CYPRESS SEMICONDUCTOR INTERNAL CORRESPONDENCE Subject: Date: To: Author: File: Category: Cc:
XO Negative Resistance Measurement System Senior Design Proposal 17 Jan 06 GBO AJK DOCUMENT2 XOCOE
1 Summary This memo describes a proposed Senior Design project that will provide important test capability for the Cypress XO COE located at CCDID. Negative resistance is an important data sheet specification that can determine whether a Cypress chip with an integrated oscillator is accepted or rejected for use with a particular crystal. Historically, two different measurement methods have been used to determine negative resistance. One method is to determine the maximum resistance that can be placed in series with the crystal and still maintain reliable crystal startup. Another method is to use a Vector Network Analyzer to determine the impedance at the ports where the crystal connects to the chip. Both methods have known problems that make correlation with simulation problematic. The proposed Senior Design product is to design an automated measurement system for crystal oscillator negative resistance or gm that is proven to provide measurement results with less than 10% error. The proposed system consists of designing an automated measurement methodology, building a test fixture, procuring required test equipment, and creating automated test software. The final deliverable will be a prototype test system that can be installed or replicated in a Cypress characterization laboratory. In addition to determining a robust test methodology, the team will also need to use RF design techniques to design a test fixture (test board) that can accept a Cypress chip because the frequency range of interest is 1MHz to 100MHz and the oscillator is not a 50Ω system. This is an important problem to our design group that Cypress has tried to solve internally. Although improvements have been made, the desired correlation between simulation and measurement results has not been achieved. If successful the test system will be used in future characterization work.
2 Background Crystal oscillator negative resistance and its relation to oscillator startup and steady state oscillation is explained the attached JSSC article and presentation written by Vittoz. Cypress oscillators are not completely linear. As a result, the average gm or negative resistance decreases as the amplitude increases as explained in the two papers from the two UFFC articles on dipole analysis. Note that the article titled “Modeling of Quartz Crystal Oscillators by Using Nonlinear Dipolar Method”, describes negative resistance characterization of a Colpitts oscillator using an Agilent 4395A. This is a promising approach, but Cypress oscillators use the Pierce (transconductance) topology and a direct impedance measurement may not be possible. The dipole method is easy to implement using an RF simulator that utilizes HB or PSS. However, Cypress specifies negative resistance for a small signal input so the bulk of the characterization is done using AC analysis. As described in the summary, current characterization has used a series resistor or the network analyzer method described in the attached application note from EPSON. A method to measure gm instead of negative resistance is provided in the Motorola application note. This method is not directly applicable to Cypress oscillators because the load capacitors are integrated and the feedback resistor is typically hard wired between the pins that connect to the crystal.
1
CYPRESS SEMICONDUCTOR INTERNAL CORRESPONDENCE 3 Proposed Deliverables The proposed deliverables are listed in Table 1. Each phase of the project will require a review with Cypress personnel. Cypress wants the Senior Design Team to provide an integrated, turnkey, test solution for measuring negative resistance. The system can be demonstrated using U of I test equipment if equivalent test equipment can be purchased by Cypress. An example would be the Agilent 4395A spectrum/network/impedance analyzer. Cypress can provide samples of oscillator chips to be used for development and verification. Table 1 Proposed Deliverables Category Deliverable Test System Automated test system. The user should be able to place the part in the test fixture, push a button, and get raw data results as a text file that can be imported into an analysis program (e.g. Excel or Matlab). Detailed BOM, schematics, pcb cad files, and software required to build the measurement system Compatibility with the Thermonics temperature control chamber and I2C programming interface utilized by Cypress.. Less than 10% measurement error. Compatible with multiple oscillator chips. C program that runs required test equipment from Unix/Linux command prompt. National Instruments gui interface. Phase 1 Initial proof of concept hardware that shows viability of proposed measurement method. Does not need to be automated. Does Feasibility not need to use an actual oscillator chip. Documentation of calculations, simulations, measured data, and documentation that show feasibility of chosen method. Final agreement between Cypress and Senior Design Team on measurement system specifications Phase 2 Documentation of final design. Simulation or measurement data to show the designed system will meet all requirements. Design Documentation of potential risks and steps taken to reduce risk as much as possible. Documentation of any hardware or software developed by the team. Phase 3 Show that the measurement system is capable of meeting accuracy requirements with at least two different parts provided Verification by Cypress. Provide data clearly showing simulated vs. measured results and explain any discrepancies. Cypress representative will operate the system to validate that it meets expectation. Document final results and create a simple user’s manual.
Required/Desired Required
Required Required Required Desired Required Desired Desired Required Required Required Required Required Required
Required Required
2
LOWPOWER HIGHPRECISION CRYSTAL OSCILLATORS Eric A.Vittoz CSEM Centre Suisse d'Electronique et de Microtechnique SA JaquetDroz 1, CH 2007 Neuchâtel, Switzerland.
• Crystal resonator. • General theory of crystal oscillators • splitting for analysis • oscillation: condition and frequency • amplitude limitation • startup time.
• Theory of the 3point oscillator • linear analysis with and without losses • nonlinear behaviour • amplitude and energy of oscillation • frequency stability, frequency tuning • phase noise • elimination of unwanted modes • loading by amplifier.
• Practical implementations • grounded drain oscillator • grounded source oscillators • amplitude regulator • circuit examples.
page 1
Crystal Oscillators
OSC0
APPLICATIONS OF CRYSTAL OSCILLATORS • Timekeeping (realtime clocks, RTC) • precision and stability: 106 ( ≅ 30s/year) • low power watches: ≤ 0.5 µW • Radio communication • precision and stability: 106 to 105 (1 to 10kHz at 1GHz) • low power («1mW) • phase noise if :  no VCO (direct RF generation)  injection synchronized VCO  wideband PLL loop otherwise negligible (~1/Q2) • Clock of analog systems (filters) • precision and stability : 104 (better than component mismatch) • Clock of digital systems • no high precision required • beware of overtone oscillation! CSEM E.Vittoz2001
page 2
Crystal Oscillators
OSC1a
BASICS ON OSCILLATORS • Frequencydependent nonlinear loop. ω = (angular) frequency A = amplitude
G(ω, A)
• strongly nonlinear: relaxation oscillator • weakly nonlinear: harmonic oscillator • Stable oscillation at frequency ω0: G(ω0, A0) = 1 with:
d(Arg(G)) ω=ω0 < 0 dω
(phase stability)
A=A0
and:
dG ω=ω < 0 dA A=A 0
(amplitude stability)
0
Alternative representation:
• Resonator and sustaining circuit complex resonator Zr (ω, A) impedance
complex Zc(ω,A) circuit impedance
• Stable oscillation at frequency ω0: Zr(ω0, A0) + Zc(ω0,A0) = 0 CSEM E.Vittoz2001
page 3
Crystal Oscillators
OSC1
CRYSTAL RESONATOR equivalent circuit: Li
Ci
Zmi motional impedance of mode i
Ri
ii v
1 C10
C12
2 C20
2
1 0
case
0 • Mechanical resonant frequency : ωmi = • Mechanical quality factor: • • •
1 LiCi 1 Qi = »1 ωmiCiRi
ii ~ velocity ~amplitude of mode i Ci ~ electromech. coupling of mode i , Ci«C12 ωmi of different modes are not exact multiples of each other, and Qi»1; therefore: even if v(t) is strongly distorted all branches other than Zmi are negligible (for a single mode i of oscillation), and the motional current ii is always sinusoidal CSEM E.Vittoz2001
page 4
Crystal Oscillators
OSC2
MECHANICAL POWER AND ENERGY
• Mechanical energy of oscillation: ^2 ^2 LI Em = = I2 2 2ωmC must be limited to avoid destruction, aging, and nonlinear effects. • Mechanical power dissipation: ^I2 ^2 RI Pm = = 2ωmQC 2
• Can be calculated as soon as the peak value ^I of sinusoidal current i(t) is known.
CSEM E.Vittoz2001
page 5 OSC3
Crystal Oscillators
GENERAL FORM OF CRYSTAL OSCILLATOR • The resonator is combined with an active circuit to sustain oscillation (compensate R) i 2 L C20 L C12 0 C v Z C c(1) C10 R 1 R Zc(1) Nonlinear Zm Resonator circuit • Frequency of oscillation ω slightly different of ωm (effect of circuit) ω  ωm • Frequency pulling p = «1 ωm • The system must be conceptually split into: 2p Motional impedance Zm= R + j ωC (linear, strongly dependent on p) Circuit impedance Zc(1), independent of p Since no energy can be exchanged at harmonic frequencies (i sinusoidal), nonlinear effects are included by defining the circuit impedance at fundamental frequency:
V(1) where V(1)is the complex value Zc(1)= of the fundamental of v for complex I value I of sinusoidal current i. CSEM E.Vittoz2001
page 6
Crystal Oscillators
OSC4
OSCILLATION • Stable oscillation: Zm + Zc(1) = 0
ReZc(1) = R 2p0 Im Zc(1) = ωC
V(1) where Zc(1)= circuit imped. at fundam. frequency I Im complex plane R Re nonlinear I effects 2p0 exact () I) on p 1 ( ωC frequency Zc p I=0 stable Zc(1) = Zc oscillation (linear) Zm(p) • Growth of oscillation: exponential, with time constant τ =
2L ReZc(1)  R
• Critical condition for startup of oscillation: Zm + Zc = 0 (linear circuit) CSEM E.Vittoz2001
page 7 OSC5
Crystal Oscillators
AMPLITUDE LIMITATION • Necessary to define amplitude (an oscillator is always non linear) • Instantaneous limitation by distortion: → creates harmonics → intermodulation → addit. fundam. component of current → frequency change → poor stability → power dissipated in harmonics. • Nondistorting amplitude limitation: ReZc
slow reduction of bias current (lowpass filter)
R equilibrium
amplitude I
stable amplitude + improved stability + reduced power dissipation in circuit (just above critical condition).
CSEM E.Vittoz2001
page 8
Crystal Oscillators
OSC5a
STARTUP TIME OF OSCILLATOR • Equivalent circuit: L
R
I
Zc
C
I) I
( ) 1 (
ReZc10

gmopt
2p ωC
p
Zc(gm) unstable sol. B (gmmax) Zm(p) gm= ∞ • No oscillation if gm too small or too large. CSEM E.Vittoz2001
page 11
Crystal Oscillators
LOSSLESS LINEAR ANALYSIS Zi = 1 Im jωCi R Rmax =
gm
1
C +C 2ωC3(1+C3 1 2 ) C 1C 2
A gmcrit
gmopt =
OSC8a
Re
1 C 1C 2 ω(C3 + ) C1+C2
gm = 0
p
ω(C1+C2+C1C2/C3)
gmmax
Zc(gm)
B Zm(p)
gm = ∞  1
ωC3
QC > 2(1+C C1+C2 ) • Oscillation only possible if: C 3 C C 3 1 2
If large margin to minimize dp/dR, then: ω0 − ωm min. for C • p0 = ω ≅ C =C C 1C 2 m tradeoff 1 2 2(C3+
) C1+C2
(C1+C2)2 (C1C2+C2C3+C3C1)2 ωC = C1C2 Qp02 4C1C2 2 for C3« C1 and C2 • gmmax ≅ ωC C1C2Q/C3
• gmcrit ≅ ω QC
2 gmmax C C1C2 Q ≅ ≅ QC thus: gmcrit C3 C3(C1C2+C2C3+C3C1) CSEM E.Vittoz2001
2
page 12
Crystal Oscillators
OSC9
EFFECT OF LOSSES • Example of linear analysis with: C = 3fF f = 32KHz
C1 = C2= 15pF C3 = 2pF
10
20 p0
gmcrit [µA/V]
105 p0 lossless
5
10
with G1 or G2=2µA/V
gmcrit 0 0
0 100K Q C2 C1 ∆gmcrit = G + G C1 1 C2 2 50K
• Causes of losses: • biasing circuitry • loading by amplifier • series resistance of capacitors (HF) • output conductance of active device • input conductance of active device (bipolar) • external load (moisture) CSEM E.Vittoz2001
page 13
Crystal Oscillators
OSC9a
RELATIVE VOLTAGE AMPLITUDES Zm Zc Z12
V3 Z3
V1
1 1 1 = + Zc Z3 Z1+Z2+gmZ1Z2 Z2
Z1
V2
• Critical oscillation: Zm = Zc thus: 1/Z12 = 1/Zm +1/Z3 = 1/Zc + 1/Z3 yields: Z12 = (Z1+Z2+gmcritZ1Z2) 1+Z12/Z1 =
V2/V1 = Z2(1/Z1 + gmcrit) V3/V1 = V2/V1 1
• Losselss circuit (Zi =1/jωCi) with R«Rmax ω (C1C2+C2C3+C3C1)2 gmcrit = QC C1C2 V2 C C C C =  1 + j 1 (1 + 3 + 3 )2 V1 C2 QC C1 C2 usually < or « 1 Then: V2/V1≅ C1/C2 and V3/V1≅  (1+C1/C2) yields:
CSEM E.Vittoz2001
page 14
Crystal Oscillators
OSC10
EXAMPLE OF NONLINEAR ANALYSIS [L.Astier, 1987]
R 600
400
Im 100
Zm
complex plane Zc(1)[KΩ] I0 = 0.14µA
Re 400
600
800 0.4µA
V+ 1000
I0 Zc(1)
1200
V
1.0µA
Zc (I0) ; linear
1400 8.9µA
^
3.4µA
increasing amplitude I stable oscillation for particular value of R CSEM E.Vittoz2001
page 15
Crystal Oscillators
OSC10a
DISTORTION OF GATE SIGNAL Zm Zc
• Drain current is distorted C3 V2 is distorted V1
C2
V2
C1
• Drain to gate attenuation: V • for fundamental frequency: F = 1 V2 (as shown before)
C ≅ 2 C1
fund.
C3 V • for harmonic components: H = 1 ≅ V2 C1+C3 (Zm = ∞ ) harm. C3C1 • relative attenuation H = F (C1+C3)C2 usually « 1, thus V1 approximately sinusoidal • Effect of residual distortion of V1: intermodulation of harmonics in transistor creation of out ofphase fund. in drain current change in p, ⇒ frequency instability • C3 must therefore be minimized (Z3 large) CSEM E.Vittoz2001
page 16
Crystal Oscillators
AMPLITUDE OF OSCILLATION OSC11a • Limitation by nonlinear iD(vG) only → Very small effect on frequency pulling p (none if Z3 = ∞).
• Goal: minimum current to produce gmcrit: → transistor operated in weak inversion: iD = A exp(vG/nUT) • Assumption: AC signal at gate sinusoidal: vG = V0 + V1 sin(ωt) results in: iD = I0 + I1sin(ωt) + harmonics 2IB1 (x) where: I1 = I0 with x=V1/nUT IB0 (x) and IBk (x) are modified Bessel functions of order k • Transonductance gm1 for the fundamental: = gmcrit =I0crit/(nUT) gm1 = I1/V1 ⇓
2I 0
/I 0
cr
it
stable oscillation Yields bias current I0 as a function of amplitude V1: 10 IB0 (x) I0 V1 =x I0crit 2IB1 (x) nU T 5 (peak) I0 V1 = 2I0/gmcrit I0crit 0 0 1 2 3 4 5 CSEM E.Vittoz2001
page 17
Crystal Oscillators
LARGER AMPLITUDES
OSC11b
• Limited overdrive to avoid excessive distortion • Use capacitive input attenuator CaCb:
C1
Ca Cb
Ca • Attenuation 1/k= Cb+Ca C2 • Cb includes CG • In weak inversion: I0 CG = Cox(11/n)
equivalent to transistor with UT ⇒ kUT • Result for transistor in weak inversion: gmequ. = I0/(knUT), thus V1 and I0 amplified by k. 10 V1 nUT (peak) 5 k= 2
4
8
I0 I0crit(k=1)
1 0 1 5 10 • Alternative solution: strong inversion • might be necessary for f»10MHz (large W/L large C1 and C2) 0
CSEM E.Vittoz2001
page 18
Crystal Oscillators
OSC11c
AMPL. LIM. BY iD(vG) IN STRONG INVERSION • Assumption: AC signal at gate sinusoidal: vG = V0 + V1 sin(ωt) results in: iD = I0 + I1sin(ωt) + harmonics • I1/V1 = gm1(V1) = gmcrit can be calculated numerically by using a continuous model for saturated MOS transistor with zero source voltage: 1994] iD = IS ln2(1 +ev/2) [Vittoz, [Enz et al., 1995] 2
weak 2 inversion
itm 0cr
4
2I 0 /I
V1 8 nUT 6
in
where v= (vGVT0)/(nUT) and IS = 2nβUT this yields [L.Astier, 1987, von Kaenel et al.1995]:
IC= 8
«1
0 012
32 4
250
64 125
6
8 10 I0crit/I0critmin ≅ IC
I0
20
I0critmin
V1 = peak voltage amplitude at gate of transistor I0 = DC bias current of transistor I0critmin = nUTgmcrit (weak inversion) IC = I0crit/IS = inversion coefficient at I0crit CSEM E.Vittoz2001
page 19
Crystal Oscillators
CHART FOR LARGE AMPLITUDES
OSC11d
[L.Astier, 1987]
• Assumptions: • gate voltage sinusoidal (peak V1) • transistor always saturated • constant mobility • constant Q, linear Z1,Z2 and Z3 • Definitions: • I0critmin=I0crit in weak inversion • IC = I0crit/IS with IS = 2nβUT2 x = V1/(nUT)
7 0
6 0
y = x / 2 thus V1= 2I0 / gmcrit (Dirac current pulses)
50
30
10
500
250
2 0
1000
0 01
64 125
2 0 16 32 8 1 0 4
parameter IC
2000
40
weak inversion: 2 y =~ x − 2x 1
30
40
CSEM E.Vittoz2001
60 5 y = I00/I0critmin
page 20
Crystal Oscillators
BASIC DESIGN PROCEDURE
OSC11e
main criteria ref.OSC 1. Select crystal resonator frequency temp. stability cost, size 12 2. Choose value of C1=C2 precision power 8a "circle" 8a 3. Calculate p0 and ωm=ω0(1p0) 8a 8a, 9 4. Calculate gmcrit 11c I0critmin too small: 5. Fix ampl. of oscill. V1  phase noise 13b  amplification 14a  jitter of amplif. too large:  power  aging 6. Fix amount of overdrive too small: 11d  large I0/V1 too large:  poor stability 12  risk of overtone 14 7. Select IC=I0crit/IS from 5. and 6. 11d 8. Calculate I0crit, I0, IS=2nβUT2, β,W/L 11d 13a,b 9. Calculate energy Em and phase noise 10.Return to 2, 5 or 6, or detailed design. CSEM E.Vittoz2001
page 21
Crystal Oscillators
OSC13a
ENERGY OF MECHANICAL OSCILLATION Zm I ZC3 I = I1 Zm+ZC3 2p Zm = R + j ωC
V1
Zc C3 C2
I1 C1
(I , I1and V1 are complex RMS values) Thus:
j/ωC3 I = I1 R + 2pj/ωC  j/ωC3
Assumptions: ligible neg
do
mi
na
tes
C « C 2p » ωCR =1/Q p = 2(C3+C1C2/(C1+C2) 2C3 C C C Then: I ≅ I1 1 + 3 where Cs = 1 2 C1 + C2 Cs current through I ≅ jωC1V1 1 + C 3 motional impedance Z m Cs
2 2IV I2 C C 2 • Mechanical energy: Em≅ III = 1 1 1 + 3 Cs C ω2 C CSEM E.Vittoz2001
page 22
Crystal Oscillators
PHASE NOISE
OSC13b
• Simple model [Leeson, 1966](linear, time invariant) • Equivalent circuit at stable oscillation noise spectral density of resonator circuit stable oscillation 4kTR 4kTγR Z
R C L
ReZc(1) = R I
Zm Zc(1) • Impedance loading the noise sources: f − f0 f − f0 for ff0« f0 Z = 2jωL = 2jQR f0 f0 where f0 is the frequency of oscillation • Noise current IN circulating in the loop: 2 dIN + γ )R (1 + γ )kT f0 2 4kT(1 = = f−f 2 2 df Z Q R 0 • Phase noise spectral density: 2 / df 2 dI dϕ N (half phase noise, =1 N half amplitude noise) 2 I2 df 2 dϕ N (1 +γ )kT f0 2= (1 +γ )kT f0 2 = 2ωQE ff df 2Q2Pm ff0 m 0 • Nonlinear, timevariant: noise may added, including 1/f [HajimiriLee,1999] CSEM E.Vittoz2001
page 23
Crystal Oscillators
FREQUENCY INSTABILITY Cause a. Crystal resonator • Aging • Temperature
OSC12
Remedy
• Preaging. • Better cut • analog or digital compensation. b. Nonlinear effects in circuit (variations with VB, VT, T)
• Nonlinear Z1, Z2 or Z3 • Nonlinear ID(VG)
• Keep trans.in saturation • avoid C(V) effects. • Reduce overdrive • stabilize amplitude • increase Z3.
c.Variation of linear effects • Variation of R~1/Q ∆R Zc gm
2∆ω ω2 C • Variation of losses • Variation of C1, C2, C3
• Increase Q • reduce losses in circuit. C1C2 • increase C3(C1+C2) • Reduce losses • Decrease pulling p • avoid C(V) effects • stabilize VB.
CSEM E.Vittoz2001
page 24
Crystal Oscillators
OSC13
FREQUENCY TUNING • On the resonator: precision limited to a few 105. • By C1 and/or C2 in the circuit: tuning range: fmax  fmin = pmax f C1,C2 small large effect of circuit
pmin gm large
degrades stability increases power consumption • Digital tuning:  adjust the ratio of subsequent counters  inhibit an adequate percentage of pulses requires a few bits of memory: pad bondings RAM E2PROM.
CSEM E.Vittoz2001
page 25
Crystal Oscillators
OSC14
ELIMINATION OF UNWANTED MODES • A resonator has always several mech. modes (parallel series resonator in model). • One mode is wanted (WM) • All other modes are unwanted (UM). ω (C1+2C3)2 (for C2=C1) gmcrit = ~ QC same for all modes = ω2R activity different for each mode a. Nondistorting amplitude limitation • No interaction between modes; gm decreases until the most active reaches critical amplitude. • WM ensured if ω2R WM < ω2R UM b. Distorting amplitude limitation • Possible interaction between modes (very complicated). • Safe solution: gmcrit WM < gm < gmcrit UM requires ω2R WM « ω2R UM c. Selection of WM of lesser activity • frequency selective Zc → degrades stability.  low or highpass for large difference in ωm  LC bandpass to select a particular overtone CSEM E.Vittoz2001
page 26
Crystal Oscillators
OSC14a
LOADING BY OUTPUT AMPLIFIER • Elementary voltage amplifier, gain A =V0/V1 + CM V1 I1 V0 C0 from gate V1 V0 (or drain) of oscillator
• Capacitive load: V0 = j AV1 (complex gain) • Miller capacitance CM, thus, for A»1 I1=jωCMV0 = ωACMV1, in phase with V1 • Input conductance G1 = I1/V1 = ωCMA may be large if CM and A large. significant loss in oscillator • Increasing V1 requires more current in lossless oscillator, but reduces loss due to G1: tradeoff. CSEM E.Vittoz2001
page 27
Crystal Oscillators
OSC15
GROUNDEDDRAIN OSCILLATOR [Luescher, 1968, SantosMeyer, 1984]
V+
output
T2 I0
C1
V (ground)
regulator
T1 C3
C2
R
T1 active, biased by R and current source T2 . + One single pin for resonator ("1pin oscillator"). + Doubled output amplitude. – Increased C3 : decreases stability and/or increases power. – T1 must be put in a separate well connected to its source; otherwise an additional conductance gm(n1) is added to G2 (large increase of losses).
CSEM E.Vittoz2001
page 28
Crystal Oscillators
OSC15a
ONEPIN OSCILLATOR WITH GROUNDED C's [van den Homberg, 1998/99]
gm • Principle:
Zc
C1 (Z1)
+C2 (Z2)
Zc =
Z1 + gmZ1Z2 1+ gm (Z1 − Z 2 )
bilinear function of gm C1 / C 2 − ω (C1− C2 )
• For Zi = 1 : jωCi • Necessary condition for oscill.: C1 / C 2 C −C Q> 1 2 R< C.C1/ C2 ω(C1− C2 )
Zplane
Im R
Re 1/ωC1
gm = 0 p
• If realized with large margin: g m gm= ∞ C p0 = 2C1 Zc(gm) 1 C2 ω ωC − gmcrit = C1C2 = ω − QC 4Qp02 C1 Z (p) (C1 C2 ) m
• Condition for stability: (pole of Zc with negative real part)
C1 > C2
radius of circle reduced for increased margin CSEM E.Vittoz2001
page 29
Crystal Oscillators
OSC16
CMOSINVERTER OSCILLATOR
VB
with out r eson ator
 a simple but poor solution V2 (peak to peak)
I
VB
V2
I
C2
C1
VB
0 VBstop VBstart Putative Zc1 plane
Im
R
Zc(VB)
Re
p
Zc1(VB,amplitude) VBstop loc us of Z
c
• Poor frequency stability. • Waste of power. • Risk of overtone.
amp
litude
Zm
VBstart
• Possible improvement by resistors in the drains. CSEM E.Vittoz2001
page 30
Crystal Oscillators
OSC17
GROUNDEDSOURCE OSCILLATOR (noncomplementary) V+ bias I0 Q2
Q1 output (to amplifier) C1
(regulation) C2 V
• For fixed bias current I0: Margin needed for variations of Q and process possible overdrive waste of power
limitation by distortion
Best: • Lowlevel amplitude regulation+ output amplifier
CSEM E.Vittoz2001
page 31
Crystal Oscillators
OSC18
AMPLITUDE REGULATOR [VittozFellrath, 1977]
V + 1 T3 = T1
T1
T5=A.T1 I0
filter
τ
V1 sinωt T 2 (ωτ » 1)
T4 =K.T2 (weak inversion) R
V
• No AC input voltage (V1=0): AU I0 = I0start = T lnK (startup current) R • For V1>0: General solution For R negligible ID2 y = I0/I0start ID1 K ln 1.0 VG2 IB0(x) t y= ln K ID4/K 0order modified Bessel funct.
0.5
0
K= 10 0
2
stable for ID1=ID4 t
20
4
40
6
V x= 1 nUT
CSEM E.Vittoz2001
V1
page 32
Crystal Oscillators
OSC18a
os ci lla to r
AMPLITUDE REGULATING LOOP V1 T1 T3 I0 T2 V1 oscillator
reg
T4 R regulator
stable amplitude ula
tor I 0
I0cri
I0start
t • Effect of T2T4 entering strong inversion: distortion of the regulator's characteristics V1 B V1 A
t I0 Case A : unstable relaxation with time constant stable but may go Case of the B: order of Q/ω through one cycle of relax. t after a perturbation 2 • Design criterion: iD2max< IS2 = 2nβ2UT(too strict) 2β1/β3 semiempirical: R > β2nUT CSEM E.Vittoz2001
page 33
Crystal Oscillators
OSC19
MICROPOWER CRYSTAL OSCILLATOR [Vittoz, 1979]
V+
out
R2
D1
I0
D2 R3
R1
^ VV1 output ampli. oscillator amplitude regul. R1,R2,R3,D1,D2 : lateral diodes in poly layer or ^ t V1 lla i c s o regu stable lator oscillation I0 I0crit
I0oscil
I0start
Example: f=32KHz VB=1 to 3 V Itot= 20 to 100nA (depends on C1,C2,Q) no external component except crystal. CSEM E.Vittoz2001
page 34
Crystal Oscillators
OSC20
CURRENT DRAIN of micropower crystal oscillator f C Q C1 C3
= 32KHz = 2.9fF = 80'000 = C2 =10pF = 3pF
Crystal resonator removed
106 I [A]
107
108
0
1
2
3 VB [V]
CSEM E.Vittoz2001
page 35 OSC19a
Crystal Oscillators
VERY LOWPOWER 2MHZ OSCILLATOR [Aebischer et al, 1997]
IB2
from amplitude regulator
I0 TR2
T2
C4
IB1 C1
TR1
C2 T1
to amplitude regulator and frequency divider
• Currrent controlled CMOS inverter T1T2 • gates separately biased by TR1TR2  controlled by bias IB1IB2 • source of T2 AC grounded by C4, with ωC4 » gms2 • 2.1 MHz ZT cut quartz, C=0.5fF, Q=300900K C1=C2=2.5pF,C3=0.7pF, C4= 10pF I = 60180nA (core oscillator only) I < 500nA (oscill.+freq. divider+ dig. tuning) @ 1.8 to 3.5V CSEM E.Vittoz2001
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Crystal Oscillators
OSC21
COMPACT PUSHPULL OSCILLATOR V+ [Thommen, 1999]
T2
T3
T5 out
Cc
I0/A
• Cc large enough for negligible discharge by VI0/A during one period
T4 T1
C2
C1
300
t ou
w
A =16 C1 =12.3 pF C2 =24.6 pF C3 = 1pF VBmin = 0.7V
ith
C
c
• Average current I0 through T2 and T3: • imposed by T3 • from amplitude regulator • Instantaneous current in T3 • proportional to that in T2 • creates a loss conductance, thus: • effect. trans. of T2 for fundamental: gm2(1)(11/A) • Output amplifier T4T5 directly coupled to T1T2 • Experimental results: 450 ^ f =32 kHz V 1 [mV] R =35 kΩ Cc
h
wit 150
I0 [nA]
• DrawbacK:poor PSRR
0 0
30
CSEM E.Vittoz2001
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90
120
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Crystal Oscillators
REFERENCES General
E.Vittoz et al, "Highperformance crystal oscillator circuits: theory and application", IEEE J. SolidState Circuits, vol.23, pp.774778, June 1988.
OSC1a
V.Uzunoglu, Semiconductor Network McGraw Hill 1964, p.245. L.Astier, unpublished work, 1987 L.Astier, unpublished work, 1987.
OSC10 OSC11c/d
OSC13b
OSC15
OSC15a
OSC18
OSC19 OSC19a
OSC21
Analysis
and
Design,
E.Vittoz, "Micropower techniques", in Design of VLSI Circuits for Telecommunication and Signal Processing, Ed. J.Franca and Y. Tsividis, Prentice Hall, 1994. C. Enz et al.,"An analytical MOS transistor model valid in all regions of operation and dedicated to lowvoltage and lowcurrent applications", Analog integrated Signal Processing, vol.8, pp.83114, 1995. V. von Kaenel and E.Vittoz, "Crystal oscillators", in Analog Circuit Design, Kluwer, Boston 1996, pp.369382. D.B.Leeson, "A simple model of feedback oscillator noise spectrum", Proc. IEEE, vol.54, pp.329330, Feb. 1966. A.Hajimiri and T.H. Lee, "The Design of Lownoise Oscillators",Kluwer Academic Publishers, 1999. J.Luescher, "Oscillator circuit including a quartz crystal operating in parallel resonance",US patent 3,585,527, filed in 1968, J.Santos and R.Meyer, "A one pin oscillator for VLSI circuits", IEEE J. SolidState Circuits, vol. SC19, pp.228236, April 1984. J.A.T.M. van den Homberg, "A universal 0.03mm2 onepin crystal oscillator in CMOS", IEEE J. SolidState Circuits, vol.34, pp.956961, July 1999. E.Vittoz and J.Fellrath, "CMOS analog circuits based on weak inversion operation", IEEE J. SolidState Circuits, vol.SC12, pp.224231, June 1977. E.Vittoz, "Quartz oscillators for watches", Proc. 10th International Congress of Chronometry, Geneva, 1979. D.Aebischer et al.,"A 2.1 MHz crystal oscillator time base with a current consumption under 500nA", IEEE J. SolidState Circuits, vol.32, pp.9991005, July 1997. W.Thommen,"An improved low power crystal oscillator", Proc. ESSCIRC’99, pp.146149, Sept. 1999.
CSEM E.Vittoz2001
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487
Modeling of Quartz Crystal Oscillators by Using Nonlinear Dipolar Method Mahmoud Addouche, R´emi Brendel, Daniel Gillet, Nicolas Ratier, Member, IEEE, Franck LardetVieudrin, and J´erˆome Delporte Abstract—A quartz crystal oscillator can be thought of as a resonator connected across an ampliﬁer considered as a nonlinear dipole the impedance of which depends on the amplitude of the current that ﬂows through it. The nonlinear ampliﬁer resistance and reactance are obtained by using a time domain electrical simulator like SPICE (Simulation Program with Integrated Circuit Emphasis): the resonator is replaced with a sinusoidal current source of same frequency and a set of transient analyses is performed by giving the current source a larger and larger amplitude. A Fourier analysis of the steadystate voltage across the dipolar ampliﬁer is performed to calculate both real and imaginary parts of the dipolar impedance as a function of the current amplitude. From these curves, it is then possible to accurately calculate the oscillation amplitude and frequency without having to perform unacceptably long transient analysis needed by a direct oscillator closed loop simulation. This method implemented in the Analyse Dipolaire des Oscillateurs ` a Quartz or Quartz Crystal Oscillators Dipolar Analysis (ADOQ) program calculates the oscillation startup condition, the oscillation steadystate features (oscillation amplitude and frequency), and the oscillator sensitivity to various parameters. The oscillation nonlinear diﬀerential equation is solved by using the slowly varying function method so that the program quickly and accurately calculates the current amplitude and frequency transients. Measurements performed on an actual ampliﬁer show a very good agreement with the results obtained by the simulation program.
I. Introduction or many years a large number of analytical and numerical methods have been developed to predict the behavior of quartz crystal oscillators with higher and higher accuracy. Because of the very high resonator’s quality factor, a number of numerical time domain methods fail to converge in a reasonable computer time or lack of accuracy. In the method presented here, the oscillator is considered as a resonator connected across an ampliﬁer that behaves like a dipole whose impedance is evaluated at the resonator’s frequency. This method is not new and has
F
Manuscript received June 13, 2002; accepted November 5, 2002. This work has been supported by CNES under contracts #714/CNES/99/7671/00 and #714/01/CNES/8738/00. M. Addouche, R. Brendel, D. Gillet, N. Ratier, and F. LardetVieudrin are with the Laboratoire de Physique et M´etrologie des Oscillateurs du CNRS associ´e ` a l’Universit´ e de FrancheComt´e—Besan¸con 25044 Besan¸con, Cedex, France (email:
[email protected]). J. Delporte is with the Centre National d’Etudes Spatiales 31055 Toulouse, Cedex, France.
been widely used in the past [1]–[3]; but it often was limited by the lack of knowledge of the ampliﬁer nonlinear behavior. Nevertheless, when used together with an electrical nonlinear time domain simulator like SPICE, it should become an accurate and powerful tool for the quartz crystal oscillator design. To describe the nonlinear behavior of the dipolar ampliﬁer, the resonator is replaced by a sinusoidal current source of a given amplitude at the oscillator resonant frequency, and a transient analysis is performed by using an electrical simulator like SPICE. Real and imaginary parts of the dipolar impedance for the actual current amplitude then are obtained by performing a Fourier analysis on the steadystate voltage across the dipole. Nonlinear ampliﬁer resistance and reactance are obtained by giving the sinusoidal current source a larger and larger amplitude. Afterward, oscillation condition reduces to state that, when steady state is reached, the resonator impedance is exactly equal and of opposite sign to the ampliﬁer impedance [1]. The real part of this complex identity gives the oscillation amplitudes, and the imaginary part gives the frequency shift with respect to the resonator frequency. This method is quite simple to implement in a simulation program and can be used to quickly and accurately obtain steadystate oscillation conditions, amplitude and frequency transients, oscillator sensitivity to various parameters (component value, temperature, supply voltage, etc.), amplitude and phase noise spectra, etc. It should be emphasized that the oscillator behavior is fully described only by analyzing the ampliﬁer impedance thus avoiding unacceptably long transient analysis often dealt with oscillator closedloop simulation. Eventually, by using slowly varying functions method, amplitude, and frequency transients can be easily obtained. II. Dipolar Representation The method used can be summarized by looking at the oscillator represented in Fig. 1(a). The resonator is regarded as an impedance strongly varying with the frequency and slightly varying with amplitude (because of the isochronism defect); and the ampliﬁer is considered to be a nonlinear dipole, the impedance of which strongly depends on the current amplitude and weakly varying with frequency. It results from these properties that the oscillator frequency is mainly ﬁxed by the resonant frequency of the
c 2003 IEEE 0885–3010/$10.00
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Resonator ZQ
i +
Amplifier ZD
Cq Lq
vd
Ld
Amplifier ZD
i Rd
Sinusoidal current source
vd
Rq –
– (b)
(a)
where
(c)
Fig. 1. (a) Dipolar representation of an oscillator, (b) resonator and ampliﬁer impedance, and (c) ampliﬁer dipolar impedance characterization.
resonator, and the amplitude is determined by the ampliﬁer nonlinearities. The representation assumes that the resonator is reduced to its series resonant branch: Lq , Cq , Rq [Fig. 1(b)], and all the other elements such as parallel capacitance or pulling capacitance are included in the ampliﬁer dipole. The ampliﬁer dipole [Fig. 1(b)] can be represented by a nonlinear resistance Rd in series with a nonlinear inductance Ld that varies with the amplitude of the current and may have negative or positive value. As will be shown later, when using highQ series resonant circuit such as quartz crystal resonator, the loop current i, in the oscillator is almost perfectly sinusoidal; and the voltage across the resonator may have a large harmonic distortion. This feature leads us to analyze the circuit, assuming that the main variable is the loop current, and to calculate the nonlinear dipolar impedance by keeping only the ﬁrst harmonic term of the voltage Fourier series expansion. This ﬁrst harmonic approximation is fully justiﬁed by the fact that, when dealing with a highQ series resonant circuit, the loop current is nearly sinusoidal and the higher the Qfactor the closer the current from a purely sinusoidal function. Furthermore, as soon as the closed loop current amplitude is determined, a transient analysis of the open loop ampliﬁer performed by giving the current source the closed loop amplitude enables us to obtain the timedomain characteristics (amplitude, phase, waveform, etc.) of all voltages and currents in the circuit without loss of information.
III. Oscillation Condition When oscillation occurs in the circuit represented in Fig. 1(a), the voltage vd across the resonator is identical to the one across the ampliﬁer. Calling i the loop current, the oscillation condition is straightforward: vd = ZD i = −ZQ i
⇒
where s is the Laplace operator. Replacing (2) and (3) in (1) leads to the oscillation equation: Ld 1 2 2 = 0, (4) (Rq + Rd ) s + ωq 1 − s + Lq Lq
(ZD + ZQ ) i = 0.
(1)
The impedance ZQ of the resonator represented in Fig. 1(b) can be written as: Lq Rq ZQ = s2 + s + ωq2 . (2) s Lq And the impedance ZD of the ampliﬁer is: ZD = Rd + Ld s,
(3)
ωq2 =
1 , Lq Cq
(5)
is the series resonant frequency of the resonator. It should be recalled that Rd and Ld depend on the amplitude a of the loop current i.
IV. Starting Condition and Steady State Expression (4) is a nonlinear second order diﬀerential operator that may produce an increasing amplitude solution only if the ﬁrst order term has a negative value. If Rds is the value of the nonlinear dipolar resistance at very low current amplitude (a ≈ 0), starting condition reduces to: Rq + Rds < 0.
(6)
As the oscillation amplitude increases, the dipolar resistance increases so that the value of the ﬁrst order coeﬃcient increases, the steady state is reached when this coeﬃcient becomes null, thus the steadystate dipolar resistance Rd (a0 ) is given by: Rq + Rd (a0 ) = 0,
(7)
where a0 is the steadystate amplitude of the oscillation; (4) also gives the steadystate frequency of the oscillation that takes the form: Ld (a0 ) ω02 = ωq2 1 − , (8) Lq where Ld (a0 ) is the value of the steadystate dipolar inductance. As we shall see in a next section, (6), (7), and (8) give a simple way to determine the starting condition and steadystate amplitude and frequency as soon as the dipolar resistance and inductance are expressed as nonlinear function of the current amplitude.
V. Amplifier Dipolar Impedance Except in a limited number of cases, it is not possible to derive an analytical expression of the nonlinear functions Rd (a) and Ld (a) from the analysis of the ampliﬁer. However, electrical simulator like SPICE can be used to accurately obtain these functions from a set of transient analyses. To this end, the resonator in Fig. 1(a) is merely replaced by a sinusoidal current source of frequency close to the resonator’s frequency ωq [Fig. 1(c)].
addouche et al.: modeling of quartz crystal oscillators by using nonlinear dipolar method
ZQ
i –
vd
+
489
A + vi
+ vo
R
–
–
Fig. 2. Van der Pol Oscillator.
By giving the amplitude a several values, a set of transient analyses with suﬃcient duration to reach the steady state are performed. Then a Fourier analysis of the voltage vd across the ampliﬁer dipole allows one to calculate both real and imaginary parts of the dipolar impedance and derive both dipolar resistance and inductance of the ampliﬁer as a function of the current amplitude a.
Fig. 3. Dipolar resistance of the Van der Pol’s ampliﬁer obtained by analytical method (solid line) and by simulation (+) using: G = 4, ε = 0.02 V−2 , R = 100 Ω, Rq = 126 Ω.
This impedance has no imaginary part so that it is reduced to a nonlinear resistance that is a quadratic function of the current amplitude a (Fig. 3). The starting condition (6) becomes: Rds = (1 − G)R < −Rq .
VI. Demonstrating Examples A. Van der Pol’s Oscillator Fig. 2 shows a simple behavioral oscillator the loop current of which obeys the classical Van der Pol equation [4]. This circuit can be analyzed either theoretically or by using an electrical simulator. In this example, the ampliﬁer has a nonlinear voltage gain that is taken under the form: vd = Gvi 1 − ε vi2 . (9) There is no current through the ampliﬁer so that: vi = R i.
Expression (14) shows that the ampliﬁer must have an initial negative resistance of magnitude larger than the resonator’s series resistance for the oscillation to occur. Steadystate amplitude is given by (7) that has the form: (1 − G)R +
3Gε R3 2 a0 = −Rq , 4
(15)
so that a20 =
4RM , 3Gε R3
(16)
where (10)
Substituting (10) in (9), the dipolar voltage vd takes the following form: vd = vi − vo = (1 − G)R i + Gε R3 i3 .
(14)
(11)
Replacing the resonator by a sinusoidal current source i of amplitude a at the same frequency: i = a sin ωq t enables us to express the dipolar voltage under the form:
RM = −Rds − Rq = (G − 1)R − Rq .
(17)
RM could be called the resistance margin of the oscillator (Fig. 3). Note that, because the ampliﬁer gain is purely real, Ld = 0 and the oscillation frequency is only ﬁxed by the resonator (8). It can be shown in Fig. 3 that the value of Rd obtained by using the simulation procedure described in Section V gives the same result as obtained by analytical method. B. Transconductance Oscillator
vd = (1 − G)Ra sin ωq t + Gε R3 a3 sin3 ωq t. (12) The dipolar impedance is obtained by expanding (12) in Fourier series, retaining only the fundamental term, and dividing by the current. This leads to the following result: ZD = (1 − G)R +
3Gε R3 2 a . 4
(13)
Fig. 4 represents another simple example of behavioral transconductance oscillator. Many quartz crystal oscillator structures, such as the Pierce family or gate oscillators, can be reduced to this simpliﬁed form. Here, the voltagecontrolled current source (transconductance) is given an analytical expression under the form: io = −Gvi 1 − ε vi2 , (18)
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ZQ +
i
G + vi
–
vd io
R
R C
C
–
+ vo –
Fig. 4. Transconductance oscillator.
so that the dipolar impedance ZD = Rd + jLd ωq takes the form: R 1 − α2 3ε R2 a2 Rd = 2+ RG 1 − , 1 + α2 1 + α2 4 (1 + α2 ) (19) −2Rα RG 3ε R2 a2 L d ωq = 1 + 1 − , 1 + α2 1 + α2 4 (1 + α2 ) (20) with α = ωq RC. Here the dipolar resistance and the dipolar inductance are both quadratic functions of the current amplitude a (Fig. 5). Eq. (7) gives the steadystate amplitude, here we have: 3 −4 1 + α2 RM 2 , (21) a0 = 3ε GR4 (1 − α2 ) where the resistance margin RM is deﬁned by: RM = −Rds − Rq 2 2R 2 1−α = −GR 2 − (1 + α2 ) − Rq . (1 + α2 )
Fig. 5. Dipolar resistance and dipolar inductance of the transductance ampliﬁer using: G = 22 mA/V, ε = 1/3, C = 200 pF, R = 10 kΩ, Rq = 126 Ω, Lq = 1 mH.
VII. Amplitude and Frequency Transients (22)
As a0 is determined by the resistance condition (7), the actual value of the dipolar inductance can be obtained by giving the current amplitude a the value a0 in the imaginary part of the dipolar impedance (20): ωq Ld (a0 ) =
d Substituting the diﬀerential operator dt for the Laplace variable s in (4) and applying the operator to the loop current i leads to a nonlinear second order diﬀerential equation that takes the form: d2 i 1 di Ld 2 + (Rq + Rd ) + ωq 1 − i = 0. dt2 Lq dt Lq (25)
RM 1 + α2 −2α R RG 1+ + . (23) (1 + α2 ) (1 + α2 ) R (1 − α2 ) In most cases LdL(aq 0 ) 1 so that the relative frequency shift with respect to the resonator frequency can be obtained from (8) under the form:
Because the loop current is a quasi sinusoidal function, (25) can be considered as a linear second order diﬀerential equation with a small nonlinear perturbation term in the righthand side:
∆f Ld (a0 ) =− . fq 2Lq
Because of the smallness of the righthand side of (26), the solution of this equation should be close to the solution of the linear equation obtained when the righthand side is null. This assumption leads to seek a solution under the form:
(24)
In the example shown in Figs. 4 and 5, (21) and (24) −3 give a0 = 7.38 mA and ∆f . These results fq = 1.2826 10 are quite close to those obtained by using simulated curves (Fig. 5).
d2 i 1 di ωq2 Ld 2 + ω i = − (R + R ) + i. q d q dt2 Lq dt Lq
i = a(t) cos (ωq t + ϕ(t)) = a(t) cos ψ(t),
(26)
(27)
addouche et al.: modeling of quartz crystal oscillators by using nonlinear dipolar method
a˙ =
−a Lq
(Rq + Rd (a)) sin2 ψ −
v = ϕ˙ =
a˙ =
1 2π
2π 0
v = ϕ˙ =
−1 Lq
aLd (a) ωq Lq
(Rq + Rd (a)) sin ψ cos ψ −
cos ψ sin ψ Ld (a) ωq Lq
cos2 ψ
491
.
aL (a) ω (Rq + Rd (a)) sin2 ψ − dLq q cos ψ sin ψ dψ 2π −1 Ld (a) ωq 2 (R + R (a)) sin ψ cos ψ − cos ψ dψ q d Lq Lq
(28)
−a Lq
1 2π
.
(29)
0
10
where the amplitude a(t) and the phase ϕ(t) are slowly varying function of time so that their time derivative are small. Diﬀerentiating (27) with respect to time enables us to transform (26) in the ﬁrst order diﬀerential system given in (28) (see above). The averaging method described in [7] states that, because a and ϕ are slowly varying quantities, in the ﬁrst order approximation these quantities keep the same value for one period of the signal so that (28) is approximately equivalent to (29) (see above). The associated system then reduces to [5], [6]:
−a a˙ = 2L (Rq + Rd (a)) , q (30) L (a) ω v = ϕ˙ = − d2Lq q ,
VIII. Colpitts Oscillator Fig. 7 shows a more realistic example of a Colpitts oscillator. By using the same procedure as described in Section V, the nonlinear resistance and reactance are obtained (Fig. 8) so that the steadystate amplitude and relative frequency shift can be calculated using (7) and (24), respectively. In the present case we obtain a0 = 552 µA and ∆f f = 3475 ppm. By solving the associated system (30), the amplitude and frequency transients are easily obtained and look like the curves obtained for the transconductance oscillator shown in Fig. 6.
6
Current envelope (mA)
4 2 0 –2 –4 –6 –8 –10 0
0.2
0.4
0.6
0.8 1 Time (ms)
1.2
1.4
1.6
1.8
2
0
0.2
0.4
0.6
0.8 1 Time (ms)
1.2
1.4
1.6
1.8
2
1285
Relative frequency shift transient (ppm)
where Rd (a) and Ld (a) are the previously obtained nonlinear functions of the current amplitude a. Solving system (30) by the numerical method allows us to quickly obtain the amplitude and frequency transients. Fig. 6 shows the result obtained with the transconductance oscillator described in Fig. 4. It should be noted that the steadystate amplitude and frequency shift are in perfect agreement with those obtained in the previous section. To compare results obtained by the dipolar impedance method and by the direct closedloop method, the resonator has been given a rather low quality factor (Q ≈ 500), such a weak Qfactor explains the quite large frequency shift obtained.
8
1284
1283
1282
1281
1280
Fig. 6. Amplitude and frequency transients of the transconductance oscillator.
IX. Sensitivity Among the various information the simulation program can provide, one of the most useful is the sensitivity calculation that is in fact the initial step toward a design optimization process [8]. To obtain the sensitivity as a function of the circuit component variation, each component value is given a small deviation, and the resulting eﬀect on the oscillator operating conditions (amplitude, frequency, etc.) is calculated. For instance, the loop resistance vari
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VCC
Re1 Re2 Rc
Rb1 Rb2 RL
C1
C2
C3
C5
C7
Vcc T
0.03
C5 Relative frequency shift (ppm)
RB1
C3 RC
C6 C1 Xt
RE1 RB2 C2
0.01 0 –0.01 –0.02 –0.03 –0.04
C7 RE2
0.02
RL
Fig. 7. Colpitts oscillator.
Re1 Re2 Rc
Rb1 Rb2 RL
C1
C2
C3
C5
C7
Vcc T
Minimum value of the parameter
Fig. 9. Frequency sensitivity of the Colpitts oscillator.
ation in startup condition for a shift of ±10% of each oscillator component values can be calculated to identify the most critical parameters; in fact, the loop resistance value should remain negative (6) to ensure the startup of the oscillation. This parametric analysis allows us to choose the circuit components so that oscillation starts under any condition. The steady state is reached when the real part of the loop impedance is zero (7). The oscillation amplitude and frequency sensitivities are obtained by varying several oscillator components. For example, Fig. 9 shows the frequency shift induced by a variation of ±10% of each parameter around their nominal value. This parametric analysis of the amplitude and frequency oscillation allows us to identify the most sensitive component that should be carefully chosen.
X. Experimental Validation A. Experimental Setup
Fig. 8. Dipolar resistance and dipolar inductance of the Colpitts oscillator.
The experimental setup is based on an impedance analyzer 4395A (Agilent Technologies Co, Engelwood, CO) that is able to perform harmonic impedance measurements on a dipolar device as a function of the input signal power. The impedance analyzer is connected to a personal computer (PC) via a IEEE488 interface bus. A software program was developed to monitor the impedance analyzer so that the frequency and other analysis parameters, as well as the operating statements, are controlled by the program. In the case of this particular impedance analyzer design, the power range is limited between −50 dBm and +15 dBm. In practice, the dipolar impedance is measured by using ﬁve steps of 13 dBm power range span. A calibration of the analyzer is required for both power and frequency ranges. The precision of the dipolar method is strongly dependent on the modeling accuracy of the ampliﬁer. Before comparing the simulation results with the experimental data, it is necessary to assess the inherent modeling uncertainties due to the scattering of the components used
addouche et al.: modeling of quartz crystal oscillators by using nonlinear dipolar method –310
Vcc = 15V, fq = 12 MHz
–40
–320
Reactance
Dipolar resistance (Ohms)
–60
–330 Q1
–80
Q2
Q3
Q4
–340
–100
–350
–120
–360
–140 –160
–370
Resistance Q1 Q2 Q4 Q3
–380
–180
–390
–200
–400
–220 0
Dipolar reactance (Ohms)
–20
493
–410 0.2
0.4
0.6 0.8 1 1.2 Current amplitude (mA)
1.4
1.6
1.8
2
Fig. 10. Dipolar impedance scattering of the Colpitts oscillator.
Fig. 11. Measured and simulated dipolar resistance of the Colpitts oscillator for diﬀerent supply voltages.
(that is, the devicetodevice variation) as well as those due to the printed circuit board parasitic elements. The choice of the MAT03 transistor (Analog Device, Norwood, MA) used is motivated because of the very low characteristics scattering and the availability of an accurate SPICE model. However, the low characteristics scattering of this transistor family minimizes the dissonance between simulation and experimental results when the transistor is changed; and, the availability of the SPICE model guarantees an optimal accuracy in the oscillator ampliﬁer modeling. The oscillator used is a 10 MHz to 20 MHz Colpitts oscillator proposed in [9] (Fig. 7). B. Dipolar Impedance Measurements Several dipolar impedance measurements were ﬁrst performed using diﬀerent transistors (Fig. 10). The spreading that is larger than 5% for lowlevel amplitude dipolar resistance reduces to less than 1% around the steady state, and the dipolar reactance spreading is close to 1% over the whole analysis current amplitude range. These measurements give an order of magnitude of the discrepancies that could be expected between simulation and experimental results. The oscillator circuit is described in SPICE ﬁle format in which the component values are given the measured ones. By carefully analyzing the diﬀerence between simulation and experiment, it has been evidenced that they mainly ﬁnd their origin in the inﬂuence of the printed circuit parasitic capacitances that should not be neglected. Thus, those stray capacitances have been measured and included in the ﬁle describing the circuit so that the simulation result now ﬁt the experimental data very well. The supply voltage source used is monitored by the PC. Several dipolar impedance measurements then are performed for diﬀerent values of the supply voltage. The results are shown in Fig. 11 for a Colpitts oscillator operating at 12 MHz at ambient temperature of 25◦ C. It can be observed that the diﬀerence between simulation and measurements remains within the expected limits. In the same way, measurements have been performed by varying the analysis frequency. The calculation of the oscil
Fig. 12. Measured and simulated dipolar resistance of the Colpitts oscillator for diﬀerent resonator frequencies.
lation frequency requires several simulations in the neighboring of the intrinsic resonator frequency. The measurements shown in Fig. 12 are performed for several analysis frequencies. It seems that the results are less good than the previous measurements. The spreading is close to 3% at low level amplitude measurement but remains below 1.5% at higher amplitude level. It is even so heartening results. The Colpitts oscillator ampliﬁer built is mounted in a temperaturecontrolled oven. An example of measurements of the dipolar impedance obtained for two temperatures is shown in Fig. 13. The shift that is about 5% at lowlevel amplitude reduces to less than 1% at higher values. When performing the simulation, the curve seems to rotate around the −75 Ω point. This rotation also appears on the measured dipolar resistance. For the three monitored parameters, simulation and measurement results show a pretty good agreement. C. Oscillator Measurements The startup condition of an oscillator can be checked easily by plotting the loop resistance Rq + Rds and check
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Experiment 0.1 Simulation
0 –0.1 –0.2 –0.3 –0.4 –0.5 0
Fig. 13. Measured and simulated dipolar resistance of the Colpitts oscillator for diﬀerent temperatures.
5
10
15
20
25
30
Time (s)
Fig. 15. Simulated and experimental starting of the Colpitts oscillator.
ing the starting condition (6) as a function of a given parameter. For example, Fig. 14 shows the value of the loop resistance Rq + Rds as a function of the supply voltage; it can be seen that the oscillator should no longer start if the supply voltage drops below 13 V because the loop resistance becomes positive. Actually, this fact has been experimentally observed. Another veriﬁcation has been achieved by comparing the simulated amplitude transient (30) with the oscillator output voltage. Fig. 15 shows that the agreement between simulation and experiment is quite satisfactory.
tained without having to simulate a timeconsuming, highQ circuit. In addition, oscillation amplitude and frequency transients can be obtained quickly by using a slowly varying function method. Remaining small discrepancies between simulation and experimental results could be attributed to unavoidable diﬀerences between modeled and actual transistor and other circuit parameters that often suﬀer from an important scattering; however, the agreement is good enough to validate the approach used in the nonlinear dipolar method and demonstrates the reliability of the sensitivity analysis results. Although the nonlinear dipolar method is mainly devoted to the analysis of radio frequency (RF) quartz crystal oscillators, it can be applied to any highQ oscillator circuit as long as the lumped element description of the circuit remains justiﬁed. The accuracy of the oscillator features obtained from the simulation can be suﬃcient when the inherent dissonances are widely assumed and the circuit modeling is precisely performed. It should be emphasized that all the steps of the method presented in this paper are currently being automated to develop an eﬃcient and powerful oscillator computer aided design tool. A perturbation method analog to the one described in [7] can be implemented to obtain amplitude and phase noise spectra and are currently being implemented.
XI. Conclusions
References
By using an electrical simulation program like SPICE, it is possible to describe the nonlinear behavior of the oscillator ampliﬁer by looking at it as a nonlinear dipolar impedance whose real and imaginary parts can be obtained by performing a set of transient analyses. Applying simple oscillation condition, both steadystate amplitude and frequency can be easily and accurately ob
[1] K. Kurokawa, “Some basic characteristics of broadband negative resistance oscillator circuits,” Bell Syst. Technical J., pp. 1937– 1955, July–Aug. 1969. [2] E. A. Vittoz, “Quartz oscillators for watches,” in Proc. 10th Int. Congr. Chronometry, 1979, pp. 131–140. [3] B. Parzen and A. Ballato, Design of Crystal and other Harmonic Oscillators. New York: Wiley Intersciences, 1983, pp. 1–21. [4] G. Birkhoﬀ and G. C. Rota, Ordinary Diﬀerential Equations. New York: Wiley, 1978, p. 134.
Fig. 14. Starting loop resistance of the Colpitts oscillator vs. supply voltage.
addouche et al.: modeling of quartz crystal oscillators by using nonlinear dipolar method [5] N. Kryloﬀ and N. N. Bogoliuboﬀ, Introduction to Nonlinear Mechanics. Princeton, NJ: Princeton Univ. Press, 1943. [6] A. A. Andronov, A. A. Vitt, and S. E. Khaikin, Theory of Oscillators. New York: Pergamon, 1966, pp. 585–613. [7] R. Brendel, N. Ratier, L. Couteleau, G. Marianneau, and P. Guillemot, “Slowly varying functions method applied to quartz crystal oscillator transient calculation,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 45, no. 2, pp. 520–527, Mar. 1998. [8] M. Addouche, N. Ratier, D. Gillet, R. Brendel, F. LardetVieudrin, and J. Delporte, “ADOQ: A quartz crystal oscillator simulation software,” in Proc. Int. Freq. Contr. Symp., 2001, pp. 753–757. [9] M. E. Frerking, Crystal Oscillator Design and Temperature Compensation. New York: Van Nostrand Reinhold, 1978, pp. 76–84.
495
Nicolas Ratier (M’98) was born in Paris, France, in 1965. He graduated in microelectronics at Toulouse III University, (Toulouse, France). In 1993, he received the Ph.D. degree in microelectronics from the same university for his work on the simulation of micromechanical capacitive pressure sensors. Since 1994 he has been an assistant professor at the graduate engineering school in Mechanics and Microtechniques (ENSMM, Besan¸con, France) and at the Laboratoire de Physique et M´ etrologie des Oscillateurs (LPMOCNRS, Besan¸con, France). His primary research activities are the simulation of ultrastable quartz crystal oscillators.
Franck LardetVieudrin was born in 1964 at Lyon, France, where he received the Licence of Theoretical Physics from the University Claude Bernard. In 1991, he joined the Laboratoire de Physique et M´etrologie des Oscillateurs (LPMOCNRS, Besan¸con, France). He currently is working on radio frequency synthesis, ultra low phase noise design, metrology and characterization of ultra stable oscillators. He is also quality engineer of the LPMO in charge of the LA BNM 2.13 (Associated Laboratory with the Bureau Na
Mahmoud Addouche was born in Alger, Algeria in 1975. He received the engineer degree from the Institute of Electronics of the University of TiziOuzou (TiziOuzou, Algeria) in 1997. He is currently working at the Laboratoire de Physique et M´etrologie des Oscillateurs (LPMOCNRS, Besan¸con, France), on automatic computer modeling for quartz crystal oscillators in order to obtain the Ph.D. degree in engineering science. tional de M´etrologie).
R´ emi Brendel was born in Besan¸con, France, in 1945. He received the engineer de´ gree from the Ecole Nationale Sup´erieure de M´ ecanique et des Microtechniques (ENSMM, Besan¸con, France) in 1971 and the Ph.D. degree in physics from the University of FrancheComt´e (Besan¸con, France) in 1983. Since 1972 he has been with the Laboratoire de Physique et de M´etrologie des Oscillateurs (LPMOCNRS, Besan¸con, France) where he is working on the problems of quartz crystal oscillators. At the same time he is a professor of electronics in the ENSMM. His current research interests are mainly in the environmental eﬀects and the modeling of quartz crystal oscillators.
Daniel Gillet was born in ArclesGray, France, in 1945. He received the Doctorat `es Sciences degree from University of Besan¸con for his contribution to the development of a molecular beam interferometer in 1970. Since 1973 he has been an assistant professor at ´ the Ecole Nationale Sup´erieure de M´ecanique et des Microtechniques (ENSMM, Besan¸con, France) where he teaches electronics. At the Laboratoire de Physique et M´etrologie des Oscillateurs (LPMOCNRS, Besan¸con, France), he is working on simulation and development of digital electronics dedicated to oscillators.
J´ erˆ ome Delporte received the engineer de´ gree and the Diplˆ ome d’Etude Approfondie Microwaves and Optical Transmissions from ´ the Ecole Nationale de l’Aviation Civile (ENAC, Toulouse, France) in 1997. Since 1999, he has been with the Centre National ´ d’Etudes Spatiales (CNES, Toulouse, France), the French Space Agency, in the time and frequency department. He currently is working on high stability quartz crystal oscillators, space atomic clocks for navigation and accurate GPS phase frequency transfer.
Proceedings of the 2003 IEEE International Frequency Control Symposium and PDA Exhibition Jointly with the 17th European Frequency and Time Forum
QUARTZ CRYSTAL OSCILLATOR CLASSIFICATION BY DIPOLAR ANALYSIS R. Brendel1, F. Chirouf1, D. Gillet1, N. Ratier1, F. LardetVieudrin1, M. Addouche2, J. Delporte3 1
Laboratoire de Physique et Métrologie des Oscillateurs du CNRS associé à l’Université de FrancheComté 32, Avenue de l’Observatoire – 25044 Besançon Cedex – France 2
Laboratoire d’Astrophysique de l’Observatoire de Besançon 41, Avenue de l’Observatoire – 25030 Besançon Cedex – France 3
Centre National d’Études Spatiales 18, Avenue E. Belin – 31055 Toulouse Cedex – France Abstract – The dipolar method associated with a nonlinear time domain simulation program make up a powerful tool to analyze high Qfactor circuits like quartz crystal oscillators. After a brief remainder of the dipolar method, the paper will attempt to identify the main amplifier characteristics such as limitation mechanism, input and output impedances, etc. and to point out their influence on the amplifier dipolar impedance. The effect of the amplifier nonlinearities on the oscillator characteristics, as well as the particular role of the crystal parallel capacitance are particularly emphasized. Keywords – Oscillators, modeling, dipolar method
An implementation of the dipolar method achieved in the ADOQ program (“Analyse Dipolaire des Oscillateurs à Quartz”) is able to compute the steady state features of the oscillator: oscillation frequency, amplitude, drive level, signal shape and gives the startup condition for oscillation to occur [2]. Also the program performs accurate oscillator sensitivity calculation to various parameters (component value, supply voltage, etc.) as well as amplitude and phase noise spectra [8].
I. INTRODUCTION
The nonlinear dipolar method consists in representing an oscillator by the quartz motional branch connected across a nonlinear dipole amplifier (Fig. 1). The parallel capacitance and, if needed, the pulling capacitance, are included in the amplifier dipole. The impedance of the nonlinear dipole amplifier strongly depends on the current amplitude and weakly depends on the current frequency. It can be represented by a nonlinear resistance in series with a nonlinear reactance, symbolized for convenience in Fig. 1 by an inductance, that vary with the amplitude of the current.
II. SIMULATION PRINCIPLE
The dipolar analysis is a nonlinear time domain method well suited to describe the behavior of high Qfactor crystal oscillators [13]. In this method, to overcome the unacceptably long transient time needed to reach the steady state of highQ circuits, the oscillator amplifier is separated from the resonator and replaced by a sinusoidal current source. The amplifier then behaves like a dipole the impedance of which is evaluated at the resonator frequency. An electrical simulation program like SPICE is used to perform a set of transient analyses of higher and higher amplitude so as to obtain the variation of both the dipolar resistance and reactance as a function of the current source amplitude. When such a dipole is connected to a crystal resonator, oscillation occurs when the nonlinear dipolar impedance is equal and opposite to the resonator impedance [45]. This leads to a nonlinear equation that can be used to obtain the oscillation startup condition, the steady state current amplitude and oscillation frequency or a time domain nonlinear differential equation whose solution is the oscillation loop current. Because of the high resonator Qfactor, the oscillation loop current is almost perfectly sinusoidal so that asymptotic method, like slowly varying functions method [67] can be used to obtain both amplitude and frequency transients without having to solve the initial nonlinear differential equation. The effect of noise can also be calculated by using a perturbation method in the vicinity of the steady state [8], and for noise Fourier components close to the carrier, perturbation results in both amplitude and frequency modulation from which amplitude and phase noise spectra are easily calculated.
Lq
Cp
Rq
Amplifier
Resonator
Cq Lq
Rd
Rq
Ld
Fig. 1. Dipolar representation of a quartz crystal oscillator.
Cq Lq Rq
+ Rd
Rd i
vd Ld
Ld – Fig. 2. Dipolar analysis of the oscillator amplifier.
Because of the resonator’s high quality factor, the loop current in the oscillator is almost perfectly sinusoidal. Thus, the nonlinear behavior of the dipolar amplifier can be ob587
0780376889/03/$17.00 © 2003 IEEE
Amplifier
Cq
Dipolar resistance Rd
tained by replacing the resonator motional branch by a sinusoidal current source, and performing a set of transient analyses with increasing amplitudes by using an electrical simulator like SPICE (Fig. 2). The complex impedance of the nonlinear dipole amplifier is obtained, for each current amplitude value by performing a Fourier analysis on the steady state voltage across the dipole. Nonlinear amplifier resistance and reactance are obtained by giving the sinusoidal current a larger and larger amplitude (Fig. 3).
ω 02 = ω q2 1 −
(6) The steady state frequency is then given by (6) where Ld(a0) is the value of the steady state dipolar inductance deduced from the curve in Fig. 3. III. OSCILLATOR AMPLIFIER CLASSIFICATIONS A large number of amplifier circuits can be used to build an oscillator each having its own advantages and drawbacks. The choice among the different circuits is of course dictated by the application the oscillator will be used in, and also by technical considerations such as: frequency range, output power range, frequency stability, output waveform, phase noise, etc. [9]. Even for a given set of specifications there are many different circuits able to meet them and a particular choice is often a matter of personal or collective experience or skill rather than the conclusion of a methodical analysis. It is not our claim to give a methodical reasoning leading to an optimal circuit, but more modestly, to give the designer an efficient tool to help him to make a good choice.
–Rq
Current amplitude a a0 Dipolar inductance Ld
Ld ( a0 ) Lq
Ld(a0)
ZQ
i vd
+
– Z3
a0
Current amplitude a
Amp
+
Fig. 3. Dipolar analysis of the oscillator amplifier.
u1
The nonlinear differential equation of the oscillator is given by (1), where ωq is the quartz series resonant frequency and a is the amplitude of the current fundamental component i(t). In quartz crystal oscillator Lq >> Ld so
–
Z1
+ u2
Z2
–
Fig. 4. Amplifier representation. ZQ
ZQ
i vd
+
that:
–
i vd
+
Z3
L (a) d 2i di 1 i = 0 + Rq + Rd ( a ) + ω q2 1 − d 2 Lq dt Lq dt 1 ω q2 = Lq C q
(
)
(1)
+ G·u1 Z’1
(2)
Z’2
–
+
+ A·u1
u2 Z’1 –
u1 –
(a)
The solution of this equation is taken under the form shown in (3) where a(t) and ϕ(t) are slowly varying functions of time. (3) i = a (t ) cos ω q t + ϕ (t )
(
G0
u1
– Z3
+ –
R0 Z’2
+ u2 –
(b)
Fig. 5. Controlled current or voltage source representation of the amplifier with current (a) or voltage (b) controlled source.
)
ZQ +
At low current amplitude level, the damping term of (1) should have a negative value to insure increasing amplitude solution. If Rds is the value of the nonlinear dipolar resistance at very low current amplitude, the startup condition takes the form (4). (4) Rq + Rds < 0
–
vd
–
G·u1
+ u1
i
Z1
+ Z2
u2 –
Fig. 6. Reduced amplifier representation.
From the dipolar point of view, the amplifiers used in the oscillator circuits can be classified according several ways described in the following sections. So as to explain the classification method, the black box amplifier shown in Fig. 1 will be taken under the form shown in Fig. 4 where Z1 and Z2 are the amplifier input and output impedances in par
As the oscillation amplitude increases, the dipolar resistance increases so that the value of the damping term increases. The steady state amplitude a0 is reached when this term becomes zero as given by (5) and Fig. 3. (5) Rq + Rd ( a 0 ) = 0 588
The nonlinear transfer function of the amplifier is determined by the conditions (9). ⇒ u 2 = Au1 − u 0 < u1 < + u 0 (9) ⇒ u 2 = −Vsat u1 ≤ − u 0 u1 ≥ u 0 ⇒ u 2 = Vsat
allel with possible external impedances, while Z3 mainly represents the effect of the resonator’s parallel capacitance considered as a part of the amplifier circuit as stated in section II in parallel with a possible biasing impedance. Although it is possible, in a linear small signal analysis, to include Z3 into Z1 and Z2 by a simple circuit transformation, in some case it is wiser to keep it as a separate impedance because of the particular role it plays in the dipolar impedance as it will be shown later. The amplifier itself can be represented either by a voltage controlled current source (Fig. 5.a) or by an equivalent voltage controlled voltage source (Fig. 5.b). In the linear case, by a simple circuit transformation, either one of the two representations in Fig. 5 can be reduced to the form shown in Fig. 6. Note that in Figs. 4, 5, and 6, the voltage reference node is not necessarily the ground node. A general expression of the small signal dipolar impedance is obtained by replacing the resonator in Fig. 6 by a current source of same frequency as shown in Fig. 7.
i
vd
u1
Z1
Z2
–0.5
–
–1 –0.5 –0.4 –0.3 –0.2 –0.1 0 0.1 0.2 0.3 0.4 0.5 Input voltage (V)
(b)
For small value of the current i, the dipolar impedance Zds given by (10) shows that it is constant, real and negative if the gain A is greater than the unity. Z ds = Rds = R (1 − A) (10) As the current amplitude a increases, the output voltage u2 reaches the saturation level and becomes square shaped (Fig. 9.a) while the dipolar voltage vd becomes distorted (Fig. 9.b).
u2 –
1.0 Output voltage (V)
Fig. 7. Dipolar impedance characterization.
By expressing the dipolar voltage vd as a function of the current i, it is simple to obtain the small signal dipolar impedance under the form (7). From the dipolar point of view, the transconductance G (or the voltage gain A) can be real or complex, linear or nonlinear moreover all variables of the right hand side in (7) might be function of the current amplitude a, so that the nonlinear dipolar impedance take the form (8). Z ds = Z 1 + Z 2 − GZ 1 Z 2 (7)
(a)
0.5 0 –0.5 –1.0 29.90 1.0
Dipolar voltage (V)
Z d = R d ( a ) + jX d ( a )
0
u2
R (a)
+
–
0.5
Fig. 8. Simple behavioral oscillator with saturation limiting. Values used are: A = 4, R = 100 Ω, Vsat = 1 V.
–
G·u1
+
+
–
i +
Amp
+ u1
–
Output voltage (V)
1
vd
+
(8)
IV. AMPLITUDE LIMITATION It is well known that the oscillation amplitude is determined by the nonlinear behavior of the amplifier. From this point of view, the amplifier circuits can be split into several categories such as: saturation or cutoff limitation, hard or soft limitation, symmetrical or nonsymmetrical limitation, with possible combination of these various limitation mechanisms.
29.95
30.00 Time (us)
30.05
30.10
29.95
30.00 Time (us)
30.05
30.10
(b)
0.5 0 –0.5 –1.0 29.9
Fig. 9. Output and dipolar waveforms (hard saturation case).
Dipolar resistance (Ohms)
0
A. Hard Saturation Limiting In the simple behavioral oscillator circuit shown Fig. 8.a, the amplifier is an ideal circuit having a real input impedance R, and a linear real positive gain A limited by symmetrical saturation as shown in Fig. 8.b. In this case, because the amplifier is assumed ideal with a zero output impedance , it cannot be reduced to a transconductance amplifier.
–50 –100 –150 –200 –250 –300 –350 0
2
4 6 8 10 12 Motional Current Amplitude (mA)
14
Fig. 10. Dipolar resistance (hard saturation case).
589
In this case, the output and dipolar voltages are no longer symmetrical (Fig. 12). As in the saturation case, for small current amplitude, the dipolar impedance is given by (10) as long as the cutoff amplitude is not reached. For increasing current amplitude, the dipolar impedance increases as shown in Fig. 13, while the dipolar reactance remains null if there is no reactive part in the amplifier.
By calculating the first harmonic of the dipolar voltage, it is possible to calculate the dipolar impedance as a function of the current amplitude a. In Fig. 10, it can be seen that the dipolar impedance Zd becomes nonlinear when the current amplitude is larger than the limit aL given by (11). V (11) a L = sat AR In the present case, because there is no reactive part in the amplifier circuit, the dipolar impedance remains purely real so that Xd = 0. Moreover, according to the expression of Rds given by (10), it is obvious that an inverting amplifier (A < 0) with real input and output impedances cannot be used in an oscillator circuit because Rd is always positive.
Dipolar resistance (Ohms)
0
B. Cutoff Limiting Another limiting mechanism often involved in the oscillator circuits is the cutoff limitation an example of which is shown Fig. 11.
–50 –100 –150 –200 –250 –300 –350
0
10
20 30 40 50 60 70 80 90 100 Motional Current Amplitude (mA)
Fig. 13. Dipolar resistance (cutoff case).
C. Soft Saturation Limiting
i 12
+ u2
R
–
i
10
–
15
8
+
6
2 0
–2
–1.5
–1
(b)
–0.5 0 0.5 1 Input voltage (V)
1.5
2
u1
–
15
Output voltage (V)
10
–10 –15 –6
–4
(b)
–2
0 2 Input voltage (V)
4
6
(a)
5 0 –5
–15 49.90
6
8
4 2 39.95
40.00 Time (us)
40.05
40.10
(b)
6
49.95
50.00 Time (us)
50.05
50.10
49.95
50.00 Time (us)
50.05
50.10
(b)
4 2 0 –2 –4
8
–6
6
–8 49.90
Fig. 15. Output and dipolar waveforms (soft saturation case).
4
Let us examine now the case of a soft limiting mechanism. A simple example of such a circuit is given by the Van der Pol oscillator that has the same representation as the previous ones, but in this case the limitation due to the nonlinear DC transfer function represented in Fig. 14.b is given by (13) where A is the small signal gain and ε is the
2 0 39.90
0 –5
–10
8
10
5
Fig. 14. Simple Van der Pol oscillator. Values used are: A = 4, R = 100 Ω, ε = 2 10–2 V–2.
Dipolar voltage (V)
Output voltage (V)
u2
R (a)
(a)
0 39.90
10
+
–
The oscillator shown has the same features as the previous ones except that the nonlinear transfer function is now given by the conditions (12) and has the shape represented Fig. 11.b. u1 ≤ −u 0 ⇒ u2 = 0 (12) u1 ≥ −u 0 ⇒ u 2 = A(u1 + u 0 )
Dipolar voltage (V)
Amp
+
Fig. 11. Simple behavioral oscillator with cutoff limiting. Values used are: A = 4 , R = 100 Ω, u0 = 0.6 V.
10
–
4
(a)
12
vd
Output voltage (V)
Amp
+ u1
– Output voltage(V)
vd
+
39.95
40.00
40.05
40.10
Time (us)
Fig. 12. Output and dipolar waveforms (cutoff case).
590
nonlinear coefficient (A and ε are assumed real and positive) (13) u 2 = Au1 (1 − ε u12 ) It is possible to show that in this case, the dipolar impedance can be derived under the form given by (14). 3 Aε R 3 a 2 (14) Z d = (1 − A) R + 4 For small current amplitude, the dipolar impedance has the same expression (10) as in the two previous cases. As the current amplitude increases, the output and dipolar waveforms become distorted (Fig. 15), but, if there is no reactive part in the amplifier circuit, Zd remains real and increases with amplitude according to the parabolic law (14) (Fig. 16).
the oscillation frequency, it is quite simple to demonstrate that in any case, the small signal dipolar impedance take the form (16) where it is obvious that the dipolar reactance is no longer null. A0 1 (15) ωc = >> ω 0 A = 1 + jωτ c τc R (1 − A0 + ω 2τ c2 ) Rds = 1 + ω 2τ c2 (16) X = RA0ωτ c ds 1 + ω 2τ c2 When the amplifier has a limited bandwidth, it can be shown that the nonlinear part of the dipolar impedance does not depend strongly on the limiting mechanism, so it will be demonstrated only in the soft saturation case. Fig. 17 shows that the dipolar resistance remains practically unaffected while the dipolar reactance is now different from zero and decreases with the current amplitude.
Dipolar resistance (Ohms)
0 –50 –100 –150 –200
B. Parallel Capacitance
–250 –300 –350
0
10
20
30 40 50 60 70 Current Amplitude (mA)
As explained in section II, the parallel capacitance is included in the amplifier part of the oscillator circuit so that it can be considered in parallel with the dipolar impedance of the amplifier alone. Nevertheless, it is not so simple to express the equivalent dipolar impedance of these two components associated in parallel because the voltage across the dipole is no longer linear.
80
Fig. 16. Dipolar resistance (soft saturation case).
V. FREQUENCY LIMITATION AND PARALLEL CAPACITANCE The simple models presented so far are only ideal behavioral models without any reactive component. Any actual amplifier has a reactive part at least because they have a limited bandwidth or because of the parallel capacitance that is included in the amplifier part as pointed out in section II.
Dipolar resistance (Ohms)
0
A. Amplifier Limited Bandwidth –50 –100 –150 Limited bandwidth
–200
Dipolar reactance (Ohms)
Dipolar reactance (Ohms)
Dipolar resistance (Ohms)
0
–250 –300 –350
0
50 45 40 35 30 25 20 15 10 5 0 0
10
20 30 40 50 60 Current Amplitude (mA)
70
80
Limited bandwidth
–50 –100 –150 –200 –250 Cp = 0 –300 –350 0 45 40 35 30 25 20 15 10 5 0 –5 –10 –15 –20 0
Cp = 5 pF 10
20 30 40 50 60 Current Amplitude (mA)
70
80
70
80
Cp = 0
Cp = 5 pF
10
20
30 40 50 60 Current Amplitude (mA)
Fig. 18. Influence of the parallel capacitance on the dipolar impedance. 10
20 30 40 50 60 Current Amplitude (mA)
70
In Fig. 18 are compared the dipolar impedances of a limited bandwidth Van der Pol oscillator, as described in section IV C, with and without a parallel capacitance. As for the bandwidth effect, the dipolar resistance is practically not affected while the reactance exhibits a more important distortion.
80
Fig. 17. Dipolar impedance of a limited bandwidth amplifier. Oscillation frequency: 10 MHz, cutoff frequency: 100 MHz.
Assuming that the linear part of the amplifier gain A used in the previous cases in section IV has the form given by (15) where the cutoff frequency ωc is much larger than 591
VI. INVERTING AMPLIFIER
filled: the tranconductance G must be larger than the limit Gc given by (20) and the frequency must be higher than the limit ωc given by (21). The small signal dipolar impedance for a given pair of impedances Z1 and Z2 is represented in Fig. 20 where oscillation cannot start if Rds is in the shaded area. Because Z1 and Z2 play symmetrical roles in the expression of the small signal dipolar impedance, the same curves are obtained when they are reversed. Unlike the case of the non inverting amplifier with real input and output impedances, here the parallel capacitance strongly modify both the real and imaginary parts of the small signal dipolar impedance as shown in Fig. 20 where it can also be seen that the critical frequency ωc below which the oscillation cannot start, is not affected by the parallel capacitance. In addition, as for the case where the parallel capacitance is null, the same small signal dipolar impedance is obtained when the input and output impedances are reversed. It should be emphasized that the results presented so far in this section do not depend on the limiting mechanism but only on the small signal transconductance as well as the input and output impedance values.
A. Representation i
+
–
Z1
vd
+
G·u1
+ u1
vd
Z2
+
+
u2
u1
–
–
–
–
G·u1 R1
+
C1
R2
(a)
u2
C2
–
(b)
Fig. 19. Inverting amplifier.
Dipolar resistance Rds (Ohms)
All the amplifier circuits presented so far had a positive gain with real input and output impedances so that, even at low frequency, they may have a negative dipolar resistance. This is the reason why an oscillator using such an amplifier type is often called “negative resistance oscillator.” Nevertheless, most of the current crystal oscillators are using an inverting amplifier that can be represented as in Fig. 19.a, the small signal dipolar impedance of which is given by (17) where the transconductance G is real and positive. Z ds = Z 1 + Z 2 + GZ 1 Z 2 = R ds + jX ds (17) Obviously, this expression may have a negative real part only if the input and output impedances Z1 and Z2 both have an imaginary part. A simple form of such a case is represented in Fig. 19.b where Z1 and Z2 are parallel combinations of a resistance and a capacitance. B. Small Signal Analysis
Rds = X ds =
R1 1 + ω 2τ 12 − R1ωτ 1 1+ω τ 2
2 1
+ −
R2 1 + ω 2τ 22 R2ωτ 2 1+ω τ
G > Gc =
2
2 2
C1
τ2
+ −
Dipolar reactance Xds (kOhms)
The small signal dipolar impedance (17) can be obtained by replacing Z1 and Z2 by their expression (18), thus the real part Rds and imaginary part Xds of the small signal dipolar impedance take the form (19) where τ1 and τ2 will be called the input and output time constant respectively. R1 τ 1 = R1C1 Z1 = 1 + ω 2τ 12 (18) R2 Z = = R C τ 2 2 2 2 1 + ω 2τ 22
1000
500
0 Cp = 5 pF –500
Cp = 0
–1000 3.0MHz
10MHz Frequency
–4 Cp = 5 pF –8 Cp = 0 –12
3.0MHz
30MHz
100MHz
C. Transconductance Amplifier with Cutoff Limiting i
(1 + ω τ )(1 + ω τ ) (19) C2 (20) + τ1 2
10MHz
Fig. 20. Small signal dipolar impedance of an inverting amplifier. G = 100 mA/V, C1 = C2 = 75 pF, R1 = 100 Ω, R2 = 1000 Ω.
(1 + ω 2τ 12 )(1 + ω 2τ 22 ) GR1 R2 (ωτ 1 + ωτ 2 ) 2 1
100MHz
Frequency
GR1 R2 (1 − ω 2τ 1τ 2 )
2
30MHz
0
2 2
+
–
1 1 + R1 R2 1 2 2 (21) ω > ωc = τ 1τ 2 C C G − 1 + 2 τ1 τ2 By looking at the expression of Rds it should be demonstrated that it can be negative only if two conditions are fulG +
–
iG
+ u1
vd
1.0
Z1
+ Z2
(a)
u2 –
Output current (A)
i
0.8 0.6 0.4 0.2 0
–4
–2
(b)
0
2 4 6 Input voltage (V)
8
10
Fig. 21. Transconductance amplifier with cutoff limiting. G = 100 mA/V, u0 = 0.6 V.
Several simulations performed on transconductance amplifiers with different limitation mechanisms have shown that the input and output impedances (or time constants) 592
have similar effects on the nonlinear dipolar impedance. Thus, the attention will be focused only on the cutoff limiting mechanism often involved in the oscillator circuits. In this case, the nonlinear transconductance represented in Fig. 21.b is defined by the conditions (22). u1 ≤ −u 0 ⇒ iG = 0 (22) u1 ≥ −u 0 ⇒ iG = G (u1 + u 0 ) Output voltage (V)
5 0
resistance and reactance keep the same value for small current amplitude but, as the amplitude increases, the two curves may have a quite different location. As for the small dipolar impedance, the parallel capacitance strongly modify both the dipolar resistance and reactance of the amplifier (Fig. 23). By looking at (6) it is obvious that a negative reactance corresponds to a positive frequency shift that can become very large if the resonator motional inductance Lq is not much greater than the dipolar inductance Ld. In such a case, the oscillation frequency may be close to the resonator antiresonant frequency, this is the reason why these oscillators are often improperly called “parallel resonance oscillators.”
(a)
–5 –10 –15 –20
D. Input and Output Impedances
–25 –30
5
39.96 40.00 Time (us)
40.04
0
40.08
(b)
–5 –10 –15 –20 –25 –30 –35 39.88
39.92
39.96 40.00 Time (us)
40.04
Fig. 22. Output and dipolar waveforms (transconductance amplifier with cutoff limiting). –50
Cp = 5 pF
–100 –150
–200
R1 = 20 R2 R1 = 30 R2
–300 –400 –500 –600 0
10
–300 –350 0
10
20
30 40 50 60 70 80 Current Amplitude (mA)
–500
Cp = 5 pF
–1500
10
20
Solid lines: input Z1, output Z2 Dashed lines: input Z2, output Z1 30 40 50 60 70 80 Current Amplitude (mA)
R1 = 10 R2 R1 = 20 R2 R1 = 30 R2
–1000 –1500 –2000 –2500 0
10
20 30 40 50 60 70 Current Amplitude (mA)
80
90 100
R2 =
–500
R1 10
2R1 10 3R1 R2 = 10 R2 =
–1000 –1500 –2000 –2500 0
10
20 30 40 50 60 70 Current Amplitude (mA)
80
90 100
0
90 100 Dipolar reactance (Ohms)
–25000
Cp = 0
90 100
0
90 100
–1000
–2000
80
Fig. 24. Influence of the input resistance R1. C1 = C2 = 75 pF, R2 = 100 Ω.
Solid lines: input Z1, output Z2 Dashed lines: input Z2, output Z1
Cp = 0
20 30 40 50 60 70 Current Amplitude (mA)
–500
–250
0 Dipolar reactance (Ohms)
R1 = 10 R2
–200
Dipolar resistance (Ohms)
Dipolar resistance (Ohms)
0
–100
0
40.08 Dipolar reactance (Ohms)
Dipolar voltage (V)
0
39.92
Dipolar resistance (Ohms)
–35 39.88
Fig. 23. Dipolar impedance of a transconductance amplifier. Z1: R1 = 100 Ω // C1 = 75 pF, Z2: R2 = 1000 Ω // C2 = 75 pF.
The output and dipolar waveforms of the transconductance amplifier with cutoff limiting for a given pair of impedances Z1 and Z2 are shown in Fig. 22, and its dipolar impedance is represented in Fig. 23. Note that the dipolar resistance looks like the one of a non inverting amplifier having the same limiting mechanism (Fig. 13) but with a much larger reactance value. As demonstrated in section VI B, when the two impedances Z1 and Z2 are reversed, the dipolar
–500 R1 10 2R1 R2 = 10 3R1 R2 = 10
–1000
R2 =
–1500 –2000 –2500 –3000 0
10
20 30 40 50 60 70 Current Amplitude (mA)
80
90 100
Fig. 25. Influence of the output resistance R2. C1 = C2 = 75 pF, R1 = 1000 Ω.
593
0 Dipolar resistance (Ohms)
Dipolar resistance (Ohms)
0 –50 –100 –150 –200 –250 –300 –350 –400 –450 –500 0
3C2 C1 = 2 C C1 = 2 2 C1 = C2
10
20 30 40 50 60 70 Current Amplitude (mA)
80
90 100
C2 = 3 C1
–400 C2 = 4 C1
–600 –800 –1000 0
10
20 30 40 50 60 70 Current Amplitude (mA)
80
90 100
0 3C2 C1 = 2 C1 = C2
–500 –1000 –1500
C1 =
Dipolar reactance (Ohms)
Dipolar reactance (Ohms)
0
C2 = 2 C1
–200
C2 2
–2000 –2500 –3000 0
10
20 30 40 50 60 70 Current Amplitude (mA)
80
90 100
Fig. 26. Influence of the input capacitance C1. R1 = 1000 Ω, R2 = 100 Ω, C2 = 100 pF.
C2 = 4 C1
–500
C2 = 3 C1
–1000 –1500
C2 = 2 C1
–2000 –2500 –3000 0
10
20 30 40 50 60 70 Current Amplitude (mA)
80
90 100
Fig. 27. Influence of the output capacitance C2. R1 = 1000 Ω, R2 = 100 Ω, C1 = 50 pF.
Figures 24 to 27 show how the dipolar impedance of the transconductance amplifier previously used is modified when one of the four components defining the input and output impedances Z1 and Z2 is modified, the three others being kept constant. Of course, these figures do not cover all the possible combinations but only demonstrate how it is possible to use the dipolar method to optimize a circuit. It is obvious in Figs. 24 and 25, for example, that modifying the input or output resistance R1 or R2 drastically changes the dipolar resistance and therefore the motional current amplitude in the crystal, while the frequency shift due to the dipolar reactance is more sensitive to a change in the output resistance than in the input resistance. On the other hand, the dipolar resistance appears more sensitive to a change in the output capacitance than in the input capacitance, while both have an important effect in the dipolar reactance as shown in Figs. 26 and 27.
REFERENCES [1]
R. Brendel, D. Gillet, N. Ratier, F. LardetVieudrin, and J. Delporte, “Nonlinear dipolar modeling of quartz oscillators,” Proc. of the 14th EFTF, Torino, Italy, pp. 184188, March 2000. [2] M. Addouche, R. Brendel, D. Gillet, N. Ratier, F. LardetVieudrin, and J. Delporte, “Modeling of quartz crystal oscillators by using nonlinear dipolar method,” IEEE Trans. UFFC, vol. 50, pp. 487495, May 2003. [3] M. Addouche, N. Ratier, D. Gillet, R. Brendel, F. LardetVieudrin, and J. Delporte, “ADOQ: a quartz crystal oscillator simulation software,” Proc. of the 55th IEEE IFCS, Seattle, USA, pp. 753757, June 2001. [4] K. Kurokawa, “Some basic characteristics of broadband negative resistance oscillator circuits,” Bell System Technical Journal, pp. 19371955, July–August 1969. [5] E.A. Vittoz, “Quartz oscillators for watches,” Proc. of the 10th International Congress of Chronometry, pp. 131140, Geneva, Switzerland, September 1979. [6] N. Kryloff, and N.N. Bogoliuboff, Introduction to Nonlinear Mechanics, Princeton University Press, 1943. [7] A.A. Andronov, A.A. Vitt, and S.E. Khaikin, Theory of Oscillators, New York, Pergamon Press, 1966, pp. 585613. [8] R. Brendel, D. Gillet, N. Ratier, M. Addouche, and J. Delporte, “Oscillator noise simulation by using nonlinear dipolar method,” Proc. of the 15th EFTF, Neuchâtel, Switzerland, pp. 123128, March 2001. [9] B. Parzen, A. Ballato, Design of Crystal and other Harmonic Oscillators, New York, Wiley Intersciences, 1983. [10] M. Addouche, N. Ratier, D. Gillet, R. Brendel, and J. Delporte, “Experimental validation of the nonlinear dipolar method,” Proc. of the 16th EFTF, St. Petersburg, Russia, March 2002.
VII. CONCLUSION In this work, the most important amplifier parameters that play a part in the behavior of quartz crystal oscillators have been studied by using the nonlinear dipolar that is a powerful and well suited method to analyze the characteristics and performance of high Qfactor circuits. The dipolar analysis has been successfully implemented in a dedicated software (ADOQ) whose efficiency and accuracy have been checked experimentally [10]. In addition to the main features described in the introduction, the program is currently being completed by a optimization module intended to help the designer in choosing the right circuit for a given purpose or to choose the right components for a given design.
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Motorola Semiconductor Application Note
AN1783
Determining MCU Oscillator Startup Parameters By Stuart Robb & David Brook, East Kilbride, Scotland Andreas Rusznyak, Geneva, Switzerland
Rev 1.0, December 1998
Introduction Many microcontrollers (MCUs) incorporate an inverting amplifier for use with an external crystal or ceramic resonator in a Pierce oscillator configuration. This paper describes how to calculate the minimum gain (transconductance) of the amplifier required to ensure oscillation with specific external components, and also how to measure the amplifier transconductance to establish whether the minimum gain requirement is met.
Oscillator Circuit STOP Internal to MCU OSC1 External Components
R0
OSC2
Q C1
C2
Figure 1 Standard Pierce Oscillator for > 1MHz Operation Figure 1 shows the standard Pierce oscillator configuration typically used on MCUs for frequencies in the range 1MHz to 20MHz. The oscillator pins are labelled OSC1, OSC2 on the MC68HC05 and
© Motorola, Inc., 1999
AN1783 Rev. 1.0
Oscillator Circuit
MC68HC08 families of MCUs and EXTAL, XTAL, respectively, on the MC68HC11 and MC68HC12 families. On some MCUs (e.g. the MC68HC05B and MC68HC05X families), the resistor R0 is integrated onchip, in which case the external resistor is not required. This circuit is not applicable to some members of the MC68HC12 family of MCUs which employ a low power oscillator, e.g. MC68HC12D60.
Internal Circuit
The circuit internal to the MCU is shown in simplified form as a NAND gate followed by an inverter. The NAND gate has two inputs; one is connected to the MCU pin called OSC1 and the other input is connected to the inverted internal STOP signal. There are two conditions under which the oscillator is required to start oscillating; one is when power is applied to the MCU (called poweron reset) and the other is when the STOP signal is deasserted. Following a poweron reset, the oscillation will start as soon as the MCU supply voltage, VDD, has reached a level where the oscillator loop gain is greater than unity. For reliable operation, the oscillator must be oscillating by the time VDD has reached the minimum specified operating value. Most MCUs have a low power STOP mode. STOP mode is entered when the software executes the STOP command and as a result the STOP signal is asserted to stop the oscillator. The MCU is no longer clocked and the only current consumed by the MCU is due to ‘leakage’. An external interrupt or a reset can release the STOP signal and allow the oscillator to restart. The remainder of this paper will ignore the STOP input and treat the NAND gate as a simple inverter. The output signal at the pin OSC2 is typically a distorted sine wave whose amplitude may even exceed the supply rail voltages. The following inverter provides additional voltage gain to produce an approximately square wave signal which in turn drives the internal clock generation circuitry.
External Circuit
In current designs the pchannel and the nchannel transistors in the inverter contribute approximately equally to the total gain provided that Vin ≈ Vout ≈ VDD/2. Resistor R0 ensures that this optimal condition is met at oscillation startup. For the circuit to oscillate, there must be positive feedback and the closed loop gain must be greater than unity. Resistor R0 results in negative feedback which increases the open loop gain requirement of the amplifier. R0 is usually made as large as possible to minimise the feedback whilst still overcoming leakage currents at startup. For operation between 1MHz and 20MHz a value in the range of 1MΩ  10MΩ is typically used. In humid or dirty environments it is good practice to lacquer the oscillator components and tracks after they have been cleaned to prevent leakage
AN1783 Rev. 1.0 MOTOROLA
2
Application Note currents due to condensation or dirt accumulating on the printed circuit board (PCB). Care should be taken when laying out the components on the PCB. The components should be positioned as close as possible to the MCU and the traces should be kept as short as possible. All other traces should be kept as far away as possible to avoid coupling. It is often worthwhile surrounding the components with a shield trace connected to ground (be careful not to create any loops) or a ground plane. The IC designer should ensure that the input pin OSC1 and preferably also the output pin OSC2 are placed between ‘quiet’ pins carrying DC signals. If a ceramic resonator is used with capacitors C1, C2 integrated into a common package, the manufacturer may recommend an optimal value of R0. The resonator Q, and capacitors C1 and C2 form the resonant circuit. C1 and C2 represent the external capacitors and any stray capacitance in parallel. The stray capacitance should be measured or estimated and included in the values used for C1 and C2 in Equations 1 to 7. Q
R
L
C
C0
Figure 2 Crystal Equivalent Circuit A crystal or ceramic resonator has the small signal equivalent circuit shown in Figure 2. R is called the ‘series resistance’, L and C are called the motional or series inductance and capacitance, respectively. C0 is the shunt capacitance, it represents the sum of the lowfrequency parallel plate capacitance of the resonator and the stray capacitance of the crystal holder. In Equations 1 to 7 any additional stray capacitance between the OSC1 and OSC2 pins should be included into this value. Values for R, L, C and C0 for a particular crystal are specified on a data sheet usually available from the crystal manufacturer. In order to measure these values, the manufacturer must apply a signal to the crystal, i.e. the values are obtained at a particular level of power dissipation in the crystal. However, at the startup of the oscillator, the only signal across the crystal is due to thermal (Johnson) noise so the power dissipation in the crystal is extremely low. It is known that the effective value of R may increase as the power dissipated in the crystal decreases to low levels. The maximum value of R is therefore estimated by the crystal manufacturer. It is this estimated maximum value which should be used in equations 1 to 7.
AN1783 Rev. 1.0 3
MOTOROLA
Calculating the Minimum Required Transconductance
Calculating the Minimum Required Transconductance R0 R
L
C V1
V1.gm
gds
C2
C0
C1
Figure 3 Simplified Oscillator Equivalent Circuit Figure 3 shows a simplified small signal equivalent circuit to the oscillator. The inverter is modelled as a current source with an output current equal to V1.gm where V1 is the input voltage and gm is the transconductance of the inverter. gds is the total output conductance i.e. the sum of the output conductances of the pchannel and the nchannel transistors in the inverter at startup. The components of the resonant circuit have been described above. As developed in [1] the impedance at resonance of the circuit comprising of the resonator Q and capacitors C1, C2 is given by: (Eqn 1)
1 R PQ = 2 ( ωC t ) R
where ω=2πƒ, ƒ being the frequency of resonance. Ct represents the total capacitance in parallel with series components R, L and C of the resonator: C1 C2 C t = C 0 + C1 + C2
(Eqn 2)
The frequency of oscillation is given to a good approximation by (Eqn 3)
1 1 1 1 ƒ =    + 2π L C C t
For quartz resonators the term Ct can be neglected. The minimum transconductance required of the inverter to sustain oscillation in this circuit is given approximately by: 2
( C1 + C2 ) C1 1 1 gm min ≈   +  + gds C1 C2 C1 + C2 R PQ R 0
2
(Eqn 4)
AN1783 Rev. 1.0 MOTOROLA
4
Application Note If gm >> gds, this can be simplified to: 2
( C1 + C2 ) 1 1 gm min ≈   + C1 C2 R PQ R 0
(Eqn 5)
The validity of this simplification can be checked by measuring gds, as described later in this paper. If R0 >> RPQ, equation 5 can further be reduced to: 2
( C1 + C2 ) 1 gm min ≈  R PQ C1 C2
or, if C1 = C2, to
C1 4 2 gm min ≈  = 4Rω C 0 + R PQ 2
(Eqn 6) 2
(Eqn 7)
Measuring Amplifier Characteristics The recommended circuit for measuring the transconductance of the amplifier is shown in Figure 4 [2]. The circuit is simple to implement and should be powered up with the MCU reset pin held at 0V to ensure the amplifier stays active and the MCU does not execute any code. All unused inputs should be connected to 0V or VDD and not left floating. Note that the diagram correctly indicates that OSC1 and OSC2 are connected together. The transconductance does not vary significantly at frequencies below the oscillator’s maximum design frequency. However, it is not recommended to measure the transconductance at the intended operating frequency, as the effects of stray capacitances will make the measurements inaccurate. A frequency in the range of 10kHz to 100kHz is recommended. A signal of around 500mVpp or less should be used with a 50Ω terminating resistor and a 1µF coupling capacitor to ensure that the amplifier input and output remain in their linear region. It is essential that a high impedance measuring instrument, such as an oscilloscope with a low capacitance, high input resistance probe (10MΩ) is used to measure Vin and Vg with respect to ground. The value for the resistance Rm should be 100Ω to 1kΩ.
OSC1
OSC2
Rm
Vin
50 ohm signal generator 500mVpp
1µ
Vg
50
Figure 4 Measuring Amplifier Transconductance AN1783 Rev. 1.0 5
MOTOROLA
Measuring Amplifier Characteristics
The transconductance of the amplifier depends on the process parameters and varies with supply voltage and temperature. Measurements should be taken on worst process parameter devices (if available) over the expected range of supply voltage and temperature. The worst case (lowest) figure can be expected at the combination of the minimum expected supply voltage and the highest expected operational temperature. Based on the measurement results, the sum of the transconductance and of the output conductance is: (Eqn 8)
Vg – Vin gm + gds = Rm × Vin
At this stage gds is unknown and a separate measurement must be made to determine it. If gds