Page - 50 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

3. ModelingMicrowaveHeating Thenthere is∥∥∥−→En∥∥∥2=EnTEn = Enx 2+Eny 2+Enz 2 = ( EnxV )T EnxV+ ( EnyV )T EnyV+ ( EnzV )T EnzV =VTEnx TEnxV+V TEny TEnyV+V TEnz TEnzV =VT ( Enx TEnx+E n y TEny+E n z TEnz ) V (3.26) The original microwave heating power Pnmw can be rewritten using equation 3.26as Pnmw= 1 2 l3σe(T) · ∥∥∥−→En∥∥∥2 = 1 2 l3σe(T) ·VT ( Enx TEnx+E n y TEny+E n z TEnz ) V =VT [Φnc(T)]V, (3.27) where thematrix [Φnc(T)] isdefinedby [Φnc(T)] := 1 2 l3σe(T) · ( Enx TEnx+E n y TEny+E n z TEnz ) . (3.28) The matrix [Φnc(T)] is aM×M symmetric square matrix, which is the effective heating matrix at this location n, and V is the control input vector that has to be calculated by a controller. In contrast to thecaseofscalarsuperposition, thevectorsuperpositionrule ismuch more complicated due to its high computation complexity. However, it is accurate and effective to describe the power superposition of mi- crowave heating systems with multiple feeding sources, as shown by thesimulationresults in [BPJD99]and[THN01]. 3.2.3. FormulationandDiscretization In this section, above approximations 3.15, 3.18 and 3.27 will be further formulated and the resulted equations will be discretized, to 50