All the noise in resistors

Every resistor contributes some noise to your circuit. The noise is tiny, and often negligible, so why bother about it? Well, RF people have long recognised that it can dominate receiver performance if it gets out of hand in front-ends. Audio amplifier designers can also find it troublesome. Now, as 12-bit Analogue-to-Digital Converters (ADCs) give way to 16-, 20- and even 24-bit types, resistor noise can be important in any device that collects digital information, and that catches lots of applications. I recently built an instrument in which resistor noise reduced my beautiful 20-bit ADC to a mere 14 Effective Number Of Bits (ENOB), and I had to learn a good deal more about noise to get it working properly. This was annoying, as I was under the fond illusion that I'd already done my noise sums properly.

Resistor noise comes in two basic flavours, Thermal (or Johnson) noise, and Flicker (also known as excess, contact, or 1/f) noise. Thermal noise afflicts all resistors equally. Flicker noise, by contrast, afflicts resistors according to their mode of construction. Consulting that great authority of the 21st century, the internet, one can readily obtain the impression that flicker noise is an historic problem with the almost-extinct carbon composition resistor, and no concern to the modern engineer. Dream on. Flicker noise is alive and well, and emanating from your nice modern surface-mount resistors. In the appropriate circumstances, which are by no means rare, flicker noise can be orders of magnitude larger than thermal noise.

[center][resize]http://www.hartmantech.com/codesign/blog_files/Res_noise_fig_1.png[/resize][/center]

As an engineer I treat noise as being defined by its effect on circuits. If you are interested in the underlying quantum physics, have a look at reference [6] for thermal noise, and reference [2] for flicker noise.

[b]Thermal Noise[/b]

This is the one all the textbooks cover.

Thermal noise, as one might reasonably expect, rises with absolute temperature. Since few of us have the luxury of operating our circuitry at absolute zero temperature, it is always present. It does not depend on the electrical operating conditions of the resistor. The classic formula, derived from [url=http://en.wikipedia.org/wiki/Johnson_noise]experimental work by Johnson and theoretical work by Nyquist[/url], is:

$$E_t^2 = 4KTRB$$[right](1)[/right]
or
$$E_t &= \sqrt{4KTRB}$$[right](2)[/right]

Where:
[list]
[*]$$E_t$$ is the mean thermal noise voltage
[*]$$K$$ is Boltzmann's constant, $$1.38 \times 10^{-23}$$ Joules/Kelvin
[*]$$R$$ is resistance, in Ohms
[*]$$T$$ is absolute temperature in Kelvins, or (°C + 273.15)
[*]$$B$$ is bandwidth, in Hz[/list]
Because [url=http://en.wikipedia.org/wiki/Ludwig_Boltzmann]Herr Doktor Professor Boltzmann[/url] was kind enough to keep his eponymous constant nice and small, this noise is not too great at room temperature, even for large values of R and B. For example, a 10k resistor at 20°C in an audio circuit covering 20Hz-20kHz will have a mean noise voltage of about 1.8 µV. If your signal is above 1.8 mV, this leaves you with a Signal to Noise (S/N) ratio better than 60dB, which used to be acceptable. CDs, with their 16-bit representation, promise (but rarely deliver) a S/N of 98dB. At this resolution, one would need a signal of 246 mV just to keep the thermal noise from this one resistor down to half a Least Significant Bit (lsb). Here one starts to see the glimmering of a problem - most circuits have a lot more then one resistor. Worse, audio circuits must often deal in signals of only a few millivolts. In a 20-bit ADC application, the minimum signal level becomes 3.89 volts, and the problem is seriously in your face. There is no place for a 10k resistor in the signal path of an audio 20-bit converter operating at 3.3V.

The bad news is that, short of moving into an igloo, there is not a lot we can do about this. Any decent circuit designer will, of course, maintain her cool at all times. It important to remember that the T in the equation is the temperature of the resisive element, not the ambient temperature. When a resistor is in use, it dissipates power, and gets hot. A tiny resistor it can get surprisingly hot on just a few milliwatts [1]. So, size matters; choose a larger resistor to reduce the temperature rise for a given dissipation.

Circuit bandwidth is often dictated by factors we cannot control. Where it is available, bandwidth restriction is clearly a powerful tool, which accounts for the lingering popularity of Morse code amongst radio operators. Slow Morse needs only a few Hz of bandwidth, as against perhaps 3kHz for voice, providing an improvement in S/N of perhaps 30dB. Clearly, at the least, we should be wary of providing more bandwidth than we really need. Filters are your friend, but beware - they are often full of resistors...

Notice that the noise formula takes no account of the operating frequency, only the bandwidth. Thermal noise is uniform from the lowest frequencies to somewhere around 80GHz. Shifting frequency will not alleviate thermal noise in the slightest. The only remaining parameter is resistance, and this is often entirely under the control of the designer. Keeping down resistor values is every bit as powerful as reducing bandwidth, and the only downside is a little more power consumption. Even in a battery-powered device, low resistor values in the small-signal stages can be a lesser evil than high noise levels. Just be quite sure your op-amps can drive the low value resistors without running out of current capability in the output stage. Notice that a low-valued resistor will dissipate more power at a given signal level, and hence run hotter, and noisier; size it up to reduce this effect. Yes, Fred, that's a board spin.

Finally, note that whatever method you use to battle thermal noise, that square root in equation (2) will greatly hamper you.

[b]Flicker noise[/b]

This is the forgotten one, which sneaks up and kicks you right in the BOM (Bill Of Materials).

Because Flicker noise is not a fundamental property of a resistor, it must be measured empirically for each type. It is calculated as:

$$E_f &= N_i V_b$$[right](3)[/right]
or
$$N_i &= \frac{E_f}{ V_b}$$[right](4)[/right]

Where:
[list]
[*]$$E_f$$ is the mean flicker noise voltage [i]for each decade of frequency.[/i]

[*]$$N_i$$ is the noise index of the resistor, in Volts/Volt.

[*]$$V_b$$ is the voltage across the resistor.[/list]
Equation (4) defines the Noise Index $$N_i$$. You will often see a noise index given in dB, in which case what is meant is dB relative to one microvolt/Volt.

In equation (3) it is easy to see that the noise is proportional to the inverse of frequency. There is the same amount of noise in the decade band from 1Hz to 10Hz as in the band from 10Hz to 100Hz, so clearly the noise per Hz in the former band is ten times larger. While this equation is simple, it is clumsy to use if your bandwidth is not an integer number of decades. In that case the fuller form is handier:

$$E_f &= N_i V_b \log_{10}{(\frac{F_h}{F_l})}$$[right](5)[/right]

Where:
[list]
[*]$$E_f$$ is the mean flicker noise voltage in band
[*]$$F_h$$ is the high-frequency limit of the band of interest
[*]$$F_l$$ is the low-frequency limit of the band of interest[/list]
Flicker noise is strongly dependent on the size of a resistor. In resistors of similar construction, larger resistors have less flicker noise.

The construction of the resistor also has a profound influence. In general, wirewound and metal-foil types have the lowest flicker noise. Carbon-composition resistors have long been regarded as the type most prone to flicker noise, which turns out to be one of those urban myths.

Carbon-composition resistors were the mainstay of electronics for many years, but young engineers may never have seen one. They are cheap, low inductance, and withstand enormous surges gracefully. On the other hand, they have a manufacturing tolerance of 20%, 10% or at best 5%, and change value, nonlinearly, by 20% or more between -55°C and 105°C, and by 10% or more in response to humidity changes. Their voltage coefficient of resistance is enough to introduce audible distortion in an audio signal, when the signal level is above a few volts. They also suffer, outrageously, from flicker noise. Commonly quoted figures suggest that a noise index of -12dB to +6db is to be expected. Oddly, the few manufacturers still making this type of resistor [2] do not give noise specifications. The resistive element is a rod of finely-divided carbon powder mixed with a ceramic filler and a resin binder, fired to stabilise it. Since it seemed that the high flicker noise related to the granular nature of the resistive material, it was sometimes called contact noise. Modern physicists seem to prefer explanations based on tunnelling between particles [3].

Because of its poor stability, the carbon-composition resistor is no longer popular, except amongst vintage guitar-amplifier fans. Now we have surface-mount thick-film resistors based on a ceramic chip, which are commonly sold with 1% or better manufacturing tolerance and a temperature coefficient of 200 ppm, meaning that an 80 degree temperature change leads to only a 1.6% change in resistance. Humidity causes about a 3% change. The resistive element is a screen-printed film of finely-divided metal oxide powder mixed with a glass or ceramic binder, fired to stabilise it -- in other words, it's a 21st-century carbon-composition resistor. If we believe that a granular resistive material distributed in a binder is noisy, we might guess that it would suffer outrageously from flicker noise, which turns out to be the case.

The best published data seem to be by Panasonic [4], who have produced a graph of noise index by size and value for their thick-film and thin-film resistors. This shows that thick-film types can have a noise index in the range of -33db to +12dB, so that at their worst they are a good deal noisier than the much-maligned carbon-composition types. For a 10k resistor the noise index is only -12dB for the 1 Watt 6432 (2512 Imperial) size, but as high as +3dB for the 1005 (0402 Imperial) size. Murphy must be sleeping on the job here, because sizing up will simultaneously reduce thermal and flicker noise. This carries over into thin-film types, in which 2012 (0805 Imperial) types show a fairly uniform -38dB, with 1608 (0603 Imperial) types a hair noisier at -37dB. Noise in thin-film resistors is very low, almost comparable with bulk-foil and wirewound types, which are often characterised as having a -40dB noise index or a little better.

Panasonic's graphs also show the strong dependence of noise index on resistance value, for the thick-film types. In the common 1608 (0603 Imperial) size, noise index ranges from -31dB at 10 Ohms to +9dB (eek!) at 1 MOhm. In thin-film types, the noise is largely independent of value.

To combat flicker noise, our most powerful weapon is simply to use a metal foil, wirewound or thin-film resistor wherever possible, in decreasing order of preference (and cost). In critical positions, size up. We should also strive to minimise the voltage across the resistor. In a perfect world, zero volts gives zero noise, but don't forget you need to take account of the signal as well as DC bias. If the signal is much smaller than the DC bias, flicker noise will be almost constant, but if the bias is zero, the flicker noise becomes multiplicative.

[b]Total Noise[/b]

Remember that uncorrelated noise sources do not add directly. We need

$$E_{tot} &= \sqrt{E_t^2 +E_f^2}$$[right](6)[/right]

Where:
[list]
$$E_{tot}$$ is the total mean noise voltage.
[/list]

The practical upshot of this is that once one of the noise sources is more than about three times the amplitude of the other, it dominates so strongly that it can safely be treated as the only noise source.

This is also the correct way to combine the noise contributions of all the resistors in the circuit, with due attention to circuit gain. Calculate the total noise in each resistor from equation (6), then multiply it by the circuit gain between the resistor and the point of measurement. Now square each of these noise contributions, add the squares, and take the square root. Voila! - total circuit noise. Caution, this number may be suitable for adults only...

[b]Noise in real circuits[/b]

How do these noise sources compare in a real circuit? One way to get a grip on this is to calculate a "noise corner frequency" for a resistor, the frequency at which the thermal and flicker noise contributions are equal. This approach is commonly used to characterise operational amplifiers. Above the corner frequency it is safe to ignore flicker noise: at lower frequencies, flicker noise is dominant, and thermal noise can be ignored. If the corner frequency is within or above the frequency band of interest, you neglect flicker noise at your peril. Precision ADCs often operate at very low sample rates, so that the effective frequency band is also very low, and that is where flicker noise is all too inclined to eat your lunch.

For example, that 10k 1005 (0402 Imperial) thick-film resistor has a noise index of 3dB. With just 3V of bias, it has a corner frequency of a little over 48kHz, which means that flicker noise dominates right through the audio band. I graphed this at the start of the article, just to horrify you.

Change to a 2012 (0805 Imperial) thin-film type, with a -38dB noise index, and the corner frequency drops to 4 Hz. This is below the audio band, and noise in the 20Hz-20kHz audio band has dropped a useful 12.5dB. For slow, precise (6sps, 24 bit) ADCs the improvement is even more impressive. Noise in the 1-10Hz band has fallen from 4.2uV to 54nV, a 38dB drop.

Here is a [url=http://www.hartmantech.com/codesign/blog_files/Resistor_noise.xls]simple little spreadsheet[/url] which will allow you to calculate the resistor corner frequency and characterise the noise. User input goes in the cells with a yellow background, all else is calculated. A free download, licensed under [url=http://www.gnu.org/licenses/gpl.html]GNU General Public license version 3[/url]. Let me know in the comments if you find it useful.

[b]References:[/b]
[list=1]
[*][url=http://www.vishay.com/docs/28844/tmismra.pdf]Thermal Management in Surface-Mounted Resistor Applications[/url] by Vishay. Thermal imaging in figure 6, the graph in Figure 9, and thermal resistances in Table one, are particularly relevant here.

[*]At least [url=http://www.ohmite.com]Ohmite[/url], [url=http://www.seielect.com/]Stackpole[/url], and [url=http://www.tycoelectronics.com/]Tyco[/url] still include carbon composition resistors in their current product offerings as of May, 2014. The well-respected Allen-Bradley carbon composition resistors went out of production after Rockwell purchased Allen-Bradley in 1985. These resistors are still traded as NOS (New Old Stock) parts.

[*][url=http://arxiv.org/abs/physics/0204033]"1/f Noise: a Pedagogical review"[/url] by Eduardo Milotti, 2002.

[*][url=http://www.hartmantech.com/codesign/blog_files/Surface_mount_resistors_tech_guide.pdf]Surface mount resistors technical guide by Panasonic[/url], 1999. Shaky English, but the graphs tell the whole tale. Flicker noise data are on page 12.

[*][url=http://www.hartmantech.com/codesign/blog_files/Resistor_noise.xls]Resistor noise calculator[/url] by Hartman Technica, 2014

[*][url=http://link.aps.org/abstract/PR/v32/p110]Thermal Agitation of Electric Charge in Conductors[/url] by H. Nyquist, 1928, is the canonical paper on thermal noise theory[/list]