Resistor Values

Why resistors have odd values.

New: 1:06 PM 29/12/10

There are quite a few things that seem a bit odd when you are learning about electronics.

One of the first odd encounters is with the values of resistors which seem to have a range of values that don't make much sense. Why not just have a selection of resistors and capacitors in linear value steps like 1, 2, 3, 4 ...?

Well, initially that was how they did it, and in pre-WW2 equipment you will find resistor values like 250 and 750 ohms, and capacitor values like 0.5uF.

Today resistors are pretty good, but they still come with a tolerance or error band on their value. This is given as a percentage variation on the marked value, so a 1000 ohm resistor with a tolerance of 10% could have an actual value between 900 (1000-10%) and 1100 (1000+10%) ohms.

{In the old days a given resistor picked from the bin could be anywhere in its tolerance range, and if you sampled enough of them you would get the classic Gaussian Distribution, the Normal or Bell curve. Modern manufacture is much tighter, so while a particular batch may be anywhere within the tolerance range they are typically much closer to nominal value, and come in long runs with a very tight spread, say a thousand with a spread of only 1% on a centre value say 3% low. This has implications for buying components where you expect to sort for tighter values, and you might not get the spread that the tolerance implies.}

If our values are a simple linear number line, 1, 2, 3..., there are gaps between the low values while the higher values start to overlap. It doesn't make a lot of sense to manufacture 900 ohm resistors when your 1000 production effectively covers that value.

So after the war the value system was revised and rationalised into the E-series, E12, E24, and so on. These are the 10%, 5%, etc., tolerance ranges. In each range new values appear to fill in the gaps created by the tighter tolerance. Resistors are now commonly available down to the 1% range

It was decided to divide the scale between 1 and 10 on a percentage basis so that all the values would just meet at top and bottom tolerance, no gap and no overlap in the range. Constant percentage results in a curve of resistor values that tend to be bunched at low values and spread at high values.

This distribution follows a law called the “12th root of N”.

z=10^(1/12) ... 12th root of 10
z = 1.211528

R=z^0	R = 1.00	1.0	1	0
R=z^1	R = 1.21	1.2	1	2
R=z^2	R = 1.47	1.5	1	5
R=z^3	R = 1.78	1.8	1	8
R=z^4	R = 2.15	2.2	2	2
R=z^5	R = 2.61	2.7	2	7
R=z^6	R = 3.16	3.3	3	3
R=z^7	R = 3.83	3.9	3	9
R=z^8	R = 4.64	4.7	4	7
R=z^9	R = 5.62	5.6	5	6
R=z^10	R = 6.81	6.8	6	8
R=z^11	R = 8.25	8.2	8	2
R=z^12	R = 10.00	10	1	0

Like any manufacturer guitar amp builders tried to rationalise their component stocks as much as possible, so the resistor values actually encountered in a typical guitar amp will be a sub-set of the values above.

The Resistor Colour Code

Value	Colour	Zeros - Multiplier
0	Black	none “10's”
1	Brown	0 “100's”
2	Red	00 “K”
3	Orange	,000 “10's K”
4	Yellow	0,000 “100's K”
5	Green	00,000 “megohms”
6	Blue	000,000 “tens of megs”
7	Violet	*
8	Grey	*
9	White	*

* You simply don't encounter multipliers this size.

Tolerance Bands

Unmarked 20%
Silver 10% also used as divide by ten sub-multiplier for very low values
Gold 5% also used as divide by one-hundred sub-multiplier for very low values (rare)
Red 2% at far end means High Stability
Brown 1%

Old Resistor Marking System

Anyone working on vintage gear will eventually encounter resistors colour-coded in a way that doesn't fit with the current method. These were typically fairly large, being from about 3/4-inch to 2 inches long, and with the leadout wires obviously wrapped around each end under the coating.

{These are always found in association with the dreaded waxed-paper capacitors which are a replace-on-sight these days. Electrolytics of this vintage are also highly suspect.}

Prior to resistors being marked in bands the “body-end-dot” method was used. The colours have the same values but they are placed differently;

1st digit is the general body colour
2nd digit is the colour of one end
multiplier is the dot in the middle

The other end usually has the tolerance colour, none (same as the body) - 20%, silver - 10%, and gold - 5%.

Musical law

This has a direct parallel in music where the notes are spaced according to the twelfth-root of two (because the notes are distributed across an octave, a doubling of frequency, not a factor of ten as with resistors), however they both follow the same basic mathematical law. This naturally means that the frets on a guitar are spaced according to the same law into the desired fretboard scale length.

z=2^(1/12) ... 12th root of 2
z = 1.059463

base=440 Hz

R=z^0*base	R = 440.00
R=z^1*base	R = 466.16
R=z^2*base	R = 493.88
R=z^3*base	R = 523.25
R=z^4*base	R = 554.37
R=z^5*base	R = 587.33
R=z^6*base	R = 622.25
R=z^7*base	R = 659.26
R=z^8*base	R = 698.46
R=z^9*base	R = 739.99
R=z^10*base	R = 783.99
R=z^11*base	R = 830.61
R=z^12*base	R = 880.00

Note that this is in agreement with the note frequency table.

Tenor guitar

E	82.41
A	110.00
D	146.83
G	196.00
B	246.94
e	329.63
12th fret	659.26

Notice that the fundamental range of the tenor guitar is quite limited, from 82Hz to only 660 Hz at the 12th fret, anything above that arising from harmonics. Guitar amplifiers are built to cover from just below this range to somewhere above 5000 Hz. The exact top end of the amplifiers is a moot point since most guitar amps use 12-inch (300mm) or larger speakers which typically cut off towards 5kHz.

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