Design Guidelines for Quartz Crystal Oscillators. R 1 Motional Resistance L 1 Motional Inductance C 1 Motional Capacitance C 0 Shunt Capacitance

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Cecil Marsh
6 years ago
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1 TECHNICAL NTE 30 Dsign Guidlins for Quartz Crystal scillators Introduction A CMS Pirc oscillator circuit is wll known and is widly usd for its xcllnt frquncy stability and th wid rang of frquncis ovr which thy can b usd Thy ar idal for small, low currnt and low voltag battry opratd portabl products spcially for low frquncy applications [,] Whn dsigning with miniaturizd quartz crystals, carful considration must b givn to th frquncy, gain and crystal driv lvl In this papr, th dsign quations usd in a typical crystal controlld pirc oscillator circuit dsign ar drivd from a closd loop and phas analysis Th frquncy, gain and crystal driv currnt quations ar drivd from this mthod Basic Crystal scillator Th basic quartz crystal CMS Pirc oscillator circuit configuration is shown on Figur Th crystal oscillator circuit consists of an amplifying sction and a fdback ntwork For oscillation to occur, th Barkhausn critria must b mt a) Th loop gain must b qual to or gratr than on, and b) Th phas shift around th loop must b qual to an intgral multipl of π Th CMS invrtr provids th amplification and th two capacitors, C D and C G, and th crystal work as th fdback ntwork A stabilizs th output voltag of th amplifir and is usd to rduc th crystal driv lvl V Figur - Basic Pirc scillator Circuit Crystal Charactristics In ordr to analyz th quartz crystal oscillator, w must first undrstand th crystal itslf Figur shows th lctrical quivalnt circuit of a quartz crystal Th L, C and ar gnrally rfrrd to as th lctrical quivalnt of th mchanical paramtrs; inrtia, rstoring forc and friction, rspctivly Ths paramtrs can b masurd using a crystal impdanc mtr or a ntwork analyzr C 0 is th shunt capacitanc btwn trminals and th sum of th lctrod capacitanc of th crystal and packag capacitanc Motional sistanc L Motional Inductanc C Motional Capacitanc C 0 Shunt Capacitanc Figur Crystal Elctrical Equivalnt Circuit V STATEK Corporation, 5 N Main St, rang, CA FAX: wwwstatkcom

2 TN-30 Pag of 8 This quivalnt circuit can ffctivly b simplifid as a rsistanc ( ) in sris with a ractanc (X ) at a frquncy f as shown in Figur 3 Figur 3 - Effctiv Elctrical Circuit of a Quartz Crystal From quation () and (), an xampl of th magnitud of and X as a function of frquncy ar shown in Figurs 4 and 5 rspctivly for f s =3768kHz, C =4fF, and =8kΩ Th frquncy is xprssd in trms of part pr million (ppm) abov th sris rsonant frquncy (f s ) of th crystal ( f/f) Ths two graphs ar vry usful in th analysis of th crystal oscillator E+06 (f) and X (f) as a function of frquncy ar as follows: (f) = X ( m X ) ( X ) + - () (ohms) C 0 =35pF C 0 =4pF E+05 E f/f abov f s (ppm) Figur 4 - (ohms) vs f/f (ppm) () E+08 whr (3) E+07 Co=35pF Th sris rsonant frquncy of th crystal is dfind as (4) X (ohms) E+06 E+05 Co=4pF E+04 Th quality factor Q is dfind as Q = ω S L = ωs C (5) E f/f abov fs (ppm) Figur 5 - X (ohms) vs f/f STATEK CPATIN 5 N MAIN ST, ANGE, CA FAX: mail:sals@statkcom

3 TN-30 Pag 3 of 8 Crystal scillator Dsign Th AC quivalnt circuit of th amplifir and fdback ntwork of a pirc oscillator is shown in Figur 6 For th following analysis, A is omittd and will b rintroducd latr + X X X θ I X X G G X = X - X Figur 6 - Pirc scillator AC Equivalnt Circuit sin θ =, + X cos θ = X + X Figur 8 - Impdanc Phas Diagram From Figur 6;,,, and Frquncy Equation From th imaginary part of th currnt phas diagram (y-axis), FVθ, ;, VLQθ (6) Th phas and amplitud rlationship of th oscillator voltag, currnt and impdanc ar shown in Figurs 7 and 8 Assum that th oscillator is oscillating at a frquncy f and th amplifir output currnt I D is 80 out of phas with th oscillator input voltag V VLTAGE and from th quations drivd from th quivalnt circuit, th voltag and impdanc phasor diagram quation (6) bcoms X X D from X I = I + I X X X = X - X G V = I X + X I X V θ V X V X V Thn (7) 9 ;H 9 ;H9 Ã 9 9 ;H9 9 5H Assuming ( ) Xm << and - X X X X Xm ( - X ) m << Figur 7 - Currnt and Voltag Phas Diagram STATEK CPATIN 5 N MAIN ST, ANGE, CA FAX: mail:sals@statkcom

4 TN-30 Pag 4 of 8 Equation () bcoms s X(f) = whr X = ωc X m Xm - X and C = C + C s (7a) s Thn (0) C s is th circuit stray capacitanc across th crystal Lt and From q 7a and 7b on can obtain X m X = X (X - X m) X = m C L X C L X X + X C L (7b) Thn (8) From q (3) and (4) From quation (8) whr C L = C S + C L Equation 0 is th oscillating frquncy of th crystal oscillator C L is calld th load capacitanc of th oscillator With a spcifid C L, th crystal manufacturr can thn match th crystal to th customrs circuit to obtain th dsird oscillation frquncy From th C L quation, th rlationship btwn th othr circuit paramtrs can b stablishd (i C D, C G, and C S ) as it rlats to th oscillation frquncy of th crystal oscillator In a typical CMS oscillator gnrally dcrass as th supply voltag incrass This causs a dcras in load capacitanc and an incras in th oscillation frquncy Figur 9 shows th ffctiv load capacitanc (C L ) changs as th output rsistanc ( ) changs CL (pf) C D = 0pF C G = 30pF C& SM V = 0pF C = 4pF = 8kΩ (MΩ) Figur 9 - Effctiv Load Capacitanc (C L ) vs utput sistanc ( ) (9) STATEK CPATIN 5 N MAIN ST, ANGE, CA FAX: mail:sals@statkcom

5 TN-30 Pag 5 of 8 Gain Equation From th ral part of th currnt phas diagram (x-axis); I D = I cos θ + I sin θ () and from th quation drivd from th voltag, and impdanc phas diagram quation () bcoms It is important to not hr that in most analyss, only th first trm of quation () is usd Th scond trm must b takn into account spcially for low frquncy application wr th scond trm bcoms largr than th first trm as shown in Figur 0, whn is lss than MΩ gm (u mhos) g m g m (MΩ) q (7), and from X = X X G and Figur 0 - Comparison of minimum gm rquirmnts vs Amplifirs output rsistanc ( ) Whr g m = first trm and g m = nd trm of quation() For C D =0pF, C G =30pF, C S =pf, C =4pF, =8KΩ, f =3768kHz and C L =3pF whr C ~ + C ( ) L () Equation () givs th minimum g m rquird for th oscillator to maintain oscillation In practic, 5 to 0 tims th calculatd valu is rquird to insur fast start of oscillation This quation also aids th dsignr in slcting th componnt valus for C D and C G to match th CMS amplifir and th crystal Using quation (), Figurs and show th chang in th minimum g m rquirmnts du to chang in ithr C D or C G, whil maintaining th othr capacitor constant For a 3768kHz oscillator, as shown in Figur, trimming th output capacitor (C G ) will produc mor chang in g m than th input capacitor (C D ) As shown in Figur, a dcras in th amplifirs output rsistanc ( ) incrass th minimum g m rquirmnt STATEK CPATIN 5 N MAIN ST, ANGE, CA FAX: mail:sals@statkcom

6 TN-30 Pag 6 of 8 5 CG= 30pF Whr; and, gm (u mhos) 5 05 CD=30pF CD or CG (pf) Figur - For = 5MΩ g m comparison btwn C D and C G, whr C S =pf, C =4pF, =8KΩ, f =3768kHz Figur 3 - scillator AC quivalnt circuit with th crystal lctrical quivalnt circuit Th crystal voltag, currnt and impdanc phas rlationships ar shown in Figur 4 and 5; VLTAGE CUENT gm (u mhos) CD= 30pF CG= 30pF CD or CG (pf) Figur - For = 500kΩ g m comparison btwn C D and C G, whr C S =pf, C =4pF, =8KΩ, f =3768kHz Figur 4 - Voltag and currnt phas rlationship with th circuit quivalnt CYSTAL IMPEDANCE Crystal Driv Currnt In ordr to analyz th currnt flowing through th crystal, th AC quivalnt circuit from Figur 6 is rdrawn to show th crystals lctrical quivalnt circuit as shown in Figur 3 Th crystal driv currnt is i b, and i a is th currnt through th shunt capacitanc C Figur 5 Crystal impdanc phas diagram STATEK CPATIN 5 N MAIN ST, ANGE, CA FAX: mail:sals@statkcom

7 TN-30 Pag 7 of 8 V From; i a = X and, ( ) i a = I + X X X whr X = = ω(c ) ω(c + C sm) (3) From th currnt phas diagram of Figur 4 and th rlationship and from th crystal impdanc phas diagram Figur 5 X sin φ = ; cos φ = + X + X Substituting sin φ and cos φ and i a from quation (3) Typical Effcts f A In Th scillator Circuit In many cass, a rsistor A is introducd btwn th amplifir output trminal and th crystal input trminal as shown in Figur Th us of A will incras th frquncy stability, sinc it provids a stabilizing ffct by rducing th total prcntag chang in th amplifir output rsistanc and also incrass th ffctiv output impdanc by A as shown on Figur 9 A also stabilizs th output voltag of th oscillator and is usd to rduc th driv lvl of th crystal Th complt AC quivalnt circuit of Figur is shown in Figur 6, whr X d is th total output capacitanc of th amplifir Using th sam analytical approach, th frquncy, gain and crystal driv currnt quations with A ar drivd or i b = Figur 6 - Pirc oscillator AC quivalnt circuit with A includd Substituting; From th frquncy quation (0); (0) (4) whr; C L = C S + C L and, whr X = X X G From q (4) th crystal driv can b calculatd from; (in Watts) whr = crystals motional rsistanc Th gain quation is; g > 4π f C m [ ] G (C D+ C d) + C d + C d A ( ) STATEK CPATIN 5 N MAIN ST, ANGE, CA FAX: mail:sals@statkcom

8 TN-30 Pag 8 of 8 whr Th crystal driv currnt; X V ( + X ) +( X ) i b = X [ + ( - + X ) A ] [ X + X ] A D frncs: [] SS Chuang and E Burntt, Analysis of CMS Quartz scillator, Proc 9 th Int Congrss Chronomtry (Stuttgart, W Grmany), Spt 974 papr C [] E Vittoz, High-Prformanc Crystal scillator circuits: Thory and Application IEEE Journal of Solid stat circuits, vol 3, No Jun 988 pp A vrsion of this papr was prsntd at th 8 th Pizolctric Dvics Conf in Aug 996 by Jim Varsovia C whr and, ~ ( + ) C L Summary By using th closd loop and phas diagram mthod, w wr abl to driv th frquncy, gain and crystal driv currnt quations for a simpl quartz crystal pirc oscillator From th quations drivd hrin, it can b shown that th stray capacitanc, minimum gain rquirmnts and th output rsistanc of th amplifir must b carfully considrd to obtain optimum oscillator prformanc Th minimum gain rquirmnts should includ considration for th full rang of oprational tmpratur and voltag Th stray capacitanc (C S ) is spcially critical du to ngativ fdback ffcts and will incras th minimum gain rquirmnts of th oscillator [] As crystal manufacturrs continu to miniaturiz th crystal rsonator, th oscillator dsignr must tak into account th trad off in th crystal, amplifir and th circuit layout strays in ordr to slct th appropriat componnt valus to achiv propr crystal driv, start up, and a stabl oscillation STATEK CPATIN 5 N MAIN ST, ANGE, CA FAX: mail:sals@statkcom

Exam 1. It is important that you clearly show your work and mark the final answer clearly, closed book, closed notes, no calculator.

Exam 1. It is important that you clearly show your work and mark the final answer clearly, closed book, closed notes, no calculator. Exam N a m : _ S O L U T I O N P U I D : I n s t r u c t i o n s : It is important that you clarly show your work and mark th final answr clarly, closd book, closd nots, no calculator. T i m : h o u r