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A multimode-quartz-crystal oscillator was developed to excite stable dual-mode resonance at different frequencies: The oscillation of the 3rd harmonic resonance of the principle C-mode and an additional resonance B-mode of SC-cut crystal. Harmonic combinations of the 3rd and fundamental mode of B-mode with the 3rd harmonics of C-mode are demonstrated. The measurement of the temperature dependence of the oscillation frequency is demonstrated along with the stability determined by root Allan variance. Dependence on the open conductance of the active circuit and the dependence on the coupling capacitors are discussed.

Piezoelectric sensing is increasingly attracting attentions, employing frequency-based measurement depending on the surface loading, mass, and a variety of force acting on the piezoelectric resonator. The quartz crystal resonator determines a stable electrical resonance frequency corresponding to the mechanical resonance frequency. The quartz crystal resonator shows a variety of resonance modes. For example, in an AT-cut quartz crystal, the harmonic combination of the different temperature characteristics of the fundamental, 3rd and 5th harmonic resonances of provides a high precision thermometric system. In the past history, dual-mode oscillation was attempted by several groups and is reviewed in References [

Local resonance circuits consist of L_{02} and C_{0x}, L_{12} and C_{1x}. C_{0y} and C_{1y} are inserted to the connection to a crystal resonator. Feedback resistors R_{02}, R_{03}, R_{12}, and R_{13} are inserted into the power line between CMOS inverter integrated circuits IC_{1} and IC_{2} and ground aimed at suppression of current and gain control. C_{05}, C_{06}, C_{15}, and C_{16} are pass capacitors. C_{02}, C_{03}, C_{12}, and C_{13} are necessary for the generation of negative resistance. This circuit synthesizes LC oscillation mode determined by L_{02} and combined capacitors C_{0x}, C_{02}, C_{03}, and C_{0y}, and LC oscillation mode determined by L_{12} and combined capacitors C_{1x}, C_{12}, C_{13}, and C_{1y}. The oscillation frequency is settled in the vicinity of the quartz crystal oscillation frequency. Coupling capacitors C_{04} and C_{14} are omitted in the following analysis. Coupling capacitors C_{0y} and C_{1y} are connected to the quartz crystal resonator. Appropriate choice of the local resonance circuit and the coupling capacitors are necessary to realize stable dual-mode quartz crystal oscillation. Two oscillator circuits OSC_{1} and OSC_{2} are connected to the resonator by capacitors C_{0y} and C_{1y}. The SC-cut quartz crystal resonator shows C-mode a primary oscillation mode with good stability, and B-mode, a side mode, linearly proportional to the change of ambient temperature. CMOS Inverters IC_{1} and IC_{2} are replaced with a constant current source. In the equivalent circuit on the left side is OSC_{1} and on the right side is OSC_{2}. I_{out1} and I_{out2} are output current, and V_{in1} and V_{in2} are input voltage of each inverter: r_{02} and r_{12} are internal resistance of inductance L_{02} and L_{12}, respectively. C_{0s} and C_{1s} are stray capacitance parallel to capacitance C_{0x} and C_{1s}. Impedance Z_{01} consists of L_{02}, r_{02}, C_{0x}, and C_{0s}, similarly, impedance of the circuit Z_{11} which consists of L_{12}, r_{12}, C_{1x}, and C_{1s}, and Z_{xt} for the parallel circuit of a quartz resonator and C_{p}, reduced impedance is found. Capacitance of leads and connected lines on the circuit board are modeled by C_{p}. Applying Kirchhoff’s law, homogeneous Equation (1) is found.

The open conductance G_{m} is adjusted by negative feedback resistors R_{02}, R_{03}, R_{12}, and R_{13}. Equivalent trans- conductance is defined by the open conductance and a feedback resistance, as in (2).

The open conductance of the CMOS inverter is expressed with drain current I_{d} and conductance coefficient K defined in the terms of electron and hole mobility μ and the unit gap capacitance C_{g}. W/L is the ratio of width over length of the gate:

Feedback resistance:

Solving the determinant of the coefficient matrix, relation (3) is found. The second term on the left side of this relation indicates the impedance of OSC_{1}, and the third term indicates the impedance of OSC_{2}. The impedance of individual oscillator is connected in parallel to Z_{xt}.

_{1} as Z_{1} and the impedance of OSC_{2} as Z_{2}, R_{1}, R_{2}, C_{1}, and C_{2} are equivalent resistance and capacitance of the imped-

ance. R_{1}, C_{1}, R_{a}, L_{a}, C_{0xs}, ω_{0x}, ω_{0s}, and ω_{1} are found in (4). R_{2}, C_{2}, R_{b}, L_{b}, C_{1xs}, ω_{1x}, ω_{1s}, and ω_{2} are found in (5).

Similarly,

where

Including the equivalent circuit of Z_{xt}, equivalent circuit-3 is found in _{1}, C_{1}, and R_{1} are the motion arm and C_{0} is the parallel capacitance. C_{p} is stray capacitance on the circuit board connected to the reso- nator. Solving for Z_{xt}, equivalent resistance R_{ci} and equivalent reactance C_{ci} of parallel connection of Z_{1} and Z_{2}

are found.

where

_{0p} consists of C_{0} and C_{p}, and C_{cci} consists of C_{ci} and C_{z}. The whole circuit consists of elements C_{0}, L_{1}, C_{1}, and R_{1} representing the equivalent circuit of the selected mode of the SC-cut quartz crystal resonator (values are in

Including C_{0}, C_{p} and C_{z} into C_{ci} and R_{ci} composed equivalent impedance C_{cci} and R_{cci} are found, as in (7).

Supposing i(ω) in the closed circuit of this diagram, the resistance condition is found from the matching condition for the real part where the real part is equal to zero. Oscillation frequency is determined from the matching condition of the capacitance, where the imaginary part is equal to zero.

Oscillation conditions are given as the resistance condition and frequency condition, in the following forms (8) and (9), respectively:

Mode | Equivalent Circuit Constant | |||||
---|---|---|---|---|---|---|

C_{0} (pF) | L_{1} (mH) | C_{1} (fF) | R_{1} (Ω) | f_{1} (MHz) | f_{r} (MHz) | |

C-mode, 3rd harmonics | 3.431 | 1297 | 0.1954 | 70.5 | 9.997418 | 9.999807 |

B-mode 3rd harmonics | 3.244 | 1757 | 0.12 | 264 | 10.96083 | 10.952481 |

B-mode fundamental | 3.899 | 495.3 | 3.710 | 61.8 | 3.712778 | 3.712836 |

Inequality in condition (8) indicates that the negative resistance larger than damping is necessary at the beginning of oscillation. The series resonance frequency f_{1} is determined by L_{1} and C_{1} determined in relation (10), where ω_{1} is angular frequency and f_{r} is the series resonance frequency.

The resistance condition is estimated for the oscillation of each mode from _{1c} and R_{1b }determined by the impedance analyzer. Relation (11) shows resistance conditions for the 3rd harmonic resonance modes. The frequency condition shows the resonance frequency determined by L_{1c} and C_{1c }or L_{1b} and C_{1b}. C_{1c} and C_{1b} are extremely smaller than C_{cci}.

In the case of the fundamental resonance of B-mode, the series resistance is reduced to 61.77Ω. Negative resistance of the active circuit is indicated by the absolute value. In the physical meaning, this value balances with the damping; this value is expressed in the following form:

_{0x} and C_{1x} combined with L_{02} and L_{12}, which determine the oscillation frequency of the individual oscillation mode.

The frequency dependence of negative resistance is tailored considering the influence of resonator capacitors C_{0x} and C_{1x}. The oscillation condition of the 3rd harmonics of C-mode is fulfilled at C_{1x} = 20.5 pF, where the oscillation frequency is close to the series resonance frequency. The maximum absolute value of negative resistance is approximately 1.99 kΩ, at 9.992 MHz. The frequency range varies from 0.28 to 0.3 MHz. Similarly, the oscillation condition of the 3rd harmonics B-mode is fulfilled at C_{0x} = 13.5 pF, close to the series resonance frequency f_{1b}. The maximum absolute value of negative resistance is approximately 1.9 kΩ, at 10.9 MHz. The oscillation condition is fulfilled from 10.84 to 10.96 MHz. The frequency range varies from 0.11 to 0.12 MHz. In the combination of the fundamental oscillation of B-mode and the 3rd harmonic resonance of C-mode, the dual mode oscillation is realized at C_{0x} = 31 pF and C_{1x} = 20.5 pF.

In the combination of the 3rd harmonic resonance modes, the multimode-oscillation is observed provided that the oscillation condition is fulfilled simultaneously. In the stable oscillation, gain G_{1M} decreases compared with

the initial phase. At G_{1M} = 1 mA/V, the damping resistance becomes larger than negative resistance. The maximum absolute value of negative resistance is approximately equal to 2 kΩ and the damping resistance of the 3rd harmonic resonance of B-mode is 264 Ω. The negative resistance of the oscillator is shown for various values of gain G_{1M}, in the vicinity of 10.9 MHz and 10 MHz.

In _{1M} = 1 mA/V, the negative resistance becomes smaller than R_{1b} then the oscillation condition is no longer fulfilled, while the oscillation condition for C-mode still holds. Transition of oscillation mode occurs from dual-mode oscillation to the single oscillation. _{2M}. G_{2M} is settled to 5 mA/V initially, and the gain of the counterpart oscillator B-mode oscillator G_{1M} is settled to 5 mA/V. The maximum absolute value of negative resistance is approximately 2 kΩ. Generally, the transconductance G_{M} of the oscillator circuit decreases in the magnitude on the growth of signals, and the frequency dependence of negative resistance varies depending on G_{M}. In the case where G_{1M} of OSC_{1} is equal to 0 mA/V, negative resistance of OSC_{2} shows wide-frequency characteristics and decreases in the frequency according to the increase in the magnitude of G_{1M}. Negative resistance of OSC_{1} shows an abrupt change in the vicinity of G_{1M} = 1 mA/V. Negative resistance of OSC_{2} shows corresponding change but the oscillation condition is fulfilled. Similar tendency is observed in the dependence of negative resistance of OSC_{1} according to the decrease in conductance G_{2M} of OSC_{2}.

The gain at the initial stage of oscillation is set at 5 mA/V, along with the growth of the signal, B-mode shows a maximum value of negative resistance 4.5 kΩ at G_{1M} of approximately 2 mA/V, then steeply decreases. In _{1M} approximately 1.8 mA/V. As the negative resistance of C- mode is larger than 1 kΩ, a maximum value 3.4 kΩ is observed at G_{1M} approximately 0.5 mA/V, the oscillation condition is fulfilled for this mode, and the oscillation mode transfers to C-mode and B-mode becomes incapable. Similarly in _{2M} approximately 1.0 mA/V. As the negative resistance of B-mode is larger than 0.7 kΩ, a maximum 4.7 kΩ at G_{1M} of 1.3 mA/V, the oscillation mode transfers to the single mode oscillation of B-mode when the condition of C-mode becomes unfulfilled. When the 3rd harmonic resonance of B-mode is selected, the resistance is 264 Ω, and the oscillation region becomes narrower, because larger negative resistance is needed due to the dependence on G_{1M}. Because the dependence on G_{2M} is fulfilled, the oscillation region is limited by the the B-mode oscillator.

_{1M} and G_{2M} are settled to 5 mA/V. When the gain G_{1M} decreases to 2.6 mA/V, negative resistance for B-mode shows 4.8 kΩ. Then it shows a maximum value 0 Ω at G_{1M} = 2.1 mA/V. Similarly, C-mode shows 2.9 kΩ, at G_{2M} = 1.8 mA/V. Then, it shows a maximum 0 Ω at G_{1M} = 1.15 mA/V. The maximum value of negative resistance is located at a certain value of gain, which is not necessarily large. In another view of the oscillation conditions, quartz crystal oscillation is stabilized at minimum negative resistance. The minimum value of negative resistance |R_{cci}|_{min} is 264 Ω for B-mode and 70.5 Ω for C-mode, from _{cci}|_{min }larger than

70.5 Ω the oscillation condition of C-mode is fulfilled before the oscillation condition of B-mode is achieved, if negative resistance is smaller than 264 Ω. In this case, the oscillation mode transfers to the single oscillation of C-mode.

_{0y} and C_{1y}.

In _{0y} is

fixed at 10 pF. The case of C_{0y} = 0 pF means that the oscillator is disconnected from the quartz resonator and only OSC_{1} is connected to the resonator. The oscillation condition for B-mode (OSC_{1}) is fulfilled in a narrow region in the vicinity of the resonance frequency, in the case where C_{1y} = 30 pF:

In _{1y} is fixed at 10 pF. The case C_{0y} = 0 pF means that OSC_{1} (B-mode) is disconnected and only OSC_{2} (C-mode) is connected with the quartz resonator. The oscillation condition for C-mode is fulfilled from 9.7 to 10.1 MHz. For C_{0y} = 10 pF, the negative resistance of OSC_{1} and OSC_{2} is approximately equal to 2 kΩ. The oscillation con-

dition for C-mode is fulfilled from 9.8 to 10.1 MHz and for B-mode from 10.8 to 10.9 MHz. If C_{0y} is set to 30 pF, the oscillation condition for OSC_{2} is fulfilled, but the negative resistance for OSC_{1} is lower than the critical value. _{2}. C_{1y} is varied while the counterpart C_{0y} is fixed at 10 pF. The peak for the OSC_{2} steps out of the resonance frequency region at C_{1y} equal to or larger than 30 pF.

_{0y} and C_{1y}, where C_{0y} either C_{1y} is variable and counterpart is fixed at 10 pF. C_{1y} = 10 pF, the oscillation condition of B-mode is fulfilled from 3.56 to 3.8 MHz, and the oscillation condition of C-mode is fulfilled from 9.8 to 10.1 MHz. The relation between negative resistance |R_{cci}| and coupling capacitor C_{0y} shows that negative resistance of C-mode oscillator fulfills the oscillation condition for wide range of C_{0y}.

On the other hand, negative resistance of the B-mode oscillator fulfills the oscillation condition only in a narrower region from 6 to 18 pF.

The oscillation condition of C-mode is fulfilled, at C_{0y} and C_{1y} from 2 to 48 pF, and stable oscillation of C-mode is available. The oscillation condition of B-mode is fulfilled only at narrower region of C_{0y} and C_{1y} from 6 to 13 pF. Stable dual mode resonance oscillation is available in this region. _{0y} and C_{1y}. The value of connecting capacitors C_{0y} and C_{1y} is a parameter that enables simultaneous dual mode resonance oscillation. In summary, the best coupling capacitances are C_{0y} = 10 pF and C_{1y} = 10 pF. If C_{0y} is larger than 30 pF, the oscillation condition for OSC_{1} is not fulfilled, while the oscillation condition for OSC_{2} is fulfilled from 9.8 to 10.2 MHz. This discussion on the choice of coupling capacitors must be reviewed in the case of the lower limit of the transconductance.

The short-time stability of the C-mode oscillation is compared. The bottom level of the σ versus τ dependence is employed as the measure of the stability, where σ is the root Allan variance defined as (12) and τ is the sampling interval (gate time) [

f_{k} is the moving average of 10 sequential of frequency data. τ is the sampling time interval. M is the number of samples per measurement. The average of frequency is calculated over the gate time. Measurement was carried out by universal frequency counters Agilent 53230A (Agilent Technologies, Santa Clara, Ca, USA) synchronized with a rubidium oscillator.

The stability of the single C-mode oscillation is compared with the simultaneous multimode oscillation of different harmonic combinations. In this experiment, the measure of the stability is the bottom level of the σ versus τ dependence, where σ is the root Allan variance defined as (12) and τ is the sampling interval (gate time).

_{y}(τ) < 10^{−10} at τ of 100 - 1000 ms and the 3rd harmonics of B-mode σ_{y}(τ) < 10^{−10} for the gate time below 500 ms. _{y}(τ) < 10^{−10} for τ from 20 to 1000 ms, and the fundamental B-mode σ_{y}(τ) < 10^{−10} for τ from 20 to 1000 ms, indicating high stability of the simultaneous multimode oscillation comparable to the single mode oscillation.

Temperature dependence of the oscillation frequency was measured in the simultaneous oscillation. Allan standard deviation was observed to avoid spontaneous mode change of additional resonance mode.

The regression curve of C-mode shows a cubic function with an inflection point at 85.7˚C (Designed value). The frequency drift of C-mode is 0.32 × 10^{−6} per degree Celsius. This result suggests that the stability of the oscillator remains at the level of typical quartz crystal oscillator. The regression curve of B-mode indicates a linear dependence of approximately −26.5 ppm/degree. The dual-mode oscillation direct thermometry is applicable over this temperature range.

The common part of circuit constants for the analysis are as follows: Oscillator 1: C_{02} and C_{03} = 47 pF; C_{0s} = 5 pF; C_{0y} = 10 pF; G_{1M} = 5 mA/V. Circuit constant of Oscillator 2: C_{12} and C_{13} = 47 pF; C_{1s} = 5 pF; C_{1y} = 10 pF; G_{2M} = 5 mA/V; C_{z} = 100 pF; C_{p} = 2 pF; C_{0c} = 3.431 pF; C_{0b} = 3.244 pF. L_{02} = 5.6 μH; L_{12} = 5.6 μH; C_{0x} = 9 pF; C_{1x} = 17 pF; L_{02} = 33 μH for the fundamental mode or L_{02} = 5.6 μH for the 3rd harmonic resonance of B-mode.

The common part of circuit constants for this experimental part is as follows. Oscillator 1: C_{02} and C_{03} = 47 pF; C_{0y} = 10 pF; L_{02} = 5.6 μH; C_{0x} = 9 pF for the 3rd harmonic resonance and L_{02} = 33 μH; C_{0x} = 30 pF for the

fundamental resonance of B-mode. Oscillator 2: C_{12} and C_{13} = 47 pF; C_{1y} = 10 pF; L_{12} = 5.6 μH; C_{1x} = 17 pF for the 3rd harmonic resonance of C-mode. IC_{1} and IC_{2} TC7SHU04F, V_{cc} = 5 V.

The SC-cut resonator is made of double-rotated cut crystals and it shows a principal resonance C-mode and B-mode, additional resonance mode. High stability of the principal mode was observed in both single and multimode resonance oscillation. The linear dependence of the B-mode on the temperature can realized the thermometric method with the assurances of the stability. Design rules were ensured by the measurement of the stability of σ versus τ curve.

The narrow-band wide variable crystal oscillator circuit was proved to be an efficient experimental maneuver in the simultaneous measurement of the stability and the oscillation frequency dependence on the ambient temperature. The frequency separation is narrower than the combination of the 3rd harmonic resonance of C-mode and the fundamental resonance of B-mode which is approximately 6.2 MHz. The combination of the 3rd harmonic resonance oscillation shows quasi-sinusoidal wave forms with negligible distortion with Allan standard deviation σ_{y}(τ), 10^{−11} < σ_{y}(τ) < 10^{−10}, sufficiently stable for standard time base and sensor applications. In the present stage of the experiment, the demonstrated result was obtained by precise choice of the circuit constant and bias current. It is necessary to monitor the unlocking from crystal resonance by the real-time monitoring of the Allan standard deviation. Although the reproducibility of the circuit and the oscillation condition is not thoroughly identified and appropriate circuit design is still necessary, we risk concluding that the dual-mode quartz crystal resonance oscillation can be applied to piezoelectric sensing and the production of a stable frequency standard.

We express acknowledgement to Ms Ruzaini Izyan binti Ruslan and Ms Izzati Syazwani binti Othman for their collaboration in the early stage of experiment, and Mr. Masahide Marumo, River Eletec Company (Japan) for fruitful discussions.