Oscillating Voltage Waveforms Case(t):Vs-A =82,V2V .=1.0V (Case 2)case(2):V2A =0.5V (Case 4)Case(3):-A< Case(4):0<V v-Vk 0enCnaAnm =1,'饰>V>Vm -'4 A =82,'≥V (2) mdCmid Auid) where e satisfies AA1.When V=0,6=1,and V=Vde,three equations of(2)satisfy -0.5V (Case 1) I-C Aid=n A (3) T,=T-TTs 15i… =-0.25V(Case3) where Imax is the maximum current in the inductor. The oscillation period is a sum of three intervals 0.20.4 0.60.8 1.2 Time(ns) T=T+T2+T3,shown in Fig.2.T1,T2 and T3 are respectively a time interval of the first,second and third Fig.2.Oscillation voltage waveforms segmental sinusoids.From(3),we obtain the amplitude ratio where Vosn and Vsp are C-V voltage offsets.Ven and Veudp (4) are the control voltages,and we define Vem=Vmin+Vos.m and Vp-Vmp+Vasp effective control voliages (ECV)of When the oscillation voltage equals Vemp(or Verm),we define positive-step and negative-step varactors.The common- the inductor current I=Iemp(or Iem).Substituting (4)in the mode tuning voltage is Vcom=(Vem+Vemp)/2, and the first(or last)two equations of(2)and eliminating Iemp(or Iem) differential tuning voltage is Ve Thus,the positive tuning lead to the ESF ECV Vem is Vcom+Vten,and the negative tuning ECV Vep is Vcom-Vtune.For the sake of the symmetric tuning characteristics,Veom normally equals Vde,and Cmin.n=Cminp, (5) A Cmax,n=Cmax.p Thus,the oscillation period is A.Period Calculation Technique In [6],a period calculation technique was first introduced π-2sim 2sin to analyze a single-ended tuned LC oscillator.Here,it is T=T+T+T= mid 十 adopted to calculate the oscillation period of a differentially tuned LC oscillator. (6) Fig.2 shows oscillation voltage waveforms of a series where T=2元√LC and T-2元√DCna LC tank.The ECV voltages Vem and Vemp control the small- signal capacitance Csn and Css.p to be a minimum or B.Advantage of Suppressing AM-PM Conversion maximum.Therefore,each waveform consists of three segmental sinusoids of different sizes,which join at ECVs. At any control voltage,in Fig.2,the first segmental With the differential tuning voltage Vune changes from low sinusoid is symmetric to the third one.The oscillation voltage waveform in a differentially tuned LC VCO always to high,there exist four cases. For example,Case(3)is -Amin<Vtune<0,Vde-Amin< remains symmetric.However,a single-ended tuned LC Vemn<Vde,and Vdc+AminVemp>Vde,Where Amin is the VCO has asymmetric waveform in the whole tuning range minimum oscillation amplitude.When the oscillation voltage [6].Therefore,the differentially tuned application has an is above Vemp the equivalent capacitance of the LC tank is advantage of suppressing the up-conversion by AM-to-PM Cmid=Cmax.n+Cminp;when the oscillation voltage is below mechanism from low-frequency flicker noise at power Vemp and above Verm,the equivalent capacitor is supply and tail current [7]. Cma=Cmax.n+Cmaxp:when the oscillation voltage is below In a single-ended tuned LC VCO,the oscillation Vep,the equivalent capacitor is Cmid=Cminn+Cmax.p. Thus the frequency sensitivity to the common-mode noise is the same oscillation waveform comprises three segmental sinusoids as the voltage-to-frequency gain Kvco.So the low-frequency joined at Ver and Vemp ECVs.One is over Verp with phase noise converted by the AM-FM conversion from the amplitude A(is an ellipse similar factor.ESF[6)and common-mode noise can only be filtered by the low- bandwidth PLL closed loop.From (6),we can conclude that frequency the second is below Vemp and above Vem the oscillation period is insensitive to the common-mode with amplitude Amin and frequency the third is below voltage.So the sensitivity is Kvco.coMVcoM=0. Ve with amplitude A and frequencyThe I-V locus Therefore,a differentially tuned LC VCO itself has an of three segmental sinusoids holds advantage of suppressing the common-mode noise from control voltages,power supply,and tail current.where Vos,n and Vos,p are C-V voltage offsets. Vctrln and Vctrlp are the control voltages, and we define Veffn=Vctrln+Vos,n and Veffp=Vctrlp+Vos,p effective control voltages (ECV) of positive-step and negative-step varactors. The common￾mode tuning voltage is Vcom=(Veffn+Veffp)/2, and the differential tuning voltage is Vtune. Thus, the positive tuning ECV Veffn is Vcom+Vtuen, and the negative tuning ECV Veffp is Vcom-Vtune. For the sake of the symmetric tuning characteristics, Vcom normally equals Vdc, and Cmin,n=Cmin,p, Cmax,n=Cmax,p. A. Period Calculation Technique In [6], a period calculation technique was first introduced to analyze a single-ended tuned LC oscillator. Here, it is adopted to calculate the oscillation period of a differentially tuned LC oscillator. Fig. 2 shows oscillation voltage waveforms of a series LC tank. The ECV voltages Veffn and Veffp control the small￾signal capacitance Css,n and Css,p to be a minimum or maximum. Therefore, each waveform consists of three segmental sinusoids of different sizes, which join at ECVs. With the differential tuning voltage Vtune changes from low to high, there exist four cases. For example, Case(3) is −Amin<Vtune<0, Vdc−Amin< Veffn<Vdc, and Vdc+Amin>Veffp>Vdc, where Amin is the minimum oscillation amplitude. When the oscillation voltage is above Veffp, the equivalent capacitance of the LC tank is Cmid=Cmax,n+Cmin,p; when the oscillation voltage is below Veffp and above Veffn, the equivalent capacitor is Cmax=Cmax,n+Cmax,p; when the oscillation voltage is below Veffp, the equivalent capacitor is Cmid=Cmin,n+Cmax,p. Thus the oscillation waveform comprises three segmental sinusoids joined at Veffn and Veffp ECVs. One is over Veffp with amplitudeθ Amid (θ is an ellipse similar factor, ESF [6]) and frequency ω mid ; the second is below Veffp and above Veffn with amplitude Amin and frequencyω min ; the third is below Veffn with amplitudeθ Amid and frequencyω mid . The I-V locus of three segmental sinusoids holds 2 2 2    −    + =    dc mid mid mid mid V V I A CA θ ω , V V≥ effp 2 2 1    −    + =    dc min min max min V V I A CA ω , V VV effp effn > > 2 2 2    −    + =    dc mid mid mid mid V V I A CA θ ω , V V effn ≥ (2) where θ satisfies ≤ ≤ 1 A A min mid θ .When Vt=0, 1 θ = , and V=Vdc, three equations of (2) satisfy Imax mid mid mid min max min = = ω ω CA CA (3) where Imax is the maximum current in the inductor. The oscillation period is a sum of three intervals T=T1+T2+T3, shown in Fig. 2. T1, T2 and T3 are respectively a time interval of the first, second and third segmental sinusoids. From (3), we obtain the amplitude ratio = min mid mid max A C A C (4) When the oscillation voltage equals Veffp (or Veffn), we define the inductor current I=Ieffp (or Ieffn). Substituting (4) in the first (or last) two equations of (2) and eliminating Ieffp (or Ieffn) lead to the ESF θ 2 2 1    =− +       tune tune min mid V V A A θ (5) Thus, the oscillation period is 1 1 123 2 2 − −   − − −     =++= + tune tune mid min mid max V V sin sin A A TTTT T T π θ π π (6) where = π2 T LC mid mid and = π2 T LC max max . B. Advantage of Suppressing AM-PM Conversion At any control voltage, in Fig. 2, the first segmental sinusoid is symmetric to the third one. The oscillation voltage waveform in a differentially tuned LC VCO always remains symmetric. However, a single-ended tuned LC VCO has asymmetric waveform in the whole tuning range [6]. Therefore, the differentially tuned application has an advantage of suppressing the up-conversion by AM-to-PM mechanism from low-frequency flicker noise at power supply and tail current [7]. In a single-ended tuned LC VCO, the oscillation frequency sensitivity to the common-mode noise is the same as the voltage-to-frequency gain KVCO. So the low-frequency phase noise converted by the AM-FM conversion from the common-mode noise can only be filtered by the low￾bandwidth PLL closed loop. From (6), we can conclude that the oscillation period is insensitive to the common-mode voltage. So the sensitivity is K V0 VCO,COM 0 COM = ∂ω ∂ = . Therefore, a differentially tuned LC VCO itself has an advantage of suppressing the common-mode noise from control voltages, power supply, and tail current. Vdc Vtune=1.0V (Case 2) Vtune=0.5V (Case 4) Vtune=-0.5V (Case 1) Vtune=-0.25V (Case 3) T3 T1 T T2=T-T1-T3 ≤ − ≥ −< < < < tune min tuen max min tune tune max Case(1) : V A Case(2) : V A Case(3) : A V 0 Case(4) : 0 V A A max Amin Veffp Veffn Fig. 2. Oscillation voltage waveforms