Page - 49 - in Adaptive and Intelligent Temperature Control of Microwave Heating Systems with Multiple Sources

3.2. Grey-boxModeling −→ En,m= En,m :=    E n,m x En,my En,mz    , (3.21) with −→ex= ex :=   10 0   , −→ey= ey :=   01 0   , −→ez= ez :=   00 1   . Therefore the overall superposed electric field at the location −→ En can bereplacedbythevectorEn, suchas En= M∑ m=1 vm ·En,m =    E n x Eny Enz    , (3.22) where Enx=v1E n,1 x +v2E n,2 x + · ··+vMEn,Mx , Eny =v1E n,1 y +v2E n,2 y + · ··+vMEn,My , Enz =v1E n,1 z +v2E n,2 z + · ··+vMEn,Mz , (3.23) and vm (1≤m≤M) is the portion of amplitude of the electric field fromthem-th feedingsource. Definingthenewvectorsas V= [ v1, v2, . . ., vM ]T , Enx= [ En,1x ,E n,2 x , . . .,E n,M x ] , Eny = [ En,1y ,E n,2 y , . . .,E n,M y ] , Enz = [ En,1z ,E n,2 z , . . .,E n,M z ] , (3.24) thecomponents in thevectorEn canbeexpressedas Enx=E n xV, E n y =E n yV, E n z =E n zV. (3.25) 49 To make the electric field easier to use in the following derivations, it is rewritten inavector form,suchas