These are some notes on calculating the characteristic impedance
of a coaxial line with a square cross section outer conductor
and a circular inner conductor. They were written in response to
comments by Zack Lau, KH6CP, in May 1995 QEX
that some
handbooks had a ``theoretically determined'' formula for the characteristic
impedance of coaxial cable with a square outer conductor that did
not agree with the empirically determined result
138*log*_{10}(1.08*D*/*a*)
where *D* is the side of the square, and *a* is the diameter of the inner
conductor. These are some notes that I sent Zack in September 1996
showing that transmission line theory predicts the 1.08 factor.
Apparently,
some handbooks suffered from a propagation of misprints.

The TEM mode characteristic impedance of coaxial lines is given by

Z_{0} = ,
| (1) |

c = ,
| (2) |

Z_{0} = .
| (3) |

One way to calculate the capacitance per unit length of a cylindrical structure is to realize that since there is no charge between the inner and outer conductors, the potential must be a solution of Laplace's equation

= 0. | (4) |

For the coaxial case, the DC field does not change along the length of the coax. Laplaces equation therefore reduces to the two-dimensional equation,

+ = 0, | (5) |

= b_{1}ln(r) + (b_{m + 1}r^{ 4m} + a_{m + 1}r^{ - 4m})cos(4m),
| (6) |

I take the case where the radius of the inner conductor is 1, and
half the side length of the outer conductor is *D* . Enforcing
the boundary condition of zero volts on the inner conductor
makes = 0 at *r* = 1 , which means *a*_{i} = - *b*_{i} . The
solution for is then

= b_{1}ln(r) + b_{m + 1}cos(4m)(r^{ 4m} - r^{ - 4m}).
| (7) |

b_{1}ln(r_{i}) + b_{m + 1}cos(4m)(r_{i}^{4m} - r_{i}^{- 4m}) = 1,
| (8) |

= , | |||

r_{i} = .
| (9) |

Once the *b*_{i} values are solved, I need to calculate the charge
per unit length. The charge density on a conductor
is the normal electric field times the dielectric constant.
In our case this is . The normal electric field
is most easily calculated for the inner conductor. Integrating
around the circular inner conductor immediately gives zero
contribution for all but the *b*_{1} term. The *b*_{1} term
give a charge per unit length of

q = 2b_{1},
| (10) |

Z_{0} = = .
| (11) |

b_{1}ln 1,
| (12) |

= 1.08, | (13) |

Z_{0} = 60ln (1.08 D),
| (14) |

Z_{0} = 60ln((D) D).
| (15) |

*Z*_{0} can only depend on the ratio of *D*/*a* where *D* is the
outer conductor halfside and *a* is the inner conductor radius.
This ratio is the same as the outer conductor side divided by
the inner conductor diameter, so that can be substituted as well.
Substituting *D*/*a* for D gives the final result.

The table shows the calculated and *Z*_{0} with N=10,
for *D*/*a* from 1.1 to 6.0. The value starts at 1.06 at *D*/*a* = 1.1
where *Z*_{0} = 9 Ohms. becomes 1.08 for *D*/*a* = 1.275
where *Z*_{0} = 19 Ohms. It remains at 1.08 thereafter. The asymptotic
value of is actually 1.0787. I have repeated the
calculation with N=20, with no change in the results indicating
good convergence.

The empirical value of 1.08 should work fine.

A final note for the compulsive nitpickers. The value 60 is really
two times the numerical value of the speed of light times the
appropriate power of ten. That is it is really
2 `x` 29.9792458
or 59.9584916.

The calculated characteristic impedance *Z*_{0} , for a coaxial
air line with a square cross section outer conductor of side *D* and a
circular cross section inner conductor of diameter a, as a function
of D/a. The value of where
*Z*_{0} = 60ln(*D*/*a*)
is also shown.

Z_{0} |
Z_{0} |
Z_{0} |
||||||

1.10000 | 1.06422 | 9.45326 | 2.40000 | 1.07868 | 57.07262 | 3.70000 | 1.07870 | 83.04562 |

1.12500 | 1.06724 | 10.97142 | 2.42500 | 1.07869 | 57.69448 | 3.72500 | 1.07870 | 83.44966 |

1.15000 | 1.06947 | 12.41565 | 2.45000 | 1.07869 | 58.30996 | 3.75000 | 1.07870 | 83.85100 |

1.17500 | 1.07118 | 13.80151 | 2.47500 | 1.07869 | 58.91918 | 3.77500 | 1.07870 | 84.24968 |

1.20000 | 1.07250 | 15.13895 | 2.50000 | 1.07869 | 59.52227 | 3.80000 | 1.07870 | 84.64572 |

1.22500 | 1.07355 | 16.43478 | 2.52500 | 1.07869 | 60.11936 | 3.82500 | 1.07870 | 85.03917 |

1.25000 | 1.07439 | 17.69392 | 2.55000 | 1.07869 | 60.71056 | 3.85000 | 1.07870 | 85.43005 |

1.27500 | 1.07507 | 18.92012 | 2.57500 | 1.07869 | 61.29598 | 3.87500 | 1.07870 | 85.81840 |

1.30000 | 1.07563 | 20.11629 | 2.60000 | 1.07869 | 61.87575 | 3.90000 | 1.07870 | 86.20425 |

1.32500 | 1.07609 | 21.28477 | 2.62500 | 1.07869 | 62.44996 | 3.92500 | 1.07870 | 86.58764 |

1.35000 | 1.07647 | 22.42752 | 2.65000 | 1.07870 | 63.01873 | 3.95000 | 1.07870 | 86.96860 |

1.37500 | 1.07679 | 23.54617 | 2.67500 | 1.07870 | 63.58215 | 3.97500 | 1.07870 | 87.34715 |

1.40000 | 1.07705 | 24.64213 | 2.70000 | 1.07870 | 64.14033 | 4.00000 | 1.07870 | 87.72333 |

1.42500 | 1.07728 | 25.71661 | 2.72500 | 1.07870 | 64.69336 | 4.02500 | 1.07870 | 88.09716 |

1.45000 | 1.07747 | 26.77070 | 2.75000 | 1.07870 | 65.24134 | 4.05000 | 1.07870 | 88.46868 |

1.47500 | 1.07763 | 27.80537 | 2.77500 | 1.07870 | 65.78436 | 4.07500 | 1.07870 | 88.83791 |

1.50000 | 1.07777 | 28.82147 | 2.80000 | 1.07870 | 66.32250 | 4.10000 | 1.07870 | 89.20489 |

1.52500 | 1.07789 | 29.81980 | 2.82500 | 1.07870 | 66.85587 | 4.12500 | 1.07870 | 89.56963 |

1.55000 | 1.07799 | 30.80108 | 2.85000 | 1.07870 | 67.38452 | 4.15000 | 1.07870 | 89.93217 |

1.57500 | 1.07808 | 31.76596 | 2.87500 | 1.07870 | 67.90857 | 4.17500 | 1.07870 | 90.29253 |

1.60000 | 1.07815 | 32.71506 | 2.90000 | 1.07870 | 68.42807 | 4.20000 | 1.07870 | 90.65074 |

1.62500 | 1.07822 | 33.64894 | 2.92500 | 1.07870 | 68.94311 | 4.22500 | 1.07870 | 91.00683 |

1.65000 | 1.07827 | 34.56815 | 2.95000 | 1.07870 | 69.45377 | 4.25000 | 1.07870 | 91.36081 |

1.67500 | 1.07832 | 35.47317 | 2.97500 | 1.07870 | 69.96012 | 4.27500 | 1.07870 | 91.71272 |

1.70000 | 1.07837 | 36.36448 | 3.00000 | 1.07870 | 70.46222 | 4.30000 | 1.07871 | 92.06258 |

1.72500 | 1.07840 | 37.24250 | 3.02500 | 1.07870 | 70.96016 | 4.32500 | 1.07871 | 92.41040 |

1.75000 | 1.07844 | 38.10766 | 3.05000 | 1.07870 | 71.45401 | 4.35000 | 1.07871 | 92.75623 |

1.77500 | 1.07847 | 38.96036 | 3.07500 | 1.07870 | 71.94382 | 4.37500 | 1.07871 | 93.10007 |

1.80000 | 1.07849 | 39.80095 | 3.10000 | 1.07870 | 72.42966 | 4.40000 | 1.07871 | 93.44195 |

1.82500 | 1.07851 | 40.62980 | 3.12500 | 1.07870 | 72.91160 | 4.42500 | 1.07871 | 93.78189 |

1.85000 | 1.07853 | 41.44725 | 3.15000 | 1.07870 | 73.38970 | 4.45000 | 1.07871 | 94.11992 |

1.87500 | 1.07855 | 42.25361 | 3.17500 | 1.07870 | 73.86402 | 4.47500 | 1.07871 | 94.45606 |

1.90000 | 1.07857 | 43.04919 | 3.20000 | 1.07870 | 74.33462 | 4.50000 | 1.07871 | 94.79032 |

1.92500 | 1.07858 | 43.83428 | 3.22500 | 1.07870 | 74.80155 | 4.52500 | 1.07871 | 95.12273 |

1.95000 | 1.07859 | 44.60917 | 3.25000 | 1.07870 | 75.26488 | 4.55000 | 1.07871 | 95.45331 |

1.97500 | 1.07860 | 45.37412 | 3.27500 | 1.07870 | 75.72466 | 4.57500 | 1.07871 | 95.78208 |

2.00000 | 1.07861 | 46.12939 | 3.30000 | 1.07870 | 76.18094 | 4.60000 | 1.07871 | 96.10906 |

2.02500 | 1.07862 | 46.87523 | 3.32500 | 1.07870 | 76.63378 | 4.62500 | 1.07871 | 96.43426 |

2.05000 | 1.07863 | 47.61187 | 3.35000 | 1.07870 | 77.08322 | 4.65000 | 1.07871 | 96.75771 |

2.07500 | 1.07864 | 48.33954 | 3.37500 | 1.07870 | 77.52933 | 4.67500 | 1.07871 | 97.07943 |

2.10000 | 1.07864 | 49.05846 | 3.40000 | 1.07870 | 77.97214 | 4.70000 | 1.07871 | 97.39943 |

2.12500 | 1.07865 | 49.76884 | 3.42500 | 1.07870 | 78.41171 | 4.72500 | 1.07871 | 97.71773 |

2.15000 | 1.07865 | 50.47089 | 3.45000 | 1.07870 | 78.84807 | 4.75000 | 1.07871 | 98.03436 |

2.17500 | 1.07866 | 51.16479 | 3.47500 | 1.07870 | 79.28129 | 4.77500 | 1.07871 | 98.34932 |

2.20000 | 1.07866 | 51.85074 | 3.50000 | 1.07870 | 79.71141 | 4.80000 | 1.07871 | 98.66264 |

2.22500 | 1.07867 | 52.52892 | 3.52500 | 1.07870 | 80.13846 | 4.82500 | 1.07871 | 98.97433 |

2.25000 | 1.07867 | 53.19950 | 3.55000 | 1.07870 | 80.56249 | 4.85000 | 1.07871 | 99.28440 |

2.27500 | 1.07867 | 53.86266 | 3.57500 | 1.07870 | 80.98355 | 4.87500 | 1.07871 | 99.59289 |

2.30000 | 1.07868 | 54.51856 | 3.60000 | 1.07870 | 81.40167 | 4.90000 | 1.07871 | 99.89979 |

2.32500 | 1.07868 | 55.16735 | 3.62500 | 1.07870 | 81.81690 | 4.92500 | 1.07871 | 100.20514 |

2.35000 | 1.07868 | 55.80920 | 3.65000 | 1.07870 | 82.22927 | 4.95000 | 1.07871 | 100.50894 |

2.37500 | 1.07868 | 56.44424 | 3.67500 | 1.07870 | 82.63883 | 4.97500 | 1.07871 | 100.81120 |

5.00000 | 1.07871 | 101.11196 |