Accurate Phase Noise Prediction in PLL Synthesizers - Lance Lascari

Accurate Phase Noise Predictionin PLL SynthesizersHere is a method that uses more complete modeling for wireless applicationsBy Lance LascariAdaptive Broadband CorporationIn modern wireless communicationssystems,the phase noise characteristicsof the frequencysynthesizer play a criticalrole in system performance.Higher thandesired phase noise cancause degraded system performanceby reducing thesignal to noise ratio,increasing adjacent channelpower, and reducingadjacent channel rejection.While many of the factorsthat affect phase noisein phase-locked frequencysynthesizers are well understood, designersoften overlook others. Neglecting these additionalfactors can cause frustration and overdesign,when a more complete up-front analysismay have yielded more elegant solutions.The goal of this article will be to first reviewthe models for standard noise sources and howthese are analyzed, then to do the same for noisecaused by the often forgotten resistors andamplifiers within the loop filter.This is Part 1 of a two-part article. Part 2 willbe published in the May 2000 issue of AppliedMicrowave & Wireless magazine.▲ Figure 1. Z fil3 (f) with optional amplifier inserted.Standard phase noise sources andanalysis techniquesThe basic equations describing the loop’s frequencyresponse will be given with a briefdescription of each. Plots that accompany theequations are from analysis of the test cases presentedlater in the article, and are typical for allPLL designs. The complete equations in contextcan be found in [1]. The equations presentedhere and in [1] have been drawn in part from[4], as well as [3], [6] and [7]. The amplifierwithin the loop will be ignored for all analysispresented here, as the implications of thisamplifier will be discussed in the text.Equation 1 describes the transfer function ofthe third-order loop filter. Equation 2 describesthe transfer function of the second order loopfilter and is applicable to systems that do notrequire the third pole for additional referencesuppression.Zfil31Zfil( f)×2 × π × f × i × C3( f)=⎛ 1 ⎞Zfil( f)+ R3+ ⎜⎟⎝ 2 × π × f × i×C ⎠3(1)30 · APPLIED MICROWAVE & WIRELESS

R f i CZfil ( f ) ( 1+ 2 × 2× π × × × 2)=[ 2× π × f × i( C + C + R × 2× π × f × i× C × C )]2 1 2 1 2(2)The total forward loop response, G(f), includes everythingfrom phase detector to VCO, and is represented byEquation 3. The equation for G_ pd (f) is not included, butit represents the transfer function of the discrete-samplingphase-frequency detector. Details on this functioncan be found in [7].▲ Figure 2. Open loop frequency response.Gf ( ) =Kφ× Zfil3( f) × Kvco× amp_gain× G2× π × f × ipd( f)(3)The reverse loop, or feedback gain, is defined as thereciprocal of the total frequency multiplication factor inthe loop, N, as shown in Equation 4.H= 1NEquation 5 illustrates the open loop gain, G ol (f). Theopen loop gain is the product of equations 3 and 4, showingthe response around the loop. This equation is particularlyuseful for stability analysis. A plot of thisresponse is shown in Figure 2.(4)G ol (f):=G(f)×H (5)Because this is a simple negative feedback system, theclosed loop response, G cl (f), at the VCO output is representedby the familiar feedback relationship seen inEquation 6. A plot of this response is shown in Figure 3.GfGcl ( f ) ( )= 1 + Gf ( ) × H(6)▲ Figure 3. Closed loop frequency response.The well known noise sources (expressed in terms ofequivalent noise voltage) are specifically crystal reference(TCXO) noise; phase detector noise; and VCOphase noise.Crystal reference oscillator noise — The crystal oscillatornoise is “amplified” in the loop by the gain of theclosed loop transfer function. A simple approximationfor this noise source due to the crystal reference itself, aswith any oscillator, is that it is inversely proportional tooffset frequency. Higher order approximations or actualdata could be used if it were available, but the 1/fapproximation is a good starting point.Within the loop bandwidth of the synthesizer, theclosed loop transfer function, G cl (f) is very large in magnitude,hence it increases the level of the reference oscillatornoise. This gain is flat until it reaches the loopbandwidth, after which it drops off rapidly. This functionrepresents amplification of the noise within the loopbandwidth, but attenuation of the noise above this frequency.The gain within the loop bandwidth comeslargely from the ratio of the loop division ratio, N, by thereference division ratio, R. If this noise can be observedat all, it is generally seen very close to the carrier (whereit is visible above the other major noise sources).If a TCXO is used, phase noise data should beobtained from the manufacturer so that reference valuescan be used with the models. Measuring the noise ofa crystal oscillator can be quite difficult since the noiseis significantly below the noise of most readily availabletest equipment. In fact, it may be easier to work backwardsfrom measured PLL data to determine the TCXOnoise if the data are not available from the manufacturer.Equation 7 illustrates the noise due to the referenceoscillator, N tcxo (f), at the synthesizer output.Ntcxo⎛ Ntcxo_ref ⎞10⎜⎝ 20⎟⎠ ⎛ 1 ⎞( f) =× ⎜Gcl( f)× ⎟f ⎝ R⎠ftcxo_refPhase detector noise — N p_d (f), is a form of noise thatrepresents the internal noise floor of the phase/frequencydetector and frequency dividers within the PLL. This(7)32 · APPLIED MICROWAVE & WIRELESS

noise is modeled as flat versus frequency and thespecific value for N pd_ref can be obtained from themanufacturer of the synthesizer IC. For NationalSemiconductor synthesizers (the one used in thisarticle, for example), the phase detector noise flooris given for an effective reference frequency of 1Hz. The actual noise floor of the phase detectordegrades proportional to 10*log(F ref /1 Hz). Thisnoise source is flat with respect to frequency, but itis shaped by the closed loop transfer function of thesynthesizer, as shown in Equation 8.N ( f) = 10 20 × G ( f)p_d⎛ fref⎞N pd_ref + 10 log⎜⎟⎝ Hz ⎠cl▲ Figure 4. Loop error response.(8)Free-running VCO noise — This tends to have thetypical noise profile. The noise is inversely proportionalto offset frequency from the carrier. The synthesizedoutput, however, is the result of the PLL “cleaning up”the close in phase noise of the VCO. This compositenoise is calculated in Equation 9. The noise of the VCOis effectively high-pass filtered by the PLL, providingrejection of phase noise or phase error within the bandwidth,but leaving VCO noise well outside of the loopbandwidth unaffected. A plot of this high-pass filterfunction provided by the PLL is shown in Figure 4.Nvco( f)=A plot of the commonly analyzed phase noise sourcesdescribed above at the synthesizer output is shown inFigure 5.Models for noise in a typical third order loop filterPrior to delving into the circuit equations, it is importantto outline the methodology that will be employed toanalyze the problem. The approach used will be to firstdetermine the voltage at the input of the VCO due toeach of the noise sources in the loop. This voltage willthen be used to calculate the equivalent FrequencyModulation (FM) (i.e. phase noise) at the VCO outputdue to each of these sources using the VCO gain, K vco .A key parameter used to analyze frequency modulationis the modulation index, m, calculated usingEquation 10.m =df m⎛ Nvco_ref ⎞10⎜⎝ 20⎟⎠ ⎛ 1×f ⎜⎝ 1 + Gfvco_ref⎞⎟( f)⎠(9)(10)where m = modulation index=peak phase deviation inradians; d = frequency deviation; and f m =message orol▲ Figure 5. TCXO, phase detector and VCO noise.modulating frequency.In the case of a “direct” FM system, where the modulation/messageis fed directly to the VCO in the form ofvoltage, the frequency deviation is simply the product ofthe modulation voltage and the tuning sensitivity (volts× Hz/volt = Hz).If the peak phase deviation is well below one radian,as is the case with all that will be studied here, the higherorder Bessel functions can be ignored and the relativelevel of the sidebands on the carrier due to the modulatingfrequency can be calculated using Equation 11, seereference [5] for further detail on this subject.sideband_level⎛ m⎞= 20log⎜⎟⎝ 2 ⎠(11)These simple equations for frequency/phase modulationform the basis for the analysis of phase noise due tovoltage noise sources within the loop hardware.Calculating the noise voltages in the loopSince the noise sources appear in different circuitnodes throughout the loop, the frequency response ofthe transfer function between each source and the VCOinput will be different. This, in turn, results in modulationat the VCO output unique to each source in frequencyand magnitude.It is important to realize that in a system such as this,the resistor or op-amp noise modulates the VCO even ifa PLL were not connected — so the “corruption” due to34 · APPLIED MICROWAVE & WIRELESS

the resistors and op-amp can be consideredan open-loop phenomenon.However the net result in the synthesizedoutput requires the closed loopresponse, and the analysis is identicalto the analysis used to show the effectof the PLL on the stand-alone VCOphase noise in Equation 9. Valuableinsight can be gained by observing theopen and closed loop SSB phase noisecurves rather than just looking at thetotal output phase noise of the closedloop system. The connection betweenthe open and closed loop responses, isthe high-pass transfer function plottedin Figure 4, sometimes referred to asthe error response of the loop.The validity of the open-loop analysistechnique requires the assumptionthat the charge pump does not loaddown the loop filter excessively. Thefollowing key points support thisassumption. First, when in lock, thecharge pump does very little but providevery minor corrections to keep theloop locked. Second, the charge pumpis made up of a current source and acurrent sink, both of which represent ahigh impedance when operating in alinear region. The closed-loop action ofthe synthesizer fights the resistor andop-amp noise, but that is adequatelydescribed by the analysis of the closedloop system.Noise found in resistorsRandom electron motion inside of ordinary resistorsmakes for a “built-in” noise source in every circuit containingresistors. The noise is white in nature [8], withit’s power in watts dissipated in an equivalent 1 ohmresistor is shown in Equation 12, and the units arevolts 2 /Hz.P ( R)= 4 × k× Temp × B×Rresistor_noise(12)Thus, Equation 13 computes the equivalent RMSnoise voltage generated in a known resistanceV ( R)= 4 × k× Temp × R×Bresistor_noise(13)where k = Boltzmann’s constant; Temp = temperaturein kelvins; B = bandwidth in Hz; and R = resistance inohms. For most of our analysis we will use a bandwidthof 1 Hz.These equations simply show the equivalent noisevoltage source that appears in series with each resistor.▲ Figure 6. (a) R2 noise model with or without op-amp; (b) R3 noise modelwithout op-amp; (c) R3 noise model with op-amp, assuming that the op-amphas very low output impedance.Note that the noise could just as easily be modeled as anoise current source in parallel with the resistor.References [2] and [8] provide details on the noise inresistors and op-amps. When crunching the numbers fortypical resistor values, one may initially dismiss the voltagesas minuscule. As we will show later, even nanovoltsof noise can cause several dB of degradation in the single-sidebandphase noise of a synthesizer because K vco isoften a very large number.Figure 6 shows the basic PLL loop filter and wherethe noise sources appear. The charge pump connectionto the PLL has been left out for these models because itrepresents a very high impedance when the loop islocked. As previously stated, the initial analysis of theresistor noise will be done “open loop,” and the effects ofthe PLL on shaping this noise further will be investigatedafter the basic models are developed.Since the noise sources are uncorrelated, each resistoris analyzed separately and the effects are added at alater stage. Deriving the actual noise voltage versus frequencyat the input to the VCO tuning port is a matterof basic circuit analysis using the models in Figure 6.36 · APPLIED MICROWAVE & WIRELESS

Resistor noise analysis for the“case 2” example PLL in this articlethat will be discussed later is shownin Figure 7 and Figure 8. Figure 7shows the baseband noise voltages atthe VCO input in the open loop case,and Figure 8 shows the noise contributionto the synthesizer output. ■References1. Complete MathCAD Analysisused in this article is available inMathCAD and PDF formats athttp://home.rochester.rr.com/lascari/lancepll.zip.2. W.P. Robins, Phase Noise inSignal Sources: Theory andApplications, W.P. Robins, 1984.3. James A. Crawford, FrequencySynthesizer Design Handbook,Artech House, 1994.4. Dean Banerjee, PLLPerformance, Simulation, andDesign, http://www.national.com/appinfo/wireless/deansbook.pdf.5. A. Bruce Carlson, CommunicationSystems: An Introduction toSignals and Noise in ElectricalCommunication, McGraw-Hill, 1986.6. William O. Keese, “An Analysisand Performance Evaluation ofPassive Filter Design Technique forCharge Pump Phase-Locked Loops,”Application Note 1001, NationalSemiconductor.7. Jeff Blake, “Design of WidebandFrequency Synthesizers,” RF Design,May 1988.8. “Noise Specs Confusing,”Application Note 104, NationalSemiconductor.Figure Analysis Conditions Comments6a R2 Without op-amp Assume the op-amp is not presentand the R2 noise must be calculatedthrought the entire network.6a R2 With op-amp Assume for R2 noise analysis the opampinput impedance is infinite, andthe R2 noise out of the network consistingof R2, C1 and C2 is determined.This noise is then amplified byamp_gain before being passedthrough the network consisting of R3and C3.6b R3 Without op-amp In this case, the R3 noise source is“floating” in the loop filter network,and the transfer function between thissource and the output of the networkincludes the impedances of all the filtercomponents.6c R3 With op-amp Since the op-amp output impedancewill be assumed to be very low, wemodel this as a short. Hence, this isthe simplest model of all, R3 shortedto ground.▲ Table 1. Description of the resistor noise models shown in Figure 6.Noise in microvolts▲ Figure 7. Baseband noise voltages at the VCO input in the open-loop state.▲ Figure 8. Resistor noise contributions at the VCO output after the PLL’s highpasserror function is included.Author informationLance Lascari is a PrincipalEngineer at Adaptive BroadbandCorporation in Rochester, NY. He hasbeen working as an RF designer onthe company’s QAM Point-Point andFSK Point-Multipoint products forthe past five years. He earned aBSEE from the RensselaerPolytechnic Institute in 1995. Hisprofessional interests include lownoisesynthesizer and VCO design,design for high linearity, and low-cost transceiverdesign. He may be reached via email at llascari@adaptivebroadband.com,or through his web page:http://home.rochester.rr.com/lascari.38 · APPLIED MICROWAVE & WIRELESS