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The Science Behind Smartphone Accelerometers

The accelerometer has become one of the most important sensors in our cell phones. In this article we find out in detail what it is used for and how it works.


What is the accelerometer of a smartphone used for?

Until the introduction of sensors in mobile phones, the accelerometer was a little-known scientific instrument, confined to very specialized applications such as the inertial stations of planes and submarines, the detection of shocks for the triggering of airbags, or the study of earthquakes .

With progress in miniaturization which has made it possible to have accelerometer chips smaller than a millimeter, this sensor has taken on another dimension and is now present in all smartphones. With the release of the iPhone in 2007 it was only used to manage the automatic orientation of the screen , then over the years it has taken an increasing place in our digital devices.

Today, accelerometers are used in the field of prevention , to alert emergency medical services in the event of falls. It is also used to analyze our physical activity , for example, to recognize if we walk and how many steps we have taken, or if we climb stairs. Coupled with other sensors such as the gyroscope or the magnetometer, they are used to know the position of a mobile at any time and are used in games for example.

Different ways to calculate acceleration

Acceleration is a physical quantity that describes the variation in the speed of an object over time. It corresponds to the measurement of the increase or decrease in the speed of an object per unit of time. Acceleration can be positive or negative, depending on the direction of change in speed, and is expressed in meters per second squared (m/s²) in the international system of units (SI).

Mathematically, the acceleration is given by the formula:

a = δ v/ δ t

where a is the acceleration, v the speed and t the time.

To measure acceleration it is therefore necessary to have a means of accurately calculating the speed. unfortunately in most cases it is difficult or even impossible to calculate the speed of an object with sufficient precision .

We then resort to another way of calculating the acceleration: using Newton's second law or fundamental principle of dynamics which states that a resultant force exerted on an object is always equal to the product of the mass of this object by its acceleration . If we are able to measure the force exerted on a mobile, then we can deduce the acceleration to which it is subjected.

Using physical constructions like springs, it is easier to determine the force exerted on a body and to determine its speed. We then have a way to measure the acceleration. Most accelerometers use this second method to determine acceleration.

Accelerometer working principle

Let's imagine that we connect a small mass of sufficiently low weight to a spring itself connected to the frame of the device whose acceleration we want to know, as shown in the graph below.

Consider the components of the diagram above. If we move the smartphone, the small mass will initially remain in its position by inertia, and the length of the spring will change by a value that we note x. This deformation of the spring creates a restoring force which is proportional to its elongation: F = kx with k the stiffness of the spring, and x the displacement.

According to Newton's second law this force creates an acceleration of the mass such that F = ma where a is the acceleration of the mass and m its weight. We deduce that kx = ma from which a, the acceleration of the mass: a = kx/m.

Solving the differential equation shows that the acceleration of the laptop is equal to the sum of two terms: a term depending on the displacement x and an oscillation on which it depends (k/m)^0.5. If the stiffness of the spring k is large compared to m, the oscillation term is negligible and the acceleration of the smartphone is directly proportional to the displacement x .

Displacement measurement

How to measure the displacement x? Direct reading can only be used in cases where the acceleration is continuous. For example, to calculate the acceleration experienced by an astronaut in a centrifuge or a pilot in an airplane. But if this value values quickly this method is not suitable.

One of the methods used in modern sensors uses the characteristics of capacitors. A capacitor is made up of two conductive plates separated by an insulator. Its main property is to be able to store opposite electrical charges on its reinforcements. It turns out that the storage capacity of a capacitor is inversely proportional to the distance between the conducting plates.

There are many ways to electronically calculate the capacitance of a capacitor, if we connect one plate to the mobile and another to the ground connected to the spring, we can then estimate the spacing of the plates by calculating the capacitance of the capacitor .

The combination of a spring and a capacitor is the most commonly used method for calculating acceleration.

MEMS technology

One of the difficulties that engineers encountered is to reduce the size of the sensor so that this measuring instrument fits into a laptop. This is where MEMS technology, which stands for Micro Electro Mechanical System, comes into play. A MEMS is a small integrated circuit in which we have completely integrated mechanical parts and electronic parts. The first MEMS were developed in the 1970s.

What does a MEMS accelerometer look like? This is the MEMS photo of an iPhone 4 ( We see in this photo the springs, the mass which surrounds the object, and the capacitors which are oriented in two directions, X and Y. These two series of capacitors are oriented at right angles to measure the acceleration in two directions. If we want to know the acceleration in the three directions, we must add a third accelerometer in the direction of the face of the smartphone. As these are generally not very thick, the engineers modified the design and in the photo we see this sensor above the other two.

Absolute acceleration and linear acceleration

Now that we know how the accelerometer in our smartphones works, let's try to understand what exactly they measure. When we place our sensor vertical, the mass of the accelerometers is attracted by gravity, and so the sensor will indicate a force and therefore an acceleration, that of gravity, g. This is why the sensor displays gravity when it is at rest. If I drop my laptop, then the laptop is weightless for a short time, and the absolute acceleration is zero. You can check this with FizziQ by dropping your phone on a (soft) bed and recording the absolute acceleration. This experiment will also allow you to calculate the gravity g by measuring the duration of the fall.

The acceleration which is calculated by the accelerometer is called absolute acceleration , it is also called acceleration with g because it includes gravity. It is distinguished from linear acceleration or acceleration without g , which is the acceleration of a mobile when we remove the vector of gravity. We will see in another article how to calculate this value which is essential for certain applications.

Accelerometer accuracy and calibration

The acceleration measured by the MEMS of our laptops is by construction affected by gravity, and therefore at rest the accelerometer will display the value equal to the acceleration of gravity, i.e. 9.81 m/s². Using the different components of absolute acceleration, we can determine the orientation of the laptop. If I select absolute vertical acceleration, I will find the projection of the acceleration due to gravity on the vertical axis of my laptop. My flat laptop displays zero, but vertically, the measurement is 9.81 m/s²...

Accelerometer calibration is a process that corrects possible accelerometer measurement errors by recalibrating it. This may be necessary if the device is not detecting movement correctly or recording incorrect measurements. If we measure the absolute acceleration of a laptop at rest, we find that the number displayed is not exactly equal to 9.81 m/s² but at a value close to this value.

In fact, all laptops will show different values because the sensors are not precisely calibrated to give this information. Is this a problem? Not really because accuracy better than 1% is not necessarily necessary for usual motion recognition applications. It would be a different story if we used these sensors to calculate our position like nuclear submarines do...

Note: Thanks to Daniel Rouan for his contribution on the theoretical calculations.

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