Created on 2013-12-17 20:25:00

*Over the course of several issues of Audio.TST in 2013, Audio Precision Co-founder Bruce Hofer wrote several columns about the impact of resistors upon ultra-low noise analog design. These articles are collected below for readers of Audio.TST.*

### Passive Imperfect

Perhaps the most important lesson I have learned since the beginning of my career in high performance analog design is that the quality of the components is paramount. The most clever circuit design cannot make up for certain component imperfections.

Consider the simple resistor. Real world resistors exhibit both thermal and voltage coefficient effects. The first is commonly known as temperature coefficient (or *tempco*), and usually has units of %/C or ppm/C. It causes gain and bias variations as the operating temperature is varied. Less well known is the fact that temperature coefficient can also result in distortion as the signal causes thermal modulation of the resistor value. This is typically seen at low frequencies where the resistor temperature can begin to follow the changes in instantaneous power dissipation.

Resistor voltage coefficient effects are just as real and can also cause unwanted distortion. Here the typical unit is in ppm/V. Voltage coefficient is a bit more mysterious because few vendors publish data or give a specification. DVM designers are quite sensitive to this effect because it limits accuracy with higher voltage inputs. In the world of audio, this same characteristic can result in measurable distortion at typical line levels. The move towards smaller and smaller components has been insidious because it exacerbates both thermal and voltage coefficient effects. The distortion performance in circuits built using 0402 or 0201 surface mount components will be substantially greater than using larger sizes.

### Resistance is Noisy

Several months ago I wrote about non-linearity in resistors. This month I would like to comment about resistors being noise generators. Almost any good electronics textbook will either contain or derive the formula:

where” Vn” is the rms noise voltage measured over the bandwidth “BW”, “k” is Boltzmann’s constant, “T” is absolute temperature, and “R” is the value of the resistor. There are several important insights revealed by this formula.

First, noise voltage is proportional to .

An alternative way to look at this is that noise “power” is proportional to BW, or the noise power density is constant per Hertz of bandwidth. In the world of audio design, there is usually very little than can be done with this portion of the equation.

Second, noise voltage is proportional to .

Room temperature is normally taken to be about +23C which is 296K in the absolute scale. However, the temperature that really matters is the temperature of the resistive element inside the resistor. This can be significantly higher than room ambient depending upon power dissipation and temperature rises inside the product. A 30C rise roughly causes about 5% or 0.4 dB higher noise voltage, all other factors being equal.

Finally, noise voltage is proportional to .

Larger valued resistors simply produce larger amounts of noise. Now, to be honest, one needs to calculate the transfer function from a given resistor to the circuit output node to make a fair assessment. However, the lesson here is that one should design with lowest value of resistors that a circuit will tolerate. Unfortunately way too many audio engineers seem to overlook this fact.

The design of extremely low-noise, high performance analog continues to be a challenge that no number of advances in the digital realm can fully address, as everything audio starts and ends as analog data. The pursuit of as much perfection as physics will allow remains my goal here at Audio Precision, and I welcome your feedback about how our equipment can help you along this shared path.

### Shot Noise

In previous months I have been sharing some thoughts regarding noise in resistors. However no discussion of noise would be complete without including the topic of “shot noise”. Shot noise is quite simply the random fluctuation in current flow caused by the fact that electronic charge comes only in discrete steps, i.e. the charge of an electron.

If we build a circuit and cause 1 mA of dc current to flow through a resistor, there will also be a small ac noise component superimposed. The magnitude of this noise current can be derived:

…where q is the charge of an electron, or 1.6022e-19 coulombs.

One immediate observation is that shot noise is independent of temperature where resistor noise is highly dependent upon temperature. Both are proportional to sqrt(BW), where BW is the measurement bandwidth of interest. There is also an interesting similarity in that shot noise if proportional to sqrt(Idc) while resistor noise is proportional to sqrt(R).

In the above example where Idc = 1 mA and BW = 20 kHz, the associated shot noise is 2.53 nA (rms). This may seem like a negligibly small value (about -111.9 dB in comparison to the 1 mA dc component), but the designer must be careful to analyze how this affects a given circuit in relation to the intended signal levels.

Shot noise tends to dominate the input noise current of bipolar op-amps. Shot noise also contributes to the noise voltage developed across semiconductor junctions, for example in the familiar emitter coupled pair. Indeed, deriving the equation for the noise voltage of an emitter coupled pair is surprisingly difficult. Beware that many such published formulas are either over-simplified or just plain wrong.