IMPEDANCE CONCEPT AND APPLICATION 



25 



The algebraic signs preceding the ratios of components with different 

 subscripts are assigned as follows. If a right-hand screw is rotated 

 through 90° from the positive axis indicated by the subscript in the 

 numerator toward the positive axis indicated by the subscript in the 

 denominator, it will advance either in the positive or in the negative 

 direction of the remaining axis. In the former case the ratio is given 

 the positive sign and in the latter the negative sign. This convention 

 happens to be particularly convenient in expressions for the Poynting 

 vector. 



Thus two impedances are associated with any pair of perpendicular 

 directions, the A;-axis and the j-axis, let us say ; these impedances are : 



H' 





If these two impedances are equal, then we define the impedance in 

 the direction of the positive z-axis as follows : 



7 = r^ = — 



H. 



Similar definitions hold for the impedances in other directions. 



While the impedances as now defined possess an attribute of direc- 

 tion, they are neither vectors nor tensors because they do not add in 

 the proper fashion. However, in practical applications this lack of 

 vectorial properties does not seem to be a drawback. 



The above definitions can be extended to other systems of coordi- 

 nates. Let r be the distance of a point P {r, 6, <p) from the origin of 

 the spherical coordinate system, 6 the polar angle or colatitude and 

 (p the meridian angle or the longitude (Fig. 1). Then the "radial" 

 impedance in the outward direction is defined as 



7 — ^ — —^ 

 Hu> Ha 



(6) 



provided the two ratios of the field components are equal. The radial 

 impedance looking toward the origin is defined as the negative of (6). 

 Similarly the "meridian" impedance in the direction of increasing d 



