3RF Design: Concepts and Technologythe asterix denotes conjugation.RF designers call this an input impedancemismatch. Consequently, the power available from the source (Ps.au) is not thesame as the power delivered at the input (Pi.del). Let us consider this in moredetail and assume that the impedances in Figure 1.I can be treated asresistors.TheinputvoltageV,isderivedasR;VV,= R + Rs(1.2)andconsequentlyR;1(1.3)Pi.del(R, +R)ThepowerPe.gyisthepowerthatthesourceV,woulddelivertoaconjugatematched circuit, and canbeexpressed asV?(1.4)Ps.au=4R.We can also define the available power at the output Po.aw,i.e., the power thatV。would deliver to a conjugate matched circuit, asv?R;V3A2_1(1.5)Po.uv4Rv4R.R,+RsThemismatchbetweenR, and R,influences theavailablegainGa,Po.auR;R.Po,auGa:(1.6)ARoPiedelPs,avR+R.Obviously,for a perfect input impedance match Ps.u = Pidet and the availablegainincreases.In a similarway,wecan consider animpedancemismatchbetween Z, and Zr.The available power at the output of system S, Po,uu ishigher than the delivered power at the load, Pi.del which is equal toV=VPASRL(1.7)Pidet(RL+R.)2RLand thepower delivered at the input is equal toV?(1.8)Pidel =R,The resulting gain, while assuming a perfect inpnt match, is called (delivered)power gain and isdefined asPo.auPi.deltRLR;A2(1.9)Gp=PidelPidel(RL+R.)2
4.CIRCUITDESIGNFORRFTRANSCEIVERSWe obtain the maximum gain for system S when we have perfect input andoutputimpedancematches:Po.auPi,del(1.10)GmaxPidelPs,au(1.11)Z,=Zi,Zo=Zuand therefore yieldsR;(1.12)Gmax4R.whichfollowsfrom(1.6)withR;=R,andfrom(1.9)withRz=R,.Similarlythe lowest gain, called the transducer gain is obtained if we have a mismatchon bothinputandoutput,R,Pi.del4RLR.(1.13)G.=42(RL + R,)2Ps.avR+RsHowever,this transducergain is the actual gain for system S when driven bya sourceV,and loaded by an impedance Zz.Note that from (1.13)we indeedobtain GaforRz=R,and GpforR,=R..We end up with five possible gain definitions and it is therefore important tospecify the used gain.There are two main reasons for powermatching in RF applications.Onereason is when reflections of signalsoccur.Signals arereflected whenthedistancebetweenthe sourceand loadis largerthanwhereisthewavelengthi.e.= where c is the speed of light and f is the frequency of interest.Infree space, for 1 GHz, > is approximately equal to 30 cm, but under realisticconditions it can reduce to a few centimeters.Hence a can only be a fewmillimeters in integratedRFapplications.Thedistancebetween theantennaoutput (i.e.the source)and the input of the low noise amplifier (LNA) (i.e. theload) is normally large and reflections can therefore take place. A general rulemight be that reflections are only of interest for the “"on-off"chip connections;connections between blocks which are both on-chip will notbedeteriorated byreflections, due tothe small sizeof theICThesecondreason tofocus on powermatching in RFis when input impedancemismatchdrasticallyinfluences theperformanceof thecircuit.Thishappensat off-chip antenna filters, for instance.They normally are designed for 50 2input/output impedance.Iftheon-chipLNAhasan inputimpedanceotherthan50 2,thefilter characteristic will change and hencetheperformance of the
RFDesign:ConceptsandTechnology5overall system will change. In such cases, it is important to have a 50 s2 inputimpedancematch.Therefore,even forRF applications it is notalways necessary to focus onoptimalpowertransfer.Microwavedesigners normallyuse scatter (s)-parameterstodefinethesmallsignal performance of their circuits [1]. These scatter parameters are based on:voltage waves entering system S,i.e. the incident waves and waves leaving Satthesameport,thescatteredwaves.Theratiosbetweenthepowers oftheseincident and scattered waves at the input and output are the s-parameters.Letus define Eu as the incident voltage wave and Er as the scattered voltage waveforport1oftwo-portsystemS.InasimilarwaywedefineEi2andEr2forport2.If we normalize these waves by the square root of Zo,i.e.aj =Eij/Zoand b, = Eri/Zo then the two-port relations may be written asbr = S11ai + S12a2(1.14)(1.15)b2=S21ai+S22a2The fivegain definitions can all beformulated in terms of s-parameters andthereflectioncoefficientF,definedasZ- ZoT(1.16)Z + ZowhereZois thecharacteristic impedance.Equation(1.16)expresses thequalityofmatching and 20 log|F/is defined as the"return loss3.In fact, itmeans thatthe load is reflecting back some of the power delivered by the source.You canalso view it as that the load not taking up all of the availablepowerfrom thesource.Asanexample,assumeZ=R+RwhereSisa502system.Foran impedance variation between20% and 30%,(1.16)results in a return lossof-15to-20dB,i.e.atleast1%of thepower isreflectedback.Therewillbenoreflectionforaperfectmatch.Theformulationsforthegains,expressed3Note that s11=F and s22=F2
6CIRCUITDESIGNFORRFTRANSCEIVERSins-parametersandreflection coefficients,aregivenbyS21 (1 +)(1.17)A,=(1 S22F) + S1 (1 S22F) + S21F/S121 - /F,/21(1.18)Gu=/s21/I1 s117,/21 |Fou/21- //21(1.19)GpI1 - s27/2 /521/121-Fin/21 - [F/21-//2/s21/2(1.20)Gr[1S/21-[rout?1.1.2NoiseOne of themost importantparameters in analogand RF circuit design is theratio between the signal powerto the total noise power,i.e.the signal-to-noiseratio (SNR). The noise factor (F), a measure normally used in RF design, isstrongly related to this definition,SNRin(1.21)F-SNRoutThe noise factor4 is a ratio between the SNR at the input and the SNR at theoutput ofa system.Clearly,(1.21)canberewritten asNoutPinNoutA(1.22)GauPinNin-GauNinshowing that the noise factor is the ratio between the total noise at the outputof the system Nour to the total noise at the output of the system due to the noiseat the input of the system Nin.The noise factor is a measure of how much theSNRdegrades as the signalpassesthrougha systemThenoisefactorcanbe calculatedbyreferring thenoise sources of the systemto the input of the system. If the system can be considered as a two-port system,the noise of the system can be modeled by two input noise generators, a parallelcurrent source I? and a series voltage source V2. Here we define X2 as themean squareof X over a certainbandwidth f.This stems from theformaldefinitionofthepowerspectraldensity(PSD)ofX,Sx(f)IXT(F))Sx(f)=limT104The noisefigure NF is normally defined as 10log(noise factor)
RF Design:Concepts and TechnologyV2n,rsgain: A ynoisysystemSFigure 1.2.Noisy system S driven by voltage source Vs.The internal resistance R of V.produces noise, represented by Vn,rs.The voltage gain of S is cqual to Au.where Xr(f) is the one-sided Fourier transformation of signal x(t). Normally,the one-sided PSD is used in circuit design, so that the thermal noise powerspectraldensitySx(f)4kTRforaresistor,wherekistheBoltzmanncon-stant and T is the absolute temperature. The dimension of mean square voltageper unit bandwidth stems from thefact that the power spectral density referstothe(rms)voltageacrossaresistorwithresistanceRtogenerateapowerof4kT R in a 1 Hz bandwidth.The mean square noise voltage of this resistor isthen given by V2 =4kTR-△f.Again we can seethe difference between analog and RFdesign.Assuming analmostinfiniteinputimpedanceinanalogdesignresultsinonlyaninputreferredvoltage noise source.This assumption no longer holds at radio frequenciesand, consequently,the input referred current noise source also has to be takeninto account.It is important tokeep inmind that these two noise sources arecorrelated.Thenoiseinfluenceofthesesourcestothetotal systemcanonlybefound after adding their influences.We can now use these input referred noise sources to calculate the noisefactor.Assumea voltage sourceV,witharesistance R,drivinga noisysystemwith voltage gain Au, as depicted in Figure 1.2. This source resistance producesnoise, modeled as V2.rs. The noise of the system can be referred to the inputmaking the system itself noiseless, as depicted inFigure 1.3.The SNRincannowbefoundasV2SNRin:(1.23)Ve.rs