Why
50 Ohms?
Updated January
13, 2009
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New for September 2008! Check
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The "50 ohm question"
comes up from time to time. Most microwave hardware is specified
to run in a fifty ohm system (OK, some stuff is 75 ohms, and we'll
talk about that as well.) Why was this standard chosen?
The standardization of fifty
ohm impedance goes back to developing coax cables
for kilowatt radio transmitters in the 1930s. A good explanation
for the choice of fifty ohms is given in Microwave
Tubes, by A. S. Gilmour, Jr. The quick answer is that 50 ohms
is a great compromise between power handling and low loss, for
airdielectric coax. Let's look at the math that proves this,
just for kicks.
Here is another thought that
recently came in from Mike:
Another thing to consider
for reason for why CATV systems use 75 ohm coax. A 2 turn to 1
turn balun changes the impedance of 300 ohm twin lead from an
antenna to 75 ohms very nicely and with a relatively broad band.
Cable loss versus impedance
For RF signals, resistance
per unit length of coax cables is determined by circumferential
area of the conductor surface due to skin
depth effect, not crosssectional area. Here's the solution
for loss/length for coax cables of arbitrary dielectric constant
and metal properties:
The details of this equation
are derived on this page.
You'd think a fat conductor would
always give the lowest insertion loss because it has the most circumferential
area (the 1/d component of the above equation decreases loss for
increasing d), but you'd be wrong! The characteristic impedance
of the cable (Z_{0}) throws that log(D/d) function into
the denominator, it increases for increasing d.
In order to plot loss/length
versus characteristic impedance, let's review the coax impedance
calculation. The impedance of coax for a given outer diameter and
dielectric is solely a function of the diameter of the inner conductor
and the dielectric constant of the filler material:
Now we can plot loss/length versus
characteristic impedance. It turns out that insertion loss has a
minimum around 77 ohms, for any cable with Er=1 (air dielectric).
In our example, we chose 10 mm inner diameter of the outer conductor,
and calculated loss at 10 GHz.
Peak power handling
The peak power handling
for air coax is limited by voltage breakdown (as opposed to heating
effects which limit average power handling). You'd think
that you'd want maximum separation between the opposing conductors
(inner wire and outer sheaf) to avoid arcing, so you'd make the
inner conductor as thin as possible, but you'd be wrong again! The
maximum voltage field in a coaxial cable if quite different than
between parallelplane conductors. Here's the equation for "field
enhancement", which is a measure of how much worse the fields
are than in parallel plate:
Beta=(a/r)/[ln(1+a/r)]
Here a is the is
the gap between the conductors and r is the radius of the inner
conductor. We took this from Gilmour's book. Once again, characteristic
impedance has to be considered because power depends on V^{2}/Z_{0}.
The way to calculate
maximum power handling is to assume a critical electric field can't
be exceeded to avoid breakdown. We'll assume 100,000 volts/meter
(actually it can exceed 1,000,000 volts per meter, but the whole
topic of voltage breakdown deserves a lot more attention so we'll
be conservative here for the time being). Next, calculate the field
that would be generated across the gap in the coax cable, without
regard to the geometry (assume the center and outer conductors are
parallel plates). Then apply the field enhancement equation above
(which is a number greater than 1). Then the maximum power is equal
to Vcritical^2/(2Z_{0}). Why the "2" in the denominator?
That's because Vcritical is a peak value, not an RMS value.
The best peak power
handling occurs at Z_{0}=30 ohms. We'll add some prettier
equations on this page soon, or go to our page on
coax power handling for more information.
The voltage breakdown
of air coax is a function of atmospheric pressure (or altitude),
temperature, humidity, and even surface roughness. How do you increase
the power handing of air coax? that's easy, fill it with a dielectric
such as PTFE! Typical "solid" dielectric withstanding
voltage is much higher that the breakdown voltage of air, by a factor
of 10 or more. Foamed dielectrics used in cables don't provide much
of an increase in voltage handling compared to air, but semirigid
coax (solid PTFE) can handle 10s of kilowatts, the overall voltage
limitation is usually the connectors that are attached to the cables.
The 50Ohm compromise
The arithmetic mean between 30
ohms (best power handling) and 77 ohms (lowest loss) is 53.5, the
geometric mean is 48 ohms. Thus the choice of 50 ohms is a compromise
between power handling capability and signal loss per unit length,
for air dielectric.
Why 75
Ohms?
For cheap commercial cables such
as those that bring CATV to your home, 75 ohms is the standard.
These cables don't have to carry high power, so the key characteristic
that should be considered is low loss. The answer to the "why
75 Ohms?" question seems obvious. We just saw that 77 ohms
gives the lowest loss for air dielectric coax, so 75 ohms might
be just an engineering roundoff. We know of one text book that
will tell you that is why RG cables are 75 ohms... but they are
wrong!
Here's the problem. Commercial
CATV cables are filled with PTFE foam, which has a dielectric constant
around 1.43. Guess what? The loss characteristic is a function of
the dielectric constant (~SQRT(ER)), while impedance is a different
function of dielectric constant (~1/[SQRT(ER)]). The opposing contibutions
of Er muddy the waters quite a bit.
It turns out that the minimum
loss impedance for ER=1.43 is around 64 ohms, as shown in the plot
below (purple trace). For the record, for solid PTFE (ER=2.2, yellow
line) the minimum loss occurs near 52 ohms. So it's serendipity
that when we use 50 ohm semirigid coax cables with solid PTFE, they
give nearly the lowest possible loss for ER=2.2! PTFE was invented
by Roy
Plunkett in 1938, well after the 50 ohm standard was in place.
So why 75 ohms? Here's our guess.
Often the center conductor of cheap cables is made of a steel core,
with some copper plating. The lower the impedance, the bigger the
diameter of the center core. An impedance 75 ohms probably was a
compromise between low loss and cable flexibility.
Another microwave myth debunked
here at Microwaves101.com! Are we not nerds?
