1526 半导体学报 第26卷 where n(b,is the function of b and too.Until 2.2 Loop parameters design now,the procedure to determine the loop parame- The target of loop parameters design is to a- ters is chieve the desirable loop performance.With the (P.1)Find Kvco from simulation. closed-loop analysis,it is possible to design the (P.2)Select capacitor ratio b and damping damping factor and nature frequency directly.The factor o,then calculate m,n,and k. nature frequency is always dependent on the damp- (P.3)Select loop bandwidth we according to ing factor;consequently,a damping factor could be reference clock,a good start-point:we=wre/25. set as a target at the beginning of parameter de- (P.4)Calculate zero @,@,and @s. sign,for example,=1.Capacitor ratio b should al- (P.5)Choose charge pump current Icr,and so be predefined.On the one hand,a small b makes dividing ratio N. the pole p close to the zero and results in the (P.6)Calculate R from Eq.(12),if R is too low frequency closed-loop pole @p.This can better large,increase the pump current Icp and recalculate filter high frequency noise from input and divider, R until its value is reasonable. and better attenuate clock injection spurs from the (P.7)Calculate C1,C2,if the value is too charge pump.On the other hand,b must be larger large,decrease pump current Icp,and go to P.6. than 8 for stability s.Then,might be set to 9 or 2.3 Parameters variation,stability,and margins 10 for the best possible noise performance at the beginning. Basically,loop transfer function changes with Taking on=mo:and wp=(b+1)wz into N to synthesize various frequencies.A configurable Egs.(4)and (5),gain factor K'can be expressed charge pump is used to keep the ratio N/Ker to as maintain the loop performancet7.The undeter- K'=k(b,)o (10) mined parameters variation still arises from where k(b,)is also a function of b and t,it is ex- process,temperature,and voltage.Traditionally, pressed as phase margin optimization is simple and always k(b,5)=(1-4)m2+25(b+1)m(11) used to determine the loop parameters.A prima- For K'=bIcp Kvco/2NC1,then k(b,)=bIcp X ry phase margin should be pre-defined for calcula- Kvco/2NC1,resistor R can be expressed as tion,which is usually set to about 50.However, R=2πNk(b, variation of loop parameters changes the phase bIcp Kvco - (12) margin.Phase margin optimization technique can- Loop bandwidth @e is an important parameter. not predict how much variation it can tolerate.Root As mentioned early,s-domain model is accurate in locus technique could be adopted to illustrate the the whole bandwidth only if the loop bandwidth is influence of parameters'variation.The parameters no larger than 1/20 reference frequencyt9).Because such as Icp,R,C1,C2,and Kvco will vary with PVT it is difficult to design a pretty low noise VCO in variation.What are the influences of the variation? CMOS technology,it is preferable to have a high Figure 2 is the root locus of the loop.Poles loop bandwidth.High loop bandwidth also has the migrate with the factor K'.Points B,C are the advantage of a fast switching speed.A good initial boundary to avoid under-damping behavior.To loop bandwidth is 1/25 reference frequency.From margin the loop parameter variation,for example, Eq.(2),let s=joe and H (jo)=1,with Eq. point A can be selected to identify the system,and (10),zero can be calculated as the damping factor is 1.If the gain factor K'de- 1 viates smaller than the designed value,poles will n(b,5“ (13) migrate towards point B and its conjugate pole,as半 导 体 学 报 第!#卷 +Y+ 599DDCACR@?@A>L@>J=G /G=F7DJ=FEH9EEMM7D76=F=D<R=<8J?8<FE70 :G8=X=FG=R=<8D7K9=9EEM M=DHED67?:=>S8FGFG= :9E<=R09EEM7?79L<8<$8F8<ME<<8K9=FE R=<8J?FG= R76M8?JH7:FED7?R?7FND=HD=^N=?:LR8D=:F9L>/G= ?7FND=HD=^N=?:L8<79@7L<R=M=?R=?FE?FG=R76M0 8?JH7:FED,:E?<=^N=?F9L$7R76M8?JH7:FED:EN9RK= <=F7<7F7DJ=F7FFG=K=J8??8?JEHM7D76=F=DR=0 <8J?$HED=\76M9=$+g$>)7M7:8FEDD7F8EH<GEN9R790 <EK=MD=R=H8?=R>W?FG=E?=G7?R$7<6799H67a=< FG=ME9=*M:9E<=FEFG=]=DE*]$7?RD=<N9F<8?FG= 9E@HD=^N=?:L:9E<=R09EEMME9=*M4>/G8<:7?K=FF=D H89F=DG8JGHD=^N=?:L?E8<=HDE68?MNF7?RR8X8R=D$ 7?RK=FF=D7FF=?N7F=:9E:a8?;=:F8E?<MND<HDE6FG= :G7DJ=MN6M>W?FG=EFG=DG7?R$H6N<FK=97DJ=D FG7?&HED<F7K898FL-&. >/G=?$H68JGFK=<=FFE%ED $"HEDFG=K=<FME<<8K9=?E8<=M=DHED67?:=7FFG= K=J8??8?J> /7a8?J*? g 8*] 7?R*M g %Hf$&*] 8?FE ’^<>%2&7?R %5&$J78?H7:FEDNl:7?K==\MD=<<=R 7< N\ VL%H$+&*! ] %$"& @G=D=L%H$+&8<79<E7HN?:F8E?EHH7?R+$8F8<=\0 MD=<<=R7< L%H$+&V %$Y2+ !&8! 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