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Solution of the equations

We notę that the equation above gives two values for y , of which one must be in the flrst quadrant, and the other in the third. We denote the value in the first quadrant by y , and its real and imaginary parts by a and fi. The value in the third quadrant is called

-y •

It is always uscful to obscrvc similarities and differenccs between various elements of notation. The variables Vrand Vrwhich have been just introduced may be contrastcd with the time domain variables Viand Vrin that the frequency domain variables just introduced are COmplex constants with no spatial or time variation whereas the time domain variables Vfand Vrare not constants but are arbitrary functions of the single implied argument z -ctor z + ct. Moving on. we notę that the subscripts f and r are chosen to reflect the fact that, as we will show latcr, Vireprescnts the amplitudę andthe phase (at the origin) of a forward wave, while Vrrepresents the amplitudę and phase (both at the origin) of a revcrsc wave. As already noted we cali y = a+jfi the complex

propagation constant. We cali a the attentuation constant and fi the phase constant. The

currcnt I(z) which accompanies this voltage can be found from cquation 2.36 to be. Notę that Zo is complex, but

if we examine carefully the derivation above we see that it flrst emerged as a notation for

the quantity Z/ y . Since both Z and y are in the first quadrant of the Argand diagram, Zo must be in the right half piane. Thus, when we dcrivc Zo from cquation 2.45, where

we appear to have two Solutions at ourdisposal. we must take the one with a positive real

part to be consistent with our choice of y as the value in the first quadrant.