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 [
Essential circuit constants R2, C10, and C0 determines the resonance condition, where C0 is the parallel capacitance of the quartz crystal resonator. R2 settles the bias in the initial stage of the oscillation. C10 stores the ground potential at the activation of the Vcc voltage, inserted between the node connecting two inverters. The oscillation frequency is determined by a recharging-time constant R2 multiplied by C10. Capacitors C2 and C3 are load capacitors which is necessary for the generation of negative resistance. C5 and C6 are pass-capacitors between the bus-line and the circuit ground. C0 and C1 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 Rf = R3, R4, R5, and R6.
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.
Applying Kirchhoff’s law, the relations for Iout and Vin are found. Vin is the input voltage of IC1 and Iout is the output current of IC2.
Solving for the relation between Iout and Vin, total conductance GM is found.
Then the following relation is found. Current I2, I3 are expressed in the terms of I1.
Rearranging the expression, relation (11) is found.
Z2 is the impedance of a quartz crystal resonator (Zxt), and impedance for other components is defined as in (12). The composed impedance Zcc 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 Lcc or capacitance Ccc is determined depending on sign of reactance Xcc.
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.
GM is separated into real and imaginary parts.
Introducing (13) and (19) into Zcc, the impedance of the active circuit is found.
is found. Composed equivalent resistance Rcci and capacitance Ccci are found.
Negatively signed capacitance is converted to an active inductance by relation (21).
The denominator of negative resistance Rcci has quadratic dependence on Rcc. The maximum value of the absolute value is reached at a specific value of Rcc determined by C0s and Ccc. The following relation is fulfilled.
The active inductance appears in the vicinity of the resonance frequency, while capacitance Ccc is negative. The resonance frequency is determined by Lcc, C0S, and the sum of C0 and Cs. In this simplified form, the absolute value of negative resistance Rcci becomes infinitely large, if Ccc approaches −C0S and condition (23) is fulfilled.
At the resonance frequency determined by Lcc and C0S, the absolute value of negative resistance determines the growth of signal. The suppression of negative resistance by inductance L1 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.
Resonance frequency | Equivalent circuit constant | ||||
---|---|---|---|---|---|
L1 | C1 | R1 | C0 | Q1 | |
39.992508 kHz | 12 kH | 1.326 fF | 12.6 kΩ | 803.279 fF | 239,149 |
In
In this analysis, the terminal impedance at a - b is expressed with Rcc and Rcci. The parallel capacitance C0 and stray capacitance Cs included in the impedance Rcci. From relation (22), Rcci becomes infinitely large at Ccc = −C0s. 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, Rcc 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) [
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 R2.
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.
In
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 Ccc = −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.