## Why 50 Ohms?

**Q: ***Why do most engineers use
50-Ω pc-board transmission lines (sometimes to the
extent of this value becoming a default for pc-board layout)? Why not 60-
or 70-Ω?-Tim Canales*

**A: **Given a fixed trace width, three factors
heavily influence pc-board-trace impedance decisions. First, the
near-field EMI from a pc-board trace is proportional to the height of the
trace above the nearest reference plane; less height means less radiation.
Second, crosstalk varies dramatically with trace height; cutting the
height in half reduces crosstalk by a factor of almost four. Third, lower
heights generate lower impedances, which are less susceptible to
capacitive loading.

All three factors reward designers who place their traces as close as possible to the nearest reference plane. What stops you from pressing the trace height all the way down to zero is the fact that most chips cannot comfortably drive impedances less than about 50 Ω. (Exceptions to this rule include Rambus, which drives 27 Ω, and the old National BTL family, which drives 17 Ω).

It is not always best to use 50 Ω. For example, an old NMOS 8080 processor operating at 100 kHz doesn't have EMI, crosstalk, or capacitive-loading problems, and it can't drive 50 Ω anyway. For this processor, because very high-impedance lines minimize the operating power, you should use the thinnest, highest-impedance lines you can make.

Purely mechanical considerations also apply. For example, in dense, multilayer boards with highly compressed interlayer spaces, the tiny lithography that 70-Ω traces require becomes difficult to fabricate. In such cases, you might have to go with 50-Ω traces, which permit a wider trace width, to get a manufacturable board.

What about coaxial-cable impedances? In the RF world, the considerations are unlike the pc-board problem, yet the RF industry has converged on a similar range of impedances for coaxial cables. According to IEC publication 78 (1967), 75 Ω is a popular coaxial impedance standard because you can easily match it to several popular antenna configurations. It also defines a solid polyethylene-based 50-Ω cable because, given a fixed outer-shield diameter and a fixed dielectric constant of about 2.2 (the value for solid polyethylene), 50-Ω minimizes the skin-effect losses.

You can prove the optimality of 50-Ω
coaxial cable yourself from basic physical principles. The skin-effect loss, *L*, (in decibels per unit length) of the cable
is proportional to the total skin-effect resistance, *R*, (per unit length) divided by the
characteristic impedance, *Z*_{0}, of
the cable. The total skin-effect resistance, *R*,
is the sum of the shield resistance and center conductor resistances. The
series skin-effect resistance of the coaxial shield, at high frequencies,
varies inversely with its diameter* d*_{2} . The series skin-effect resistance of the coaxial inner
conductor, at high frequencies, varies inversely with its diameter *d*_{1} . The total series resistance, *R*, therefore varies proportionally to
(1/*d*_{2} +1/*d*_{1}). Combining these facts and given fixed values of *d*_{2} and the relative electric
permittivity of the dielectric insulation,* ε*_{R}, you can minimize the skin-effect loss, *L*, starting with the following
equation:

In any elementary textbook on electromagnetic fields and waves,
you can find the following formula for *Z*_{0} as a function of *d*_{2}, *d*_{1}, and* ε*_{R}:

Substituting Equation [2] into Equation
[1] , multiplying numerator and denominator by *d*_{2}, and rearranging terms:

Equation [3] separates out the constant terms (*ε*_{R}^{½}/60)×(1/*d*_{2})) from the operative terms ((1+*d*_{2}/*d*_{1})/ln(*d*_{2}/*d*_{1})) that control the
position of the minimum. Close examination of Equation 3 reveals that the position of the minima is a function only of the ratio *d*_{2}/*d*_{1} and not of either* ε*_{R} or
the absolute diameter *d*_{2}.

A plot of the operative terms as a function of the argument *d*_{2}/*d*_{1} shows a minimum at *d*_{2}/*d*_{1} =3.5911. Assuming a solid polyethylene insulation with a
dielectric constant of 2.25 corresponding to a relative speed of 66% of
the speed of light, the value *d*_{2}/*d*_{1} =3.5911, when plugged into Equation
2, produces a characteristic impedance of 51.1 Ω. A long time ago, radio engineers decided to simply
round off this optimal value of coaxial-cable impedance to a more
convenient value of 50 Ω. It turns out that the
minimum in *L* is fairly broad and flat, so as
long as you stay near 50 Ω, it doesn't much matter
which impedance value you use. For example, if you produce a 75-Ω cable with the same outer-shield diameter and
dielectric, the skin-effect loss increases by only about 12 percent. Different
dielectrics each posses their own slightly
different optimal impedance.