Accelerometers Accelerometers are devices that produce voltage signals proportional to the acceleration experienced. There are several techniques for converting acceleration to an electrical signal. The most general technique is described first and more recent techniques will be considered later.

1. 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 = ∂2
x/∂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.

2. • 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

3. • 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
k
k
damping
b
x
X
sensor

4. • 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
k
k
damping
b
x
X
sensor

5. k (X-x) + b d (X-x) = m d2
x
dt dt2
• Take Z = X-x or x = X - Z, then
m d2
X/dt2
= m d2
Z/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
-mω2
X0eiωt
= -mω2
Z0eiωt
+ kZ0eiωt
+ iωbZ0eiωt
.

6. • If we assign
ω0 = (k/m)½
ζ = b/[2(km)½
] the damping ratio
(another definition of the damping term)
A0 = -ω2
X0
and Substitute and rearrange,
Z0 = -A0 / [ω2
– ω0
2
- i2ωω0ζ].

7. Z0 = -A0 / [ω2
– ω0
2
- 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 ω0
2
term is much greater
than the other terms in the denominator) as,
Z0 = A/ ω0
2
= -ω2
X0/ω0
2
In this case, the displacement of the mass is proportional to the
acceleration of the frame, and is the response needed for an accelerometer.

8. (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 5.0
ω/ω0
x0
/X0
0
b=
0
0.25
0.5
0.75
1
Accelerometer Seismometer
Figure 2

9. • 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

10. • 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.

11. 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

12. • 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

13. • 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 C2
Fig 3

14. 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.

15. 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.

16. 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,
40
200µ
2µ x 2µ

17. ADXL50 Structure
electrodes
x0
w
0⁰ 180⁰
Gap x0 = 2 µm, Electrode width w = 2 µm,
40
200µ

18. 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,
40
200µ
2µ x 2µ

19. 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

20. 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
+
-

21. • 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
+
-

22. • 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
+
-

23. • Using the electrostatic equation q = u C (where u denotes voltage) and
substituting for qc, qinp and qf,
qin = uinp(Cc + Cinp) + ufCfwhere 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
uinp 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
+
-

24. 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
+
-