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A new closed-loop drive scheme which decouples the phase and the gain of the closed-loop driving system was designed in a Silicon Micro-Gyroscope (SMG). We deduce the system model of closed-loop driving and use stochastic averaging to obtain an approximate “slow” system that clarifies the effect of thermal noise. The effects of mechanical-thermal noise on the driving performance of the SMG, including the noise spectral density of the driving amplitude and frequency, are derived. By calculating and comparing the noise amplitude due to thermal noise both in the opened-loop driving and in the closed-loop driving, we find that the closed-loop driving does not reduce the RMS noise amplitude. We observe that the RMS noise frequency can be reduced by increasing the quality factor and the drive amplitude in the closed-loop driving system. The experiment and simulation validate the feasibility of closed-loop driving and confirm the validity of the averaged equation and its stablility criterion. The experiment and simulation results indicate the electrical noise of closed-loop driving circuitry is bigger than the mechanical-thermal noise and as the driving mass decreases, the mechanical-thermal noise may get bigger than the electrical noise of the closed-loop driving circuitry.

The Silicon Micro-Gyroscope (SMG) is an important MEMS inertia sensor with a broad application in the national economy and defense fields [

The effects of mechanical-thermal noise on the sense-mode have been presented in the literature [

As shown schematically in _{xi}≫ K_{x}) is very large, the outer-frame and the inner-frame are driven together to vibrate along the x-axis by the alternating drive force, which causes the alternating capacitance between the outer-frame and fixed drive-sense electrode. We can capture the drive displacement by detecting the alternating capacitance. When the rotation rate along the z-axis is input, according to the Coriolis effect, the Coriolis force along the y-axis will be loaded on both the outer-frame and the inner-frame. Because the stiffness of the outer support beam along the y-axis (K_{yo}≫K_{y}) is very large, only the inner-frame is driven to vibrate along the y-axis by the Coriolis force, which induces the alternating capacitance between the inner-frame and fixed sense electrode. We can obtain the rotation rate along the z-axis by detecting the alternating capacitance.

The simplified motion equations of SMG are described by:
_{x} (m_{x}=m_{1}+m_{2}) and m_{y} (m_{y}=m_{2}) the drive proof mass and the sense proof mass in kilograms, R_{x} and R_{y} the damping in Newtons/meter/second, K_{x} and K_{y} the stiffness in Newtons/meter, and −2_{x}_{e} (F_{e}=F_{d}sinω_{d}t) is the electrostatic force used to maintain the drive-mode vibration at a specified amplitude in terms of displacement, and at a resonant frequency of the drive-mode. Mechanical thermal noise on the drive axis is represented by the random force n(t), in units of force.

Ignoring the influence of the random force n(t), the drive axis displacements and sense axis displacements in the steady state are described by:
_{nx} =(K_{x}/m_{x})^{(1/2)}; ω_{ny} =(K_{y}/m_{y})^{(1/2)}; Q_{x}=m_{x}ω_{nx}/R_{x}, Q_{y}=m_{y}ω_{ny}/R_{y}.

When ω_{d}=ω_{nx}=ω_{ny}, the maximum drive axis displacements and sense axis displacements are described by:

Consider the damped harmonic oscillator:

The presence of damping in the system suggests that any oscillation would continue to decrease in amplitude forever. Inclusion of the fluctuating force n(t) prevents the system temperature from dropping below that of the system's surroundings. The damper provides a path for energy to leave the mass-spring system. This is the essence of the Fluctuation-Dissipation Theorem. According to Equipartition, if any collection of energy storage mode is in thermal equilibrium, then each mode will have an average energy equal to (1/2)k_{B}T where k_{B} is Boltzmann's constant(1.38×10^{-23}J/K) and T is the absolute temperature in degrees Kelvin. A mode of energy storage is one in which the energy is proportional to the square of some coordinate; e.g., kinetic and spring potential.

When this system is in thermal equilibrium, the probability distribution of x and

For the oscillator, the energy is the sum of the kinetic and spring potential energy:

From here, the equipartition theorem can be derived, namely that the mean energy in any energy storage mode is equal to(1/2)k_{B}T. Thus:
_{x} and K_{x}, the thermal noise n(t) must be a white Gaussian noise with two sided spectral density [

The spectral density of drive displacements due to thermal noise is:

So the noise power spectrum is:

The RMS noise displacements due to thermal noise is:

According to

The above equation indicates that we can improve the signal-to-noise ratio by increasing the quality Q_{x} and driving force amplitude F_{d}, or by reducing the stiffness and temperature.

As is known in the art of Coriolis force sensors, in order to achieve an acceptable response from the sensor, the proof mass vibration of the drive-mode should have a frequency at, or close to, the resonant frequency of the proof mass. At the same time, in order to improve the entire performance of the SMG, a high stability of the driving frequency and the amplitude of the drive-mode are needed. To satisfy those demands, the closed-loop driving of the drive-mode must be achieved. To this end, the drive signal has a frequency equal to the resonant frequency of the proof mass. However, parasitic capacitances between the drive electrode and the drive-sense electrode can cause significant errors. That is, when the drive signal capacitively couples into the drive-sense electrode, the accuracy of amplitude control by the feedback circuit is degraded and the harmonic frequency of the closed-loop system departs from the resonant frequency of the proof mass, resulting in less than optimum sensor performance, so we must eliminate the capacitive coupling. Various techniques are generally utilized in an effort to reduce capacitive coupling. In this paper, such a technique is utilized as follows: the drive electrode is arranged on the left, the drive-sense electrode is arranged on the right and the anchor of the SMG is connected with the ground or the virtual ground, which is shown in ^{-1} Torr and the quality factor of the drive-mode above 2,500, which can also reduce these capacitive coupling by decreasing the drive voltage. The modulation-demodulation method through applying high-frequency carrier to the proof mass can also reduce these capacitive couplings.

First, we need to extract the resonance signal of the drive-mode. The simplified interface circuitry is shown in _{0}+ΔC, C_{0} is constant capacitance. The part of signal sense can be equivalent to a current supply I(t) and a internal resistance C_{0} [see

In _{xc}
_{0} is very minute and generally has hundreds of fF, thus the impedance of C_{0} is very large and we can ignore the influence of the impedance C_{0}. The resistance R_{1} generally has a few MΩ, the capacitance C_{1} hundreds of nF, ω/ 2ᴫ a few kHz (ω≈ω_{nx}), so R_{1}≫1/ωC_{1}, the output voltage P(t) is described by:

_{s}, V_{ref} and V_{sup} are the direct current biases. L and J are the zero and the pole of the integrator separately. G is the gain of the integrator. In order to improve the precision and the stability of the closed-loop driving, the Q-factor of the drive-mode should be increased (the SMG is executed in vacuum encapsulation), while the well closed-loop control should be achieved. The closed-loop control must meet such two conditions: 1. The phase of the whole loop θ=2nπ (n is an integer); 2. The gain of the whole loop A>1.

A new closed-loop drive scheme which decouples the phase and the gain of the closed-loop driving system is adopted in the SMG, so that the phase and the gain can be optimized, respectively. The gain of the whole closed-loop system is controlled by the branch circuit above, and the phase is controlled by the branch circuit below. These two branch circuits respectively fulfill the two conditions of closed-loop control, adjusting and optimizing the closed-loop parameter separately. The “voltage comparator” is the key component of the closed-loop driving. The output of the “voltage comparator”, with an invariable output amplitude, only reserves the phase information of the input signal, so the phase conditions of the closed-loop are isolated from the gain conditions. Except the drive mode of SMG and “voltage comparator”, with the suppose that the phase of the other parts is fixed, when the phase of the “voltage comparator” is changed, the vibrating frequency of the closed-loop system will depart from the resonant frequency of the proof mass, so the phase condition controls the frequency of the driving displacement. It is obvious that the gain branch controls the amplitude of the driving displacement. According to _{1}(t) and n_{2}(t), we can know that average amplitude

To make sure that the harmonic frequency of the closed-loop system equals, or gets close to the resonant frequency of the proof mass, that is to say, the phase of the whole loop θ = 2nπ(n is an integer). When ω_{d}=ω_{nx}, the phase-shift of the drive displacement x(t) comparing to the drive force F_{e} is -π/2(See _{cp}(t) comparing to output voltage of preamplifier P(t) is −π. The other parts, which in fact all have tiny phase errors, have no phase-shifts, so the closed-loop control meets the phase conditions. In this way, it is hoped that the above closed-loop phase errors should be as tiny as possible. Various techniques are generally utilized in an effort to reduce closed-loop phase error, or drift, in servo circuits, such as amplifier circuits utilizing an operational amplifier. One such technique includes the addition of one or more zeros (i.e., a lead filter) in cascade with the open-loop gain of the operational amplifier in order to flatten the open-loop gain over a portion of the frequency band, generally resulting in only moderate closed-loop error reduction and also compromising stability. Another technique for reducing gain and phase errors is to increase the gain-bandwidth product associated with the operational amplifier. However, use of this technique is limited by the gain-bandwidth product of commercially available operational amplifiers as well as by the acceptable increased power dissipation associated with higher performance operational amplifiers. However, a Phase-Corrected Amplifier Circuit can be used to remove the closed-loop phase error [

In _{0} is the permittivity, h the thickness of the comb fingers, x_{0} the overlap length of the fingers, and d the width of the gap between fingers. According to _{cp}(t) is a AC with the frequency near the resonant frequency ω_{nx}. The first term of

According to _{n}(u) is electrostatic drive force

Simplifying the integrator (See

So the output of the “voltage comparator” is:

In summary, the entire closed-loop driving system, shown in _{eq}
_{eq} through adjusting the output of the integrator z(t). When the equivalent damper R_{eq} is bigger than the zero(e.g.R_{eq}>0), the gain control loop reduces the output of the integrator z(t) and then the equivalent damper R_{eq} will decrease near the zero (R_{eq}≈0,
_{xc}). When the equivalent damper R_{eq} is smaller than the zero (e.g.R_{eq}<0), the gain control loop enhances the output of the integrator z(t) and thus, the equivalent damper will increase near the zero (R_{eq}≈0). So the closed-loop system shows approximately an undamped-free vibration with the invariable amplitude and frequency(e.g. the resonant frequency ω_{nx}). However, the random process n(t) will influence the stability of the driving amplitude and frequency.

After the vacuum encapsulation, the quality factor of the drive-mode Q_{x} becomes very big. The drive-mode of the SMG can be equivalent to a band-pass filter. Only the displacement with the resonant frequency ω_{nx} can be magnified and the other harmonics are attenuated greatly. So the transient displacement can be simplified into a pure sin component. In order to analyze the transient behavior of the system, the driving displacement of the SMG is defined as [

where

Differentiating

One of the equations used to determine

Thus, the velocity equation becomes:

The acceleration is obtained by differentiating

Substituting

Substituting

It should be noted that _{nx}t evolves much faster than the other variables, such as _{nx}t+_{nx}t+_{nx}t change very little. Hence, it is possible to apply the averaging method to the non-autonomous system described by

As pointed out above, instantaneous phase ω_{nx}t is regarded as an independent variable and the differential equations _{nx}t, over the interval [-π, π]. The averaged autonomous equations are:
_{1}(t) and n_{2}(t) are independent white noise with the same intensity as n (t) [

Ignoring the influence of the random processes n_{1}(t) and n_{2}(t), the equilibrium of the averaged system described by

When the power is switched on, the system stabilizes finally in equilibrium. According to _{o}_{x}, so the change of quality factor resulting from the variety of temperature and pressure does not impact on the equilibrium of the average amplitude _{o}

Ignoring the influence of the random processes n_{1}(t) and n_{2}(t), the Jacobian matrix of the nonlinear dynamic system of _{xc}. In order to make sure that the square root is in existence, the inequation hereinafter must be satisfied:
_{ref} is the referring DC voltage used as a reference to the amplitude of the pre-amplifier output voltage. The V_{refo} is the criterion voltage. When V_{ref} < V_{refo}, the gain control branch works normally. When V_{ref} >V_{refo}, the gain control branch loses the control ability. Because the _{n}_{o}, z_{n}_{o}_{n}_{n}_{n}_{1}(_{1}, so the steady-state spectral density for the noise component of the _{n}

So the noise power spectrum is:

The RMS noise amplitude due to thermal noise is:

Comparing _{x}.

According to _{o}_{n}_{o}_{n}_{n}

Suppose the work bandwidth is f_{B}Hz, the noise power is:

The RMS noise frequency due to thermal noise is:

It is useful to reduce RMS noise frequency by increasing quality factor Q_{x} and drive amplitude _{0}

In order to validate the feasibility of closed-loop driving and confirm the validity of the averaged equation and its stable criterion, the closed-loop driving in

_{ref}_{refo}_{ref}_{refo}_{ref}_{refo}_{ref}_{ref}_{refo}_{ref}_{ref}_{refo}_{ref}_{ref}_{refo}_{ref}_{ref}_{refo}

_{ref}_{refo}_{ref}_{refo}_{ref}_{refo}_{ref}_{refo}

In _{x}=2,500 and Q_{x}=5,000 are compared. From _{nx}= 25,120 (rad/s), T=300 K, f_{B}=100 Hz, Q_{x}=2,500, _{o}

The whole experiment circuit is constructed on the idea of _{d}=ω_{nx}, the amplitude of driving displacement is proportional with the quality factor Q_{x}, so the change of quality factor, due to the variety of temperature and pressure, impacts directly on the amplitude of driving displacement in the open-loop driving. But the new closed loop driving is immunized to the change of the quality factor [see _{x} decreases about 2.6 times [see

The experiment results of closed-loop driving are shown in _{x}=2.89×10^{-7}(kg), _{o}_{nx}= 25,120 (rad/s), we can find that the relative noise of drive frequency is approximately 0.0119 ppm and the relative noise of the amplitude is approximately 0.8 ppm, respectively. The noise comparison is shown in

The mechanical thermal noise on drive-mode is discussed, and then stochastic averaging is used to develop a model for the “slow” dynamics that represent the driving amplitude and frequency of the SMG. Both the steady-state and transient response of the model are obtained by stochastic averaging. The spectral density of the random error due to thermal noise on drive-mode is also derived. By calculating and comparing the RMS noise amplitude due to thermal noise both in the opened-loop driving and in the closed-loop driving, we find that the closed-loop driving does not reduce the RMS noise amplitude. We observe that the RMS noise frequency can be reduced by increasing the quality factor and drive amplitude in the closed-loop driving system. The experiment and simulation validate the feasibility of closed-loop driving and confirm the validity of the averaged equation. The experiment and simulation results indicate the electrical noise of closed-loop driving circuitry is bigger than the mechanical-thermal noise and with the driving mass decreasing, the mechanical-thermal noise may get bigger than the electrical noise of closed-loop driving circuitry.

(A) The frame of the SMG. (B) The simple model of SMG. (C) The picture of the processed SMG.

(A) The interface circuit of drive-sense signal. (B) The equivalent circuit.

The frame of the closed-loop driving.

The Matlab simulation frame of the closed-loop driving.

The simulation curves of closed-loop driving when _{ref}_{refo}

The simulation curves of closed-loop driving when _{ref}_{refo}

Closed-loop response (gray) and averaged equation simulation (black) when _{ref}_{refo}

Closed-loop response (gray) and averaged equation simulation (black) when _{ref}_{refo}

The simulation curve of closed-loop driving in Q_{x}=2,500 (black) and Q_{x}=5,000 (gray) when _{ref}_{refo}

The influence of thermal noise to driving performance when ω_{nx}= 25,120 (rad/s), T=300 K, f_{B}=100 Hz, Q_{x}=2,500, _{o}

The experiment of closed-loop driving. (A) The PCB circuitry of SMG. (B) The test setup environment. (C)The vibrating waveshape of closed loop driving signal.

The experiment results of amplitude control with temperature. (A) The curve of the quality factor of the drive-mode Q_{x} with temperature. (B) The signal amplitude of closed loop driving with temperature.

The experiment results of closed-loop driving.(A) Frequency drift of driving signal for 1 h. (B) Amplitude drift of driving signal for 1h. (C) Frequency spectrum of driving signal.

The value of simulation parameter.

Qx | 2,500 |

G | 999 |

Kxc | 2.92×10^{-8} (F/m) |

Vs | 10 (V) |

R | 10 (MΩ) |

Vsup | 5 (V) |

Vrefo | 0.596 (V) |

L | -50 |

J | -55 |

|∂C/∂x| | 1.46×10^{-6} (F/m) |

ωnx | 25120 (rad/s) |

mx | 2.89×10^{-7} (kg) |

The noise comparison.

mx = 2.89×10^{-7}(kg), _{o} | |||||

Absolute value | Relative value | Absolute value | Relative value | ||

Thermal noise frequency | 0.0000476 Hz | 0.0119 ppm | Noise frequency | 0.0205 Hz | 5.0 ppm |

Thermal noise amplitude | 4×10^{-12} m |
0.8 ppm | Noise amplitude | 0.0165 mV | 14.7 ppm |