Here, we follow an approach posted by Tim Healy. The line is modeled by a series of differential segments with differential series elements ( R d ⁡ ⁡ x , L d ⁡ ⁡ x ) and shunt elements ( C d ⁡ ⁡ x , G d ⁡ ⁡ x ) (as shown in the figure at the beginning of the article). The characteristic impedance is defined as the ratio of the input voltage to the input current of a semi-infinite length of line. We call this impedance Z 0 . That is, the impedance looking into the line on the left is Z 0 . But, of course, if we go down the line one differential length d ⁡ ⁡ x , the impedance into the line is still Z 0 . Hence we can say that the impedance looking into the line on the far left is equal to Z 0 in parallel with C d ⁡ ⁡ x and G d ⁡ ⁡ x , all of which is in series with R d ⁡ ⁡ x and L d ⁡ ⁡ x . Hence: Z 0 = ( R + j ω ω L ) d ⁡ ⁡ x + 1 ( G + j ω ω C ) d ⁡ ⁡ x + 1 Z 0 Z 0 = ( R + j ω ω L ) d ⁡ ⁡ x + Z 0 Z 0 ( G + j ω ω C ) d ⁡ ⁡ x + 1 Z 0 + Z 0 2 ( G + j ω ω C ) d ⁡ ⁡ x = ( R + j ω ω L ) d ⁡ ⁡ x + Z 0 ( G + j ω ω C ) d ⁡ ⁡ x ( R + j ω ω L ) d ⁡ ⁡ x + Z 0