The 50 Ω story

After an idea by ON7YD

Transmission-lines are available in a large range of characteristic impedances and in many different sizes and qualities. In radio-amateur environment the 50 Ohm type may almost be regarded as a 'standard' type and the question could arise why this is so?

As it happens, the 50 Ohm value is not an accidental type but is exhibiting somewhat lower loss than other impedance types. This story will discuss the background to this extend.

A coaxial transmission-line usually is constructed according to the picture below with a centre conductor, dielectric insulator that also provides mechanical stability, a metallic shield as an outer conductor and a protecting jacket. Sometimes the jacket as well as the metallic shield will be called 'sleeve'.

A stretch of conductive material may be regarded as an inductor and two conductors with some insulating material in between may easily be recognize as a capacitor. The inductor as well as the capacitor may dissipate a little energy since no practical components is ideal. Al together we may regard a coaxial transmission-line as a succession of inductors, capacitors and resistors (the loss mechanism), as in the next picture.

At very small system loss, characteristic impedance of this coaxial transmission-line may be defined as:

R₀ = √L/C

This time L and C are lossless components and without loss signals may be transported along indefinitely long lines without degradation. Signals will never-the-less be delayed by the L-C charging / discharging mechanism, but this is outside this little discussion.

The inductor value of L is related to the (inside) diameter of the metallic shield (D) and the (outside) diameter of the centre conductor (d). The capacitor value is also determined by the dielectric constant (ε_r) of the dielectric insulator. We therefore may describe the characteristic impedance of the coaxial cable in 'mechanical' terms as in:

R₀ = 60 * ln (D/d) * 1/√ε_r

We already discussed loss mechanisms that to a large extend are related to the intrinsic resistance of the conductive materials, but also to the amount of material that is taking part in transporting the energy. The higher the frequency, the less material is participating since the electromagnetic field is 'forced' out of the material (skin effect). Through this effect loss is increasing with frequency and may no longer be neglected.

To still keep loss low at low skin depth, we may attempt to increase the material surface. Since surface is proportional to diameter of the conductors we may write:

R ~ 1/d + 1/D, after some reworking: R ~ (D/d +1) * 1/D

When comparing this formula with the earlier one, we find both characteristic impedance R₀ and loss resistance R be related to the diameter of the coaxial conductors. We like to keep loss as low as possible and may further regard the relation R₀ / R as some kind of 'characteristic efficiency' of this transmission-line,

ή = R₀ / R

Since R₀ and R have a different relation to diameters, we could investigate the possibility of an optimal relation between the coaxial conductors diameters to arrive at a high efficient transport system. By rewriting the formulas for R₀ en R to the diameter-relation only, we will end up with:

ή ~ ln (D/d) / (D/d +1)

We may now calculate this 'characteristic efficiency' for different ratio's of D/d, to find the following figure:

In the figure above we indeed find the blue curve with an optimal diameter ratio at D/d = 3,59. When we next enter this optimal diameter ratio into the formula for characteristic impedance for coaxial transmission-lines, we will find:

R₀ = 60 * ln (D/d) = 60 * ln (3,59) = 76,7 Ohm.

This type of coaxial transmission-line (with air as a dielectric) is being manufactured for high power applications or at special request.

To better guarantee mechanical integrity many coaxial cables are constructed with a dielectric fill material between the inner and outer conductor. When applying the low loss polyethylene dielectric material (ε_r = 2,25), characteristic impedance of this optimal diameter-ratio cable will come to:

R₀ = 60 * ln (D/d) * 1/√ε_r = 60 * ln (3,6) * 1/√2,25 = 51,2 Ω

This specific characteristic impedance indeed is an industrial standard and may be found at coaxial types: RG9 (51 Ohm) and RG8 (52 Ohm). The above value may also be rounded to 50 Ohm, to come to a more general 'standard'.

In the graph it may further be found D/d may vary between 3 and 4,25 to still be within 2 % of the optimal efficiency value; the optimal efficiency apparently is not too sensitive for the ratio of the diameters.

We also calculated the green curve for the characteristic impedance of a coaxial transmission-line at different D/d ratio's, this time filled with polyethylene. This curve is showing characteristic impedance of the transmission-line will vary between 44 and 57,5 Ω when D/d is changing between 3 and 4,25 .

In an analogue way we may derive an optimal D/d ratio for the maximum amount of power a coaxial transmission-lines may 'transport'. This optimal ratio turns out to be D/d = 1,65, with a characteristic impedance of 30 Ohm at air dielectric.

The selection of 50 Ohm for coaxial cable may therefore be regarded as an optimal value between the optimization for minimal loss and maximum power transport capabilities.

Bob J. van Donselaar, on9cvd@veron.nl