18CIRCUITDESIGNFORRFTRANSCEIVERSgate口R+Ringso-gdoRSRdsourcedrainARolRb2Rp5Rp4Rp3口bulkFigure1.6.MOST model including RF effects.whereCJ,MJ,and,VJareconstants,definedbythetechnologyFora bipolar device,the small-signal common-emitter currentgain β is nowgiven by1 + s/z1β(o)=βo(1.47)+s/p1with the pole and zero defined as1gm(1.48)Z1piCur元(Cm+Cμ)andwiththestaticcommon-emittergainβodefinedasβo=gm元(1.49)
19RFDesign:Concepts and TechnologyCRYBaseCollector+VbeREmitterFigure 1.7.Equivalent small-signal circuit for common-emitter configuration with a shortcircuit load.For highfrequencies and normal operation, oCμ<< gm.The current gainIβ drops to unity at a frequency fr,bipogm(1.50)fr.bipo= 2元 (Cμ+Ck)In a similar way, we can define the cut-off frequency fr.mosr for a MOsdevice,gm(1.51)fr,most2元(Cg+Cpar)where wehaveused the small-signal equivalent circuitfrom Figure 1.8 in com-bination with theRF extentions mentioned inFigure1.6.Here, Cgg representsthe intrinsic gate capacitance and hence reflects the total sum of Cgs and Cgd.Capacitance C pur includes overlap capacitance and interconnect capacitance tothe substrate. For a MOS device, the gate length has a major impact on fr.mort;a small gate length is needed for a high cut-off frequencyIl The cut-off frequency can also be obtained from the chain matrix k, defined for a two-port as)-(k1k12)(u2)“1k21k22Thefr isnowmeasured for uz=O,whilethe inputport is only loaded bya current source.According tothematrixdefinition,weobtain→2-1-ijk22and for low frequencies for a bipolar device, k22 = --1/βo
20CIRCUITDESIGNFORRFTRANSCEIVERSCgdRgd-gatedrain1/gRgs9m'gs9mbbsOCgbgssourcesourceCosCbdRbdbulkbulkFigure1.8.Small equivalent circuit of a MOS deviceCRbBaseCollectorA+VpeRheoEmitterFigure l.9.Small-signal equivalent circuit of a bipolar device to calculate the voltage gain ofthe device.MaximumOscillationFrequencyThe maximum oscillation frequency fmax has been defined to include the effectof the base resistance for bipolar transistors and similarly thegate, source andchannel resistanceforMOs devices.Atthisfrequency,themaximumavailablepower gain, i.e. Gmax, becomes unity. This implies that both the voltage gainand the current gain are important.For a bipolar device the voltage gain can be obtained using the small signalequivalent circuit of Figure 1.9.The input is short-circuited while the output isloaded by a current source. The frequency dependent voltage gain a, is derived
21RFDesign:ConceptsandTechnologyas1 + s/z1(1.52)a(o)=a+s/piwithgmgm(1.53)au=-gmRLZ1 =PICulaulCuFrom this expression together with the current gain expression, fmaxbipo can befoundtobe1/2fr,bipo(1.54)fmax.bipo8元Cwhere C is the depletion layer capacitance of the collector-base junction. In asimilar way, the fmax.mos can be found to be 12fr,most(1.55)fmax.m(4元Cjunc/(Rg+Rin+Rs)Rest）where Cjunc is the drain junction capacitance and Reff the effective bulk resis-tance. Clearly, fmax,mos is more influenced by parasitic elements than fr.mosand thedevicelayout isthereforeextremelyimportant.Input Limited FrequencyThe input limited frequency f,is relatedto thefrequencydependency of thetransconductance of the device.If we use the small signal equivalent circuit ofFigure 1.10, then the transconductance is derived as1 + s/z.1(1.56)gm(a) = gm1+ s/pi12The maximum oscillation frequency can also be obtained using the chain matrices.The Gmax is definedas Gmax = dg aj under the condition that the two port is perfectly terminated, Zu =Zou, and Zy =ZinThe voltage gain is defined asZLdu=k12+Zzklland the current gain is defined as1dj=k22+ZLk21where ZL is a function of Zin-An analytical expression of Gmax is difficult to obtain because of thematching property
22CIRCUITDESIGNFORRFTRANSCEIVERSCRpBaseCollector1+VbeoutcEEmitterFigure I.10.Small signal equivalentcircuitofabipolardevicetocalculatethetransconductanceasfunctionofthefrequency.with1gm(1.57)Z1piCurb(Cu+C)The parameter f, is defined as the frequency at which gm(o) is dropped by3 dB. For a bipolar device, fu,bipo can then be derived as1(1.58)fu.bipo~2元h(Cu+C元)In a similar way,we can obtain the input limitedfrequencyfora MOS device,1fo.mos~(1.59)2元rg (Cgg +Cpar)OutputLimitedFrequencyIf the device isloadedwith a resistor, we can define thevoltagegain, as wehavedonetocalculatethemaximum oscillationfrequency.Thefrequencyatwhichthe voltage gain drops with 3dB is know as the output limited frequency, foutFor a bipolar device this yields1four.hipo=2元RLCu(1.60)