Page - 47 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

3.2. Grey-boxModeling Approximationsof themicrowaveheatingterm Compared with the heat loss part, the approximation of the mi- crowave heating term is more difficult, because of the complex mi- crowaveheatingpowersuperpositions. Ingeneral, therearetwotypes of microwave power superposition principles. The first power su- perposition principle corresponds to the situation when the EM fields from different sources are incoherent. In this case, the instantaneous microwave heating power of each source is varying, but the aver- age heating power is constant. Therefore the superposed heating power is equivalent to the addition of heating power of individual sources. For instance, assuming there are two microwave heating sources, the electric fields from two sources at the same locationn are −→ En,1 and−→ En,2, respectively. Theheatingpowerfromthetwosourcesarerepre- sentedby Pn,1mw∝ ∥∥−→En,1∥∥2, Pn,2mw∝∥∥−→En,2∥∥2. Thenthesuperposedheatingpower [BPJD99] is Pnmw=P n,1 mw+P n,2 mw∝ ∥∥−→En,1∥∥2+∥∥−→En,2∥∥2. (3.16) For HEPHAISTOS, considering multiple heating sources and control- lable feeding power, the total effective microwave heating power at the locationncanbewrittenas Pnmw= M∑ m=1 umP n,m mw , (3.17) whereum (1≤m≤M) stands for the portion of heating power from them-thfeedingsourceandPn,mmw isthemaximumheatingpowerfrom them-thsourcetothen-thlocation. Theaboveequation 3.17isrewrit- ten inamatrixmultiplicationformas Pnmw=P n mwU, (3.18) with U := [u1, u2, . . ., uM] T, Pnmw := [P n,1 mw, P n,2 mw, . . ., P n,M mw ], 47