# Accelerometers

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Last updated : 03/18/2009

### Introduction

This article presents a brief review on micro machined accelerometers. Different device structures and designs are discussed alongside with their specification and principle of operation. Also a simple design of a single axis accelerometer tutorial is presented.

### MEMS Accelerometers

Fig.1: Three axis linear accelerometer from ST

The application of micro-accelerometers covers a wide range of fields due to their small size, high performance and low cost. This clearly confirms its second largest sensor market share after pressure sensors. Micro-accelerometers are commonly used tool in automotive, biomedical, industrial, military and numerous consumer applications since it is crucial for safety, measurement and control. Micro-machined gyroscopes (for rate or angle of rotation measurement) are also an emerging technology and it is paving its path into commercialisation as a small companion device with accelerometer. It is becoming an important tool for automotive and different other applications for the low cost, small size compared to conventional gyroscopes. Both micro-accelerometers and micro-gyroscopes are a type of micro-machined inertial sensors.

### Micro-machined Accelerometers:

Structure, Specification and Operation:

Fig.2: Mechanical model of an accelerometer

The general structure of an accelerometer consists of a proof mass suspended by compliant beams anchored to a fixed frame. This structure can be modelled by means of a second order mass-dumper-spring system Fig. 2. The fixed frame displacement relative to proof-mass is sensitive to external acceleration which in turn changes the internal stress in the suspension spring. These two values can be used to measure the external acceleration using the Newton’s second law and the accelerometer model to obtain the mechanical transfer function:

$H(s)=\frac{x(s)}{a(s)}=\frac{1}{s^2+\frac{D}{M}s+\frac{K}{M}}=\frac{1}{s^2+\frac{\omega_n}{Q}s+\omega_n^2}$

Where:
a is the external acceleration and x is the proof mass displacement
M : is the proof mass
D : is the dumping factor
K : is the spring constant
$\omega_n=\sqrt{K/M}$ : is the natural resonant frequency
$Q=\sqrt{KM}/D$ :is the quality factor
At a static state (s = 0) the sensitivity of the accelerometer is:
$\frac{x_{static}}{a}=\frac{M}{K}=\frac{1}{\omega^2_n}$

The total noise equivalent acceleration due to Brownian motion of mass proof suspensions or gas molecules surrounding it is:

$TNEA=\frac{\sqrt{4k_BTD}}{M}=\sqrt{\frac{4k_bT\omega_n}{QM}}$

The final micro-machined accelerometer design can be simulated and optimized using commercially available finite element method or dedicated MEMS software packages. The specification of micro-accelerometer is concerned with its sensitivity, frequency range, off-axis sensitivity and other application specific parameters.

### Classification for Micro-machined Accelerometers:

The classification of accelerometers is based on transduction mechanisms, the most useful ones are:

• #### Piezoresistive Accelerometers

• A silicon piezoresistor is generally placed at the edge of the rim and proof mass where stress variation is maximum. This causes change in the resistivity as the beam changes its mechanical state. The structure, fabrication process and circuitry of these devices are simple. However, they have a larger temperature sensitivity and smaller overall sensitivity which drops its accuracy.

• #### Capacitive Accelerometers:

• This is based on the gap variation between the proof mass and a fixed electrode which in turn changes the capacity. The capacitive based accelerometers are preferred for several reasons: simple structure, high performance, low cost, low power dissipations, high sensitivity and low temperature sensitivity. Although it is susceptible to electromagnetic interference, good packaging and shielding prove to be the solution.

• #### Tunnelling Accelerometers:

• This type of devices uses a constant tunnelling current between a tip attached to a movable microstructure and its counter electrode to sense the displacement. The tunnelling current is maintained constant as long as the distance and tunnelling voltage are unchanged. Once the proof mass moves due to acceleration, the circuit responds by adjusting the deflection voltage to bring the tip back to its place. Measurement of deflection voltage in this closed loop system can be used to calculate acceleration. Its drawback is that it is sensitive to low frequency noise.

• #### Resonant Accelerometers

• This is based on transferring proof mass inertial force to axial force on the resonant beams and hence shifting its frequency. The output is digital and highly sensitive.

• #### Thermal Accelerometers:

• The temperature flux between a heater and heat sink is proportional to the inverse of their separation; hence by measuring the temperature, the displacement can be also measured.

• #### Other Accelerometers:

• In addition to the above mentioned accelerometers types, there are other devices based on optical, electromagnetic and piezoelectric principles. The reason behind is to use advantages of both micro-machined and physical principle like optics are immune to noise and linear.

### How to design an Accelerometer?

The main steps to design a simple single-axis capacitive accelerometer bulk micromachined are covered in this tutorial.

The design specifications and parameters are required to take in consideration to design an accelerometer are:

The Bandwidth (###) Hz,

The Sensitivity (#. #) pF/G,

The Dynamic range +/- (##) G,

The Minimum detectable acceleration (#) mG.

#### Mechanical study

##### Design approach

Fig.3.1: Single axis accelerometer

The design to be fabricated is a single -axis capacitive accelerometer (See Fig. 3.1), this system contains:

• a mass (m),
• a spring (with constant k),
• and a dumper (with coefficient b).

This system can be translated to a simple mechanical system as it can be seen in Fig.3.2:

Fig.3.2: The Behaviour of an accelerometer

This system is just an approximation to the real approach behaviour. This mechanical system gives a second order system given by:

$m\frac{\partial^2x}{\partial t}+b\frac{\partial x}{\partial t}+kx=F_{ext} = ma$

dividing by m, thus:

$\frac{\partial^2x}{\partial t}+\frac{\omega_n}{Q}\frac{\partial x}{\partial t}+\omega^2_n=a$…….(1)

witch gives a transfer function (Lapalce domain):

$\frac{x(s)}{a(s)}=\frac{1}{s^2+\frac{b}{m}s+\frac{k}{m}}$ or $\frac{x(s)}{a(s)}= \frac{1}{s^2+\frac{\omega_n}{Q}s+\omega_n^2}$…….(2)

where:
$\omega_n=\sqrt{\frac{k}{m}}$: is the resonant frequency, and
$Q=\frac{\omega_nm}{b}$: is the quality factor.

#### Assumptions and limitations

The basic limitation are needed to look at it is the damping, where the accelerometer has to be critically damped, hence this permits to get the least amplitude distortion. This means that  $Q=2 \sqrt{2}$ therefore [3]:

$\frac{b}{2m\omega_n}=\frac{1}{\sqrt{2}}$…….(3)

In order to characterise the dumping we need to solve  the dominator’s equation by calculating the Δ of the transfer function (equation (2)) of  our system.

$s^2+\frac{b}{m}s+\frac{k}{m}=0$

$\Delta =(\frac{b}{m})^2-4\frac{k}{m}$              for Δ=0 thus:

$b=2\sqrt{km}$……..(4)

Three different cases can be distinguished then:

• Under dumped system  where $b<2\sqrt{km}$,
• Critically dumped system where $b=2\sqrt{km}$,
• Over dumped system where $b>2\sqrt{km}$.

The three different cases are illustrated by the graphs below:

3.3 Dumping characteristics

In Order to get a  maximum bandwidth, the sensing element should be critically damped [2].
Note also, that the mass should be big enough to conform to our sensitivity requirements,  and at the same time it has to be small enough to be compatible with “b ” in such a way we can  get critical damping.
Another  important assumption  which can   help us to find the right parameters  for designing our capacitive accelerometer; is to assign the sensitive gap “d” , since it is limited by the fabrication processes.

##### Bandwidth

The mechanical resonance frequency of a suspended mass is given by:

$\omega_n= \sqrt{\frac{k}{m}}$

This means that in an open loop arrangement a high sensitivity   yields to a small bandwidth. In a closed loop arrangement the resonance peak can be suppressed by the control circuit. The bandwidth is no longer limited by the mechanical resonance of the sensor but is limited by the transition frequency of the control circuit [6].

##### Minimum detectable acceleration and mechanical noise

The given specifications for our design are the  bandwidth, sensitivity, dynamic range and the minimum detectable acceleration. The minimum acceleration that the system can detect must be higher than the noise level, this means that the minimum acceleration is limited by the noise boundary.

The noise affects the system is a combination of two different noises come  from the mechanical sensor and electronic readout circuit.

In the mechanical study of the design,   we need to focus on the mechanical noise only.   However   in the electrical/electronic study which will follow later on we will  neglect  the mechanical noise and focus only on the electric noise since it is the dominant one in the electrical system.

The mechanical noise of the accelerometer is mainly caused due to the damping, which is called Brownian motion noise. This is used to specify the noise in terms of acceleration noise. Therefore the noise or the minimum acceleration can be detected, is given by equation (5) [4]:

$a_{min}= \sqrt{\frac{8 \pi k_B Tf_nB}{Qm}}$ or $a_{min}= \sqrt{\frac{4k_B TbB}{m}}$…..(5)

#### Sensitivity

The sensitivity in a capacitive accelerometer is defined by the difference of variation in the capacitance divided by the difference in variation in the displacement, in which the sensitivity equation is given by [4]:

$S_o=\frac{Am \varepsilon}{kd^2}=\frac{C_om}{kd}$…..(6)

Where:
ε: is the electric permittivity of air,
A: is the overlap area of electrodes ,
d: is the gap between the electrodes. However the gap between electrodes should be as small as possible and it is defined by the process of fabrication.

#### Dynamic Range

In an open loop arrangement the operating range is limited by the maximum deflection of the seismic mass. Since a small spring constant k yields a high sensitivity, seismic masses in high resolution accelerometers are suspended softly. Therefore, the operating range of these accelerometers is small. In our design the dynamic range of operation which is given equals to  ±##G. So the maximum measurable acceleration $a_{max}''$ is determined by [4]:

$a_{max}=\frac{kd_{max}}{m}$…..(7)

#### Spring constant

As it can be seen clearly in the equations above, that the spring constant  “k ” affects directly the resonant frequency, bandwidth, sensitivity and also the pull-in voltage. Instead in the real design the spring constant is related directly to the beam characteristics,  which are the length (L), the thickness (t), the width (W) and the elasticity of material  coefficient ( Young modulus (E)).

Note that the spring constant changes in a beam due to the tonsil and compressive stresses. However we assume that there is no variation in spring constant and the following equation can be applied :

$k=\frac{16Wt^3}{L^3}E$…..(8)

where:

E=190 GPa  (Young’s Modulus for the silicon).

#### Mass & Damping Factor

In our design we aim to get a critical damping for our system.  The damping force in the accelerometer arises from the so-called squeeze-film effect, i.e., the interaction of the silicon mass and the air-film trapped in the gap between the mass and the electrodes.  Provided that the “squeeze number” $\sigma=12\mu A\omega/(pd)^2<<1$ within the bandwidth of the accelerometer, the damping coefficient can be calculated from [2]:

$b=0.42 \mu A^2/d^3$…..(9)

$\mu = 1.85 X 10^{-5} N/m^2.s$   is the dynamic viscosity of air, and $p =1.013X10^5Pa$   is the atmospheric pressure,  $A$ is the area of the air film, and $\omega$  is the driving frequency of a sinusoidal excitation.

#### Parameters calculation

Since the dumping is directly related to the mass “m” so it is required that the mass should be big enough to confirm to our sensitivity given and small enough to be compatible with “b” so that we get critical damping.

Equation used  are:

The minimum acceleration is given by equation (5):

$a_{min}= \sqrt{\frac{4k_B TbB}{m}}$

The damping coefficient is given by equation (9) as:

$b=D=0.42 \mu A^2/d^3$

From equation  (6) the capacitance $C_o$ is:

$C_o=\frac{ k.d}{mS}= \frac{ A\epsilon_o}{d}$

combining both equations gives:

$D= \frac{ 0.42 \mu A^2}{d^3} \Rightarrow\frac{D.d}{ 0.42 \mu }=\frac{A^2}{ d^2}$

therefore:

$\frac{A }{ d }=\sqrt{\frac{D.d}{ 0.42 \mu }}$

#### Nominal capacitance $C_o$:

In order to calculate the nominal capacitance we need to combine equations (6) and (7) for the maximum value of distance “d” which is given in the assumption, thus we get:

$C_o= \frac{S}{ a_{max} }$ (In the range of picofarad (##pF))

#### The area (A) of the accelerometer:

The area “A”  can be calculated form the nominal capacitance “$C_o$“:

$C_o=\frac{ A\epsilon_o}{d}\Rightarrow A=\frac{C_od}{ \epsilon_o}$

#### The mass “m”:

In order to calculate the mass “m” we need to use a combination between equations (4) and (5) thus we get:

$m=\frac{\sqrt{4k_B.T.b.B}}{ a_{min}}$

#### The effective spring constant K:

In order to calculate k the spring constant we need to use equation (7):

$latex a_{max} =\frac{K.d}{m}\Rightarrow K=\frac{a_{max}.m}{ d} &s=2$

$a_{max}=\frac{K.d}{ m}\Rightarrow K=\frac{a_{max}.m}{ d}$

#### The resonance frequency:

$\omega_{n}=\frac{K.}{ m}$

### Design the sensing element

At this stage we need to define the sensing elements which  consists of the proof mass and suspension system in such a way, our design will meet any specifications given (Bandwidth, sensitivity, dynamic range, minimum detectable acceleration).  By calculating the length of the beam “L” (as it can be seen in Fig.3.4) and finding out the appropriate values for the width “w” and thickness ” t”  for the beams, the size of the proof mass has  to be  calculated also  and defining the thickness $T_m$,  the width $W_m$; then the Length $L_m$.

### Beam geometry

The dimensions of  thickness  and the width of the beam can be initially  chosen by the designer although   sometimes are limited by microfabrication capabilities. However the relationship between the thickness “t” , width “w”, and length “L”  have to obey the following equation [2]:

$K=\frac{16WT^3}{ L^3}E\Rightarrow L=\sqrt[3]{\frac{16WT^3E}{ K}}$

This capacitive accelerometer contains a cantilevered beam, shown in Figure 3.5, with the following dimensions, where L is length, W is width, t is thickness. The proof mass with  a thickness $T_m$,  a width $W_m$; and a  Length $L_m$. A gap $d_o$ which is the spacing between the substrate and the underside of the beam, $\rho$ is material density, and E is the material Young’s modulus (eg. Young’s modulus for silicon is $E=190.10^9 N/m^2$ )

### Proof mass geometry

We know that our proof mass volume is  $V=L_mT_mW_m$  since it is homogeneously parallelepiped with rectangular area A. volume should be calculated from the Volumic mass density  $\rho$ so:

$\rho=\frac{m}{ V}\Rightarrow V=\frac{m}{\rho}$

therefore to find the thickness $T_m$:

$V=L_mW_m T_m=A.T_m\Rightarrow T_m=\frac{V}{A}$

Volumic mass density for silicon for example is:$2300kg/m^3$

Fig.3.5:Sensing element dimensions

## Electrical & Electronic Studies

• ### Capacitive signal pick-off

Fig. 4.1: Shows the function of the designed accelerator with the electrical model including the parasitic capacitors [10

The third stage in our design is the conversion of the difference on capacitance to voltage. In a typical capacitive accelerometer, the proof mass is suspended above a substrate by compliant springs. When the substrate undergoes acceleration, the proof mass displaces from the nominal position, causing an imbalance in the capacitive. This imbalance can be measured using an integrator or using a charge amplifier which has been used in this design for this purpose in response for example to 5V sinusoidal signal with a frequency of 100 kHz applied to the sense capacitors. The derivation below confirms the relationship between the capacitance and the voltage applied.

Fig.4.2: difference in capacitance (equivalent electrical model without parasitic capacitor)

The values of both capacitances at the equilibrium  position$x_o$ are:

$C_1=C_2=C$

However if the mass “m” of the accelerometer moves , the values   capacitances will vary to:

$C_1=C\frac{x_o}{x_o+\delta x}$ $C_2=C\frac{x_o}{x_o-\delta x}$

For small displacement the difference in the capacitance is:

$C_1-C_2=C(\frac{x_o}{x_o-\delta x}-\frac{x_o}{x_o-\delta x})=C(\frac{-2x_o\delta x}{{x_o}^2-\delta x^2})$

since $\delta x^2\ll {x_o}^2$ therefore:

$C_1-C_2\simeq-C\frac{2x_o}{\delta x}$

On the other side:

$C_1+C_2=2C$

To calculate the voltage, use the voltage divider circuit in Fig.4.2:

$V_o=-V_s+\frac{C_1}{C_1+C_2}2V_s$ $V_o=\frac{C_1-C2}{C_1+C_2}V_s$

we calculated previously  $C_1-C_2$ and $C_1+C_2$ for small displacement, using these formulas, therefore:

$V_o\simeq-\frac{\delta x}{x_o}V_s$…..(10)

Equation (10) related the displacement to the voltage, which is defined the operation of the accelerometer described before. We can deduce from this equation that the output voltage is linearly proportional to the displacement.
As it is mentioned before that in order to make the measurement of the acceleration, an interface circuit is required for the accelerometer. A charge amplifier has been used in this design for this purpose. Figure 3.4 represents the entire block diagram that shows the function of the circuit proposed for an open loop configuration[8].

Fig.3.4: Block diagram for open loop configuration.

The open-loop block diagram is shown in Figure 3.4 consists of charge amplifier where the output signal from the accelerometer is amplified by an the op-amp of gain Av, and followed by a demodulator with the reference signal which is generated by the oscilloscope , and finally a low pass filter. Each block will be characterised later.

# the rest is coming soon .

### Fabrication process of a bulk micromachined accelerometer:

Microfabrication process of the mass and beams of the acceletometer

## References

[1] Yazdi, N., Ayazi, F. and Najafi, K. Micromachined inertial sensors. Proc. IEEE, Vol.86, No. 8, pp. 1640-1659, 1998.
[2] Michael Kraft, Notes, university of Southampton 2004.
[3] Senol Mutlu, Surface Micromachined Capacitive Accelerometer With Closed-Loop Feedback
[4] N. Malauf. An Introduction to Microelectronical System Engineerin. Aretech House. 2000.
[5] Kovacos, micromachined Transducers Sourcebook
[6] Beißner, S.1; Puppich, M.2; Bütefisch, Analog Force Feedback Circuit for Capacitive Micromechanical Acceleration Sensors.
[8] M. C. Wu.Case Study I: Capacitive Accelerometers. Chapter 19 of S. Senturia, Microsystem Design.
[9] Michael Kraft’s Thesis, http://www.ecs.soton.ac.uk/~mk1/. 13.05.2004
[10] Joseph I. Seeger and Bernhard E. DYNAMICS AND CONTROL OF PARALLEL-PLATE ACTUATORS BEYOND THE ELECTROSTATIC INSTABILITY. Sendai, Japan, June 7-9, 1999, pp. 474-477

1. [...] A: At first it was just a water tube containing air bubble which indicates the direction of the acceleration. Nowadays MEMS accelerometer can be classified to several types based on its sensing elements and principle of operation. The one that I am using is based on polysilicon surface-micromachined sensor. A good further reference about how an accelerometer works can be found here. [...]

2. Georg Tenckhoff says:

Hi,

as I understand the MEMS acelerometers (as all other acclerometers for use below their resonance frequency) measure a displacement of an elasticaly bound mass, which is in first order proportional to the acceleration. How does this fit to the fact that for constant acceleration amplitude the oscillations way amplitude is inversly proportional to the square of the circular frequency (it’s the second integral of it)? Where is my wrong assumption? Thanks for your reply!

George

3. hosung eom says:

just incredible thing!
really good thing! i’ll talk about accelerometer with my children this evening.. thanks for your kind explanation!

4. Ghuge Shivshankar says:

Help me regarding question 3,4 and 5
(Question)
The mechanical transfer function of an accelerometer relates the displacement of the mass (the output) to the applied acceleration (the input).
(i)Draw schematically the spring-mass-frame model of an accelerometer and illustrate the forces that act in this system when an external acceleration is applied.
(ii)Write the equation for the displacement of the mass under the applied forces.
(iii)What is the cross axis sensitivity of an accelerometer?
(iv)Consider that the applied acceleration is a sinusoidal function of time, producing vibrations of the mass. Illustrate schematically the amplitude of these vibrations as a function of the frequency (from zero to a frequency higher than the resonance frequency of the system).
(v)Illustrate the effect of the damping on this amplitude.
(vi)Illustrate schematically a capacitive surface micromachined accelerometer and explain its principle.
(vii)Name a few critical steps in the fabrication process of such an accelerometer

5. dinesh says:

hey would u help me out in converting the analog output in digital form

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