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W6KWF > TECHNI 20.09.90 22:33l 119 Lines 4302 Bytes #-12969 (0) @ WW
BID : 54207_N0ARY
Subj: THE RESISTOR E SERIES
Path: OK0NBR<OK2PEN<CX2SA<VE3CGR<WG0A<N2MH4<N2MH<N0ARY
Sent: 260921/0113z @:N0ARY.#NCA.CA.USA.NOAM [San Jose, CA] #:54207 $:54207_N0AR
THE RESISTOR E SERIES: GEOMETRIC PROGRESSION IN COMPONENT VALUES
==================================================================
In electronics, you cannot buy a resistor of every possible resistance
value. Instead, manufacturers produce standard values organized into "E
series" -- E3, E6, E12, E24, E48, E96, and E192 -- where the number
indicates how many values exist per decade (factor of 10). The spacing
between adjacent values follows a geometric progression, which
guarantees that any desired resistance can be approximated by a standard
value within a known maximum proportional error.
THE GEOMETRIC SERIES BASIS
--------------------------
For an E series with N values per decade, the ratio between successive
values is the N-th root of 10:
r = 10^(1/N)
The standard values in one decade (1.0 to 10.0) are then:
V_k = 10^(k/N) for k = 0, 1, 2, ..., N-1
Rounded to the appropriate number of significant figures, these become
the familiar E series tables. For example, E12 (N=12) gives
r = 10^(1/12) ~ 1.21, yielding the classic 1.0, 1.2, 1.5, 1.8,
2.2, 2.7, 3.3, 3.9, 4.7, 5.6, 6.8, 8.2 sequence (multiplied by
powers of 10 for other decades).
WHY GEOMETRIC SPACING MATTERS
-----------------------------
If values were spaced linearly (e.g., 1, 2, 3, 4... ohms), the relative
error would be huge at low values and tiny at high values. A 1-ohm step
is 100% error at 1 ohm but only 1% at 100 ohms. Geometric spacing makes
the *relative* (proportional) error constant across all decades.
For any target value R_target, the nearest standard value R_std
satisfies:
R_std / R_target = 10^(m/N) for some integer m
The worst-case proportional error occurs exactly halfway between two
standard values on a logarithmic scale. The maximum ratio error is:
max_error = 10^(1/(2N)) - 1 ~= ln(10) / (2N) for large N
MAXIMUM PROPORTIONAL ERROR BY SERIES
-------------------------------------
E3 (N=3) : r=2.15 max error ~= 36%
E6 (N=6) : r=1.47 max error ~= 20%
E12 (N=12) : r=1.21 max error ~= 10%
E24 (N=24) : r=1.10 max error ~= 5%
E48 (N=48) : r=1.05 max error ~= 2.4%
E96 (N=96) : r=1.024 max error ~= 1.2%
E192(N=192) : r=1.012 max error ~= 0.6%
This means with E12 (10% tolerance parts), you can always find a
standard value within +/-10% of any target. With E24 (5% parts), within
+/-5%. The proportional guarantee holds whether you are designing a
10-ohm pull-up or a 10-megohm voltage divider.
PRACTICAL EXAMPLE
-----------------
Suppose you calculate a needed resistance of 3,742 ohms. Using E12
values in the 1k-10k decade (1.0, 1.2, 1.5, 1.8, 2.2, 2.7, 3.3, 3.9,
4.7, 5.6, 6.8, 8.2 kohm):
3.742k falls between 3.3k and 3.9k.
Ratio to 3.3k: 3.742/3.300 = 1.134 (+13.4%)
Ratio to 3.9k: 3.900/3.742 = 1.042 (+4.2%)
Choose 3.9k -- error is +4.2%, well within the E12 guarantee of
+/-10%.
The same math applies at any scale. Need 37.42 ohms? Choose 39 ohms.
Need 374.2k? Choose 390k. The geometric series makes the *proportional*
decision identical in every decade.
TOLERANCE AND THE E SERIES
--------------------------
The E series number roughly matches the component tolerance class:
E3 -- 50% tolerance (historical)
E6 -- 20% tolerance
E12 -- 10% tolerance
E24 -- 5% tolerance
E48 -- 2% tolerance
E96 -- 1% tolerance
E192 -- 0.5% tolerance (and 0.25%, 0.1%)
A 5% resistor from the E24 series will have its actual value within
+/-5% of the *marked* value. Since the E24 spacing guarantees +/-5%
coverage, *any* target value can be met by some E24 part whose *actual*
value (within its own +/-5% tolerance) could hit the target. The
geometric series is what makes this coverage complete without gaps.
SUMMARY
-------
The E series uses a geometric progression (constant ratio between steps)
so that the *relative* spacing is uniform on a logarithmic scale. This
guarantees that for any desired resistance, a standard value exists
within a predictable maximum proportional error -- approximately 1/(2N)
decades, or ln(10)/(2N) fractionally. The series number N directly
tells you the worst-case percentage error: roughly 50%/N. This elegant
mathematical property is why we can design circuits with standard parts
and know exactly how close we can get.
---
Kenneth, W6KWF @ N0ARY.#nca.ca.usa.noam
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