Nonlinear Dielectric Heating and more

[iluvlhc]

Nonlinear Dielectric Heating and more!

Hello all,

Here is an interesting query for some work I am trying to figure out. Everyone knows about dielectric heating, which is the principle on which microwave ovens operate. Without deriving the formula for the power dissipated into an object using dielectric heating, the equation is:

power = $$\omega$$ $$\epsilon$$0 $$\epsilon$$r" E^2

where $$\omega$$ is the angular frequency (2 pi f), $$\epsilon$$0 is the permittivity of free space, $$\epsilon$$r" is the imaginary part of the relative permittivity and E is the applied electric field. This formula assumes a sinusoidal signal.

In reality, $$\epsilon$$r" for a given material depends on several things, like the frequency and the temperature, which is in part why microwave ovens work more efficiently on water than ice. However, most materials are treated as being linear, that is to say, that $$\epsilon$$r" is CONSTANT with electric field, and this is usually the case. Air, for instance is linear until about 1 MV/inch, when dielectric breakdown occurs.

But what about materials where $$\epsilon$$r" depends on E very strongly, for instance $$\epsilon$$r" = $$\sqrt{E}$$? (where E is just the magnitude, not a vector) An example of this would be materials that have an saturated ionic polarization along with the electronic component. Several ceramics have this behavior.

Let's get even more complicated, what if $$\epsilon$$r" isn't a single-valued function of E? The analog to this in magnetism is $$\mu$$r" for a ferromagnetic material. Because ferromagnets have hysteresis, one can not simply explicitly write an equation for $$\mu$$r" as a function of magnetic field.

So getting back to the electric field side of things, this hysteresis and nonlinearity comes into play with ferroELECTRICS, a fascinating type of material that has the same sort of "memory" as ferromagnets (but with electric fields, of course).

So, how does one calculate the power dissipation term for a nonlinear dielectric (paraelectric), and then, how does one do the same for a nonlinear and hysteretic dielectric (ferroelectric)?

Any and all help would be appreciated. Textbooks generally don't deal with nonlinearities unless they are addressing optical effects (Kerr effect, Pockel's effect, Faraday Effect, etc.), but this is a real problem for applications including piezoelectric sensors and transducers, along with a variety of other materials.

Thanks!

 

[eljose79]

Hello,

Thank you for bringing up this interesting topic! Nonlinear dielectric heating is indeed a complex and important phenomenon in materials science and engineering. As you mentioned, most materials are treated as linear and have a constant \epsilonr", but when dealing with nonlinear materials, the equations become more complicated.

To calculate the power dissipation in a nonlinear dielectric, the equation you provided can still be used, but the \epsilonr" term will be a function of the electric field instead of a constant. In general, this equation can be solved numerically using computer simulations or analytical approximations can be made for specific cases.

For a nonlinear and hysteretic dielectric, the power dissipation term becomes even more complex due to the hysteresis loop. In this case, the equation for power dissipation may involve integrals over the hysteresis loop or other more complicated methods. Again, computer simulations and analytical approximations are commonly used to solve these equations.

One way to approach these calculations is to use the Landau-Ginzburg-Devonshire (LGD) theory, which is commonly used to describe the behavior of ferroelectric materials. This theory takes into account the nonlinear and hysteretic behavior of these materials and can be used to calculate the power dissipation term.

I hope this helps in your research and understanding of nonlinear dielectric heating. Best of luck in your work!