It is easy to assume, when wrestling with electronic design, that the active devices will cause most of the trouble. This, like so much in electronics, is subject to Gershwin’s Law; it ain’t necessarily so. Passive components cannot be assumed to be perfect, and while their shortcomings are rarely discussed in polite company, they are all too real. In this chapter, I have tried to avoid repeating basic stuff that can be found in many places to allow room for information that goes deeper.
Normal metallic conductors, such as copper wire, show perfect linearity for our purposes and, as far as I am aware, for everybody’s purposes. Ohm’s Law was founded on metallic conductors, after all; not resistors, which did not exist as we know them at the time. George Simon Ohm published a pamphlet in 1827 entitled “The Galvanic Circuit Investigated Mathematically” while he was a professor of mathematics in Cologne. His work was not warmly received except by a perceptive few; the Prussian minister of education pronounced that “a professor who preached such heresies was unworthy to teach science”. This is the sort of thing that happens when politicians try to involve themselves in science, and in that respect, we have progressed little since then.
Although the linearity is generally effectively ideal, metallic conductors will not be perfectly linear in some circumstances. Poorly made connections between oxidised or otherwise contaminated metal parts are capable of generating harmonic distortion at the level of several percent, but this is a property of the contact interface rather than the bulk material and usually means that the connection is about to fail altogether. A more subtle danger is that of magnetic conductors –the soft iron in relay frames causes easily detectable distortion at power-amplifier current levels.
From time to time, some of the dimmer audio commentators speculate that metallic conductors are actually a kind of “sea of micro-diodes” and that non-linearity can be found if the test signal levels are made small enough. This is both categorically untrue and physically impossible. There is no threshold effect for metallic conduction. I have myself added to the mountain of evidence on this by measuring distortion at very low signal levels.  Renardsen has some more information at .
Copper is the preferred metal for conducting electricity in almost all circumstances. It has the lowest resistance of any metal but silver, is reasonably resistant to corrosion, and can be made mechanically strong; it’s wonderful stuff. Being a heavy metal, it is unfortunately not that common in the earth’s crust and so is expensive compared with iron and steel. It is, however, cheap compared with silver. The price of metals varies all the time due to changing economic and political factors, but at the time of writing, silver was 100 times more expensive than copper by weight. Given the same cross-section of conductor, the use of silver would only reduce the resistance of a circuit by 5%. Despite this, silver connection wire has been used in some very expensive hi-fi amplifiers; output impedance-matching transformers wound with silver wire are not unknown in valve amplifiers. Since the technical advantages are usually negligible, such equipment is marketed on the basis of indefinable subjective improvements. The only exception is the moving-coil step-up transformer, in which the use of silver in the primary winding might give a measurable reduction in Johnson noise.
Table 2.1 gives the resistivity of the commonly used conductors, plus some insulators to give it perspective. The difference between copper and quartz is of the order of 10 to the 25, an enormous range that is not found in many other physical properties.
|Material||Resistivity ρ (Ω-m)||Temperature coefficient per degree C||Electrical usage|
|Gold||2.2 ×10–8||0.0041||inert coatings|
|Tungsten||5.6 ×10–8||0.0045||lamp filaments|
|Mild steel||15 ×10–8||0.0066||busbars|
|Solder (60:40 tin/lead)||15 ×10–8||0.006||soldering|
|Lead||22 ×10–8||0.0039||storage batteries|
|Manganin (Cu, Mn, Ni)**||48.2 ×10–8||0.000002||resistances|
|Constantan (Cu, Ni)**||49–52 ×10–8||+/–0.00002||resistances|
(Ni, Fe, Cr alloy)
|100 ×10–8||0.0004||heating elements|
|Carbon (as graphite)||3–60 ×10–5||–0.0005||brushes|
|Fused quartz||More than 1018||…||insulators|
* A barreter is an incredibly obsolete device consisting of thin iron wire in an evacuated glass envelope. It was typically used for current regulation of the heaters of RF oscillator valves to improve frequency stability.
** Constantan and manganin are resistance alloys with moderate resistivity and a low temperature coefficient. Constanan is preferred, as it has a flatter resistance/temperature curve and its corrosion resistance is better.
There are several reasonably conductive metals that are lighter than copper, but their higher resistivity means they require larger cross-sections to carry the same current, so copper is always used when space is limited, as in electric motors, solenoids, etc. However, when size is not the primary constraint, the economics work out differently. The largest use of non-copper conductors is probably in the transmission line cables that are strung between pylons. Here minimal weight is more important than minimal diameter, so the cables have a central steel core for strength, surrounded by aluminium conductors.
It is clear that simply spending more money does not automatically bring you a better conductor; gold is a somewhat poorer conductor than copper, and platinum, which is even more expensive, is worse by a factor of six. Another interesting feature of this table is the relatively high resistance of mercury, nearly 60 times that of copper. This often comes as a surprise; people seem to assume that a metal of such high density must be very conductive, but it is not so. There are many reasons for not using mercury-filled hoses as loudspeaker cables, and their conductive inefficiency is just one. The cost and the insidiously poisonous nature of the metal are two more. Nonetheless … it is reported that the Hitachi Cable company has experimented with speaker cables made from polythene tubes filled with mercury. There appear to have been no plans to put such a product on the market. RoHS compliance might be a bit of a problem.
We also see in Table 2.1 that the resistivity of solder is high compared with that of copper –nine times higher if you compare copper with the old 60/40 tin/lead solder. This is unlikely to be a problem, as the thickness of solder the current passes through in a typical joint is very small. There are many formulations of lead-free solder, with varying resistivities, but all are high compared with copper.
One other point about solder. I have known people to tin the ends of mains cables (i.e. solder them) so that the end slides easily into the screw terminals. (I should say I am talking about the magnificent UK mains plug here.) This sounds harmless, but it is not. Under pressure, the solder will creep and the connection can come loose, possibly causing overheating and fire. Overheating increases the rate of creep, and we’re talking positive feedback here. This danger is not the issue it was, because nowadays almost all mains cables come with a moulded-on plug, but it remains a subtle and non-obvious hazard.
Copper is a good conductor because the outermost electrons of its atoms have a large mean free path between collisions. The electrical resistivity of a metal is inversely related to this electron mean free path, which in the case of copper is approximately 100 atomic spacings.
Copper is normally used as a very dilute alloy known as electrolytic tough pitch (ETP) copper, which consists of very high purity metal alloyed with oxygen in the range of 100 to 650 ppm. In view of the wide exposure that the concept of oxygen-free copper has had in the audio business, it is worth underlining that the oxygen is deliberately alloyed with the copper to act as a scavenger for dissolved hydrogen and sulphur, which become water and sulphur dioxide. Microscopic bubbles form in the mass of metal but are completely eliminated during hot rolling. The main use of oxygen-free copper is in conductors exposed to a hydrogen atmosphere at high temperatures. ETP copper is susceptible to hydrogen embrittlement in these circumstances, which arise in the hydrogen-cooled alternators in power stations. Many metals are subject to hydrogen embrittlement, including steel, which makes it a difficult gas to deal with.
It has been stated many times that oxygen-free copper makes superior cables, etc. In the world of subjectivist hi-fi, this is treated as a fact, but I am not aware that any hard evidence has ever been put forward, and I do not believe it. More information on the metallurgy of copper is given in , with an account of a comparison between copper and mercury, but the results are just those of “listening tests”, and I do not agree at all with the conclusions drawn.
As stated, gold has a higher resistivity than copper, and there is no incentive to use it as the bulk metal of conductors, not least because of its high cost. However, it is very useful as a thin coating on contacts because it is almost immune to corrosion, though it is chemically attacked by fluorine and chlorine. (If there is a significant amount of either gas in the air, then your medical problems will be more pressing than your electrical ones.) Other electrical components are sometimes gold plated simply because the appearance is attractive. A carat (or karat) is a 1/24 part, so 24-carat gold is the pure element, while 18-carat gold contains only 75% of the pure metal. 18-carat gold is the sort usually used for jewellery, as it retains the chemical inertness of pure gold but is much harder and more durable; the usual alloying elements are copper and silver.
18-carat gold is widely used in jewellery and does not tarnish, so it is initially puzzling to find that some electronic parts plated with it have a protective transparent coating, which the manufacturer claims to be essential to prevent blackening. The answer is that if gold is plated directly onto copper, the copper diffuses through the gold and tarnishes on its surface. The standard way of preventing this is to plate a layer of nickel onto the copper to prevent diffusion, then plate on the gold. I have examined some transparent-coated gold-plated parts and found no nickel layer; presumably the manufacturer finds the transparent coating is cheaper than another plating process to deposit the nickel. However, it does not look so good as bare gold.
Electrical cable is very often specified by its cross-sectional area and current-carrying capacity, and the resistance per meter is seldom quoted. This can, however, be a very important parameter for assessing permissible voltage drops and for predicting the crosstalk that will be introduced between two signals when they unavoidably share a common ground conductor. Given the resistivity of copper from Table 2.1, the resistance R of L metres of cable is simply:
Note that the area, which is usually quoted in catalogues in square millimetres, must be expressed here in square metres to match up with the units of resistivity and length. Thus 5 metres of cable with a cross-sectional area of 1.5 mm2 will have a resistance of:
It is also useful to be able to calculate the resistance of a PCB track for the same reasons. This is slightly less straightforward to do; given the smorgasbord of units that are in use in PCB technology, determining the cross-sectional area of the track can present some difficulty.
In the USA and the UK, and probably elsewhere, there is inevitably a mix of metric and imperial units on PCBs, as many important components come in dual-in-line packages which are derived from an inch grid; track widths and lengths are therefore very often in thousandths of an inch, universally (in the UK at least) referred to as “thou”. Conversely, the PCB dimensions and fixing-hole locations will almost certainly be metric, as they interface with a world of metal fabrication and mechanical CAD that (except in the United States) went metric many years ago. Add to this the UK practice of quoting copper thickness in ounces (the weight of a square foot of copper foil) and all the ingredients for dimensional confusion are in place.
Standard PCB copper foil is known as one-ounce copper, having a thickness of 1.4 thou (= 35 microns). Two-ounce copper is naturally twice as thick; the extra cost of specifying it is small, typically around 5% of the total PCB cost, and this is a very simple way of halving track resistance. It can of course be applied very easily to an existing design without any fear of messing up a satisfactory layout. Four-ounce copper can also be obtained but is more rarely used and is therefore much more expensive. If heavier copper than two-ounce is required, the normal technique is to plate two-ounce up to three-ounce copper. The extra cost of this is surprisingly small, in the region of 10% to 15%.
Given the copper thickness, multiplying by track width gives the cross-sectional area. Since resistivity is always in metric units, it is best to convert to metric at this point, so Table 2.2 gives area in square millimetres. This is then multiplied by the resistivity, not forgetting to convert the area to metres for consistency. This gives the “resistance” column in the table, and it is then simple to treat this as part of a potential divider to calculate the usually unwanted voltage across the track.
For example, if the track in question is the ground return from an 8 Ω speaker load, this is the top half of a potential divider, while the track is the bottom half (I am of course ignoring here the fact loudspeakers are not purely resistive loads), and a quick calculation gives the fraction of the input voltage found along the track. This is expressed in the last column of Table 2.2 as attenuation in dB. This shows clearly that loudspeaker outputs should not have common return tracks, or the interchannel crosstalk will be dire.
|Weight||Thickness||Thickness||Width||Length||Area||Resistance||Atten ref 8 Ω dB|
It is very clear from this table that relying on thicker copper on your PCB as means of reducing path resistance is not very effective. In some situations, it may be the only recourse, but in many cases, a path of much lower resistance can be made by using 32/02 cable soldered between the two relevant points on the PCB.
PCB tracks have a limited current capability because excessive resistive heating will break down the adhesive holding the copper to the board substrate and ultimately melt the copper. This is normally only a problem in power amplifiers and power supplies. It is useful to assess if you are likely to have problems before committing to a PCB design, and Table 2.3, based on MIL-standard 275, gives some guidance.
Note that Table 2.3 applies to tracks on the PCB surface only. Internal tracks in a multi-layer PCB experience much less cooling and need to be about three times as thick for the same temperature rise. This factor depends on laminate thickness and so on, and you need to consult your PCB vendor.
Traditionally, overheated tracks could be detected visually because the solder mask on top of them would discolour to brown. I am not sure if this still applies with modern solder mask materials, as in recent years I have been quite successful in avoiding overheated tracking.
|Track temp rise||10°C||20°C||30°C|
|Copper weight||1 oz||2 oz||1 oz||2 oz||1 oz||2 oz|
|Track width thou|
|10||1.0 A||1.4 A||1.2 A||1.6 A||1.5 A||2.2 A|
|15||1.2 A||1.6 A||1.3 A||2.4 A||1.6 A||3.0 A|
|20||1.3 A||2.1 A||1.7 A||3.0 A||2.4 A||3.6 A|
|25||1.7 A||2.5 A||2.2 A||3.3 A||2.8 A||4.0 A|
|30||1.9 A||3.0 A||2.5 A||4.0 A||3.2 A||5.0 A|
|50||2.6 A||4.0 A||3.6 A||6.0 A||4.4 A||7.3 A|
|75||3.5 A||5.7 A||4.5 A||7.8 A||6.0 A||10.0 A|
|100||4.2 A||6.9 A||6.0 A||9.9 A||7.5 A||12.5 A|
|200||7.0 A||11.5 A||10.0 A||16.0 A||13.0 A||20.5 A|
|250||8.3 A||12.3 A||12.3 A||20.0 A||15.0 A||24.5 A|
The previous section described how to evaluate the amount of crosstalk that can arise because of shared track resistances. Another crosstalk mechanism is caused by capacitance between PCB tracks. This is not very susceptible to calculation, so I did the following experiment to put some figures to the problem.
Figure 2.1 shows the setup; four parallel conductors 1.9 inches long on a standard piece of 0.1-inch pitch prototype board were used as test tracks. These are perhaps rather wider than the average PCB track, but one must start somewhere. The test signal was applied to track A, and track C was connected to a virtual-earth summing amplifier A1.
The tracks B and D were initially left floating. The results are shown as Trace 1 in Figure 2.2; the coupling at 10 kHz is -65 dB, which is worryingly high for two tracks 0.2 inch apart. Note that the crosstalk increases steadily at 6 dB per octave, as it results from a very small capacitance driving into what is effectively a short circuit.
It has often been said that running a grounded screening track between two tracks that are susceptible to crosstalk has a beneficial effect, but how much good does it really do? Grounding track B, to place a screen between A and C, gives Trace 2 and has only improved matters by 9 dB, not the dramatic effect that might be expected from screening. The reason, of course, is that electric fields are very much three dimensional, and if you could see the electrostatic “lines of force” that appear in physics textbooks, you would notice they arch up and over any planar screening such as a grounded track. It is easy to forget this when staring at a CAD display. There are of course two-layer and multi-layer PCBs, but the visual effect on a screen is still of several slices of 2-D. As Mr Spock remarked in one of the Star Trek films, “He’s intelligent, but not experienced. His pattern indicates two-dimensional thinking”.
Grounding track D, beyond receiving track C, gives a further improvement of about 3 dB; (Trace 3) this would clearly not happen if PCB crosstalk was simply a line-of-sight phenomenon.
To get more effective screening than this, you must go into three dimensions too; with a double-sided PCB, you can put one track on each side, with ground plane opposite. With a four-layer board it should be possible to sandwich critical tracks between two layers of ground plane, where they should be safe from pretty much anything. If you can’t do this and things are really tough you may need to resort to a screened cable between two points on the PCB; this is of course expensive in assembly time. If components, such as electrolytics with their large surface area, are talking to each other, you may need to use a vertical metal wall, but this costs money. A more cunning plan is to use electrolytics not carrying signal, such as rail decouplers, as screening items.
The internal crosstalk between the two halves of a dual opamp is very low according to the manufacturer’s specs. Nevertheless, avoid having different channels going through the same opamp if you can, because this will bring the surrounding components into close proximity and will permit capacitive crosstalk.
There are of course no three-layer PCBs as such; PCBs come in two-layer, four-layer, and so on upwards in multiples of two. Standard PCB technology first gave us one-layer, then two-layer, and after some time four-layer. At the time of writing the maximum in common use appears to be 16-layer, though 70-plus is possible.  The cost difference between two and four-layer is much less than it once was. Odd numbers of PCB layers are not used because four-layer and up PCBs are made by joining together two-layer PCBs, and there is no cost saving on leaving one layer unused.
The desirability of three layers on a PCB appeared when people began to make flat mixers all on one big PCB; there were no longer screened-cable connections to and from the faders, as these are time consuming and costly to fabricate and connect. All the connections had to be on the PCB for economy, but a four-layer solution would have been wholly impractical due to the great cost at the time. Since the routing switches on a channel are almost always above the fader, the tracks to the fader had to cross the mix buses on both send and return. This naturally meant that on a two-layer PCB, there was significant capacitive crosstalk from the fader connections to the mix buses.
I decided an answer to this problem was needed; at the time, it was called the three-layer PCB because it provided (to a limited extent) three layers of connectivity. Figure 2.3 shows how. The mixing buses were placed on the bottom side of a two-layer PCB. The top copper layer was a ground plane completely covering the mix bus area, and on top of that, two wire links provided the send and return paths to the fader. At the time, this sort of thing was going on, solder resist was not as tough as it is now, and so a band of solid screen-print ink (as used for the component ident) was placed under each link to prevent short circuits at zero cost. Two wire links can be fitted much more quickly than a twin screened-cable connection; indeed, they can be auto-inserted.
This technology is typically only required where signals cross the mix buses; no other part of the system is so vulnerable to capacitive crosstalk.
Capacitive crosstalk between two opamp outputs can be surprisingly troublesome. The usual isolating resistor on an opamp output is 47 Ω, and you might think that this impedance is so low that the capacitive crosstalk between two of these outputs would be completely negligible, but … you would be wrong.
A stereo power amplifier had balanced input amplifiers with 47Ω output isolating resistors included to prevent any possibility of instability, although the opamps were driving only a few cm of PCB track rather than screened cables with their significant capacitance. Just downstream of these opamps was a switch to enable biamping by driving both left and right outputs with the left input. This switch and its associated tracking brought the left and right signals into close proximity, and the capacitance between them was not negligible.
Crosstalk at low frequencies (below 1 kHz) was pleasingly low, being better than -129 dB up to 70 Hz, which was the difference between the noise floor and the maximum signal level. (The measured noise floor was unusually low at -114 dBu because each input amplifier was a quadruple noise-cancelling type as described in Chapter 18, and that figure includes the noise from an AP System 1.) At higher frequencies, things were rather less gratifying, being -96 dB at 10 kHz, as shown by the “47R” trace in Figure 2.4. In many applications, this would be more than acceptable, but in this case, the highest performance possible was being sought.
I therefore decided to reduce the output isolating resistors to 10 Ω, so the inter-channel capacitance would have less effect. (Checks were done at the time and all through the prototyping and pre-production process to make sure that this would be enough resistance to ensure opamp stability; it was.) This handily reduced the crosstalk to -109 dB at 10 kHz, an improvement of 13 dB at zero cost. This is the ratio between the two resistor values.
The third trace, marked “DIS”, shows the result of removing the isolating resistor from the speaking channel, so no signal reached the biamping switch. As usual, this reveals a further crosstalk mechanism, at about -117 dB, for reducing crosstalk is proverbially like peeling onions. There is layer after layer, and even strong men are reduced to tears.
In the past, there have been many types of resistor, including some interesting ones consisting of jars of liquid, but only a few kinds are likely to be met with now. These are usually classified by the kind of material used in the resistive element, as this has the most important influence on the fine details of performance. The major materials and types are shown in Table 2.4.
These values are illustrative only, and it would be easy to find exceptions. As always, the official data sheet for the component you have chosen is the essential reference. The voltage coefficient is a measure of linearity (lower is better), and its sinister significance is explained later.
It should be said that you are most unlikely to come across carbon composition resistors in modern signal circuitry, but they frequently appear in vintage valve equipment, so they are included here. They also live on in specialised applications such as switch-mode snubbing circuits, where their ability to absorb a high peak power in a mass of material rather than a thin film is very useful.
Carbon-film resistors are currently still sometimes used in low-end consumer equipment but elsewhere have been supplanted by the other three types. Note from Table 2.4 that they have a significant voltage coefficient.
Metal-film resistors are now the usual choice when any degree of precision or stability is required. These have no non-linearity problems at normal signal levels. The voltage coefficient is usually negligible.
|Type||Resistance tolerance||Temperature coefficient (ppm/°C)||Voltage coefficient (ppm)|
|Carbon composition||±10%||+400 to –900||350|
|Carbon–film||±5%||–100 to –700||100|
|Metal oxide||±5%||+300||variable but too high|
|Wirewound||±5%||±70% to ±250%||1|
Metal oxide resistors are more problematic. Cermet resistors and resistor packages are metal oxide and are made of the same material as thick-film SM resistors. Thick-film resistors can show significant non-linearity at opamp-type signal levels and should be kept out of high-quality signal paths.
Wirewound resistors are indispensable when serious power needs to be handled. The average wirewound resistor can withstand very large amounts of pulse power for short periods, but in this litigious age, component manufacturers are often very reluctant to publish specifications on this capability, and endurance tests have to be done at the design stage; if this part of the system is built first, then it will be tested as development proceeds. The voltage coefficient is usually negligible.
Resistors for general PCB use come in both through-hole and surface-mount types. Through-hole (TH) resistors can be any of the types tabled above; surface-mount (SM) resistors are always either metal film or metal oxide. There are also many specialised types; for example, high-power wirewound resistors are often constructed inside a metal case that can be bolted down to a heat sink.
These are too familiar to require much description; they are available in all the materials mentioned above –carbon film, metal film, metal oxide –and wirewound. There are a few other sorts, such as metal foil, but they are restricted to specialised applications. Conventional through-hole resistors are now almost always 250 mW 1% metal film. Carbon film used to be the standard resistor material, with the expensive metal-film resistors reserved for critical places in circuitry where low tempco and an absence of excess noise were really important, but as metal film got cheaper so it took over many applications.
TH resistors have the advantage that their power and voltage rating greatly exceed those of surface-mount versions. They also have a very low voltage coefficient, which for our purposes is of the first importance. On the downside, the spiral construction of the resistance element means they have much greater parasitic inductance; this is not a problem in audio work.
Chip surface-mount (SM) resistors come in a flat tombstone format, which varies over a wide size range; see Table 2.5.
MELF surface-mount resistors have a cylindrical body with metal endcaps, the resistive element is metal film, and the linearity is therefore as good as that of conventional resistors, with a voltage coefficient of less than 1 ppm. MELF is apparently an acronym for “metal electrode face-bonded”, though most people I know call them “metal-ended little fellows” or something quite close to that.
Surface-mount resistors may have thin-film or thick-film resistive elements. The latter are cheaper and so are more often encountered, but the price differential has been falling in recent years. Both thin-film and thick-film SM resistors use laser trimming to make fine adjustments of resistance value during the manufacturing process. There are important differences in their behaviour.
|Size L ×W||Max power dissipation||Max voltage|
|2512||1 W||200 V|
|1812||750 mW||200 V|
|1206||250 mW||200 V|
|0805||125 mW||150 V|
|0603||100 mW||75 V|
|0402||100 mW||50 V|
|0201||50 mW||25 V|
|01005||30 mW||15 V|
Thin-film (metal-film) SM resistors use a nickel-chromium (Ni-Cr) film as the resistance material. A very thin Ni-Cr film of less than 1 μm thickness is deposited on the aluminium oxide substrate by sputtering under vacuum. Ni-Cr is then applied onto the substrate as conducting electrodes. The use of a metal film as the resistance material allows thin-film resistors to provide a very low temperature coefficient, much lower current noise and vanishingly small non-linearity. Thin-film resistors need only low laser power for trimming (one-third of that required for thick-film resistors) and contain no glass-based material. This prevents possible micro-cracking during laser trimming and maintains the stability of the thin-film resistor types.
Thick-film resistors normally use ruthenium oxide (RuO2) as the resistance material, mixed with glass-based material to form a paste for printing on the substrate. The thickness of the printing material is usually 12 μm. The heat generated during laser trimming can cause micro-cracks on a thick-film resistor containing glass-based materials, which can adversely affect stability. Palladium/silver (PdAg) is used for the electrodes.
The most important thing about thick-film surface-mount resistors from our point of view is that they do not obey Ohm’s Law very well. This often comes as a shock to people who are used to TH resistors, which have been the highly linear metal-film type for many years. They have much higher voltage coefficients than TH resistors, at between 30 and 100 ppm. The non-linearity is symmetrical about zero voltage and so gives rise to third-harmonic distortion. Some SM resistor manufacturers do not specify voltage coefficient, which usually means it can vary disturbingly between different batches and different values of the same component, and this can have dire effects on the repeatability of design performance.
Chip-type surface-mount resistors come in standard formats with names based on size, such as 1206, 0805, 0603, and 0402. For example, 0805, which used to be something like the “standard” size, is 0.08 in. by 0.05 in.; see Table 2.5. The smaller 0603 is now more common. Both 0805 and 0603 can be placed manually if you have a steady hand and a good magnifying glass.
The 0402 size is so small that the resistors look rather like grains of pepper; manual placing is not really feasible. They are only used in equipment in which small size is critical, such as mobile phones. They have very restricted voltage and power ratings, typically 50 V and 100 mW. The voltage rating of TH resistors can usually be ignored, as power dissipation is almost always the limiting factor, but with SM resistors, it must be kept firmly in mind.
Recently, even smaller surface-mount resistors have been introduced; for example, several vendors offer 0201, and Panasonic and Yageo offer 01005 resistors. The latter are truly tiny, being about 0.4 mm long; a thousand of them weigh less than a twentieth of a gram. They are intended for mobile phones, palmtops, and hearing aids; a full range of values is available from 10 Ω to 1 MΩ (jumper inclusive). Hand placing is really not an option.
Surface-mount resistors have a limited power-dissipation capability compared with their through-hole cousins because of their small physical size. SM voltage ratings are also restricted for the same reason. It is therefore sometimes necessary to use two SM resistors in series or parallel to meet these demands, as this is usually more economic than hand-fitting a through-hole component of adequate rating. If the voltage rating is the issue, then the SM resistors will obviously have to be connected in series to gain any benefit.
Resistors are widely available in the E24 series (24 values per decade) and the E96 series (96 values per decade). There is also the E192 series, which, you might have guessed, has no less than 192 values per decade, but this is less freely available and not commonly used in audio design. Using the E96 or E192 resistor series over a product range means you have to keep an awful lot of different values in stock, and when non-standard values are required, it is usually more convenient to use a series or parallel combination of two or three E24 resistors. There is much more on this later in the chapter. Some types of resistor, such as wirewound, may belong to more restricted series like E6.
It is common in USA publications to see something like: “Resistor values have been rounded to the nearest 1% value”. which usually means the E96 series has been used. E96 values seem to have been in frequent use in the USA decades before they became so in Europe, leading to much head-scratching over “odd resistor values”. The value series from E3 to E96 is given in Appendix 1 at the back of this book.
There is a subtle trap built into the resistor series. If a resistor value exists in E6 or E12, then it will exist in E24. However, this does not follow for the E48 and E96 series. As an example, the 120 Ω value appears in both E12 and E24 but does not occur in E48 or E96, where the nearest value is 121 Ω in both cases. Mysteriously, 120 Ω reappears in E192, which has 121 Ω as well. You need to be aware of this if you are changing resistor series. There actually seems to be little or no interest in manufacturing E48 resistors as such; they are a subset of E96 resistors.
There are some other aspects of the resistor series that are worth knowing. Circuit design often requires specific ratios of components; as an example, if you are designing a second-order high-pass filter, you will need resistors in a 1:2 ratio; there are no such pairs in the E3, E6, or E12 series. The E24 series is much more helpful here because there are no fewer than six values which have an exact 2:1 ratio, such as 1 kΩ and 2 kΩ. All six pairs are shown in Table 2.6.
|E24: 6 value pairs in 1:2 ratio|
The E24 series also has four value pairs in an exact 1:3 ratio (e.g. 1 kΩ–3 kΩ) two pairs in a 1:4 ratio (e.g. 75 Ω–300 Ω), and six pairs in a 1:5 ratio (e.g. 2 kΩ–10 kΩ), which can be very useful in design. All these values are given in Appendix 1.
The E96 series sounds as though it would be even more fruitful of resistor pairs in integer ratios, and things start well with no less than 16 pairs in a 1:2 ratio. However, there are no 1:3 pairs at all, and also no 1:4 pairs – none. You will recall that I said earlier E24 was not a subset of E96. There are, however, in E96 a full 16 pairs in a 1:5 ratio. These are likewise listed in Appendix 1. I have not examined larger integer ratios, as it seems relatively unlikely they will be useful in the design process.
As noted in Table 2.4, the most common tolerance for metal-film resistors today is 1%. If you want a closer tolerance, then the next that is readily available is 0.1%; while there is considerable variation in price, roughly speaking, the 0.1% resistors will be 10 times as expensive.
When non-E24 values are required, it is usually more convenient to use a series or parallel combination of two E24 resistors. I call this the 2xE24 format.
Using two or more resistors to make up a desired value has a valuable hidden benefit. If it is done correctly, it will actually increase the average accuracy of the total resistance value so it is better than the tolerance of the individual resistors; this may sound paradoxical, but it is simply an expression of the fact that random errors tend to cancel out if you have a number of them. This also works for capacitors, and indeed any parameter that is subject to random variations, but for the time being, we will focus on the concrete example of multiple resistors. Note that this assumes that the mean (i.e. average) value of the resistors is accurate. It is generally a sound assumption, as it is much easier to control a single value such as the mean in a manufacturing process than to control all the variables that lead to scatter about that mean. This is confirmed by measurement.
Component values are usually subject to a Gaussian distribution, also called a normal distribution. It has a familiar peaky shape, not unlike a resonance curve, showing that the majority of the values lie near the central mean and that they get rarer the farther away from the mean you look. This is a very common distribution, cropping up wherever there are many independent things going on that affect the value of a given component. The distribution is defined by its mean and its standard deviation, which is the square root of the sum of the squares of the distances from the mean –the RMS sum, in other words. Sigma (σ) is the standard symbol for standard deviation. A Gaussian distribution will have 68.3% of its values within ±1 σ, 95.4% within ±2 σ, 99.7% within ±3 σ, and 99.9% within ±4 σ. This is illustrated in Figure 2.5, where the X-axis is calibrated in numbers of standard deviations on either side of the central mean value.
If we put two equal-value resistors in series or in parallel (see Figure 2.6a and 2.6b), the total value has proportionally a narrower distribution that of the original components. The standard deviation of summed components is the rms sum of the individual standard deviations, as shown in Equation 2.2. σsum is the overall standard deviation, and σ1 and σ2 are the standard deviations of the two resistors in series or parallel.
Thus if we have four 100 Ω 1% resistors in series, the standard deviation of the total resistance increases only by the square root of 4, that is 2 times, while the total resistance has increased by 4 times; thus we have made a 0.5% close-tolerance 400 Ω resistor for 4 times the price, whereas a 0.1% resistor would be at least 10 times the price and might give more accuracy than we need. There is a happy analogue here with the use of multiple amplifiers to reduce electrical noise; we are using essentially the same technique of RMS summation to reduce “statistical noise”.
You may object that putting four 1% resistors in series means that the worst-case errors can be 4 times as great. This is obviously true –if they are all 1% low, or 1% high, the total error will be 4%. But the probability of this occurring is actually very, very small indeed. The more resistors you combine, the more the values cluster together in the centre of the range.
The mathematics for series resistors is very simple (see Equation 2.2) but in other cases gets complicated very quickly. Equation 2.2 also holds for two parallel resistors as in Figure 2.6b, though this is mathematically much less obvious. I verified it by the use of Monte Carlo methods.  A suitable random number generator is used to select two resistor values, and their combined value is calculated and recorded. This is repeated many times (by computer, obviously), and then the mean and standard deviation of all the accumulated numbers is recorded. This will never give the exact answer, but it will get closer and closer as you make more trials. For the series and parallel cases, the standard deviation is 1/√2 of the standard deviation for a single resistor. If you are not wholly satisfied that this apparently magical improvement in average accuracy is genuine, seeing it happen on a spreadsheet makes a convincing demonstration.
In an Excel spreadsheet, random numbers with a uniform distribution are generated by the function RAND(), but random numbers with a Gaussian distribution and specified mean and standard deviation can be generated by the function NORMINV(). Let us assume we want to make an accurate 20 kΩ resistance. We can simulate the use of a single 1% tolerance resistor by generating a column of Gaussian random numbers with a mean of 20 and a standard deviation of 0.2; we need to use a lot of numbers to smooth out the statistical fluctuations, so we generate 400 of them. As a check, we calculate the mean and standard deviation of our 400 random numbers using the AVERAGE() and STDEV() functions. The results will be very close to 20 and 0.2 but not identical and will change every time we hit the F9 recalculate key, as this generates a new set of random numbers. The results of five recalculations are shown in Table 2.7, demonstrating that 400 numbers are enough to get us quite close to our targets.
To simulate two 10 kΩ resistors of 1% tolerance in series, we generate two columns of 400 Gaussian random numbers with a mean of 10 and a standard deviation of 0.1. We then set up a third column, which is the sum of the two random numbers on the same row, and if we calculate the mean and standard deviation using AVERAGE() and STDEV() again, we find that the mean is still very close to 20, but the standard deviation is reduced on average by the expected factor of √2. The result of five trials is shown in Table 2.8. Repeating this experiment with two 40 kΩ resistors in parallel gives the same results.
If we repeat this experiment by making our 20 kΩ resistance from a series combination of four 5 kΩ resistors of 1% tolerance, we generate four columns of 400 Gaussian random numbers with a mean of 5 and a standard deviation of 0.05. We sum the four numbers on the same row to get a fifth column and calculate the mean and standard deviation of that. The result of five trials is shown in Table 2.9. The mean is very close to 20, but the standard deviation is now reduced on average by a factor of √4, which is 2.
I think this demonstrates quite convincingly that the spread of values is reduced by a factor equal to the square root of the number of the components used. The principle works equally well for capacitors or, indeed, any quantity with a Gaussian distribution of values. The downside is the fact that the improvement depends on the square root of the number of equal-value components used, which means that big improvements require a lot of parts, and the method quickly gets unwieldy. Table 2.10 demonstrates how this works; the rate of improvement slows down noticeably as the number of parts increases. The largest number of components I have ever used in this way for a production design is five. Constructing a 0.1% resistance from 1% resistors would require a hundred of them and is hardly a practical proposition. It would cost a lot more than just buying a 0.1% resistor.
You might be wondering what happens if the series resistors used are not equal. If you are in search of a particular value, the method that gives the best resolution is to use one large resistor value and one small one to make up the total, as this gives a very large number of possible combinations. However, the accuracy of the final value is essentially no better than that of the large resistor. Two equal resistors, as we have just demonstrated, give a √2 improvement in accuracy, and near-equal resistors give almost as much, but the number of combinations is very limited, and you may not be able to get very near the value you want. The question is, how much improvement in accuracy can we get with resistors that are some way from equal, such as one resistor being twice the size of the other?
|Mean kΩ||Standard deviation|
|Mean kΩ||Standard deviation|
|Mean kΩ||Standard deviation|
|Number of equal-value parts||Tolerance reduction factor|
The mathematical answer is very simple; even when the resistor values are not equal, the overall standard deviation is still the RMS sum of the standard deviations of the two resistors, as shown in Equation 2.2; σ1 and σ2 are the standard deviations of the two resistors in series or parallel. Note that this equation is only correct if there is no correlation between the two values; this is true for two separate resistors but would not hold for two film resistors on the same substrate.
Since both resistors have the same percentage tolerance, the larger of the two has the greater standard deviation and dominates the total result. The minimum total deviation is thus achieved with equal resistor values. Table 2.11 shows how this works; using two resistors in the ratio 2:1 or 3:1 still gives a worthwhile improvement in average accuracy.
|Series resistor values Ω||Resistor ratio||Standard deviation|
|19.9K + 100||199:1||0.1990|
|19.5K + 500||39:1||0.1951|
|19K + 1K||19:1||0.1903|
|18K + 2K||9:1||0.1811|
|16.7K + 3.3K||5:1||0.1700|
|16K + 4K||4:1||0.1649|
|15K + 5K||3:1||0.1581|
|13.33K + 6.67K||2:1||0.1491|
|12K + 8K||1.5:1||0.1442|
|11K + 9K||1.22:1||0.1421|
|10K + 10K||1:1||0.1414|
The entries for 19.5k + 500 and 19.9k + 100 demonstrate that when one large resistor value and one small are used to get a particular value, its accuracy is very little better than that of the large resistor alone.
The relatively small number of combinations of E24 resistor values that can approximate a given value means that it is difficult to pursue good nominal accuracy and effective tolerance reduction simultaneously. This can be completely solved by using three E24 resistors in parallel; I call this the 3xE24 format. As with 2xE24, a parallel rather than series connection makes PCB tracking easier. Resistors are relatively cheap, and it can be highly economical to use three rather than two to approach very closely a few critical values. The extra PCB area required is modest. However, the design process is significantly harder. I used the Willmann Tables to design the 3xE24 combinations in this book; this process is fully described in Chapter 9, where it is particularly relevant. The Willmann tables can be downloaded free of charge from my website. 
So far, we have looked at serial and parallel combinations of components to make up one value, as in Figures 2.6a and 2.6b. Other important combinations are the resistive divider in Figure 2.6c (frequently used as the negative-feedback network for non-inverting amplifiers) and the inverting amplifier in Figure 2.6f, where the gain is set by the ratio R2/R1. All resistors are assumed to have the same tolerance about an exact mean value.
I suggest it is not obvious whether the divider ratio of Figure 2.6c, which is R2/ (R1 + R2), will be more or less accurate than the resistor tolerance, even in the simple case with R1 = R2. However, the Monte Carlo method shows that in this case, partial cancellation of errors still occurs, and the division ratio is more accurate by a factor of √2.
This factor depends on the divider ratio, as a simple physical argument shows:
- If the top resistor R1 is zero, then the divider ratio is obviously one with complete accuracy, the resistor values are irrelevant, and the output voltage tolerance is zero.
- If the bottom resistor R2 is zero, there is no output, and accuracy is meaningless, but if instead R2 is very small compared with R1, then the R1 completely determines the current through R2, and R2 turns this into the output voltage. Therefore, the tolerances of R1 and R2 act independently, and so the combined output voltage tolerance is worse by their RMS-sum √2.
Some more Monte Carlo work, with 8000 trials per data point, revealed that there is a linear relationship between accuracy and the “tap position” of the output between R1 and R2, as shown in Figure 2.7. Plotting against division ratio would not give a straight line. With R1 = R2, the tap is at 50%, and accuracy improved by a factor of √2, as noted above. With a tap at about 30% (R1 = 7 kΩ, R2 = 3 kΩ), the accuracy is the same as the resistors used. This assessment is not applicable to potentiometers, as the two sections of the pot are not uncorrelated.
The two-tap divider (Figure 2.6d) and three-tap divider (Figure 2.6e) were also given a Monte Carlo-ing, though only for equal resistors. The two-tap divider has an accuracy factor of 0.404 at OUT 1 and 0.809 at OUT 2. These numbers are very close to √2/(2√3) and √2/(√3), respectively. The three-tap divider has an accuracy factor of 0.289 at OUT 1, of 0.500 at OUT 2, and 0.864 at OUT 3. The middle figure is clearly 1/2 (twice as many resistors as a one-tap divider so √2 times more accurate), while the first and last numbers are very close to √3/6 and √3/2 respectively. It would be helpful if someone could prove analytically that the factors proposed are actually correct.
For the inverting amplifier of Figure 2.6f, the accuracy of the gain is always √2 worse than the tolerance of the two resistors, assuming the tolerances are equal. The nominal resistor values have no effect on this. We therefore have the interesting situation that a non-inverting amplifier will always be equally or more accurate in its gain than an inverting amplifier. So far as I know, this is a new result.
At this point, you may be complaining that this will only work if the resistor values have a Gaussian (also known as normal) distribution with the familiar peak around the mean (average) value. Actually, it is a happy fact that this effect does not assume that the component values have a Gaussian distribution, as we shall see in a moment. An excellent account of how to handle statistical variations to enhance accuracy is found in . This deals with the addition of mechanical tolerances in optical instruments, but the principles are just the same.
You sometimes hear that this sort of thing is inherently flawed, because, for example, 1% resistors are selected from production runs of 5% resistors. If you were using the 5% resistors, then you would find there was a hole in the middle of the distribution; if you were trying to select 1% resistors from them, you would be in for a very frustrating time, as they have already been selected out, and you wouldn’t find a single one. If instead you were using the 1% components obtained by selection from the complete 5% population, you would find that the distribution would be much flatter than Gaussian and the accuracy improvement obtained by combining them would be reduced, although there would still be a definite improvement.
However, don’t worry. In general, this is not the way that components are manufactured nowadays, though it may have been so in the past. A rare contemporary exception is the manufacture of carbon composition resistors , where making accurate values is difficult, and selection from production runs, typically with a 10% tolerance, is the only practical way to get more accurate values. Carbon composition resistors have no place in audio circuitry because of their large temperature and voltage coefficients and high excess noise, but they live on in specialised applications such as switch-mode snubbing circuits, where their ability to absorb high peak power in bulk material rather than a thin film is useful, and in RF circuitry where the inductance of spiral-format film resistors is unacceptable.
So, having laid that fear to rest, what is the actual distribution of resistor values like? It is not easy to find out, as manufacturers are not exactly forthcoming with this sort of sensitive information, and measuring thousands of resistors with an accurate DVM is not a pastime that appeals to all of us. Any nugget of information in this area is therefore very welcome.
Hugo Kroeze  reported the result of measuring 211 metal-film resistors from the same batch with a nominal value of 10 kΩ and 1% tolerance. He concluded that:
- The mean value was 9.995 kΩ (0.05% low)
- The standard deviation was about 10 Ω, i.e. only 0.1%. This spread in value is surprisingly small (the resistors were all from the same batch, and the spread across batches widely separated in manufacture date might have been less impressive).
- All resistors were within the 1% tolerance range
- The distribution appeared to be Gaussian, with no evidence that it was a subset from a larger distribution.
I decided to add my own morsel of data to this. I measured 100 ordinary metal-film 1 kΩ resistors of 1% tolerance from Yageo, a Chinese manufacturer, and very tedious it was too. I used a recently calibrated 4.5-digit meter.
- The mean value was 997.66 Ω (0.23% low).
- The standard deviation was 2.10 Ω, i.e. 0.21%.
- All resistors were within the 1% tolerance range. All but one was within 0.5%. with the outlier at 0.7%.
- The distribution appeared to be Gaussian, with no evidence that it was a subset from a larger distribution.
These are only two reports, and it would be nice to have more confirmation, but there seems to be no reason to doubt that the mean value is very well controlled, and the spread is under good control as well. The distribution of resistance values appears to be Gaussian, with no evidence of the most accurate specimens being selected out. Whenever I have attempted this kind of statistical improvement in accuracy, I have always found that the expected benefit really does appear in practice.
As I mentioned earlier, improving average accuracy by combining resistors does not depend on the resistance value having a Gaussian distribution. Even a batch of resistors with a uniform distribution gives better accuracy when two of them are combined. A uniform distribution of component values may not be likely, but the result of combining two or more of them is highly instructive, so stick with me for a bit.
Figure 2.8a shows a uniform distribution that cuts off abruptly at the limits L and –L and represents 10 kΩ resistors of 1% tolerance. We will assume again that we want to make a more accurate 20 kΩ resistance. If we put two of the uniform-distribution 10 kΩ resistors in series, we get not another uniform distribution but the triangular distribution shown in Figure 2.8b. This shows that the total resistance values are already starting to cluster in the centre; it is possible to have the extreme values of 19.8 kΩ and 20.2 kΩ, but it is very, very unlikely.
Figure 2.8c shows what happens if we use more resistors to make the final value; when 4 are used, the distribution is already beginning to look like a Gaussian distribution, and as we increase the number of components to 8 and then 16, the resemblance becomes very close.
Uniform distributions have a standard deviation just as Gaussian ones do. It is calculated from the limits L and –L as in Equation 2.3. Likewise, the standard deviation of a triangular distribution can be calculated from its limits L and –L as in Equation 2.4.
Applying Equation 2.3 to the uniformly distributed 10 kΩ 1% resistors in Figure 2.8a, we get a standard deviation of 0.0577.
Applying Equation 2.4 to the triangular distribution of 20 kΩ resistance values in Figure 2.8b, we get 0.0816. The mean value has doubled, but the standard deviation has less than doubled, so we get an improvement in average accuracy; the ratio is √2, just as it was for two resistors with a Gaussian distribution. This is also easy to demonstrate with Monte Carlo methods on a spreadsheet.
It is well known that resistors have inductance and capacitance and vary somewhat in resistance with temperature. Unfortunately, there are other less obvious imperfections, such as excess noise and non-linearity; these can get forgotten, because parameters describing how bad they are, are often omitted from component manufacturer’s data sheets.
Being components in the real world, resistors are not perfect examples of resistance and nothing else. Their length is not infinitely small and so they have series inductance; this is particularly true for the many kinds that use a spiral resistive element. Likewise, they exhibit stray capacitance between each end and also between the various parts of the resistive element. Both effects can be significant at high frequencies but can usually be ignored below 100 kHz unless you are using very high or low resistance values.
It is a sad fact that resistors change their value with temperature. Table 2.4 shows some typical temperature coefficients. This is not likely to be a problem in small-signal audio applications, where the temperature range is small and extreme precision is not required unless you are designing measurement equipment. Carbon-film resistors are markedly inferior to metal film in this area.
All resistors, no matter what their resistive material or mode of construction, generate Johnson noise. This is white noise, its level being determined solely by the resistance value, the absolute temperature, and the bandwidth over which the noise is being measured. It is based on fundamental physics and is not subject to negotiation. In some cases, it places the limit on how quiet a circuit can be, though the noise from active devices is often more significant. Johnson noise is covered in Chapter 1.
Excess resistor noise refers to the fact that some kinds of resistor, with a constant voltage drop imposed across them, generate excess noise in addition to their inherent Johnson noise. According to classical physics, passing a current through a resistor should have no effect on its noise behaviour; it should generate the same Johnson noise as a resistor with no steady current flow. In reality, some types of resistors do generate excess noise when they have a DC voltage across them. It is a very variable quantity, but it is essentially proportional to the DC voltage across the component, a typical spec being “1 μV/V”, and it has a 1/f frequency distribution. Typically it could be a problem in biasing networks at the input of amplifier stages. It is usually only of interest if you are using carbon- or thick-film resistors –metal-film and wirewound types should have very little excess noise. 1/f noise does not have a Gaussian amplitude distribution, which makes it difficult to assess reliably from a small set of data points. A rough guide to the likely specs is given in Table 2.12.
The level of excess resistor noise changes with resistor type, size, and value in Ohms; here are the relevant factors:
- Thin-film resistors are markedly quieter than thick-film resistors; this is due to the homogeneous nature of thin-film resistive materials, which are metal alloys such as nickel-chromium deposited on a substrate. The thick-film resistive material is a mixture of metal (often ruthenium) oxides and glass particles; the glass is fused into a matrix for the metal particles by high-temperature firing. The higher excess noise levels associated with thick-film resistors are a consequence of their heterogeneous structure due to the particulate nature of the resistive material. The same applies to carbon-film resistors, in which the resistive medium is finely divided carbon dispersed in a polymer binder.
- A physically large resistor has lower excess noise than a small resistor, because there is more resistive material in parallel, so to speak. In the same resistor range, the highest-wattage versions have the lowest noise. See Figure 2.9.
- A low-ohmic-value resistor has lower excess noise than a high ohmic value. Noise in μV per V rises approximately with the square root of resistance. See Figure 2.9 again.
- A low value of excess noise is associated with uniform constriction-free current flow; this condition is not well met in composite thick-film materials. However, there are great variations among different thick-film resistors. The most readily apparent relationship is between noise level and the amount of conductive material present. Everything else being equal, compositions with lower resistivity have lower noise levels.
- Higher resistance values give higher excess noise since it is a statistical phenomenon related to the total number of charge carriers available within the resistive element; the fewer the total number of carriers present, the greater will be the statistical fluctuation.
- Traditionally at this point in the discussion of excess resistor noise, the reader is warned against using carbon-composition resistors because of their very bad excess noise characteristics. Carbon-composition resistors are still made –their construction makes them good at handling pulse loads –but are not likely to be encountered in audio circuitry.
- One of the great benefits of dual-rail opamp circuitry is that is noticeably free of resistors with large DC voltages across them. The offset voltages and bias currents are far too low to cause trouble with resistor excess noise. However, if you are getting into low-noise hybrid discrete/opamp stages, such as the MC head amplifier in Chapter 10, you might have to consider it.
- To get a feel for the magnitude of excess resistor noise, consider a 100 kΩ 1/4 W carbon-film resistor with 10 V across it. This, from the graph above, has an excess noise parameter of about 0.7 μV/V, and so the excess noise will be of the order of 7 μV, which is -101 dBu. This definitely could be a problem in a low-noise preamplifier stage.
Ohm’s Law strictly is a statement about metallic conductors only. It is dangerous to assume that it also invariably applies to “resistors” simply because they have a fixed value of resistance marked on them; in fact, resistors –whose main raison d’etre is packing a lot of controlled resistance in a small space –do not always adhere to Ohm’s Law very closely. This is a distinct difficulty when trying to make low-distortion circuitry.
Resistor non-linearity is normally quoted by manufacturers as a voltage coefficient, usually the number of parts per million (ppm) that the resistor value changes when 1 volt is applied. The measurement standard for resistor non-linearity is IEC 6040.
Through-hole metal-film resistors show perfect linearity at the levels of performance considered here, as do wirewound types. The voltage coefficient is less than 1 ppm. Carbon-film resistors are quoted at less than 100 ppm; 100 ppm is, however, enough to completely dominate the distortion produced by active devices if it is used in a critical part of the circuitry. Carbon-composition resistors, probably of historical interest only, come in at about 350 ppm, a point that might be pondered by connoisseurs of antique equipment. The greatest area of concern over non-linearity is thick-film surface-mount resistors, which have high and rather variable voltage coefficients; more on this in what follows.
Table 2.13 (calculated with SPICE) gives the total harmonic distortion (THD) in the current flowing through the resistor for various voltage coefficients when a pure sine voltage is applied. If the voltage coefficient is significant, this can be a serious source of non-linearity.
A voltage-coefficient model generates all the odd-order harmonics at a decreasing level as order increases. No even-order harmonics occur, as the model is symmetrical. This is covered in much more detail in my Active Crossover book. 
|Voltage||THD at||THD at|
|Coefficient||+15 dBu||+20 dBu|
My own test setup is shown in Figure 2.10. The resistors are usually of equal value to give 6 dB attenuation. A very low-distortion oscillator that can give a large output voltage is necessary; the results in Figure 2.11 were taken at a 10 Vrms (+22 dBu) input level. Here thick-film SM and through-hole resistors are compared. The gen-mon trace at the bottom is the record of the analyser reading the oscillator output and is the measurement floor of the AP system I used. The THD plot is higher than this floor, but this is not due to distortion. It simply reflects the extra Johnson noise generated by two 10 kΩ resistors. Their parallel combination is 5 kΩ, and so this noise is at -115.2 dBu. The SM plot, however, is higher again, and the difference is the distortion generated by the thick-film component.
For both thin-film and thick-film SM resistors non-linearity increases with resistor value and also increases as the physical size (and hence power rating) of the resistor shrinks. The thin-film versions are much more linear; see Figures 2.12 and 2.13. Sometimes it is appropriate to reduce the non-linearity by using multiple resistors in series. If one resistor is replaced by two with the same voltage coefficient in series, the THD in the current flowing is halved. Similarly, three resistors reduces THD to a third of the original value. There are obvious economic limits to this sort of thing, but it can be useful in specific cases, especially where the voltage rating of the resistor is a limitation.
Capacitors are diverse components. In the audio business, their capacitance ranges from 10 pF to 100,000 μF, a ratio of 10 to the tenth power. In this they handily out-do resistors, which usually vary from 0.1 Ω to 10 MΩ, a ratio of only 10 to the eighth. However, if you include the 10 GΩ bias resistors used in capacitor microphone head amplifiers, this range increases to 10 to the eleventh. There is, however, a big gap between the 10 MΩ resistors, which are used in DC servos, and 10 GΩ microphone resistors; I am not aware of any audio applications for 1 GΩ resistors.
Capacitors also come in a wide variety of types of dielectric, the great divide being between electrolytic and non-electrolytic types. Electrolytics used to have much wider tolerances than most components, but things have recently improved, and ±20% is now common. This is still wider than for typical non-electrolytics, which are usually ±10% or better.
This is not the place to reiterate the basic information about capacitor properties, which can be found from many sources. I will simply note that real capacitors fall short of the ideal circuit element in several ways, notably leakage, equivalent series resistance (ESR), dielectric absorption and non-linearity:
Capacitor leakage is equivalent to a high value resistance across the capacitor terminals, and allows a trickle of current to flow when a DC voltage is applied. It is usually negligible for non-electrolytics but is much greater for electrolytics.
ESR is a measure of how much the component deviates from a mathematically pure capacitance. The series resistance is partly due to the physical resistance of leads and foils and partly due to losses in the dielectric. It can also be expressed as tan-δδ (tan-delta). Tan-delta is the tangent of the phase angle between the voltage across and the current flowing through the capacitor.
Dielectric absorption is a well-known phenomenon; take a large electrolytic, charge it up, and then make sure it is fully discharged. Use a 10 Ω WW resistor across the terminals rather than a screwdriver unless you’re not too worried about either the screwdriver, the capacitor, or your eyesight. Wait a few minutes, and the charge will partially reappear, as if from nowhere. This “memory effect” also occurs in non-electrolytics to a lesser degree; it is a property of the dielectric and is minimised by using polystyrene, polypropylene, NPO ceramic, or PTFE dielectrics. Dielectric absorption is invariably simulated by a linear model composed of extra resistors and capacitances; nevertheless, dielectric absorption and distortion correlate across the different dielectrics.
Capacitor non-linearity is undoubtedly the least known of these shortcomings. A typical RC low-pass filter can be made with a series resistor and a shunt capacitor, and if you examine the output with a distortion analyser, you may find to your consternation that the circuit is not linear. If the capacitor is a non-electrolytic type with a dielectric such as polyester, then the distortion is relatively pure third harmonic, showing that the effect is symmetrical. For a 10 Vrms input, the THD level may be 0.001% or more. This may not sound like much, but it is substantially greater than the mid-band distortion of a good opamp. Capacitor non-linearity is dealt with at greater length in what follows.
Capacitors are used in audio circuitry for three main functions, where their possible non-linearity has varying consequences:
- Coupling or DC blocking capacitors. These are usually electrolytics and, if properly sized, have a negligible signal voltage across them at the lowest frequencies of interest. The properties of the capacitor are pretty much unimportant unless current levels are high; power amplifier output capacitors can generate considerable mid-band distortion. 
- Much nonsense has been talked about mysterious coupling capacitor properties, but it is all nonsense. For small-signal use, as long as the signal voltage across the capacitor is kept low, non-linearity is not normally detectable. The capacitance value is non-critical, as it has to be, given the wide tolerances of electrolytics.
- Supply filtering or decoupling capacitors. These are electrolytics if you are filtering out supply rail ripple, etc., and non-electrolytics, usually around 100 nF, when the task is to keep the supply impedance low at high frequencies and so keep opamps stable. The capacitance value is again non-critical.
- Setting time-constants, for example the capacitors in the feedback network of an RIAA amplifier. This is a much more demanding application than the other two. First, the actual value is now crucially important, as it defines the accuracy of the frequency response. Second, there is by definition significant signal voltage across the capacitor and so non-linearity can be a serious problem. Non-electrolytics are normally used; sometimes an electrolytic is used to define the lower end of the bandwidth, but this is a bad practice likely to introduce distortion at the bottom of the frequency range. Small-value ceramic capacitors are used for compensation purposes.
In subjectivist circles, it is frequently asserted that electrolytic coupling capacitors (if they are permitted at all) should be bypassed by small non-electrolytics. There is no sense in this; if the main coupling capacitor has no signal voltage across it, the extra capacitor can have no effect.
Capacitors are available in a much more limited range of values than resistors, often restricted to the E6 series, which runs 10, 15, 22, 33, 47, 68. There are no values in that series which have an exact 2:1 ratio, which complicates the design of second-order low-pass active filters. If you can source capacitors in the E12 series, this gives much more freedom of design.
When attempting the design of linear circuitry, everyone knows that inductors and transformers with ferromagnetic core material can be a source of non-linearity. It is, however, less obvious that capacitors and even resistors can show non-linearity and generate some unexpected and very unwelcome distortion. Resistor non-linearity has been dealt with earlier in this chapter; let us now examine the shortcomings of capacitors.
The definitive work on capacitor distortion is a magnificent series of articles by Cyril Bateman in Electronics World . The authority of this work is underpinned by Cyril’s background in capacitor manufacture. (The series is long because it includes the development of a low-distortion THD test set in the first two parts.)
Capacitors generate distortion when they are actually implementing a time-constant –in other words, when there is a signal voltage across them. The normal coupling or DC-blocking capacitors have no significant signal voltage across them, as they are intended to pass all the information through, not to filter it or define the system bandwidth. Capacitors with no signal across them do not generally produce distortion at small-signal current levels. This was confirmed for all the capacitors tested in what follows. However, electrolytic types may do so at power amplifier levels where the current through them is considerable, such as in the output coupling capacitor of a power amplifier. 
It has often been assumed that non-electrolytic capacitors, which generally approach an ideal component more closely than electrolytics and have dielectrics constructed in a totally different way, are free from distortion. It is not so. Some non-electrolytics show distortion at levels that are easily measured and can exceed the distortion from the opamps in the circuit. Non-electrolytic capacitor distortion is primarily third harmonic, because the non-polarised dielectric technology is basically symmetrical. The problem is serious, because non-electrolytic capacitors are commonly used to define time-constants and frequency responses (in RIAA equalisation networks, for example) rather than simply for DC blocking.
Very small capacitances present no great difficulty. Simply make sure you are using the COG (NP0) ceramic type, and so long as you choose a reputable supplier, there will be no distortion. I say “reputable supplier” because I did once encounter some allegedly COG capacitors from China that showed significant non-linearity. 
Middle-range capacitors, from 1 nF to 1 μF, present more of a problem. Capacitors with a variety of dielectrics are available, including polyester, polystyrene, polypropylene, polycarbonate, and polyphenylene sulphide, of which the first three are the most common. (Note that what is commonly called “polyester” is actually polyethylene terephthalate, PET.)
Figure 2.14 shows a simple low-pass filter circuit which, in conjunction with a good THD analyser, can be used to get some insight into the distortion problem; it is intended to be representative of a real bit of audio circuitry. The values shown give a pole frequency, or -3 dB roll-off point, at 710 Hz. Since it might be expected that different dielectrics give different results (and they definitely do), we will start off with polyester, the smallest, most economical, and therefore the most common type for capacitors of this size.
The THD results for a microbox 220 nF 100 V capacitor with a polyester dielectric are shown in Figure 2.15, for input voltages of 10, 15, and 20 Vrms. They are unsettling.
The distortion is all third harmonic and peaks at around 300 to 400 Hz, well below the pole frequency, and even with input limited to 10 Vrms, it will exceed the non-linearity introduced by opamps such as the 5532 and the LM4562. Interestingly, the peak frequency changes with applied level. Below the peak, the voltage across the capacitor is constant, but distortion falls as frequency is reduced, because the increasing impedance of the capacitor means it has less effect on a circuit node at a 1 kΩ impedance. Above the peak, distortion falls with increasing frequency, because the low-pass circuit action causes the voltage across the capacitor to fall.
The level of distortion varies with different samples of the same type of capacitor; six of the above type were measured, and the THD at 10 Vrms and 400 Hz varied from 0.00128% to 0.00206%. This puts paid to any plans for reducing the distortion by some sort of cancellation method. The distortion can be seen in Figure 2.15 to be a strong function of level, roughly tripling as the input level doubles. Third-harmonic distortion normally quadruples for doubled level, so there may well be an unanswered question here. It is, however, clear that reducing the voltage across the capacitor reduces the distortion. This suggests that if cost is not the primary consideration, it might be useful to put two capacitors in series to halve the voltage and the capacitance and then double up this series combination to restore the original capacitance, giving the series-parallel arrangement in Figure 2.16. The results are shown in Table 2.14, and once more it can be seen that halving the level has reduced distortion by a factor of three rather than four. The series-parallel arrangement obviously has limitations in terms of cost and PCB area occupied, but it might be useful in some cases.
Clearly polyester gives significant distortion, despite its extensive use in audio circuitry of all kinds.
|Input level (Vrms)||Single capacitor||Series-parallel capacitors|
An unexpected complication was that every time a sample was remeasured, the distortion was lower than before. I found a steady reduction in distortion over time if a test signal was left applied; 9 Vrms at 1 kHz halved the THD over 11 hours. This is a semi-permanent change, as some of the distortion returns over time when the signal is removed. This effect may be of little practical use, but it does demonstrate that polyester capacitors are more complicated than they look. For much more on this, see .
The next dielectric we will try is polystyrene. Capacitors with a polystyrene dielectric are extremely useful for some filtering and RIAA-equalisation applications because they can be obtained at a 1% tolerance at up to 10 nF at a reasonable price. They can be obtained in larger sizes at an unreasonable, or at any rate much higher, price.
The distortion test results are shown in Figure 2.17 for a 4n7 2.5% capacitor; the series resistor R1 has been increased to 4.7 kΩ to keep the -3 dB point inside the audio band, and it is now at 7200 Hz. Note that the THD scale has been extended down to a subterranean 0.0001%, and if it was plotted on the same scale as Figure 2.15, it would be bumping along the bottom of the graph. Figure 2.17 in fact shows no distortion at all, just the measurement noise floor, and the apparent rise at the HF end is simply due to the fact that the output level is decreasing because of the low-pass action, and so the noise floor is relatively increasing. This is at an input level of 10 Vrms, which is about as high as might be expected to occur in normal opamp circuitry. The test was repeated at 20 Vrms, which might be encountered in discrete circuitry, and the results were the same: no measurable distortion.
The tests were done with four samples of 10nF 1% polystyrene from LCR at 10 Vrms and 20 Vrms, with the same results for each sample. This shows that polystyrene capacitors can be used with confidence; this is in complete agreement with Cyril Bateman’s findings. 
Having settled the issue of capacitor distortion below 10 nF, we need now to tackle it capacitor values greater than 10 nF. Polyester having proven unsatisfactory, the next most common dielectric is polypropylene, and I might as well say at once that it was with considerable relief that I found these were effectively distortion free in values up to 220 nF. Figure 2.18 shows the results for four samples of a 220 nF 250 V 5% polypropylene capacitor from RIFA. Once more the plot shows no distortion at all, just the noise floor, the apparent rise at the HF end being increasing relative noise due to the low-pass roll-off. This is also in agreement with Cyril Bateman’s findings. Rerunning the tests at 20 Vrms gave the same result –no distortion. This is very pleasing, but there is a downside. Polypropylene capacitors of this value and voltage rating are physically much larger than the commonly used 63 or 100 V polyester capacitor, and they’re more expensive.
It was therefore important to find out if the good distortion performance was a result of the 250 V rating, and so I tested a series of polypropylene capacitors with lower voltage ratings from different manufacturers. Axial 47 nF 160 V 5% polypropylene capacitors from Vishay proved to be THD free at both 10 Vrms and 20 Vrms. Likewise, microbox polypropylene capacitors from 10 nF to 47 nF, with ratings of 63 V and 160 V, from Vishay and Wima proved to generate no measurable distortion, so the voltage rating appears not to be an issue. This finding is particularly important, because the Vishay range has a 1% tolerance, making them very suitable for precision filters and equalisation networks. The 1% tolerance is naturally reflected in the price.
The only remaining issue with polypropylene capacitors is that the higher values (above 100 nF) appear to be currently only available with 250 V or 400 V ratings, and that means a physically big component. For example, the EPCOS 330 nF 400 V 5% part has a footprint of 26 mm by 6.5 mm, with a height of 15 mm. One way of dealing with this is to use a smaller capacitor in a capacitance multiplication configuration, so a 100 nF 1% component could be made to emulate 470 nF. It has to be said that the circuitry for this is only straightforward if one end of the capacitor is connected to ground.
When I first started looking at capacitor distortion, I thought that the distortion would probably be lowest for the capacitors with the highest voltage rating. I therefore tested some RF-suppression X2 capacitors, rated at 275 Vrms, which translates into a peak or DC rating of 389 V. These are designed to be connected directly across the mains and therefore have a thick and tough dielectric layer. For some reason, manufacturers seem to be very coy about saying exactly what the dielectric material is, normally describing them simply as “film capacitors”. A problem that surfaced immediately is that the tolerance is 10% or 20%, not exactly ideal for precision filtering or equalisation. A more serious problem, however, is that they are far from distortion free. Four samples of a 470 nF X2 capacitor showed THD between 0.002% and 0.003% at 10 Vrms. Clearly a high voltage rating alone does not mean low distortion.
Cyril Bateman’s series in Electronics World  included two articles on electrolytic capacitor distortion. It proved to be a complex subject, and many long-held assumptions (such as “DC biasing always reduces distortion”) were shown to be quite wrong. Distortion was in general a good deal higher than for non-electrolytic capacitors.
My view is that electrolytics should never, ever, under any circumstances, be used to set time-constants in audio. There should be a time-constant early in the signal path, based on a non-electrolytic capacitor, that determines the lower limit of the bandwidth, and all the electrolytic-based time-constants should be much longer so that the electrolytic capacitors can never have significant signal voltages across them and so never generate measurable distortion. There is of course also the point that electrolytics have large tolerances and cannot be used to set accurate time-constants anyway.
However, even if you obey this rule, you can still get into deep trouble. Figure 2.19 shows a simple high-pass test circuit designed to represent an electrolytic capacitor in use for coupling or DC blocking. The load of 1 kΩ is the sort of value that can easily be encountered if you are using low-impedance design principles. The calculated -3 dB roll-off point is 3.38 Hz, so the attenuation at 10 Hz, at the very bottom of the audio band, will be only 0.47 dB; at 20 Hz, it will be only 0.12 dB, which is surely a negligible loss. As far as frequency response goes, we are doing fine. But … examine Figure 2.20, which shows the measured distortion of this arrangement. Even if we limit ourselves to a 10 Vrms level, the distortion at 50 Hz is 0.001%, already above that of a good opamp. At 20 Hz, it has risen to 0.01% and by 10 Hz a most unwelcome 0.05%. The THD is increasing by a ratio of 4.8 times for each octave fall in frequency, in other words increasing faster than a square law. The distortion residual is visually a mixture of second and third harmonic, and the levels proved surprisingly consistent for a large number of 47 uF 25 V capacitors of different ages and from different manufacturers.
Figure 2.20 also shows that the distortion rises rapidly with level; at 50 Hz, going from an input of 10 Vrms to 15 Vrms almost doubles the THD reading. To underline the point, consider Figure 2.21, which shows the measured frequency response of the circuit with 47 uF and 1 kΩ; note the effect of the capacitor tolerance on the real versus calculated figures. The roll-off that does the damage, by allowing an AC voltage to exist across the capacitor, is very modest indeed, less than 0.2 dB at 20 Hz.
Having demonstrated how insidious this problem is, how do we fix it? Changing capacitor manufacturer is no help. Using 47 uF capacitors of higher voltage does not work; tests showed there is very little difference in the amount of distortion generated. An exception was the sub-miniature style of electrolytic, which was markedly worse.
The answer is simple: just make the capacitor bigger in value. This reduces the voltage across it in the audio band, and since we have shown that the distortion is a strong function of the voltage across the capacitor, the amount produced drops more than proportionally. The result is seen in Figure 2.22 for increasing capacitor values with a 10 Vrms input.
Replacing C1 with a 100 uF 25 V capacitor drops the distortion at 20 Hz from 0.0080% to 0.0017%, an improvement of 4.7 times; the voltage across the capacitor at 20 Hz has been reduced from 1.66 Vrms to 790 mVrms. A 220 uF 25 V capacitor reduces the voltage across itself to 360 mV and gives another very welcome reduction to 0.0005% at 20 Hz, but it is necessary to go to 1000 uF 25 V to obtain the bottom trace, which is indistinguishable from the noise floor of the AP-2702 test system. The voltage across the capacitor at 20 Hz is now only 80 mV. From this data, it appears that the AC voltage across an electrolytic capacitor should be limited to below 80 mVrms if you want to avoid distortion. I would emphasise that these are ordinary 85°C-rated electrolytic capacitors and in no sense special or premium types.
This technique can be seen to be highly effective, but it naturally calls for larger and somewhat more expensive capacitors and larger footprints on a PCB. This can be to some extent countered by using capacitors of lower voltage, which helps to bring back down the CV product and hence the can size. I tested 1000 μF 16 V and 1000 μF6 V3 capacitors, and both types gave exactly the same results as the 1000 μF 25 V part in Figure 2.21, with useful reductions in CV product and can size. This does, of course, assume that the capacitor is, as is usual, being used to block small voltages from opamp offsets to prevent switch clicks and pot noises rather than for stopping a substantial DC voltage.
The use of large coupling capacitors in this way does require a little care, because we are introducing a long time-constant into the circuit. Most opamp circuitry is pretty much free of big DC voltages, but if there are any, the settling time after switch-on may become undesirably long.
More information on capacitor distortion in specific applications can be found in Chapter 19.
For several reasons, inductors are unpopular with circuit designers. They are relatively expensive, often because they need to be custom-made. Unless they are air-cored (which limits their inductance to low values), the core material is a likely source of non-linearity. Some types produce substantial external magnetic fields, which can cause crosstalk if they are placed close together, and similarly they can be subject to the induction of interference from other external fields. In general, they deviate from being an ideal circuit element much more than resistors or capacitors.
It is rarely, if ever, essential to use inductors in signal-processing circuitry. Historically, they were used in tone controls before the Baxandall configuration swept all before it, and their last applications were probably in mid EQ controls for mixing consoles and in LCR filters for graphic equalisers. These too were gone by the end of the seventies, being replaced by active filters and gyrators, to the considerable relief of all concerned (except inductor manufacturers).
The only place inductors are essential is when the need for galvanic isolation, or enhanced EMC immunity, makes input and output transformers desirable, and even then they need careful handling; see Chapters 18 and 19 on line-in and line-out circuitry.
 Self, Douglas “Ultra-Low-Noise Amplifiers & Granularity Distortion” JAES, Nov 1987, pp 907–915
 Borwick, John, ed Loudspeaker and Headphone Handbook 2nd edition. Focal Press, pp 246–247
 Sanmina Co www.sanmina.com/contract-manufacturing-design/printed-circuit-boards/technology/ Accessed Jun 2019
 Smith, Warren J. Modern Optical Engineering. McGraw-Hill, 1990, p 484
 Kroeze, Hugo www.rfglobalnet.com/forums/Default.aspx?g=posts&m=61096 Mar 2002
 Self, Douglas The Design of Active Crossovers. Newnes.
 Self, Douglas Audio Power Amplifier Handbook 5th edition. Newnes, p 43 (amp output cap)
 Bateman, Cyril Capacitor Sound? Parts 1–6. Electronics World Jul 2002–Mar 2003
 Self, Douglas Audio Power Amplifier Design 6th edition, Newnes, 2013, p 299 (COG cap)
 Self, Douglas Self-Improvement for Capacitors, Linear Audio, Vol. 1, Apr 2011, p 156
 Bateman, Cyril Capacitor Sound? Part 3. Electronics World, Oct 2002, pp 16, 18
 Bateman, Cyril Capacitor Sound? Part 4 Electronics World, Nov 2002, p 47