Deep Brain Supplement - You Must Get These Noise Relationships of Amplifiers!

The general characteristics of operational amplifier current or voltage noise are shown in Figure 1 below.

Figure 1: Frequency characteristics of op amp noise

At high frequencies, the noise is white (i.e., its spectral density does not vary with frequency). This is true for most of the frequency range of the op amp, but at low frequencies, the noise spectral density rises at 3 dB/octave, as shown in Figure 1 above.

The power spectral density is inversely proportional to frequency in this region, so the voltage noise spectral density is inversely proportional to the square root of the frequency. For this reason, this noise is often called "1/f noise." Note, however, that some textbooks still use the old term "flicker noise." The frequency at which this noise begins to increase is called the "1/f corner frequency" (F _C ), and is one of the quality factors—the smaller the better. The 1/f corner frequencies of voltage noise and current noise are not necessarily the same for a particular amplifier, and some current feedback op amps may have three 1/f corner frequencies: one for its voltage noise, another for its inverting input current noise, and another for its non-inverting input current noise.

The general calculation formula used to describe the voltage or current noise spectral density in the 1/f region is as follows:

where k is the “white” current or voltage noise level and F _C is the 1/f corner frequency.

The best low frequency, low noise amplifiers have corner frequencies in the 1-10 Hz range, while JFET devices and more common op amps are in the 1-100 Hz range. However, very high speed amplifiers may compromise processing power in order to achieve high speed performance, resulting in quite poor 1/f corner frequency characteristics of hundreds of Hz or even 1-2kHz. This is usually not very important for the wideband applications such devices are intended for, but it may affect their use in audio conditions, especially in equalization circuits.

RMS Noise Considerations As mentioned above, the noise spectral density is a function of frequency. To obtain the rms noise, the noise spectral density curve must be integrated over the entire bandwidth of interest.

In the 1/f region, the rms noise in the bandwidth _{FL to} FC is given by

Where vnw is the voltage noise spectral density in the “white” region, FL is the lowest target frequency in the 1/f region, and _{FC is} the 1/f corner frequency.

The next region of interest is the “white” noise region from F _C to F _H. The rms noise in this bandwidth is given by

The total rms noise from F _L to F _H is given by :

In many cases, low frequency peak-to-peak noise is specified in a 0.1 Hz to 10 Hz bandwidth, measured with a 0.1 to 10 Hz bandpass filter between the op amp and the device being measured. The measurement is often presented as an oscilloscope plot with a time scale of 1 s/div, as shown in Figure 2 below.

Figure 2: 0.1Hz to 10 Hz input voltage noise of the OP213

Figure 3: Input voltage noise of the OP177

The 1/f noise measured in a 0.1 to 10 Hz bandwidth can be related to the voltage noise spectral density. The figure above shows the OP177 input voltage noise spectral density on the left and a 0.1 to 10 Hz peak-to-peak noise oscilloscope on the right.

Note that at higher frequencies, the terms involving the natural logarithm become insignificant and the rms noise expression becomes

If F _H >> F _L

However, some op amps have a voltage noise characteristic that increases slightly at high frequencies. So When using this approximation to calculate high frequency noise, the flatness of the op amp voltage noise vs. frequency curve should be carefully checked.

At very low frequencies, when operating only in the 1/f region, _{F C >> ( F H – FL} ) _, the rms _noise expression simplifies to

Note that this 1/f noise cannot be reduced by filtering if the operating range is extended to dc. Letting F _H = 0.1 Hz and F _L = 0.001 Hz still yields an rms 1/f noise of approximately 18 nV rms or 119 nV pp. The problem is that averaging a large number of measurements over a long period of time has virtually no effect on the rms value of the 1/f noise. A way to further reduce the 1/f noise is to use a chopper-stabilized op amp, which removes the low frequency noise.

In practice, it is almost impossible to measure noise within a specific frequency limit without being affected by noise outside of the limit because of the finite roll-off characteristics of real filters. Fortunately, the measurement error caused by a single-pole low-pass filter is easy to calculate. Noise in the spectrum above the cutoff frequency, _{fc, of a single-pole low-pass filter extends the corner frequency to 1.57fc} . Similarly, the apparent corner frequency of a two-pole filter is about 1.2fc _. For filters with more than two poles, the error correction factor is usually negligible. The net bandwidth after correction is called the "equivalent noise bandwidth" of the filter (see Figure 4 below).

Figure 4: Equivalent noise bandwidth

It is often necessary to convert RMS noise measurements to peak-to-peak values. To do this, we must have some understanding of the statistical properties of noise. For Gaussian noise and a given RMS noise value, statistics tells us that the probability of exceeding a certain peak-to-peak value decreases dramatically as that value increases, but the probability will never be zero.

Therefore, for a given value of rms noise, it is possible to predict the percentage of time that a given peak-to-peak value will be exceeded, but there is no peak-to-peak value that can never be exceeded, as shown in Figure 5 below.

Figure 5: RMS-to-peak ratio

Therefore, the peak-to-peak noise specification must be written with a time limit. A good value is 6.6 times the RMS value, meaning that the value is exceeded only 0.1% of the time.

Well, that's all for today. Do you understand the relationship between the 1/f noise, RMS noise, and equivalent noise bandwidth of an operational amplifier?