Measurement and Modelling of Self-Heating in Piezoelectric Materials and Devices

Abstract

There are many uses of piezoelectric ceramics where the desire for increased power output means increased drive levels, which subsequently can lead to thermal problems within the device.

Appendix A: Thermal Property Data of Piezoelectric Materials

In order to be able to make any predictions of how a piezoelectric device will perform in terms of self induced temperature rise, some knowledge of the thermal properties are needed, typically the thermal conductivity k, but also the thermal diffusivity $\alpha$, and the specific heat capacity Cp. Data for these properties is scarce in the literature, with the most often quoted source coming from Berlincourt et al. [36], where a single value for thermal conductivity and specific heat capacity is given for all PZT materials with no information regarding the temperature dependence.

1.1 A.1 Specific Heat Capacity, $C_{p}$ (J/kg K )

The specific heat capacity, $C_{p}$, is the amount of energy needed to raise the temperature of 1kg of the material by one degree. This can be used to determine the temperature rise of a volume of a material since the change in energy stored is proportional to the specific heat through the following equation;

$$\begin{aligned} \dot{E}_{st} = \frac{d}{dt}(\rho V C_{p} T) \end{aligned}$$

(A.1)

where $\rho$ and V are the density and volume of the material. If the internal power generation rate of the material is known, then the resultant temperature rise can be determined.

As discussed the most quoted value for the specific heat capacity of PZT is 420 J/kgK [36], but Table 2 gives a summary of other sources that give values for the room temperature value ranging from 350 to 491 J/kg K. Yarlagadda et al. [37] made a comprehensive set of measurements of heat capacity and thermal conductivity over the temperature range 20–300 K and show that the specific heat capacity varies greatly over this range, and is different for hard and soft materials. The soft material PZT-5H increases 15 fold over the temperature range, whilst the hard composition only increases by around 4 times. These values are not necessarily inconsistent with the room temperature values reported in the literature as some of the temperature dependence may be due to phase changes, indeed the authors attribute deviations at around 50–80 K to transition type behaviour.

Table 2 Specific Heat Capacity values for PZT type piezoelectric materials

The most common operating regime for piezoelectric actuators is from ambient temperature to around 300 $^\circ$C and there is little published information on the change in specific heat over this temperature range. Measurements of the specific heat capacity of a soft and a hard PZT composition were made at NPL, from room temperature to 200 $^\circ$C, using a Perkin Elmer DSC 7 Differential Scanning Calorimeter. The basis of DSC measurements is that a sample of known weight is placed in a holder, and the amount of energy provided by an electrical heater, which is required to increase the temperature of the sample per degree is measured. The test is carried out at a constant heating rate during which the heater power is adjusted continuously to compensate for heat absorbed or evolved by the sample, to keep the sample holder temperature identical to that of the reference holder. The energy input by the electrical heater is measured and this is proportional to the specific heat.

Fig. 26

Specific heat variation with temperature, for PZT 4D and PZT 5H

Measurements were repeated on two samples of each composition and the results are plotted in Fig. 26. The room temperature values are the order of 20 % lower than the most often quoted value, although they do increase by approximately 15 % over the measured temperature range. The soft material has a slightly higher specific heat capacity than the hard, but the difference is not much greater than the sample-to-sample variation.

1.2 A.2 Thermal Conductivity, k (W/m K)

In order to determine the equilibrium state of a device, i.e. a steady state thermal solution, only the thermal conductivity of the material is required, which has units W/m K. Thermal conductivity is defined as the rate at which heat flows through a certain area of a body. The precise definition is given by Fourier’s equation:

$$\begin{aligned} \frac{dq}{dt}=-kA\frac{dT}{dx} \end{aligned}$$

(A.2)

where q is the heat energy flowing in the x direction through the area, A during time, t. dT/dx is the temperature gradient, and k is the thermal conductivity of the material.

Table 3 Thermal conductivity values for PZT type piezoelectric materials

Table 3 summarises the published values of thermal conductivity of PZT type piezoelectric materials, where values range from 0.8 to 2.3 W/m K. Again Yarlagadda et al. [37] have determined the thermal conductivity below 300 K and found that PZT-4S was almost twice as conductive as PZT-5H, and that there is a significant temperature dependence, however their room temperature thermal conductivity results fall well below the generally accepted value of between 1 and 2 W/mK. In general most thermal models assume the thermal conductivity is temperature independent, most probably because of a lack of published data on the temperature dependence.

A.2.1 Errors

As can be seen from Table 3, the published values of thermal conductivity varies by around 100 %, and the errors in the measured value will be reflected in the predicted thermal behaviour, depending on the exact details of the case. In general the errors in the thermal conductivity are propagated linearly to the results. For example, Sherrit et al. [24] predict an internal temperature rise of 36 $^\circ$C in a 20 mm diameter, 2 mm thick disc, using a value of k = 1.25 W/m K, however this will be reduced to 23 $^\circ$C when a value of k = 2 W/m K is used. This is evident from the first term in Eq. (15) where the temperature rise is proportional to the reciprocal of the thermal conductivity. In contrast, Hu [25] has shown that varying the thermal conductivity from 1 to 1.5 W/m K has little effect on the predicted temperature rise, and that power input was the dominant factor in modelling temperature rise in piezoelectric transformers.

1.3 A.3 Thermal Diffusivity, $\alpha$ (m$^{2}$/s)

Thermal diffusivity is the material property governing heat flow when the temperature is varying with time, and has the dimension length$^{2}$/time, with units of m$^{2}$/s. The thermal diffusivity can be determined experimentally, and it is related to the thermal conductivity k, through the following relationship

$$\begin{aligned} \alpha = \frac{k}{\rho \cdot C_{p}} \end{aligned}$$

(A.3)

where $\rho$ is the density and $C_{p}$ is the specific heat capacity.

Using the published values for k, $\rho$ and $C_{p}$ the thermal diffusivity of PZT is around 5 $\times$ 10$^{-7}\mathrm {m{^2}/s}$. Lang [45] has measured the thermal diffusivity of bulk piezoelectric materials using the Laser Intensity Modulation Method (LIMM), and obtained values close to this estimate, Table  4.

Table 4 Thermal diffusivity values for PZT type piezoelectric materials

The thermal diffusivity of PZT was also measured at NPL, using a Netzsch Laser Flash 427. The measurement involves heating the front face of a disc-shaped sample of known dimensions, usually 12.5 mm diameter and 1.5–2.5 mm thickness using a high intensity Nd:YAG laser with a pulse width of between 0.2 and 1.2 ms. The temperature rise on the rear face is monitored using an InSb infrared detector. From the temperature rise with time, the thermal diffusivity can be calculated, applying appropriate corrections for radiation and the finite laser pulse length. Because of limited samples of the correct dimension only a soft PZT 5H composition was measured. The measured value for the thermal diffusivity of PZT-5H was around 4.5 $\times$ 10$^{-7}$ m$^{2}$/s, with little measurable change over the range room temperature to 350 $^\circ$C, this compares favourably with measured values reported in the literature, Fig. 27.

Fig. 27

Thermal diffusivity measurements of PZT-5H using laser flash

The thermal conductivity can also be derived from the measured thermal diffusivity and the measured values of specific heat capacity, and lie between 1.1 and 1.3 W/mK over the temperature range room temperature to 200 $^\circ$C.