Inertial_Sensors
- 2. MEMS Inertial Sensors • Fundamental mechanical system underlying inertial sensors – The mass spring system model • Energy metric for evaluating sensor performance
- 3. ELECTROMECHANICAL MASS/SPRING SYSTEMS • The fundamental concept of an inertial sensor. • In this system, a proof mass, m, is suspended on a mechanical frame by a spring, km, and responds to an input force, F, representative of a quantity to be measured. • The input force causes a displacement, x, of the mass, and the displacement is measured to sense the force. • For example, the input force may result from acceleration of the mass, as would be the case in an accelerometer. • The force may relate to Coriolis acceleration, – resulting from angular rotation of the mass. – This is the case of the vibratory Gyroscope. +- VB ke F km b CS(x) X m
- 4. ELECTROMECHANICAL MASS/SPRING SYSTEMS • The Various sensors employ different transduction methods to transform the quantity of interest into a orce acting on the proof mass. • A high-quality sensor generally possesses high transduction gain, while rejecting the effects of parasitic forces on the mass that may degrade sensor accuracy. • Such parasitic forces vary by application, but may generally include stresses induced by packaging or thermal expansion and forces • acting on the proof mass due to motions of a type other than what is to be sensed. • For example,: – angular rate gyroscopes should reject forces – due to linear acceleration. +- VB ke F km b CS(x) X m
- 5. ELECTROMECHANICAL MASS/SPRING SYSTEMS: Damping Forces • The proof mass is also subjected to a damping force proportional to its velocity, represented by a damping coefficient, b. • Damping results from many sources, but for inertial sensors air damping is typically dominant. • Thus, if a high quality factor is required for the sensor, it is • important to hermetically seal the mechanical elements to allow operation of the sensor at low pressure. • Hermetic sealing also prevents contaminants: – particles, and moisture from interfering with the sensor • Additional forces +- VB ke F km b CS(x) X m
- 6. ELECTROMECHANICAL MASS/SPRING SYSTEMS: Electromechanical Transduction • Additional forces acting on the proof mass pertain to the electromechanical transduction • Electromechanical transduction: is necessary to provide an operable sensor. • Types of Electromechanical transduction: – Electrostatic transduction, – Piezoelectric transduction • All sensors require sensing of the proof mass displacement. • Some sensors additionally require that the mass be driven into motion for generating the necessary input force. • For example, in vibratory rate gyroscopes, – The Coriolis force resulting from angular rotation is proportional to proof mass velocity. – Thus, the proof mass must be moving at a known velocity for a given rate of rotation to manifest itself as a known force acting on the mass.
- 7. ELECTROMECHANICAL MASS/SPRING SYSTEMS: Capacitive Sensing of the displacement • To sense the displacement of the proof mass by electrostatic means, a capacitor, CS(x), is used with one terminal residing on the mass and the other terminal on the fixed frame. • With a fixed bias voltage, VB, a change-in-charge, ∆Q, on the • capacitor is created by a change-in-position, ∆x, according to • Note that the first derivative with respect to position of CS(x) determines the charge sensitivity along with the bias voltage. • The charge sensitivity can further be expressed in terms of the geometrical parameters of the capacitor and the permittivity of free space.
- 8. ELECTROMECHANICAL MASS/SPRING SYSTEMS: Capacitance Charge Sensitivity, Parallel Plate • The table provides a summary for parallel plate capacitor geometry. • A key observation is that the charge sensitivity generally increases for greater bias voltage, VB, and smaller initial gap size, g0. Capacitance Charge transduction Force transduction Spring constant C(x) ∆Q(∆x) ∆F(∆v) kE Parallel-plate capacitor ε0 A g0 −x ε0A V ∆x g0 B2 ε0A V ∆v g0 B2 − ε0A V2 g0 B3 VB X g0 A: plate area
- 9. ELECTROMECHANICAL MASS/SPRING SYSTEMS: Capacitance Charge Sensitivity, Comb Capacitance Capacitance Charge transduction Force transduction Spring constant C(x) ∆Q(∆x) ∆F(∆v) kE Comb-finger capacitor VB g0 w: comb width l: comb overlap @ x=0 ε0w(l + x) g0 ε0w V ∆x g0 B ε0w V ∆v g0 B 0
- 10. ELECTROMECHANICAL MASS/SPRING SYSTEMS: Capacitors as Source of Electrostatic Force on Proof mass • Force Transduction • In addition to providing charge transduction, capacitors can be used to apply electrostatic forces to the proof mass. • The electrostatic force is related to the gradient of the potential energy • of the charge stored on the capacitor. • With a fixed bias, VB, and assuming small-signal operating conditions, the change in force, ∆F, relates to the change-in-voltage on the capacitor terminals, ∆v, as follows: • As in the case of charge transduction, we see that the first derivative of CS(x) with respect to position determines the force along with the bias voltage.
- 11. ELECTROMECHANICAL MASS/SPRING SYSTEMS: Force Transduction • Force transduction is important: – For sensors that require the proof mass to be driven to a known motion (such as vibratory rate Gyroscopes) – For evaluating forces due to parasitic electric fields.
- 12. ELECTROMECHANICAL MASS/SPRING SYSTEMS: Electrostatic Spring Softening • Origin of Electrostatic Spring Softening – Although not explicitly used for force transduction, the sense capacitance, CS(x), nonetheless has associated with it an electrostatic force that — for parallel-plate capacitors — varies with the proof mass position. – For small displacements, the effect of this electrostatic force on the system dynamics may be modeled by an equivalent spring constant, denoted kE – Because electrostatic forces are attractive in nature, the electrostatic – spring constant is actually negative, leading to the phenomenon known as electrostatic spring softening. • Electrostatic Srping Softening – Electrostatic forces reduce the effective spring constant of the system. • Pull-In, Mechanical Instability: – Spring softening can lead to outright mechanical instability, – This is known as pull-in – Mechanical instability happens, if the magnitude of the – electrostatic spring constant, kE, is allowed to – exceed that of the mechanical spring constant, kM. • +- VB ke F km b CS(x) X m
- 13. ELECTROMECHANICAL MASS/SPRING SYSTEMS: Electrostatic Spring Softening • Comb-finger capacitors do not suffer from this Electrostatic Spring Softening problem, as the electrostatic force is not position-dependent, to first order. • The strong gap and bias dependence of the electrostatic spring constant, kE is worth noting. • Relation between Trimming and Temperature Compensation to Electrostatic Spring Softening: – As variations in the parameters such as the gap and the bias, over manufacturing process and temperature can strongly influence the sensor. – It is possible that there is a need for trimming and/or temperature compensation to correct.
- 14. ELECTROMECHANICAL MASS/SPRING SYSTEMS: As second-order dynamical system • The electromechanical system is a second-order dynamical system, and the frequency-domain output charge is related to the sensor input signal as shown: • GF is the transduction of: the sensor input signal to force acting on the proof mass (which varies according to each type of sensor). • The mechanical dynamics relate the change in position to the applied force, accounting for both mechanical, kM and electrostatic springs, kE. • The charge transduction gives the conversion from position to output charge. +- VB ke F km b CS(x) X m S F 1 ms2+bDs+kM+kE X V ∂Cs Q Sensor output GF B Mεχhanical dynamics Force transduction Charge transduction ∂x Decreasing damping (1) (2) (3) GFVB .∂Cs Q(s) S(s) kM+kE kM+kE 2πf ∂x m