10-11] TYPES OF TRANSMISSION LINES, MODES OF PROPAGATION 537 



In air or in air-filled transmission lines, the product of phase velocity and 

 group velocity is constant and is equal to c^, i.e. 



v^v, = c\ (10-21) 



In a refractive medium or in a transmission line filled with a refractive 

 medium this is modified so that 



'^'^ = (v^) 



(10-22) 



the relative permeability of the medium, 



er is the relative permittivity or dielectric constant of the medium, 

 and 



/^T-Cr is the refractive index of the medium. 



The propagation constant of a transmission line is 



7 = a+j/3 (10-23) 



where a, the attenuation constant, expresses the loss per unit length of 

 transmission line, and /3, the phase constant, expresses the phase change 

 per unit length of line. The phase constant /S is related to the phase velocity 

 by the equation 



B = — (10-24) 



Vp 



where co is the angular frequency, lirf. In general, each of the foregoing 

 parameters will have a different value for each mode and for each type of 

 transmission line. 



Coaxial Lines. Operating in the principal (TEM) mode (Fig. 10-1 la), 

 a coaxial transmission line is nondispersive and will transmit a wave of any 

 frequency. Higher-order modes can propagate, however, when the mean 

 internal circumference of the coaxial line is greater than a wavelength. As 

 one approaches microwave frequencies, coaxial lines become less practical 

 because of increased losses and reduced power-handling capability incident 

 to reduced dimensions. Consequently, coaxial transmission lines find 

 greatest application below 3000 Mc. 



Rectangular Waveguides. The most important line for microwave 

 use, the rectangular waveguide operating in the T£io mode (Fig. 10-1 Ic), 

 is a high-pass transmission medium having a low-frequency cutoff charac- 

 teristic which is governed by its greater width a, as illustrated in Fig. 10-12. 

 The height b of the waveguide has no influence on the cutoff wavelength of 

 the dominant mode, but does affect impedance, breakdown power, and 

 transmission loss. For each possible mode of propagation in a waveguide 



