VSWR=1 and X≠0

Another of those threads has broken out on eHam illustrating that lots of hams do not understand the complex nature of impedance and cannot see the consequences of the formula to calculate VSWR from load impedance and transmission line characteristic impedance.

Most methods of measuring VSWR are indirect, and they are based on an assumed Zo which is purely real (ie Xo=0Ω), and we speak loosely of that as the VSWR even though the standing wave that might exist on a practical transmission line is a little different as a consequence of that assumption being a little bit in error.

In that context, discussion is not about the actual standing wave, but the value calculated from the indirect observation and the assumed purely real Zo.

We can speak of Z at any point along a transmission line (in this case, one with Zo purely real) being the ratio V/IΩ, and Z having complex components R and X.

For VSWR=1 under those conditions, R=Ro=|Zo|Ω and X=Xo=0Ω.

In that case, there is NO value for X other than zero that will coincide with VSWR=1.

If you think otherwise, try the calculator at Calculate VSWR and Return Loss from Zload and Zo but you MUST leave Xo blank or zero to comply with the assumption that an instrument is calibrated for a nominal purely real Zo.

Whilst it is possible that there are points along a line with standing waves that X=0, R at that point cannot be equal to Zo because there are standing waves, ie there is no point on a line of characteristic impedance Zo with standing waves where Z=Ro+j0.

Real world calculations that do account for line loss will give slightly different results. For example, 50m of RG58/CU with a 50+j0Ω load will have an input impedance at 1MHz of 49.94-j4.86 Ω and actual VSWR at the source end will be 1.05, VSWR(50) at the source end is 1.10, all this a consequence of the fact that Zo of the nominal 50Ω line is actually 50.07-j2.64Ω.

There is nothing wrong with calibration of instruments for a nominal purely real Zo, in fact it is the only practical option since to do otherwise would be to calibrate for Zo which depends on frequency, conductor loss and dielectric loss. You just need to remember that the assumption is made that Zo is purely real, and consequences of that assumption.