# Coaxial Cable Impedance Calculations

#### John Kewley

Joined Aug 2, 2018
7
The formula for calculating the characteristic impedance of a coaxial cable is:
Zo = 138/sqrt(Er) (log10 D/d)

Can anyone explain where the constant 138 in this formula is derived from?
Thanks
John

#### John Kewley

Joined Aug 2, 2018
7
Thanks Eric, but being a bear-of-very-little-brain I'm still confused ! The dielectric constant of air is 1.00059. How does this relate to 138?
J

#### ericgibbs

Joined Jan 29, 2010
13,208
hi John,
Been scouring the web for a derivation, no luck so far, will keep trying.
Someone may have the simple answer.?
E
@WBahn

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#### John Kewley

Joined Aug 2, 2018
7
We are bespoke cable designers and manufacturers and we struggle to hit accurate designs on Coaxial Cables.
Using our own calculators which, for impedance, is based on the formula above (which is the same as the calculator in the tool section of this site) our characteristic impedance turns out about 8% - 10% higher than calculated.
To confirm that this isn't the quality of our manufacture ( as if !!) we took a good quality, certified RG179, tested it to confirm a Zo of 75 Ohm and then fed the construction details into the design formula. This resulted in a calculated Zo of 67 Ohm (8% below).
By adjusting the constant 138 to 158 the design figures come out accurate to the actuals.
Can anyone explain what's going on?
Thanks, John

#### John Kewley

Joined Aug 2, 2018
7
Thanks Eric!
John

#### ebp

Joined Feb 8, 2018
2,332
The formula is an approximation and almost certainly assumes lossless dielectric and lossless conductors. That will not accurately predict impedance. Many coaxial cables are manufactured with copper clad steel centre conductor and often copper clad steel for the shielding braid. The constant will be derived from the constants which describe the inductance and capacitance of a perfectly round centre conductor inside a perfectly round shield conductor.

EDIT: The capacitance depends directly on the dielectric constant of the insulator. Polyethylene is a common insulator in co-ax. The actual dielectric constant will vary by ±2-4% depending on the specific formulation (molecular weight). For precision cable, a high-quality plastic with highly consistent dielectric constant is essential.

Concentricity and longitudinal consistency in dielectric thickness are critical to decent performance. This is important not only for coaxial cables but twisted pairs and something that small manufacturers seem to commonly have problems with. Some of the lines I've seen left me quite amazed that they could produce even moderate quality in terms of impedance.

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#### John Kewley

Joined Aug 2, 2018
7
The formula is an approximation and almost certainly assumes lossless dielectric and lossless conductors. That will not accurately predict impedance. Many coaxial cables are manufactured with copper clad steel centre conductor and often copper clad steel for the shielding braid. The constant will be derived from the constants which describe the inductance and capacitance of a perfectly round centre conductor inside a perfectly round shield conductor.

EDIT: The capacitance depends directly on the dielectric constant of the insulator. Polyethylene is a common insulator in co-ax. The actual dielectric constant will vary by ±2-4% depending on the specific formulation (molecular weight). For precision cable, a high-quality plastic with highly consistent dielectric constant is essential.

Concentricity and longitudinal consistency in dielectric thickness are critical to decent performance. This is important not only for coaxial cables but twisted pairs and something that small manufacturers seem to commonly have problems with. Some of the lines I've seen left me quite amazed that they could produce even moderate quality in terms of impedance.
I have to agree that the lack of control and consistency of not only some of the smaller manufacturers but some of the major producers absolutely amazes me!! As a small cable designer and manufacturer for the subsea, defence and aerospace industries I am pleased to say that generally our quality is second to none and, as a UK manufacturer, the virgin compounds that we use are specifically designed for the application.
Thank you for the comment that the formula, as it stands, is not likely to predict the impedance completely accurately as at least this gives me some comfort that we aren't missing a trick somewhere!! A lot of the cables we produce are one-offs in relatively short runs for specific projects but with this knowledge we will be able to fine-tune the design calculations to achieve the required results.

#### Tesla23

Joined May 10, 2009
473