1. BACKGROUND
The dimensioning process of electronic circuits often results in non-standard
values for passive components like resistors, capacitors, or inductors. This can
be a critical issue particularly in connection with electronic filters, in which
case even small relative deviations of component values from their theoretical
values can significantly affect the circuit behaviour.
This raises the basic question how, for example, an arbitrary resistance value
can be realised or at least closely approximated with standard resistors. Two
common strategies are to use multiple standard resistors that are either con-
nected in series or in parallel; and when using just two resistors, this is in
fact everything that is possible. With more than two components, however, there
are a few more generalised topologies that are of practical interest, at least
as long as the number of components is not too large.
Unfortunately, the necessary calculations are not as simple or straightforward
as it might seem at first glance, because the underlying problem belongs to the
class of so-called integer optimisation problems, which are inherently hard to
solve and for which no efficient general-purpose algorithms are known.
Hence, a relatively straightforward way to deal with this is to provide concise,
pre-calculated tables of feasible combinations, and this is exactly the purpose
of this collection.
2. DESCRIPTION
This collection of tables deals with combinations of values from the so-called
E series of preferred values. These values are commonly used in connection with
passive components such as resistors, capacitors, or inductors. The different
E series are specified in IEC Standard 60063.
The following two basic types of combinations are addressed:
- Combinations of type A: X = x1 + x2 + ... + xn
- Combinations of type B: 1/X = 1/x1 + 1/x2 + ... + 1/xn
Here, x1, x2, ..., xn are n arbitrary values taken from a particular E series,
and X is the resulting value of the respective combination.
Thus, combinations of type A apply to series connections of resistors, to series
connections of inductors, and to parallel connections of capacitors. Similarly,
combinations of type B apply to parallel connections of resistors, to parallel
connections of inductors, and to series connections of capacitors; the latter
two cases, however, are of minor practical importance. Note that combinations of
less than n values are seamlessly included, in which case some of the x values
are either zero (type A) or infinite (type B).
In addition, for the special case of exactly three values, the following two
mixed combinations are also addressed:
- Combinations of type C: X = x1 + 1/(1/x2 + 1/x3)
- Combinations of type D: 1/X = 1/x1 + 1/(x2 + x3)
These two combinations are primarily of interest for resistors, in which case
combinations of type C apply to extended series connections and combinations of
type D to extended parallel connections of one plus two resistors.
In order to limit the potentially infinite number of possible combinations to a
reasonable amount, combinations where the ratio between the largest finite value
and the smallest non-zero value would exceed an upper limit of 100 are excluded.
Such an upper limit corresponds to nominal component tolerances in the order
of 1% for the dominant values of a combination, which should be sufficient for
most cases of interest.
For convenience and ease of use, the resulting tables of feasible combinations
are sorted in ascending order of the resulting values X, and these values are
normalised such that they generally fall into the base decade from 100 to 1000,
thus 100 <= X < 1000. Of course, all table entries can be arbitrarily scaled up
or down by factors of 10, 100, 1000, and so forth.
This is best illustrated with three resistors that have resistance values of r1,
r2, and r3, a total resistance value of R, and a maximum allowed ratio of M.
- Combinations of type A:
____ r1 ____ r2 ____ r3
o-----|____|----|____|----|____|-----o
R = r1 + r2 + r3
- Combinations of type B:
____ r1
+----|____|----+
| ____ r2 |
o-----+----|____|----+-----o
| ____ r3 |
+----|____|----+
1/R = 1/r1 + 1/r2 + 1/r3
- Combinations of type C:
____ r2
____ r1 +----|____|----+
o-----|____|----+ ____ r3 +-----o
+----|____|----+
R = r1 + 1/(1/r2 + 1/r3)
- Combinations of type D:
____ r1
+---------|____|---------+
o-----+ r2 ____ ____ r3 +-----o
+----|____|----|____|----+
1/R = 1/r1 + 1/(r2 + r3)
Of course, all four combinations are subject to the constraints 100 <= R < 1000
and 1 <= max(r1,r2,r3) / min(r1,r2,r3) <= M. Note also that the number of values
that is actually used in a basic combination of type A or B can vary from n = 1
(just a single value) up to n = 5, depending on the considered E series.
The following E series are explicitly addressed here:
- E6 series: 6 values per decade, combinations up to n = 5
- E12 series: 12 values per decade, combinations up to n = 4
- E24 series: 24 values per decade, combinations up to n = 3
- E48 series: 48 values per decade, combinations up to n = 2
- E48+E24 series: 48 + 21 = 69 values per decade, combinations up to n = 2
- E96 series: 96 values per decade, combinations up to n = 2
- E96+E24 series: 96 + 18 = 114 values per decade, combinations up to n = 2
As regards the combined E48+E24 and E96+E24 series, it is important to note that
most E24 values are, for historical reasons, neither contained in the E48 nor in
the E96 series. Therefore, these combined pseudo-series are explicitly addressed
in separate tables to make this collection as versatile as possible.
All tables also contain a supplementary column holding the normalised relative
uncertainty (NRU) value of the respective combination. It is calculated via the
common error propagation formula for independent variables and gives an estimate
for the mean relative uncertainty of the resulting value X of a combination in
relation to the mean relative uncertainties of its x values. This estimate is
valid under the simplifying assumption that the x values are statistically inde-
pendent and normally distributed about their nominal values and that they have
identical relative uncertainties, in which case 1/sqrt(n) <= NRU <= 1. Inciden-
tally, these theoretical NRU values represent upper bounds for all symmetrical
distributions that are strictly bounded by the given tolerance bands, which
should normally be the case in the current context.
In effect, this means that the relative dispersion of the resulting value X of
a combination is somewhat reduced compared to the relative dispersions of its
individual x values. The worst-case behaviour in the sense of a strict tolerance
value for X is not improved, though. Note, however, that the above independence
assumption may already be violated when a combination is made up of components
from the same production batch, just to name one potential caveat.
3. PACKAGES
The tables contained in this collection are made available in five separate
packages, each of which containing the set of files for a particular E series.
For simplicity, however, the E48 and the E48+E24 files share the same package,
just like the E96 and the E96+E24 files. Each such package is stored in a ZIP
archive to save space as well as download time. Besides, this also guarantees
the integrity of the contained text files when transferred across different
platforms.
The whole collection consists of the following seven files:
- Read_Me.pdf
- Read_Me.txt
- E6_Combinations.zip
- E12_Combinations.zip
- E24_Combinations.zip
- E48_Combinations.zip
- E96_Combinations.zip
The read-me file is included in PDF and in text form, for convenience. Unpacking
the ZIP files on current operating systems such as macOS, Unix, or Windows ought
to be straightforward. Instructions on how to do this can be found, for example,
at .
After unpacking, each of the above ZIP files expands to a single folder (or
directory). The corresponding five folders should then contain the following
files:
Folder 'E6_Combinations' (1.7 MB):
- E6_Read_Me.pdf
- E6_Read_Me.txt
- E6_Combinations_2_A.txt
- E6_Combinations_2_B.txt
- E6_Combinations_3_A.txt
- E6_Combinations_3_B.txt
- E6_Combinations_3_C.txt
- E6_Combinations_3_D.txt
- E6_Combinations_4_A.txt
- E6_Combinations_4_B.txt
- E6_Combinations_5_A.txt
- E6_Combinations_5_B.txt
- GNU_GPL_V3.txt
- GNU_LGPL_V3.txt
Folder 'E12_Combinations' (4.3 MB):
- E12_Read_Me.pdf
- E12_Read_Me.txt
- E12_Combinations_2_A.txt
- E12_Combinations_2_B.txt
- E12_Combinations_3_A.txt
- E12_Combinations_3_B.txt
- E12_Combinations_3_C.txt
- E12_Combinations_3_D.txt
- E12_Combinations_4_A.txt
- E12_Combinations_4_B.txt
- GNU_GPL_V3.txt
- GNU_LGPL_V3.txt
Folder 'E24_Combinations' (8.9 MB):
- E24_Read_Me.pdf
- E24_Read_Me.txt
- E24_Combinations_2_A.txt
- E24_Combinations_2_B.txt
- E24_Combinations_3_A.txt
- E24_Combinations_3_B.txt
- E24_Combinations_3_C.txt
- E24_Combinations_3_D.txt
- GNU_GPL_V3.txt
- GNU_LGPL_V3.txt
Folder 'E48_Combinations' (1.1 MB):
- E48_Read_Me.pdf
- E48_Read_Me.txt
- E48_Combinations_2_A.txt
- E48_Combinations_2_B.txt
- E48_E24_Combinations_2_A.txt
- E48_E24_Combinations_2_B.txt
- GNU_GPL_V3.txt
- GNU_LGPL_V3.txt
Folder 'E96_Combinations' (3.1 MB):
- E96_Read_Me.pdf
- E96_Read_Me.txt
- E96_Combinations_2_A.txt
- E96_Combinations_2_B.txt
- E96_E24_Combinations_2_A.txt
- E96_E24_Combinations_2_B.txt
- GNU_GPL_V3.txt
- GNU_LGPL_V3.txt
All read-me files are again included in PDF and in text form. In each folder,
the tables of feasible combinations are stored in text files that have system-
atically coded file names of the form:
- _Combinations__.txt
The string specifies the underlying E series or pseudo-series, respec-
tively, specifies the (maximum) number of values used in a combination, and
specifies the type of combinations (see the above file names).
Each of these files also contains information about its original size in bytes,
and it is recommended to cross-check this if a problem with one of the files is
encountered, before contacting the author.
The last two files in each folder are the GNU General Public License (GPL) and
the GNU Lesser General Public License (LGPL). These files contain the detailed
license agreements under which the accompanying tables are being released.
All text files included in these packages are Unix plain text files which use
single line-feed (LF) characters as line terminators; the various tables also
assume a fixed tab spacing of eight characters. Thus, a monospaced font and
proper tab settings are recommended for best readability. On Windows, it may be
necessary, too, to replace LF with CR LF (which will increase the file sizes),
or to use a compatible text viewer or editor.
4. FEEDBACK
This collection of tables is developed and maintained by Gert Willmann. Please
send comments, questions, or bug reports to my e-mail alias at 'ieee.org'; the
mailbox or user-name, respectively, is 'gert.willmann' (this indirect specifi-
cation merely serves to prevent spam).
Alternatively, or if e-mail doesn't work, my postal address is:
Ecklenstrasse 27 B
70184 Stuttgart
Germany
5. LICENSE
Copyright (C) 2017 Gert Willmann
All files included in this collection are free software; they can be redis-
tributed and/or modified under the terms of the GNU Lesser General Public
License (LGPL) as published by the Free Software Foundation, either Version 3
of the License or (at your option) any later version.
These files are distributed in the hope that they will be useful, but without
any warranty; without even the implied warranty of merchantability or fitness
for a particular purpose. See the GNU Lesser General Public License for more
details.
A copy of the GNU Lesser General Public License should be included in each
package of this collection. If not, see .