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Accelerometer Design

# Accelerometer Design

Now we know enough to understand  the basic function of an accelerometer which we want to look at in this chapter.  We also will see some design considerations which are dictated by the environmental constraints and then the basic set-ups using the different working modes of the piezo-element. Finally we will have a look at the electric output of the sensor with a short overview of common practice electronics used.

## Basic Principles

### Functional Principle

We retain:

• The piezoelectric accelerometer can only measure dynamic acceleration.
Below a certain minimal frequency the output becomes smaller and smaller.

• At static conditions the output becomes zero.

• The sensitivity is proportional to the mass of the inertial mass and the piezoelectric constant of the sensing element.

Functional Priciple

### Two Fundamental Properties

#### Discharge Curve

We have seen that a charged piezo element left on it’s own will discharge slowly due to the leak current  between the plus and the minus pole of the piezo element.

The discharge curve is a natural exponential function. The curve is characterized by the so called time constant T.

The time it takes for the discharge is directly proportional to the time constant. The discharge curve is a natural exponential function

The tangent to this curve at any point of time intersects the abscissa always exactly one time constant later.

The value of q(t) is decreasing continuously but theoretically never reaches zero.

If q(t=0) =Q  then the value of q is 63.2% of Q at the time t = T .

At t = 3T  the value of q(t) will be about at 5% and after 5 time constants it will be less that 1% of the initial value Q

How exactly is the discharge curve defined?

The time constant T equals to the internal resistance R times the capacitance C of the piezo element

T = R·C     [T] = sec

The shape of the exponential discharge curve is characterized by the function

-t/T                                 -t/RC

q(t) = Q ·  e          or    q(t) = Q ·  e

The capacity is determined by the shape and the material of the piezo element:

C = ɛ · A/t

​​RC  is called time constant  because in general electronics it is assumed that R and C are constants as well. However the dielectric “constant” ɛ  of a piezo material is normally quite variable with temperature up to a factor 2 or 3.

Even more the internal resistance is varying decades with temperature or external effects such as humidity or surface pollution. It is therefore fair to say that the time constant is dominated by the resistance and therefore strongly influenced not only by the piezo material itself but also by the direct environment, cleanliness, humidity and temperature !

#### Accelerometer Output Signal

m    = mass
a     = acceleration

dik   = piezoelectric constant

For a piezo ceramic in compression mode dik  would be d33.

The compression mode constant for other materials could well be d11  or d22 .  The output charge ​Q  of the accelerometer in general is

​Q     = dik·m·a

In the case shown here the acceleration is a= A∙ sin ωt

We remember that in a sinusoidal motion the acceleration is out of phase by 180° with respect to the displacement. If we are only interested in the magnitude of the vibration this is irrelevant.

However if we need the phase information, for example for balancing an engine the phase is of prime importance.  It is also a good idea to have the phase of the signal identical over all individual sensors of the same type. That's why we want to place the sensing element always in the same orientation.

The sensitivity is only depending on the piezoelectric constant and the mass. All technical materials show a certain variation of their characteristics and so does the piezoelectric constant. In practice we see often up to 10% variation. If we want the final sensitivity to be in a closer tolerance than that, we can adapt the mass accordingly. This is why the mass is designed to be about 10 to 15% heavier than really required so all sensitivities come out high. The final sensing element can then be calibrated by removing weight from the mass.

#### Accelerometer Output at Low Frequencies

The self discharge of our piezo element may lead to a problem at low frequencies. As the discharge happens not only for static charges but under all conditions it’s easy to imagine that  the refreshing of the oscillating charge signal must be faster than the discharging otherwise the charge signal will disappear. Although the signal will not disappear suddenly, the amplitude will be reduced more and more when the frequency is decreased. We will illuminate this relation between the oscillation frequency and the discharge time constant in the chapter  Accelerometer Properties

Accelerometer Output
Discarge Curve
positive or negative?

### Functional Modes

We find typically two function modes or principles used with the classic piezo accelerometers i.e. the compression mode and shear mode. In some particular design we find also the transverse mode.

#### Compression Mode

The figure shows a compression mode accelerometer.

Let’s define the parts between the base and the inertial mass as the sensing element. The sensing element in this case consists of two insulation rings, two electrodes with wires and finally the piezo element  in compression mode, which means the polarization is vertical through the thickness of the piezo-element (as indicated by the green arrow). The parts are normally held together by a central bolt but they might as well be glued or soldered.

This is an electrically insulated design and it is also electrically balanced (symmetric) when the mass is connected to the base for example through the bolt. Compression-mode accelerometer

A similar but simplified design we get when we omit the two insulators. One pole is then on the base and the other on the inertial mass. Hereby the sensing element becomes electrically asymmetric or single ended. Of course we have to omit the central bolt as well otherwise we have a short circuit from the mass to the base.

The compression mode accelerometer is probably the most common design as it is most simple and has very good basic characteristics. Due to the compact design it is also a good choice when we want a high resonance for a wide frequency response.

A possible drawback of the compression mode is the potential occurrence of the pyroelectric effect. We remember that certain piezo materials, especially all piezo ceramics, have the peculiarity that the remnant polarization varies with the temperature and the piezo element produces a charge. The pyroelectric charge appears in the poling axis i.e. we collect it entirely with our electrodes and it will be fully added to the acceleration signal.

#### Shear Mode

Another most popular design is the shear mode accelerometer. The variant shown on the right is a symmetric design with two identical sensing elements. The base part includes a central support plate. Two identical sensing elements are mounted left and right with the respective inertial weights at the ends.

A characteristic of the shear-mode is that the piezo elements are upright (in the direction of the sensitivity) with the electrodes side-wards.

Sometimes only one half of the set-up is used which results in a more simple design with only one sensing element. Shear mode accelerometer

The poling direction is upright as shown by the green arrows.

The contact wires are connecting the corresponding electrodes of left and right in parallel to increase the charge sensitivity. This design as shown is electrically balanced only if the two inertial masses are connected to the base!

Shear mode designs become normally very compact because the piezo electric constant of the shear mode is often higher than in the compression mode.

We will see later that a high sensitivity is a good precondition for a high resonance however elastic modules of the shear mode are normally smaller that in the compression mode.

With respect to the pyro effect the shear mode is a very good choice. The pyroelectric charge appears in the poling direction i.e. on the insulation face between the electrodes and is therefore not transmitted to the output.

#### Stacked Sensing Element

In order to increase the sensitivity we can use several piezo elements on top of each other in a stack. By using for example 3 piezo rings instead of one we will get 3 times the sensitivity. All the plus poles and the minus poles are connected together in parallel to one output.

By flipping every second piezo element upside down, the plus sides and the minus sides come together and we can connect them to the same electrode.

When we talk about increasing the sensitivity we have to distinguish between charge sensitivity and voltage sensitivity. To increase the charge sensitivity we have to connect the piezo elements in parallel, to increase the voltage the elements must be in series. Compression mode accelerometer with multiple elements

Mechanically the piezo elements are mounted in series which means that the same (inertial) force is carried by all the elements in series.

The use of stacks is possible in compression and shear mode designs.

#### Non-Insulated Designs

There are also designs where the sensing element is not insulated with respect to the housing which simplifies the design considerably.

A popular shear-mode design variant is using a tube shaped piezo element and inertial mass mounted around a center post. This results in a very compact and rigid design. The parts are normally soldered together. The base (together with the housing) is then one electrical pole of the sensing element while the inertial mass the other.

This kind of electrical layout is also called single ended as it has only one live pole, the other pole being the housing.

As we can already suppose from the sketch on the right this design is very rigid and together with the high sensitivity of the shear mode it is prone to deliver a very good frequency response.

It gets a bit more complex when we want an insulated design.

Also the thermal expansion of the different material parts needs consideration.  Accelerometer with shear tube element

#### Transverse Mode

A few designs use piezo elements in the transverse mode. That means that the force is applied on the small end faces to a piezo plate which is polarized through the thickness.

For an accelerometer an upright plate between the inertial mass and the base would not be stable enough, particularly with respectto lateral forces or shocks. However in some famous piezo-electric pressure transducers (where the mechanical load is truly uniaxial) the transverse elements are used. Piezo plate in transverse mode

If the transverse effect is used on accelerometers the piezo plate is normally bonded to a flat structure to support it.

For example this mushroom-design uses a round horizontal plate supported in the center to which a piezo disc of the same size is bonded to. Axial acceleration has tendency to flex the plate and thereby stretch the piezoelectric plate radially. Mushroom accelerometer

As we have seen in the last chapter the transverse mode sensitivity notation contains a form factor of the piezo element: The longer the piezo plate (in this case the diameter of the disk) and the smaller the thickness the bigger gets the force-charge conversion of the element.

So let’s increase the surface to a maximum…

However when using the transverse mode, depending on the piezo material, we also have to consider the pyro effect. The pyro electric charge appears directly on the electrode plated surface and increases linearly with the surface area.

The technique to bond a piezo element to a flat structure allows for very small thicknesses and large area of the piezo. It is therefore also used in MEMS applications.

Yet another (macroscopic) design is the so called bimorph. Here two piezoelectric plates are bonded together. The piezo plates are poled in opposite direction and used electrically in series with just one common electrode on the top and one the bottom. When the assembly is bent one of the plates is stretched while the other is compressed. Piezoelectric Bimorph element

This arrangement generates also pyro electric charges however the charge on one side has the opposite polarity to the other, so in first approximation the effect cancels out but this is only true when the temperature evolution over both side is exactly identical.

Functional Modes

This is to contunue on the yellow, more advanced path

This is the continuation of the green, easy path

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