## Resistor Tutorial - Part Two, film resistor frequency effects

### Resistor Tutorial - Part Two, film resistor frequency effects

In my earlier post, I covered the basics on how resistors
were manufactured.  I also briefly mentioned that the spiral
cuts on film resistors created inductance, and that the
encapsulation and physical shape of a resistor also creates
shunt capacitance.

a typical 1/4 watt film resistor.  A companion graph showing
the effective impedance of different values of resistors
over a broad frequency range has been posted to the
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~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

To answer the question about how much inductance can be found in
a spiral cut metal film resistor, I calculated the inductance for
a ribbon coil which would be the same dimensions as the film on a
1/4 watt film resistor.  These calculations were based on:

5 complete turns for the winding
Length = 0.15 inches
Diameter = 0.07 inches
Winding Pitch = 0.03 inches
Strip Width = 0.025 inches
Coating Thickness = 0.001 inches
1/4 inch leads on each end

To calculate the inductance, I used equations for "Helices of
Rectangular Strip" from:

Grover, F. W.: "Inductance Calculations", Instrument Society
of America, 1946, ISBN 0-87664-557-0, pp. 164-166.

The equations present a correction term to the equivalent
cylindrical current sheet inductance.  The correction for edge
insulation is based on geometric mean differences for rectangles.
In addition to solving some monstrous equations, the solution
required several one dimensional and one two dimensional cubic
spline interpolation of tabular data.

I obtained the following results:

Sheet Inductance = 36.36 nH
Correction = -13.40 nH
Inductance = 22.96 nH

We can compare this result to that given by traditional formulas
for solenoid coils:

ARRL Equation = 16.87 nH  (ARRL Handbook)
Nagaoka's Equation = 16.98 nH  (Radiotron Designers
Handbook)

The increase in inductance is due to the width of the strips
produced by the spiral t*** of the resistor.

To model this resistor, we place this inductance in series with
the resistance.  We also need to add the shunting capacitance of
the ceramic body and the conformal insulation.  This is typically

We can also include the inductance of the resistor leads using an
equation by Rosa.

Rosa, E. B.:"Bureau of Standards Paper", 169, 1912.

of 0.6 mm in diameter) an added inductance of 2.92 nH on each
end.

The equivalent circuit for this 1/4 watt resistor (forgiving the
crude ASCII representation) I came up with is:

Resistor
----UUUU--+-/\/\/\/\-+--UUUU----
3 nH  |          |  3 nH
+----|(----+
2.5 pF

If you calculate the impedance of this equivalent circuit, you
will find that the capacitive reactance dominates the response
for resistor values over approximately 100 ohms.  For resistances
below 100 ohms, the inductive effect dominates.

To get an idea of how good this film resistor behaves over a
frequency range, the following table gives the approximate
frequency for which the resistance increases or decreases by 10
percent over its DC value.

Resistance, Ohms        Frequency for 10%
Change in Resistance
1    3   MHz  increase
10    30  MHz  increase
100    300 MHz  increase
1000    30  MHz  decrease
10K    3   MHz  decrease
100K    300 KHz  decrease
1Meg    30  KHz  decrease

My conclusion is that for resistor values over 100 ohms, film
resistors can replace carbon composition units with no worries.
The inductance from the spiral design is just not significant
unless you go to very low resistor values.