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In practice, it is virtually impossible to measure noise within specific frequency limits with no contribution from outside those limits, since practical filters have finite rolloff characteristics. Fortunately, measurement error introduced by a single pole lowpass filter is readily computed. The noise in the spectrum above the single pole filter cutoff frequency, f

**Operational Amplifier Noise**
Popcorn noise is so-called because when played through an audio system, it sounds like cooking popcorn. It consists of random step changes of offset voltage that take place at random intervals in the 10+ millisecond timeframe. Such noise results from high levels of contamination and crystal lattice dislocation at the surface of the silicon chip, which in turn results from inappropriate processing techniques or poor quality raw materials.

When monolithic op amps were first introduced in the 1960s, popcorn noise was a dominant noise source. Today, however, the causes of popcorn noise are well understood, raw material purity is high, contamination is low, and production tests for it are reliable so that no op amp manufacturer should have any difficulty in shipping products that are substantially free of popcorn noise. For this reason, it is not even mentioned in most modern op amp textbooks.

**RMS Noise Considerations**

As was discussed above, noise spectral density is a function of frequency. In order to obtain the RMS noise, the noise spectral density curve must be integrated over the bandwidth of interest.

In the 1/f region, the RMS noise in the bandwidth F

_{L}to F_{C}is given by
where v

_{nw}is the voltage noise spectral density in the "white" region, F_{L}is the lowest frequency of interest in the 1/f region, and FC is the 1/f corner frequency.
The next region of interest is the "white" noise area which extends from F

_{C}to F_{H}.
The RMS noise in this bandwidth is given by

Eq. 1-25 and 1-26 can be combined to yield the total RMS noise from F_{L}to F_{H}:
In many cases, the low frequency p-p noise is specified in a 0.1 to 10Hz bandwidth, measured with a 0.1 to 10Hz bandpass filter between op amp and measuring device.

Figure 1-70: The peak-to-peak noise in the 0.1Hz to 10Hz bandwidth for the OP213 is less than 120nV

The measurement is often presented as a scope photo with a time scale of 1s/div, as is shown in Figure 1-70 above for the OP213.

Figure 1-71: Input voltage noise for the OP177

It is possible to relate the 1/f noise measured in the 0.1 to 10Hz bandwidth to the voltage noise spectral density. Figure 1-71 above shows the OP177 input voltage noise spectral density on the left-hand side of the diagram, and the 0.1 to 10Hz peak-to-peak noise scope photo on the right-hand side. Equation 1-26 can be used to calculate the total RMS noise in the bandwidth 0.1 to 10Hz by letting F

_{L}= 0.1Hz, F_{H}= 10Hz, F_{C}= 0.7Hz, v_{nw}= 10nV/√Hz. The value works out to be about 33nV RMS, or 218nV peak-to-peak (obtained by multiplying the RMS value by 6.6— see the following discussion). This compares well to the value of 200nV as measured from the scope photo.
It should be noted that at higher frequencies, the term in the equation containing the natural logarithm becomes insignificant, and the expression for the RMS noise becomes:

And, if F

_{H}>> F_{L},
However, some op amps (such as the OP07 and OP27) have voltage noise characteristics that increase slightly at high frequencies. The voltage noise versus frequency curve for op amps should therefore be examined carefully for flatness when calculating high frequency noise using this approximation.

At very low frequencies when operating exclusively in the 1/f region, F

Note that there is no way of reducing this 1/f noise by filtering if operation extends to DC. Making F_{C}>> (F_{H}–F_{L}), and the expression for the RMS noise reduces to:_{H}=0.1Hz and F_{L}= 0.001 still yields an RMS 1/f noise of about 18nV RMS, or 119nV peak-to-peak.
Figure 1-72: Equivalent noise bandwidth

The point is that averaging results of a large number of measurements over a long period of time has practically no effect on the RMS value of the 1/f noise. A method of reducing it further is to use a chopper stabilized op amp, to remove the low frequency noise.

In practice, it is virtually impossible to measure noise within specific frequency limits with no contribution from outside those limits, since practical filters have finite rolloff characteristics. Fortunately, measurement error introduced by a single pole lowpass filter is readily computed. The noise in the spectrum above the single pole filter cutoff frequency, f

_{c}, extends the corner frequency to 1.57f

_{c}. Similarly, a two pole filter has an apparent corner frequency of approximately 1.2f

_{c}. The error correction factor is usually negligible for filters having more than two poles. The net bandwidth after the correction is referred to as the filter equivalent noise bandwidth (see Figure 1-72 opposite).

It is often desirable to convert RMS noise measurements into peak-to-peak. In order to do this, one must have some understanding of the statistical nature of noise. For Gaussian noise and a given value of RMS noise, statistics tell us that the chance of a particular peak-to-peak value being exceeded decreases sharply as that value increases— but this probability never becomes zero.

Thus, for a given RMS noise, it is possible to predict the percentage of time that a given peak-to-peak value will be exceeded, but it is not possible to give a peak-to-peak value which will never be exceeded as shown in Figure 1-73 below.

Figure 1-73: RMS to peak-to-peak ratios

Peak-to-peak noise specifications, therefore, must always be written with a time limit. A suitable one is 6.6 times the RMS value, which is exceeded only 0.1% of the time.

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