Operational Amplifier Noise

Q: What should I know about op amp noise?

A: First, it is important to note the difference between the noise generated by the operational amplifier and the components in its circuits, and the noise generated by external interference or unwanted signals and the voltage or current noise generated at one end of the amplifier or its related circuits.
Interference can appear as spikes, steps, sine waves or random noise and the sources of interference exist everywhere: machinery, near power lines, RF transmitters and receivers, computers and internal circuits of the same equipment (for example, digital circuits or switching power supplies). Understanding interference, preventing interference from appearing near your circuits, knowing how it comes in and how to eliminate it or find ways to deal with it is a big topic.

If all interference is eliminated, there is still random noise associated with the operational amplifier and its resistive circuits. It constitutes the ultimate limitation of the control resolution capability of the operational amplifier. Our discussion below starts with this topic.

Q: OK, then please tell me about random noise in op amps. How is it generated?

A: The noise that appears at the output of an op amp is measured as voltage noise. But both voltage and current noise sources can contribute noise. All of the internal noise sources of an op amp are usually referred to the inputs, that is, they are considered to be uncorrelated or independent random noise generators in series or parallel with the two inputs of the ideal noise-free amplifier. We consider op amp noise to have three basic sources:
A noise voltage generator (similar to the offset voltage, usually appears in series with the noninverting input).
Two noise current generators (similar to the bias current, draining current through the two differential inputs).
Resistor noise generators (if there are any resistors in the op amp circuit, they will also contribute noise. This noise can be considered to come from current sources or voltage sources, whichever form is common in a given circuit).

Op amps can have voltage noise as low as 3 nV/Hz. Voltage noise is often emphasized as a specification, but current noise is often the limiting factor in system noise performance at high impedances. This is similar to offset, where offset voltage is often responsible for output offset, but bias current is the real responsibility. Bipolar op amps have lower voltage noise than traditional FET op amps, but current noise is still relatively high. Today's FET op amps can achieve bipolar op amp voltage noise levels while maintaining low current noise.

Q: Where does the voltage noise unit of 3 nV/Hz come from? What does it mean?
A: Let's talk about random noise. In practical applications (i.e., within the bandwidth that the designer is concerned about), many noise sources are white noise and Gaussian noise. White noise refers to noise whose noise power is independent of frequency within a given bandwidth. Gaussian noise refers to noise whose probability of occurrence of a specified amplitude X follows a Gaussian distribution. Gaussian noise has the following property: When the effective values ​​(rms) of noise from more than two sources are combined, and provided that these noise sources are uncorrelated (i.e., one noise signal cannot be converted into another noise signal), the total noise of the combination is not the arithmetic sum of these noises but the square root of their square sum (rss) (which means that the noise power is linearly superimposed, i.e., the square sum is added). For example, there are three noise sources V1, V2 and V3, and their rms sum is:
V0 = V21 + V22 + V23

Since the different frequency components of the noise signal are uncorrelated, the RSS synthesis result is: if the white noise with unit bandwidth (brick-wall bandwidth) Δf is V, then the noise with bandwidth 2Δf is V2+V2=2V. More generally, if we multiply the unit bandwidth by a factor K, then the noise with bandwidth KΔf is KV. Therefore, the function defined by the effective value of the noise with bandwidth Δf=1Hz over any frequency range is called the (voltage or current) noise spectral density function, and the unit is nV/Hz or pA/Hz. For white noise, the noise spectral density is a constant, and the total effective value noise can be obtained by multiplying the spectral density by the square root of the bandwidth.
A useful result about the RSS sum is that if there are two noise sources that contribute to the system noise, and one is 3 or 4 times larger than the other, the smaller one is often ignored because
42=16=4, but 42+12=17=4.12
The difference between the two is less than 3%, or 0.26 dB.
32 = 9 = 3, but 32 + 12 = 10 = 3.16
The difference is 6% less, or 0.5 dB.
Therefore, the larger noise source dominates the noise.

Q: What about current noise?
A: The current noise of simple (i.e., without bias current compensation) bipolar and JFET op amps is usually within 1 or 2 dB of the bias current shot noise (sometimes called Schottky noise). It is not usually given in the product specifications. Shot noise is the current noise caused by the random distribution of charge carriers through the PN junction in the form of current. If the current flowing is I, then the shot noise In within the bandwidth B can be calculated as follows:
In = 2IqB
where q is the electron charge (1.6×10-19 C). It should be noted that 2Iq is the noise spectral density, i.e., this noise is white noise.
This tells us that the current noise spectral density of a simple bipolar op amp is about 250 fA/Hz at Ib=200 nA and does not vary much with temperature, while the current noise spectral density of a JFET input op amp is lower (4 fA/Hz at Ib=50 pA) and doubles for every 20 °C increase in temperature because the bias current doubles for every 10 °C increase in temperature.
The actual current noise of an op amp with bias current compensation is much larger than the current noise predicted from its input current. The reason is that its net bias current is the difference between the input bias current and the compensation current source, while its noise current is derived from the rss sum of these two noise currents. A traditional
voltage feedback op amp with balanced inputs always has equal (but unrelated) current noise at its non-inverting and inverting inputs. Current feedback or transconductance op amps have different input structures at the two inputs, so their current noise is also different. Please refer to their product data sheets for details on the current noise at the two inputs of these two op amps.
The noise of an op amp follows a Gaussian distribution, with a constant spectral density, or "white" noise, over a wide frequency band, but as the frequency decreases, the spectral density begins to rise at 3 dB/octave. This low-frequency noise characteristic is called "1/f noise" because the noise power spectral density is inversely proportional to the frequency. It has a slope of -1 on a logarithmic scale (the slope of the noise voltage or current 1/f spectral density is -1/2). The frequency corresponding to the intersection of the -3 dB/octave spectral density line and the mid-band constant spectral density line is called the 1/f corner frequency, which is the quality factor of the amplifier. Early monolithic op amps had a 1/f corner frequency above 500 Hz, but today's op amps are common with a corner frequency of 20 to 50 Hz, and the best amplifiers (such as the ADO-27 and ADO-37) have a corner frequency as low as 2.7 Hz. 1/f noise has equal increments for frequency intervals of equal ratio (e.g., per octave or per decade).
Q: Why don't you publish the noise figure?
A: The noise figure (NF) of an amplifier is the ratio of the amplifier noise to the thermal noise of the source resistor. It is expressed in dB and can be expressed as follows:
NF = 20logVn(amp) + Vn(source)Vn(source)
where Vn(amp) is the amplifier noise and Vn(source) is the thermal noise of the source resistor.
NF is a useful specification for RF amplifiers, which are usually driven by the same source resistor (50 or 75 Ω), but it is misleading when applied to op amps, because op amps have a wide range of source impedances (not necessarily resistive) in many different applications.

Q: How does source impedance affect noise?

A: All resistors are noise sources above absolute zero, and their noise increases with resistance, temperature, and bandwidth (we will discuss basic resistor noise or thermal noise later). Reactance does not generate noise, but noise current passing through reactance will generate noise voltage.

If we drive an op amp from a source resistor, then the equivalent input noise will be the rss sum of the op amp's noise voltage, the source resistor's noise voltage, and the amplifier's noise current In flowing through the source resistor. If the source resistor is low, then the source resistor's noise voltage and the amplifier's noise current In flowing through the source resistor will not contribute significantly to the total noise. In this case, the total noise at the amplifier input will be dominated by the op amp's voltage noise.

If the source resistance is high, then the thermal noise generated by the source resistance dominates both the voltage noise of the op amp and the voltage noise caused by the current noise. However, it is important to note that since thermal noise only increases with the square root of the resistance, and the noise voltage caused by the current noise is directly proportional to the input impedance, the amplifier's current noise always dominates for sufficiently high input impedances. When both the amplifier's voltage noise and current noise are high enough, there is no question of at what value of input resistance the thermal noise dominates.

Figure 8.1 Relationship between thermal noise and source resistance

This is illustrated in Figure 8.1, which shows the voltage noise and current noise of several typical op amps from Analog Devices for a range of source resistances. The diagonal line in the figure shows the relationship between thermal noise on the ordinate and source resistance on the abscissa. Let's look at the ADOP-27 in the figure: the horizontal line shows the voltage noise of about 3 nV/Hz for source resistances less than 500 Ω. It can be seen that reducing the source impedance by 100 Ω does not reduce the noise, but increasing the source impedance by 2 kΩ does increase the noise. The vertical line of the ADOP-27 shows that when the source resistance is above about 100 kΩ, the noise voltage generated by the amplifier's current noise exceeds the thermal noise generated by the source resistance, so the current noise becomes the dominant noise source.
It should be remembered that any resistor at the noninverting input of the amplifier has thermal noise, which in turn converts the current noise into noise voltage. In addition, the thermal noise of the feedback resistor is very prominent in high-resistance circuits. All possible noise sources must be considered when evaluating the performance of an op amp.

Q: Could you please explain thermal noise?
A: At temperatures above absolute zero, all resistors have noise due to the thermal motion of charge carriers. This noise is called thermal noise, also known as Johnson noise. This property is sometimes used to measure freezing temperatures. At temperature T (Kelvin), bandwidth B Hz, the voltage noise Vn and current noise In of a resistor R Ω are given by:
Vn = 4kTRB and In = 4kTB/R
where k is the Boltzmann constant (1.38×10 -23 J/K). A rule of thumb is that a 1 kΩ resistor has a noise of 4 nV/Hz at room temperature.
The noise generated by all resistors in a circuit and the effect they have are always a consideration. In practice, only the resistors in the input circuit, feedback circuit, high gain circuit, and front-end circuit are likely to have such a significant effect on the total circuit noise.
Noise can generally be reduced by reducing resistance or bandwidth, but reducing temperature usually has little effect unless the temperature of the resistor is very low, because noise power is proportional to absolute temperature, which is T = °C + 273°.

Q: What is "noise gain"?
A: So far we have only discussed the sources of noise, but not the gain of a circuit that exhibits noise. One might think that if the noise voltage at a given input of an amplifier is Vn and the signal gain of the circuit is G, then the noise voltage at the output should be GVn. But this is not always the case.
Now consider the basic op amp gain circuit shown in Figure 8.2. If the op amp is connected as an inverting amplifier (connected to terminal B), the noninverting input is grounded, and the signal is applied to the free end of resistor Ri, then the gain is -Rf/Ri. Conversely, if the op amp is connected as a noninverting amplifier (connected to terminal A), the signal is applied to the noninverting input, and the free end of resistor Ri is grounded, then the gain is (1+Rf/Ri).

Figure 8.2 Signal gain and noise gain

The voltage noise of the amplifier itself is always amplified in the manner of a non-inverting amplifier. So when the op amp is connected as an inverting amplifier with a signal gain of G, its own voltage noise is still amplified by the noise gain (G+1). This behavior may be problematic for precision attenuation (G < 1). A common example of this is an active filter circuit, where the stopband gain may be small, but the stopband noise gain is at least 1.
Only the voltage noise generated at the amplifier input and the noise generated by any impedance flowing through the amplifier's non-inverting input current noise (for example, the noise generated by the bias current compensation resistor) are amplified by the noise gain. The noise generated by resistor Ri (whether thermal noise or voltage noise caused by the noise current at the inverting input) is amplified by G in the same way as the input signal, but the thermal noise voltage generated by feedback resistor Rf is not amplified and is buffered to the output with unity gain.

Q: What is "popcorn" noise?
A: More than 20 years ago, people spent a lot of effort to study the problem of "popcorn" noise. It is a typical low-frequency noise that occurs occasionally and manifests as low-amplitude (random) jumps in the offset voltage. When speaking through a speaker, this noise sounds like the sound of frying popcorn, hence the name.
Before the formation of integrated circuit technology, this problem did not exist at all. "Popcorn" noise was caused by surface processing problems (such as contamination) on the integrated circuit. Today, the cause of its generation is completely clear, and no well-known op amp manufacturer will have "popcorn" noise become a major concern for users.

Q: Peak-to-peak noise voltage is the most convenient way to tell me if there is a problem with noise. But why don't amplifier manufacturers like to specify noise in this way?
A: As pointed out earlier, because noise generally follows a Gaussian distribution. For a Gaussian distribution, it is meaningless to say that the noise has a maximum value, that is, if you wait long enough, it can theoretically exceed any value. In addition, the concept of effective noise value is often used in practice. In a sense, it is an invariant, that is, using the Gaussian probability distribution curve of this noise, we can predict the noise value greater than any given value.

Table 8.1 Probability of peak-to-peak value greater than specified noise level

The probability that the peak-to-peak value is greater than the specified peak-to-peak value

Assuming that the effective value of the noise source is V, since the probability of any given value of the noise voltage follows a Gaussian distribution, we can obtain: the probability of the noise voltage being greater than 2 V peak-to-peak is 32%, greater than 3 V is 13%, and so on, as shown in Table 8.1.
If we use the probability of the occurrence of the noise peak-to-peak value to define the peak-to-peak value, then the peak-to-peak value can be used as a technical indicator, but the effective value is more appropriate because it is easier to measure. When the peak noise voltage is specified, it is often 6.6 times the effective value (i.e., 6.6×rms), and the probability of its occurrence is less than 0.1%.

Q: How do you measure the effective value of low-frequency noise within the usually specified bandwidth (0.1 to 10 Hz)? This must take a long time. Isn't time in the production process precious?
A: Time is indeed precious. Although it is necessary to make many fine measurements during the characterization of the device, it is not necessary to spend so much time measuring its effective value later in the production process. The method we use is to measure its peak value within the range of 1 to 3 times the 30 s period in the very low frequency range of the 1/f region (down to 0.1 to 10 Hz), and it must be lower than a certain specified value. Although this is not a satisfactory method in theory, because some good devices may be excluded and some noise will be missed, it is actually the best method within the test time range that can be achieved. And if it is close to the appropriate threshold limit, it is also an acceptable method. From a conservative point of view, this is a reliable method to measure noise. Those devices that do not meet the highest level standard can still be sold as devices that meet this indicator level.

Q: Have you encountered other op amp noise effects?
A: There is a common noise effect that is often seen as missing codes due to op amp noise. This serious effect can be caused by modulation of the input impedance of the analog-to-digital converter (ADC). Let's look at how this effect occurs.
Many successive approximation ADCs have a finite input impedance that is modulated by the converter clock. If a precision op amp is used to drive this ADC, and the bandwidth of the op amp is much lower than the clock frequency, the op amp will not be able to generate sufficient feedback to provide a very stable voltage source at the ADC input, and missing codes may occur. This problem typically occurs when using an op amp such as the OP-07 to drive the AD574.
The solution to this problem is to use an op amp with a wide enough bandwidth to have a low output impedance despite the ADC clock frequency, or to use an ADC with an internal input buffer, or to use an ADC whose input impedance is not modulated by its internal clock (many sampling ADCs do not have this problem). In cases where the op amp can stably drive capacitive loads and the reduction in system bandwidth is not important, adding a bypass decoupling capacitor to the ADC input will solve this problem.

Q: Are there other important noise phenomena in high-precision analog circuits?
A: The tendency of high-precision circuits to drift over time is a noise-like phenomenon (it can actually be shown that this time drift is at least the same as the low-frequency end of 1/f noise). When we specify long-term stability, we usually use units of μV/1 000 h or ppm/1 000 h. Since there are 8 766 hours (h) per year (Y), the user assumes that the instability of x/1 000 h is equal to 8.8x/Y.
This is not the case. Long-term instability (assuming that a component inside the device is damaged and its performance is not long-term stable degradation) is like a "drunkard's walk", that is, the performance of the device in the first 1 000 hours is not representative of the performance in the last 1 000 hours. This long-term instability is measured as a square root relationship of the elapsed time. This means that the annual drift of an instability of x/1 000 h should actually be multiplied by 8.766, or by about 3 per year, or by about 9 per decade. This specification should be expressed in μV/1 000 h.
In practice, the long-term stability of many devices is a little better than this. As mentioned above, this "drunkard's walk" assumes that the device characteristics have not changed. In practice, as the device ages, the manufacturing stresses tend to decrease, making the performance more stable (except for the original fault source). Since it is difficult to quantitatively describe this long-term stability of the device, it is better to say that the long-term drift rate tends to decrease over the lifetime of the device, assuming that the device is operated in a low stress environment. The limit of this drift rate may be determined by 1/f noise, which can be calculated using the square root of the natural logarithm of the time ratio formula, for example, a time ratio of 8.8 x/Y corresponds to a drift rate of ln8.8 = 1.47, or a drift of 1.47x per year. Similarly, the drift of 8.8 years is 2.94x, the drift of 7.7 years is 4.4x, and so on, the drift of n years is xln(8.8n)/ln8.8.

Reader Mailbox

Q: A reader wrote to me, which is too long to quote directly, so I will summarize the content of the letter here. He expressed his views on the issue of shot noise or Schottky noise (Schottky first correctly explained the shot effect in vacuum electron tubes) in this column (Analog Dialogue 24-2, pp. 20-21). The reader specifically objected to the definition of shot noise as a junction phenomenon and commented on the problems caused by our combining operational amplifiers with other semiconductor devices as brothers to form a complete device. He specifically proposed the shot noise formula:

In = 2qIB, unit A
, where In is the effective value of the shot noise current, I is the current flowing through a certain junction area, q is the electron charge, and B is the bandwidth. This formula does not seem to contain any physical quantity that depends on the physical properties of a specific junction area. Therefore, he pointed out that shot noise is a universal phenomenon, which is related to the following fact: any current is a flow of electrons or holes, which carries discrete charges, so the noise calculated by the above formula just represents the particle nature of this current.

He believed that ignoring this noise component in any circuit that carries current (including purely resistive circuits) could lead to serious design problems. He calculated the noise generated by a DC current through any ideal resistor to illustrate the role of this noise current. If only 52 mV is applied to the resistor, the noise current generated is equal to the thermal noise current at room temperature; if more than 200 mV is applied, this noise current becomes the dominant current noise source.

A: Because the low-noise op amp designer has disregarded this subjective speculation, so where is he wrong? The above reasoning assumes that the above shot noise formula is valid for conductors.

In fact, the shot noise formula is derived from the assumption that the carriers are independent of each other. Although this shot noise is indeed a current formed by discrete charges passing through a potential barrier (composed of a junction diode or a vacuum tube), it is not a true metal conductor. Since the current in a conductor is composed of a very large number of carriers (the flow of a single carrier is very slow), the noise related to the flow of current is correspondingly very small, so the thermal noise in the circuit is generally negligible.
Here is a quote from Horowiz and Hill in their paper: "The current is the flow of discrete charges, not a continuous flow like a fluid. The finiteness of the charge quantum gives rise to the statistical fluctuation theory of the current. If the actions of these charges are independent of each other, then the fluctuating current is:

In(rms)=I nR =(2 qI dc B) 1/
2where q is the electron charge (1.60×10 -19 C), B is the measurement bandwidth, and rms represents the effective value. For example, for a 1 A "stable" current, the effective value of the fluctuating current is 57 nA, and the measurement bandwidth is 10 kHz. This means that the degree of fluctuation is about 0.000006%. This relative fluctuation is relatively large for small currents. For example, within a 10 kHz bandwidth, for a "stable" current of 1 μA, the actual fluctuation of the effective value of the current noise is 0.006%. That is -85 dB. For a 1 pA DC current, the effective value of its current fluctuation within the same bandwidth is 56 fA, that is, the relative fluctuation is 5.6%. It can be seen that shot noise is extremely small. Shot noise, similar to resistor thermal noise, belongs to Gaussian noise and white noise. "

"Earlier shot noise formulas assumed that the charge carriers had the ability to independently form currents. This is actually the case for charges across a barrier, such as a junction diode current, which is formed by diffusion of charge. In contrast, the importance of shot noise in metal conductors is unrealistic because there are large-scale correlations between the charge carriers. Therefore, the current noise in simple resistive circuits is much smaller than that calculated by the shot noise formula. We provide an important alternative to the shot noise formula in the standard transistor current source circuit, where negative feedback acts to reduce shot noise."