I have done a little searching in the library, and I have found references for some previous calculations that are essentially the same as the ones I did for the equivalent diameters of rectangular cross section conductors.

The
basic transformation for the
self impedance equivalent diameter was written down in
1929 by W.G. Bickley, *Two Dimensional Problems Concerning a Single
Closed Boundary*, Phil. Tran. **228A**, 235 (1929).
He describes the conformal mapping and the result
but doesn't actually write down the result.

Seymour B. Cohn, *Problems in Strip Transmission Lines*,
IRE Transactions on Microwave Technology and Techniques,
**3**, 119 (1955),
uses the result for the self impedance equivalent
diameter to calculate the equivalent resistance diameter by taking the
derivative numerically. He apparently thought the analytic
calculation was too difficult, but that obviously isn't true
since I was able to do it.
The Bickley result had
apparently been reinvented and tabulated by C. Flammer for a
Stanford Research Institute report. Cohn was also at the Stanford
Research Institute so he had easy access to this report. Most of
these reports were not publically available since they were generally
done under contract. Cohn also cites (his reference 8)
that the results were
published by N. Marcuvitz, *Waveguide Handbook*, M.I.T.
Radiation Laboratory Series, Vol. 10, McGraw-Hill, 1951.
Sure enough, the elliptic function results for the
self impedance diameter are calculated for
the effective diameter of rectangular cross section obstructions in
waveguides.

You can verify that Cohn's graphs of the
two effective diameters are the same as the ones I calculated.
His *d*_{0} graphed in figure 2 is my *d*_{self} . By
looking at his Eq. 19 for the attenuation of microstrip with
a rectangular cross section conductor,

= | (1) |

d_{resistance} = .
| (2) |

Cohn's graph in figure 4 is therefore the ratio
*d*_{0}/*d*_{resistance} . Here's a quick comparison of my calculations
and Cohn's results showing they are identical to the extent I can
read the graphs.

t/w = d''/d' |
Figure 2 | my d_{self} |
Figure 4. | Figure 2/Figure 4 | my
d_{resistance} |

1.0 | 1.18 | 1.18 | 1.18 | 1.0 | 1.0 |

0.5 | 0.875 | 0.875 | 1.195 | 0.732 | 0.732 |

0.2 | 0.672 | 0.672 | 1.256 | 0.535 | 0.535 |

0.1 | 0.595 | 0.595 | 1.330 | 0.447 | 0.449 |

0.05 | 0.55 | 0.553 | 1.412 | 0.390 | 0.391 |