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Resistors in Analog Circuitry

A resistor is conceptually simple – it has two terminals and a single property: the resistance between them. Traditionally electronic engineers visualise resistors as colour-coded cylinders with axial wires, although today they are more often rectangular or cylindrical surface-mount devices (SMDs).

Figure 1

Practically a resistor is a lot more complex. It has capacitance and inductance, its resistance will vary with temperature and may vary with applied voltage too, its connections, whether wire or SMD pads, contain thermocouples, it has a fundamental noise source which cannot be eliminated and quite possibly others which may be current dependent, and it has a power rating.

Figure 2

All these characteristics can affect the behaviour of high precision or high frequency analog circuitry. This article discusses the effects of the various imperfections of resistors on circuit function, and how they may be minimised. It is not intended as a dissertation on various types of resistor, although we shall briefly summarise the characteristics of the commoner types in the last section.

Resistor precision
Precision analog circuitry uses resistors to program the gain of amplifiers and to convert current to voltage. For current to voltage conversion the precision of the conversion depends on the absolute accuracy of the resistance, and its stability over temperature. It is therefore necessary to have resistors which are both precise and with low temperature coefficients (TCs). Some examples are shown in Fig 3.

Figure 3

Some 2- and 4-terminal current sense resistors.

Where high currents are involved the resistance required will be low but even then the dissipation may be quite large so resistors intended for high current measurements may be mounted in packages designed for mounting on a heatsink. Some types have heatsinks which are electrically isolated from the resistance – if these are required the insulation breakdown voltage is another important parameter.

If the current sense resistance is low the lead resistance may be a significant source of error and it is common to have "Kelvin" connections where the current flows through one pair of leads and the voltage across the resistor is sensed with a second set in which no significant current will flow. Such devices are, of course, more complex and so more expensive – Fig 4 shows how careful board layout can often make them unnecessary.

Figure 4

Kelvin connection of a two-terminal resistor.

Medium accuracy current sense resistors often use thick film or wire-wound technology, with resistance accuracy of 1-5% and TC of 20-80 parts per million (ppm) per K. High precision sense resistors use metal foil resistors and can meet specifications as high as 0.1% accuracy and 5 ppm TC.

Resistors with accuracies of 0.1% are quite expensive. However if we wish to make an amplifier with a gain of 1000±1 this is the matching accuracy that we shall need. Suppose that the amplifier in Fig 5 is to be made with a pre-calibration accuracy of 0.1%, then R1 and R2 must be matched to 0.1% (note that this does NOT mean that two 0.1% parts will do – if R2 is 0.1% high and R1 is 0.1% low then they are in fact matched to 0.2% and that is approximately what the gain error will be). To obtain guaranteed accuracy of 0.1% without testing or calibration requires 0.05% resistors or better.

Figure 5

Non-inverting amplifier circuit with gain, G, of 1 +

To simplify our analyses in this article we shall assume that all op-amps and other active devices are perfect and do not contribute any errors to the circuits being analysed.

0.1% accuracy is only 10-bits. Today "high precision" is unlikely to be less than 12-bits and will probably be higher – 14- or 16-bits or even more. 16-bits requires matching of better than 15 ppm. Even if our resistors are perfectly matched at one temperature they will not necessarily still be matched at another. Consider the result of R1 and R2 having TCs which differ by 5 ppm/°C. If the temperature changes by only 20°C then the matching of the resistors will change by 100 ppm, over 6½ LSBs at 16-bits. Not only is the resistance match important – so is the TC match, and the TCs of resistors may not be very precisely specified.

Even if the TCs are matched the temperatures of individual resistors may not be and this is another source of potential error. When designing precision circuitry critical resistors should be located well away from heat sources.

Figure 6

Resistor self-heating.

Even without external heat sources resistor self-heating can also affect matching. Consider a non-inverting amplifier with R1 and R2 quarter watt resistors of 100 Ω and 10 kΩ respectively, with an exact (no errors) resistance ratio and perfectly matched TCs of 10 ppm/°C. If the amplifier input voltage is zero then the voltage across each resistance is also zero and they will have no dissipation. However if the input voltage is 100 mV the output will be 10 V and the dissipation in R1 and R2 will be 100 µW and 9.8 mW respectively. A ¼W resistor has a thermal resistance to ambient of the order of 200°C/W, so 9.8 mW in R2 will produce a temperature rise of about 2°C, giving rise to 20 ppm resistance change, while the rise, and change, in R1 is negligible. The matching will change by 11/3 LSB at 16-bits.

Figure 7

A resistor is a thermocouple.

But thermal effects can go even further. If the opposite ends of a resistor are at different temperatures there is yet another phenomenon to consider – the Seebeck effect. When two dissimilar conductors are in contact there is a small potential difference between them. This potential difference is a function of temperature so if we arrange for two conductors to join at two points which are at different temperatures there will be a net electromotive force (emf) in the circuit. Such a device is called a thermocouple.

Since a resistor consists of a length of resistive material with metal leads or contacts it will behave as a thermocouple and if its ends are at different temperatures there will be a potential difference across it. For copper and nichrome this thermocouple has an output of about 40 µV/°C but for carbon resistors it may be as high as 400 µV/°C. There are specially-made resistors with very low Seebeck potentials, but they are somewhat expensive.

Figure 8

Correct placement of high precision resistors with respect to a heat source.

The placing of resistances with respect to heat sources and air flow from fans is obviously important. As is illustrated in Fig 8 critical high precision resistors should not be close to a heat source; the two terminals of a resistor must not be at different distances from it as this will produce a temperature differential and cause thermocouple effects; resistors whose matching is important must be at equal distances from it; and leaded resistors should not be mounted with their axis normal to the PCB as this results in different thermal properties for opposite ends of the resistor and again may cause a temperature differential. This last precaution applies even in the absence of heat sources if there is any possibility of the resistor self-heating.

Figure 9

Correct placement of high precision resistors with respect to air flow.

Fig 9 shows correct and incorrect alignment of high precision resistors with respect to airflow from cooling fans. Again the governing principle is to avoid temperature gradients between terminals and to ensure that all matched resistors are at the same temperature.

The precautions in Figs 8 & 9 apply only to high precision resistors and are not necessary in low precision applications, such as bias, load or current-limiting resistance circuits.

From these analyses we see that if the precision of an analog circuit depends on resistors it is necessary to consider their accuracy, the accuracy with which two or more resistors are matched, the matching of their TCs, the matching of their actual temperatures, and temperature differences between their terminals.

When resistors are used in subtractor amplifiers and digital-analog converters (DACs) the absolute accuracy of the resistors does not necessarily need to be particularly high – but the accuracy of the ratio (or "matching") of two or more resistors may need to be within a few ppm.

In order to achieve high matching accuracy with a number of discrete resistors it is necessary to have high absolute accuracy, which is expensive, or to select resistors, which is expensive, time-consuming and impracticable for mass production.

Figure 10

A network of resistors on a common substrate.

The usual way of achieving accurate matching of resistance, TC and resistor temperature is to use integrated resistor networks in which all the resistors are made using the same process, which matches TCs, are deposited on the same substrate, which tends to match resistor temperatures, and may be trimmed to optimise resistance matching.

A number of technologies may be used to make such networks: thick film, thin film, and foil. Commonly-used substrate materials include plastics, glass, ceramics, sapphire and silicon. For applications where there is a relatively complex circuit involving active analog circuitry and precision resistance networks the best technology is silicon-chromium (SiCr) thin film resistors fabricated directly on the oxide of the silicon chip carrying the analog circuitry. These resistors have quite good initial matching precision (>> 12-bits) and may be laser trimmed for even better performance.

The films are of the order of 50 atoms thick and so statistical variation in thickness affects the achievable matching accuracy of very (dimensionally) small resistors. In larger ones this variation averages out – but of course larger resistors increase chip size, and so cost. At some point it becomes more economic to laser trim rather than to make larger resistors.

It is also less demanding (and so cheaper) to optimise matching and TC and not the absolute accuracy of the resistors – this is why it is not uncommon for resistors in a DAC to be matched to a few ppm but have a specified tolerance of as much as 5% or even 10%.

Figure 11

A network of resistors on an analog IC – the AD5543 16-bit DAC.

This is a mature technology and delivers high precision and stability at quite low cost. It is ideal for devices such as digital-analog converters and analog-digital converters (DACs and ADCs) and instrumentation amplifiers ("in-amps"). Using this technology Analog Devices Inc. makes DACs and ADCs with resolutions to 18-bits and in-amps with common-mode rejection ratio (CMRR – which is a measure of resistance matching) of more than 110 dB.

Figure 12

Typical precision resistor networks.
(Not all to the same scale.)

Thin film on silicon is less suitable for simple passive networks required in small quantities as the design and set up costs overwhelm the low per device manufacturing cost. For such purposes simple passive networks, made in both thick and thin film on ceramic substrates are available in a variety of surface mount and through hole packages from a number of resistor manufacturers. For up-to-date details of the limits of their technologies you should visit their websites.1 A summary of resistor technologies will be found in Appendix I.

High frequency effects
The inductance of resistors can vary quite widely. Carbon rod resistors have quite low inductance, but some types of film resistor have a spiral track around the cylindrical resistor substrate and have more substantial inductance. Wire-wound resistors are nothing more or less than a coil – although some do have separate sections wound in opposite directions to minimise their inductance.

In high frequency circuits resistors can have two different functions – to provide a well-defined resistance at high frequency for purposes of termination or matching, or to limit DC current in the HF circuit. In the latter case the capacitance of the resistor may be important but the inductance will probably not matter and spiral film or wire wound resistors may still be suitable.

Where HF reactance does matter we must use low inductance and low capacitance resistors designed for high frequency applications. Resistor manufacturers rarely specify the inductance of their resistors, even of parts specifically sold as "low inductance", but do specify the VSWR, over some frequency range, of low value (usually 50 Ω or 75 Ω but sometimes a bit higher) resistors sold for terminating transmission lines and RF loads. The data sheets have "low inductance" on the front page but, in general, no actual value of inductance is given in the detailed specifications.

However given that 1 cm of thin wire has an inductance of about 7 nH we can reasonably suppose that small resistors are unlikely to have inductances below this, and if they are "low inductance" a maximum of 100 times this, 0.7 µH, seems unlikely to be exceeded. So for a working frequency of F MHz it is probable that the inductive impedance of a "high frequency" resistor will lie between 0.044 F Ω and 4.4 F Ω. To a first approximation this adds to the resistance so at 100 MHz our worst-case inductance will add only 440 Ω to the total impedance.

Similarly the shunt capacitance seems likely to be between 100 fF and 1 pF and so the capacitive reactance will lie between 1.6 MΩ/F and 0.16 MΩ/F. The capacitance, of course, shunts the resistance, so with worst-case capacitance and 100 MHz the resistance will be shunted by 1600 Ω. Furthermore in many cases the stray capacitances of the printed circuit board (PCB) on which the resistor is mounted will be far larger than the capacitance of the resistor itself – but this is less likely to be true of its inductance.

These assumptions are very crude, but they do show that there is quite a wide range of resistance and working frequency where it is not necessary to be concerned about the exact reactances of our resistors, so long as we are careful not to use spirally-cut or wirewound resistors in applications where inductance may be a concern.

Resistor noise

Figure 13

Noise in resistors.

The basic physics of resistance shows2 that the random thermal movement of charge carriers in a conductor always produces electrical noise of value where K is Boltzmann's Constant (1.38065 x 10-23 J/K), T is the absolute temperature, B is the bandwidth and R the resistance. We often express this noise in terms of spectral density, making a resistor's voltage noise

Such noise is known as thermal noise, or Johnson noise, after John B Johnson (Bell Telephone Laboratories – 1928) who was the first person to observe it.3 It might more properly be named Johnson/Nyquist noise as it was Harry Nyquist (the Nyquist of "the Nyquist Criterion" – also at Bell Laboratories) who explained the physical basis of Johnson's observation.4

There is another fundamental type of noise known as "shot noise". Shot noise is caused by random fluctuations in the rate at which discrete charge carriers cross a potential barrier (such as a semiconductor junction or the cathode/anode circuit of a thermionic tube) and is proportional to the square root of the current across the potential barrier. The current noise spectral density when a current of I amperes is flowing is

where q is the electronic charge (1.602 × 10-19 C). Thermal noise and shot noise are both the result of quantum fluctuations, but resistors do not exhibit significant shot noise since they do not have discrete potential barriers. This statement is a simplification, and indeed it is possible to analyse thermal and shot noise together as a single more complex phenomenon, but it is true for almost all practical resistor applications.

So anywhere at all that there is a resistor or resistance in a circuit, whether it is carrying current or not, there is a noise generator with an output voltage noise spectral density of . We can reduce the noise by reducing the resistance (this may increase current and/or power), but reducing the temperature is not usually practicable (if we cool a resistor from room temperature (298 K) to liquid nitrogen temperature (77 K) its noise voltage is still more than half its room temperature value). Thus, despite occasional requests from the ill-informed for "good" resistors (i.e. ones without thermal noise) thermal noise in resistors is a fundamental phenomenon with which electronic engineers must live. It is not possible to change Boltzmann's Constant because Professor Boltzmann is dead5.

Figure 14

Professor Ludwig Boltzmann.

Metal and oxide thin film and foil resistors usually have noise which is very close to their theoretical thermal noise, but some carbon composition, carbon film and thick film resistor materials, especially very high resistance ones, may have additional noise (known as, and often specified as, "excess noise") which is a function of current For all applications where low resistor noise is important it is sensible to use metal or oxide thin film or metal foil resistors despite their slightly greater cost.

Resistor technologies
Wire wound resistors

Figure 15

Basic structure of wire wound resistors.

The very earliest resistors used resistance wire, lengths of thin wire of an alloy ("Nichrome" – a group of alloys of nickel (≈80%) and chromium (≈20%) with a resistivity of the order of 105 µΩcm – is one of the most commonly used). Such resistors are still used today.

The wire, insulated with enamel, is wound on a former (usually ceramic, but sometime glass or plastic) with wire ends and protected with a further coating of paint, plastic or enamel. Such resistors are quite inductive, even if adjacent sections of winding are wound in opposite directions to reduce inductance, but can be made very accurate and very stable. They have no noise beyond their basic thermal noise and can be made large, with very high power (dissipation) ratings. To increase dissipation still further some high power wire wound resistors are assembled in (but insulated from) a metal tube which can be attached to a heat sink. In general wire wound resistors can be high power or high precision, but rarely need to be both. Because the resistivity of alloys is never very high the maximum resistance of wirewound resistors rarely exceeds 100 kΩ.

Carbon composition resistors

Figure 16

Carbon composition resistors.

The first low cost mass-produced resistors used this technology, which has today largely been replaced by film resistors. The resistive element consists of a rod made from finely divided carbon and an inert filler, bound by a resin. The ratio of carbon to filler determines the resistivity. Connection is made either by metal end caps, or by embedding axial wires at each end of the rod during manufacture. The resistance may be adjusted during manufacture by removing material, either by abrasion or by cutting a shallow spiral along the rod. Originally the resistors were sealed in ceramic protective tubes, but later plastic encapsulation was more common.

Carbon composition resistors have poor accuracy and stability and tend to be noisy. Some, especially very high resistance types (≥4.7 MΩ) have non-linear resistance, their resistance decreasing with applied voltage. But they are cheap and easy to manufacture and they dominated the resistor market for most of the first half of the 20th Century.

Film Resistors

Figure 17

Two common trimmable film resistor architectures.

Thick film and thin film are two different technologies where resistive films are attached to insulating substrates – most commonly ceramic, but sometimes glass or plastic, and, in the case of thin films, silicon integrated circuits. The difference between them is not, as the name would suggest, the actual film thickness but the means by which it is applied. Although thick films are generally thicker than thin films this is by no means always the case.

Both thick and thin films are usually applied to flat substrates or to cylindrical rods. Both may be trimmed by laser, chemical etching or abrasion, and both technologies may be used to make networks of two or more resistors on flat substrates. Small flat substrates carrying a single resistor have their ends metallised to make surface-mount resistors, while the larger networks may be configured as multi-connection surface-mount or wired devices.

Cylindrical film resistors may have axial leads embedded in the substrate to make classical wire-ended axial resistors, or have their ends metallised to form surface-mount MELF (Metal Electrode Leadless Face) resistors. When cylindrical film resistors are made with a spiral groove to increase resistance they will have appreciable inductance, but low inductance types are also available.

Figure 18

Cylindrical film resistors.


Figure 19

Rectangular surface mount film resistors.

Thick film

Thick film resistance material consists of a binder in a liquid carrier with conductive/resistive particles of carbon and a filler as in carbon composition resistors; or ceramic conductors such as bismuth iridate (Bi2Ir2O7), bismuth ruthenate (Bi2Ru2O7), lead oxide (PbO), ruthenium dioxide (RuO2) or tantalum nitride (TaN); or nichrome. It is applied by screen printing or spraying and then dried and cured. After curing precision resistors may be adjusted by abrasion or laser trimming but many thick film networks are then used without adustment.

In the case of carbon the binder may be a resin which is cured at relatively low temperatures, and yields a cheap resistor or network with relatively poor accuracy, stability and noise. The other materials are often bound with a low melting point glass which is fused at temperatures above 800°C. These resistors, though not as good as thin film or foil resistors, can be made with good accuracy, stability and noise and quite low TCs.

Thin film

Thin films of resistive alloys or the conductive compounds mentioned above are applied to their substrates in a vacuum or a low pressure plasma by sputtering or various other vacuum deposition techniques. The resulting films, which again may be used as is or may be trimmed by laser, etching or abrasion, are more stable but more expensive than thick films and have lower TCs. Thin film networks can have resistors with matching to a few ppm and TC matching as low as 0.1 ppm/°C.

As was mentioned previously, this is the technology used to manufacture thin film resistors on integrated circuit chips.

Metal foil

Figure 20

Metal foil resistors.

This technology is another where a metallic resistive material lies on a ceramic substrate. Its particular feature is that its resistive TC is balanced by the strain gauge effect of differing thermal expansion of the foil and the substrate, giving a resultant TC which can be as low as 0.05 ppm/°C. Coupled with resistor accuracies of 0.001% this makes such resistors the highest performance parts available – but they are not cheap.

Since the foil is thick compared with thin films the resistivity is comparatively low so the resistors have high aspect ratios (length/width ratios) and cannot go into the smallest packages.

Type Benefits Disadvantages
Bulk metal foil Very high precision (0.001%)
Very low TC (<0.2 ppm/°C)
Extremely low matching TC (<0.02 ppm/°C)
Low inductance
Quite large
Thin film on a silicon IC Very accurate matching (0.001 %)
Very low matching TC (<0.05 ppm/°C)
Well-matched to its associated circuitry
Very small size
Low cost
Relatively poor absolute accuracy
(often ≈5% but can be 0.5%)
Expensive for low volume specials
Thin film on ceramic or glass High precision (0.005%)
Low TC (<50 ppm/°C)
Low matching TC (<1 ppm/°C)
Larger than monolithic ICs
Thick film Low cost
Medium precision (0.05%)
Medium TC matching (5 ppm/°C)
High TC (>100 ppm/°C)
Can be noisy
Note that figures are for the best of each type, there are networks for applications where high precision is not necessary which are much lower performance – and much cheaper. Never over-specify, it wastes money.

Table 1
Comparison of resistor networks

RTDs, thermistors and resistors with well-defined TC

There are a number of applications for resistors whose resistance varies with temperature. This is properly the subject of a separate paper but it is worth mentioning that there are three common types: RTD (or platinum resistance thermometers) which are resistors made of platinum, which has a very well-defined resistance/temperature characteristic6 and is used for accurate temperature measurement over a wide temperature range; thermistors, which are ceramic resistors with large non-linear resistance/temperature variation (which may be positive or negative depending on type) widely used for inexpensive temperature measurement, overheating and inrush protection, and as simple thermostats 7; and positive TC resistors which have well-defined and fairly linear temperature coefficients and are used in a variety of analog circuits to stabilise gain over temperature and perform other similar functions.8

Light Dependent Resistors (LDRs)

A film resistor made of selenium or some other materials and packaged in a transparent encapsulation has a resistance which is a function of light level and acts as a type of photocell. As with thermal resistors this is properly the subject of a separate paper, but since light levels vary very widely it should be noted that the resistance of such devices also varies widely. The conductance (1/R – see appendix 3) of an LDR is usually proportional to the incident light level.

Power dissipation
The final rating of a resistor which we must consider is its power rating. All resistors have a steady-state power rating, which is the amount of power that they can dissipate for an indefinite time without damage caused by overheating. The exact specifications depend on the individual manufacturer and resistor type but the general rule is that below some specified ambient temperature the permitted dissipation is a fixed value, but above that temperature it must be derated linearly to some maximum temperature. Sometimes the threshold is room temperature, 25°C, sometimes it is higher. In the chart in Fig 21 the threshold is 70°C and the maximum temperature is 155°C.

Figure 21

Typical resistor thermal derating chart.

Axial wire-ended resistors often have their size defined by their power rating – one buys a 1/8 W, ¼ W, ½ W, 1 W or whatever (approximate sizes are listed in appendix 2), but surface mount are usually defined by chip size and their power rating determined from their data sheet. In general the larger the chip the higher the power rating – 0603 chips usually have a rating about 1/8 W.

High power resistors may have integral thermal fins, or means of thermal attachment to a heat sink, or both – and their dissipation rating will be affected by the heat sink used. There are even water-cooled ones – but these are rarely used in precision circuitry!

Resistors have significant thermal mass, so the dissipation rating may be exceeded briefly if the temperature of the resistor does not rise beyond its specified limits. With small resistors this will happen quickly since its thermal mass is small but with larger ones the data sheet may specify a thermal time constant such that if the mean power in that time does not exceed the maximum the instantaneous power may do so. Some resistors may also have a one-off pulse rating for the amount of energy they can handle in a single brief event, such as the charge or discharge of a capacitor or the dumping of energy from an abruptly switched inductor – such an event should not be repeated until the resistor has returned to near ambient temperature.

Voltage rating
In addition to their power rating resistors have a voltage rating. If this is exceeded the resistor structure may break down, resulting in permanent damage to the device.

This rating has nothing to do with power dissipation - an axial 1 MΩ ¼W resistor needs 500 V to achieve a quarter of a watt dissipation, but the rating of such a device is usually 100-200 V which is probably less than US line voltage (≈170 V pk) and certainly much less than European (≈325 V pk).

This rating is even more important with small surface-mount resistors – their rating can be as low as a few tens of volts and breakdown may be to a low resistance, damaging other components. Always check resistor voltage ratings in any application where a resistor will see more than 50 V pk, and especially in ones where it may have line voltage applied to it.

We have seen that the common resistor is not quite such a simple component as it appears. To optimise any circuit containing resistors (and they almost all do) we must understand the strengths and weakness of the many different types and select the one best suited for our application.

James M. Bryant
Midsomer Norton
November 2007


Preferred Resistor & Capacitor Values

There is an international standard (IEC60063) which specifies the values of resistors and capacitors by subdividing each decade into a number of steps of roughly equal ratios. If we take the six values in the E6 series any arbitrary value can be replaced by a standard value which will be within 20%.

Similarly the twelve values in the E12 series allow 10% and the twenty four in E24, 5%. The E24 series contains the values for the E6 and E12 series unaltered.

The E6, E12 and E24 series
E6, E12 & E24 E24 E12 & E24 E24
10 11 12 13
15 16 18 20
22 24 27 30
33 36 39 43
47 51 56 62
68 75 82 91

Table 2
The E6, E12 and E24 component values

Table 3
E12 and E24 resistor ratios

Table 3 shows the ratio of pairs of E12 and E24 resistors and may be used to select the pair which is closest to a particular ratio. The optimum is found by inspecting the diagonal band of values close to the required ratio.

Above E24 there are the E48, E96 and E192 series, which allow 2%, 1% and 0.5% (approximately) but these series do not contain the exact values from the smaller series but closer approximations.

The E48, E96 & E192 series
E48, E96 & E192 E192 E96 & E192 E192 E48, E96 & E192 E192 E96 & E192 E192
100 101 102 104 105 106 107 109
110 111 113 114 115 117 118 120
121 123 124 126 127 129 130 132
133 135 137 138 140 142 143 145
147 149 150 152 154 156 158 160
162 164 165 167 169 172 174 176
178 180 182 184 187 189 191 193
196 198 200 203 205 208 210 213
215 218 221 223 226 229 232 234
237 240 243 246 249 252 255 258
261 264 267 271 274 277 280 284
287 291 294 298 301 305 309 312
316 320 324 328 332 336 340 344
348 352 357 361 365 370 374 379
383 388 392 397 402 407 412 417
422 427 432 437 442 448 453 459
464 470 475 481 487 493 499 505
511 517 523 530 536 542 549 556
562 569 576 583 590 597 604 612
619 626 634 642 649 657 665 673
681 690 698 706 715 723 732 741
750 759 768 777 787 796 806 816
825 835 845 856 866 876 887 898
909 920 931 942 953 965 976 988

Table 4
The E48, E96 and E192 component values


Resistor Marking & Colour Coding

Figure 22

Surface-mount resistor marking.

Surface mount resistors may be so small that they are not marked. When they are marked they are usually marked with three or four printed digits. The first two or three are the value of the resistor and the last is the multiplier – that is power of ten by which the first two or three digits is multiplied (putting it another way, it is the number of zeros after the digits). When there are no zeros (three digit resistors in the range 10-91 Ω or four zero types between 100-988 Ω) there is a risk of ambiguity since, for example, 100 might be a hundred ohms or ten ohms. This is often avoided by using R, K or M and two or three digits instead of a multiplier, for example 10R instead of 100 for 10 Ω. The letter does not need to be the last digit: The forms R68 (0.68 Ω), 6R8 (6.8 Ω), 68R (68 Ω), M47 (470 kΩ), 33K6 (33.6 kΩ) are all permissible. R is used rather than the Ω sign because Ω might be confused with a slightly worn zero.

Surface mount resistors marked with one to four zeros (0 - 0000) are usually short-circuiting "zero-ohm links" or "jumpers".

Cylindrical resistors are most commonly marked with their values by means of a colour code, although the details are sometimes simply printed on them. The usual colour scheme is defined in the EIA9 standard EIA-RS-279 and IEC 60425. The marking consists of from three to six coloured bands around the resistor. These are numbered from the band nearer to an end. IEC 60425

The simplest code, with just three bands, gives the value of the resistor by means of two digits and a multiplier – such resistors have resistance tolerance of ±20%. Four band resistors have an additional band defining the resistance tolerance, and five bands are used with higher precision resistors to specify resistance with three digits, a multiplier and a tolerance band. The most complex coding, with six bands, adds the TC to the other parameters.

The scheme is summarised in Table 5. Note that the multiplier indicates different ranges depending on whether the resistor is three or four band or five or six band (for example a red multiplier on a three or four band resistor indicates resistance in the range 1.0 - 9.1 kΩ, but on a five or six band resistor the range with this multiplier is 10.0 - 98.8 kΩ.

Table 5
Resistor colour codes

In military standard resistors with printed coding the tolerance may be indicated by a letter instead of a percentage tolerance. These may be decoded as follows

Letter K J G F D C B
Tolerance 10% 5% 2% 1% 0.5% 0.25% 0.1%

Table 6
Tolerance letter equivalents.

Rating ¼ W ½ W 1 W 2 W
Length 3.8 mm / 0.15" 6.2 mm / 0.244" 9.0 mm / 0.354" 14.3 mm / 0.56"
Diameter 1.68 mm / 0.066" 2.3 mm / 0.091" 3.0 mm / 0.118" 4.83 mm / 0.23"

Table 7
Axial resistor sizes v power ratings (approximate).


Basic resistance formulae

Ohms Law is


where V is voltage, I is current and R is resistance.

The resistance of a bar of length L and uniform cross-sectional area A made of a material with resistivity ρ is


this formula may be used to calculate the resistance of wires and PC tracks if we know the resistivity of the material involved.

From this we can deduce that the resistance of a square of a sheet conductor measured between two opposite edges is


where T is the thickness of the sheet. This is why the resistance of sheet material of defined thickness (such as PC metallisation or thin film resistor material) is expressed in "ohms per square" since the size of the square is immaterial.

The power dissipation in a resistor is given by


Where the voltage and current are AC the values of V and/or I must be the root mean square (RMS) values, not the simple arithmetic mean. Remember that it is possible to exceed the voltage rating of a resistor without exceeding its power rating – don't!


R = R1 + R2 + R3[5]

Figure 23

Resistors in series

When resistors are connected in series the total resistance is simply the sum of the individual resistances. When resistors are connected in parallel their conductances are added.

Conductance, G, is simply the reciprocal of resistance


Its SI unit is the siemens [note the lower-case s in the name] after the German inventor Ernst Werner von Siemens, the symbol is S [in upper-case – just to be difficult – and not to be confused with the symbol for a second, which is lower-case]. The siemens used to be called the mho ["ohm" spelt backwards] with an inverted omega, , as its symbol but it was renamed by the International Committee of Standards in 1971.

G = G1 + G2 + G3[7A]


Figure 24

Resistors in parallel

For the simplest case of two parallel resistors, R1 and R2, the resulting resistance is given by


but this formula cannot simply be extended to larger numbers of resistors.

Kirchoff's Current Law (often called "Kirchoff's Law" – but there are others, he was a versatile man) states in its simplest form that the net current at any node in a circuit must be zero ("what goes in must come out again"). We can use this to analyse networks of resistors. While complex networks are beyond the scope of this appendix, they are all extensions of the simple case of three resistors and three voltages shown in Fig 25.

Figure 25

A network of three resistors

The voltage at the junction of the three resistors is V. Since the net current at this node must be zero by Kirchoff's Current Law


1 The websites of a number of manufacturers of precision networks are given below:


3 Phys. Rev. 32, 97 (1928)

4 Phys. Rev. 32, 110 (1928)

5 Ludwig Boltzmann died on the 5th September 1906 at Diuno near Trieste, and is buried at Zentralfriedhof, Vienna.

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8 Precision Resistor Co positive TC resistors


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