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 fundamental question how, for example, an arbitrary resistance
value can be realised or at least closely approximated with standard resistors.
Two commonly applied strategies are based on multiple standard resistors which
are either connected in series or in parallel; and in case of just two resis-
tors, this is already everything that is possible. When allowing more than two
resistors, however, there are also a few more generalised topologies which can
be of practical interest.
Another, closely related question arises when a given ratio between the values
of two components is to be realised. This is of interest, for example, in con-
nection with voltage dividers, in which case the ratio between two resistance
values is the crucial factor rather than the resistance values themselves. Such
an arbitrary ratio can be realised or approximated by simply picking an appro-
priate pair of standard values or, if this is not accurate enough, by using one
or two combinations of standard values.
Unfortunately, the necessary calculations to find optimum solutions are not as
simple or straightforward as it might seem at first glance. This is because the
underlying problems belong to the class of so-called integer optimisation prob-
lems, which are inherently hard to solve and, even worse, 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 ratios, and this is exactly
the purpose of this collection.
2. TABLES OF COMBINATIONS
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 combinations of three resistors that have resis-
tance 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
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:
- 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. TABLES OF RATIOS
This additional set of tables deals with ratios between two values or combina-
tions of values from the E series of preferred values. In order not to render
these tables too complex, however, combinations are restricted to the two basic
types A and B described above, and ratios involving both types of combinations
are not taken into consideration.
The ratio between two such values or combinations of values, respectively, can
then be expressed by the following fraction:
- Z = X / Y
The numerator X and the denominator Y of this fraction are in turn combinations
of type A or B. Thus, for combinations of type A:
- X = x1 + x2 + ... + x_nx
- Y = y1 + y2 + ... + y_ny
And for combinations of type B:
- 1/X = 1/x1 + 1/x2 + ... + 1/x_nx
- 1/Y = 1/y1 + 1/y2 + ... + 1/y_ny
Here, x1, x2, ..., x_nx are nx arbitrary values from a particular E series, and
X is the resulting value of the respective combination. Similarly, y1, y2, ...,
y_ny are ny arbitrary values from the same E series with the resulting value Y.
Z is the ratio between the two resulting values.
Combinations of type A apply again 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 are of minor practical importance, though. Combinations of less than
nx or ny values are seamlessly included, in which case some of the x or y values
are either zero (type A) or infinite (type B).
In order to limit the potentially infinite number of possible numerator or
denominator 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 again excluded.
For convenience and ease of use, the resulting tables of feasible ratios are
sorted in ascending order of the resulting ratios Z, and these ratios are
normalised such that they generally fall into the base range from 1 to 10, thus
1 <= Z < 10. Moreover, the numerators X are further normalised such that they
fall into the base decade from 100 to 1000, thus 100 <= X < 1000. This does not
introduce any loss of generality because X and Y can be independently scaled up
or down by factors of 10, 100, 1000, and so forth, which allows to adjust the
magnitudes of the numerators and denominators and, in turn, those of the result-
ing ratios as well.
All this is best illustrated with ratios that are realised using three resis-
tors in total, one single resistor and a two-resistor combination. The under-
lying resistance values are rx1 and rx2 for the numerator and ry1 and ry2 for
the denominator, respectively, with resulting resistance values of RX and RY.
The maximum allowed ratio between the largest and smallest value of a combi-
nation is M.
- Ratios involving combinations of type A:
____ rx1 ____ ry1 ____ ry2
o-----|____|-----o and o-----|____|-----|____|-----o
RX = rx1
RY = ry1 + ry2
____ rx1 ____ rx2 ____ ry1
o-----|____|-----|____|-----o and o-----|____|-----o
RX = rx1 + rx2
RY = ry1
- Ratios involving combinations of type B:
____ ry1
____ rx1 +-----|____|-----+
o-----|____|-----o and o-----+ ____ ry2 +-----o
+-----|____|-----+
RX = rx1
1/RY = 1/ry1 + 1/ry2
____ rx1
+-----|____|-----+ ____ ry1
o-----+ ____ rx2 +-----o and o-----|____|-----o
+-----|____|-----+
1/RX = 1/rx1 + 1/rx2
RY = ry1
The resulting ratios Z = RX / RY and their numerators RX are subject to the
constraints 1 <= Z < 10 and 100 <= RX < 1000. Moreover, the combinations are
subject to the additional constraints 1 <= max(rx1,rx2) / min(rx1,rx2) <= M and
1 <= max(ry1,ry2) / min(ry1,ry2) <= M, respectively. Note, however, that the
total number of values that is actually used for realising an arbitrary ratio
can vary from n = 2 (just one pair of values) up to n = nx + ny = 4, depending
on the considered E series; nx and ny are the number of numerator and denomina-
tor values, respectively.
The following E series are explicitly addressed:
- E6 series: ratios up to n = nx + ny = 4
- E12 series: ratios up to n = nx + ny = 3
- E24 series: ratios up to n = nx + ny = 3
- E48 series: ratios of two values
- E48+E24 series: ratios of two values
- E96 series: ratios of two values
- E96+E24 series: ratios of two values
The combined E48+E24 and E96+E24 series are treated as described above.
4. 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' (2.8 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
- E6_Ratios_2.txt
- E6_Ratios_3_A.txt
- E6_Ratios_3_B.txt
- E6_Ratios_4_A.txt
- E6_Ratios_4_B.txt
- GNU_GPL_V3.txt
- GNU_LGPL_V3.txt
Folder 'E12_Combinations' (4.8 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
- E12_Ratios_2.txt
- E12_Ratios_3_A.txt
- E12_Ratios_3_B.txt
- GNU_GPL_V3.txt
- GNU_LGPL_V3.txt
Folder 'E24_Combinations' (12.8 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
- E24_Ratios_2.txt
- E24_Ratios_3_A.txt
- E24_Ratios_3_B.txt
- GNU_GPL_V3.txt
- GNU_LGPL_V3.txt
Folder 'E48_Combinations' (1.3 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
- E48_Ratios_2.txt
- E48_E24_Ratios_2.txt
- GNU_GPL_V3.txt
- GNU_LGPL_V3.txt
Folder 'E96_Combinations' (3.7 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
- E96_Ratios_2.txt
- E96_E24_Ratios_2.txt
- GNU_GPL_V3.txt
- GNU_LGPL_V3.txt
All read-me files are again included in PDF and in text form. The individual
tables of feasible combinations or ratios are stored in text files that have
systematically coded file names of the form:
- _Combinations__.txt
- _Ratios_.txt
- _Ratios__.txt
The string specifies the underlying E series or the combined pseudo-
series, specifies the number or the maximum number, respectively, of values
used in a combination or for a ratio, and , if applicable, specifies the
type of combinations.
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.
5. 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
6. 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 .