Derivation of this mho circle equation?

Question

I'm picking through Protective Relaying Principles and Applications by Blackburn and Domin (4th ed), and am currently figuring out distance relaying. Intuitively, I understand their operation. However, I cannot figure out how they derived the equation (6.4) for the mho circle. I expected Zop = (Zr/2 ∠ θr) - (Zr/2 ∠ θ+θr) and tried to see if it was equivalent, but could not.
I also believe (6.3) is incorrect; since the the line before desribed the voltage as line-neutral, there should be no sqrt3 in the denominator. This would also mean that (6.6) and (6.8) are incorrect.
Routledge.com has a pdf of errata available for download, but the link returned 403 Forbidden. I emailed customer service to ask about it. Are these actual typos, or am I missing something?

relayman357 · Answer

I'm looking in the old version (yellow cover) and it defines VLN as line-to-line in the text underneath the equation.  One of the later revisions injected a bunch of errors (can't remember which one as i stick with old yellow).  You have found one such error and you are correct.  Equation 6.3 is only correct when you recognize VLN is line-line voltage.  Sloppy work in this edition.
Just use equation 6.5 to plot the circle.  ZR is the reach at maximum torque angle (φr) and φx is the angle at which you are finding a point on the circle (ZX).
If i recall correctly, you can find out if your version of the text is the garbage version by going to the list of device numbers.  Many of them were wrong (i think including device 21 which should be distance relay).

Derivation of this mho circle equation?

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