here to go to our main page on resistors
here to go to our resistor math page
here to go to our page on skin depth
here to go to our main page on conductivity
New for April 2010! This
page gathers together a several others so they are easier to find.
Most of the material was recently split off from our "conductivity"
page. Yes, we know, it still needs some consolidation, come back
DC sheet resistance
The "DC" explanation
of sheet resistance ignores skin depth. For resistor calculations,
99% of the time this is a reasonable approximation. For attenuation
in metal traces at microwave frequencies, this assumption will lead
you astray, and you need to read our content on RF
Bulk resistivity is a the property
which is independent of frequency and geometry. In microwaves, often
we are dealing with thin films of conductors, which have been applied
at a controlled thickness. A more convenient property to deal with
in this case is sheet resistance. The sheet resistance of a metal
film is often expressed in ohms/square. What's a square? Exactly
on first? I Don't Know's on second.
Recall the equation for calculating
resistance from bulk resistivity:
If you consider
the quantity L/w, it is unitless. It can be considered as a measure
of how many squares of area your conductor or resistor has.
For example, a thin-film resistor with length 30 mils and width
10 mils is three squares. A smaller resistor of 3 microns length
and 1 microns width also has three squares (thanks Jack!)
If they both have the same thickness and bulk resistivity, they
both have the same value in ohms. They will have far different power
ratings, and the smaller resistor will have a higher usable frequency
response. Be careful not to mix up length and width, a resistor
with 10 microns length and 30 microns width measures 1/3 square,
not three squares!
Rsh, is equal to bulk resistivity divided by thickness.
It can be used to conveniently calculate resistance values from
number of squares, as follows:
As in all engineering,
you will need to keep units consistent in order to make the calculation
correctly (if rho is in ohm centimeters, the thickness must also
be in centimeters). One last thing to consider: sheet conductivity
is the inverse of sheet resistivity. When is sheet conductivity
useful? When you have more than one metal layer. The sheet conductivities
of the layers can be added, because the conduction paths are in
The concept of sheet resistance
is critical to an understanding of thin film resistors. The equation
for resistance, based on bulk resistivity, is:
where R is the resistance in
is the bulk resistivity in micro-ohm centimeters
L is the length of the resistor
W is the width of the resister
and t is the thickness of the
resistor in centimeters
In practice, you will have to
convert units that you are given until they are all consistent with
each other, but bulk resistivity is often expressed in micro-ohm
For convenience, the quantity
is used to express "sheet resistance", the units are ohms/square.
For a given thin-film network (or MMIC), the sheet resistance should
remain constant for all resistors, because the resistor material
was uniformly deposited across the circuit. We can rewrite the equation
x # squares
The quantity L/W is the number
of "squares" that a resistor has, and total resistance
is proportional to number of squares. In the figure below, we see
a resistor with six squares (the light blue represents resistor
material). If the sheet resistivity of the thin-film resistor is
50 ohms/square, we are looking at a 300 ohm resistor.
Note that squares have no units.
So you can measure length and width in microns, centimeters, mils,
etc., and then determine the number of squares by dividing length
by width. Also note that there are an infinite number of solutions
to achieving the same ohm value for a thin-film resistor. A 1 mil
x 1 mil resistor will have the same resistance as a 1 inch x 1 inch
resistor. However, the one-mil-square resistor will have better
high-frequency performance (because it is smaller), while the one-inch-square
resistor will have a much higher power handling capability (because
it spreads heat out more). These two resistors should have the same
resistance value, even though they are of different size:
Some other things
to consider when designing resistors are that because of skin
depth, the RF sheet resistance might be higher than the DC sheet
resistance (but usually it is very close), and the tolerance of
the resistor due to edge definition and other things might force
you to laser trim it.
A mistake to avoid
is to count the number of squares as W/L instead of L/W. The resistor
below might look like it has five squares, when in reality it has
only 0.2 squares:
Microwaves101 Rule of Thumb
When you are counting the number
of squares in a meandering resistor, the squares at each bend should
be counted as 1/2 square. In the figure below, if you count up all
of the resistor squares, you'll get 43. But when you measure the
resistor, it will behave like it has only 40 squares. That is because
you need to reduce the corner squares by 50%.
Remember, the resistance
calculated this way does not account for skin
depth effects. It is accurate if your conductor thickness is
small compared to a skin depth.
RF sheet resistance
Because of skin depth, the effective
sheet resistance changes with frequency, but only in a meaningful
amount if the resistor's thickness is a less than a few skin depths.
The content links below will help you understand the increase in
sheet resistance due to the skin effect.
sheet resistance (separate page)
sheet resistance examples (separate page)