Start-Up Acceleration of Quartz Crystal Oscillator Using Active Inductance Double Resonance and Embedded Triggering Circuit

Low-frequency double-resonance quartz crystal oscillator was developed with active inductance circuit aiming the start-up of stable oscillation of tuning fork-type quartz crystal resonator at 32.768 kHz within 0.37 ms. The initial oscillation is triggered by a part of crystal oscillator forming a CR oscillator. The negative resistance ranges to 4 MΩ at gmf of 4.1 μA/V. In a limited frequency range, the circuit shows negative reactance Ccci = −3.4 pF equivalent to inductance Lcc = 9.8 H. The Allan standard deviation indicated 10−11 to 10−10, showing high stability comparable to general quartz crystal oscillator.


Introduction
Piezoelectric quartz crystal oscillators have widely expanded in sensing of the environmental data such as static pressure and temperature.Acceleration of the piezoelectric oscillator enables the intermittent operation of the piezoelectric sensor for power management.Engineering issue in the acceleration of the start-up of low frequency quartz crystal oscillator includes 1) triggering circuit, 2) generation of large negative resistance, and 3) linearity of the active device in large amplitude oscillation.In this work, we aim at the acceleration of the startup of low frequency, tuning fork quartz crystal resonator within several oscillation periods, which enables the intermittent operation of the sensor system.Acceleration of the start-up is studied by the gain control in the quartz crystal oscillator using a cascade circuit in the frequency region of several Mega Hertz [1]- [3].In recent works, double-resonance quartz crystal oscillator was reported for the enhancement of the frequency pulling [4] and the mode separation of the multimode quartz crystal resonator [5].Stability of the oscillation frequency is generally discussed based on the moving average of the variance determined for the discrete samples following the proto call [6]- [8].The modified Allan standard deviation for moving average of finite length data (k = 10) was employed as the measuring rule of the short range stability.

Acceleration of the Start-Up of a Quartz Crystal Oscillator
Figure 1 shows a circuit diagram of the active inductance double resonance oscillator circuit.The initial oscillation is generated by a part this oscillator acting as a CR oscillator, and after the oscillation of the quartz crystal resonator current starts, the double resonance is established between the quartz crystal and an active inductance combined with the parallel capacitance resulting in the generation of negative resistance.Essential circuit constants R 2 , C 4 , and C 0 satisfy this resonance condition, where C 0 is the parallel capacitance of the quartz crystal resonator.R 2 settles the bias in the initial stage of the oscillation.C 4 stores the ground potential at the activation of the V cc voltage, inserted between the node connecting two inverters.The oscillation frequency is determined by a recharging-time constant R 2 multiplied by C 4 .Capacitors C 2 and C 3 are load capacitors which is necessary for the generation of negative resistance.C 5 and C 6 are pass-capacitors between the bus-line and the circuit ground.C 0 and C 1 are reserved for the parallel capacitance of the resonator and the series capacitor of the motion arm.The maximum negative resistance is generated at specified value of the conductance g mf of the active circuits: CMOS (Complementary Metal Oxide Semiconductor) inverters IC 1 and IC 2 .The conductance is controlled by negative feedback resistors R f = R 3 , R 4 , R 5 , and R 6 .
. 1 Figure 2 shows simplified equivalent circuit-1.CMOS inverter IC 1 and IC 2 is replaced by two current sources controlled by the gate voltage V in and V g .Applying Kirchhoff's law, the relations for I out and V in are found.V in is the input voltage of IC 1 and I out is the output current of IC 2 .( ) 0, where 2 . 1 2 . .
Solving for the relation between I out and V in , total conductance G M is found.
( ) Then the following relation is found.Current I 2 , I 3 are expressed in the terms of I 1 .
Z 2 is the impedance of a quartz crystal resonator (Z xt ), and impedance for other components is defined as in (12).The composed impedance Z cc of the active circuit is found, substituting the impedance.From the condition for the non-zero solution of current, the oscillation condition results in (13).The impedance of the circuit is divided into resistive and reactance parts.
The equivalent resistance and the reactance of the circuit are found.Equivalent inductance L cc or capacitance C cc is determined depending on sign of reactance X cc .
Factors "a", "b", "c" and "d" are introduced for the simplicity of the expression, where factors "c" and "d" have the dimension of Ω and factors "a" and "b" are dimensionless numbers.
Introducing ( 13) and ( 19) into Z cc , the impedance of the active circuit is found.( Negatively signed capacitance is converted to an active inductance by relation (21), The active inductance appears in the vicinity of the resonance frequency, while capacitance C cc is negative.The resonance frequency is determined by L cc , C 0s , and the sum of C 0 and C s .In this simplified form, the absolute value of negative resistance R cci becomes infinitely large, if C cc approaches-C 0s and condition (23 At the resonance frequency determined by L cc and C 0s , the absolute value of negative resistance determines the growth of signal.The suppression of negative resistance by inductance L 1 establishes the stability and inhibitory action against the signal growth.Table 1 shows the equivalent circuit constant of the quartz crystal resonator. Figure 5 compares the absolute value of negative resistance R cci for parameters L cc and C cc .Figure 6 shows the absolute value of negative resistance R cc as functions of frequency and g mf .The enhancement of negative resistance and the correlation with active inductance is explained in the following part.Resonance occurs in the inductive region of the motion arm, while the reactance of the active circuit is capacitive.The maximum absolute value of negative resistance approximately 1.3 MΩ is obtained with infinitely large C cc .The maximum value is limited at 0.8 MΩ for C cc = 5 pF, a practical value.Larger negative resistance is generated in the region where the active circuit is inductive.When the circuit reactance is inductive, the maximum value increases.The maximum absolute value of negative resistance approximately 13 MΩ for C cci = −2 pF equivalent to active   inductance L cx = 12 H.In Figure 6(a), parameter g mf is selected at 4 μA/V.At the lower gain, g mf = 2 μA/V, the active inductance appears in the lower frequency region and disappears in the higher frequency region.For example, it appears at 10 kHz and disappears at 27 kHz.The frequency limit is 55 kHz for g mf = 4 μA/V, and 110 kHz for g mf = 8 μA/V.In the lower frequency region, at approximately 9 kHz, the active inductance disappears.Figure 6(b) shows the dependence on conductance g mf .At frequency of 32.768 kHz, the active inductance appears for g mf = 2.4 μA/V.The limit varies depending on the oscillation frequency.The frequency limit is 20 kHz for g mf = 1.5 μA/V, 40 kHz for g mf = 3 μA/V and 110 kHz for g mf =8 μA/V.The negative resistance and the active inductance appear from the low frequency side, and the resonance condition with CR oscillation is established before the motion arm appears.Figure 8 shows the absolute value of negative resistance for different resonance frequency.This result tells that the active inductance is generated with certain value of gain in a narrow range corresponding to the required frequency.This result indicates that high gain is not necessarily for better performance.
This result suggests a design principle of the circuit: Higher resonance frequency needs higher g mf .In this analysis, we take a look at the circuit impedance from the resonator terminal.The parallel capacitance C 0 and stray capacitance C s are included in the impedance of the active circuit.Thus, the relation between R cci and R cc is presented, As a part of the final solution, the result that R cci becomes infinitely large at C cc = −C 0s , must be interpreted carefully in the context of the actual circuit design.The minimum idea given here is that the active inductance can generate large negative resistance compared to the capacitive region.Actually, R cc is determined by number of circuit constants and angular frequency of the oscillation, and the strength of the oscillation is limited within the linear region of the active circuit.

Modelling of the Start-Up of Initial Oscillation
The crystal current through the motion arm is not generated in the initial stage of the oscillation.In another  expression, this branch does not exists in the circuit.Because of high Q, the start-up needs reasonable acceleration system.Figure 9 shows simplified circuit diagrams of the oscillation Mode-1 and Mode-2.Before the establishment of the resonance oscillation, parallel capacitance C 0 is the existing circuit component and the motion arm is disconnected.Based on the result of analysis, the active circuit is indicated with C cci and the composed equivalent negative resistance R cci .The oscillation frequency is determined by the equivalent reactance of the active circuit and C 0 .The motion arm appears after the certain growth of the crystal current.This figure also explains the relation between R cc and R cci .The circuit in Figure 9(b) is closed with the motion arm, and oscillation current i(t) is excited in the closed loop.V x and V c may have different initial values, frequency and polarity depending on the switching sequence.
The start-up mode of the oscillator depends on the rise of V cc .When the bias current increases the CR oscillation as in Mode-1 starts before the establishment of the crystal current.The oscillation frequency of Mode-1 is determined by R 2 multiplied by the composed capacitance.When the quartz crystal resonator is activated sufficiently, the motion arm appears in the circuit, as in Model-2.Computer simulation was carried out using LTspice for Windows (Linear Technology Corporation, 1630 McCarthy Blvd., Milpitas, CA, USA) [9]. Figure 10 shows the transient excitation of the crystal current with matched frequency setting: the oscillation frequency of the CR oscillator is slightly higher than the resonance frequency.The crystal current in the motion arm grows up faster and the oscillation frequency of the entire oscillator circuit is locked to the resonance frequency.Figure 12 shows the circuit diagram for the computer simulation.Here, CMOS inverter IC 1 and IC 2 are replaced with pairs of complementary MOSFETs (Metal Oxide Semiconductor Field Effect Transistor).
The circuit constants of the motion arm are not corresponding to the values assigned in the analysis and experiment.Also, the delayed connection of the motion arm is not considered in this simulation.
When the motion arm is removed, this circuit forms a CR oscillator.The oscillation frequency is determined by the reactance of the parallel capacitance of the quartz resonator and feedback resistor R 2 .Figure 13 shows a typical wave form of the CR oscillator and FFT spectral analysis, for the matched frequency peak in the vicinity of the resonance frequency of the motion arm.The CR oscillation peak appears in the vicinity of the quartz resonance frequency.

Experimental Result and Discussions
The start-up and the stability of the stable oscillation of the double resonance oscillator is experimentally evaluated.The stability of the oscillation frequency is analyzed with 53,230 A universal frequency counter (Agilent Technologies, Santa Clara, Ca, USA) synchronized with external rubidium oscillator with long period stability < 2 × 10 −11 /month and short period stability < 1 × 10 −11 /s.

Start-Up of the Double Resonance Quartz Crystal Oscillator
The proposed quartz oscillator circuit is activated with V cc and the minimum start-up time marked 0.37 ms, as shown in Figure 14.The start-up time and the CR oscillation frequency are shown as functions of the resistance R 2 .In the proposed oscillator circuit, the output of inverter IC 2 is positively fed back to the input of inverter IC 1 through the quartz crystal resonator.Replacing the quartz resonator with a capacitor equivalent to the parallel capacitance C 0 , the circuit exhibits CR oscillation and this frequency is determined by the time constant: R 2 multiplied by C 4 .In the case of R 2 = 2.3 MΩ, the frequency of the CR oscillation f CR is approximately equal to the resonance frequency.The quartz crystal oscillation is triggered by the CR oscillation and the start-up time shows its minimum.The entire oscillator circuit is synchronized after short transient, showing the rapid start demonstrated with typical waveform: the initiation of the oscillation within 0.37 ms, several oscillation cycle.Figure 15 shows a typical example for the accelerated start-up of the quartz crystal oscillator.

Stability of the Double Resonance Quartz Crystal Oscillator
In this experiment, modified two-sample Allan standard deviation is employed as a measure of the short-time frequency stability.This protocol is defined in (25), following IEEE Standard 1139 [9].The frequency of oscillator circuit f k is the discrete sample of oscillation frequency.τ is the gate time and n is the sequential number of samples.Dimensionless parameter is defined from frequency deviation normalized by the moving average over 10 sequential samples.Figure 16 shows the modified Allan standard deviation showing the short range stability of the order of 10 −10 to 10 −11 .The quartz crystal oscillator was isolated in a shield box with isolated DC power supply.This result satisfies the industrial requirement for the standard quartz sensor.
( ) Table 2 shows the stability of the quartz crystal oscillation in this experiment.This result shows Allan standard

Conclusion
Environmental sensing awaits solutions to reduce the electric-power, in continuous monitoring.The quick start of quartz crystal oscillators allows excitation of stationary oscillation established after short transient meeting the request for the power management in the environmental sensing such as the pressure and temperature.In this work, we resolved the engineering issues for the rapid start-up: 1) Large negative resistance; 2) Low distortion and linearity; 3) Triggering circuit.The start-up of a low frequency quartz oscillator is triggered with a CR oscillator and transferred to a quartz crystal oscillator.The maximum negative resistance ranges to 4 MΩ at specified gain of the active CMOS inverter circuit g mf = 4.1 μA/V.The composed reactance of the active circuit C cci shows negative value, −3.4 pF which acts as inductance of 9.8 H and generates large negative resistance.Rapid start-up of the oscillation was established by the energy transfer by the initial CR oscillation of the active circuit and the minimum start-up time was realized.The oscillation condition was examined by the analysis and the start-up in the initial stage was examined by the computer simulation and experiment.The result shows corresponding dependence of the start-up time on circuit parameter R 2 .The stability performance of the double-resonance oscillator showed that short range stability of 10 −11 satisfies the industrial requirement for the standard quartz oscillator circuit.

Figure 3
Figure3shows simplified diagram of equivalent circuit-2 of the oscillator.The active circuit is indicated with R cc and reactance C cc or L cc depending on the sign.The resonator consists of parallel capacitance C 0 and the motion arm, L 1 , C 1 , and R 1 , the equivalent series inductor, capacitor, and resistor respectively.C S is a stray capacitance.Calculating the parallel composition of C 0 and C s with the active circuit, equivalent circuit-3 in Figure4is found.Composed equivalent resistance R cci and capacitance C cci are found.

Figure 4 .
Figure 4. Equivalent circuit-3.The denominator of negative resistance R cci has quadratic dependence on R cc .The maximum value of the absolute value is reached at a specific value of R cc determined by C 0s and C cc .The following relation is fulfilled.0 max 0 0 0 1 1 , at 1 .2 1

Figure 5 .
Figure 5.Comparison of the absolute values of negative resistance for L cc and C cc , as functions of the negative resistance R cc .Circuit constants: C 0 = 1.14 pF; C S = 1 pF.

Figure 7
Figure7shows the dependence of the absolute value of negative resistance R cci on frequency and g mf .Figure 7(a) shows the frequency dependence of R cci and reactance L cci , C cci .The maximum value of negative resistance R cci ranges to 4 MΩ.This result is obtained in the case of inductive reactance L cci = 9.8 H.The composed reactance is capacitive C cci = 3.4 pF. Figure 7(b) shows the dependence of the absolute value of negative resistance R cci and reactance C cci on g mf .For g mf = 4.1 μA/V, the absolute value of negative resistance is 4 MΩ.The composed circuit reactance C cci = −3.4pF is equivalent to L cc = 9.8 H.Figure8shows the absolute value of negative resistance for different resonance frequency.This result tells that the active inductance is generated with certain value of gain in a narrow range corresponding to the required frequency.This result indicates that high gain is not necessarily for better performance.This result suggests a design principle of the circuit: Higher resonance frequency needs higher g mf .In this analysis, we take a look at the circuit impedance from the resonator terminal.The parallel capacitance C 0 and stray capacitance C s are included in the impedance of the active circuit.Thus, the relation between R cci and R cc is presented, As a part of the final solution, the result that R cci becomes infinitely large at C cc = −C 0s , must be interpreted carefully in the context of the actual circuit design.The minimum idea given here is that the active inductance can generate large negative resistance compared to the capacitive region.Actually, R cc is determined by number of circuit constants and angular frequency of the oscillation, and the strength of the oscillation is limited within the linear region of the active circuit.

Figure 10 .
Figure 10.Initial stage with frequency match conditions.Upper track: Current I(L 1 ) fowing through the series inductance L 1 .Lower track: Output voltage.

Figure 11
Figure11the initial stage of the mismatched case.If the frequency of the CR oscillation is too high, and the oscillation apparently starts at different frequency.The oscillation frequency abruptly locked to the crystal resonance frequency by the growth of the crystal current.The faster growth of crystal resonance occurs with frequency mated pumping by the CR oscillation.The similar result of the matched and mismatched case was observed in the experiment.Figure12shows the circuit diagram for the computer simulation.Here, CMOS inverter IC 1 and IC 2 are replaced with pairs of complementary MOSFETs (Metal Oxide Semiconductor Field Effect Transistor).The circuit constants of the motion arm are not corresponding to the values assigned in the analysis and experiment.Also, the delayed connection of the motion arm is not considered in this simulation.When the motion arm is removed, this circuit forms a CR oscillator.The oscillation frequency is determined by the reactance of the parallel capacitance of the quartz resonator and feedback resistor R 2 .Figure13shows a typical wave form of the CR oscillator and FFT spectral analysis, for the matched frequency peak in the vicinity of the resonance frequency of the motion arm.The CR oscillation peak appears in the vicinity of the quartz resonance frequency.

Figure 11 .
Figure 11.Comparison of growth of the crystal current and output with a mismatch conditions at R 2 = 0.8 MΩ.Upper track: Current I(L 1 ) flowing through the series inductance L 1 ; Lower track: Output voltage.

Figure 12 .
Figure 12.Equivalent circuit for the computer simulation.Courtesy of LTspice, Linear technology corporation.

Figure 13 .
Figure 13.Typical example for the waveform of the oscillator and the FFT spectral analysis R 2 = 3.0 MΩ.

Figure 14 .
Figure 14.Experimental comparison of the start-up time and the frequency of the CR oscillation.

Figure 15 .
Figure 15. of the double-resonance quartz crystal oscillator.Horizontal scale: 100 μs/div.Vertical scale 200 mV/div.Circuit constants: C 2 , C 3 and C 4 = 10 pF.Start-up of the stable oscillation 0.37 ms after the activation of V cc .deviation of 10 −11 satisfy the requirement for the standard sensing.Probably, for the further improvement of the stability, the improvement of the Q-value of the resonator is necessary.

Figure 16 .
Figure 16.Short range stability of the double-resonance quartz crystal oscillator.

Table 1 .
Equivalent circuit constant of the quartz resonator.

Table 2 .
Stability: Averaged Allan standard deviation of the proposed circuit.