Page 160 - Maxwell House
P. 160
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