-established harmnic-balance methd 5

53 A simplified NL lcal stability analysis: The linear/nn-linear stability analysis may nw be easily perfrmed with cmmercially available sftware packages The figure belw shws the principle f the nn-linear pen lp cncept applied t a HEMT scillatr circuit, fr stability analysis purpses Vgs Ids=f(Vgs,Vds) Initial circuit Vds Stp band filter ω = Ω E(Ω): small signal perturbatin Vgs Eext=Vgs Stp band Ω E(Ω) new Vgs Ids=f(new Vgs,Vds) Vds Mdified transistr mdel fr stability analysis 53

54 The pen lp gain Ω = is calculated and its mdule and G ~ phase are pltted as a functin f frequency V gs Ω E Ω Nte that these plts are equivalent t the plt drawn in the linear case fr the determinatin f the scillatin frequency (See sectin: Mdern linear design apprach) 54

55 Steady-state analysis - Mdern harmnic balance sftware packages include: Mdified ndal analysis frmulatin t describe the circuit Krylv subspace methds t slve the netwrk matrices -Cnditins fr a successful steady-state simulatin: High number f spectral cmpnents High number shuld lead t high accuracy, but 55

56 Are the mdels f passive elements and active devices valid at very high frequencies? Slutin : Shrt-circuit at the active device prts fr f > Nf N: depending n the fundry mdels Are the number f samples f the time-dmain state variables sufficient? Slutin : Accurate results need versampling Cnclusin: Special care is needed in chsing the number f spectral cmpnents as well as the number f time dmain samples (See example belw) 56

57 Simulatin f phase nise in scillatrs - Accurate simulatin needs t take int accunt: All kind f nise surces lcalized at every nde f the circuit: - at lw frequency - r RF frequency linear, and cyclstatinary nise surces must take als int accunt the lw-frequency dynamics f the scillatr circuit and bias netwrk (see example belw) 57

58 - Accurate mdeling f the nise surces: Current mdels are generally nt sufficient There is a cnfusin between : - Mdels valid nly fr linear applicatins - Mdels whse validity extends t nnlinear applicatins -The lcalizatin f the nise surces in the nnlinear device mdels is still nly apprximated 58

59 Oscillatr phase nise calculatin in the frequency dmain by means f cnversin matrices frmalism - utput wavefrm: V( t) = V + v t ( ) cs ω t t ϕ ϕ V V ϕ v(t) ϕ t Peak amplitude f the carrier ω A cnstant phase Amplitude nise mdulatin Phase nise mdulatin 59

60 60 IRCOM UMR CNRS n University f Limges - At a frequency ffset Ω, frm carrier: AM and PM nise mdulatins are expressed as pseudsinusids Φ + Ω = A t V t v cs Φ + Ω = φ ϕ ϕ t t cs Where t v and t ϕ dente their assciated spectral densities; φ φ φ ϕ,,, A V are randm variables We can write V(t) as : ( ) ( ) ( ) Σ + +Ω Σ + + Ω + + = ϕ ω ϕ ω ϕ ω t V V t V V t V t V cs cs cs

61 Let s write We btain : j e φ V A ~ V = V V j e φ ϕ ϕ = ~ ϕ jφ ~ V e = V jφ V e Σ ~ = V Σ Σ AM nise = PM nise = P AMtt P carrier P PMtt P carrier = = ~ V V ~ ϕ V where = S A = = S = φ V = V ω Ω V = V ω +Ω Σ ~* jϕ V e ~* jϕ V e ~* + V e Σ V ~* V e Σ V jϕ jϕ 61

62 The AM and PM nise are defined as: AM nise C N at = 10lg Ω dbc P AM ssb P carrier by Hertz PM nise at Ω = 10lg dbc P PM ssb P carrier by Hertz Sφ = ~ V, ~ V, Σ V, ϕ are calculated using the harmnic balance and its related technique: the cnversin matrices frmalism 6

63 RELATION BETWEEN STEADY-STATE ACCURACY (ε) ε = f (NH, Nt, δω 0 ) ε < ε < ε < ε AND PHASE NOISE RESULTS Accurate PM nise calculatin is directly related t the large steady-state accuracy It is nly a numerical prblem which is slved by an apprpriate chice f : - number f harmnics - versampling 63

64 Cnversin matrices frmalism allws t handle large signal/small signal interactins in nnlinear devices with gd accuracy very efficient fr nise calculatin in nnlinear circuits driven by a peridic/quasi-peridic large signal : mixers cnverters scillatrs 64

65 - Principle f Cnversin matrices Let us cnsider a nnlinear element described by its cnstitutive equatin: y t = f x t Examples: - Nnlinear cnductance I/V - Nnlinear capacitance Q/V In small-signal applicatins arund a bias pint X : δy t = df δ x t = h x t δ dx X h : cnstant, time invariant 65

66 - If X becmes a peridic time varying bias pint when a large signal f fundamental frequency ω is applied: X X t + N = N X e k jkω t Then: h h t + N = N H e k jkω t - If the small signal δx t cntains an input frequency: Ω Intermdulatin cmpnents are generated at frequencies: kω ±Ω 66

67 67 IRCOM UMR CNRS n University f Limges - In the frequency dmain x t δ and y t δ are expanded as: ( ) +Ω +Ω Ω Ω Ω = N X k X X k X N X X ω δ ω δ δ ω δ ω δ ω δ * * r and ( ) +Ω +Ω Ω Ω Ω = N Y k Y Y k Y N Y Y ω δ ω δ δ ω δ ω δ ω δ * * r Then by identificatin with: = t x h t t δy δ

68 68 IRCOM UMR CNRS n University f Limges A matrix relatin is fund in the frequency dmain: ( ) ( ) ω δ ω δ X H Y r r = with: = H N H N H N H H H H N H H H H H N H H H H H 1 * 1 * 1 * 1 1 * * * 1 is the cnversin matrix f the nnlinear element H

69 - The variable δx(ω) and the resulting functin δy(ω) at each mixing frequency are related by the cnversin matrix - Since the relatin between δx r ( ω ) and δy r ( ω ) is linear (Matrix frm), it results that the nnlinear devices appear as a linear peridic time varying element: r y () t = f x( t) = Nnlinear element ( ) δy( ω) [ H] δx( ω) becmes r LPTV element Nise analysis can be perfrmed using the cnversin matrices frmalism when the large signal is applied at the scillatin frequency ω and its harmnic frequencies 69

70 Fr example, we recall the phase nise expressin : ~* V e jϕ ~* V e Σ jϕ S = φ V -first : V, ϕ f the niseless scillatr are cmputed by harmnic balance -secndly : ~ V at ω Ω ~ andv Σ at ω+ω due t internal nise surces, are cmputed by the cnversin matrices methd 70

71 Nise crrelatin matrix cncept - Intrduced fr several linear nise surces by HA Haus and RB Adler : 1956 I n1 (Ω) Physical Nise surce I n3 (Ω) I n (Ω) - One internal physical surce at a frequency Ω gives rise t several equivalent external crrelated nise surces at the same frequency Ω 71

72 7 IRCOM UMR CNRS n University f Limges The crrelatin matrix at the given frequency Ω writes: Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω Ω = Ω 3 * 3 1 * 3 * 3 * 1 * 3 1 * 1 1 n I n I n I n I n I n I n I n I n I n I n I n I n I n I n I C The crrelatin matrix f several linear nise surces describes the aut-crrelatin and crss-crrelatin between them at a frequency Ω

73 Nise crrelatin matrix cncept (cntinued) Extensin t cyclstatinary nise surces: - The crrelatin matrix f a cyclstatinary nise surce describes the aut-crrelatin and crss crrelatin between the nise cmpnents at different frequencies : kω ± Ω and pω ± Ω, f a single nise surce - The crrelatin matrix f several nise surces describes the aut-crrelatin and crss crrelatin f nise cmpnents at different frequencies f several crrelated cyclstatinary nise surces 73

74 Tw examples f phase nise calculatin: A basic scillatr circuit : C dec C dec1 P-HEMT L in L chke L chke V gs0 V ds0 N 1 N N 3 N 4 N 6 N 5 R r L r C r R lad V lad Phase nise (dbc/hz) Q = 167 Q = 1670 Gate R D Drain L G R G Z c/c Intrinsic nnlinear Mdel Z c/c L D C pg δi 1/f C pd Frequency ffset frm carrier (Hz) R S L S Surce 74

75 An scillatr circuit shwing the influence f the lw-frequency dynamics f the bias netwrk Gate R D Drain L G C pg R G Z c/c Intrinsic nnlinear Mdel R S L S Z c/c δi 1/f Lw-frequency filter L lf, R plf C lf L D C pd Phase nise (dbc/hz) Q=1670 Surce Frequency ffset frm carrier (Hz) 75

76 Breadbard example Input matching netwrk IRCOM UMR CNRS n University f Limges Practical examples ΗΕΜΤ Output matching Islatr cupler G Lad netwrk G 0, θ 1 G 0, θ DR V C I n 1 : n L G Lad G 0 G M0 V C G ut 1 : n 1 n : 1 G R C R L R G 0, θ 1 G 0, θ Linear representatin 76

77 The analytical expressin shws that S Φut varies peridically with θ 1 Simulatin f a cmplete circuit, including a nnlinear mdel f HEMT and accurate mdels f passive elements shws the same variatin Finally, experimental results perfrmed n a breadbard circuit, made with cmmercially available cmpnents give the fllwing cnclusin KHzfrmcarrier(dBc) relative input phase shift (degrees) 77

78 In this experiment : the laded Q f the passive circuit is maintained cnstant the variable is θ 1, electrical length f the input resnatr cupling line Fllwing the mdified Leesn frmula, we have : S = K φin 1 1 G p jθ e 1 where: - G p is the transistr gain - K is a cefficient independent f θ 1 78

79 P scil =-6dBm et F=9145GHz cnstants -60 LPMO Phase nise measurement setup -30dB/décade the reference scillatr nise 100Hz (dbc/hz) θ 1 (degrees) Output phase nise spectral density versus the electrical length θ 1 f transmissin line in the feedback path S 100Hz=-80dBc 79

80 MMIC-based scillatrs fr high vlume prductin Circuits presented have been designed in an industrial envirnment using the tls presented in this talk DRO with the external DR measure simulatin Auxiliary utput pwer (dbm) Oscillatin frequency (GHz)

81 DRO PHASE NOISE SPECTRUM Phase Nise L f (dbc/hz) SIMULATION MEASUREMENT -115 dbc/hz Offset frm 19 GHz 81

82 VCO MMIC 385 GHZ (I) RÉSONATEUR (imprimé) Q 0 80 Tuning vltage OCT :175 GHz GHz VCO with an external printed resnatr 4 Transistrs : 30 µm, 4 50 µm ( ), 4 75 µm 8

83 VCO MMIC 385 GHZ (II) Output pwer (dbm) Measures Simulatin 0 0,5 1 1,5 Tuning Vltage (V) 39 38,9 38,8 38,7 38,6 38,5 38,4 38,3 38, 38,1 38 Frequency (GHz) Measures Simulatin 0 0,5 1 1,5 Tuning Vltage(V) 83

84 Phase nise f (dbc/hz) IRCOM UMR CNRS n University f Limges VCO MMIC 385 GHZ (III) Simulatin Measurement -79 dbc/hz Offset frm 385 GHz 84

85 CONCLUSION -The key-aspects f the design f lw phase-nise RF, micrwave, and millimetre-wave transistr-scillatrs have been presented -The mdern linear analysis f scillatr circuits has been detailed -Accurate nn-linear mdels f transistrs have been described -Mdeling f lw-frequency nise surces has been discussed -We have presented advanced design tls and methds, used in an industrial envirnment, making design reliable and leading t reprducible electrical characteristics -In cnclusin, rapid develpments f new methds, mdels, and tls, cnstrain the designer t systematically revisit his wn design methdlgy 85

86 GENERAL REFERENCES [1] M ODYNIEC, Editr, RF and micrwave scillatr design, Artech huse, 00 [] J Obrégn et al Mdeling and Design f lw Nise Micrwave Oscillatrs, IEEE MTT-S Internatinal Micrwave Sympsium, Wrkshp Ntes In Micrwave Oscillatrs Lking back and lking frward, BOSTON, MA, June 16, 000 [3] D B LEESON : A simple mdel f feedback scillatr nise spectrum, Prc IEEE, vl 54, February 1966 [4] JC NALLATAMBY, M PRIGENT, M CAMIADE, J OBREGON, Extensin f the Leesn frmula t phase nise calculatin in transistr scillatrs with cmplex tank, IEEE Trans n MTT, vl 51, n 3, mars 003 [5] JC NALLATAMBY, M PRIGENT, M CAMIADE, J OBREGON, Phase nise in scillatrs Leesn frmula revisited, IEEE Trans n MTT, vl 51, n 3, mars 003 [6] K Kurkawa, Nn Linear Micrwave Oscillatrs, Prc Of the IEEE, Vl 61, N 10, 1973 [7] S Pérez, T Gnzalez, SL Delage Micrscpic Analysis f Generatin-Recmbinatin Nise in Semicnductrs Under DC and Time-Varying Electric Fields, Jurnal f Applied Physics, Vl 88, N, 000, pp

87 [8] E Ngya et al, Steady State Analysis f Free r Frced Oscillatrs by Harmnic Balance and Stability Investigatin f Peridic and Quasiperidic Regimes,, Vl 5, N 3, 1995 [9] V Rizzli and A Neri, State-f-the-Art Present Trends in Nnlinear Micrwave CAD Techniques, IEEE Trans Micrwave Thery and Tech, Vl 36, 1988, pp [10] P Blcat, J-C Nallatamby et al, Efficient Algrithm fr Steady-State Stability Analysis f Large Analg/RF Circuits, Prc IEEE MTT-S Digest, Phenix, 001 [11] J Jug, J Prtilla et al Clsed-Lp Stability Analysis f Micrwave Amplifiers, IEE Electrnics Letters, Vl 37, N 4, 001, pp6-8 [1] S Mns, J-C Nallatamby et al A Unified Apprach fr the Linear and Nnlinear Stability Analysis f Micrwave Circuits Using Cmmercially Available Tls, IEEE Trans On Micrwave Thery and Tech, Vl47, N 1, 1999, pp [13] C Dragne, Analysis f thermal and sht nise in pumped resistive dides, The Bell System technical jurnal, nvember 1968 [14] H Siweris and B Schiek, Analysis f Nise Upcnversin in Micrwave FET Oscillatrs, IEEE Trans Micrwave Thery Tech, Vl 33, N 3, 1985, pp

88 [15] P Penfield Circuit Thery f Peridically Driven Nnlinear Sytems, Prc IEEE, Vl 54, N, 1966, pp66-80 [16] JM Paillt, JC Nallatamby, et al, A General Prgram fr the Steady-State,Stability and FM Nise Analysis f Micrwave Oscillatrs, Prc IEEE MTT-S Digest, Dallas, TX, 1990, pp [17] P Blcat and R Pujis, A New Apprach f Nise Simulatin in Transient Analysis, Prc ISCAS, 199, pp [18] P Blcat, J-C Nallatamby et al, A Unified Apprach f PM Nise Calculatin in Large RF Multitne Autnmus Circuit, Prc IEEE MTT-S Digest, Bstn, MA, 000 [19] Haus, HA and RB Adler, Invariants f Linear Nisy Netwrks, IRE,Vl 54, N, 1956, pp