8 BELL SYSTEM TECHNICAL JOURNAL 



line, and at the receiving end is reflected in the ratio of the difference 

 between the receiving impedance and characteristic Hne impedance 

 to the sum of these impedances; propagated back toward the trans- 

 mitting end with continued attenuation; reflected there if the trans- 

 mitting end impedance is not equal to the characteristic line impedance, 

 the doubly reflected wave propagated toward the receiving end, and 

 so on in an infinite series of propagations and reflections. 



In both power and communication work one quantity which is of 

 great importance in considering the characteristics of the transmission 

 line is the ratio of the voltage at the transmitting end to that at the 

 receiving end. This is, of course, readily derived from equation 2, and 

 is presented in equation 3 in the beautifully compact form offered by 

 the use of hyperbolic functions of the propagation constant of the line. 



j^ = cosh yl -\- -^ sinh 7/. (3) 



Considering first the application of this general transmission line 

 theory to power transmission circuits, it would appear to be of great 

 value for the student to appreciate the relationship between the general 

 formula and the approximations used for short lines. This is brought 

 out clearly by equation 4 and the diagrams and equations pre- 

 sented under 5. 



F, ~ ^Z,"^ 2! ^ 3!Z, ^ 4! "^ 5!Z, ^ ••• ^' 



Equation 4 is simply the development of the general formula of 

 equation 3 into a series of terms of ascending powers of Zl, the 

 total resistance and reactance of the line, and of Yl the total shunt 

 admittance of the line. The comparison of this series expansion with 

 the results indicated by various approximate methods is of consider- 

 able interest. The first term, unity, is naturally the ratio of trans- 

 mitting and receiving voltages with no transmission line, as indicated 



I Zl \ 

 in 5a. With the addition of the second term ( y- \ one has the result 



obtained by entirely ignoring the capacity of the Hne as indicated in Sh. 

 The first three terms of the series give the result obtained by assuming 

 that one half of the capacity of the line is concentrated at each end of 

 the line as indicated in 5c, namely, a simple Pi network. The simple 

 T network, assuming the capacity all concentrated at the middle as 

 indicated in Sd, gives 4 terms, but you will note that the fourth term 



