Resistor Tutorial - Part Two, film resistor frequency effects

Resistor Tutorial - Part Two, film resistor frequency effects

Post by Dr. Barry L. Ornit » Thu, 25 Jan 2001 16:30:15



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.

This article goes into more detail on an electrical model for
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
"alt.binaries.pictures.radio" newsgroup.  If your site does
not subscribe to the binary groups, please send me email and
I can forward the file directly to you.  Note that the graph
is in Adobe Acrobat format.  You can download a free Acrobat
reader from Adobe's website and many other places like
Tucows, Winfiles.com, or Shareware.com.  Use your favorite
search engine to look for Adobe Acrobat Reader.

This article and the companion graph are both copyrighted.  
While you are free to make a copy for yourself, I ask that
these not be placed on any websites without my permission.

If you have questions about the calculations, please send
them directly to me at the address shown.  If I feel the
question is of general interest, I shall try to answer it
with a post to this group.  I do tend to ignore "golden-
eared audiophools" with their silver cables and green magic
markers.


~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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
about 2 to 3 picofarads.

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

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

For 1/4 inch leads on each end, this adds (for non-magnetic leads
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.

     Copyright 1999    B. L. Ornitz    All Rights Reserved