Coaxial cable impedance calculator: diameter ratio, velocity factor and cut-off

What sets a coax cable's impedance

A coaxial line is two conductors sharing an axis: a round inner conductor of diameter d, a tubular outer conductor whose inside diameter is D, and a dielectric filling the space between them. Its characteristic impedance is the ratio of voltage to current in a wave travelling along it, and for the TEM wave a coax carries it depends on exactly two things. Huber+Suhner's connector guide puts it in a box: "The impedance of a transmission (coaxial) line is determined by the ratio of outer conductor diameter to inner conductor diameter and by the dielectric constant εr of the insulation material", and "The impedance is independent of the line length and frequency (at approximately > 1 MHz)".

The formula, H+S Equation 6 and Rohde & Schwarz's Equation 5-1, is in the reference note above: Z₀ is 60 Ω divided by √εr, times the natural logarithm of D/d. Two consequences follow from its shape. The size of the line does not appear, only the ratio: a 3.5 mm connector and a cable ten times its diameter have the same impedance if their ratios and dielectrics match. And the ratio enters through a logarithm, so a fixed step in impedance takes a fixed multiple of the ratio. Doubling D/d adds 41.6 Ω in air and 27.5 Ω in solid PE, whatever the starting point; getting from 50 Ω to 100 Ω in air takes the ratio from 2.30 to 5.30. The figure's chart plots Z₀ against D/d on a logarithmic axis, where that behaviour is a straight line.

The dielectric divides the impedance by √εr. H+S sums both effects in two lines: "A reduction of the inner conductor diameter increases impedance" and "An increasing dielectric constant reduces impedance". Filling an air line with PE, at the εr of 2.28 H+S uses, divides its impedance by 1.510, so a line that is 50 Ω in air becomes 33.1 Ω; to stay at 50 Ω its inner conductor has to shrink from D/2.30 to D/3.52.

The 60 Ω in the formula is not a fitted constant. The University of Houston's course notes derive the impedance from the fields as η/(2π)·ln(b/a), with a and b the radii and η the wave impedance of the dielectric, η₀/√εr. The SI Brochure gives the impedance of vacuum as μ₀c, so the constant is μ₀c/2π = 59.958 Ω, and H+S's 60 Ω is that rounded. The 138 Ω both vendors print for the base-10 form is the same number times ln 10, 138.06 Ω, also rounded. The calculator uses the exact constant and shows both printed forms beside it; all three agree to better than 0.1 %.

Coax impedance table: D/d for 50, 75 and 93 Ω

The diameter ratio for a given impedance in the four dielectrics H+S uses: air (εr 1), foam PE (1.5), PTFE (2.05) and solid PE (2.28). Each cell is the calculator's ratio; beside it, in brackets, the value H+S prints in its Table 2 for 50 and 75 Ω. The 93 Ω column is computed only: H+S's cable chapter says "The most common impedance values are 50 and 75, but also 93/95 and others are sometimes used for special applications", but its table stops at 75 Ω.

Every printed ratio puts its line within 0.07 Ω of the nominal impedance, and seven of the eight match the calculator to the second decimal. The eighth is 75 Ω in PE, which H+S prints as 6.60: that ratio gives 74.93 Ω, and 75.00 Ω needs 6.611. H+S's rounded 60 Ω constant happens to give 6.60 there, and gives 2.77 rather than the printed 2.78 for 50 Ω in foam PE, so no single constant reproduces the whole table; the differences are in the last printed digit and far inside any cable's tolerance.

Velocity factor, delay and capacitance per metre

A wave on a coax travels at c/√εr, H+S Equation 13, and nothing else about the line changes it: "The velocity is only determined by the dielectric constant (εr)." The velocity factor is therefore 1/√εr, the delay per metre √εr/c, and the wavelength at a frequency f is c/(f·√εr), H+S Equation 11. The capacitance and inductance per metre do depend on the ratio. The Houston notes give them as C = √(με′)/Z₀ and L = Z₀·√(με′); substituting their Z₀ gives C′ = 2πε₀εr/ln(D/d) and L′ = (μ₀/2π)·ln(D/d), which is H+S's statement in words that "The cable inductance L′ is determined by the ratio" and the capacitance "by the ratio and the dielectric constant εr". The table is the 50 Ω line in each dielectric.

The bracketed wavelengths are the ones H+S draws in its Figure 16 for 1 GHz, with c rounded to 300 000 km/s: 30, 24.5, 21 and 20 cm. At a fixed impedance the capacitance per metre depends on εr alone, since C′ = √εr/(c·Z₀): a 50 Ω line has 66.7 pF/m in air and 100.7 pF/m in PE, whatever its size. The inductance per metre goes the other way, because the solid dielectric forces a larger ratio for the same impedance.

Why coax is 50 Ω or 75 Ω

Rohde & Schwarz's connector note gives the reason for 50 Ω in three sentences. "In an air-filled coaxial transmission line, the minimum insertion loss per unit length occurs when the ratio of D/d ≈ 3.6, giving a characteristic impedance of 77 Ω." "For maximum power handling, the geometry is different, resulting in a calculated ratio of D/d ≈ 1.65, and a characteristic impedance of 30 Ω in air." And "50 Ω is the compromise between power handling and attenuation per unit length and is the nominal characteristic impedance used in most RF and microwave applications."

Both ratios come out of the same sources' equations, and the calculator derives them rather than quoting them. H+S's conductor-loss equation, Equation 34, divides (1/d + 1/D) by Z. With the outer diameter and the dielectric fixed, that is proportional to (1 + x)/ln x with x = D/d, a thicker inner conductor lowering its resistance but also the impedance the current flows into. It is least at x·ln x = 1 + x, which is x = 3.591, 76.65 Ω in air. The Houston notes give the power a line carries with a given field at the inner conductor's surface, the place the field is strongest and breakdown starts, as a²·ln²(b/a)/2Z₀ times that field squared. With the outer diameter fixed, that goes as ln x/x², which peaks at x = √e = 1.649, 29.98 Ω in air.

The calculator reports where any line sits against both. A 50 Ω air line, at D/d = 2.30, has 10 % more conductor loss than the best line of its outer diameter and carries 86 % of the most power one could at the same breakdown field. The 75 Ω air line of H+S's table, at 3.49, is within 0.0 % of the loss minimum but carries 56 % of the power. Of its 75 Ω TNC connectors, H+S says: "The 75 ohm designs for impedance matching are suitable for precision video and computer cables."

One consequence of the equations that the sources do not spell out: the minimum-loss ratio is a property of the geometry, so it does not move with the dielectric, but the impedance it gives does. In PTFE the ratio 3.59 is 53.5 Ω, and in solid PE it is 50.8 Ω. A 50 Ω line in PE, at D/d = 3.52, is within 0.01 % of the lowest conductor loss its outer diameter allows.

Worked example: R&S's 3.5 mm connector

R&S works its cut-off equation on one example: "an air filled 3.5 mm connector has the following dimensions: d = 1.52 mm, D = 3.5 mm, εr ~ 1", and "Therefore, the 'cut off frequency', fc, of the first higher mode is approximately 38 GHz. Some manufacturers rate 3.5 mm precision connectors up to 33GHz." The same dimensions give the impedance, which the note does not state. The left column is the calculator's arithmetic; the right, what R&S prints.

ratio      D/d = 3.5 / 1.52                             = 2.303       
Z0         59.958 Ω × ln(2.303) / √1                    = 50.01 Ω     
Eq 5-1     138 Ω × log10(2.303) / √1                    = 49.99 Ω     
cut-off    c / (π × (3.5 + 1.52) mm / 2 × √1)           = 38.02 GHz    R&S: ≈ 38 GHz
usable     0.90 to 0.95 × f_c                           = 34.2–36.1 GHz
rating     33 GHz / f_c                                 = 86.8 %      
C′         2π × ε0 / ln(2.303)                          = 66.7 pF/m   
L′         (μ0 / 2π) × ln(2.303)                        = 166.8 nH/m  
delay      √1 / c                                       = 3.336 ns/m  

The cut-off agrees with R&S's 38 GHz, and the dimensions are a 50 Ω line to within 0.01 Ω, as a 3.5 mm connector for a 50 Ω system should be. The 33 GHz rating some makers give is 86.8 % of the cut-off, a little below the 90–95 % R&S calls typical. These are the calculator's defaults, with 33 GHz as the working frequency.

Worked example: the same outer diameter filled with PTFE

A derived example, not one from the sources: keep D = 3.5 mm, fill the line with PTFE at H+S's εr of 2.05, and ask for 50 Ω. The inner conductor has to shrink, and the cut-off falls.

ratio      D/d = exp(50 × √2.05 / 59.958)               = 3.300       
d          3.5 mm / 3.300                               = 1.061 mm    
cut-off    c / (π × (3.5 + 1.061) mm / 2 × √2.05)       = 29.23 GHz   
usable     0.90 to 0.95 × f_c                           = 26.3–27.8 GHz
v/c        1 / √2.05                                    = 0.698       
C′         2π × ε0 × 2.05 / ln(3.300)                   = 95.5 pF/m   

The cut-off drops by 23 %, from 38.0 GHz to 29.2 GHz. Two things move it: the √εr in the denominator of the cut-off equation, and the smaller inner conductor, which reduces D + d. The velocity factor falls to 0.698 and the capacitance per metre rises by the same √εr, from 66.7 to 95.5 pF/m. R&S's note on the cure applies in either direction: "the cut-off frequency of the first higher order mode can be increased through reducing the physical dimensions of the inner and outer conductors."

Where the coax impedance formula stops being valid

Above the TE11 cut-off. The impedance describes the TEM wave, the one whose fields lie entirely across the line. H+S: "The cut-off frequency is the frequency at which other waves than TEM waves can take place, i.e. a field component in the direction of propagation appears, causing significant changes in the characteristics of the RF line (e.g. resonances)." Its glossary adds that "The transmission characteristics of cables above their cutoff frequency may be unstable." R&S draws the working limit lower still, at 90–95 % of the cut-off, and the calculator reports both that band and the share of the cut-off a working frequency reaches. The equation itself is an approximation; the Houston notes set up the exact Bessel-function condition and label this form the approximate solution.

At low frequencies. H+S writes the general impedance with the conductor resistance and the dielectric's conductance in it, and notes that "At higher frequencies R′ and G′ have no influence on the impedance". Its approximate floor for that is the "> 1 MHz" quoted above. Below it, the formula gives the line's high-frequency impedance, not what an audio or DC circuit sees.

Loss. The Houston notes flag their impedance formula plainly: "This formula does not account for conductor loss." H+S gives the attenuation separately, conductor loss rising as √f and dielectric loss as f·√εr·tanδ (Equations 34 and 35, in the reference note), and says "Up to about 10 GHz the conductor losses are dominant. From about 10 GHz the dielectric losses are dominant." This calculator does not compute attenuation: Equation 34 needs the conductors' resistivity and Equation 35 the dielectric's loss tangent, and neither is in the sources for the four materials. What it reports is the conductor loss relative to the best ratio, which needs neither.

Braid and screening. The model's outer conductor is a solid tube. H+S, on cut-off and operating range: "Due to the much higher RF leakage on higher operating frequencies, the insertion loss will be influenced by increased radiation, especially in single braided cables." On screening: "On cable designs with woven braids (single or double), the screening will also depend on various parameters of the braid (i.e. braiding angle, number of wires per spindle, etc.)." A braid's inside diameter is also less well defined than a tube's, which makes D an effective value.

Discontinuities. The impedance is a property of a uniform line. H+S lists what breaks it at a connector: "Changes in conductor diameters", "Change in insulator diameter", "Change in interface dimensions" and "Space between parts (gaps)". Each is a short section of a different D/d, or a different εr, and the formula gives its local impedance, not how the step reflects.

Common coax impedance mistakes

• Radii instead of diameters. The ratio is the same either way, b/a = D/d, so the impedance survives the mix-up. The cut-off does not. The Houston notes work their example in radii, a = 0.035 in and b = 0.116 in with εr 2.2, and get 16.8 GHz; the calculator gives 16.77 GHz with the diameters 2a and 2b. Put the radii into the diameter formula of H+S or R&S and it returns 33.5 GHz, twice the real cut-off.
• Mixing the log and its constant. 60 Ω goes with the natural log, 138 Ω with log₁₀. On the 3.5 mm connector, 60 Ω with log₁₀ gives 21.7 Ω and 138 Ω with ln gives 115.1 Ω, against the real 50.0 Ω. H+S's Equation 6 writes the base-10 form as plain "log", which leaves the base to the reader.
• Measuring D in the wrong place. R&S defines it as "the inside diameter of the outside conductor or shield", and d as "the outside diameter of the inner conductor". The diameter over the braid or over the jacket is not D; the diameter of the dielectric is.
• Trusting a nominal εr. The impedance and the velocity factor both follow √εr, so an error in εr moves them together. H+S's foam PE value of 1.5 is one figure for one material. As an illustration, a line built to 50 Ω at εr 1.5 is 50.9 Ω if the foam is really 1.45 and 49.2 Ω if it is 1.55. For a real cable, take εr or the velocity factor from its datasheet rather than from a table of materials.
• Using the cut-off as the limit. The TE11 cut-off is where the trouble starts, not where the line stops being usable; R&S's working limit is 90–95 % of it, and the makers it mentions rate the 3.5 mm connector lower still.
• Expecting the impedance to set the velocity. Two lines of the same impedance in different dielectrics have different velocity factors and different capacitance per metre; two lines of different impedance in the same dielectric have the same velocity. For matching delays, the dielectric is the variable that matters.

Further reading

• Huber+Suhner, RF Connector Guide, 4th edition — the coaxial impedance equation (p.16), the table of diameter ratios for 50 and 75 Ω (p.18), the cut-off frequency (p.18), velocity and wavelength (p.20) and the attenuation equations (p.32).
• Rohde & Schwarz 1MA99, Guidance on Selecting and Handling Coaxial RF Connectors — the appendix on coaxial line properties: the impedance, the minimum-loss and maximum-power ratios, the TE11 cut-off and the 3.5 mm connector example (pp.16–17).
• University of Houston ECE 5317, Notes 10: Coaxial Cable — the field derivation of the impedance, the per-unit-length parameters, the conductor and dielectric attenuation, power flow and the higher-order modes, all in radii.
• BIPM, The International System of Units (SI Brochure, 9th edition) — the exact c and the relations ε₀ = 1/μ₀c² and Z₀ = μ₀c for the vacuum.
• Microstrip and stripline impedance calculator — the same quantity for a trace over a plane, where the field is partly in air and the dielectric constant becomes an effective one.
• Coplanar waveguide calculator — the PCB line that connects most often to a coaxial connector's pin.
• Critical length calculator — whether a connection is long enough for its impedance to matter at all.
• VSWR and return loss calculator — what a mismatch between a line's impedance and its load costs.
• Attenuator pad calculator — pi and T pads for 50 and 75 Ω lines, and the minimum-loss pad between them.