Resistor math

Updated January 26, 2006

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Ohm's law

Even at microwave frequencies, resistors obey Ohm's law:

V=IxR

However, in the real world, there is often some non-linearity associated with the term "R". This might be because you are dealing with a semiconductor, which might have a saturation effect, or perhaps because of the temperature effect on resistance due to power dissipation. Watch out!

Sheet resistance

The concept of sheet resistance is critical to an understanding of thin film resistors. The equation for resistance, based on bulk resistivity, is:

R=L/Wt

where R is the resistance in ohms

is the bulk resistivity in micro-ohm centimeters

L is the length of the resistor in centimeters

W is the width of the resister in centimeters

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 centimeters.

For convenience, the quantity /t 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 this way:

R=L/Wt=(/t)x(L/W)=Rsheet 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%.

Skin depth

Skin depth considerations: In a microstrip transmission line, the part of the conductor and ground plane that carry the most current is the closest to the substrate. In the case of a thin film that has a resistor layer below the gold, guess what? In this case you want the resistor material to be a fraction of a skin depth, while the gold to be at least three skin depths.

Resistance temperature coefficient

All resistors exhibit some degree of variation with temperature. Usually the variation is close to linear. The temperature coefficient of resistance, , is usually expressed in ppm/degrees Celsius:

The temperature coefficient of resistance can be negative or positive. In the former case the resistance is decreased with temperature. In the latter case it increases with temperature. Pure metals have a positive coefficient. Some alloys have been formulated to have a near-zero temperature coefficient (constantin and maganin for example). Carbon and its associated binders usually has a negative temperature coefficient.

Thermistors are resistors that are built specifically to exploit the temperature coefficient, and are often used as temperature control elements. A thermistor with negative temperature coefficient is called "NTC", while a positive temperature coefficient thermistor is called "PTC".

Did we mention that the temperature coefficient is always at least a slight function of temperature? Something you need to consider, thermistors are not perfectly linear.

Power derating

Power rating of a resistor specifies the most power that a resistor can dissipate up to a maximum temperature, which will not damage the resistor. Power rating usually implies that a maximum hot-spot temperature must not be reached, surpassing the limit may result in permanent damage

Power rating specifies two temperatures; the first is the temperature up to which the maximum power rating applies, the second temperature is where the rating must be derated to zero dissipation, in between the rating derated linearly with temperature. Two things can be inferred from these ratings: the maximum storage temperature is equal to the derated temperature. Also, the slope of the derating can be used to calculate an equivalent thermal resistance in degrees C per watt. The difference between the no-load and maximum full load temperature is less than or equal to the temperature rise at full load.

Let's run through the math, for the component who's derating curve is shown. The calculated thermal resistance is:

Thermal resistance=(150C/85C)/1W=65 degrees C/W

Nothing could be easier!