**Journal of Sensor Technology**

Vol.05 No.01(2015), Article ID:54245,12
pages

10.4236/jst.2015.51003

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

Tomio Sato^{1}, Akira Kudo^{2}, Tetsuya Akitsu^{1*}

^{1} Department of Human, Environment and Medical Engineering, Interdisciplinary
Graduate School of Medicine and Engineering, University of Yamanashi, Kofu, Japan

^{2}SEIKO EPSON Corporation, Suwa, Japan

Email: ^{*}akitsu@yamanashi.ac.jp

Copyright © 2015 by authors and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Received 3 February 2015; accepted 20 February 2015; published 26 February 2015

ABSTRACT

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.

**Keywords:**

Double Resonance Quartz Crystal Oscillator, Active Inductance

1. 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 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 [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.

2. Design and Analysis of Quartz Crystal Oscillator

2.1. 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}.

Figure 1. Circuit diagram
of the quartz crystal oscillator. Circuit constants: Bias resistors R_{3},
R_{4}, R_{5} and R_{6} = 3.3 kΩ, feedback resistor R_{2}
= 3 MΩ, varied for the optimum setting ; C_{2}, C_{3}, and C_{4}
= 10 pF; C_{5} = 0.1 μF; C_{6} = 10 μF; C_{7} = 100 pF;
Inverter IC_{1} and IC_{2} TC7SHU04F; V_{cc}= 3 V. Equivalent
circuit constant of the quartz crystal resonator: f_{1} = 32.768 kHz; L_{1}
= 11,797 H; R_{1} = 47.6 kΩ; C_{1} = 2fF.

Figure 2. Simplified circuit: equivalent circuit-1.

(2)

(3)

(4)

(5)

Solving for the relation between I_{out} and V_{in}, total conductance
G_{M} is found.

(6)

Then the following relation is found. Current I_{2}, I_{3} are expressed
in the terms of I_{1}.

(7)

(8)

(9)

Rearranging the expression, relation (11) is found.

(10)

(11)

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.

(12)

(13)

(14)

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}.

(15)

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.

(17)

(18)

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

(19)

Figure 3 shows 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
Figure 4 is found. Composed equivalent resistance R_{cci} and capacitance
C_{cci} are found.

(20)

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

(21)

Figure 3. Equivalent circuit-2.

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.

(22)

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.

(23)

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. 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

Table 1. Equivalent circuit constant of the quartz resonator.

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 6. Absolute value
of negative resistance R_{cc} and reactance C_{cc} as functions
of frequency and gain. Circuit constant: C_{2} and C_{3} = 14 pF;
C_{4} = 18 pF; R_{2} = 1.0 MΩ, (a) g_{mf} = 4 μA/V, (b)
f_{r} = 32.768 kHz.

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.

(24)

Figure 7 shows 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} = −3.4 pF is
equivalent to L_{cc} = 9.8 H.

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_{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.

2.2. 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

Figure 7. Absolute value
of negative resistance R_{cci} and reactance as functions of frequency and
g_{mf}. Ciruit constant: R_{2} = 1.9 MΩ; C_{2} and C_{3}
= 14 pF; C_{0} = 1.14 pF; C_{4} = 18 pF; C_{s} = 1 pF. (a)
g_{mf} = 4 μA/V; C_{x} infinite. (b) f_{r} = 32.768 kHz.

Figure 8. Comparison of
negative resistance for different values of g_{mf} and f_{r}. (a)
Frequency f_{r} = 20, 32.768, and 40 kHz. Circuit constants: C_{2}
and C_{3} = 14 pF; C_{4} = 18 pF; R_{2} = 1.9 MΩ. (b) g_{m}
= 2, 4, and 8 μA/V. Circuit constants: C_{2} and C_{3} = 10 pF;
C_{4} = 18 pF. R_{2} = 1.9 MΩ.

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 LT- spice 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 9. Equivalent circuit diagram of quartz crystal oscillator. (a) Model-1; (b) Model-2.

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 shows the 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.

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.

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

Figure 13. Typical example
for the waveform of the CR oscillator and the FFT spectral analysis R_{2}
= 3.0 MΩ.

3. 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.

3.1. 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.

3.2. 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.

(25)

Table 2 shows the stability of the quartz crystal oscillation in this experiment. This result shows Allan standard

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

Figure 15. Start-up 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. Short range stability of the double-resonance quartz crystal oscillator.

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

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

Acknowledgements

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.

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