Novel Gyroscopic Mounting for Crystal Oscillators to Increase Short and Medium Term Stability under Highly Dynamic Conditions

In this paper, a gyroscopic mounting method for crystal oscillators to reduce the impact of dynamic loads on their output stability has been proposed. In order to prove the efficiency of this mounting approach, each dynamic load-induced instability has been analyzed in detail. A statistical study has been performed on the elevation angle of the g-sensitivity vector of Stress Compensated-cut (SC-cut) crystals. The analysis results show that the proposed gyroscopic mounting method gives good performance for host vehicle attitude changes. A phase noise improvement of 27 dB maximum and 5.7 dB on average can be achieved in the case of steady state loads, while under sinusoidal vibration conditions, the maximum and average phase noise improvement are as high as 24 dB and 7.5 dB respectively. With this gyroscopic mounting method, random vibration-induced phase noise instability is reduced 30 dB maximum and 8.7 dB on average. Good effects are apparent for crystal g-sensitivity vectors with low elevation angle φ and azimuthal angle β. under highly dynamic conditions, indicating the probability that crystal oscillator instability will be significantly reduced by using the proposed mounting approach.


Introduction
Nowadays oscillators are widely used in huge number of electronic communication, measurement and testing devices. In particular, crystal oscillators with high-Q factors are intended for use as frequency and low-phase noise standards [1]. On the other hand, crystal oscillator output stability is influenced by environmental conditions such as cleanliness and contamination, and the electromagnetic, thermal and mechanical environments [1][2][3][4][5][6][7][8][9]. Under highly dynamic conditions, crystal oscillators are exposed to a variety of dynamic loads, thus their short and medium-term stability is degraded during missions [4,5,10]. Under these conditions, the performance of the total system will be deteriorated when the short-term instability of the oscillator exceeds a specified boundary [11] determined by the host vehicle and frequency application [12].
Dynamic load-induced oscillator instability can be regarded as clock bias and drift [13]. These deviations are influenced by the magnitude and angular orientation of an inherent characteristic, i.e., a g-sensitivity vector. Even among a series of carefully designed crystals with the same cut, same vibration mode and same overtone resonant frequency each crystal will have its own g-sensitivity vector. In addition, for a specified crystal, different measurement methods give different results for the magnitude and especially for the angular orientation of this vector [14]. Thus, reduction of oscillator instability is a function of an unknown parameter, which is the direction of its g-sensitivity vector. In general, two approaches are adopted to reduce the effect of dynamic loads on crystal oscillator stability [15,16]: 1-Passive control approach: • Use of mechanical vibration isolation [17]. • Use of multiple, unmatched oppositely-oriented resonators [18,19]. • Reduction of resonator vibration sensitivity via resonator design [20][21][22][23][24][25].
In this paper, a novel mounting method, named gyroscopic mounting, is introduced for crystal oscillators. This mounting is a self-aligned passive component, thus it does not need sensor measurement, special installation, periodical inspection, maintenance or tune ups. The most important feature is that it does not need a power supply. It senses and responds to the load and remains perpendicular to the applied load at any given moment. Furthermore, the analysis and simulations show that the proposed mounting can provide up to 30 dB improvement in phase noise that is equal to the best reduction possible with the costly and complicated accelerometer feedback method [28]. In addition, this mounting is able to suppress the attitude change of host vehicle-induced instability. In order to reduce or ideally suppress the drawbacks of the gyroscopic mounting method which could occur in the case of any dynamic load, it is proposed to use a combined method of employing the aforementioned mounting with an active vibration noise control approach using accelerometer feedback [16,[26][27][28].
In this paper, all possible dynamic loads which affect the short and medium term stability of crystal oscillators under highly dynamic conditions will be introduced in Section 2. The main concept of using the gyroscopic mounting will be described in Section 3. A statistical study on the g-sensitivity vector of SC-cut crystals, which are commonly used in highly dynamic conditions, will be introduced in Section 4. In Section 5, all dynamic loads-induced instabilities and the gyroscopic mounting efficiency will be analyzed. In Section 6, the conclusions of the paper will be provided.

Medium-term instability (Clock drift/T)
where Δƒ is frequency disturbance; Δφ is phase disturbance; £(ƒ) is phase noise;  is g-sensitivity vector of crystal blank (1/g) and Γ is its magnitude; φ is elevation angle of  ; θ is azimuthal angle of  ; A is the applied dynamic load (g) and A is its magnitude; α is the angle between  and A ; β is the angle between pages passing through  and A ; ƒ0 is oscillator frequency in Hz, T is the host vehicle mission duration. According to Equations (1)-(4) and Figure 2, frequency and phase disturbances are functions of α and β cosine. Therefore, when A moves away from  , the oscillator output will be improved. Ideally, we tend to fix the applied dynamic load perpendicular to  . However, this method is not feasible in practice because each crystal blank has its own  , even among crystals with the same cut, same vibration mode and same overtone resonant frequency. On the other hand, for a specified crystal, different measurement methods give different results for the magnitude and especially for the angular orientation of this vector. Also in the case of random vibration, the angular orientation of dynamic load A can be in any direction.
In this paper, the main idea to reduce these disturbances is to hold the applied dynamic load perpendicular to crystal surface in any given time instant. For a specified range of angle β shown in Figure 2 (where │β│ < cos -1 (sinφ/cosα)), when α is increased, the dynamic load-induced disturbances can be reduced and consequently oscillator output stability can be improved.

Gyroscopic Mounting
In order to implement the aforementioned idea, a gyroscopic mounting method is proposed in the installation of crystal oscillator on electronic board, which gives it the freedom to rotate freely around roll, pitch and yaw. By this way, the resultant applied load will be perpendicular to oscillator surface in each time instant. Modeling and prototype of this instrument are shown in Figures 3 and 4.
This mounting is a simple and cheap passive component and does not need to any power supply. It must be as quick as possible to respond to dynamic loads. We manufactured it from low-density materials and the softness of surfaces should be high as well.  After installation of the crystal oscillator on it, dynamic load-induced disturbances can be expressed as: where Δƒgyro: frequency disturbance; Δφgyro: phase disturbance; £gyro (ƒ): phase noise. The maximum effect of gyroscopic mounting appears in the critical state shown in Figure 2b. In this case, gyroscopic mounting protects the system from the maximum probable instabilities.

Impacts of Dynamic Loads on Stability of Crystal Oscillator and Proof of Gyro Efficiency
Commonly GPS disciplined crystal oscillators are used as primary frequency standards for calibration and metrology in laboratories [40]. Therefore, in this paper this kind of oscillator is used to do the numerical analysis. To simulate highly dynamic conditions, the Ariane launch vehicle has been assigned as host vehicle. The parameters of this oscillator can be found in Table 1. The parameters of the Ariane launch vehicle will be mentioned in Sections 5.2-5.4.

Attitude and Altitude Changes of Host Vehicle
As shown in Figure 6, the attitude of the host vehicle changes nπ (i.e., 0 < δ + φ < nπ) during the mission. On the other hand, as altitude of host vehicle increases, magnitude of gravity acceleration g decreases (gh < g). Thus the angle between  and g , and the magnitude of g are time variant parameters. Therefore in either case, the gravity acceleration g plays the role of dynamic load and causes disturbances in oscillator output in the form of clock bias or drift.
Note: the altitude gravity gradient is not significant, as shown in Figure 7, therefore in calculations it has been substituted by gavr on the trajectory: where: 0 < δ + φ < nπ, r = the Earth's radius, h = altitude above the Earth's surface, gh = acceleration gravity at altitude h.

Gyro Effect
Due to the use of the gyroscopic mounting, the angle between  and g , shown in Figure 6c remains constant during the mission. Therefore attitude change-induced disturbances are completely suppressed.

Steady State Acceleration (g)
For the Ariane 5 launch vehicle, the longitudinal steady state acceleration Along changes between 0 and 4.2 g on its trajectory, as shown in Figure 8, and the lateral acceleration Alat changes between 0 and 0.2 g. The impact of this load on the stability of crystal oscillator output directly depends on how it is installed on the host vehicle as shown in Figure 9. To calculate the maximum gyro efficiency, analyses have been performed by assuming vertical installation.

Results and Analysis
Statistically the elevation angle φ of SC-cut crystals follows the distribution of Figure 5. According to Figure 10, gyroscopic mounting shows its positive effects for the most probable crystals i.e., |φ| < 38° (84%) and its drawbacks for low probability ones. The average steady state load-induced frequency deviation and phase noise are shown in Figure 11 and summarized in Table 2.  Gyroscopic mounting efficiency is as shown in Figures 12 and 13 and summarized in Tables 3 and 4 for the most probable crystals.   According to Figures 12 and 13 and Table 3, for a statistically minimum φ value i.e., φ = 2.6°, gyroscopic mounting has a great effect on frequency deviation which is reduced near the Allan deviation safety margin (10 −4 Hz). Only when the β angle is very close to 90°, this mounting causes negligible drawbacks in oscillator output, such that frequency deviation is still less than the Allan deviation (10 −4 Hz). Gyroscopic mounting shows its great effects on phase noise for β < 70°. Its drawbacks appear only for β values too close to 90°. However, this drawback is negligible because phase noise is still less than the Allan deviation (−70.05 dBc/Hz).
Based on Figures 12 and 13 and Table 4, for a low elevation angle φ, gyroscopic mounting shows its positive effect on frequency deviation; however, it is still more than the Allan deviation (10 −4 Hz). Its drawback on frequency deviation appears only for β angles too close to 90°. The best effects on phase noise appear for β < 45°. Drawbacks with small values start from β = 60°. Big drawbacks only occur for β close to 90°.

Sinusoidal Vibrations (fv,si = 2-100 Hz)
Sinusoidal vibrations applied on the payload installed inside the fairing of Ariane 5 are as shown in Figure 14. The magnitude and angular orientation of these vibrations are illustrated in Figure 15 by dividing the frequency into three bands. That is why, in this section, two jumps are seen in the analyses figures corresponding to sinusoidal vibration-induced disturbances. The conic area in Figure 15 points out that this load can be applied on the crystal surface with any angle 0° < β < 360°, and it directly depends on the installation of the oscillator on the host vehicle.

Impact on Crystal Oscillator Short-Term Stability
Sinusoidal vibrations appear as frequency jitter and phase noise in oscillator output and degrade its short-term stability. These disturbances are calculated as: Gyro Effect on frequency and phase noise: where: 0 < |β| < π; 2 <ƒv,si < 100; angle α and Aztp as indicated in Figure 15. Critical state and maximum effect of gyroscopic mounting appears when β = 0.

Results and Analysis
In Figure 16 the gyroscopic mounting effect on frequency jitter and phase noise is illustrated. According to this figure and Figure 5, this mounting shows its greatest advantages especially for the most probable crystals i.e., |φ| < 38° (84%). The average frequency jitter and phase noise are shown in Figure 17 and summarized in Table 5, with respect to the distribution specified in Figure 5 for angle φ.   According to Figure 5, the gyro efficiency is shown in Figures 18 and 19 and summarized in Tables 6 and  7 for elevation angle |φ| < 2σ (84%).   According to Figures 18 and 19 and Table 6, for a statistically minimum φ value i.e., φ = 2.6°, gyroscopic mounting has great effect and reduces frequency jitter close to the safety margin of the Allan deviation (10 −4 Hz). For 2 < ƒv,si < 100, this mounting reduces phase noise −23.25 dBc at most. Its drawbacks appear only when the β value is too close to 90°.
According to Figure 18 and 19 and Table 7, for 2 < ƒv,si < 100, gyroscopic mounting reduces the phase noise −3.97 dBc at most. The best effects on phase noise appear for β < 45°. Small drawbacks start from β = 60° and big ones only occur for β values close to 90°.

Random Vibration (20 < ƒRV < 2000 Hz)
Random vibration is a band-limited noise with Gaussian distribution and could be created by different sources and applied in any direction i.e., 0 < |ξ| < π and 0 < |β| < π as shown in Figure 20a. Mechanical random vibration of the Ariane 4 launch vehicle is shown in Figure 20a [32]. Generally, this load is defined by Acceleration Spectral Density (ASD) in (g 2 /Hz). Since random vibration is a combination of all the frequencies at the same time, it is necessary to configure this load in the time domain. According to the Parseval's law, grms is equal to 1σ of random vibration. Thus the random vibration in time domain can be shown according to Figure 21b by calculation of grms from ASD.
On the other hand, there could be an infinite number of input ASD curves which have the same area, and therefore the same grms. Therefore, beside analysis in time domain, it is necessary to analyze the random vibration in the frequency domain.

(b) Impact on crystal oscillator short-term stability
Random vibration causes short-term instability as frequency jitter and phase noise in oscillator output.

Frequency jitter, Phase jitter and Phase Noise
If ξ = φ and β = 0, thus  and A coincide each other, therefore this mounting shows the best positive effects for low elevation angle φ. If ξ -φ = π/2 or β = π/2, then  and A are perpendicular to each other, in this case gyroscopic mounting shows the worst performance. These results are shown in Figures 22 and 23. According to Figure 22 and 23 and Table 8, the maximum effects of gyroscopic mounting on frequency jitter can be obtained when β = 0 and φ = 2.6°. In this case, gyroscopic mounting reduces the frequency jitter to near the safety margin of the Allan deviation. Maximum effect on phase noise appears for β = 0 and φ = 2° and this positive effect continues up to β = 60°. Drawbacks start from β > 60° and its maximum value occurs for β = 90° and φ = 38°. With regard to the distribution shown in Figure 5 for elevation angle φ, the average frequency jitter and phase noise are as shown in Figure 24 and summarized in Table 9.    According to Figure 20, in practice random vibration could be applied in any direction i.e., 0° < |ξ| < 180°. In this case, the dynamic load induces instability in the oscillator output, which is shown in Figure 26a,c and the gyroscopic mounting presents its maximum effect, which is shown in Figure 26b,d. In this case, when ξ − φ = 90° or β = 90°, gyroscopic mounting induces its maximum drawback which is limited to the angles close to the specified margin of ξ − φ = 90°. As shown in Figure  26d, better effects appear for low elevation angle φ.

Conclusions
According to the analysis results, gyroscopic mounting protects systems from instabilities caused by dynamic loads. It shows benefits on reduction of clock drift caused by the attitude change of the host vehicle. The mounting can also reduce the disturbances induced by steady state acceleration. The results of 27 dB maximum and 5.7 dB on average in phase noise improvement have been validated in our analyses. When considering sinusoidal vibrations, gyroscopic mounting reduces disturbances close to the safety margin of the Allan deviation. The phase noise improvement is 24 dB maximum and 7.5 dB on average. The random vibration-induced instability can be reduced 30 dB maximum and 8.7 dB on average in phase noise. The gyroscopic mounting for crystal oscillators has been proven to be able to improve the frequency deviation and phase noise caused by dynamic loads. It is promising for use in low cost frequency standard improvement for civil aviation and military purposes.