Accelerometers

• Accelerometers are devices that produce a voltage signal proportional to the acceleration experienced.

Definitions and units: • Acceleration is the time rate of change of velocity or the time rate of

change of the time rate of change of distance. • If we denote v and x as velocity and distance respectively, acceleration a = ∂v/∂t = ∂2x/∂t2. • The units of acceleration are measured in (m/s)/s or m/s2. • A “g” is a unit of acceleration equal to earth’s gravity at sea level and has

the value 9.81 m/s2. • Example of some g reference points are

1g = Earth’s gravity, 2 g = Passenger car in corner, 3 g = Race car driver in corner, 7 g = Human unconsciousness, 10 g = Space shuttle.

• The most general approach to acceleration measurement is to take advantage of Newton’ law,

• which states that any mass m that undergoes an acceleration a is responding to a force given by F = ma.

• The most general way to take advantage of this force is to suspend a mass on a linear spring from a frame which surrounds the mass as shown.

Frame

m

mass

spring

k

k

damping

b

x

X

sensor

• b denotes damping introduced to the system• When the frame is shaken, it begins to move, pulling the mass along with

it. • If the mass is to undergo the same acceleration as the frame, there will be

a force exerted on the mass, which will lead to an elongation of the spring.

• Any type of displacement transducer, such as a capacitive transducer, can be used to measure this deflection.

Frame

m

mass

spring

kk

damping

b

x

X

sensor

• Let X denote the position of the frame and x the position of the mass.• b denotes damping introduced to the system and k the stiffness constant

for the spring.• For the case shown in the Figure, the sum of the forces on the mass are

equal to the acceleration of the mass and given by , k (X-x) + b d (X-x) = m d2 x dt dt2

Frame

m

mass

spring

kk

damping

b

x

X

sensor

k (X-x) + b d (X-x) = m d2 x dt dt2

• Take Z = X-x or x = X - Z, then

m d2X/dt2 = m d2Z/dt2 + kZ + b dZ/dt.

• Since X is the position of the frame, we impose an acceleration on this problem by forcing X to take the form X = X0eiωt.

• We also assume that all the time varying quantities also oscillate, so Z = Z0eiωt.

Substituting we have -mω2X0eiωt = -mω2Z0eiωt + kZ0eiωt + iωbZ0eiωt.

• If we assign ω0 = (k/m)½

ζ = b/[2(km)½] the damping ratio (another definition of the damping term) A0 = -ω2X0

and Substitute and rearrange,

Z0 = -A0 / [ω2 – ω02 - i2ωω0ζ].

Z0 = -A0 / [ω2 – ω02 - i2ωω0ζ]

• There are three regions for this equation: (1) If b = 0 (no damping), Z0 will be infinite for ω = ω0 .

This means that the signal at the resonance of an undamped accelerometer can lead to infinitely large signals. Thus accelerometer designers generally impose finite damping on the system.

(2) If ω < ω0,

this expression can be simplified to (assuming the ω02 term is much greater

than the other terms in the denominator) as, Z0 = A/ ω0

2 = -ω2X0/ω02

In this case, the displacement of the mass is proportional to the acceleration of the frame, and is the response needed for an accelerometer.

(3) If ω > ω0 , the expression simplifies to: Z0 = X0 .

This is the case for high frequency signals, during which the mass remains stationery, and the accelerometer frame shakes around it. In this case, the displacement between the mass and the frame is the same size as the motion of the frame. This mode of operation is generally referred to as the ‘seismometer mode’. Seismometers are instruments that measure ground motion rather than ground acceleration.

1.0

1.0

2.0

2.0

3.0

3.0

4.0

4.0 5.0ω/ω0

x0/X0

0

b=

0

0.25

0.5

0.75

1

Accelerometer Seismometer

Figure 2

• Thus the general accelerometer consists of a mass, a spring and a displacement transducer.

• The overall performance of an accelerometer is generally limited by the mechanical characteristics of the spring (linearity, dynamic range, cross-axis sensitivity) and the sensitivity of the displacement transducer

• Recent advances in microelectronic engineering have given us a class of devices known as Micro ElectroMechanical Systems, generally described by the acronym MEMS. The dimensions of these devices are in micrometers (microns).

• The most common MEMS accelerometer is the ADXL50.• The ADXL50 is based on differential capacitance changes due to changes

in acceleration. • The principle of operation of the device is still based on the General

Accelerometer described above.

Capacitive Sensing

• Capacitance measurement can be used to detect the motion of a sensor element. • A simple example would involve the motion of one electrode in a plane parallel to

the electrodes as shown in Fig 1. • Electrodes dimensions are length L and width W. • If one electrode moves laterally a distance x, the capacitance changes from εε0LW/d to εε0(L-x)W/d. • Thus the capacitance changes linearly with displacement. • To implement such a sensor, it is necessary to guarantee that the lateral motion

does not also affect the separation d between the plates. • This approach is difficult to use for measurement of very small lateral

displacements.

W

L

d

L

xFig 1

• The most common use of capacitive detection for sensors is based on signals which are coupled to changes in the electrode separation d as shown in Fig 2.

• A physical signal causes the separation to increase by a small quantity ∆. • The capacitance changes from εε0A/d to εε0A/ (d+∆)• The relationship between the displacement and change in capacitance is

now not linear, but for small changes in separation we can approximate the capacitance by using the Taylor series expansion to give

C = (εε0A/d) [1 -∆/d +½ ∆2/d2]• So for ∆ << d, the capacitance change is linear with respect to

displacement

L

W d

L

d+ΔFig 2

• A technique for reducing the effect of the non-linearity is to use a differential capacitor shown in Fig 3.

• In this case, the capacitance measuring circuit is set up to measure the difference between the two capacitances which can be expressed as:

∆C = C2 – C1 = εε0A/ (d-∆) - εε0A/ (d+∆)

= εε0A/d [1 +∆/d + ½∆2/d2] - εε0A/d [1 -∆/d + ½∆2/d2] = εε0A/d [2∆/d]• In this case the non-linearity associated with the ∆2/d2 term is subtracted away,

and the first non-linearity appears as a cubic ∆3/d3 term, which would be much smaller than the squared term.

• The linearity in capacitor sensors are important as generally, capacitive measuring techniques are only applied in cases where precision measurement is necessary.

dd

Δ

C1 C2Fig 3

ADXL50

• Analog Devices ADXL50 is the first commercially available surface-micromachined accelerometer with integrated signal processing.

• It measures acceleration in a bandwidth from DC to 1 kHz with 0.2% linearity, and it outputs a scaled DC voltage.

• It is fabricated on a 9 mm2 chip. • It also is a force-balance device which uses electrostatic force

to null the acceleration force on the “proof” (seismic) mass, with advantages in bandwidth, self test and linearity.

ADXL50

• The ADXL50 uses an exceptionally small (1mm2) capacitive sensor element.

• The full scale range is ±50 g, compatible with automotive airbag deployment requirements, and the accuracy is 5% over temperature and power supply extremes.

• It uses a 5 V supply and includes a calibrated high level output and a self testing feature, at a high volume price of US $5 (or less).

• A more sensitive version, the ADXL05 is also available, spanning the range of ±5 g with 0.005g resolution and a typical nonlinearity of 0.3% full scale.

ADXL50 Structure

tether

anchor

proof mass

amp

electrodes

x0

w

0⁰ 180⁰

Gap x0 = 2 µm, Electrode width w = 2 µm, Proof mass = 0.1 x 10-6 g,

40200µ

2µ x 2µ

ADXL50 Structureelectrodes

x0

w

0⁰ 180⁰

Gap x0 = 2 µm, Electrode width w = 2 µm,

40200µ

ADXL50 Structure

tether

anchor

proof mass

amp

electrodes

x0

w

0⁰ 180⁰

Gap x0 = 2 µm, Electrode width w = 2 µm, Proof mass = 0.1 x 10-6 g,

40200µ

2µ x 2µ

ADXL50 Block Diagram

fixed plate

fixed plate

moving plate

1 MHz

0⁰

1 MHz

180⁰

3.4 V DC

0.2 V DC

demod LPF

built-in test

3M

x0

+

-

E0

Force-balance negative feedback

Accelerometers with charge output: Charge Amplifiers

• Accelerometers with charge output generate an output signal in the range of picocoulombs (pC) with very high impedance.

• To process this signal by standard AC measuring equipment it needs to be transformed into a low impedance voltage signal.

• Charge amplifiers are used for this purpose.

GND

Sensor Cc Cinp

qin

qc qinp uinp

qf Cf

Rf

uout

+

-

• The input stage of a charge amplifier features a capacitive feedback circuit which balances the effect of the applied charge input signal.

• The feedback signal is then a measure of input charge.

GND

Sensor Cc Cinp

qin

qc qinp uinp

qf Cf

Rf

uout

+

-

• The input charge qin is applied to the summing point (inverting input) of the amplifier.

• It is distributed to the cable capacitance Cc, the amplifier input capacitance Cinp and the feedback capacitor Cf.

• The node equation of the input is therefore, qin = qc + qinp + qf .

GND

Sensor Cc Cinp

qin

qc qinp uinp

qf Cf

Rf

uout

+

-

• Using the electrostatic equation q = u C (where u denotes voltage) and substituting for qc, qinp and qf ,

qin = uinp(Cc + Cinp) + ufCf where uf is voltage across Cf. • Since the voltage difference between the inverting and the non-inverting

input of a differential amplifier becomes zero under normal operating conditions, we can assume that the input voltage of the charge amplifier u inp will be equal to GND potential.

• With uinp = 0 we can simplify the equation to qin = ufCf

and solving for the output voltage uout = uf = qin / Cf.

GND

Sensor Cc Cinp

qin

qc qinp uinp

qf Cf

Rf

uout

+

-

uout = uf = qin / Cf

• This shows that the output of a charge amplifier depends only on the charge input and feedback capacitance. Input and cable capacitance has no influence on the output signal.

• This is an important fact when measuring with different cable length and types.

GND

Sensor Cc Cinp

qin

qc qinp uinp

qf Cf

Rf

uout

+

-