# Piezoelectric Technology: A Primer

Piezoelectric ceramics are used in myriad applications ranging from generating exquisitely controlled forces at the microscopic level to developing tremendous forces at the macro level. Piezoelectric ceramics, and in particular PZT ceramic, are used in inexpensive but rugged sensors as well as multi-layer transformers for generating high voltages without any EMI. In fact, there are too many applications to cover in a single article, but the basic principals in this primer can be used to both understand and design piezoelectric structures and devices.

Piezoelectric ceramics have unusual properties in both the electrical and mechanical domains. So if it's been a while since you shook hands with the dielectric constant or Hooke's Law, see Reviewing the Equations for High-Permittivity Dielectrics and Bridging the Mechanical and Electrical Worlds.

Several ceramic materials exhibit a piezoelectric effect. These
include lead-zirconate-titanate (PZT), lead-titanate
(PbTiO_{2}), lead-zirconate (PbZrO_{3}), and
barium-titanate (BaTiO_{3}). Strictly speaking, these
ceramics are not actually piezoelectric but rather exhibit a
polarized electrostrictive effect.

A material must be formed as a single crystal to be truly piezoelectric. Ceramics have a multi-crystalline structure made up of large numbers of randomly orientated crystal grains. The random orientation of the grains results in a net cancelation of the piezoelectric effect. The ceramic must be polarized to align a majority of the individual grains' effects. Nonetheless, the term "piezoelectric" has become interchangeable with "polarized electrostrictive effect" in most literature.

**Dielectric Constant**

_{r FREE}(1 – k²) =

_{r CLAMPED}

The dielectric constant of a material reduces with mechanical load. "Free" stands for a state when the material is able to change dimensions with applied field. "Clamped" refers to either a condition where the material is physically clamped or is driven at a frequency high enough above mechanical resonance that the device can't respond to the changing E field.

**Elastic Modulus (Young's Modulus)**

_{OPEN}(1 – k²) = Y

_{SHORT}

The elastic modulus means that the mechanical stiffness of the material reduces when the output is electrically shorted. This is important in that both the mechanical QM and resonant frequency will change with load. Young's Modulus is also the property that is used in the variable dampening applications, one of the many interesting applications for these materials.

D_{i} =
d_{ij}
_{j} |
where: | D_{i}
Electric Displacement (or Charge Density)d _{ij}
Piezoelectric Modulus, the ratio of strain to applied field or
charge density to applied mechanical stress |

Stated differently, d measures charge caused by a given force or deflection caused by a given voltage. We can also use this to define the piezoelectric equation in terms of field and strain:

Electric displacement is defined as:

therefore,

and

which results in a new constant:

This constant is known as the piezoelectric constant and is equal to the open circuit field developed per unit of applied stress or as the strain developed per unit of applied charge density or electric displacement. The constant can then be written as:

Fortunately, many of the constants in the formulas above are equal to zero for PZT piezoelectric ceramics. The non-zero constants are:

_{11}= s

_{22}, s

_{33}, s

_{12}, s

_{13}= s

_{23}, s

_{44}, s

_{66}= 2 (s

_{11}– s

_{12})

d

_{31}= d

_{32}, d

_{33}, d

_{15}= d

_{24}

During poling the material permanetly increases in dimension between the poling electrodes and decreases in dimensions parallel to the electrodes. The material can be de-poled by reversing the poling voltage, increasing the temperature beyond the material's Currie point, or by inducing a large mechanical stress.

Applied voltage:

Voltage applied to the electrodes at the same polarity as the original poling voltage results in a further increase in dimension between the electrodes and decreases the dimensions parallel to the electrodes. Applying a voltage to the electrodes in an opposite direction decreases the dimension between the electrodes and increases the dimensions parallel to the electrodes.

Applied force:

Applying a compressive force in the direction of poling (perpendicular to the poling electrodes) or a tensile force parallel to the poling direction results in a voltage generated on the electrodes that has the same polarity as the original poling voltage. A tensile force applied perpendicular to the electrodes or a compressive force applied parallel to the electrodes results in a voltage of opposite polarity.

Shear:

Removing the poling electrodes and applying a field perpendicular to the poling direction on a new set of electrodes will result in mechanical shear. Physically shearing the ceramic will produce a voltage on the new electrodes.

**Piezoelectric Benders**

Piezoelectric benders are often used to create actuators with
large displacement capabilities. The bender works similarly to the
action of a bimetallic spring. Two separate bars or wafers of
piezoelectric material are metallized and poled in the thickness
expansion mode. They are then assembled in a +–+– stack
and mechanically bonded. In some cases, a thin membrane is placed
between the two wafers. The outer electrodes are connected together
and a field is applied between the inner and outer electrodes.

As a result, for one wafer the field is in the same direction as the poling voltage while the other is opposite to the poling direction. Therefore, one wafer is increasing in thickness and decreasing in length while the other wafer is decreasing in thickness and increasing in length. This results in a bending moment.

Mechanical loss: Q |
Mechanical stiffness reactance or mass
reactanceMechanical resistance |

Electrical loss: tan = |
Effective series resistanceEffective series reactance |

R_{i} = Electrical resistance

C_{i} = Input capacitance =

_{O}= 8.85 x 10-12 farads / meter

A = Electrode Area

t = Dielectric Thickness

L

_{M}= Mass (kg)

C

_{M}= Mechanical compliance = 1 / Spring rate (M / N)

N = Electro-mechanical linear transducer ratio (newtons/volt or coulomb/meter)

This model has been simplified and it is missing several
factors. It is only valid up to and slightly beyond resonance. The
first major problem with the model is related to the mechanical
compliance (C_{M}). Compliance is a function of mounting,
shape, deformation mode (thickness, free bend, cantilever, etc.),
and modulus of elasticity. The modulus of elasticity is, however,
anisotropic and it varies with electrical load. The second issue is
that the resistance due to mechanical Q_{M} has been left
out. Finally, there are many resonant modes in the elements, each
of which has its own C_{M} as shown below.

Simple Beam—Uniform end load

Simple Beam—Uniform load—End mounts

Simple Beam—Uniform load—Cantilever mount

The various elements that have been explained can now be combined into the design of a complete piezoelectric device. The simple piezoelectric stack transformer will be used to demonstrate the way they are combined to create a functional model.

_{1}) resulting in a thickness mode vibration. This vibration is coupled into the upper half and the output voltage is taken across the thinner dimension (d

_{2}).

The equivalent circuit model for the transformer (shown above)
can be thought of as two piezoelectric elements that are assembled
back to back. These devices are connected together by an ideal
transformer representing the mechanical coupling between the upper
and lower halves. The input resistance (R_{i}) and the
output resistance (R_{O}) are generally very large and have
been left out in this model. The resistor (R_{L})
represents the applied load. Determining the values of the various
components can be calculated as shown previously.

Input / Output capacitance:

similarly,

Mechanical compliance:

The mechanical compliance (C_{M}) can be represented by
a simple beam subjected to a uniform axial load. This is because
the thickness expansion mode will apply uniform stress across the
surface. It should be noted that the beam length is measured with
respect to the vibration node. The vibration node is used as this
is the surface that does not move at resonance and can, therefore,
be thought of as a fixed mounting surface.

**Note:** Even if nd_{2}
d_{1} the vibration node will still be located in the
mechanical center of the transformer.

Mass:

L_{M1} =
A W d_{1}

L_{M2} =
A W nd_{2} =
A W d_{1}

Resistance:

The resistances in the model are a function of the mechanical QM and Q of the material at resonance and will be calculated later.

Ideal transformer ratio:

The transformer ratio (N_{1}) can be thought of as the
ratio of electrical energy input to the resulting mechanical energy
output. This term will then take the form of newtons per volt and
can be derived from the piezoelectric constant (g).

as before:

therefore:

or

The output section converts mechanical energy back to electrical
energy and the ratio would normally be calculated in an inverse
fashion to N_{1}. In the model, however, the transformer
ratio is shown as N_{2} : 1. This results in a calculation
for N_{2} that is identical to the calculation of
N_{1}.

or

The transformer 1 : N_{C}, represents the mechanical
coupling between the two halves of the transformer. The stack
transformer is tightly coupled and the directions of stress are the
same in both halves. This results in N_{C}
1.

Model Simplification:

The response of the transformer can be calculated from this model, but it is possible to simplify the model through a series of simple network conversions and end up in an equivalent circuit whose form is the same as that of a standard magnetic transformer.

where, due to translation through the transformer,

_{M2}' = N

_{C}² C

_{M2}and L

_{M2}' = L

_{M2}/ N

_{C}²

but N_{C}²
1, therefore

_{M2}' = C

_{M2}= C

_{M1}and L

_{M2}' = L

_{M2}= L

_{M1}

which allows the next level of simplification

here

_{M1}+ L

_{M2}' = 2 L

_{1}= 2 A W d

_{1}

Final simplification

where

_{1}² and L = L' / N

_{1}²

and, from before

therefore

The last value we need to calculate is the motional resistance. This value is based upon the mechanical QM of the material and the acoustic resonant frequency.

Resonant frequency

therefore

_{O}= c

_{PZT}/ d

_{1}

The equation shown above states that the resonant frequency is equal to the speed of sound in the material divided by the acoustic length of the device. This is the definition of acoustic resonance and acts as a good check for the model. The final derivation is the value of resistance:

_{M}1 /

_{O}R C

or

**Note:** C_{M} and R are both functions of
Y_{33} and Y_{33} is a function of
R_{L}

It should be noted that the model is only valid for transformers driven at or near their fundamental resonant frequencies. This is because the initial mechanical model assumed a single vibration node located at the center of the stack, which is only true when the transformer is driven at fundamental resonance. There are more nodes when the transformer is driven at harmonic frequencies.

**Note:** Stress is 90° out of phase from
displacement

There are no fixed nodes at frequencies other than resonance. This means that the transformer must be designed with the resonate mode in mind or phase cancellations will occur and there will be little or no voltage gain. It is often difficult to understand the concept of nodes and phase cancellation, so a simple analogy can be used. In this case, waves created in a waterbed will be used to explain the effect.

Pressing on the end of a waterbed creates a wave of displacement that travels down the length of the bed until it reaches the opposite end and bounces back. The water pressure (stress) is the lowest, or negative with respect to the water at rest, at a point just in front of the wave and highest at a point just behind the wave. The pressures at the crest and in the trough are at the same pressure as the bed at rest.

The wave will reflect back and forth until resistance to flow causes it to damp out. The average pressure over time at any point in the bed will be exactly the same as the pressure at rest. Similarly, the average stress in a transformer off resonance will approach zero and there will be no net output.

Pressing on the end of the same bed repeatedly just after the wave has traveled down the length, reflected off the end, returned, and reflected off the driven end will result in a standing wave. This means that one half of the bed is getting thicker as the other half is getting thinner and the center of the bed will be stationary. The center is the node and the thickness plotted over time of either end will form a sine wave. There will be no net pressure difference in the center, but the ends will have a pressure wave that forms a sine wave 90° out of phase with the displacement. The piezoelectric transformer again works in the same manner with no voltage at the node and an AC voltage at the ends. It is fairly simple to expand this concept to harmonics and to other resonate shapes.