NIST Technical Note 1337: Characterization of Clocks and Oscillators

and eq (1.1) we get2m>(t) = ~T(t) = 2m>0Rearranging, we have2m>(t) - 2m>0 = ~(t)or"It) - "0 =~+ ~(t)(8.5Jspectral density of frequency fluctuations isdenoted by S (f) and is obtained by passing theysignal from an oscillator through an ideal FMdetector and performing spectral analysis on theresultant output voltage. S (f) has dimensions ofy(fractional frequency)llHz or Hzal. Differentiationof ~(t) corresponds to multiplication by -!"0in terms of spectral densities. With further calculation,one can derive thatThe quantity u(t) -va can be more convenientlydenoted as 6v(t). a change in frequency at time t.Equation (8.5) tells us that if.we differentiateth.e phase f1uctuatlons ~(t) and divide by 2n, wewill have calculated the frequencyOU (t) .fluctuationRather than specify; ng a frequency f1 uctuationin terms of shift in frequency, it ;suseful to denote O\1'(t) ....ith respect to the nominalfrequency ~O- The quantity ~ is called theVofractional freguency fluctuation"'· at time t and;s signified by the variable y(t), We have(8.6)The fractional frequency fluctuation yet) isa dimensionless quantity.Wtu!O ta1!(;ng about frequencystabil ity, its approoriateness becomesclearer if weconsider the following example.Suppose in two oscillators 6,,(t) is consistentlyequal to + 1 Hz and we have sampled this value formany times t. Are the two oscillators equal intheir ability to producefrequencies?their desired outputNot if one oscillator is .operatingat 10 Hz and the other at 10 MHz.In one case Ith@ average value of t~e fractional frequencyfluctuation is 1110, and in the second case is1/10,000,000 or 1 x 10- 7 . The 10 MHz oscillatoris then lIore precise.If frequencies are multipliedor divided IJsingfractional stability is not changed.ideal electronics, theIn ttle frequency domain. we can measure the.spectrum of frequency fluctuations yet). The~~ Some international recommendations replaceUfractional ll by " norma 11zed u •(8.7)We ~ill address ourselves primarily to S~(f). thatis, the spectral density of phase fluctuations.For no; se-measurement purposes, Slj)( f) can bemeasured with a straightforward, easily duplicatedequipment set-up.Whether one measures phase orfrequency spectral densities is of minor importancesince they bear a direct relationship. It isimportant, however, to make the distinction and touse eq (8.7) if ~ecessary.8ft 1 The Loose Phase-locked loopSection 1, 1.1, C described a method afme-asuring phase fluctuations between two phaselock.edoscillators. How we win detail the procedurefor measuring S~(f),Suppose: we have a noisy oscillator. We wishto measure the oscjllator's phase fluctuationsrelative to nominal phase. One can do this byphaselocldng another oscillator (called the referenceoscillator) to themixing the two oscillator signals 90~test oscillat.or andout of phase(phase quadrature). Th; s ; $ shown schematicallyin figure 8.9.The two oscillators are at thesame frequency in long term as guaranteed by theph.s.-loc~ loop (PLL). A low-pass filter (tofi1ter ttle R.F. sum component) is used after themixer since the difference (baseband) signal isthe one of interest.By holding the two signalsat a relative phase difference of 90~.short-termphase fluctuations between the test and referenceoscillators will appear as V"oltage fluctuationsout of ~he mi~er.21TN-34