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Transmission
line loss
Updated July 7,
2008
Click
here to go to our main transmission line page
Click
here to go to our minimum attenuation page
Click
here to go to our propagation constant page
Click here
to go to our "What's a Neper?" page
Update for June 2007! We've
cleaned this page up to the point where it's downright useful! We
also vastly improved the propagation constant
page.
Here we will review some of
the math for calculating the RF losses (attenuation) of transmission
lines over frequency. We'll guide you to some of our other pages
that show calculations of attenuation of different transmission
line geometries.
Here's a clickable index to this
page
Transmission
line loss Rule of Thumb 1
Attenuation
or rejection?
Transmission
line model
Four loss mechanisms:
Transmission
line rule of thumb 2
Attenuation
due to metal ( C)
Attenuation
due to dielectric loss tangent (D)
Attenuation
due to dielectric conductivity (G)
Attenuation
due to radiation (R)
Transmission
line loss Rule of Thumb 1
Let's propose a rule
of thumb before we even get started discussing transmission
line losses:
Transmission line loss rule of thumb
For a given frequency, waveguide
will give the lowest loss per unit length. Coax loss will be about
10X higher (in dB). Transmission line loss on MMICs (microstrip
or coplanar waveguide) is about 10X worse than coax, or 100X that
of waveguide (but the lengths of the transmission lines are really
small!) Stripline, depending on its geometry, usually will be slightly
higher in loss than coax.
Attenuation
or rejection?
We need to draw a distinction
between "real" loss (attenuation) and mismatch loss (rejection).
A filter can be used to reject a signal,
but rejection usually means reflection. The rejected
signal is returned to the generator where it is dissipated. With
an appropriate matching network, the "loss" of a filter
could be theoretically tuned out to zero dB.
Attenuation can be reduced by
increasing system characteristic impedance (usually not an option),
but cannot be completely tuned away, unless you are able to change
the characteristic impedance to infinity. This is never an option!
On this page, we are discussing
"real" loss, not mismatch loss.
Transmission
line model
The transmission line model is
used in many of the loss calculations. We recommend that you study
our page on propagation constant,
and in particular become familiar with the "attenuation
constant" before you continue reading this page on transmission
line loss. The propagation
constant is what determines the phase and amplitude of a signal
on a transmission line. It is denoted by Greek letter gamma:
The transmission line model is
for an infinitesimal section of line, the line can be composed of
four lumped elements:
Note that the "prime"
notation here means that parameters are normalized to length. C'
is in Farads/meter, L' is in Henries/meter, R' is in Ohms/meter,
and G' is in Siemens/meter. A wealth of transmission line parameters
can be expressed in terms of of these four lumped elements, including
characteristic impedance, propagation
constant and phase velocity.
Four types
of losses
To quantize the RF
losses in transmission
lines we need to calculate the attenuation constant ,
which is in the "natural" units of Nepers/meter.
The attenuation constant can be broken down into at least four components,
one representing metal loss, one representing dielectric loss due
to loss tangent, one due to conductivity of the dielectric, and
one due to stray radiation:
Below we'll deal with each of
these loss mechanisms in terms of the transmission line model.
By the way, we struggled with
the nomenclature of these subscripts for this web site. The "C"
for metal loss would be better as "M" (for metal), in
our opinion. Then the "C" subscript could be used for
the conductivity loss of the dielectric instead of "G".
The loss of the metal is not proportional to its conductivity, its
proportional to its resistivity, but the letter "R" had
to be reserved for "radiation". In the end we had to bow
to convention, there are many textbooks that use C and D for metal
loss and dielectric losses (but textbooks tend to forget about the
third and fourth components).
Each of the four components of
loss are geometry dependent, meaning the calculation is quite different
for coax than it is for waveguide for example. We'll just touch
on each subject below, and move most of the math to separate pages
on different transmission lines. But first it's time for another
Microwaves101 Rule of Thumb!
Transmission
line loss Rule of Thumb 2
Different loss mechanisms have different behaviors over frequency.
Metal loss is proportional to square-root frequency. Dielectric
loss is proportion to frequency. Dielectric conduction loss is constant
over frequency.
Attenuation
due to metal conductivity (C)
The overall loss of most transmission
lines is dominated by the metal loss at microwave frequencies. Metal
loss varies predominantly as SQRT(f). Metal loss is modeled by the
R' component in the transmission line model which is series resistance
per unit length. The R' term is a function of the geometry of the
transmission line and the RF sheet resistance of the metal system
that is used.
The RF
sheet resistance can be far more than the DC sheet resistance
(thanks, Loren!), because of the skin effect.
The skin effect says that at higher and higher frequency, the path
of the signal bunches up toward the outside of a conductor, on the
surface where the EM wave is propagating.
Metal loss varies predominantly
as SQRT(f) because RF sheet resistance varies as SQRT(f). The RF
sheet resistance for a single-metal system is calculated
as:
RF sheet resistance
u nits are Ohms/square. The calculation for RF sheet resistance
of multiple-layer metal systems is explained on our skin
depth page. Here's the RF sheet resistance versus frequency
for three mils of copper (many, many skin depths at microwave frequencies),
which you can regard as the best you'll ever do, unless someone
starts making printed wiring boards with pure silver metal traces,
silver has lower bulk resistivity
than copper.
The next step in
calculating transmission line metal loss is to find the RF resistance
per unit length R' (convert Ohms/square to Ohms/meter), based on
the physical dimensions of the structure and the RF sheet resistance.
This calculation is different for coax, waveguide, microstrip or
any other transmission line structure. The devil is in the details!
The links below will help you with this part of the calculation.
Note that with the exception of waveguide, resistance per length
has two components in transmission lines; you have to sum up the
contribution of the "hot" conductor with the ground plane
conductor (or ground plane conductors in the case of coplanar
waveguide).
The final step in calculating
RF loss due to metal: from the resistance per unit length, you find
the loss per unit length:
Notice the "natural"
units of transmission line loss are Nepers/length.
To get loss in dB/length, multiply Nepers by 8.686.
Click
here to go to our page on microstrip loss due to metal
Click
here to go to our page on coax loss due to metal
Click
here to go to our page on waveguide loss due to metal
Click
here to go to our page on minimizing attenuation due to metal
Attenuation
due to dielectric loss tangent (D)
Loss due to dielectric loss
tangent (tan )
can be very important at microwave frequencies. This term is proportional
to frequency, so the higher you go, the more likely it will dominate
overall loss (metal loss is only proportional to SQRT of frequency).
The calculation for loss due
to loss tangent is straightforward and uses the elements of the
transmission line model with one stipulation. Did we mention that
the capacitor C' in the transmission line might be lossy? The loss
tangent is a measure of the ratio of its conductance to its susceptance
(like "Q"). Once you have calculated capacitive susceptance/length
for the specific geometry, all you need to do is multiply by the
loss tangent to get the frequency-dependent conductance term that
causes loss tangent loss:
Click
here to go to our page on coax loss due to dielectric loss tangent
Other transmission line calculations
coming soon!
Attenuation
due to substrate conductivity (G)
Note: many textbooks link
substrate conduction loss term in with the loss tangent loss term.
We choose to separate them, they behave quite differently.
Owing to the great dielectrics
we have available for coax (such as PTFE) and MMICs (GaAs), the
loss due to substrate conductivity term is often ignored because
it's usually very small because the dielectric has extremely low
conductivity. However, the proliferation
of silicon into the microwave realm has brought this term back to
our attention, because silicon has relatively poor electrical insulating
properties (a true semiconductor!) Let's add another Microwaves101
Rule of Thumb to draw the line where substrate conductivity
needs to be considered:
Transmission
line loss Rule of Thumb 3
When considering the loss of a transmission line due to dielectric
conductivity, if the resistivity of the dielectric is greater than
10,000 Ohm-cm, forget it! That pretty much rules out all substrates
except silicon, which can be anywhere from 1 Ohm-cm (very lossy)
to 10,000 Ohm-cm (very expensive float-zone silicon). PTFE is 1E18
Ohm cm!
Here's the generic equation for
this loss mechanism using the G' element of the transmission line
model:
One thing you should know about
loss due to substrate conductivity: it's NOT a function of frequency!
So if you use a cheap Ohmmeter to measure DC resistance across a
transmission line's two conductors (and don't terminate the other
end, that would be stupid!), you have what you need to predict loss
due to conduction at 10 GHz!
Another thing you should know:
for transmission line media with uniform dielectric media (coax,
stripline) the calculation of G' is essentially the same calculation
as C', except you substitute the dielectric's conductivity in place
of its permittivity. We'll go into this in further detail on the
page links below.
Click
here to go to our page on microstrip loss due to substrate conductivity
Click
here to go to our page on coax loss due to dielectric conductivity
Attenuation
due to radiation (R)
This is another attenuation mechanism
that has a very small effect, if your circuits are behaving well.
It really isn't attenuation in the sense of the word that the energy
goes up in heat, it is more of a leakage loss. But the effect open
your signal is the same either way, it loses energy.
There's no way to account for
attenuation due to radiation using the transmission line model.
That's why 3D electromagnetic structure simulators
are so popular!
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