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 R₂, C₄, and C₀ satisfy this resonance condition, where C₀ is the parallel capacitance of the quartz crystal resonator. R₂ settles the bias in the initial stage of the oscillation. C₄ 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 R₂ multiplied by C₄. 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 maximum negative resistance is generated at specified value of the conductance g_mf of the active circuits: CMOS (Complementary Metal Oxide Semiconductor) inverters IC₁ and IC₂. The conductance is controlled by negative feedback resistors R_f = R₃, R₄, R₅, and R₆.

Figure 2 shows simplified equivalent circuit-1. CMOS inverter IC₁ and IC₂ 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 IC₁ and I_out is the output current of IC₂.

Solving for the relation between I_out and V_in, total conductance G_M 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 (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.

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 C₀ and the motion arm, L₁, C₁, and R₁, the equivalent series inductor, capacitor, and resistor respectively. C_S is a stray capacitance. Calculating the parallel composition of C₀ 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.

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 C₀ 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 L₁ 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

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.

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 C₀ and stray capacitance C_s are included in the impedance of the active circuit. Thus, the relation between R_cci and R_cc 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. Figure 9 shows simplified circuit diagrams of the oscillation Mode-1 and Mode-2. Before the establishment of the resonance oscillation, parallel capacitance C₀ 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 C₀. 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 R₂ 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 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 IC₁ and IC₂ 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 R₂. 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.

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 R₂. In the proposed oscillator circuit, the output of inverter IC₂ is positively fed back to the input of inverter IC₁ through the quartz crystal resonator. Replacing the quartz resonator with a capacitor equivalent to the parallel capacitance C₀, the circuit exhibits CR oscillation and this frequency is determined by the time constant: R₂ multiplied by C₄. In the case of R₂ = 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.

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.

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

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.