Static and dynamic characteristics of angular velocity and acceleration transducers based on optical tunneling effect

Theoretical and experimental analysis of quasi-linear conversion function of angular velocity and acceleration microoptoelectromechnical (MOEM) transducers based on optical tunneling effect (OTE) are conducted. Equivalent oscillating circuit is developed and dynamic characteristics of angular velocity and acceleration MOEM-transducers are investigated.


Introduction
Angular velocity and acceleration transducer, constructed with structure «optical components of total internal reflection (TIF)mediumsensing element (SE)» are suggested to apply for improving effectiveness of navigation and mobile objects control system with satisfied modern requirements [1][2]. Thus optical power, reflected from the structure «TIFmedium -SE», is received by photodetector and supports information about measuring result [3].
While developing such MOEM-transducers it is needed the complete description of actual conversion function of the measuring value (angular velocity and acceleration) in the change of optical output power in static and dynamic modes. The validity of the derived characteristics of MOEMtransducers should be based on the comparison results of theoretical calculations and experimental studies.

Correction of static characteristic of angular velocity and acceleration MOEM-transducers based on OTE according to experimental studies results
The operating principle of angular velocity and acceleration MOEM-transducers is based on dependence of optical source reflective coefficient on the value of operating gap between SE and TIF base. Therefore, the reflected part of optical power, reached to photodetector, is an informative value [4].
Experimental studies by using semi-natural setup has been conducted to verify the conversion function of MOEM-transducers based on OTE and determine the correction type of theoretical calculations (figure 1).
Semi-natural setup is comprised of FOT-930 Multifunction Loss Tester 1, fiber optic cables 2, TIF prism made up of quartz 3, piezoceramic SE in form of round plate 4, load transferring ball 5, aluminum beam 7 with hanging сable 6. FOT-903 is used as optical source in generation mode with wavelengths λ 1 =1.3µm and λ 2 =0.85µm while supplying constant optical power source P OS =2.045µW. Beam 7, fixed at one end, can change load F H by changing mass m H 8 and its hanging position. This varying load F H , acting on SE center, leads to change of operating gap d between SE and TIF base. From optical source 1, signal enters to the structure «TIF-medium-SE» via optical fiber cables 2. While reducing operating gap d from the influence of load F H most of the optical energy penetrates to SE and is absorbed, which leads to decrease the optical input signal of photodetector Р РD .
While neglecting the mass of beam 7 (light compared to the suspended mass 8) load F H acting on the SE of MOEM-transducer is calculated by formula: SE of those MOEM-transducers based on OTE is clamped on the perimeter of circular end, and the largest deflection is observed at its center. Under the influence of the central load F H magnitude of SE deflection y F (F H ) is function of parameters y F (F H )=f (R SE , h SE , ν, E SE ) [5],where R SE , h SEradius and thickness of SE; ν -Poisson's ratio; E SE -Young's modulus of SE material.
Magnitude of operating gap d(F H ) between SE and TIF base is defined by initial gap d 0 and deflection of SE y F (F H ): The output optical power with the conversion function of such MOEM-transducer based on OTE in account of optical loss is determined by the following formula [6]: where k optoptical loss; λwavelength of optical source; θangle of incidence, R[d F (F H ),θ,λ]reflectivity of structure «TIFmedium -SE». Experimental studies of conversion function is conducted with following parameters: R SE = 5mm, h SE = 1mm, ν = 0.17, E SE = 72.5GPa (figure 2). From comparing of theoretical calculation and experimental results it should follow that the analytical formula to calculate the conversion function can support to be quarsilinear and can be considered as acceptable for dependent description of output signal on the external influence to MOEM-transducer based on OTE, while ensuring not more than 15% of error.
To improve the accuracy of MOEM-transducer model is necessary to select the correction coefficient of theoretical calculations based on the experimental results, e.g., least squares method. The correction coefficient of conversion function must be satisfied in the condition with n experiments: where Theoretical calculation of conversion function uses correction coefficient, described polynomial with 3 rd order: K CORR (F Н ) = (1.0137+0.00424·F H -0.00017·F H 2 +0.00001· F H 3 )·10 -8 to provide the error not more than 2%.

Dynamic behaviors of angular velocity and acceleration MOEM-transducers based on OTE
Block diagram is composed to study the dynamic mode of angular velocity and acceleration MOEMtransducers based on OTE (figure 3.), which includes: a movable part -SE, which converts the measuring value Ψ (angular velocity Ω z or acceleration a z ) according to varying of operating gap d Ψ ; modulator (M), which converts optical power source P OS to output optical power of photodetector P PD ; converter "current-voltage" (I-U), which converts current I PD into voltage, supplied to low pass filter (LPF) and generate output voltage; У-amplifier in feedback circuit (OC); PA -piezoactuator that converts the voltage U OC with movement of d U .  Using feedback circuit can reduce the transfer ratio of conversion, which allows to increase the maximum measuring value to desired measuring range by adjusting operating gap of transducers [7]. In contravention of this part sensitivity and linearity of the MOEM-transducers conversion function can be decreased.
According to the block diagram of MOEM-transducers a transfer function W Σ (p) in form of operators is: SE of MOEM-transducers based on OTE can be considered as a mechanical oscillating system which can be described with second order differential equation [8]: where J, G stiffinertial magnitude and stiffness of SE, K GDcoefficient of gas dynamic damping due to viscous friction of the medium of structure «TIFmedium -SE», yoscillating velocity.
Note that an arbitrary external factor Q(t), acting on the SE of MOEM-transducers, can be expanded in Fourier series [9]: where A k , kω, θ k -amplitude, circular frequency, initial phase k times harmonics oscillation, a 0coefficient of Fourier series. Let us assume that the investigated system is satisfied by filter condition, which do not effect with higher harmonics significantly the basic, i.e. can be neglected. Thus, the equation (7) can be written as: By following equation (8), the external factor Q(t) according to the principle of superposition can be separately considered as a constant and harmonious action. Then, the external factor Q(t), acting on the acceleration MOEM-transducer can be assumed constant, and for angular velocity MOEMtransducer -harmonic.
Angular velocity and acceleration MOEM-transducers are acoustomechanical oscillating system with distributed parameters, consisting of SE and the air gap between SE and prism base. To analyze the dynamic characteristics of MOEM-transducers, which are defined by properties of acoustomechanical oscillatory system, conveniently and efficiently by using well known methods of electrical engineering and circuit theory. The criteria of equivalence are governed by the laws of energy conservation. In addition, each factor of the differential equation (6) is replaced by combination of equivalent elements of electrical oscillating RLC-circuit, in which the value of inertial J and, gas dynamic damping coefficient K GD and stiffness of SE G stiff are corresponded to inductive L E , capacitive C E and active electrical resistance R E . Equivalence, acting on external factor of SE corresponds to EMF U E (t), and oscillating velocity of system y -current I E (t) (figure 4). where μ=f(р c ,Т 0 ,ρ c ,v c ,k В ) -coefficient of dynamic viscosity, depend on density ρ c , pressure р c , average flow rate v c , ambient temperature Т c and Boltzmann constant k В (at temperature Т 0 =20 о С, here μ≈1,83·10 -6 kg/m·s can be selected for air); l a , h a , b a , m a , E a -length, thickness, width, mass of SE and Young's modulus of four-beam fused silicon SE, m add -additional mass accordingly; and, electrical current I E (t)=dd a (t)/dt -rate of central part deflection of SE. By the Kirchhoff's voltage law: Differentiating equation (9) with respect to time and multiplying by 1/L E , obtain: where α=R E а / (2•L E а ) -attenuation coefficient of oscillation in RLC-circuit, ω 0 = 1/√(L E а •С E а )natural oscillating frequency in RLC-circuit. At the initial condition: current I E а | t=0 =U a /R E а and the rate of change of current is different from zero L E а •dI E а (t)/dt| t=0 = -U a , the total current I E a (t) in an oscillatory RLC-circuit varies according to formula [10]: Consequently, the transfer function W SE a (p) of SE of acceleration transducer based on OTE may be defined by formula: where L -Laplace transform. Under the influence of measuring angular velocity Ω z SE of angular velocity MOEM-transducers based on OTE (Figure 4b) occurs small deviations d Ω (t) due to Coriolis effect [11]. This schematic can be seen as an oscillatory RLC-circuit (Figure 4c)  displacement of SE along OY axis (for small value φ Ω (t)≈sin[φ Ω (t)] is accepted, that φ Ω (t)=2·d Ω (t)/l Ω ). Then second order differential equation for such oscillatory RLC-circuit is: Similarly, the general solution of equation (13) can be written as where ∆φ = arctan[(ω 2 -ω 0 2 )/(2αω)]phase shift. Transfer function W SE Ω (p) of SE of angular velocity MOEM-transducers based on OTE is: Transient process modelling of angular velocity and acceleration MOEM-transducers based on OTE is shown in figure 5 with the following parameters of functional elements: PD T PD =1µs,

Conclusion
In this study angular velocity and acceleration MOEM-transducers based on OTE, providing quasilinearity conversion function is investigated. An experimental study of this transducer conversion function is described. Correction coefficient of conversion functions based on comparing of theoretical calculations and experimental studies is defined and it supports to improve the accuracy of the calculation of the described model.
The block diagram is determined and equivalent oscillatory RLC-circuit is composed for the study of dynamic characteristics angular velocity and acceleration MOEM-transducers based on OTE. It is shown that the transition process of described transducers have oscillated character with permissible magnitude of transition process duration τ PP is around 1ms, which allows to apply those transducers in mobile objects control systems.