Analysis of Crystal Oscillator

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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
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

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
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
              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

voltage are calculated and compared to the measure-                           20
ment.                                                                         16

                                                                 Ci [pF]
                                Rc                                            10
                    RD                                                         8
                           Q2     Output                                       4
                         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

                                                                 Ri [pF]
                         CA                                                  -200
                         CB                                                  -300

        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
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

by the obtained equivalent parameters of the ac-
                                                                                RA · RB
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
   Recently, sophisticated simulators base on the
harmonic balance technique are available. ADS[8]                                     1
                                                                                     RA · R¼B
and Micro wave office[9] are the examples of these                              1
simulators. These simulators also have the function                           VT
of phse noise analysis.
                                                                        ¢     ln ´
                                                                                                                     µ (7)
                                                                                         ´RA · R¼ µRE
                                                                                                  B      1      RA
                                                                                              R ¼     1·
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   ·
 Ri           ω 2CmC K ´ ωCI¼x V µ · R p
                   ¼                               (3)
                   A B        A T                               where,
                 1              1             1
            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

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-
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-
                                                                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.

                                                                Table 2: Comparison of frequency stability
 dCi Ci              1      1 MJC       CJC                              estimated by approximate analysis
                  ´      ·   µ
                                       VBC0 1·MJC BC0
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 · CA CB CA V JE                                No. 1          0.485         0.543             12
                           ´              V JE     µ               No. 2          0.063         0.070             11
             ¢        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%
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
                                                                                         · 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.

             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.


 Table 3: Circuit parmeters of cascode oscillator                                     1
                                                                                               Circuit No. 2

                                                                 IX [mA]
        Parameter     No. 1        No. 2
                                                                                    1e-2                       Circuit No. 1
         CA pF         180          60
         CB pF         220          180

                                                                 Crystal current
         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
         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)
                                                            [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:
 [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,         
     March 1990.


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