082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

594 Chapter 19 Ordinary differential equations I
Thewallsofmicrowavecavitiesareusuallymadeofmetalswithgoodconductivity
such as copper, steel or aluminium but in some applications the internal surfaces
areplatedwithpreciousmetalssuchasgoldtopreventoxidationwhichwouldreduce
the surface conductivity. When used at high power levels the conductivity is important.Lowconductivity
canleadtolossofpower andheatingofthecavityduetothe
currents flowing on the internal walls. The space inside the cavity can be empty or
it can contain a dielectric insulating material such as PTFE. The analysis here will
consider an empty cavity.
The purpose of the device is to act as a narrow band filter which selects a specific
frequency. In fact, there is a set of frequencies at which the microwave cavity
will work, known as the resonant frequencies. The electromagnetic waves inside
the cavity are governed by the propagation modes within the cylinder. These modes
are described mathematically by Bessel functions. It can be shown that the resonant
frequencies, f mnp
, associated with the transverse magnetic modes of the cavity are
given by
f mnp
= c
2π
√ (Xmn
r
) 2 ( pπ
+
l
) 2
wherem,nand paremodeindexnumbers,0,1,2,...andX mn
denotesthen-thzero
oftheBesselfunctionofthefirstkindoforderm,J m
(x),wherecisthespeedofwave
propagation inthe medium.
The zeros can be observed in Figure 19.18 as the points where the graph crosses
thexaxis.UsuallysoftwaresuchasMATLAB ® hasthecapabilitytocalculatenumericallythezerosofafunction.IfweobservethatthefirstzeroofJ
0
(x)isapproximately
x = 2.5 (see Figure 19.18) we can type the following at the MATLAB ® command
prompt:
X = fzero(@(x)besselj(0,x),2.5)
which gives the resultX = 2.4048.
The second zero ofJ 0
(x) isclose to5.5,thus wetype,
X = fzero(@(x)besselj(0,x),5.5)
which gives the resultX = 5.5201.
The first zero ofJ 1
(x) isapproximately 4,so,
X = fzero(@(x)besselj(1,x),4)
which gives the resultX = 3.8317.
Table19.2showsasummaryoftheresultsofapplying thismethodform = 0,1,
and 2.
Consideramicrowavecavityresonatormadefromanaluminiumfizzydrinkscan
with dimensionsr = 33 mm and l = 120 mm. The transverse magnetic modes are
labelledTM nmp
andthefirstcommonlyusedmodeisTM 011
.Theresonantfrequency
would be