Analysis of excitation and detection capability of hemispherical resonator based on interdigital electrodes

In this paper, a theoretical model of excitation and detection is established based on the excitation and detection method of the uncoated hemispherical resonator with interdigital electrodes, which proves the feasibility of the excitation and detection of the interdigital electrodes in principle. The simulation and analysis of its influence parameters are also carried out to provide theoretical support for the design and optimization of the interdigital electrode to realize the excitation and detection of the hemispherical resonator. Finally, the excitation and detection of uncoated hemispherical resonators by interdigital electrodes were successfully verified through experiments.


Introduction
Hemispherical resonant gyro (HRG) is a classical Coriolis vibrating gyro (CVG) has attracted the interest of the world inertial navigation community with its high dynamics, high accuracy, high reliability, ultra-long lifetime, low power consumption and miniaturization [1][2] .The hemispherical resonator is an important component of HRG, and its mechanical performance parameters have a very important influence on the accuracy of HRG [3] .Hemispherical resonators are generally shaped by ultra-precision grinding and polishing.Machining errors and inherent inhomogeneities in the material can lead to imperfect resonators.Researchers have utilized Fourier series to decompose the errors.The 1st-3rd order harmonic errors are prone to induce coupled vibrations in dynamic environments (e.g., frequency-specific ambient vibrations), while the 4th harmonic errors lead to mode-frequency mismatches (a.k.a., frequency cleavage), which cause quadrature errors [4] .The premise of harmonic error detection is that a certain amplitude of operating mode vibration needs to be excited at multiple locations of the resonator.Currently frequently used excitation methods include hammering, piezoelectric and interdigital electrodes.The prerequisite for harmonic error detection is the need for a test system that enables multi-position excitation and amplitude detection.Currently, commonly used test systems include electrostatic excitation detection systems, piezoelectric excitation laser detection systems, hammering excitation laser detection systems, and interdigital electrode excitation detection systems.Among them, only the interdigital electrode excitation detection scheme can realize the multi-position excitation detection of the resonator before coating.
Interdigital electrodes are periodic electrodes structures that have a finger-like or comb-like shape, and these electrodes are often used to make sensors [5] .Based on their non-contact, non-destructive and high sensitivity, interdigital electrodes are often used in non-destructive testing, chemical sensing, biotechnology, environmental monitoring and other applications [6][7] .Interdigital electrodes are also often used as excitation and detection structures for resonators of gyroscopes, especially in MEMS gyroscopes [8][9][10] .However, for hemispherical resonators with large mass and stiffness, there are few reports on interdigital electrodes.
In this paper, we provide a detailed discussion of the interdigital electrodes excitation and detection for hemispherical resonators.First we theoretically model the excitation and detection, and then we analyze the interdigital electrodes parameters that affect the excitation and detection capabilities from simulation.

Theory
Like the electrostatic mode excitation and detection of gyros, the excitation and detection of gyros by interdigital electrodes essentially relies on the capacitance formed by the interdigital electrodes.Unlike the electrostatic excitation and detection of a gyro, where the gyro is part of a capacitive electrode plate, the interdigital electrode excitation and detection of a gyro is based on the principle that the gyro is part of a capacitive dielectric.

Excitation principle of interdigital electrodes
When the interdigital electrodes work as actuators, the interdigital electrodes generate an alternating electric field under the action of an AC voltage, and the edge of the HRG, which is a part of the dielectric in the capacitor, is subjected to vibration by the action of the electrostatic force F: where d is the distance d between the edge of the resonator and the electrodes and E is the energy stored in the interdigital capacitor: Where C is the interdigital capacitor and V(t) is the voltage loaded across the interdigital capacitor: Where Vd is the DC voltage and Va is the AC voltage.
So we end up with the electrostatic force on the edge of the harmonic oscillator as follows:   From Eq.( 4), it can be seen that the magnitude of the electrostatic force F is related to the deflection of the capacitance C with respect to the the distance d between the edge of the resonator and the electrodes and the voltage parameters Vd and Vc.The theoretical model of the interdigital capacitor consists of an electrode substrate, interdigital electrodes, air, and a resonator, as shown in Fig. 1.The gaps between electrodes have a width of b while the fingers have a width of a. a, b, d and the total width of the interdigital electrodes L are important parameters in relation to the effective capacitance magnitude C.
In a gyro, The number of groups of electrodes is usually a deterministic 8 or 16, so the total width of the interdigital electrodes L is also deterministic.For convenience, the metallization ratio is defined as follows: So the interdigital capacitance C can be defined as a function of a,  , d:

Detection principle of interdigital electrodes
The detection principle of the interdigital electrodes is shown in Fig. 2. It's a system that detects changes in the interdigital capacitance .In general, a small displacement change is proportional to the change in capacitance it induces, so that the frequency response of the resonator can be analyzed.
Fig. 2. The detection principle of the interdigital electrodes.When the capacitance C is varied, the output voltage Vout is as follows: Where C  is the change in C, Vd is the voltage loaded on the capacitor, Cp is the capacitance of the electric charge amplifier, a reasonable choice of Cp can improve the signal-to-noise ratio of the output signal.

Simulation and analysis of excitation and detection capabilities of the interdigital electrodes
Through the analysis of the interdigital electrodes structure and working principle, it is known that the parameters affecting the excitation and detection of the interdigital electrodes mainly include the finger width of the interdigital electrode (a), the metallization ratio (η), the distance between interdigital electrodes and the resonant structure (d), and the applied driving and detection voltages.In this section, numerical calculations and finite element simulations are used to analyze the influence laws of the interdigital electrode structure parameters on the driving and detection, and to optimize the design of the interdigital electrode structure.
Fig. 3. Schematic of the finite element simulation model of the interdigital electrodes.

excitation capabilities of the interdigital electrodes
In this section, the effect of different interdigital electrode parameters on its excitation capabilities is analyzed by simulating the electrostatic force F generated per unit area of the interdigital electrodes under the action of unit voltage.
The distance between the edge of the resonator and the interdigital electrodes is an important factor affecting the excitation capabilities of the interdigital electrode.The simulation results (Fig. 4) show that the smaller d is, the larger F is, and when the gap d exceeds 50 um, the driving force is close to 0, which indicates that the driving ability is less desirable when d is large.6 shows the influence of metallization ratio η on F. The simulation results illustrate that the higher the metallization ratio, the greater the excitation capacibilities, and the metallization ratio also affects the optimal fingers width mentioned above.
In summary, the smaller the distance between the edge of the resonator and the interdigital electrodes d, the adapted finger width a and metallization ratio η and the larger loading voltage will get the larger driving capacity.However, subject to the assembly process, d is usually between 10-20 um; too small fingers width can lead to large machining errors, resulting in charge buildup, which can easily lead to electrode breakdown, which contradictions with loading high voltages.

detection capabilities of the interdigital electrodes
The main factors affecting the detection ability of the interdigital electrodes are the capacitance change ΔC.Fig. 7 shows the influence of the distance between the edge of the resonator and the electrode d on Ceff, where Ceff is the effective capacitance.It can be seen that the smaller d is, the larger the change in Ceff is.And as can be seen from Figure 8, there is the same trend for the smaller a the greater the change in Ceff.From Eq.7, it can be seen that the detection capability of the interdigital electrode is also related to the loading voltage, but as mentioned in Section 3.1, smaller finger widths contradict larger loading voltages, so attention should also be paid to the electrode voltage tolerance when designing the parameters of the interdigital electrode.

Experiment
Through the above simulation and analysis, it can be seen that the parameters of the interdigital electrodes and the mounting gap have a large impact on the excitation and detection capabilities of the interdigital electrodes.
Firstly, in order to obtain the smallest and most uniform mounting gap possible, a fixture for resonator and electrode assembly was designed to adjust the mounting gap.As shown in Fig. 9, the electrode substrate is mounted on a movable platform, while the resonator is held stationary on a fixation base.The gap between the electrode and the resonator can be changed by adjusting the movable platform.Fig. 9. Gap adjustment platform.After adjusting the gap, the relationship between the breakdown voltage and the parameters of the interdigital electrodes was on this basis, and finally better set of parameters for excitation and detection was obtained.The specific parameters are shown in Table 1 below: Table 1: parameters for excitation and detection of interdigital electrodes.

Conclusion
In this paper, the excitation and detection principles based on interdigital electrodes are investigated.Through theoretical analysis and simulation, the influence of the interdigital electrode parameters on the excitation and detection capabilities is obtained, which can

CSMNT-2023
Journal of Physics: Conference Series 2740 (2024) 012026 provide a design and direction for the of the interdigital electrode used to realize the excitation and detection of uncoated resonators.

Fig. 4 .
Fig. 4. The effect of the distance between the edge of the resonator and the interdigital electrodes (d) of the resonator on F. The pattern of influence of a is shown in Fig. 5.As a increases, the excitation force F increases and then decreases, indicating that there exists an optimal a to maximize the excitation force F.

Fig. 5 .
Fig. 5.The effect of the finger width a of the resonator on F for different gap d.

Fig. 6 .
Fig.6.The effect of the finger width (a) of the resonator on F for different metallization ratio η Fig.6shows the influence of metallization ratio η on F. The simulation results illustrate that the higher the metallization ratio, the greater the excitation capacibilities, and the metallization ratio also affects the optimal fingers width mentioned above.In summary, the smaller the distance between the edge of the resonator and the interdigital electrodes d, the adapted finger width a and metallization ratio η and the larger loading voltage will get the larger driving capacity.However, subject to the assembly process, d is usually between 10-20 um; too small fingers width can lead to large machining errors, resulting in charge buildup, which can easily lead to electrode breakdown, which contradictions with loading high voltages.

Fig. 7 .
Fig. 7.The effect of the distance between the edge (d) of the resonator on Ceff.

Fig. 8 .
Fig. 8.The effect of the finger width (a) of the resonator on Ceff.From Eq.7, it can be seen that the detection capability of the interdigital electrode is also related to the loading voltage, but as mentioned in Section 3.1, smaller finger widths contradict larger loading voltages, so attention should also be paid to the electrode voltage tolerance when designing the parameters of the interdigital electrode.
and detection experiments are performed on the hemispherical resonator.And the experimental results are shown in Fig.10.The results verify the feasibility of using interdigital electrodes for excitation and detection of the uncoated hemispherical resonator.

Fig. 10 .
Fig. 10.Experimental results on excitation and detection of interdigital electrodes for uncoated hemispherical resonators.