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1. Introduction to acceleration sensors
   
   I am working on my final year project which involves the use of the Nitendo Wii Nunchuk. The Wii Nunchuk operates based on a 3-axis accelerometer, a joystick and two push buttons. I carried out a small theoretical research on the accelerometer, and in this wiki page I am going to briefly explain what an accelerometer is, how it works, types of accelerometers, characteristics of accelerometers, how to select an accelerometer for a project requirement, and shows some applications of accelerometers.
   
   Physically, acceleration is a vector quantity having both direction and magnitude that is defined as the rate at which an object changes its velocity with respect to time. It is a measure of how fast speed changes . An object is accelerating when its velocity is changing. [1] In order to measure acceleration, an acceleration sensor called accelerometer is used. Accelerometer measures in units of g. A g is the acceleration measurement for gravity which is equal to 9.81 m/s². However, depending on altitude, this measurement can be 10 m/s²  in some place.
   
   
   

2. Working principle of an accelerometer
   
   The design of an accelerometer is based on the application of physics phenomenon. In aviation, accelerometers are based on the properties of rotating masses. In the world of industry, however, the design is based on a combination of Newton's law of mass acceleration and Hooke's law of spring action. This is the most common design applied to the making of accelerometers, and therefore, in this wiki page I will focus on explaining the accelerometer's working principle  based on this combination of Newton's law and Hooke's law. Figure 1 shows a simplified spring-mass system. In figure 1a, the mass of mass m is attached to a spring at equilibrium position x0 which in turn is attached to the base. The mass can slide freely on the base. Suppose that the base friction is negligible. Figure 1b shows the mass is moving to the right by a displacement of Δx = x -  x0. Since the mass is slowing down, the direction of acceleration vector is to the left. In this case, the mass is subject to the force according Newton's second law and Hooke's law.
   
   
   Figure 1. A simplified spring-mass system accelerometer. [2]
   Reprinted from Process Control Instrumentation Technology book
   
   According to Newton's second law, if a mass, m, is undergoing an acceleration, a, then there must be a force, F, acting on the mass with a magnitude of

   
   Hooke's law states that if a spring of spring constant k is extended from its equilibrium position () by a distance of

   x is the current position with reference to 

   Then there must be a force acting on the spring given by

   
   
   Equating equations (1) and (2) yields

   
   
   
   where k is spring constant in N/m
   Δx is the displacement in m
   m is the mass in kg
   a is acceleration in m/s²
   
   Equation (3) shows the measurement of acceleration in relation to mass, spring constant and spring extension. This is equivalent to the the linear equation y = λ x where y represents acceleration a, λ = k /m being a constant, and x being the displacement of the mass. This is how the design of an accelerometer is based on. The design and types of accelerometer based on spring-mass system differ in how this displacement measurement is made, and this comes to discussion on section 3 of this wiki page.
   
   

3. Characteristics of spring-mass system based accelerometers
   
   The equation (3) as a result of the analysis from figure 1 and b holds true on the assumption that there is no friction applied to the mass m. In practice, other parameters such as natural frequency and damping coefficient need to be considered since they have a profound effect on the application of accelerometers. 
   
   The mass-spring system exhibits oscillations at some characteristic natural frequency. This natural frequency is given by

   
   where f is natural frequency in Hz
   k is spring constant in N/m
   m is seismic mass in kg
   
   If there was no friction in the spring-mass system, the mass would oscillate forever. This is, however, not the case in reality. The friction that causes the system to rest is defined by a damping coefficient that has a unit of s^-1. The effect of oscillation is described by periodic damped signal which has the equation as follows. [2,2]

   
   where XT(t) is transient mass position
   μ is damping coefficient
   f is natural frequency in Hz
   
   Now two parameters that affect the accelerometer have been described. If the spring-mass system is exposed to a vibration, then the resultant acceleration is given by

   
   Applying equation (4) to equation (3) yields

   
   
   
   
   Figure 2 a. Response of spring-mass system to vibration compared to w^2 prediction b. Effect of various peak motion [2,3]
   Reprinted from Process Control Instrumentation Technology book
   
   Define f as the applied frequency, and fN is the natural frequency of the accelerometer. When f < fN , the natural frequency has little effect on the operation of the accelerometer. when f > fN, the accelerometer output is independent of the applied frequency. An important point worth noting is that with the applied frequency much larger than the natural frequency, the accelerometer becomes a measurement of vibration displacement . At near the resonance of accelerometer's natural frequency, the output of the accelerometer becomes high non-linear.
   A rule of thumb is that with f < fN, the safe maximum applied frequency should be f < 2/5 fN.With f > fN, the minimum applied frequency should be f > 5/2 fN.[2,4] So to best avoid the severe effect of resonance, the accelerometer should not be used near the resonance of their frequency because the output will become non-linear. Furthermore, the accelerometer should not be used with the applied frequency larger than the natural frequency if the measurement system is meant to measure acceleration.This has an important implication in the selection of the accelerometer for an application.

   
   
   

4. Characteristics of an accelerometer
   
   The equation of motion of an accelerometer is given by:

   Where M (kg) is the seismic mass, x (m) is the displacement from its equilibrium, d²y/dt² is the acceleration applied to the accelerometer case, b is the damping coefficient.
   
   To describe the relationship between the accelerometer input and output, a transfer function is used. Taking Laplace transformation from the above equation yields

   where X(s) and A(s) are the Laplace transforms of x(t) and d2y/dt2, respectively. Solving the above for X(s) we receive

   Equation (5) represents a typical transfer function for accelerometer.
   
   
                                                            Figure 3. Mechanical characteristics of LIS3L02AL accelerometer
                                                     Reprinted from STMicroelectronics LIS3L02AL accelerometer datasheet [8]
   
   

   Like any other sensors, a typical accelerometer possesses several characteristics which are needed to understand. Those characteristics are explained as follows:
   
   - Sensitivity: sensitivity is the output voltage produced by a certain force measured in g's. It is usually expressed in terms of volts per unit of acceleration under the specified conditions. The sensitivity of accelerometers is typically measured at a single reference frequency of a sine-wave shape. In the United States of America, it is 100 Hz while in most European countries it is 160 Hz. This is because they are removed from the power line frequencies and their harmonics. The LIS3L02AL accelerometer which is used in the Nintendo Wii Nunchuk has a typical sensivity of 0.66 at Vdd = 3.3V and temperature = 25 degrees as can be seen from figure 3.

   - Frequency response: is the output signal over a range of frequency where the sensor should operate. Unfortunately, STMicroelectronics does not give any information about the frequency response in their datasheet for the LIS3L02AL accelerometer.

   - Acceleration rage: is the level of acceleration supported by the sensor's output signal specification, typically specified in ±g. From figure 3, we can see that the acceleration range for the LIS3L02AL accelerometer is typically ±2g.
   

   - Sensitivity change due to temperature: is specified as a % change per ºC. The value of this parameter for LIS3L02AL accelerometer is ±0.01.

   - Nonlinearity: is a measurement of the deviation of an accelerometer response from a perfectly linear response. It is specified as a percentage with respect to either full-scale range (%FSR) or ± full scale (%FS).
   

   - Zero-g level: this is measured in V, and it specifies the range of voltages that may be expected at the output under 0g of acceleration

   - Cross-axis sensitivity: is a measure of how much output is seen on one axis when acceleration is imposed on a different axis, typically specified as a percentage. The coupling between two axes results from a combination of alignment errors, etching inaccuracies, and circuit crosstalk.

   - Noise density:

   Noise Density, in ug/rt(Hz) RMS, is the square root of the power spectral density of the noise output. Total noise is determined by the equation:

   Noise = Noise Density * sqrt(BW * 1.6)

   where BW is the accelerometer bandwidth, set by capacitors on the accelerometer outputs.

   Until now, all the basic characteristics of an accelerometer have been explained. One important characteristic is the sensitivity and the question is how to improve the sensitivity of the accelerometer. For analog-output sensors, sensitivity is ratiometric to supply voltage. Therefore, the sensitivity can be improved by increasing the supply voltage for the sensor as long as it does not exceed the maximum rating power supply specified by the datasheet. So this means that doubling the supply, for example, doubles the sensitivity. [9]
   
   
   
5. Types of accelerometers
   
   So far, I have discussed mainly spring-mass based accelerometers since it is the focus of this wiki page. In addition to that, there are many other accelerometers that are designed based on other physical phenomenon. In fact, what makes a difference between the types is the sensing element and their operating principles. There are many types of accelerometers in real life applications such as capacitive, piezoelectric, piezoresistive, Hall effect, magnetoresistive, heat transfer, MEMS-based accelerometers, to name just a few. Commonly used accelerometers, however, will be represented in this wiki page.
   
   
   4.1 Potentiometric accelerometer
   
   This is a type of an accelerometer which bases its working principles on the spring-mass system. The potentiometric accelerometer employs a mass (seismic mass), a spring, a dashpot, and a resistive element. The seismic mass is connected between a spring and a dashpot. The wiper of the potentiometer is connected to the mass.The following figure illustrates the structure of the potentiometric accelerometer.
   
   
   
   
   Figure 3. Structure of a potentiometric accelerometer [3]
   
   The way it works is simple. It measures the motion of the seismic mass by attaching the wiper arm to the spring-mass system. When the mass is moving, the position of the wiper changes according, thus changing the resistance of the resistive element. Since the natural frequency fN of the potentiometer accelerometer is generally less then 30Hz, this type of accelerometer should be used in low frequency vibration measurements. 
   
   
   4.2 Hall effect accelerometer
   
   Hall effect accelerometer is based the working principles of on spring-mass system. The output voltage varies according to a change in magnetic field from the magnet which is attached on a seismic mass. The mass deflects because of the forces due to acceleration. The output Hall voltage is calibrated in terms of acceleration.
   
   
   Figure 4. Simplified structure of a Hall effect accelerometer [4,3]
   
   
   4.3 Capacitive accelerometer
   
   Capacitive accelerometer operates based on spring-mass system working principles. It differs from Hall effect accelerometer and potentiometric accelerometer in its sensing element. Figure 5 shows the structure of a capacitive accelerometer, The sensing electrodes are in stationary state, and the diaphragm which is attached to the seismic mass is sandwiched in between the two sensing electrodes creating two capacitors.
   
   Figure 5. Structure of capacitive accelerometer [5]
   
   The vibration because of the forces due to acceleration causes the seismic or proof mass to move. The motion of the mass leads to the capacitance change of the sensing electrodes so as to determine the acceleration. The movement of the diaphragm causes a capacitance shift by altering the distance between the two parallel plates, with the diaphragm itself being one of the plates.   
   
   4.4 Piezoresistive accelerometer
   
   Unlike the three types of accelerometers mentioned above, Piezoresistive accelerometers do not use a spring. Instead of that, the mass is attached to cantilever beam which in turn is sandwiched in between strain gages.
   
   
   
   Figure 6. Piezoresistive accelerometer [4,2]
   
   Piezoresistive accelerometer's working principle is based on piezoresistive effect. As far as the piezoresistive effect is concerned, applied mechanical stress changes the resistivity of a semiconductor. The force exerted by the seismic mass changes the resistance of the strain gages. Piezoresistive accelerometers are used in high shock applications, and they can also measure accelerations down to zero Hz or up to ±1000g. But, the disadvantage is that they have limited high frequency response. [6]
   
   
   4.5 MEMS-Based Accelerometers
   
   MEMS stands for Micro-Electro Mechanical System. It is the technology which is based advanced technologies used to form small structures with dimensions in micrometer scale. MEMS technology is now being employed to manufacture state-of-the-art MEMS-based accelerometers.
   Initially, MEMS accelerometers was designed using piezoresistors. Since piezoresistors are less sensitive than capacitive detection, the majority of MEMS accelerometers nowadays use capacitive sensing principle. MEMS-based accelerometer typically consists of a proof mass with plates attached through a mechanical suspension system to a reference frame. Movable plates (part of the seismic mass) and the outer plates in stationary state form differential capacitor. Because of the forces due to acceleration, the seismic mass deflects; the deflection is measured in terms of capacitance change. [4,3]
   
   
   
   Figure 7. MEMS-based accelerometer structure [4,3]
   
   
   
6. Selection of an accelerometer
   

   In order to make a decision on which accelerometer can be used based on the requirements, it is necessary to understand the specifications of the accelerometer. These specifications can be found from the datasheet of an accelerometer. They include dynamic specifications, electrical specifications and mechanical specifications. Some of the important ones are:

    

   - Dynamic range: This is the +/- maximum amplitude that the accelerometer can measure before distorting or clipping the output signal.     Dynamical range is typically specified in g's.

    

   - Sensitivity: Sensitivity is the scale factor of a sensor or system, measured in terms of change in output signal per change in input measured. Sensitivity references the accelerometer's ability to detect motion. Accelerometer sensitivity is typically specified in millivolt per g of acceleration(mV/g).

    

   - Frequency response: Frequency response is the frequency range for which the sensor will detect motion and report a true output. Frequency response is specified as a range measure in Hz.

    

   - Sensitivity axis: Accelerometers are designed to detect inputs in reference to an axis; single-axis accelerometers can only detect inputs along one plane. Tri-axis accelerometers can detect inputs in any plane and are required for most applications. The accelerometer used in the Nintendo Wii Nunchuk is an example of tri-axis accelerometer.

    

   - Size and mass: Since the size and mass of an accelerometer have an effect on the object being tested, the mass of the accelerometers should be greatly smaller than that of the system to be monitored. [4,4]

    

   In addition to understanding some of the important specifications just mentioned, it should be noted that if the measurement is meant to measure acceleration, then the applied frequency f should be less than the natural frequency fN of the accelerometer.

   As a rule of thumb, in low-frequency applications (having a bandwidth on orders from 0 to 10 Hz), position and displacement measurements generally provide good accuracy. In the intermediate-frequency applications (less than 1 kHz), velocity measurement is usually favored. In measuring high-frequency motions with appreciable noise levels, acceleration measurement is preferred. [7,327]

   
   
   
7. Applications of accelerometers
   

   Accelerometers have many different applications ranging from industry, entertainment, sports to education. These applications can be, for example, triggering airbag deployments or monitoring of nuclear reactors. Accelerometers can also be used to measure static acceleration (gravity), tilt of an object, dynamic acceleration in an aircraft, shock to an object in a car, orientation or vibration of an object. Cell phones, washing machines or computers nowadays also have accelerometers.

    

8. References: