FREQUENCY STABILITY. 6.1 Frequency Stability of a Passive Atomic Frequency Standard

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Cordelia Hardy
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1 Chapter 6 9 FRQUNCY TBILITY 6. Frequec tabilit o a Passive tomic Frequec tadard This sectio calculates the requec stabilit o a passive atomic requec stadard, which depeds o the width o the atomic resoace sigal, the sesitivit o the atomic resoace measuremet, the ree-ruig phase oise ad the badwidth o the requec cotrol loop. Ma atomic clocks use a simple itegrator to lock the voltage-cotrolled oscillator to the atomic resoace sigal (Figure 6.): Voltage-Cotrolled Oscillator tomic Resoace Noise Floor Itegrator Oscillator Phase-Noise Figure 6.: Passive tomic Frequec tadard

2 93 ssume the slope o the voltage-cotrolled oscillator requec-voltage tuig curve is K V, ad the slope o the atomic resoace error sigal is: (6.) The dierece i the requec o the voltage-cotrolled oscillator ad the ceter o the atomic resoace is: O (6.) The traser uctio o the itegrator is modeled usig: H ( s) (6.3) RCs I the igure above, the loop gai is give b the product o these three quatities: G ( s) KV (6.4) RCs s Whe the sstem is ope loop, a oise voltage v at the itegrator iput geerates a sigal V proportioal to (s) at the output o the atomic resoace measuremet. The equatio which describes the closed-loop sstem is: ( v + V ) ( s) V (6.5) ad thereore V ( s) G H ( s) (6.6) v ( s) s + G The requec deviatio o the voltage-cotrolled oscillator is:

3 94 V (6.7) The traser uctio betwee ijected oise v ad the oscillator requec deviatio is: ) ( H s v (6.8) For atomic clocks, the power spectrum o the ormalized requec deviatio is covetioal [,]: O t ) ( (6.9) The traser uctio betwee the ormalized requec deviatio ad the receiver oise is: ) ( H s v (6.0) ad, thereore, the power spectral desit o the voltage oise is related to the power spectral desit o the requec deviatio accordig to: (s) H (6.) For requec osets withi the loop badwidth, the requec stabilit is:, (6.)

4 95 The requec stabilit improves b decreasig the oise loor or icreasig the slope o the error sigal. For example, i the slope o the error sigal is µv / MHz, the receiver oise loor is V / Hz, ad the resoace requec is 00 MHz, the the power spectrum o the ormalized requec deviatios is approximatel 0 -. To take ito accout the phase oise o the voltage-cotrolled oscillator, we ca show that the overall requec stabilit is [3,4]:, O H( s) +, H ( s) (6.3) where,o is the (ope-loop) requec spectrum o the voltage-cotrolled oscillator, ad, is the itrisic stabilit o the atomic resoace deied i quatio 6.. The relatioship betwee the power spectral desit o phase ad the equivalet power spectral desit o the ormalized requec is []: ( ) θ ( ) (6.4) O where is the oset rom carrier. The uctio H (s) is the low-pass characteristic deied i quatio 6.6. The ilterig o the oscillator phase oise is a high-pass uctio H (s): G s H ( s) (6.5) s + G s + G Thereore, well outside the loop-badwidth, the phase-oise is the same as the ope-loop oise o the oscillator; iside the loop the requec stabilit is determied b the atomic resoace stabilit (6.). For oset requecies i the viciit o the loop-badwidth, the overall requec oise ca be calculated rom (6.3). We ca also see ituitivel that the eedback loop is desiged to adjust the voltage at the itegrator iput to zero. Hece, withi the loop badwidth, oise at the itegrator iput is

5 96 compesated b a deviatio i the voltage-cotrolled oscillator requec rom the ceter o the atomic resoace (Figure 6.): Voltage Istataeous Frequec Nomial Frequec Frequec Figure 6.: tomic Resoace igal showig Istataeous ad Mea Frequec This requec deviatio is determied b the slope o the atomic resoace sigal, ad thereore, withi the loop badwidth: v (6.6) This equatio leads directl to (6.). ice, as described i quatio 6.3, the best possible requec stabilit iside the loop badwidth is or a oiseless voltage-cotrolled

6 97 oscillator, (6.) ca be cosidered as the itrisic stabilit o the atomic resoace, which is see to deped o the measuremet sesitivit ad the atomic resoace width. 6. Measuremet o Frequec tabilit The log-term stabilit o a requec source is usuall characterized i the time domai. simple techique or measurig the requec stabilit is show below: Frequec Couter Log Frequec Data Reerece Oscillator DUT Oscillator Figure 6.3 Direct Frequec Couter Measuremet o lla Variace I Figure 6., a requec couter samples the requec o the device uder test ad the requec measuremet data is logged to a computer. The variatios betwee successive requec measuremets characterize the requec stabilit o the oscillator. The usual metric or the time-domai requec stabilit is the two-sample variace, or lla variace [,]:

7 98 M σ ( τ ) ( k + k ) (6.7) ( M ) k where is the ormalized requec deviatio (quatio 6.9) averaged over the samplig iterval: tk + τ τ ( t) dt tk (6.8) The direct requec couter techique assumes the reerece oscillator iteral to the requec couter is more stable tha the DUT oscillator. The estimate o the lla variace should also accout or the dead time betwee measuremets, as described i [5]. The article cotais ormulas to correct the error itroduced b dead time, depedig o the power-law characteristic o the oise processes. dditioal measuremet techiques are described i [6]. The lla variace, deied i (6.7), ad the power spectrum o the ormalized requec deviatio () are related b the trasormatio []: 4 si ( π τ ) σ ( τ ) ( ) d 0 (6.9) ( πτ ) The trasormatio betwee the lla variace ad spectral desit o the requec luctuatios is simple whe the spectral desit is a power law: α ( ) h α (6.0) White oise requec luctuatios result i a lla variace: 4 si ( πτ ) 0 σ ( τ ) h0 d 0 (6.) ( πτ ) τ h

8 ad licker requec oise results i a variace []: 99 4 h si ( ) ( ) πτ σ τ d log 0 e h (6.) ( πτ ) White oise results i τ - scalig o the lla variace, ad / oise sources result i a costat licker loor i the lla variace plot. For example, assumig the short-term stabilit o the passive atomic requec stadard is due to a white oise process, usig (6.) ad (6.) shows that the lla variace is give b: σ, ( τ ) τ (6.3) The short-term stabilit o a requec stadard is ote quoted as the lla deviatio at secod: σ (, τ ) (6.4) For example, i the slope o the error sigal is µv / MHz, the receiver oise loor is V / Hz, ad the resoace requec is 00 MHz, the lla deviatio is approximatel at secod itegratio.

9 6.3 olid-tate tomic Frequec tadard 00 The prototpe solid-state atomic requec stadard is the same as the zero-ield electro spi resoace spectrometer described i Chapter 4, with the additio o a eedback loop to lock the voltage-cotrolled oscillator to the ceter o the magetic resoace sigal (Figure 6.4): Helmholtz Coils 90 O Hbrid Modulatio Driver Voltage-Cotrolled Oscillator lectrical Resoator Lock-I mpliier Phase djuster Phase Detector Noise Filter Itegrator Figure 6.4 olid-tate tomic Frequec Reerece I the prototpe measuremets, the badwidth o the requec-lockig loop was approximatel Hz. The proo-o-cocept experimet eeds reiemet to reduce the techical oise sources to below the udametal radom oise processes; however, requec-lockig has bee demostrated as show o the ext page i Figure 6.5:

10 0.9 x 09 Frequec Lock Frequec Time (secods) ~ 30 mis Figure 6.5 Frequec-Lockig o olid-tate tomic Frequec tadard The solid-state atomic clock geerates a reerece requec at 6.5 MHz. The oscillator is locked to the magetic resoace or more tha 0 miutes, but shows cosiderable requec drit. The requec drit is due thermal drit o the aligmet o the loop-gap resoator, which results i drit i the backgroud itererece sigal rom acoustic pickup. t 600 secods, the drit is such that the zeroig is lost ad the loop comes ulocked. The short-term requec variatios are due to 60 Hz spurs close to the lock-i modulatio requec, ot the radom oise loor. With improvemets i the desig, lla deviatios at secod o the order o 0-8 or better appear easible, as will be discussed i the coclusio.

11 6.4 Frequec tabilit Thermal Noise Limit 0 The theoretical short-term requec stabilit ca be calculated assumig thermal oise o the receiver is the limitig oise source. I practice, other oise sources ma limit sstem perormace; however, with proper desig the magetic resoace measuremet will approach the thermal limit as discussed i Chapter 5. For example, i the prototpe desig, thermal drit i the aligmet o the capacitive gap results i a relativel large backgroud sigal. The variatios i the gap result i chages i sesitivit to the acoustics ad commo mode pickup. secod source o oise is spurs rom 60 Hz harmoic pickup. I the curret prototpe these sstematic oise sources mask the thermal oise loor. The requec stabilit assumig thermal oise as a limitig source is calculated usig quatio 6.4. example measuremet o the vaadium magetic resoace is show i Figure.3, Chapter. The lock-i ampliier gai was ad the peak-to-peak resoace sigal is. µv, with a width o 6 MHz. The slope o the error sigal rom the atomic resoace is 0. µv / MHz. The measured spectrum show at the ed o Chapter (Figure.5) used a mixer with a 6 db pad o the iput to improve matchig. The coversio gai o the mixer ad atteuator loss accout or 3 db sigal loss which subtracts directl rom the sesitivit. I a lowoise ampliier were used ahead o the mixer, the coversio loss would ot limit the sesitivit. With these assumptios, the sigal ma be icreased b 3 db (actor o 4.5 i voltage) to estimate the thermal limit o sesitivit. s a result, we use 0.9 µv / MHz as the error sigal slope i the estimate below. The oise o a 50 Ohm resistor at 300K is 0.9V / Hz. Usig quatio 6.4, with a reerece requec o 5 MHz, the thermal limit o the lla variace at secod is see to be This calculatio is ot a udametal limit o the requec stabilit, i that ma additioal actors, such as the cocetratio o paramagetic ios, the carrier power, ad the resoator desig, ma be optimized to improve the short-term stabilit.

12 03 BIBLIOGRPHY, CHPTR 6 [] I td , I tadard Deiitios o Phsical Quatities or Fudametal Frequec ad Time Metrolog Radom Istabilities, March 999. [] J.. Bares et al., Characterizatio o Frequec tabilit, I Trasactios o Istrumetatio ad Measuremet, vol. 0, o. 05, 97. [3] J. Vaier ad L. Berier, O the igal-to-noise ad hort-term tabilit o Passive Rubidium Frequec tadards, I Trasactios o Istrumetatio ad Measuremets, IM-30, 77, 98. [4] J. Vaier, M. Tetu ad L. Berier, Traser o Frequec tabilit rom a tomic Reerece to a Quartz-Crstal Oscillator, I Trasactios o Istrumetatio ad Measuremets, (979), IM-8, 88 [5] J.. Bares ad D. W. lla, Variaces Based o Data with Dead Time betwee the Measuremets, NIT Techical Note 38, 990. [6] J. Vaier ad M. Tetu, Time Domai Measuremet o Frequec tabilit Proc. 0th PTTI Meetig, pp. 47-9, Nov. 978.

13 Chapter 7 04 CONCLUION The dissertatio has developed the origial cocept o a solid-state atomic clock, ad preseted, i detail, a prototpe sstem to realize the cocept. The prototpe sstem uses ol low-cost radio-requec compoets which ma easil be itegrated ito a compact atomic requec stadard. The atomic resoace sigal was derived rom the zero-ield electro spi resoace o divalet vaadium i a magesium oxide lattice. The thesis also showed a origial measuremet o the zero-ield magetic resoace o vaadium i magesium oxide. s was discussed i Chapter 3, or the m F 0 to m F 0 magetic ield idepedet trasitios, there is a optimal cocetratio o paramagetic ios which icreases the paramagetic resoace sigal without sigiicatl icreasig the resoace width. The maximum cocetratio is limited b exchage couplig or secod-order dipolar broadeig. For magetic resoace lies with a irst-order Zeema shit, the short-term stabilit is idepedet o cocetratio, because the dipolar resoace width icreases i proportioal to cocetratio. With icreased dopig, the slope o the magetic resoace sigal is uchaged, ad thereore the requec stabilit is costat (Chapter 6, quatio 6.4). This is wh the existece o a secod-order magetic ield idepedet trasitio is so sigiicat or the perormace o a small solid-state atomic clock, because it implies the possibilit o icreasig the sigal itesit without irst-order broadeig o the resoace width. secod topic or urther stud is the eect o higher order terms proportioal to 3 o the apparet resoace width at zero-ield, cosidered at the begiig o Chapter. These additioal terms i the vaadium spi Hamiltoia ma accout or the asmmetr o the resoace sigal measured i Figure.3.

14 05 The temperature depedece o the zero-ield magetic resoace was ot ivestigated i this thesis. However, previous high-ield measuremets (Chapter, []) idicate the temperature coeiciet o divalet vaadium i magesium oxide is o the order o a ew parts-per-millio per degree. The hperie resoace requec would decrease with icreasig temperature. However, ver careul measuremets o the higher order terms i the isoelectroic Cr 3+ /MgO sstem showed that the actors proportioal to 3 have strog positive temperature depedece (Chapter, [6]). Hece, the combied eect o the hperie splittig ad the higher order terms ma reduce the overall temperature depedece at zero-ield. other possibilit is to look at dieret materials sstems where the Hamiltoia is a combiatio o hperie ad crstal ield terms, or to look at multiple dopats i a sigal crstal, to miimize the measured temperature coeiciet. Thereore, several research questios remai, as ar as the possibilities or exploitig arrow magetic ield idepedet trasitios to optimize the short-term stabilit, ad the possibilities o developig a materials sstem with reduced temperature depedece or eve a tur-over temperature. The best techique or measurig the magetic ield idepedet trasitios is also a ope questio, where we have suggested the possibilit o a rotatig polarizatio measuremet as oe solutio. The short-term stabilit is alwas a trade-o with the volume o the resoator, because or a larger resoator, the RF magetic ield desit is lower at a give carrier power, ad geerall, the uloaded qualit actor Q o the resoator icreases. For example, the Q o a loop-gap resoator is approximatel 500. This trade-o with sample volume is similarl the case or miiaturized rubidium atomic clocks. There is o oe aswer or the requec stabilit o the solid-state atomic requec stadard the theoretical stabilit will deped largel o the acceptable power cosumptio. I particular, or a large resoator volume, the power to drive the audio magetic ield is cosiderable. imilarl, regulatig the temperature o a large resoator uses cosiderable power. The trade-os ivolvig power cosumptio ad the volume (i.e., uloaded-q) o the resoator are characteristic o oscillator desig i geeral, provided

15 06 thermal oise is domiat ad ot excess device oise, or other oise sources which scale with carrier power. everal improvemets i the prototpe desig were also suggested i Chapters 4, 5, ad 6, which are summarized below: addig carrier suppressio or usig a secod requec discrimiator to demostrate a thermal oise limited measuremet at high carrier power miiaturizatio o the electrical resoator, ad reductio i sesitivit to acoustics ceramic mauacturig o high purit vaadium doped magesium oxide ad developig a optimal heat treatmet recipe idig the optimal dopig cocetratio or best short-term stabilit b aalsis o the secod-order dipolar broadeig o the magetic ield idepedet trasitios studies o the log-term stabilit o solid-state atomic requec stadard The goal is to demostrate a spectrometer with oise perormace limited b the receiver oise igure. I the curret demostratio, techical oise sources remai the limitig actors. We aticipate a sigiicat improvemet i the measured short-term stabilit with the use o arrow magetic-ield idepedet resoaces ad better spectrometer sesitivit. I coclusio, this work opes the possibilit or developig a commodit requec stadard with short-term stabilit ad accurac i the rage o 0-7 to 0-9. The ew requec stadard is etirel solid-state, works at room temperature, ad without large magetic ields or hermetic packagig. The possibilities or log-term stabilit better tha crstal oscillators make the solid-state atomic requec stadard a iterestig techolog to address the eeds o a umber o commuicatios ad avigatio sstems, or both militar ad ext-geeratio cosumer products.

FIR Filter Design: Part I

FIR Filter Design: Part I EEL3: Discrete-Time Sigals ad Systems FIR Filter Desig: Part I. Itroductio FIR Filter Desig: Part I I this set o otes, we cotiue our exploratio o the requecy respose o FIR ilters. First, we cosider some