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374 Microwave waveguides and antennas
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Propagation modes in waveguides 375
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The TEM wave will not propagate in a waveguide because certain boundary conditions apply Although the wave in the waveguide propagates through the air (or inert gas dielectric) in a manner similar to free-space propagation, the phenomenon is bounded by the walls of the waveguide, and that implies certain conditions must be met The boundary conditions for waveguides are 1 The electric field must be orthogonal to the conductor in order to exist at the surface of that conductor 2 The magnetic field must not be orthogonal to the surface of the waveguide In order to satisfy these boundary conditions the waveguide gives rise to two types of propagation modes: transverse electric mode (TE mode), and transverse magnetic mode (TM mode) The TEM mode violates the boundary conditions because the magnetic field is not parallel to the surface, and so does not occur in waveguides The transverse electric field requirement means that the E field must be perpendicular to the conductor wall of the waveguide This requirement is met by use of a proper coupling scheme at the input end of the waveguide A vertically polarized coupling radiator will provide the necessary transverse field One boundary condition requires that the magnetic (H) field must not be orthogonal to the conductor surface Because it is at right angles to the E field, it will meet this requirement (see Fig 19-6) The planes formed by the magnetic field are parallel to both the direction of propagation and the wide dimension surface As the wave propagates away from the input radiator, it resolves into two components that are not along the axis of propagation, and are not orthogonal to the walls The component along the waveguide axis violates the boundary conditions, so it is rapidly attenuated For the sake of simplicity, only one component is shown in Fig 19-7 Three cases are shown in Fig 19-7: high, medium, and low frequency Note that the angle of incidence with the waveguide wall increases as frequency drops The angle rises toward 90 as the cutoff frequency is approached from above Below the cutoff frequency the angle is 90 , so the wave bounces back and forth between the walls without propagating
Top view
"a" Dim
Weak H-field Strong H-field Cross sectional view at center of side view
"a" Dim /2 End view Cross sectional view /4 from end
Side "b" view Dim 19-6 Magnetic fields in waveguide
376 Microwave waveguides and antennas
A High frequency
19-7 Frequency effect on propagating wave
B Medium frequency
C Low frequency
(approaching cutoff) = = of incidence of reflection
Coordinate system and dominant mode in waveguides
Figure 19-8 shows the coordinate system used to denote dimensions and directions in microwave discussions The a and b dimensions of the waveguide correspond to the x and y axes of a cartesian coordinate system, and the z axis is the direction of wave propagation In describing the various modes of propagation, use a shorthand notation as follows: Txm,n where x is E for transverse electric mode, and M for transverse magnetic mode m is the number of half-wavelengths along the x axis (ie, a dimension) n is the number of half-wavelengths along the y axis (ie, b dimension)
Propagation modes in waveguides 377 The TE10 mode is called the dominant mode, and is the best mode for low attenuation propagation in the z axis The nomenclature TE10 indicates that there is one half-wavelength in the a dimension and zero half-wavelengths in the b dimension The dominant mode exists at the lowest frequency at which the waveguide is a half-wavelength
Velocity and wavelength in waveguides
Figures 19-9A and 19-9B show the geometry for two wave components, simplified for the sake of illustration There are three different wave velocities to consider with respect to waveguides: free-space velocity c, group velocity Vg , and phase velocity Vp The free-space velocity is the velocity of propagation in unbounded free space (ie, the speed of light c 3 108 m/s)
90 X TEM,N TMM,N 19-8 Rectangular waveguide coordinate system
378 Microwave waveguides and antennas
Antenna a
19-9A Antenna radiator in capped waveguide
/4 A a
g/ 4
a Vg C
19-9B Wave propagation in waveguide
The group velocity is the straight line velocity of propagation of the wave down the center line (z axis) of the waveguides The value of Vg is always less than c, because the actual path length taken, as the wave bounces back and forth, is longer than the straight line path (ie, path ABC is longer than path AC) The relationship between c and Vg is Vg c sin a [191]
Propagation modes in waveguides 379 where Vg is the group velocity, in meters per second c is the free-space velocity (3 108 m/s) a is the angle of incidence in the waveguide The phase velocity is the velocity of propagation of the spot on the waveguide wall where the wave impinges (eg, point B in Fig 19-9B) This velocity, depending upon the angle of incidence, can actually be faster than both the group velocity and the speed of light The relationship between phase and group velocities can be seen in the beach analogy Consider an ocean beach, on which the waves arrive from offshore at an angle other than 90 In other words, the arriving wavefronts are not parallel to the shore The arriving waves have a group velocity Vg But as a wave hits the shore, it will strike a point down the beach first, and the point of strike races up the beach at a much faster phase velocity Vp, that is even faster than the group velocity In a microwave waveguide, the phase velocity can be greater than c, as can be seen from Eq 192: Vp c sin a [192]
Example 19-1 Calculate the group and phase velocities for an angle of incidence of 33 Solution: (a) Group velocity Vg c sin a (3 (3 (b) Phase velocity Vp c/sin a (3 (3 551 For this problem the solutions are c Vp Vg 3 551 16 108 m/s 108 m/s 108 m/s 108 m/s)/sin 33 108 m/s)/(05446) 108 m/s 108)(sin 33 ) 108)(05446) 16 108 m/s
We can also write a relationship between all three velocities by combining Eqs 191 and 192, resulting in
380 Microwave waveguides and antennas c VpVg [193]
In any wave phenomenon the product of frequency and wavelength is the velocity Thus, for a TEM wave in unbounded free space we know that: c F
[194]
Because the frequency F is fixed by the generator, only the wavelength can change when the velocity changes In a microwave waveguide we can relate phase velocity to wavelength as the wave is propagated in the waveguide: Vp
[195]
where: Vp is the phase velocity, in meters per second c is the free-space velocity (3 108 m/s) is the wavelength in the waveguide, in meters is the wavelength in free space (c/F), in meters (see Eq 194) o Equation 195 can be rearranged to find the wavelength in the waveguide: Vp c
[196]
Example 19-2 A 56-GHz microwave signal is propagated in a waveguide Assume that the internal angle of incidence to the waveguide surfaces is 42 degrees Calculate (a) phase velocity, (b) wavelength in unbounded free space, and (c) wavelength of the signal in the waveguide Solution: (a) Phase velocity Vp c sin a 3 108 m/s sin 42 108 m/s 06991 (b) Wavelength in free space
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