Page - 48 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

3. ModelingMicrowaveHeating where U is the control input vector that is used to adjust the mi- crowaveheatingpowerandcorrespondingheatingrate. The scalar superposition rule is simple to implement and control, which is the main approximation approach used in former researches about temperature controller design of microwave heating, such as [RMTV85] and [SBA00]. Since the above expression is a linear super- position, it is named as the linear microwave heating power approxi- mation. Besides the scalar addition rule, there is another principle of heat- ing power superposition, which deals with the situation when all EM fields from different sources are coherent. When the EM fields havethesamefrequencyandconstantphasedifferencesbetweeneach other, the microwave heating power of different sources superposes following the rule of vector addition. Using the same example as in equation 3.16, the superposed electric field at the locationn can be representedbytheequation −→ En= −→ En,1+ −→ En,2, andthecorrespondingtotalheatingpower [BPJD99] is Pnmw∝ ∥∥−→En∥∥2=∥∥−→En,1∥∥2+∥∥−→En,2∥∥2+2 ·∥∥−→En,1∥∥ ·∥∥−→En,2∥∥cosΦ, (3.19) whereΦ isthephasedifferencebetween −→ En,1and −→ En,2. WhilecosΦ≥ 0, the superposition is constructive; otherwise while cosΦ < 0, the superposition isdestructive. Equation 3.19 can be extended to a more general case like HEPHAIS- TOS.AssumingthereareMmicrowaveheatingsources, foreachheat- ing sourcem, its radiated electric field at the locationn is denoted by−→ En,m,which isdefinedas −→ En,m := En,mx −→ex+En,my −→ey+En,mz −→ez, (3.20) where En,mx (En,my , En,mz ) and −→ex (−→ey,−→ez) are the amplitude of electric field componentand unit vector in thex (y,z) -direction, respectively. 48