Tim Healy, Santa Clara University

This note presents an alternative to the derivation of the characteristic impedance of a transmission line found in most texts. The author believes that this approach offers the student a different viewpoint, and perhaps a more intuitive perspective than the conventional derivation.
Recall the model of a transmission line which presents the line as a distributed series resistance and inductance, and shunt conductance and capacitance:

o------Rdx---Ldx------------o------Rdx----Ldx----------------o------Rdx-----Ldx---- - - - -

>>>>>>>>>>>> Cdx Gdx

o------------------------------o----------------------------------o--------------------- - - - -

where the ">>>>>>>>>>" has no technical meaning. I just put it in to overcome HTML's aversion for spaces.

In the model R, L, G and C, are respectively, the per unit length resistance, inductance, conductance and capacitance of the line.

Thus the line is modeled by a series of differential segments with differential series (Rdx, Ldx) and shunt(Cdx, Gdx) elements. 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 Zo. That is, the impedance looking into the line on the left is Zo. But, of course, if we go down the line one differential length dx, the impedance into the line is still Zo. Hence we can say that the impedance looking into the line on the far left is equal to Zo in parallel with Cdx and Gdx, all of which is in series with Rdx and Ldx. That is:

Zo = Rdx + jwLdx + Zo//(1/(Gdx + jwCdx)

where // means "in parallel with". Hence:

Zo = Rdx + jwLdx + Zo/[Zo(Gdx + jwCdx) + 1]

Zo + Zosq(Gdx + jwCdx) = (Rdx + jwLdx) + Zo(Gdx + jwCdx)(Rdx + jwldx) + Zo

The product of two parentheses involves second order terms in dx, and hence these are discarded, leaving:

Zo = sqrt[(R + jwL)/(G + jwC)]

where, in the above, Zosq means Zo squared, and sqrt means square root.

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