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140                                                                Chapter 3


        The magnitude | | always can be normalized such way that  =   �  and ∫ | � |  = 1.
                                                                            
                                                          
                       
                                                                 
                                                               0
                                                                        1  
        Then (3.65) reduces to
                                             2
                                          ⁄
                                         (1 2)
                                                        
                                     2
                                    =     ′ ∫ | x  � |         (3.66)
                                                       3
                                    0
                                          ′
                                         0    0     1
        Evidently, electric fields in  (3.66) must be the solutions of  homogeneous  (source-free)
        Maxwell’s equations that satisfy the zero boundary conditions (3.58) for tangential components
        at each frequency 
                        0
                             ′
                                    ′
                                                                           ′′
                                                              ′′
                                        ′
                                                                      ′′
                  3  =  ( , ) x  = 0 ( ∈  ),      3  =   ( , ) x  = 0  ( ∈  )   (3.67)
                                                        0
                                                                           1
                                        1
                       0
                    3
                                                    
        It follows from the above discussion that any free of loss domain completely enclosed by electric
        or magnetic walls, or a combination of such walls is capable of containing oscillating in time
        electromagnetic fields. It is customary to call such resonating domains as cavity resonators.
                  11
        Figure 3.2.1  illustrates several types of cavity resonators with metal walls ( 3  = 0 in (3.67))
        and their expected equivalent frequency response curve. It is not wrong to think about them as
                                                                                   12
        a different realization of the well-known parallel (or series) ℒC circuit shown in Figure 3.2.2
        with electric and magnetic energy level at different moment of time during the oscillation period
                                            a)                                b)

               Figure 3.2.1 a) Cavity resonators with metal walls, b) The frequency response

        T. In the same way, the electromagnetic energy in cavity resonators continually bounce back
        and forth  ↔   between the stored electric and the magnetic energy at specific frequencies
                       
                 
        defined by (3.66). After the initial portion of energy is injected into a cavity, no additional
        sources are required to support such oscillation. Such resonance circuits are certainly more
        complicated and bulky than widely used lumped ℒC circuits, but their Q-factor (see the next
        section) or quality much exceeds the quality of any ℒC circuits. The fact of the matter following
        from (3.66) is that the resonance frequency depends on the rate of electric field variation in
        cavity. Less space variations mean lower resonance frequency. This effect lets develop quite
        miniaturized resonance cavities at low frequencies around a hundred megahertz and bellow.




        11  Public Domain Image, source: http://www.tpub.com/neets/book11/44n.htm
        12  Public Domain Image, source: http://163.13.111.54/general_physics/OSC_Ch-
        30_EM_oscillations_n_currents.html
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