NIST Technical Note 1337: Characterization of Clocks and Oscillators

where V0 ;: nomi na I peak vo Itage amp I i tude,e(t) ;: deviation of amplitude from nominal,"0 ;: nominal fundamental frequency,~(t) ;: deviation of phase from nominal.Ideally "e" and "dl" should equal zero for alltime. However, in the real world there are noperfect oscillators. To determine the extent ofthe noise components "e" and "ill", we shall turnour attention to measurement techniques.The typical precision oscillator, of course,has a very stable sinusoidal voltage output with afrl!quency v and a period of oscillation T, whichis the reciprocal of the frequency (v =1/T). Onegoa1 is to measure the frequency and/or the frequencystability of the sinusoid. Instability isactually measured, but with little confusion it isoften called stability in the literature. Naturally,fluctuations in frequency correspond tofluctuations in the period. Almost all frequencymeasurements, with very few exceptions, are measurementsof phase or of the period fluctuationsin an osci 11 ator, not of frequency, even thoughthe frequency may be the readout. As an example,most frequency counters sense the zero (or nearzero) crossing of the sinusoidal voltage, which isthe point at which the voltage is the most sensitiveto phase fluctuations.One must also realize that any frequencymeasurement involves two oscillators. In someinstances, one oscillator is in the counter. Itis impossible to purely measure only one osc11­, ator. In some instances one osci 11 ator may beenough better than the other that the fluctuationsmeasured may be considered essentially those ofthe latter. However, in general because frequencymeasurements are always dual, it is useful todefine:v - vY (t)·C" 1 a (1.2)vaas the fractional frequency difference or deviationof 05cillator one, v l ' with respect to a referenceoscillator V o divided by the nominal frequency v o 'Mow, Yet) is a dimensionless quantity and uSl!fu1in describing oscillator and clock performance;e.g., the time deviation, x(t), of an oscillatorover a period of time t, is simply given by:tx(t) = f y(t')dt' (1. 3)oSince it is impossible to measure instantaneousfrequency, any frequency or fracti ona1 frequl!ncymeasurement always involves sOflle saJllPle time, Ator "t"--some time window through which the oscillatorsare observed; whether it's a picosecond, asecond, or a day, there is always some sampletime. So when determining a fractional frequency,y(t). in fact what is happening is that the timedeviation is being measured say starting at sometime t and again at a later time, t + t. Thedifference in these two time deviations, dividedby t gi ves the average fracti ana1 frequency overthat period t:yet) = x(t + t) - x(t) (1.4)tTau, t, may be called thl! sample time or averagingtime; e.g., it may be determined by the gate timeof a counter.What happens in many cases is that one samplesa number of eyc1 es of an osci 11 ation duri n9 thepreset gate time of a counter; after the gate timehas elapsed, the counter latches the value of thenumber of cyc 1es so that it can be read out,printed, or stored in some othl!r way. Then thereis a delay time for such processing of the databefore the counter anns and starts again on thenext cycle of the oscillation. During the delaytime (or process time), infonnation is lost. Wehave chosen to call it dead time and in someinstances it becomes a problem. Unfortunately fordata processing in typical oscillators the effectsof dead time often hurt most when it is the hardestto avoi d. In other words, for times that areshort comparl!d to a second when it;s very difficultto avoid dead time, that is usually wOeredead time can make a significant difference in thedata analysis. Typically for many oscillators, if3TN-16