coax-fringe-capacitance.md

Fringe Capacitance of Open-Ended Coax

Termination Type

• Thin-Radiating Open: The termination is a flush cross-section of a bare coax cable, with an infinitely-thin shield, surrounding objects near the instrument makes it somewhat unpredictable in practical setups. Radiation loss requires small corrections at high frequencies. Rarely analyzed or used for both reasons. Abbrevation: Thin rad.

• Flanged Radiation Open: The outer conductor is extended perpedicular to the axial direction of the coax, creating a ground plane. This geometry is commonly used for coaxial probes for dielectric material measurements. Radiation loss requires small corrections at high frequencies, but is usually neglected. This assumption is sometimes made as a theoretical simplification even for bare coax cables without flanges because the outer shield has non-zero thickness, creating a similar effect. Abbrevation: Flange.

• Thick-Shield Radiating Open: The termination is a flush cross-section of a bare coax cable, the outer diameter of the outer coax conductor is 3x as long as the inner conductor. This thick shield creates an effect similar to a flange. Used in openEMS simulations for consistency (for comparison with other shield diameters). Abbrevation: Thick rad.

• Waveguide Beyond Cutoff: The outer conductor is extended to the axial direction of the coax, creating an abrupt transition from coax to a circular waveguide beyond cutoff. Radiation is fully suppressed. This geometry behaves consistently, sometimes also known as a "shielded" open. This is commonly found in "Open" coax calibration standards. Abbrevation: Cutoff.

Vacuum 3.5 mm Coax

• Din = 1.52 mm, Dout = 3.5 mm, ε_cable = ε0, ε_termination = ε0

Method	Type	C (fF)	Cutoff Diff	Thin Rad Diff	Thick Rad Diff	Notes
openEMS	Cutoff	42.48	As Reference	22.03%	10.80%	[1]
Somlo (via Woods)	Cutoff	39.85	-6.19%	14.48%	3.94%	[2]
Chramiec-Piotrowski	Cutoff	39.76 (DC-1 GHz)	-6.40%	14.22%	3.70%	[3]
openEMS	Thin Rad	34.81	-18.06%	As Reference	-9.21%	[4]
openEMS	Thick Rad	38.34	-9.75%	10.14%	As Reference	[5]
Gajda-Stuchly (FEM)	Flange	37.79	-11.04%	8.56%	-1.43%	[6]
Gajda-Stuchly (MoM)	Flange	35.73	-15.89%	2.64%	-6.81%	[6]

1. Linear fit from 100 MHz to 1 GHz, assume a long extruded outer conductor, creating a waveguide beyond cutoff.
2. The pre-computed value C / (2πb) = 36.242 fF/cm by Woods et, al from Somlo's charts is used (with interpolation), not recomputed from scratch. Assume ε_cable = ε0, Dout / Din = 2.30230 (Z0 = 50 Ω), termination into waveguide beyond cutoff.
3. DC capacitance is a polynomial fit of Somlo over a range of characteristic impedances, given for comparison only. Woods's pre-computed value is considered more accurate. Paper's Equation 5 contains a typo. The value “1.73875” should be “17.3875” instead. RF correction by Chramiec-Piotrowski. Assume Z0 = 50 Ω.
4. Linear fit from 100 MHz to 1 GHz, assume flush open with no flange, outer conductor is infinitely thin.
5. Linear fit from 100 MHz to 1 GHz, assume flush open with no flange, outer conductor's outer diameter is 3x of the inner conductor diameter.
6. Assume a flush open with a flange as ground plane (Dflange = 3 x Dout), Dout / Din = 3.27.

Vacuum 7 mm Coax

• Din = 3.04 mm, Dout = 7.0 mm, ε_cable = ε0, ε_termination = ε0

Method	Type	C (fF)	Cutoff Diff	Thin Rad Diff	Thick Rad Diff	Notes
openEMS	Cutoff	82.04	As Reference	19.82%	14.97%	[1]
Somlo	Cutoff	79.70	-2.85%	16.40%	11.69%	[2]
Chramiec-Piotrowski	Cutoff	79.51-79.54 (DC-1 GHz)	-3.08%	16.17%	11.46%	[3]
Razaz-Davies	Cutoff	79.88-79.917 (DC-1 GHz)	-2.63%	16.72%	11.99%	[4]
openEMS	Thin Rad	68.47	-16.54%	As Reference	-4.05%	[5]
openEMS	Thick Rad	71.36	-13.02%	4.22%	As Reference	[6]
Gajda-Stuchly (FEM)	Flange	75.58	-7.87%	10.38%	5.91%	[7]
Gajda-Stuchly (MoM)	Flange	71.46	-12.90%	4.37%	0.14%	[7]

1. Linear fit from 100 MHz to 1 GHz, assume a long extruded outer conductor, creating a waveguide beyond cutoff.
2. Somlo's original reported value, widely used as a standard reference value. Assume ε_cable = ε0, Dout / Din = 2.30230 (Z0 = 50 Ω), termination into waveguide beyond cutoff.
3. DC capacitance is a polynomial fit of Somlo over a range of characteristic impedances, given for comparison only. Woods's pre-computed value is considered more accurate. Paper's Equation 5 contains a typo. The value “1.73875” should be “17.3875” instead. RF correction by Chramiec-Piotrowski. Assume Z0 = 50 Ω.
4. Assume waveguide beyond cutoff.
5. Linear fit from 100 MHz to 1 GHz, assume flush open with no flange, outer conductor is infinitely thin.
6. Linear fit from 100 MHz to 1 GHz, assume flush open with no flange, outer conductor's outer diameter is 3x of the inner conductor diameter.
7. Assume a flush open with a flange as ground plane (Dflange = 3 x Dout), Dout / Din = 3.27.

Vacuum 14 mm Coax

• Din = 6.21 mm, Dout = 14.2875 mm, ε_cable = ε0, ε_termination = ε0

Method	Type	C (fF)	Cutoff Diff	Thin Rad Diff	Thick Rad Diff	Notes
openEMS	Cutoff	165.47	As Reference	17.69%	13.27%	[1]
Somlo	Cutoff	162.67	-1.69%	15.70%	11.35%	[2]
Chramiec-Piotrowski	Cutoff	162.29-162.57 (DC-1 GHz)	-1.92%	15.63%	11.28%	[3]
Razaz-Davies	Cutoff	163.04-163.336 (DC-1 GHz)	-1.47%	16.17%	11.81%	[4]
openEMS	Thin Rad	140.60	-15.03%	As Reference	-3.76%	[5]
openEMS	Thick Rad	146.09	-11.71%	3.90%	As Reference	[6]
Gajda-Stuchly (FEM)	Flange	154.16	-6.84%	9.64%	5.52%	[7]
Gajda-Stuchly (MoM)	Flange	145.76	-11.91%	3.67%	-0.23%	[7]

1. Linear fit from 100 MHz to 1 GHz, assume a long extruded outer conductor, creating a waveguide beyond cutoff.
2. Somlo's original reported value, widely used as a standard reference value. Assume ε_cable = ε0, Dout / Din = 2.30230 (Z0 = 50 Ω), termination into waveguide beyond cutoff.
3. DC capacitance is a polynomial fit of Somlo over a range of characteristic impedances, given for comparison only. Woods's pre-computed value is considered more accurate. Paper's Equation 5 contains a typo. The value “1.73875” should be “17.3875” instead. RF correction by Chramiec-Piotrowski. Assume Z0 = 50 Ω.
4. Assume waveguide beyond cutoff.
5. Linear fit from 100 MHz to 1 GHz, assume flush open with no flange, outer conductor is infinitely thin.
6. Linear fit from 100 MHz to 1 GHz, assume flush open with no flange, outer conductor's outer diameter is 3x of the inner conductor diameter.
7. Assume a flush open with a flange as ground plane (Dflange = 3 x Dout), Dout / Din = 3.27.

Type 50-250 (RG401)

• Din = 1.63 mm, Dout = 5.38 mm, ε_cable = 2.05 x ε0, ε_termination = ε0

Method	Type	Z0 (Ω)	Capacitance (fF)	Radiating Diff	Cutoff Diff	Notes
openEMS	Radiating	50	35.71	As Reference	-20.93%	[1]
openEMS	Cutoff	50	45.16	26.46%	As Reference	[2]
Chramiec-Piotrowski	Cutoff	71.60	41.00-41.07 (DC-1 GHz)	15.01%	-9.21%	[3]
Gajda-Stuchly (FEM)	Flange	49.62	40.95	14.67%	-9.32%	[4]
Gajda-Stuchly (MoM)	Flange	49.62	39.10	9.49%	-13.42%	[4]

1. Linear fit from 100 MHz to 1 GHz, assume flush open with no flange
2. Linear fit from 100 MHz to 1 GHz, assume a long extruded outer conductor, creating a waveguide beyond cutoff.
3. DC capacitance is a polynomial fit of Somlo over a range of characteristic impedances, given for comparison only. Woods's pre-computed value is considered more accurate. Paper's Equation 5 contains a typo. The value “1.73875” should be “17.3875” instead. RF correction by Chramiec-Piotrowski. Assume dielectric removed to form a hypothetical vacuum cable.
4. Assume a flush open with a flange as ground plane (Dflange = 3 x Dout), Dout / Din = 3.27.

Type 50-141 (RG402)

• Din = 0.91 mm, Dout = 3.00 mm, ε_cable = 2.05 x ε0, ε_termination = ε0

Method	Type	Z0 (Ω)	Capacitance (fF)	Radiating Diff	Cutoff Diff	Notes
openEMS	Radiating	49.96		As Reference	-27.18%	[1]
openEMS	Cutoff	49.96		37.33%	As Reference	[2]
Chramiec-Piotrowski	Cutoff	71.53	22.89 (DC-1 GHz)	18.36%	-13.82%	[3]
Gajda-Stuchly (FEM)	Flange	49.62	22.95	18.67%	-13.59%	[4]
Gajda-Stuchly (MoM)	Flange	49.62	22.02	13.86%	-17.09%	[4]

1. Linear fit from 100 MHz to 1 GHz, assume flush open with no flange
2. Linear fit from 100 MHz to 1 GHz, assume a long extruded outer conductor, creating a waveguide beyond cutoff.
3. DC capacitance is a polynomial fit of Somlo over a range of characteristic impedances, given for comparison only. Woods's pre-computed value is considered more accurate. Paper's Equation 5 contains a typo. The value “1.73875” should be “17.3875” instead. RF correction by Chramiec-Piotrowski. Assume dielectric removed to form a hypothetical vacuum cable.
4. Assume a flush open with a flange as ground plane (Dflange = 3 x Dout), Dout / Din = 3.27.

Type 50-011 (RG174)

• Din = 0.48 mm, Dout = 1.58 mm, ε_cable = 2.05 x ε0, ε_termination = ε0

Method	Type	Z0 (Ω)	Capacitance (fF)	Radiating Diff	Cutoff Diff	Notes
openEMS	Radiating	49.89	11.30	As Reference	-32.74%	[1]
openEMS	Cutoff	49.89	16.80	48.67%	As Reference	[2]
Chramiec-Piotrowski	Cutoff	71.43	12.08 (DC-1 GHz)	6.90%	-28.10%	[3]
Gajda-Stuchly (FEM)	Flange	49.62	12.08	6.90%	-28.10%	[4]
Gajda-Stuchly (MoM)	Flange	49.62	11.59	2.57%	-31.01%	[4]

1. Linear fit from 100 MHz to 1 GHz, assume flush open with no flange
2. Linear fit from 100 MHz to 1 GHz, assume a long extruded outer conductor, creating a waveguide beyond cutoff.
3. DC capacitance is a polynomial fit of Somlo over a range of characteristic impedances, given for comparison only. Woods's pre-computed value is considered more accurate. Paper's Equation 5 contains a typo. The value “1.73875” should be “17.3875” instead. RF correction by Chramiec-Piotrowski. Assume dielectric removed to form a hypothetical vacuum cable.
4. Assume a flush open with a flange as ground plane (Dflange = 3 x Dout), Dout / Din = 3.27.

Conclusions

1. Extensive literature exists (beyond what's cited here), but the conventions and assumptions vary. Different numerical roundings of characteristic impedances, coax dimensions and reported results, air vs. vacuum medium, waveguide beyond cutoff vs. flush radiating open, different ground plane and flange assumptions (even applied to real cables without them) due to theoretical simplifications, etc. These differences are often neglected for practical applications as experimental uncertainty is much higher. For example, many methods assume a flange or ground plane for simplicity, but theoretical results are then applied to cables without them such as the experiment by Kraszewski et, al. On the other hand when making exact numerical comparisons, results are ambiguous since no source code is available to redo calculations based on consistent inputs.

2. openEMS appears to be 7% accurate for vacuum dielectric for matched termination type ("Cutoff" to "Cutoff", "Flange" to "Thick Rad"), but accuracy degrades to 20% for PTFE dielectric. Since the "Thick Rad" simulations for PTFE coax cables are still in progress, the main reason is likely a mismatch of the calculation assumptions rather than a systematic error of the numerical method: Gajda-Stuchly assumes a flange a ground plane. This can be seen in the results of Type 50-141 coax simulation: the Chramiec-Piotrowski value (assumes a waveguide beyond cutoff) agrees with openEMS's cutoff model within 9.21%, but the deviation increases to 15.01% for the radiation model. Furthermore, low capacitance appears to degrades the simulation's accuracy given the same mesh resolution, since even a small numerical error is now a large proportion of the value: for Type 50-011 coax, the error appears large (30%), but the difference is only 4 fF.

3. The same vacuum coax fringe capacitance analysis appears valid for PTFE coax as well. By removing the PTFE while keeping the geometry, the hypothetical equivalent vacuum coax has almost the same fringe capacitance. However, this equivalent vacuum coax's characteristic impedance would deviate significant from 50 Ω, thus pre-computed numerical constants for Z0 = 50 Ω in the literature can't be used (e.g. the constant by Gajda-Stuchly from Somlo's tables), the full Somlo tables [Somlo1967_IRE] (supposely more accurate than [Somlo1967_MTT]) must be used to recalculate normalized capacitance constants for non-standard Z0. The author is so far unable to find a copy of [Somlo1967_IRE] since it's only published in IRE's local Australian chapter journal, and the inter-library loan system of National Library of Australia is currently offline. Thus, existing Chramiec-Piotrowski polynomial fit (based on [Somlo1967_MTT]) is used here. At DC, Eq. 6 is accurate to 1% within Z0 = 30-70 Ω, and 2.5% between Z0 = 70-100 Ω, making it a satisfactory solution.

4. The "Thin Rad" simulations have poor agreement with other values reported in the literature - this geometry is not well-defined for practical purposes, thus it's rarely analyzed or used. "Thick Rad" simulations and the "Flange" values in the literature is good because both setups have identical cross-sections. The capacitance of the "Cutoff" simulations are insensitive to the outer conductor's thickness, they're either the same or with minimal difference.

   New simulation runs are planned to check cables with realistic shield thickness, not too thick and not too thin.

5. Work on more simulations are ongoing, including "Thick Rad" simulations for PTFE cables.

6. The Open standard APC-7 connectors has been modeled as an ideal waveguide-beyond-cutoff for calibration standards for impedance and network analyzers succesfully for years: HP 4191, HP 8410B, HP8542A all use a value between 81-84 fF, see citations by Bianco et, al. But the same can't be said for other connector types - initial work found the agreement is worse for 3.5 mm connectors, and much worse for SMA connectors - which behave significantly differently. This problem is still under investigation. DO NOT apply these results to SMA connectors. Connector modeling work is still ongoing.

Bibliography

The study of discontinuities in transmission lines began since World War 2 at MIT Radiation Laboratory, including contributions by noted physicists such as Julian Schwinger. Summary of the fringe capacitance of open-ended vacuum coax cables (among other results) can be found in Waveguide Handbook, Volume 10 of MIT Rad Lab Series textbooks, and still serves as a reference today, although the result is vacuum-only and the accuracy is slightly lower than modern numerical treatments.

In spite of its history, study of this problem continues based on a diverse category of methods, including the Rayleigh–Ritz variational method, Least-Squares Boundary Residual Method (LSBRM), Schwarz-Christoffel transformation, Finite Element Method (FEM), and Method of Moments (MoM). The first motivation is found for developing open-circuit and capacitance calibration standards for impedance and network analyzers, which is the focus of most of the papers cited here. The second motivation originated from material science and biology for developing coaxial probes and sample holders for non-destructive testing, and the characterization of dielectric samples and biological tissues. This made generalization of the computation from a homogeneous vacuum cable to a PTFE-filled coax probe terminated by a sample with a different permittivity. The third kind of papers originated from computational electromagnetics as a benchmark problem for comparing field solvers based on different numerical methods - which in turn is motivated by practical applications. The fourth application is found in dielectric-loaded antennas.

Due to the sheer volume of publications, a complete citation is not given in this memo. Readers are suggested to read all bibliographies contained within the following papers themselves.

• N. Marcuvitz. Waveguide Handbook, volume 10 of MIT Radiation Laboratory Series. McGraw-Hill, New York, 1951.

• Woods, D. (1972, December). Shielded-open-circuit discontinuity capacitance of a coaxial line. In Proceedings of the Institution of Electrical Engineers (Vol. 119, No. 12, pp. 1691-1692). IEE.

• Somlo, P. I. (1967). The computation of coaxial line step capacitances. IEEE Transactions on Microwave Theory and Techniques, 15(1), 48-53.

• Somlo, P. I. (1967). The discontinuity capacitance and the effective position of a shielded open circuit in a coaxial line. Proc. Inst. Radio Elec. Eng. Australia, 28(1), 7-9.

• Razaz, M., & Davies, J. B. (1979). Capacitance of the abrupt transition from coaxial-to-circular waveguide. IEEE Transactions on Microwave Theory and Techniques, 27(6), 564-569.

• Gajda, G. B., & Stuchly, S. S. (1983). Numerical analysis of open-ended coaxial lines. IEEE Transactions on microwave theory and techniques, 31(5), 380-384.

• Kraszewski, A., & Stuchly, S. S. (1983). Capacitance of open-ended dielectric-filled coaxial lines-experimental results. IEEE transactions on instrumentation and measurement, 32(4), 517-519.

• Bianco, B., Corana, A., Gogioso, L., Ridella, S., & Parodi, M. (1980). Open-circuited coaxial lines as standards for microwave measurements. Electronics Letters, 16(10), 373-374.

• Chramiec, J., & Piotrowski, J. K. (1999). Universal formula for frequency-dependent coaxial open-end effect. Electronics Letters, 35(17), 1474–1475. https://doi.org/10.1049/el:19990978

• Costamagna, E., & Fanni, A. (2002). Computing capacitances via the Schwarz-Christoffel transformation in structures with rotational symmetry. IEEE transactions on magnetics, 34(5), 2497-2500.

Appendix 1: Multigrid Accuracy Test

In all simulations, an angular resolution of 4.05° (89 mesh lines), a radial resolution of 0.05 mm, and a Z-axis resolution of 0.05 mm are used. However, in polar and cylindrical coordinates, the size of mesh cells decreases towards the origin. This creates extremely small mesh cells and tiny simulation timestep.

To mitigate the problem, openEMS's "multigrid" feature is used to remove half of the number of a-axis mesh lines at a given radius r. Multigrid is applied recursively at r = 0.4 and r = 0.2, increasing the smallest mesh cells by a factor of 4.

To study its impact on simulation, multigrid-mesh and full-mesh simulations are compared. The comparison finds the error never excedded 0.5% in all cases, thus, all subsequent simulations will use the same multigrid settings.

Vacuum 3.5 mm Coax

• Din = 1.52 mm, Dout = 3.5 mm, ε_cable = ε0, ε_termination = ε0

Method	Type	Capacitance (fF)	Diff
Full	Radiating	34.65	As Reference
Multigrid	Radiating	34.81	0.16 fF (0.46%)
Full	Cutoff	42.38	As Reference
Multigrid	Cutoff	42.48	0.10 fF (0.24%)

Vacuum 7 mm Coax

• Din = 3.04 mm, Dout = 7.0 mm, ε_cable = ε0, ε_termination = ε0

Method	Type	Capacitance (fF)	Diff
Full	Radiating	68.42	As Reference
Multigrid	Radiating	68.47	0.05 pF (0.07%)
Full	Cutoff	82.06	As Reference
Multigrid	Cutoff	82.03	-0.03pF (-0.04%)

Vacuum 14 mm Coax

• Din = 6.21 mm, Dout = 14.2875 mm, ε_cable = ε0, ε_termination = ε0

Method	Type	Capacitance (fF)	Diff
Full	Radiating	140.29	As Reference
Multigrid	Radiating	140.60	0.31 fF (0.22%)
Full	Cutoff	165.28	As Reference
Multigrid	Cutoff	165.47	0.19 fF (0.11%)

Type 50-250 (RG401)

• Din = 1.63 mm, Dout = 5.38 mm, ε_cable = 2.05 x ε0, ε_termination = ε0

Method	Type	Capacitance (fF)	Diff
Full	Radiating	35.71	As Reference
Multigrid	Radiating	35.92	0.21 fF (0.59%)
Full	Cutoff	45.16	As Reference
Multigrid	Cutoff	45.19	0.03 fF (0.07%)

Type 50-141 (RG402)

• Din = 0.91 mm, Dout = 3.00 mm, ε_cable = 2.05 x ε0, ε_termination = ε0

Method	Type	Capacitance (fF)	Diff
Full	Radiating	19.34	As Reference
Multigrid	Radiating	19.41	0.07 fF (0.36%)
Full	Cutoff	26.56	As Reference
Multigrid	Cutoff	26.45	-0.11 fF (-0.41%)

Type 50-011 (RG174)

• Din = 0.48 mm, Dout = 1.58 mm, ε_cable = 2.05 x ε0, ε_termination = ε0

Method	Type	Capacitance (fF)	Diff
Full	Radiating	11.30	As Reference
Multigrid	Radiating	11.33	0.03 fF (0.27%)
Full	Cutoff	16.80	As Reference
Multigrid	Cutoff	16.88	0.08 fF (0.48%)