^{1}

^{*}

^{2}

^{1}

Low-frequency double-resonance quartz crystal oscillator circuit was developed with active inductance aiming the quick start-up in the intermittent operation on the sensor circuit and DC isolation using a Q-MEMS sensing crystal HTS-206. Allan standard deviation indicated 5 × 10
^{–12}, showing short range stability of the sensor circuit sufficient for the ubiquitous environmental sen sor network.

Environmental sensing awaits solutions to reduce the electric-power in monitoring under the limitation of the power source. The ubiquitous sensor network is realized with varieties of sensor circuit and a wireless network. The temperature measurement in the environmental sensing is realized by several methods: thermistors, platinum wire or sheet resistor and semiconductor sensor devices. The temperature is measured as the amplitude of low level DC or modulated signals and faces the difficulty which arises from the drift. Q-MEMS crystal temperature sensor can realize high resolution in the sensing of environmental temperature. Direct digital temperature measurements have been developed by several research groups, usint crystal cut LT, SC-cut or equivalent cut [

_{1} and IC_{2} are CMOS inverter, where CMOS is Complementary Metal Oxide Semiconductor. HTS-206 Q-MEMS temperature sensing crystal is connected through coupling capacitors which are formed between metallic sheets, 20 mm square in dimension attached on both sides of a Pyrex glass plate. The value is approximately C_{16}, C_{17} = 18 pF.

Essential circuit constants R_{2}, C_{10}, and C_{0} determines the 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_{10} 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_{10}. 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 conductance is controlled by negative feedback resistors R_{f} = R_{3}, R_{4}, R_{5}, and R_{6}.

The problem is if the active inductance can generate the negative resistance, and if the negative resistance is large enough to realize the short start-up time. Practical question is the shift of the resonance frequency of the crystal sensor by the series capacitors. _{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.

_{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

is found. Composed equivalent resistance R_{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_{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) is fulfilled.

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. Temperature sensing crystal HTS-206 is a tuning-fork type resonator, 2 mm in diameter and 6 mm in length of the exterior size, produced for low power oscillation of 0.1 μW typically.

Temperature dependence of the crystal sensor is explained in the experimental part. _{cc} and R_{cci} as functions of frequency. In _{mf} is selected at 4.1 μA/V. The active inductance disappears at 55 kHz for g_{mf} = 4 μA/V, and 110 kHz for g_{mf} = 8 μA/V. The frequency limit is 40 kHz for g_{mf} = 3 μA/V and 110 kHz for g_{mf} = 8 μA/V. Larger gain is necessary for the negative resistance and the active inductance.

Resonance frequency | Equivalent circuit constant | ||||
---|---|---|---|---|---|

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

39.992508 kHz | 12 kH | 1.326 fF | 12.6 kΩ | 803.279 fF | 239,149 |

In _{cci} and the active inductance is compared as functions of on frequency. The parameter g_{mf} = 4.1 μA/V was optimized for 32.768 kHz realizes the maximum value of negative resistance R_{cci} approximately of 150 kΩ and the reactance: C_{cci} = −3 to −4 pF.

_{cc} = 2 × 10^{3} kΩ and the reactance is inductive C_{cc} = −0.6 pF at g_{mf} = 8 μA/V.

In this analysis, the terminal impedance at a - b is expressed with R_{cc} and R_{cci}. The parallel capacitance C_{0} and stray capacitance C_{s} included in the impedance R_{cci}. From relation (22), R_{cci} becomes infinitely large at C_{cc} = −C_{0s}. This result must be interpreted carefully, because the optimum condition is not realized in the context of

the actual circuit design. The idea given in this result is that the active inductance can generate large negative resistance compared to the capacitive region. Actually, R_{cc} is determined under the limitation of the circuit constants and the oscillation frequency. The strength of the oscillation is limited within the linear region of the active circuit.

The curve indicated as 32.768 kHz shows the result calculated using the equivalent circuit constant of a time- base quartz resonator analyzed in Ref. 6. Comparing the dependence on frequency and gain, larger gain is needed for the appropriate design of the active inductance and negative resistance, when the resonance frequency is higher.

Computer simulation was carried out using LTspice IV 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). Because original Q is too high for the stable simulation, the motional capacitance and inductance are scaled with the resonance frequency fixed. The motional components do not correspond to the values assigned

in the analysis and experiment, neither the delayed connection of the motion arm is considered. 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 stability of the stable oscillation of the double resonance oscillator is evaluated experimentally. The stability of the oscillation frequency is analyzed with 53230A 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.

_{k} is the discrete sample of oscillation frequency. τ is the gate time and k is the sequential number of samples. Dimensionless parameter is defined from frequency deviation normalized by the moving average of finite length data over 10 sequential samples [^{−12} indicates sufficiently high stability when the sensor is isolated in a constant temperature vessel. The Allan standard deviation shows increase in the range from 1000 to 10,000 ms indicating the increase in the environmental drift.

In _{16}, C_{17} = 18 pF. Once the regression curve is determined, temperature can be calculated from the oscillation frequency. It is necessary to calibrate the temperature dependence curve must with different standard, because the temperature dependence varies depending on the value of the coupling capacitor.

Environmental sensing awaits solutions to reduce the electric-power in monitoring under the limitation of the power source. The quick start of the crystal sensor circuit allows intermittent excitation of the sensor system meeting the request for the power management in the environmental sensing. In this work, active inductance double resonance circuit resolved the engineering issues for the quick start-up: 1) Large negative resistance; 2) Low distortion and linearity; 3) Triggering circuit. The quartz crystal oscillator is triggered with a CR oscillator, and transferred to a stable excitation within several period. The maximum negative resistance ranges to 2 MΩ at specified gain of the active CMOS inverter circuit. The composed reactance of the active circuit negative capacitance C_{cc} = −0.6 pF. Simulation showed the rapid start-up of the oscillation by the energy transfer by the initial CR oscillation. The oscillation condition was examined by the analysis, the start-up in the computer simulation and examined by the experiment. The stability of the double-resonance oscillator showed short range stability of 5 × 10^{−12} which satisfied the industrial requirement for the resolution of the standard quartz crystal sensor.

The authors acknowledge Mr. Ruan Zheng 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.