4. Microsystems in measurements of mechanical quantities- displacement,

velocity and accelerationMechanical quantities important in measurements with sensors:

x(t), (t), = x/x , - position (linear, angular), displacement, elongation (strain)

v = dx/dt, = d/dt, a = dv/dt – velocity, acceleration (linear, angular) F = ma, = dF/dA, M = F d - force, pressure, torque

dm/dt, dV/dt - mass flow, volume flow

For quantities varying in time we measure:

average values

rms values for periodic motion t = T

t

dttAt

A0

)(1

t

rms dttAt

A0

22 )(1

2

The unknown parameters can be determined from the basic relationships between quantities, e.g. from the knowledge of acceleration one obtains succesively

Bearing in mind determination of integration constants, it is necessary to do additional measurements.

Recent MST technologies allow to fabricate low cost but precise acceleration sensors.

Accelerometers, regardless of the conversion technique, require the existence of a seismic mass, which displacement with respect to the housing is rgistered. Taking into account the conversion technique of a displacement, one deals with different kinds of accelerometers: piezoelectric, piezoresistive, capacitive, thermal.

1Cdt)t(a)t(v

2Cdt)t(v)t(x

3

Mechanical model of vibration sensor

2

2

2

2

dtxdy

mk

dtdy

mb

dtyd

An equation of motion of mass m with respect to the reference frame with coordinate y, under the influence of spring force – ky, damping force – b dy/dt and inertial force – md2x/dt2 can be written as:

Sensor case moves relative to the Earth along a coordinate x

1. m – largeb – small position sensork – small

2. m – small b – large velocity sensor k – small

3. m – small b – small accelerometer k – large

Adjusting the oscillator constants one can neglect selected terms in the equation, thus obtaining different sensors

dtdxy

mb

xy

2

2

dtxdy

mk

In reality the sensor mass should be small enough to avoid influence on the investigated object. In this case one can build a sensitive accelerometer and other vibration parameters can be obtained by integration of acceleration.

4

In MEMS technology a silicon cantilever with deposited piezoelectric film, e.g. BaTiO3 is used.

Piezoelectric accelerometer

1. – seismic mass2. – piezoelectric plates (with magnification)3. – tension control4. – FET preamplifier5. – cable attachment

Experimental setup with piezoelectric accelerometer used for investigation of vibration parameters.

Piezoelectric plates are sandwiched between the casing and the seismic mass, which exerts on them a force proportional to acceleration.

5

Section of the quartz crystal in a plane perpendicular to c axis (z-axis).There exist 3 mechanical axes (perpendicular to crystal planes) and 3 electrical axes (drawn through edges).A plate cut from the crystal is also shown.

Piezoelectric effect

Applying a force to the plate along the electrical axis one generates the charge on the surfaces, to which a stress is applied (longitudinal piezoelectric effect).Acting with a force along mechanical axis y we induce a charge on the surfaces as before (transversal effect).

6Longitudinal effect

x1

x1

Piezoelectric effect, cont.

Quartz crystal structure(the first atomic layer is shown,in the second layer there are 3 O2- atoms,the third layer is identical as the first one,aso.)

Transversal effect

7

Application of stress σ generates a charge with density q

dtdk

dtdq

p

kp –piezoelectric module (for quartz 2.2 ·10-12 C/N, for ferroelectrics ca.100 times higher)

For the longitudinal effect (force Fx) one obtains for surface Ax a charge

Q’ = Axq’ = Axkpσ = kpFx - independent of Ax

For transversal effect (force Fy) one gets

Q’’ = -kpFy b/a

Piezoelectric effect, cont.

8

Generated charge on capacitance C gives the voltage

U = Q/C = Q/(Ck + Cm) = kpFx/C, Ck, Cm – cap. of crystal and cable, resp.

For n parallel connected plates one gets

U = nQ/(nCk + Cm)

This gives piezoelectric sensitivity

Sp = dU/dFx = nkp/(nCk + Cm)

Sensor discharge time constant is then equal:

τ = (Ck + Cm)/(Gk + Gm) G – conductance

This time constant limits fmin.

Piezoelectric effect, cont.

9

Capacitive sensor

l

C C

l l

0,1

Capacitance of a flat-plate capacitor C = ε0εr A/ l

A - plate area, l – distance between plates

SensitivitydC/dl = - ε0εrA/ l2 , changes with l

dC/C = - dl/l , high relative sensitivityfor small l (nonlinearity)

10

Differential capacitor

l

l

l

C1 C2

C

l l

-1 -0,6

0,6 1

ΔC = C2 – C1 = ε0εr A·2Δl/(l2 – Δl2)for Δl << l ΔC = ε0εrA·2Δl/l2, hence ΔC/C = 2Δl/l

Therefore one obtains increased sensitivity and linearity.

Differential technique decreasesthe temperature error and the influence of ε drift.

11

Cylindrical capacitive sensor with movable dielectric shaft in ratiometricconfiguration:W – common electrodeS – fixed electrodeR – variable electrode

The displacement causes moving of the shaft and is calculated from the ratio of capacitances CWR/CWS

In practice the S electrode needed carefull screening to avoid the inflence of air humidity variations on CWS capacitance(change in configuration of the electric field).

Capacitivedisplacement sensor

12 12

Angular position sensor

In the simplest case a rotating capacitorcan be used

CCC 0

13

Practical realisation of a differential rotating capacitor (Zi-Tech Instruments Corp.)

Stators 1 and 2 form with a rotor separate capacitances.The difference of those capacitances varies linearly with movement of the rotor.

Angular position sensor, cont.

14

A capacitive accelerometer with a differential capacitor fabricated in silicon bulk micromachining technology.

Capacitive accelerometer

From discussion of the mechanical model of vibration sensor it follows that for the system with high spring konstant k we can cosider the deflection, i.e. also the change in capacitance, as proportional to the acceleration.

In this case one obtains a capacitive accelerometer.

The movable mass is sandwiched between upper and base fixed electrodes forming two variable capacitances.

15

Examples of MEMS accelerometers: Analog Devices ADXL250 (on the left)and Motorola dual-structure microsystem before encapsulation (on the right)

16

Tilt (inclination) sensor based oncapacitive accelerometer

Construction of integrated Analog Devices accelerometer:

(a) scheme of interdigitated differential capacitor, (b) upper view of the sensing structure.

The central moving belt forms with static belts the interdigitated structure (46 capacitors) and deflects from the central position by inertial forces.

17

Determination of the tilt angle θ from measurements of the gravitational acceleration

Tilt angle determination for single axis and dual axis sensors (g = 1)

For single axis sensor thesensitivity decreases with θ

Inclination sensor based oncapacitive accelerometer, cont.

single axis sensor dual axis sensor