A LADDER NETWORK THEOREM 309 



The impedance across either of these terminal pairs is the parallel 

 impedance of the full-series and full-shunt iterative impedances (or 

 one-half the mid-shunt iterative impedance). The full-series and full- 

 shunt iterative impedances are given by the following formulas: 



Full-series Ki = ^[V2(2 + 423) + s] = i^2 + 2, 



Full-shunt K2 = |[V2(2 -f 423) - z] = J^'f' , (3) 



Ki -\- Z3 



where, for brevity, 2 = 2i -}- 22 — 2zi2. 



Then 



7 7 TT ^^^^ \ ^ , 423]-^/^ ... 



^12:12 = ^34:34 = A = Z^ _l_ J^ = 23 1 H • (4) 



The voltage across lines is propagated as exp (— ka) where a is the 

 section propagation constant; hence 



Zi2:34 = T = Ke-'^\ (5) 



The propagation factor exp (— a) is defined in terms of the iterative 

 impedances by 



<^- = 1^ • (6) 



The currents 4:i2 and 4:34 are given by the following formulas: 



'"""' jg.+'ji:/ ''"'""'"'- <^' 



This completes the formulation, since the remaining quantities, v 

 and Z, are given immediately by 



2i — 2i2 



2i -f 22 — 2Zi 



„ 21Z2 — 2i2^ 2i22 — 2i2^ 



Z = 2^ , ^j — = n 



k=lZ\ + 22 — 22i2 2i -f 22 — 22i2 



Figure 2B shows driving-point and transfer impedances for energiza- 

 tion between terminals, omitting certain impedances equal by symmetry. 

 Figure 2C shows the corresponding ^-section currents in side 2. 



