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Temperature challenges the accuracy and stability of even the best-made crystal oscillators. For that reason, a precision temperature sensor is typically part of any high-precision quartz frequency source, helping to provide compensation for the effects of changes in temperature. In a temperature-compensated crystal oscillator (TCXO), for example, a thermistor is used in close proximity to the oscillator’s resonator, which is usually an AT-cut quartz crystal. Unfortunately, this temperature-correction approach suffers from inaccuracies due to thermal lag stemming from a difference of effective thermal time constants in crystal and thermistor, thermal gradients, and thermistor aging.^{1}

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Another method for overcoming these limitations is resonator self-temperature-sensing using a dual-harmonic-mode crystal oscillator.^{2} This method provides for the resonator to determine its own temperature reading (rather than using external temperature sensors), and yields high-precision microcomputer-compensated crystal oscillators (MCXOs) with better precision than any other TCXO. Unfortunately, the large size of these sources, the high cost of SC-cut crystal, and the complicated dual-harmonic-mode oscillator structure of the MCXO prevent it from being widely used in electronic products.

A dual-AT-cut-crystal oscillator structure with self-temperature-sensing was first proposed in ref. 3. The current authors’ experimental results prove that this method has nearly the same level of precision as the dual-harmonic-mode resonator self-temperature-sensing. With increasing progress towards ultrasmall crystal units,^{4} this method is particularly well suited for application in miniature MCXOs. In the present report, a MCXO using a dual-AT-cut-crystal self-temperature-sensing circuit was developed, capable of providing frequency-temperature stability of better than 0.1 ppm, with smaller size and lower cost than conventional MCXO^{5,6} by employing AT-cut crystals.

*1. Frequency-temperature characteristics of AT-cut crystal.*

An AT-cut crystal for use as a resonator in an oscillator is typically cut around a nominal angle of +35^{○}15’. Furthermore, its frequency-temperature (F-T) characteristic, which can be generally expressed in terms of Eq. 1, depends on the precise shearing angle **(Fig. 1)**:

Δf/f_{0} = a + bΔT + cΔT^{2} + dΔT^{3} (1)

where the normalized frequency, Δf/f_{0}, and the difference temperature, ΔT, are referenced at room temperature. Comparing the cut angle of (+35^{○}15’) + 16’ with (+35^{○}15’) - 4’, the difference between the two normalized frequencies has a monotone decreasing from +1.10 x 10^{-4} to -1.05 x 10^{-4} in the temperature range from -40 to +85°C. When the baseline frequency is 10 MHz, the differential frequency range will be 2150 Hz, which means that the average temperature-frequency sensitivity is 0.058^{○}C/Hz, and nearly equal to the dual-harmonic-mode resonator self-temperature-sensing.

*2. Frequency-temperature characteristics of two different shearing angle AT-cut crystals.*

The F-T characteristics of two AT-cut 10-MHz quartz crystals with shearing angles of (+35^{○}15’)+4’ and (+35^{○}15’)−2’ are shown in **Fig. 2**, and their difference frequency characteristic can be described by Eq. 2 as:

Δf_{1}(ΔT) - Δf_{2}(ΔT) = -60.584 + 2.00948 x ΔT + 0.001977 x ΔT^{2 }+ 0.00000424 x ΔT^{3} ≈ -60.584 + 2.00948 x ΔT (2)

which means that the temperature-frequency sensitivity of this dual-resonators temperature sensor is 0.4976^{○}C/Hz. When the frequency resolution of this sensor is 0.1 Hz, its temperature resolution will be 0.04976^{○}C and it will be free of thermal lag in terms of temperature compensation.

*3. Architecture of the MCXO with the dual-crystals self-temperature-sensing.*

**Figure 3** shows the basic architecture of the MCXO with a dual-AT-cut-crystal resonator and self-temperature-sensing approach. For the differential frequency measurement, one of the oscillator’s outputs, f_{1}, is divided by the fixed value, N_{1}, to obtain a signal f_{N1} for gating a counter, which is fed from the other oscillator’s output, f_{2}. The counter produces a count value, C_{1}, during a period of f_{N1}, according to the relationships of Eq. 3:

N_{1}(1/f_{1}) = C_{1}(1/f_{2}) (3)

In 10-MHz crystals, the deviation range of f_{2} is typically less than 1 kHz or, as shown by Eq. 4:

[(C_{1} - N_{1})/C_{1}]10^{7 }= [(f_{1} - f_{1})/f_{2}]10^{7} ≈ f_{2} - f_{1} (4)

The computation of differential frequency between f_{2} and f_{1} has a relative precision to 10^{-4}, which means a measurement accuracy of at least 0.1 Hz. The divisor N_{1} serves to scale the C_{1} counter range consistent with the required temperature-sensing resolution and counter width.

*4. Block diagram of the divider with fractional division ratio.*

The differential frequency is read by the microprocessor to determine the actual frequency of f_{1} via the polynomial computation of Eq. 1, the coefficients of which are derived from a least-squares curve-fitting routine using data taken during a frequency-temperature characteristic measurement. The predicted frequency is subtracted from the target 10-MHz frequency to determine the deviation, Δf_{1}. The deviation Δf_{1} of approximately 400 Hz is generated from f_{0} by a divider with a fractional division ratio as shown in** Fig. 4**. The microprocessor then computes a fractional division ratio, N_{C} as in Eq. 5, which is used to control the fractional divider to produce a correction frequency equal to Δf_{1}. A voltage-controlled crystal oscillator (VCXO) is employed to establish the target 10-MHz frequency, in which the output frequency of VCXO is phase-locked to the sum of f_{1} - Δf_{1}:

f_{out} = (f_{in}/10000)(1- N_{C}/N_{2}N_{3}) (5)

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## The Small Print

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The proposed MCXO circuit was fabricated in a 0.18-μm, 1.8-VDC silicon CMOS process; it features power consumption of less than 5 mW. It can achieve frequency stability within ±0.1 ppm over a range of temperatures from -40 to +85°C. **Figure 5** shows the frequency deviation for two repeated cold to hot and hot to cold temperature runs for one proposed MCXO, which is better than for a conventional TCXO.^{5,6} For all temperature runs, the protocol initiated with a 10 minute “soak” time followed by a change of 1°C every minute. The entire testing process was automated including protocol control and data collection.

*5. Frequency stability under temperature tests.*

By employing ultrasmall AT-cut crystal units, the proposed MCXO has a smaller size of 13 x 13 x 1.75 mm, comparing with a traditional MCXO using SC-cut dual-mode crystal,^{7} which has a size of 38.1 x 38.1 x 12.7 mm, making the proposed MCXO more suitable for applications in portable productions.

**Liang Chen, Engineer**

**Yangbo Huang, Engineer**

**Yonghu Zhang, Engineer**

**Gang Ou, Engineer**

**College of Electronic Science and Engineering, National University of Defense Technology, Changsha, Hunan, China, People's Republic of China.**

### References

1. S.S. Schodowski, “Resonator self-temperature-sensing using a dual-harmonic-mode crystal oscillator,” in Proceedings of the 43rd Annual Symposium on Frequency Control, Denver, CO, May 31-June 2, 1989, pp. 2-7.

2. A. Benjaminson and S. C. Stallings, “A Microcomputer-Compensated Crystal Oscillator Using a Dual-Mode Resonator,” in Proceedings of the 43rd Annual Symposium on Frequency Control, Denver, CO, May 31-June 2, 1989, pp. 20-27.

3. T. Schmid, J. Friedman, Z. Charbiwala, Y. H. Cho, M. B. Srivastava, "XCXO: An Ultra-low Cost Ultra-high Accuracy Clock System for Wireless Sensor Networks in Harsh Remote Outdoor Environments," 2007.

4. M. Harada, K. Kubota, and H. Takahashi, “Research on Miniaturization of Low-Frequency Range SMD Crystal Unit," in Proceedings of IEEE International Frequency Control Symposium, 2007, pp. 150-153.

5. H. Rokos, "Precision, Low Power, analogue TCXO using a Single Integrated Circuit," in Proceedings of European Frequency Time Forum, 1996, pp. 515-519.

6. Mi Zhang, Wei-xun Cao, “A 0.1 ppm Successive Approximation Frequency-Temperature Compensation Method for Temperature Compensated Crystal Oscillators (TCXO),” in Proceedings of World Congress on Computer Science and Information Engineering, 2009, pp. 493-498.

7. E. Jackson and B. Rose, “The microprocessor compensated crystal oscillator-new developments,” in Proceedings of IEEE International Ultrasonics, Ferroelectrics, and Frequency Control, 1999, pp. 376-379.

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