IMPEDANCE CONCEPT AND APPLICATION 19 



all physical phenomena are essentially three-dimensional, frequently 

 all but one are irrelevant and can be ignored or are relatively unimpor- 

 tant and can be neglected. In the mathematical language, this 

 means that only one coordinate (distance, angle, etc.) is retained ex- 

 plicitly in the equations of transmission. 



The paper is divided into three parts. Part I discusses broadly the 

 ratios to which the term "impedance" can appropriately be applied 

 in a wide variety of physical fields, ranging from electric circuits and 

 heat conduction to electromagnetic radiation. In this part the con- 

 cept is gradually broadened until at the end it has acquired the prop- 

 erty of direction mentioned above. Parts II and III consider the 

 general laws governing reflection, refraction, shielding and power 

 absorption, and rephrase them as theorems regarding the generalized 

 impedances. To make the illustrations more effective, familiar ex- 

 amples are chosen. 



PART I 

 THE IMPEDANCE CONCEPT 

 Electric Circuits 

 In an electric circuit comprised of a resistance R and an inductance 

 L, the instantaneous voltage-current relation is described by the fol- 

 lowing differential equation 



L^ + RIo=Vo, (1) 



where Vo is the applied electromotive force. If Vo varies harmonically 

 with frequency /, ultimately /o will also vary harmonically with fre- 

 quency /. What happens is that the solution of (1) consists of two 

 parts, the transient part and the steady state part, the former decreasing 

 exponentially with time and the latter being periodic. 



The steady state solution of (1), or indeed of the most general linear 

 differential equation with constant coefficients, can be found by means 

 of a simple mathematical device based upon the use of complex num- 

 bers. Thus if Fo and /o vary harmonically, they may be regarded as 

 real parts of the corresponding complex expressions Ve'"^ and /e*"', 

 where/ = co/lir is the frequency. The quantities V and / are complex 

 numbers whose moduli represent the amplitudes and whose phases 

 are the initial phases (at the instant / = 0) of the electromotive force 

 and the electric current. The time rate of change of /o is then the 

 real part of the derivative of /e'"', that is, the real part of ioiIe'^K 



If we form another equation after the pattern of (1), replacing /o 

 and Vq by the imaginary part of /e'"' and Fe™', and add the new 



