326 IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 35, NO. 3, MARCH 2000Oscillator Phase Noise: A TutorialThomas H. Lee, Member, IEEE, and Ali Hajimiri, Member, IEEEAbstract—Linear time-invariant (LTI) phase noise theoriesprovide important qualitative design insights but are limited intheir quantitative predictive power. Part of the difficulty is thatdevice noise undergoes multiple frequency translations to becomeoscillator phase noise. A quantitativeunderstanding of this processrequires abandoning the principle of time invariance assumed inmost older theories of phase noise. Fortunately, the noise-to-phasetransfer function of oscillators is still linear, despite the existenceof the nonlinearities necessary for amplitude stabilization. Inaddi-tion to providing a quantitative reconciliation between theory andmeasurement, the time-varying phase-noise model presented inthis tutorial identifies the importance of symmetry in suppressingthe upconversion of 1noise into close-in phase noise, andprovides an explicit appreciation of cyclostationary effects andAM–PM conversion. These insights allow a reinterpretation ofwhy the Colpitts oscillator exhibits good performance, and suggestnew oscillator topologies. Tuned LC and ring oscillator circuitexamples are presented to reinforce the theoretical considerationsdeveloped. Simulation issues and the accommodation of amplitudenoise are considered in appendixes.Index Terms—Jitter, low-noise oscillators, noise, noise measure-ment, noise simulation, oscillators, oscillator noise, oscillator sta-bility, phase jitter, phase-locked loops, phase noise, phase-noisesimulation, voltage-controlled oscillators.I. INTRODUCTIONIN GENERAL, circuit and device noise can perturb both theamplitude and phase of an oscillator’s output. Of necessity,however, all practical oscillators inherently possess an ampli-tude-limiting mechanism of some kind. Because amplitude fluc-tuations are usually greatly attenuated as a result, phase noisegenerally dominates. Thus, even though it is possible to designoscillators in which amplitude noise is significant (particularlyat frequencies well removed from that of the carrier), we willfocus primarily on phase noise in the body of this tutorial. Theissue of amplitude noise is considered in detail in the Appendix,as are practical issues related to how one performs simulationsof phase noise.We begin by identifying some very general tradeoffs amongkey parameters, such as power dissipation, oscillation fre-quency, resonator, and circuit noise power. These tradeoffsare first studied qualitatively in a hypothetical ideal oscillatorin which linearity of the noise-to-phase transfer function isassumed, allowing characterization by the impulse response.Although linearity is defensible, time invariance fails tohold even in this simple case. That is, oscillators are lineartime-varying (LTV) systems. By studying the impulse response,Manuscript received August 16, 1999; revised October 29, 1999.T. H. Lee is with the Center for Integrated Systems, Stanford University, Stan-ford, CA 94305-4070 USA.A. Hajimiri is with the California Institute of Technology, Pasadena, CA91125 USA.Publisher Item Identifier S 0018-9200(00)00746-0.Fig. 1. “Perfectly efficient” RLC oscillator.we discover that periodic time variation leads to frequencytranslation of device noise to produce the phase-noise spectraexhibited by real oscillators. In particular, the upconversionof 1noise into close-in phase noise is seen to depend onsymmetry properties that are potentially controllable by thedesigner. Additionally, the same treatment easily subsumesthe cyclostationarity of noise generators, and helps explainwhy class-C operation of active elements within an oscillatormay be beneficial. Illustrative circuit examples reinforce keyinsights of the LTV model.II. GENERAL CONSIDERATIONSPerhaps the simplest abstraction of an oscillator that still re-tains some connection to the real world is a combination of alossy resonator and an energy restoration element. The latterprecisely compensates for the tank loss to enable a constant-am-plitude oscillation. To simplify matters, assume that the energyrestorer is noiseless (see Fig. 1). The tank resistance is thereforethe only noisy element in this model.To gain some useful design insight, first compute the signalenergy stored in the tank(1)so that the mean-square signal (carrier) voltage is(2)where we have assumed a sinusoidal waveform.The total mean-square noise voltage is found by integratingthe resistor’s thermal noise density over the noise bandwidth ofthe RLC resonator(3)Combining (2) and (3), we obtain a noise-to-signal ratio (thereason for this “upside-down” ratio is one of convention, as willbe seen shortly)(4Sensibly enough, one therefore needs to maximize the signallevels to minimize the noise-to-carrier ratio.0018–9200/00$10.00 © 2000 IEEELEE AND HAJIMIRI: OSCILLATOR PHASE NOISE 327We may bring power consumption and resonator explicitlyinto consideration by noting thatcan be defined generally asproportional to the energy stored, divided by the energy dissi-pated:(5)Therefore(6)The power consumed by this model oscillator is simply equalto, the amount dissipated by the tank loss. The noise-to-carrier ratio is here inversely proportional to the product of res-onatorand the power consumed, and directly proportional tothe oscillation frequency. This set of relationships still holds ap-proximately for real oscillators, and explains the near obsessionof engineers with maximizing resonator, for example.III. DETAILED CONSIDERATIONS: PHASE NOISETo augment the qualitative insights of the foregoing analysis,let us now determine the actual output spectrum of the idealoscillator.A. Phase Noise of an Ideal OscillatorAssume that the output in Fig. 1 is the voltage across the tank,as shown. By postulate, the only source of noise is the whitethermal noise of the tank conductance, which we represent asa current source across the tank with a mean-square spectraldensity of(7)This current noise becomes voltage noise when multiplied bythe effective impedance facing the current source. In computingthis impedance, however, it is important to recognize that theenergy restoration element must contribute an average effectivenegative resistance that precisely cancels the positive resistanceof the tank. Hence, the net result is that the effective impedanceseen by the noise current source is simply that of a perfectlylossless LC network.For