RF sheet resistance
here to go to our page on skin depth
here to go to our main page on resistors
here to go to our page on resistor math
here to go to our page on conductivity
here to go to our page on transmission line losses
This page will show you how to
calculate sheet resistance of a layer of metal versus frequency.
By the way, what we call "RF sheet resistance" is often
called "surface resistance" or "surface impedance"
or other names in textbooks.
DC sheet resistance
The term sheet resistance
should be familiar if you work with electronic thin films. It is
a function of the bulk resistivity of a metallic film, and its thickness.
We discuss it further on this
page. Sheet resistance, Rsh, is given in ohms per
square, where squares are the unitless dimension of length divided
For a more complete discussion
of direct current sheet resistance, go to our page on this
topic. In summary, for direct current, the sheet resistance
is calculated from the metal resistivity and thickness:
resistance and sheet conductance as functions of frequency
Something that is not often considered
is that sheet resistance is a function of frequency. Applied
in the RF world, errors can result from using the DC definition.
The assumption of constant sheet
resistance is only valid for conductors that are thin compared to
skin depth (which is often the case for thin-film resistors, but
never the case for transmission lines.)
Knowing RF sheet resistance versus
frequency of interconnect metals (copper or gold for examples) can
be a very useful short-cut for evaluating attenuation of strip conductors
such as microstrip.
We offer a free download that
you should grab if you want to analyze RF sheet resistance of various
metals (including stacked metals) that you specify. Look for it
here. It's near the bottom
of the page.
maximum sheet conductance (minimum sheet resistance)
Sheet conductivity is the inverse
of sheet resistance, its units are Siemen-squares, or mho-squares.
This quantity is useful when you are dealing with multiple-layer
conductors, as their conductivities can be added in parallel, then
the sum can be re-inverted and expressed in a composite sheet resistance.
When RF is involved, the so called skin-effect can become apparent,
and therefore DC calculations of conductivity and resistivity are
invalid. The skin effect is taken into account by using a decreasing
exponential factor whose exponent is inversely proportional to a
parameter called skin depth.
Maximum sheet conductance is
the best you can do, and is a function of frequency. Adding more
metal beyond five skin depths doesn't help!
The percentage of conduction
that is achieved versus depth into a metal varies as the negative
exponential of the depth expressed in skin depths. At the surface,
complete conduction takes place, and the resistivity of the metal
is 100% of its value at DC, equal to .
At one skin depth, the metal's conductivity has been reduced to
36.8% of ,
at 2 skin depths, 13.5%, etc. By the time you reach five skin depths
the metal's conductivity is reduced to just 0.7% of its full value.
That is where the rule of thumb of five skin depths comes from,
adding tons of additional metal beyond five skin depths can only
reduce your RF resistance by 0.7%, so why bother? Expressed below
is what we call the incremental RF conductivity, it is the
conductivity at a given depth, reduced by the skin depth equation:
Thanks for the correction, Ron!
And thanks to Giovanni, our use of the Greek
alphabet is now consistent for skin depth! If we integrate all
of the conductivity of the thin film from the surface to include
infinite skin depths, we arrive at the maximum sheet conductance
for a given frequency. This is different from DC sheet conductivity,
which can be quite a bit higher since every free electron in the
metal contributes to conduction during direct current. Although
we have pledged never to use calculus on this web site, integrating
an exponential function is so easy that even we can do it (but we
won't show you all of the intermediate steps that required a big
eraser). The maximum sheet conductance is:
(Thanks for the correction, Michael!)
The maximum RF sheet conductance is in units of Seimen-squares (or
mho-squares) which is the inverse of sheet resistance (units of
ohms/square). Similarly, the minimum RF sheet resistance is just
the reciprocal of the above equation:
Voila! that equation looks an
awful lot like the DC sheet resistance equation (top of page) except
the skin depth is now in the dominator instead of the conductor's
Remember, this is the best you
can achieve, no matter how much more metal you add to the transmission
line! Now let's look at what this means for various metals. Click
here to look up conductivities of various metals. The plot below
compares aluminum, gold, copper and silver. Silver is best, followed
by copper, then gold, then aluminum. At DC you can achieve nearly
zero sheet resistance, because the skin depth is infinite. But to
get truly zero sheet resistance, you'd need infinite metal thickness!
The minimum sheet resistance will have an effect on the transmission
line loss, which you can read about on this
In any case, how
about a rule of thumb?
The minimum RF sheet resistance you can achieve
is on the order of 30 milli-ohms/square at X-band, and increases
as the square-root of frequency up to 100 milliohms/square at W-band.
The exact value will depend on the metal's conductivity.
Here's some example
RF sheet resistance calculations.
Here's the special case of