# Multiple frequencies microwave heating

I was wondering how to calculate the power dissipation density (electromagnetic losses) when two waves of different frequencies are used simultaneously to heat a dielectric object. Of course, this material would present different permittivities depending on the frequency and also on the temperature. But just focusing on the frequency here:

In a sinusoidal steady state, for a point in the dielectric, the time average of the power loss can be calculated as

$$P_d = \mathbf E \cdot \frac{\partial \mathbf D}{\partial t}$$

Which, in view of the time average theorem, can be expressed as

$$P_d = \frac{1}{2}Re(jw \mathbf D)\cdot \mathbf E$$

Knowing that the displacement flux density vector is related to the electric field by a complex permittivity or dielectric constant:

$$\mathbf D = \epsilon \mathbf E=(\epsilon'-j\epsilon'')\mathbf E$$

Then:

$$P_d = \frac{\omega}{2} |\mathbf E|^2\epsilon''$$

If using the superposition principle, the combined electric field would be the sum of the electric field caused by the two waves, how can one calculate the combined flux density vector? I guess this would be the way to then use the expression for the power loss (rather than just adding up the two individual power losses), but I am not certain as I lack knowledge in the matter at hand.

Averaging might be somewhat tricky here: in one-frequency case the averaging is performed over a period of oscillations, but this is difficult to do in case of two frequencies, unless they are commensurate, $$n_1\omega_1=n_2\omega_2$$, so that combined oscillations ahve a well-defined period. Assuming that the two frequencies are commensurate is a good practical way to solve this problem. Alternative is averaging over a long period of time: $$\frac{1}{T}\int_0^Tdt\left|E_1e^{i\omega_1 t} + E_2e^{i\omega_2 t}\right|^2 = \frac{1}{T}\int_0^Tdt\left[|E_1|^2 + |E_2|^2 + 2\Re\left(E_1E_2^*e^{i(\omega_1 -\omega_2) t}\right)\right]=\\ |E_1|^2 + |E_2|^2 + 2\Re\left[E_1E_2^*\frac{1}{T}\int_0^Tdte^{i(\omega_1 -\omega_2) t}\right]= |E_1|^2 + |E_2|^2 + 2\Re\left[E_1E_2^*\frac{1}{T}\frac{e^{i(\omega_1 -\omega_2) T}-1}{i(\omega_1 -\omega_2)}\right]$$ The last term vanishes in the limit $$T\rightarrow +\infty$$, leaving us with sum $$|E_1|^2 + |E_2|^2$$.
Remark: For the question in the OP this calculation has to be generalized taking into account the time-dependence of $$D(t)$$, which in time domain is a convolution of $$\epsilon$$ and $$E$$.
• I think the simplest thing to do would be to consider $E(t)=E_1e^{i\omega_1 t}+E_2e^{i\omega_2 t} + c.c$, $D(t)=D_1e^{i\omega_1 t}+D_2e^{i\omega_2 t} + c.c$, use the formula for power in the OP and perform the derivation similar to the one that I have in my answer. Then in the end one can set $D_{1,2}=\epsilon_{1,2}E_{1,2}$. Apr 29 at 17:01