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Analysis of Crystal Oscillator Takehiko Adachi Faculty of Engineering, Yokohama National University Tokiwadai 79-5, Yokohama, Japan Abstract 2.1 Principle of analysis Fig. 1 shows an equivalent circuit of a crystal oscil- In this paper, analysis methods of a crystal oscillator lator. A crystal oscillator is composed of a crystal are explained. The SPICE base method insures the resonator and an active circuit. The oscillator cir- 10 6 10 8 frequency resolution with short calcu- cuit is usually designed to oscillate at the frequency lation time. The algebraic analysis methods presents where the crystal resonator is inductive. Therefore, the equations of negative resistance and equivalent the one-port impedance of the active circuit is ex- capacitance, the frequency stability against power pressed by the series circuit of a negative resistance supply voltage, and the start-up times. Ri and an equivalent capacitance Ci . The negative re- sistance and the equivalent capacitance are the func- tion of the frequency and the magnitude of the crys- 1 Introduction tal current. On the other hand, the equivalent circuit of a crystal resonator is composed of the series res- onance circuit of inductance L1 , capacitance C1 , re- A quartz crystal oscillator is a key device as a stable sistance R1 , and the parallel capacitance C0 . At the frequency source for communication equipments. steady state oscillation, the characteristics of a crys- Recently, there are urgent needs for improvement tal is expressed by the series circuit of equivalent re- of its performance; low power operation, low power sistance Re and inductive reactance XL obtained by supply voltage operation, high frequency operation, its equivalent circuit parameters. The steady state phase noise reduction, miniaturization, etc. Precise oscillation is attained when the negative resistance and efﬁcient analysis method of crystal oscillator is and equivalent capacitance of the active circuit bal- indispensable to meet these requirements. SPICE ance the equivalent resistance and reactance of the and its family simulators are widely used for crys- resonator, respectively. tal oscillator design[1]. Although SPICE is a pow- erful simulation tool, speciﬁc technique is effective for crystal oscillator simulation. We have been stud- ied the effective and efﬁcient analysis method and L1 -R i design technique of crystal oscillators. In this pa- C1 C0 per, we have outlined the analysis method using R1 Ci SPICE[2][3][4] and an algebraic approximate anal- ysis method[5][6]. Figure 1: Equivalent circuit of crystal oscillator. 2 Analysis method using SPICE Usually, crystal current has only fundamental os- Usually crystal oscillator is analyzed by transient cillation frequency, because the impedance of the analysis of SPICE. But its transient analysis requires crystal resonator becomes 2π 1C0 at the harmonic f large amount of calculation and obtained results are oscillation frequency and is much bigger than the not sufﬁcient in frequency resolution. Our approach crystal equivalent resistance Re . And the frequency is a kind of simpliﬁed harmonic balance technique. change of the equivalent parameters of the active 1 circuit is much smaller than that of the resonator. frequency is calculated from the obtained equiva- Therefore, when the current dependence of the lent capacitance and the crystal parameters. The fre- equivalent parameters of the active circuit are calcu- quency deviation caused by the power supply volt- lated beforehand, the oscillation frequency and the age variation can be calculated by similar calcula- crystal current can be calculated by these character- tion for the power supply voltage VCC · dVCC . By istics and the equivalent parameters of a crystal res- this method, the frequency deviation of 1ppm onator. 0.01ppm can be estimated by calculation of the ac- tive impedance with the 10 3 10 4 resolution. 2.2 Calculation method of negative resis- Ci tance and equivalent capacitance Ci B1 VCC VCC +dVCC Ci +dCi B2 The negative resistance and the equivalent capaci- tance of the active circuit can be calculated by the following procedure; At ﬁrst, the voltage response at the crystal terminals is calculated using the transient ix i x +dix i 0 analysis, when the crystal resonator is replaced by A1 - Re the sinusoidal current source with magnitude Ix and A2 frequency f in Fig. 1. Then, the magnitude Vx and phase φ of the fundamental component of the volt- VCC VCC +dVCC age response are calculated from the 1-period data - Ri of the steady state voltage response. The negative resistance and the equivalent capacitance are calcu- Figure 2: Crystal current dependence of negative lated using obtained Vx and phse φ and the crystal resistance and equivalent capacitance. current Ix , by the following equations. Ri Vx Ix cos φ (1) 2.3.1 Convergence Ix To obtain the negative resistance and the equivalent Ci (2) 2π fVx sin φ capacitance, the calculation of the transient analysis is necessary until the voltage response reaches the steady state. Judgment whether the response reaches the steady state is easily made by checking the varia- 2.3 Calculation method of steady state tion of the period of voltage response cycle by cycle. characteristics Necessary cycles to reach the steady state is roughly proportional to the number of transistors used in the The crystal current and the oscillation frequency at oscillator circuit. Although necessary cycles depend the steady state can be calculated by following pro- on the purpose of the calculation, 20 50 cycles is cedure; The crystal current dependence of the nega- sufﬁcient for the colpitts crystal oscillator composed tive resistance and the equivalent capacitance are ob- by single transistor. tained at the power supply voltage VCC and the crys- tal nominal frequency, as in Fig. 2. The steady state 2.4 Example crystal current is obtained from the current value at the point A1 where the magnitude of negative resis- Fig. 3 shows the schematic diagram of the cas- tance agrees with the crystal equivalent resistance code crystal oscillator[7] widely used as a frequency Re . The steady state equivalent capacitance is ob- source of mobile communication equipments. The tained from the value of capacitance at the current negative resistance and the equivalent capacitance, level obtained above. The steady state oscillation and frequency stability against the power supply 2 voltage are calculated and compared to the measure- 20 18 ment. 16 Ci [pF] 14 12 Rc 10 RD 8 6 Q2 Output 4 2 CD 0 VCC 0 0.2 0.4 0.6 0.8 1 RA Ic Ix [mA] Ix Cs 0 Q1 0.2 0.4 0.6 0.8 1 -100 Ri [pF] CA -200 RB CB -300 RE -400 Figure 3: Cascode crystal oscillator -500 Figure 4: Crystal current dependence of negative resistance and equivalent capacitance 2.5 Negative resistance and equivalent ca- pacitance 0.6 Fig. 4 shows the crystal current dependence of the 0.4 f/f [ppm] negative resistance and the equivalent capacitance. 0.2 The values at Ix 0 were calculated by the small 0 signal analysis (AC analysis). The calculated results 2.2 2.3 2.4 2.5 agree with the measurement results within a several -0.2 VCC [V] percent error. The margin of negative resistance for -0.4 oscillation can be estimated from the difference be- tween the values at the steady state current and null Figure 5: Frequency deviation caused by power level. supply voltage variation 2.5.1 Frequency stability Fig. 5 shows the frequency deviation caused by the grammed by C language or shell script. And the sim- power supply voltage variation. Calculated results ulation is made automatically by changing the input agree with the measurement within 10 7 order er- ﬁle of SPICE. ror in absolute values. The resolution of 10 6 In the case where the harmonic components of the 10 8 can be obtained for the estimation of frequency crystal current are not negligible, as high frequency change . Averaging is necessary for the estimation oscillator, the current source replaced crystal res- of frequency deviation in 10 8 resolution. onator must be parallel connection of the necessary harmonic component current sources. The voltage 2.6 Automatic calculation and hybrid har- responses of the active circuit and the crystal res- monic balance method onator are calculated at each harmonic frequencies, and the magnitude and phase of each harmonic cur- To obtain the crystal current dependence of the neg- rent sources are adjusted to obtain the agreement of ative resistance and the equivalent capacitance, the the voltage responses. The obtained current values transient analyses are made repeatedly by chang- are the components of the steady state crystal cur- ing the driving current. This procedure can be pro- rent and the oscillation frequency can be calculated 3 R¼BVA by the obtained equivalent parameters of the ac- RA · RB VBE tive circuit and the crystal parameters. This method RA R¼B IC0 (6) is a modiﬁed harmonic balance method based on 1 B f ´RA · R¼Bµ 1· RE · the time domain analysis and the frequency domain Bf analysis[4]. Recently, sophisticated simulators base on the harmonic balance technique are available. ADS[8] 1 RA · R¼B VBE and Micro wave ofﬁce[9] are the examples of these 1 R¼BVA · simulators. These simulators also have the function VT of phse noise analysis. ¢ ln ´ VA µ (7) ´RA · R¼ µRE B 1 RA IS R ¼ 1· Bf · Bf B 3 Algebraic analysis method where B f , VA , IS are transistor model parameters, Although simulator is a powerful tool for crystal os- and R¼B is the parallel resistance of RB and B f RE . cillator design, it is difﬁcult to grasp the relation be- K ´vµ is expressed as follows; tween circuit performances and circuit parameters by simulation. We have derived the several approxi- 2 I1 ´vµ mate equations for the fundamental performances of K ´vµ (8) v I0 ´vµ a crystal oscillator. where I0 ´vµ and I1 ´vµ are 0th and 1st order modiﬁed Bessel functions, respectively. 3.1 Negative resistance and equivalent ca- R p is the equivalent loss component of the active pacitance circuit. The loaded Q of the oscillator is determined The approximate expression of the negative resis- by the crystal equivalent resistance Re and R p . tance for the cascode oscillator shown in Fig. 3 is The equivalent capacitance is given by the follow- given by the following equations[6]. ing equations. 1 φ 1 Ci C jBC · CA¼ · CB (9) Ri ω 2CmC K ´ ωCI¼x V µ · R p g ¼ (3) A B A T where, 1 1 1 Rp RBE ´ωCA ¼ µ2 · RE ´ωCB µ2 · RAB ´ωCt µ2 gm Ix 1 1 φ 1· ¼ K ´ ωC¼ VT µ ωC¼ RBE · ωCB RE (10) (4) ωCA A A ¼ where VT is the thermal voltage, CA is the parallel 3.2 Frequency stability against power sup- capacitance of CA and the large-signal base-emitter ply voltage variation capacitance of transistor Q1 , Ct is parallel capaci- ¼ tance of CA and CB , RAB is the parallel resistance of Frequency deviation caused by power supply volt- age change is calculated by the deviation of equiv- RA and RB , RBE is the large-signal base-emitter re- alent capacitance of the active circuit using the fol- sistance. lowing equation[5]. gm is the mutual conductance of transistor Q1 at the large-signal collector DC current IC0 and is ex- df f CiC1 dCi Ci pressed by the following equations. (11) dVCC VCC 2´Ci · C0 µ2 dVCC VCC ICO The deviation of equivalent capacitance is calcu- gm (5) lated by the following equation. VT 4 Table 2: Comparison of frequency stability dCi Ci 1 1 MJC CJC estimated by approximate analysis ´ · µ VBC0 1·MJC BC0 V 1 dVCC VCC CA CB V JC and simulation V JC frequency deviation ppm Error 1 1 MJE CJE · Circuit Simulation Analysis % VBE 1·MJE 1 1 · CA CB CA V JE No. 1 0.485 0.543 12 ´ V JE µ No. 2 0.063 0.070 11 ¢ 1 VT 1 1 1 VBE0 1 VB 2VT k VB BBE0 (12) The start-up time is deﬁned by the time where the crystal current reaches the 90% of its steady state where MJC,CJC,CJE,V JE are the transistor model value. The approximate equation of start-up time is parameters, VBC0 , VBE0 , VB are the base-collector given by the following equation. bias voltage, base-emitter bias voltage, base bias RDCD ´RA · R¼B µ Ò VBE Ó voltage, respectively. 3 ln 1 RA · R¼B · RD k is the margin of the negative resistance: the ra- t90% VBmax tion of the negative resistances at the steady state oscillation and the small signal level. 2L1 Rn0 ´R p · R1 µ Calculation of the frequency deviation are made and compared to the simulated values for the cir- R1 Ri ´Ix1 µ cuits with the parameters of Table 1. Tabel 2 shows ¢ ln Ix2 Ix1 · ln R1 Ri ´Ix2 µ the comparison of the frequency deviation between ´ µ ln 10 · ln 1 2 the algebraic approximate analysis and simulation. 2L1 R p · R1 · Sufﬁciently good agreement is obtained between the R p · R1 Rn0 approximate analysis and simulation. (13) Table 1: Circuit parmeters where VBmax is DC base voltage, Rn0 is small signal negative resistance, Ix1 , Ix2 are the crystal boundary Parameter No. 1 No. 2 current used by approximate analysis. CA CB pF 36.0 108.0 The analysis of the start-up characteristics of the RE kΩ 18.0 6.0 cascode oscillator is made and compared to the sim- RA RB kΩ 128.0 42.7 ulation. Tabel 3 shows the circuit parameters of the L1 mH 5.80 5.80 circuits. Fig. 6 shows the start-up characteristics. C1 f F 11.9 11.9 Tabe 4 shows the calculated results of start-up time. R1 Ω 14.0 14.0 Sufﬁciently good agreement is obtained between the C0 pF 2.84 2.84 approximate analysis and simulation. 4 Summary 3.3 Start-up Characteristics of Cascode The analysis methods of a crystal oscillator are pre- Crystal Oscillator sented. SPICE base method gives high resolution On the design of cellular phones, the reduction of the with short simulation time. Approximate analysis start-up time of a crystal oscillator is required for re- methods can estimate the frequency stability and duction of the power consumption. We have derived start-up time with sufﬁcient accuracy. Presented the algebraic analysis method of start-up character- methods are thought to be useful for crystal oscil- istics of the cascode crystal oscillator. lator design. 5 10 Table 3: Circuit parmeters of cascode oscillator 1 Circuit No. 2 IX [mA] 1e-1 Parameter No. 1 No. 2 1e-2 Circuit No. 1 CA pF 180 60 1e-3 CB pF 220 180 Crystal current 1e-4 CD nF 10 10 CS pF 42.29 1e-5 RA kΩ 68 20.0 1e-6 RB kΩ 62 40.0 1e-7 RC kΩ 1.8 0.5 1e-8 RD kΩ 47 15.4 1e-9 Approximation Simulation RE kΩ 5.1 1.9 1e-10 0 10 20 30 40 50 Q1 Q2 2SC1359 2SC3585 t [ms] L1 mH 18.72 27.6 C1 f F 13.54 5.607 Figure 6: Start-up characteristics of crystal current R1 Ω 11.9 14.56 calculated by approximate analysis and C0 pF 2.75 1.52 simulation. VCCC V 5.0 3.0 f MHz 10 12.8 [4] Akio Ushida, Takehiko Adachi, and Leon O. Chua,“Steady-State Analysis of Nonlinear Cir- Table 4: Comparison of start-up time estimated cuits Based on Hybrid Methods,” IEEE Trans. by approximate analysis and simulation on Circuits and Systems-1, Vol. 39, no. 8, pp.649-661, Aug. 1992. start-up time ms Error Circuit simulation analysis % [5] Takahiro Shoji and Takehiko Adachi;“An Al- No. 1 36.97 35.20 4.8 gebraic Formulation of Frequency Stability No. 2 11.14 11.42 2.5 against Supply Voltage Variation of Quartz Crystal Oscillator,” IEICE Trans. on Electron- ics, J81-C-II, no.10, pp.796-804, Oct. 1998 [6] Shoji Izumiya, Jun Asaki and Takehiko References Adachi;“Algebraic Analysis Method of Start- up Characteristics of Cascode Crystal Oscilla- [1] R. W. Rhea;“Oscillator Design and Computer tor,” IEICEC Trans on Electronics Vol.J86- Simulation,” Noble Publishing Corporation, C,No.5,pp.500-510 (2003-05) 2000 [7] Jinqin Lu, Takehiko Adachi, and Yasuo [2] Takehiko Adachi, Motoyoshi Hirose, and Ya- Tsuzuki;“Cascode Crystal Oscillator Circuit,” suo Tsuzuki;“Computer Analysis of Crystal IEICE Trans. on Electronics J71-C no.12 Osicllator,” IEICE Trans. on Electronics J68- pp.1712-1713, Dec. 1988 C, no. 7, pp. 570-577, July 1985. [8] Agilent Technology: http://eesof.tm.agilent.com/products/ [3] Takehiko Adachi and Yasuo Tsuzuki;“On Op- ads2003a welcome.html eration Characteristics and Design Consider- ations of Colpitts Crystal Oscillator Circuit,” [9] Cybernet system: IEICE(C-2), J73-C-2, no. 3, pp. 154-162, http://www.cybernet.co.jp/awr/ March 1990. 6