On working with complex Zo

Some recent articles discussed some effects that in part are a result of Zo having a complex value (ie a non-zero imaginary part).

Distortionless Lines (and Lossless Lines are a special case of Distortionless Lines) have purely real Zo.

Practical transmission lines almost never have purely real Zo, Zo usually has a non-zero imaginary part, even if very small.

The maths of transmission line behavior in terms of known Zo, Zload, and propagation constant γ is sound, solutions exist for the complex reflection coefficient Γ along the line.

ρ, the magnitude of Γ, is often derived but we should remember that in discarding the phase of Γ we know less.

When Zo is purely real

However, ρ leads to some interesting inferences but only in the case where Zo is purely real.


Attenuation is uniform with displacement.

We can talk of the notion of forward and reverse power (Pfwd and Prev) as components of power at some point along the line, irrespective of the ratio V/I at that point (ie independent of Z at that point).

We can speak of Return Loss being Pfwd/Prev calculated from ρ as ρ^-2, or more commonly in dB as -20*log(ρ).

We can speak of Mismatch Loss based on ρ as -10*log(1-ρ2), but only when the Thevenin equivalent source impedance is exactly the same as Zo (and of course by virtue of that, purely real).

Additionally we can infer VSWR in that region of the line based on ρ (VSWR=(1+ρ)/(1-ρ)), but only when line loss is very low.

Within those restrictions, calculated values are sane.

When Zo is approximately real

When Zo is approximately real and approximately equal to that of the real line, we can do all those things listed under “When Zo is purely real”, but there will be some error, and if the results seem silly, it is not the formulas but the misapplication that is the problem.

The error will be less where:

  • the imaginary part of Zo is relatively very small (-Xo/Ro<<<1); and
  • ρ is very small (ρ<<1).

If results look insane, better review the basis for your approximation.

Measuring instruments

Measuring instruments are almost always calibrated for purely real Zo or Zref, and as such and within the limits of accuracy, all those things listed under “When Zo is purely real” can be done with good accuracy… but the measurer must keep in mind that the measurements are not in terms of the adjacent things and that itself is a source of error.

An article on the practical pitfalls with only a small imaginary part to Zo is given at On Witt’s calculation of Matched Line Loss from Return Loss. Notwithstanding the failure, lots of ham grade antenna analysers have a “Matched Cable Loss” function that falls foul of the discrepancy.