*(Noise, continued from the last issue, 24-2)*

*Q. What is “noise gain”?*

A. So far we have considered noise sources but not the gain of the circuits where they occur. It is tempting to imagine that if the noise voltage at the input of an amplifier is V_{n} and the circuit’s signal gain is G, the noise voltage at the output will be GV_{n}; but this is not always the case.

Consider the basic op-amp gain circuit in the diagram. If it is being used as an inverting amplifier (B), the non-inverting input will be grounded, the signal will be applied to the free end of R_{i} and the gain will be -R_{f}/R_{i}. On the other hand, in a non-inverting amplifier (A) the signal is applied to the non-inverting input and the free end of R_{i} is grounded; the gain is (1 + R_{f}/R_{i})

The amplifier’s own voltage noise is always amplified in the non-inverting mode; thus when an op-amp is used as an inverting amplifier at a gain of G, its voltage noise will be amplified by the noise gain of (G+ 1). For the precision attenuation cases, where G<1, this may present problems. (A common example of this is an active filter circuit where stopband gain may be very small but stop-band noise gain is at least unity.)

Only the amplifier voltage noise-and any noise developed by the noninverting-input current noise flowing in any impedance present in that input (for example, a bias-current compensation resistor)-is amplified by the noise gain. Noise in R_{i}, either Johnson noise or arising from inverting input noise current, is amplified by G in the same way as the input signal, and Johnson noise voltage in the feedback resistor is not amplified but is buffered to the output at unity gain.

*Q. What’s “popcorn” noise?*

A. Twenty years ago this column would have spent a great deal of space discussing popcorn noise, which is a type of low frequency noise manifesting itself as low level (but random amplitude) step changes in offset voltage occurring at random intervals. When played through a loudspeaker it sounds like cooking popcorn-hence the name.

While no integrated circuit process is entirely free from the problem, high levels of popcorn noise result from inadequate processing techniques. Today its causes are sufficiently well understood that no reputable op-amp manufacturer is likely to produce op-amps where popcorn noise is a major concern to the user. {Oat-bran noise is more likely to be an issue in situations where cereal data is concerned[:-)]}

*Q. Pk-pk noise voltage is the most convenient way to know whether noise will ever be a problem for me. Why are amplifier manufacturers reluctant to specify noise in this way?*

A. Because noise is generally Gaussian, as we pointed out in the last issue. For a Gaussian distribution it is meaningless to speak of a maximum value of noise: if you wait long enough any value will, in theory, be exceeded. Instead it is more practical to speak of the rms noise, which is more or less invariant-and by applying the Gaussian curve to this we may predict the probability of the noise exceeding any particular value. Given a noise source of V rms, since the probability of any particular value of noise voltage follows a Gaussian distribution, the noise voltage will exceed a pk-pk value of 2 V for 32% of the time, 3 V for 13% of the time, and so on:

Pk-pk value | % of time pk-pk value is exceeded |

2 × rms | 32% |

4 × rms | 4.6% |

6 x rms | 0.27% |

6.6 × rms | 0.10% |

8 × rms | 60 ppm |

10 × rms | 0.6 ppm |

12 × rms | 2 x 10^{-9} ppm |

14 × rms | 2.6 × 10^{-12} ppm |

So if we define a peak value in terms of the probability of its occurrence, we may use a peak specification-but it is more desirable to use the rms value, which is generally easier to measure. When a peak noise voltage is specified, it is frequently 6.6 x rms, which occurs no more than 0.1% of the time.

*Q. How do you measure the rms value of low-frequency noise in the usually specified band, 0.1 to 10 Hz? It must take a long time to integrate. Isn’t this expensive in production?*

A. Yes, it is expensive, but-Although it’s necessary to make many careful measurements during characterization, and at intervals thereafter, we cannot afford the time it would take in production to make an rms measurement. Instead, at very low frequencies in the 1/f region (as low as 0.1 to 10 Hz), the peak value is measured during from one to three 30-second intervals and must be less than some specified value. Theoretically this is unsatisfactory, since some good devices will be rejected and some noisy ones escape detection, but in practice it is the best test possible within a practicable test time and is acceptable if a suitable threshold limit is chosen. With conservative weightings applied, this is a reliable test of noise. Devices that do not meet the arbitrary criteria for the highest grades can still be sold in grades for which they meet the spec.

*Q. What other op-amp noise effects do you encounter?*

A. There is a common effect, which often appears to be caused by a noisy op amp, resulting in missing codes. This potentially serious problem is caused by ADC input-impedance modulation. Here’s how it happens:

Many successive-approximation ADCs have an input impedance which is modulated by the device’s conversion clock. If such an ADC is driven by a precision op amp whose bandwidth is much lower than the clock frequency, the op amp cannot develop sufficient feedback to provide a stiff voltage source to the ADC input port, and missing codes are likely to occur. Typically, this effect appears when amplifiers like the OP-07 are used to drive AD574s.

It may be cured by using an op amp with sufficient bandwidth to have a low output impedance at the ADC’s clock frequency, or by choosing an ADC containing an input buffer or one whose input impedance is not modulated by its internal clock (many sampling ADCs are free of this problem). In cases where the op amp can drive a capacitive load without instability, and the reduction of system bandwidth is unimportant, a shunt capacitor decoupling the ADC input may be sufficient to effect a cure.

*Q. Are there any other interesting noise phenomena in high-precision analog circuits?*

A. The tendency of high-precision circuitry to drift with time is a noise-like phenomenon (in fact, it might be argued that, at a minimum, it is identical to the lower end of 1/f noise). When we specify long-term stability, we normally do so in terms of µV/1,000 hr or ppm/1,000 hr. Many users assume that, since there are, on the average, 8,766 hours in a year, an instability of x/1,000 hr is equal to 8.8 x/yr.

This is not the case. Long-term instability (assuming no long-term steady deterioration of some damaged component within the device), is a “drunkard’s walk” function; what a device did during its last 1,000 hours is no guide to its behavior during the next thousand. The long-term error mounts as the square-root of the elapsed time, which implies that, for a figure of x/l,000 hr, the drift will actually be multiplied by √8.766, or about 3x per year, or 9x per 10 years. Perhaps the spec should be in µV/1,000√hr.

In fact, for many devices, things are a bit better even than this. The “drunkard’s walk” model, as noted above, assumes that the properties of the device don’t change. In fact, as the device gets older, the stresses of manufacture tend to diminish and the device becomes more stable (except for incipient failure sources). While this is hard to quantify, it is safe to say that-provided that a device is operated in a low-stress environment-its rate of long-term drift will tend to reduce during its lifetime. The limiting value is probably the 1/f noise, which builds up as the *square-root of the natural logarithm* of the ratio, i.e., √1n 8.8 for time ratios of 8.8, or 1.47 x for 1 year, 2.94 x for 8.8 years, 4.4 x for 77 years, etc.

### A Reader's Challenge:

Q. A reader sent us a letter that is just a wee bit too long to quote directly, so we’ll summarize it here. He was responding to the mention in these columns (*Analog Dialogue* 24-2, pp. 20-21) of the shot effect, or Schottky noise (Schottky was the first to note and correctly interpret shot effect-originally in vacuum tubes1). Our reader particularly objected to the designation of shot noise as solely a junction phenomenon, and commented that we have joined the rest of the semiconductor and op-amp engineering fraternity in disseminating misinformation.

In particular, he pointed out that the shot noise formula-

I_{n} = √(2q IB) amperes,

where I_{n} is the rms shot-noise current, I is the current flowing through a region, q is the charge of an electron, and B is the bandwidth-does not seem to contain any terms that depend on the physical properties of the region. Hence (he goes on) shot noise is a *universal* phenomenon associated with the fact that any current, I, is a flow of electrons or holes, which carry discrete charges, and the noise given in the formula is just an expression of the graininess of the flow.

He concludes that the omission of this noise component in any circuit carrying current, including purely resistive circuits, can lead to serious design problems. And he illustrates its significance by pointing out that this noise current, calculated from the flow of dc through any ideal resistor, becomes equal to the thermal Johnson noise current at room temperature when only 52 mV is applied to the resistor-and it would become the dominant current noise source for applied voltages higher than about 200 mV.

A. Since designers of low-noise op amps have blithely ignored this putative phenomenon, what’s wrong? The assumption that the *above shot noise equation is valid for conductors*.

Actually, the shot noise equation is developed under the assumption that the carriers are independent of one another. While this is indeed the case for currents made up of discrete charges crossing a barrier, as in a junction diode (or a vacuum tube), it is not true for metallic conductors. Currents in conductors are made up of very much larger numbers of carriers (individually flowing much more slowly), and the noise associated with the flow of current is accordingly very much smaller-and generally lost in the circuit’s Johnson noise.

Here’s what Horowitz and Hill^{2} have to say on the subject:

“An electric current is the flow of discrete electric charges, not a smooth fluidlike flow. The finiteness of the charge quantum results in statistical fluctuations of the current. If the charges act independently of each other,* the fluctuating current is ...

I *noise* (rms) = I_{nR} = (2 qI_{dc} B)^{1/2}

where q is the electron charge (1.60 x 10^{-19} C) and B is the measurement bandwidth. For example, a “steady” current of 1 A actually has an rms fluctuation of 57 nA, measured in a 10-kHz bandwidth; i.e., it fluctuates by about 0.000006%. The relative fluctuations are larger for smaller currents: A “steady” current of 1 µA actually has an rms current-noise fluctuation, over 10 kHz, of 0.006%, i.e., -85 dB. At 1 pA dc, the rms current fluctuation (same bandwidth) is 56 fA, i.e., a 5.6% variation! Shot noise is ‘rain on a tin roof.’ This noise, like resistor Johnson noise, is Gaussian and white.

“The shot noise formula given earlier assumes that the charge carriers making up the current act independently. That is indeed the case for charges crossing a barrier, as for example the current in a junction diode, where the charges move by diffusion; *but it is not true for the important case of metallic conductors, where there are long-range correlations between charge carriers. Thus the current in a simple resistive circuit has far less noise than is predicted by the shot noise formula*.* Another important exception to the shot-noise formula is provided by our standard transistor current-source circuit, in which negative feedback acts to quiet the shot noise.”

*Italics ours

^{1}Goldman, Stanford, *Frequency Analysis, Modulation, and Noise*. New York: McGraw-Hill Book Company, 1948, p. 352.

^{2}Horowitz, Paul and Winfield Hill, *The Art of Electronics*, 2nd edition. Cambridge (UK): Cambridge University Press, 1989, pp. 431-2.