Design of crystal oscillators Willy Sansen KULeuven, ESAT-MICAS Leuven, Belgium willy.sansen@esat.kuleuven.be Willy Sansen 0-05 22
Table of contents Oscillation principles Crystals Single-transistor oscillator MOST oscillator circuits Bipolar-transistor oscillator circuits Other oscillators Willy Sansen 0-05 222
The Barkhausen criterion F(jω) V f V in V ε A(jω) Σ V out V out = A(jω) V ε V f = F(jω) V out = F(jω) A(jω) V ε V f V ε = A(jω) F(jω) Oscillation if V in = 0 or if Ref. Barkhausen, Hirzel, Leipzig, 935 V f = A(jω) F(jω).0 V ε Positive FB! V f = Φ V A + Φ F = 0 o ε Willy Sansen 0-05 223
Split analysis Z resonator Z circuit Y res +Y circ Y res +Y circ = 0 Z res + Z cir = 0 Z circ +Z res Z res Z circ = 0 Oscillation if Re (Z circ +Z res ) = 0 sets the minimum gain! Im (Z circ +Z res ) = 0 sets the frequency! Willy Sansen 0-05 224
Table of contents Oscillation principles Crystals Single-transistor oscillator MOST oscillator circuits Bipolar-transistor oscillator circuits Other oscillators Willy Sansen 0-05 225
Crystal as resonator d f s =.66 d f s in MHz if d in mm quartz L s C s R s (series) C p = A ε 0 ε r d ω s 2 = Ls C s ε r 4.5 f s = 2π L s C s C p (package, parallel) L s ω s = Cs ω s L s Q ω s = Rs C s Q= Rs C s R s = Q Cs ωs Willy Sansen 0-05 226
Crystal parameters L s C s R s C p Xtal : f s = 0.000 MHz Q = 0 5 C s = 0.03 pf C p 6 pf ( 200 C s ) L s ω s = Cs ω s L s 8.4 mh R s = = 5.3 Ω QC s ω s f s L s C s R s C p Q 00.0 khz 52 H 49 ff 400 Ω 8 pf 0.8 0 5.000 MHz 2 H 6 ff 24 Ω 3.4 pf 5.3 0 5 0.00 MHz 0 mh 26 ff 5 Ω 8.5 pf.2 0 5 Willy Sansen 0-05 227
Series and parallel resonance L C R f r = 2π LC L C Z Cap. Ind. Z R R Ind. Cap. -90 o + 90 o + 90 o -90 o R f r f f r f Willy Sansen 0-05 228
Crystal impedance s 2 L s C s +sr s C s + Z s (s) = (s 2 L s C s C p R s C s C p s (C s +C p ) + s + ) C s +C p C s +C p Z s (s) R s f s f p C p s f Willy Sansen 0-05 229
Crystal impedance at resonance Z 00 kω f p f s =.998 MHz C s = 2.2 ff L s 0.52 H C p = 4.27 pf 00 Ω f s R s = 82 Ω Φ(Z) 00 o 0 o 90 o induct. Crystal operates in inductive region if circuit is capacitive! -00 o -90 o capac..97.98.99 2.00 2.0 2.02 MHz Willy Sansen 0-05 220
Series and parallel resonance Z s (ω) = -j ωc p ω 2 - ω s 2 ω 2 - ω p 2 ω s 2 = Ls C s ω p 2 = ( + ) L s C p C s Z s (ω) = R s +jωl s + jωc s series parallel Z s (ω) = R s + ( - ) j ω s C s 2p ω ω s ω s ω Frequency pulling factor p = ω - ω s ω s Z s (ω) R s + j ωcs Ref. Vittoz, JSSC June 88, 774-783 Willy Sansen 0-05 22
Series or parallel resonance? f s f p = 89.000 khz p p = 0.25 % C p f m = 88.850 khz p m = 0.25 % f s = 88.700 khz p s = 0 f m - f s C s p m = = = 0.25 % 4C p khz f p f s = + + p p = C s 2C p f s C s C p C s f p - f s 2C p = 0.25 % Willy Sansen 0-05 222
Table of contents Oscillation principles Crystals Single-transistor oscillator MOST oscillator circuits Bipolar-transistor oscillator circuits Other oscillators Willy Sansen 0-05 223
Single-transistor X-tal oscillator C 3 C 2 Basic three-point oscillator g m C C 2 C 2 C 3 C 3 C 3 C 2 g m g m g m C C C Pierce Colpitts : -pin X=D Santos : -pin X=G Willy Sansen 0-05 224
Single-transistor X-tal oscillator analysis C 3 = C p + C DG C 3 C 2 L s C s C 3 C g m R s g m C 2 Z s Z c C Barkhausen : Z s +Z c = 0 Z s = R s + j Re (Z c ) = -R s 2p Im (Z c ) = - ωc s yields g m yields f or p 2p ωc s Ref. Vittoz, JSSC June 88, 774-783 Willy Sansen 0-05 225
Complex plane for 3-pt oscillator : Design crit. g mmax g m A Im -R s 0 Re g m = 0 Im 0 - C C 2 ω(c 2p 3 + ) C Im A = - Im +C 2 0 ω s C s Ø = ωc3 + C C +C 2 3 C C 2 Small p : Large C,2 B g m = Im - ωc3 Large circle: Small C 3 Willy Sansen 0-05 226
Complex plane for 3-pt oscillator : Example -6 kω -80 Ω Im -R s 0 Re C = C 2 = 3 pf C 3 = 0.5 pf 20 MHz 80 Ω A g m = 0 Im 0-4 kω g mmax 3 ms 2p A Im A = - ωc s Ø = 2 kω p A = C s C 2(C 3 + C 2 ) C +C 2 g ma R s C C 2 ω s 2 g mb 450 ms B g m = Im - 6 kω µs Willy Sansen 0-05 227
Amplitude of oscillation i ds I ds I DSA t I ds V gs = = g ma 2 π I ds I DSA 2 I DSA g ma V GS -V T 2 kt V gs V GS -V T or 2n in wi q Large! I ds I DSA Nonlinear (Bessel) More spiked for higher C,2!!! Willy Sansen 0-05 228
Start-up of oscillation τ min occurs at g m g mmax τ min = L s Re (Z s ) + R s Re (Z s ) is half circle Ø Re (Z s ) = if C ω s C 3 <<C 2 3 R s << Re (Z s ) 2 C 3 400 τ min since C 3 200 C s ω s C s ωs or also τ min 2Q R s C 3 Willy Sansen 0-05 229
Power dissipation In MOST : g ma ω s 2 R s C C 2 R s (C ω s ) 2 I DSA g V GS -V T ma 2 µa 6 µw 2 V gs In X-tal : I c = = V gs C ω s V GS -V T C ω s Z C R s I 2 c R P c = = s V GS -V T 2 (C ω s ) 2 2 2 = V GS -V T 2 g ma 0.2 µw 2 Willy Sansen 0-05 2220
Design procedure for X-tal oscillators - X-tal : f s f p R s C p (or f s Q C s C p ) (Q = / ω s C s R s ). Take : C 3 > C p but as small as possible C s Pulling factor p = 2 C C 3 + C 2 2 C +C 2 C s C L C C L = = 2 C 2 2 C s If p < it is a series oscillator (best!) 4C p If p > it is a parallel oscillator (not stable!) Choose C L large (> C 3 ), subject to power dissipation! Willy Sansen 0-05 222
Design procedure for X-tal oscillators - 2 2. Calculate g ma R s C 2 ω L ω 2 s s ( C L C L ) C s Q and take g mstart 0 g ma 3. Choose V GS -V T, which gives the amplitude V gs g m (V GS -V T ) and current I DS = and W 2 L and power P = (V GS -V T ) 2 g m 2 4. Verify that biasing R B > / (R s C 3 2 ω s 2 ) Willy Sansen 0-05 2222
Single-transistor X-tal oscillator C 3 C 2 Basic three-point oscillator g m C C 2 C 2 C 3 C 3 C 3 C 2 g m g m g m C C C Pierce Colpitts : -pin X=D Santos : -pin X=G Willy Sansen 0-05 2223
Pierce X-tal oscillator I B V B R B C 3 C 2 g m C 32 khz.2 V 78 na Willy Sansen 0-05 2224
Colpitts X-tal oscillator I B C 2 v OUT v OUT V B g m C 3 C 2 C 3 C C Crystal grounded : single-pin : X = D Willy Sansen 0-05 2225
Santos X-tal oscillator i OUT I B (AGC) V B R B g m C v OUT C 3 C v OUT C 2 g m R B C 2 I B (AGC) C 3 Crystal grounded : single-pin : X = G Ref. Santos, JSSC April 84, 228-236 Ref. Redman-White, JSSC Feb.90, 282-288 Willy Sansen 0-05 2226
Table of contents Oscillation principles Crystals Single-transistor oscillator MOST oscillator circuits Bipolar-transistor oscillator circuits Other oscillators Willy Sansen 0-05 2227
Practical Pierce X-tal oscillator M5 M4 C C c M M6 00 MΩ C 2 C s = 0.5 ff C 3 = 0.6 pf C = C 2 = 2.8 pf 2 MHz M3 M2 g ma = 2 µs I DSA = 80 na I DS = 350 na V gs = 300 mv Ref. Vittoz, JSSC June 88, 774-783 Willy Sansen 0-05 2228
Full schematic Ref. Vittoz, JSSC June 88, 774-783 Willy Sansen 0-05 2229
Single-pin oscillator with crystal to Gate C C2 R B M OUT f s = 9.9956 MHz f p = 0.02 MHz C s = 24.3 ff C o = 7.4 pf L = 0.4 mh R = 7.2 Ω (?) p = 0.8 0-3 C = C 2 = 50 pf g ma = 350 µs I DSA = 90 µa (V GS -V T = 0.5 V) Willy Sansen 0-05 2230
Single-pin oscillator - g m + - g m g m2 2 3 + + - - C load C R B C 2 g m = R s (C s ω 0 ) 2 DC unstable! Positive FB dominant at crystal frequency! Ref. van den Homberg, JSSC July 99, 956-96 Willy Sansen 0-05 223
Single-pin oscillator - 2 2 3 0 MHz, 3.3 V, 0.35 ma Ref. van den Homberg, JSSC July 99, 956-96 Willy Sansen 0-05 2232
X-tal oscillators with CMOS inverters = Large current peaks! Bad PSRR!! Willy Sansen 0-05 2233
Table of contents Oscillation principles Crystals Single-transistor oscillator MOST oscillator circuits Bipolar-transistor oscillator circuits Other oscillators Willy Sansen 0-05 2234
Pierce X-tal oscillator I B R 2 C 3 C C 2 R L vout C 3 C 2 V B R B g m R C Willy Sansen 0-05 2235
Colpitts X-tal oscillator I B C 2 v OUT v OUT V B g m C 3 C 2 C 3 C C Crystal grounded : single-pin : X = D Willy Sansen 0-05 2236
Santos X-tal oscillator V B R B g m C 3 C C 2 v OUT I B (AGC) Crystal grounded : single-pin : X = G Buffered output Ref. Santos, JSSC April 84, 228-236 Ref. Redman-White, JSSC Feb.90, 282-288 Willy Sansen 0-05 2237
98 GHz VCO in SiGe Bipolar technology Colpitts 0.55 x 0.45 mm 2 ma at - 5 V -97 dbc/hz at MHz Ref. Prendl BCTM Toulouse 03 Willy Sansen 0-05 2238
Positive feedback circuits - R L Q Q 2 v OUT T=g m R L R L >R s Ref. Nordholt, CAS 90, 75-82 Willy Sansen 0-05 2239
Positive feedback circuits - 2 600 Ω 200 v OUT Buffered ouput! - 250 Ω 500 µa 250 Ref. Nordholt, CAS 90, 75-82 Willy Sansen 0-05 2240
Positive feedback circuits - 3 g ma = 8 ms 00 MHz Ref. Nordholt, CAS 90, 75-82 Willy Sansen 0-05 224
Table of contents Oscillation principles Crystals Single-transistor oscillator MOST oscillator circuits Bipolar-transistor oscillator circuits Other oscillators Willy Sansen 0-05 2242
Voltage Controlled Oscillator I B R L L L R L ω s = LC D C D V c C D g ma R L (C D ω s )2 v OUT v OUT dv 2 out { ω}= 4 ω s 4kTR L ( + ) ( ) 2 df 3 ω Ref. Craninckx, ACD Kluwer 96, 383-400 ; JSSC May 97, 736-744 Willy Sansen 0-05 2243
Differential crystal Oscillator R R v OUT v OUT C I B I B Willy Sansen 0-05 2244
Relaxation Oscillator R R v OUT v OUT C I B I B Ref. Grebene, JSSC, Aug.69, 0-22; Gray, Meyer, Wiley, 984. Willy Sansen 0-05 2245
RC Oscillators : 3 x 60 o = 80 o C R C R C R f c = φ 0-45 o -90 o -60 o at.73 f c 2π RC f c f - + V out Willy Sansen 0-05 2246
Wien Oscillator : 3 x Gain required C R V ε + 2τ s + τ = 2 s 2 V out 3 + 3τ s + τ 2 s 2 C R V ε + - 2R V out φ τ = RC f osc = 2π τ R 0 f Willy Sansen 0-05 2247
Voltage-controlled X-tal oscillator ± C Res. Resonator 457 khz Tuning ± 5 khz Wien bridge : R 2 = 2 R Ref. Huang, JSSC June 88, 784-793 Willy Sansen 0-05 2248
Variable capacitance ± C G m block ± C I 2 Y in = sc d G m R d 25 R d C d block ± G m with I 2 Ref. Huang JSSC June 88, 784-793 Willy Sansen 0-05 2249
G m block to generate ± G m I = 90 µa I 2 = 0 80 µa G m = B [(2βI ) /2 -(2βI 2 ) /2 ] Willy Sansen 0-05 2250
R d C d block as differentiator C d = 4 pf R d = 40 kω Ref. Huang JSSC June 88, 784-793 Willy Sansen 0-05 225
Table of contents Oscillation principles Crystals Single-transistor oscillator MOST oscillator circuits Bipolar-transistor oscillator circuits Other oscillators Willy Sansen 0-05 2252
References X-tal oscillators - A.Abidi, "Low-noise oscillators, PLL s and synthesizers", in R. van de Plassche, W.Sansen, H. Huijsing, "Analog Circuit Design", Kluwer Academic Publishers, 997. J. Craninckx, M. Steyaert, "Low-phase-noise gigahertz voltage-controlled oscillators in CMOS", in H. Huijsing, R. van de Plassche, W.Sansen, "Analog Circuit Design", Kluwer Academic Publishers, 996, pp. 383-400. Q.T. Huang, W. Sansen, M. Steyaert, P.Van Peteghem, "Design and implementation of a CMOS VCXO for FM stereo decoders", IEEE Journal Solid-State Circuits Vol. 23, No.3, June 988, pp. 784-793. E. Nordholt, C. Boon, "Single-pin crystal oscillators" IEEE Trans. Circuits. Syst. Vol.37, No.2, Feb.990, pp.75-82. D. Pederson, K.Mayaram, Analog integrated circuits for communications, Kluwer Academic Publishers, 99. Willy Sansen 0-05 2253
References X-tal oscillators - 2 W. Redman-White, R. Dunn, R. Lucas, P. Smithers, "A radiation hard AGC stabilised SOS crystal oscillator", IEEE Journal Solid-State Circuits Vol. 25, No., Feb. 990, pp. 282-288. J. Santos, R. Meyer, "A one pin crystal oscillator for VLSI circuits", IEEE Journal Solid-State Circuits Vol. 9, No.2, April 984, pp. 228-236. M. Soyer, "Design considerations for high-frequency crystal oscillators", IEEE Journal Solid-State Circuits Vol. 26, No.9, June 99, pp. 889-893. E. Vittoz, M. Degrauwe, S. Bitz, "High-performance crystal oscillator circuits: Theory and application", IEEE Journal Solid-State Circuits Vol. 23, No.3, June 988, pp. 774-783. V. von Kaenel, E. Vittoz, D. Aebischer, " Crystal oscillators", in H. Huijsing, R. van de Plassche, W.Sansen, "Analog Circuit Design", Kluwer Academic Publishers, 996, pp. 369-382. Willy Sansen 0-05 2254
Appendix: Polar diagrams Willy Sansen willy.sansen@esat.kuleuven.be Willy Sansen 0-05 2255
Amplitude, phase, Real & Imaginary Im φ Re = Re 2 + Im 2 tg(φ) = Im Re Re = cos (φ) Im = sin (φ) Willy Sansen 0-05 2256
Polar diagram of RC network - Z R Im Z Z C C R 0 Im 0 ω = ω = 0 Cω R Z = R + ω = Cjω Re Re ω = 0 ω = RC Willy Sansen 0-05 2257
Polar diagram of RC network - 2 Z C Z = R + Cjω R Im 0 Im 0 R ω = 0 R = ω = ω = ω C RC R = Re Re R = 0 Willy Sansen 0-05 2258
Polar diagram of RC network - 3 Z Im 0 R ω = ω = 0 Re R Z = R + RCjω C Im 0 R = 0 ω = R = RC ωc Re - ωc R = Willy Sansen 0-05 2259
Polar diagram of RC network - 4 Z R C Im 0 R ω = ω = 0 Re Z r R Z = R + RCjω C 0 r ω = ω = 0 R+r Re Z = r + R + RCjω ω = RC Ref. Sansen, JSSC Dec.72, 492-498 Willy Sansen 0-05 2260
Polar diagram of RC network - 5 Z R C Im 0 R = 0 Re Im 0 Re Z C 2 Z = R + RCjω - ωc 2 R = 0 R = Z = + jωc 2 R C R + RC jω - ωc R = C +C 2 - ωc C 2 R = ωc Willy Sansen 0-05 226
Circuit input impedance Zc Z c g m + 2 jωc C jωc 3 (g m + jωc ) C 3 C 3 C 2 if C 3 << C = C 2 g m = C Z c C For g m 0 Z c0 2 /ωc For g m Z c /ωc 3 Willy Sansen 0-05 2262
Complex plane for 3-point oscillator g mmax g m A Im -R s 0 Re g m = 0 Im 0 - C C 2 ω(c 2p 3 + ) C Im A = - Im +C 2 0 ω s C s C 3 C 2 C B Ø = ωc3 g m = + C 3 C +C 2 C C 2 Im - ωc3 - + C 3 Willy Sansen 0-05 2263
Calculation of g ma Z c C 3 = R S Z c = g ma C 2 = C C 3 s g m + (C +C 2 )s C C 2 g m + (C +C 2 + )s C 3 C Re (Z c ) = R s For small g m : g ma R s (C eff ω s ) 2 2C 3 C eff = C ( + ) C C Maximum negative resistance is / 2ωC 3 at g mmax = ωc C C 3 Willy Sansen 0-05 2264