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# Line Impedance

Module by: Bill Wilson. E-mail the author

Unfortunately, since we don't know what value the phasor V + V + has, these equations do not do us a whole lot of good! One way to deal with this is to simply divide this equation into this equation. That gets rid of V + V + and the eiβs β s and so we now come up with a new variable, which we shall call line impedance, Zs Z s .

ZsVsIs= Z 0 1+ Γ ν e-2iβs1 Γ ν e-2iβs Z s V s I s Z 0 1 Γ ν -2 β s 1 Γ ν -2 β s
(1)
Zs Z s represents the ratio of the total voltage to the total current anywhere on the line. Thus, if we have a line LL long, terminated with a load impedance Z L Z L , which gives rise to a terminal reflection coefficient Γ ν Γ ν , then if we substitute Γ ν Γ ν and LL into Equation 1, the ZL Z L which we calculate will be the "apparent" impedance which we would see looking into the input terminals to the line!

There are several ways in which we can look at Equation 1. One is to try to put it into a more tractable form, that we might be able to use to find Zs Z s , given some line impedance Z 0 Z 0 , a load impedance Z L Z L and a distance, ss away from the load. We can start out by multiplying top and bottom by eiβs β s , substituting in for Γ ν Γ ν , and then multiplying top and bottom by Z L + Z 0 Z L Z 0 .

Zs= Z 0 ( Z L + Z 0 )eiβs Z L e(iβs)( Z L + Z 0 )eiβs( Z L Z 0 )e(iβs) Z s Z 0 Z L Z 0 β s Z L Z 0 β s Z L Z 0 β s Z L Z 0 β s
(2)
Next, we use Euler's relation, and substitute cosβs±isinβs ± β s β s for the exponential. Lots of things will cancel out, and if we do the math carefully, we end up with
Zs= Z 0 Z L +i Z 0 tanβs Z 0 +i Z L tanβs Z s Z 0 Z L Z 0 β s Z 0 Z L β s
(3)
For some people, this equation is more satisfying than Equation 1, but for me, both are about equally opaque in terms if seeing how Zs Z s is going to behave with various loads, as we move down the line towards the generator. Equation 3 does have the nice property that it is easy to calculate, and hence could be put into MATLAB or a programmable calculator. (In fact you could program Equation 1 just as well for that matter.) You could specify a certain set of conditions and easily find Zs Z s , but you would not get much insight into how a transmission line actually behaves.

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