lEilkCTROMAGNETIC FIELD ANALYSIS 507 



Thus if we write 



^^Ze^/, H^Zhr,p\ (15) 



n=0 n=0 



and substitute in (14), we obtain 



curl eo = 0, curl Cn+i = — fi hn, 



curl ho = geo, curl hn+i = gCn+i + ee n. 



(16) 



If these equations are solved subject to the prescribed boundary conditions, 

 E and H will be expressed as power series in the oscillation constant p- 



The function theory has already been used successfully in the restricted 

 circuit theory; that is, in the theory of finite networks composed of ideal 

 (independent of the frequency) resistances, inductances and capacitances. 

 Likewise, some very general theorems have been established concerning 

 any physical input impedance. Whereas the poles and zeros of a function 

 can be anywhere in the complex ^-plane, the poles and zeros of the input 

 impedance of a passive system never lie to the right of the imaginary axis. 

 This leads to a theorem to the effect that all poles and zeros on the imaginary 

 axis are simple. The resistance components of the input impedance on 

 the imaginary axis determine the reactance component and hence the 

 complete impedance function except for a purely reactive impedance. The 

 zeros and poles of an impedance occur always in conjugate pairs. These 

 are some of the general theorems of impedance analysis. Not very long 

 ago I came across an expression for the input impedance of a spherical 

 antenna which was obtained by what appeared superficially as a straight- 

 forward conventional method; but as soon as I observed that some poles 

 were situated to the right of the imaginary axis, I knew that the expression 

 had to be false. The existence of poles in this region meant a possibility 

 of oscillations which would increase indefinitely of their own accord. 



The difference between finite and infinite networks consists in that the 

 former possess a finite number of zeros and poles. All physical structures 

 always possess an infinite number of such singularities; but a finite number 

 of them may form a cluster in the vicinity of the origin, far removed from 

 all other zeros and poles. When this happens we have a physical finite 

 network. In a reactive network all zeros and poles lie on the imaginary 

 axis. In a slightly dissipative system these zeros and poles move a little 

 to the left of the imaginary axis. This happens, for instance, in the case 

 of a thin antenna. The field in the vicinity of a thin wire is large and the 

 radiated power is only a small fraction of the stored energy. The distribu- 

 tion of poles (the solid circles) and zeros (the hollow circles) is illustrated 



