Optimization of thermal energy harvesting by PZT using shape memory alloy

Description of PZT system

Generally, the piezoelectric Element used in mode 33, where tension and stress act in direction 3, can also work with mode 31. In this case, tension acts following direction 3, stress and deformation work on direction 1. Operation in mode 31 involves the use of elements for which a significant deformation is generated by bending towards direction 1 (see Figure 1).

Figure 1:

PZT working mode.

(a) mode 33 (b) mode 31 (Ahmed-Seddik 2012).

Thirty one elements are bi-morphs with two connected piezoelectric layers sets and are the most common type used.

During bending, the upper surface is in tension while the lower is in compression or vice versa. Therefore, each layer is polarized in the same way. Similarly, with the electrodes correctly connected, the currents produced are added (parallel polarization). If the layers are reversely polarized, the voltages are added (serial polarization); in both cases, the conversion potential remains constant. In theory, the polarization and the number of layers act only on the ratio between the voltage and the current. We consider a cantilever structure. This structure has a lower stiffness for a given size. Then, for some input strength, it generates the greatest constraint.

Electromechanical analogy

This analogy verifies the classical conversion between electrical and mechanical quantities (Table 1). Thus, a system {mass, spring, damper} can be modelled by an impedance RLC, usually referred to as the emotional branch (Figure 2).

Figure 2:

Equivalence between mechanical and electrical system (Boisseau 2011).

Equivalent circuit in quasi-static mode

The generator implemented in this work is intended to recover the energy where the frequency of mechanical stress applied on the piezoelectric device is very low compared to its resonance mode. The quasi-static model is therefore solely determined from static equations of PZT. It is possible to obtain this model from the dynamic models, by simply tilting the frequency to zero. In this case, the inductive elements as well as mechanical losses will not be taken into account since their values may be ignored at low frequencies

Long piezoelectric bar deformed according to mode 33

Considering a parallelepiped PZT bar polarized according to the height mentioned in Figure 4, we assume that it is subjected to a uniform force F applied over its entire surface and that it is deformed according to mode 33 (a constraint along the axis of polarization causes deformation along the same axis) (see Figure 3). Other modes of mechanical deformation are assumed to be negligible and the height of the bar being big in front of the transverse dimensions; additionally, only the component along axis 3 is to be considered for all the studied quantities; also, losses not taken into account in this part.

Figure 3:

PZT bars in mode 33 (Defosseux 2011).

Figure 4:

Equivalent circuit to PZT bars in mode 33 (Defosseux 2011).

The ratings are as follows:

1. L₃: height of the bar;

2. A₃: area of the section of the bar;

3. P: dielectric polarization; V: speed of the end of the bar;

4. F: force applied to the end of the bar;

5. u: displacement of the end of the bar;

6. U: tension between the two ends of the bar;

7. I: current through the bar;

8. T₃: stress;

9. S₃: relative deformation;

10. E₃: electric field;

11. D₃: electrical induction.

These quantities linked by:

In addition, electromechanical coefficients are:

• β33s et β33T Inverse of the dielectric permittivity at S or T constant;

• S33D Elastic flexibility at constant D;

• g₃₃eth₃₃ Piezoelectric coefficients.

Equivalent circuits take into account the propagation of the mechanical wave in the piezoelectric material. In the case of the generator presented here, the presented model is therefore determined solely from the static equations of PZT. The equations chosen to establish the equivalent model are:

We deduce that

where C₀ is built-in ceramic capacity

where C_m is the mechanical capacity in Law frequency mode.

Equation (13) reveals the electromechanical analogy between voltage and force current – speed. The coefficient N, with F as its weight force and V as its speed, suggests the introduction of a transformer N:1 ratio; thus connecting the two mechanical and electrical branches. This transformer, associated with the negative capacitance −C₀, models the electromechanical conversion due to PZT.

This circuit can be in the form of an equivalent Norton generator, including I₃₃ current source and Z₃₃ parallel impedance:

In the time domain, the piezoelectric bar is equivalent to a current source proportional to the derivative of the applied force, in parallel with a capacitance C₃₃ (Figure 4b):

Characteristics of the piezoelectric P188

The used PZT is a cylinder with a radius of 2 cm and thickness of 0.33 mm, constituted by PZT ceramic of P188 of Quartz and Silicon.

Table 2 shows the characteristics of the materials and the parameters of the model. The frequency of the applied force is much less weaker than the natural frequency of the system vibration.

System feedback for exponential constraint

Considering the previous bars connected in parallel to RC charge.

Supposed that the mean value of Force F (t) applied on the bar section has an exponential form:

Source of courant I₃₃:

The differential equation becomes:

The solution of this equation is under the form:

Relation between temperature and mechanical force

Relation of the perfect gas:

P: Pressure in Pa.

V: Volume.

n: Number of moles.

R: 8.31 J mol⁻¹ K⁻¹.

θ: Temperature in kelvin.

Therefore, if we enclose a certain amount of gas in a constant envelope volume V, the pressure developed by the gas is proportional to the absolute temperature:

P = RT/V with R/V constant.

Pressure will be calculate as force by surface

Results

Under a temperature gradient ΔT = 40 °C which can give us/yield a force F = 0.001 N according to the volume and the surface of the used PZT. Obtained from the simulation, responses related to voltage and as function of resistance R is shown in Figure 5.

Figure 5:

Voltage variation accroding to load R.

The resulted curve represents the voltage variation as a function of the resistance and where the PZT is modelled by a current source; the voltage path follows the path required by the RC load. We can distinguish two zones: The transient zone where the linear variation demonstrates a fast response of the voltage and the second indicates the steady state of the operation after a resistance of 1.5 10⁵ Ω.

Observing the curves representing the instantaneous power applied during the compression cycle of the PZT (Figure 6), we noticed that the area under each curve corresponds to the energy supplied or received over the considered interval.

Figure 6:

PZT power variation according to load R.

During compression, the PZT supplies the mechanical energy from which a large part is stored in an elastic form by the compressed PZT, the rest of it supplied to the resistive load, the energy supplied to the internal capacitor C₀ is zero.

In the second phase, the mechanically stored energy supplies the source because the speed on the surface of the bar is negative while the applied force decreases in being positive. This highlights the importance of the applied stress form: It should be zero while the bar relaxes, in order to provide the electric charge with all the mechanical energy stored by the PZT.

Considering that the losses would not change the obtained results, it is worth mentioning that they are of three natures: mechanical, dielectric and piezoelectric; and that they depend on the frequency and intensity of the electric field.

Precise modelling goes beyond the scope of this study, especially since they were neglected in the case of quasi-static operations.

Mono dimensional model

The comportment of SMA modulation, which is based on the Martensitic variant’s phase transition, is a temperature-managed process. Müller, Achenbach, and Seelecke demonstrate that the potential energy of Martensitic variables is given by the free energy of Gibbs or, in the absence of solicitation, by the Free energy of Helmholtz (Smith 2005). By considering an uni-axial load, assumed to coincide with x1 direction, the model of SMAs is then reduced to 1D model. The stress tensor has only one non-zero component that is:

The transformation strain components given by

The demonstration replaced by the following equations to summarize the 1D thermo-mechanical model (Lagoudas 2008).

Thermo-elastic response:

Forward phase transformation:

Reverse phase transformation:

Parameters

See Table 3.

Results

Variations in the external temperature ignite the Memory Form effect, an effect followed by the phenomenon of super elasticity. A gradient of a 40 °C shows a reversible deformation of a 4% max. The applied effort helps to produce a force cycle that can reach 300 MPA where the force doesn’t return to its initial state “0 MPa” because the transformation A-M, M-A does not form a closed cycle ‘total inverse transformation’ (see Figure 7).

Figure 7:

Up) Relation stress strain – Down) Strain variation according to external temperature.

AC/DC rectifier

The full-wave AC/DC rectifier consists of a Graëtz bridge and a C_rect capacitor (Figure 15). In order that Graëtz bridge diodes conduct, it is necessary that the voltage delivered by the transducer V_p be greater than diode’s threshold voltage. This problem does not arise when we use large generator dimensions that provide higher voltages and higher powers than losses introduced by the diodes (see Figure 14).

Figure 14:

Electric schematic of AC/DC converter (Zenati 2007).

Figure 15:

Power in function of rectified volatge.

The current of the piezoelectric generator i_p is a sinusoidal signal. As long as the voltage at the terminals of C_piezo is lower than the voltage V_rect, the diodes remain blocked and the output current i_o (t) is zero. When this voltage becomes higher, the current flows to the capacity C_rect and loads.

The current i_o (t) can be described as follows:

In our case, we assume that the capacity C_rect is much larger than the capacity C_piezo which allows a good filtering of the AC voltage and that the voltage V_rect is considered constant. The expressions of the average current i_o(t) as a function of the generator current amplitude I_p and of the output voltage V_rect as well as the average power recovered P_average are given by:

The figure above gives the value of the power after conversion, It is clearly seen that the power depends on the voltage V_oc (rectified output voltage); the maximum power reached for an optimum voltage is equal to half of the no-load voltage.

DC/DC converter

In our case, we assume that the piezoelectric transducer delivers a higher voltage than the battery. To adapt the input voltage V_rect to the battery voltage V_bat, we used a voltage step-down, this circuit is represented by Figure 16. Allow also avoiding current passage into AC/DC converter. In steady state, the voltage across the battery is almost constant, and therefore the transfer of maximum power is done by electric current.

Figure 16:

BUCK converter schematic (Zenati 2007).

Two operating modes exist in step-down converter function: continuous and discontinuous mode. The discontinuous mode corresponds to the cancellation of the current in the inductance during a period of operation. This situation occurs when the average current absorbed by the load is less than half of the current ripple in the inductor. The continuous mode of operation has as characteristics a less corrugated current output, lower constraints on the devices, but a reaction time longer than the discontinuous mode. In both modes, the converter is characterized by a discontinuous input current which implies a harmonic disturbance in the power source.

The discontinuous mode of operation is interesting for applications that require a certain feedback speed to the variation of the load and/or the input voltage. Contrariwise, this mode causes a large current in inductor, and a high capacity for output filtering (Qi 2011). The transfer function of the converter operating in discontinuous mode is:

Inductor current determined by the following expressions:

The input current of the converter is

Output current

Choice of L and C

So optimal cyclic ratio calculated from the model under (see Figure 18):

Figure 17:

Studied model to cyclic ration calculation (Zenati 2007).

Figure 18:

Battery courant variation in function of cyclic ratio.

We report the current of the piezoelectric transducer curve according to the cyclic ratio, taking into account that the optimal duty cycle value for Ib equals 1*10⁻⁶ at 4.3% At this value, the Cyclic Duty Ratio is associated to the maximum value of the battery current for a V_b = 4 V.