here to go to our main transmission line page
here to go to our minimum attenuation page
here to go to our propagation constant page
to go to our "What's a Neper?" 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
Here's a clickable index to this
line loss Rule of Thumb 1
Four loss mechanisms:
line rule of thumb 2
due to metal (C)
due to dielectric loss tangent (D)
due to dielectric conductivity (G)
due to radiation (R)
Let's propose a rule
of thumb before we even get started discussing transmission
Transmission line attenuation Rule of Thumb #80
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.
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.
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.
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.
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:
line attenuation Rule of Thumb #79
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
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.
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
Sigma is the metal's
conductivity in Siemens/meter; sigma is 1/rho, where rho is resistivity
and is in units of Ohm-meters. RF sheet resistance units 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
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.
here to go to our page on microstrip loss due to metal
here to go to our page on coax loss due to metal
here to go to our page on waveguide loss due to metal
here to go to our page on minimizing attenuation due to metal
Click here to learn about surface roughness effect on metal loss
due to dielectric loss tangent (D)
Loss due to dielectric loss
can be very important at microwave frequencies. Loss tangent is also known as dissipation factor, or "DF" as an abbreviation. This term is proportional
to frequency, so the higher you go, the more likely it will dominate
overall loss (metal loss is 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:
For any TEM transmission line of any impedance, the equation for dielectric loss tangent loss can be simplified to the following:
Here the length units will be consistent with units you use for wavelength. Making the conversion from Nepers to dB (multiplying by 8.686) results in a more familiar result:
Let's go one step farther than most textbooks. Who spends a lot of time thinking about wavelength? Here is the equation in terms of frequency:
And further, let's assume frequency is in GHz, and speed of light is 29.979 cm/nanosecond. Then:
Notice that loss tangent loss is independent of geometry. Although you might reduce metal loss by going to fatter coax, you will still suffer the same dielectric loss if you are using the same dielectric material.
At the risk of repeating ourselves, loss tangent loss is proportional to frequency, as opposed to metal loss which is proportional to SQRT(frequency). What this means is, as you go higher in frequency, loss tangent starts to dominate the loss of a transmission line at some point. If you are working up at W-band, you can no longer go by the advice of your X-band predecessors which might have been "just ignore it". Further, it becomes more important than ever to obtain accurate loss tangent data. If you scour the web you will see widely-different values for loss tangent of mature substrate technology such as GaAs. This is an industry-wide problem, and maybe a good career to consider (material measurements). Note that loss tangent is often a function of frequency, temperature, material quality.... so don't ever hang your hat on a single number.
Now let's look at a plot of loss tangent loss. Here we swept tanδ from 0 to 0.01 in increments of 0.001, for ER=1. Check it out, loss is almost exactly 1 dB/cm at 110 GHz for 1% loss tangent (for the impossible condition that ER=1). If you can remember that one data point you could scale loss tangent loss in your head, if you didn't have such a hangover. But don't forget to multiply by SQRT(ER)!
Speaking of ER, what happens if you are using a quasi-TEM transmission line, like CPW or microstrip? It turns out that you can substitute Keff for ER in the above equation and get very nearly the correct result (go ahead and verify this using EDA software if you don't trust us). Here's an example: consider microstrip-on-GaAs with loss tangent=0.0016 and ER=12.9. Ignoring (for the moment) the ER component, loss at 110 GHz would be 0.16 dB/cm. An educated guess for Keff in any microstrip line is that 70% of the fields are contained in the substrate, so Keff is ~9. Take the square root (3) and multiply 0.16 dB/cm you will estimate that loss tangent loss on a GaAs MMIC ~0.5 dB/cm. Note that metal loss will be the dominant effect at most frequencies of interest....
Now it's time for another
Microwaves101 Rule of Thumb:
Loss tangent loss Rule of Thumb #116
TEM transmission line, loss tangent of 0.01 (which is pretty high) results in almost exactly 1 dB/cm loss at 110 GHz, before you scale it by SQRT(dielectric constant). Since it is linear with frequency, you should be able to scale loss tangent attenuation in your head. You can approximate attenuation in microstrip or CPW if you scale by the effective dielectric constant.
here to go to our page on coax loss due to dielectric loss tangent
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:
line loss Rule of Thumb #80
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
Here's the generic equation for
this loss mechanism using the G' element of the transmission line
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.
here to go to our page on microstrip loss due to substrate conductivity
here to go to our page on coax loss due to dielectric conductivity
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 upon
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!