A Note on the Transmission Line Equation 

 in Terms of Impedance 



By J. R. PIERCE 



INCREASED familiarity derived in handling Maxwell's equations, 

 especially in connection with problems arising at very high frequencies, 

 has resulted in a variety of forms for expressing certain laws and behavior. 

 Especially, work by Schelkunoflf in extending the impedance concept^ shows 

 that impedance can be quite as general and exact a means for expressing 

 electromagnetic relations as are current, voltage, electric and magnetic 

 fields, and vector and scalar potentials. 



In reformulating certain problems in terms of impedance the content and 

 ultimate solution must of course be equivalent. There may, however, be 

 a considerable change of procedure and sometimes a simplification. For 

 instance, in many cases a single impedance condition can replace the usual 

 two boundary conditions for voltage and current. 



One very simple case in which it is perhaps easiest to deal directly with 

 impedance is in the derivation of the transmission line equation on a dis- 

 tributed constant basis. In the usual derivation, two linear second order 

 differential equations are obtained, one for voltage and one for current. 

 The impedance, in terms of which the engineer expresses many of his results, 

 is obtained as a ratio from solutions for voltage and current. In treating 

 the transmission line from the impedance point of view, without dealing 

 with currents and voltages, a first order non-linear differential equation in 

 terms of impedance and distance is obtained. This impedance equation 

 is a Ricatti equation and could be obtained from the usual line equations. 

 It is simpler, however, to derive it directly. 



As the principal interest of such a treatment lies in the method and in 

 the fact that the line may be tapered, rather than in losses, the derivations 

 will be carried out for lossless lines. Losses can be taken into account by 

 allowing the inductance per unit length, L, and the capacitance per unit 

 length, C, to become complex quantities. 



Consider the section of line dx long, shown in the figure, having an 

 inductance L dx and a capacitance C dx. We can write immediately 



Zx + dZ = Zx+dx 



= jooL dx -\- .- 



juC dx -f- l/Zx 



= Zx + jo}[L — CZx] dx. (1) 



' "The Impedance Concept and Its Application to Problems of Reflection, Refraction, 

 Shielding, and Power Absorption," B.S.T.J. Vol. 17, pp. 17-48, January, 1938. 



263 



