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