ELECTROMAGNETIC FIELD ANALYSIS 489 



Natural Network Models of Continuous Media and 

 Maxwell's Differential Equations 



Transmission line theory represents a well known example of the applica- 

 tion of circuit theory to continuous systems. Two-wire transmission lines 

 are subdivided into infinitesimal sections by planes perpendicular to the 

 lines. Each section is replaced by a capacitor whose capacitance is so 

 chosen that, for a given voltage across the transmission line, the electric 

 charges on the plates of the capacitor are correspondingly equal to the 

 charges on the sections of the wires constituting the Hne. The leads con- 

 necting the terminals of these capacitors are then assumed to possess an 

 inductance and a resistance but no capacitance. Thus the electric flux or 

 displacement is ''swept" into tiny capacitors, and the magnetic flux or 

 displacement into tiny inductors. 



This representation is good only at low frequencies because it depends on 

 the assumption that the electric displacement is only in one direction, 

 namely at right angles to the transmission line. In effect, this representa- 

 tion neglects the capacitance between different parts of the same conductor 

 and includes only the capacitance between the opposite segments of different 

 conductors. That is, while we have recognized that the inductance and 

 capacitance are distributed in the direction parallel to the transmission line, 

 we have ignored the fact that they are also distributed at right angles to the 

 line. In the general representation we should subdivide the medium into 

 infinitesimal blocks and devise a three-dimensional network lattice of 

 infinitely small meshes. Fig. 1. The displacement current can be swept 

 equally into tiny capacitors. If the medium is dissipative, the resistors 

 may be inserted in parallel with the capacitors to take care of the con- 

 duction currents in the medium. The magnetic flux is swept equally into 

 tiny coils in the corners of each mesh. However, the resulting network is 

 not homogeneous. Besides meshes of type A consisting of four capacitors 

 and four inductors, it contains meshes of type B consisting of inductors only; 

 and yet we started with a homogeneous medium. Gabriel Kron solved the 

 difficulty by introducing ideal transformers (with one-to-one turn ratio) with 

 their windings in series with the coils at the opposite corners of each A-mesh. 

 These transformers do not affect the electrical performance of the A-meshes 

 but introduce infinite impedance into B-meshes and thus e^ectively elimi- 

 nate them. 



As a matter of fact, such transformers should properly be included in the 

 network representations of two-wire lines. In fact, by implication they are 

 included as soon as we state that the direct and return currents in the line 

 are equal and opposite. \Mthout an infinite impedance to currents flowing 

 in the same direction we cannot have the balance. Pursuing the matter 



