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c/F (3 108 m/s)/(56 109 Hz)
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Cutoff frequency (Fc ) 381 (c) Wavelength in waveguide Vp c (45 108 m/s)(0054 m) 3 108 m/s 008 m
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Comparing, we find that the free-space wavelength is 0054 m, and the wavelength inside of the waveguide increases to 008 m
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Cutoff frequency (F )
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The propagation of signals in a waveguide depends, in part, upon the operating frequency of the applied signal As covered earlier, the angle of incidence made by the plane wave to the waveguide wall is a function of frequency As the frequency drops, the angle of incidence increases toward 90 The propagation of waves depends on the angle of incidence and the associated reflection phenomena Indeed, both phase and group velocities are functions of the angle of incidence When the frequency drops to a point where the angle of incidence is 90 , then group velocity is meaningless We can define a general mode equation based on our system of notation: 1 ( c) where is the longest wavelength that will propagate c a, b are the waveguide dimensions (see Fig 19-2) m, n are integers that define the number of half-wavelengths that will fit in the a and b dimensions, respectively Evaluating Eq 197 reveals that the longest TE-mode signal that will propagate in the dominant mode (TE10) is given by
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[197]
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[198]
from which we can write an expression for the cutoff frequency: Fc where Fc is the lowest frequency that will propagate, in hertz c is the speed of light (3 108 m/s) a is the wide waveguide dimension c 2a [199]
382 Microwave waveguides and antennas Example 19-3 A rectangular waveguide has dimensions of 3 late the TE10 mode cutoff frequency Solution: Fc c 2a (3 (2) 5 cm 3 108 m/s (2)(005 m) 108 m/s 1m 100 cm 3 GHz 5 cm Calcu-
Equation 197 assumes that the dielectric inside the waveguide is air A more generalized form, which can accommodate other dielectrics, is Fc where e is the dielectric constant u is the permeability constant For air dielectrics, u uo and e eo, from which c 1 uoeo [1911] 1 2 ue m a
[1910]
To determine the cutoff wavelength, we can rearrange Eq 1910 to the form: 2
[1912] n b
One further expression for air-filled waveguide calculates the actual wavelength in the waveguide from a knowledge of the free-space wavelength and actual operating frequency:
[1913] 1 Fc F
where is the wavelength in the waveguide is the wavelength in free space o Fc is the waveguide cutoff frequency F is the operating frequency
Waveguide impedance 383 Example 19-4 A waveguide with a 45-GHz cutoff frequency is excited with a 67-GHz signal Find (a) the wavelength in free space and (b) the wavelength in the waveguide Solution: (a)
c/F 108 m/s 109 Hz 67 GHz 1 GHz 3 3 108 m/s 67 109 Hz 00448 m
Fc F
00448 m 1 45 GHz 67 GHz
00448 m 1 067 00448 033 0136 m
Transverse magnetic modes also propagate in waveguides, but the base TM10 mode is excluded by the boundary conditions Thus, the TM11 mode is the lowest magnetic mode that will propagate
Waveguide impedance
All forms of transmission line, including the waveguide, exhibit a characteristic impedance, although in the case of waveguide it is a little difficult to pin down conceptually This concept was developed for ordinary transmission lines in Chap 3 For a waveguide, the characteristic impedance is approximately equal to the ratio of the electric and magnetic fields (E/H), and converges (as a function of frequency) to the intrinsic impedance of the dielectric (Fig 19-10) The impedance of the waveguide is a function of waveguide characteristic impedance (Zo ) and the wavelength in the waveguide:
384 Microwave waveguides and antennas
Impedance
Intrinsic impedance N of dielectric
F Fc Frequency 19-10 Impedance versus frequency
[1914]
Or, for rectangular waveguide, with constants taken into consideration: Z 120
[1915]
The propagation constant B for rectangular waveguide is a function of both cutoff frequency and operating frequency: B W eu 1 Fc F
[1916]
from which we can express the TE-mode impedance:
Waveguide terminations 385 ue ZTE 1 and the TM-mode impedance: ZTM 377 1 Fc F
[1917] Fc F
[1918]
Waveguide terminations
When an electromagnetic wave propagates down a waveguide, it must eventually reach the end of the guide If the end is open, then the wave will propagate into freespace The horn radiator is an example of an unterminated waveguide If the waveguide terminates in a metallic wall, then the wave reflects back down the waveguide, from whence it came The interference between incident and reflected waves forms standing waves (see Chap 3) Such waves are stationary in space, but vary in the time domain In order to prevent standing waves, or more properly, the reflections that give rise to standing waves, the waveguide must be terminated in a matching impedance When a properly designed antenna is used to terminate the waveguide, it forms the matched load required to prevent reflections Otherwise, a dummy load must be provided Figure 19-11 shows several types of dummy load The classic termination is shown in Fig 19-11A The resistor making up the dummy load is a mixture of sand and graphite When the fields of the propagated wave enter the load, they cause currents to flow, which in turn cause heating Thus, the RF power dissipates in the sand-graphite rather than being reflected back down the waveguide A second dummy load is shown in Fig 19-11B The resistor element is a carbonized rod critically placed at the center of the electric field The E field causes currents to flow, resulting in I 2R losses that dissipate the power Bulk loads, similar to the graphite-sand chamber, are shown in Fig 19-11C, D, and E Using bulk material such as graphite or a carbonized synthetic material, these loads are used in much the same way as the sand load (ie, currents set up, and I 2R losses dissipate the power) The resistive vane load is shown in Fig 19-11F The plane of the element is orthogonal to the magnetic lines of force When the magnetic lines cut across the vane, currents are induced, which gives rise to the I 2R losses Very little RF energy reaches the metallic end of the waveguide, so there is little reflected energy and a low VSWR There are situations where it isn t desirable to terminate the waveguide in a dummy load Several reflective terminations are shown in Fig 19-12 Perhaps the simplest form is the permanent end plate shown in Fig 19-12A The metal cover must be welded or otherwise affixed through a very-low-resistance joint At the substantial power levels typically handled in transmitter waveguides, even small resistances can be important The end plate (shown in Fig 19-12B) uses a quarter-wavelength cup to reduce the effect of joint resistances The cup places the contact joint at a point that is a
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