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 stable 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. IC₁ and IC₂ 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 C₁₆, C₁₇ = 18 pF.

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 V_cc 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 R_f = 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. 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 and equivalent circuit-3. 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

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. 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. Table 1 shows the equivalent circuit constant of the quartz crystal resonator.

Temperature dependence of the crystal sensor is explained in the experimental part. Figure 4 and Figure 5 compare the absolute values of negative resistance R_cc and R_cci as functions of frequency. In Figure 4, the absolute value of negative resistance and the active inductance is compared as functions of frequency. The parameter g_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.

In Figure 5, the absolute value of negative resistance R_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.

Figure 6 shows that the active inductance is generated in narrow range of gain corresponding to the resonance frequency and higher gain is necessarily compared with the case of 32.768 kHz, for better performance. The absolute value of negative resistance ranges to R_cc = 2 × 10³ 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 C₀ 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) [18] . Figure 7 shows the circuit layout for the simulation. CMOS inverter IC₁ and IC₂ 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 R₂. Figure 8 shows a typical wave form of the CR oscillator and FFT spectral peak of the initial CR oscillation.

Figure 9 shows the transient excitation of the crystal current with miss-matched frequency setting of the CR oscillation. As the peak of the CR oscillation at 70 kHz is higher than the resonance frequency, phase mismatch can be observed in a few cycles of the initial oscillation. In the optimum setting, the crystal current in the motion arm grows faster, and the oscillation frequency of the oscillator circuit is locked to the resonance frequency within several oscillation cycles.

Figure 10 shows the modified two-sample Allan standard deviation as a measure for the short-time frequency stability showing sufficiently high short range stability. The protocol is defined in (25), following IEEE Standard 1139. The frequency of oscillator circuit f_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 [17] . As a sensor system, the frequency is measured at 100 to 1000 ms range. Allan standard deviation of 5 × 10⁻¹² 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.

Figure 11 shows the normalized frequency shift as functions of temperature.

In Figure 11, solid line shows the regression curve of the frequency shift in the direct connection normalized with respect to the oscillation frequency at 0 degree C. Cross symbols indicate the frequency shift when the crystal sensor is connected through coupling capacitors: C₁₆, C₁₇ = 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. Figure 12 shows the dependence of the resonance frequency on temperature measured by the impedance curve. The active circuit satisfies the oscillation condition for the variation of motional impedance CI over operating temperature.