^{1}

^{2}

^{1}

^{*}

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
*g*
_{mf} of 4.1 μA/V. In a limited frequency range, the circuit shows negative reactance
*C*
_{cci} = -3.4 pF equivalent to inductance
*L*
_{cc} = 9.8 H. The Allan standard deviation indicated 10
^{-11} to 10
^{-10}, showing high stability comparable to general quartz crystal oscillator.

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 start- up 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 [

_{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} 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}.

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}.

Rearranging the expression, relation (11) is found.

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.

G_{M} is separated into real and imaginary parts.

Introducing (13) and (19) into Z_{cc}, the impedance of the active circuit is found.

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 _{cci} and capacitance C_{cci} are found.

Negatively signed capacitance is converted to an active inductance by relation (21),

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.

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_{0}_{s}, 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_{0}_{s} and condition (23) is fulfilled.

At the resonance frequency determined by L_{cc} and C_{0}_{s}, 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.

_{cci} for parameters L_{cc} and C_{cc}. _{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

Quartz resonator | τ(ms) | ||||
---|---|---|---|---|---|

L_{1} | C_{1} | R_{1} | C_{0} | Q_{1} | |

Tuning fork-type (32.768 kHz) | 11,797 H | 2 fF | 47.6 kΩ | 1.14 pF | 51,023 |

inductance L_{cx} = 12 H.

In _{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. _{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.

_{cci} on frequency and g_{mf}. _{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. _{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} = −3.4 pF is equivalent to L_{cc} = 9.8 H.

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_{c}_{c} 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.

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. _{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 _{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 LT- spice for Windows (Linear Technology Corporation, 1630 McCarthy Blvd., Milpitas, CA, USA) [

_{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}.

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.

The proposed quartz oscillator circuit is activated with V_{cc} and the minimum start-up time marked 0.37 ms, as shown in _{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.

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 [_{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. ^{−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.

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.

Quartz oscillator circuit | τ(ms) | |||
---|---|---|---|---|

1 | 10 | 100 | 1000 | |

Double-Resonance Quartz Oscillator | 2.7 × 10^{−7} | 2.4 × 10^{−8} | 2.7 × 10^{−10} | 6.8 × 10^{−11} |

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-re- sonance oscillator showed that short range stability of 10^{−11} satisfies the industrial requirement for the standard quartz oscillator circuit.

The authors acknowledge Ms Ruzaini Izyan binti Ruslan and Mr. Satoshi Goto for their collaboration in the early stage of this experiment. This work was supported in part by JST A-STEP Contract No. AS251Z01794J.