Notes on the characteristic impedance of coax with a square outer conductor

Kevin Schmidt, W9CF

Abstract:

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 138log₁₀(1.08D/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

 

$$\sqrt{L/C}$$ (1)

where L and C are the inductance and capacitance per unit length. For an air dielectric and perfect cylindrical conductors, the waves travel at the speed of light so that L and C are related by

 

$${\textstyle\frac{1}{\sqrt{LC}}}$$ (2)

which when combined with Eq. 1 shows that only the capacitance per unit length is needed to calculate the characteristic impedance,

 

$${\textstyle\frac{1}{cC}}$$ (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

 

$$\nabla^{2}_{} \Phi$$ (4)

The capacitance can be calculated by solving Laplaces equation with the boundary condition that the potential is one volt on the outer conductor and zero volts on the inner conductor. The charge per unit length on either conductor is the capacitance per unit length.

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,

 

$${\frac{ \partial^2 \Phi}{\partial x^2}} {\frac{ \partial^2 \Phi}{\partial y^2}}$$ (5)

where x and y are cartesian coordinates of the cross sectional area. For a square cross section outer conductor, the potential must have the symmetry of the square. A general solution to Laplaces equation with that symmetry is,

 

$$\Phi \sum_{m=1}^{} \phi$$ (6)

where $\phi$ = $\tan^{-1}_{}$(y/x) , and r ² = x ² + y ².

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 $\Phi$ = 0 at r = 1 , which means a_i = - b_i . The solution for $\Phi$ is then

 

$$\Phi \sum_{m=1}^{} \phi$$ (7)

I must now enforce the condition that $\Phi$ = 1 on the outer conductor. I do this by point matching. I simply take points evenly spaced in the angle $\phi$ and require that $\Phi$ be one on the outer conductor at these angles. This gives a set of linear equations for the coefficients b_i that can be solved. Specifically for N coefficents, I require that

 

$$\sum_{m=1}^{N-1} \phi_{i}^{}$$ (8)

for all i values 1 to N , and
 

$$\phi_{i}^{} {\frac{(2 i-1) \pi}{8N}}$$

$${\frac{D}{\cos(\phi_i)}}$$ (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 $\epsilon_{0}^{}$ . 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 b₁ term. The b₁ term give a charge per unit length of

 

$$\pi \epsilon_{0}^{}$$ (10)

and the characteristic impedance is

 

$${\textstyle\frac{1}{ 2 \pi \epsilon_0 c b_1}} {\textstyle\frac{60}{b_1}}$$ (11)

Before looking at the numerical results for larger N, I can solve analytically for the case of N = 1 . In that case I have a purely logarithmic potential and match the potential to 1 at the point given by $\phi$ = $\pi$/8 . Eq. 8 becomes,

$$\left (\frac{D}{\cos(\pi/8)}\right)=$$ (12)

and evaluating

$${\textstyle\frac{1}{\cos(\pi/8)}}$$ (13)

the characteristic impedance is approximately

Z₀ = 60ln (1.08 D), (14)

in agreement with the empirical value. The results for larger values of N are easily calculated on the computer. One way to write the results is as

$$\alpha$$ (15)

Z₀ 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 $\alpha$ and Z₀ with N=10, for D/a from 1.1 to 6.0. The value starts at 1.06 at D/a = 1.1 where Z₀ = 9 Ohms. $\alpha$ becomes 1.08 for D/a = 1.275 where Z₀ = 19 Ohms. It remains at 1.08 thereafter. The asymptotic value of $\alpha$ 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 Z₀ , 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 $\alpha$ where Z₀ = 60ln($\alpha$D/a) is also shown.

$${\frac{D}{a}} \alpha {\frac{D}{a}} \alpha {\frac{D}{a}} \alpha$$ Z₀ Z₀ Z₀

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