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**Popcorn Noise**
We have already pointed out that any noise source which produces less than one third to one fifth of the noise of some greater source can be ignored, with little error. When so doing, both noise voltages must be measured at the same point in the circuit. To analyze the noise performance of an op amp circuit, we must assess the noise contributions of each part of the circuit, and determine which are significant. To simplify the following calculations, we shall work with noise spectral densities, rather than actual voltages, to leave bandwidth out of the expressions (the noise spectral density, which is generally expressed in nV/√Hz, is equivalent to the noise in a 1Hz bandwidth).

If we consider the circuit in Figure 1-74 below, which is an amplifier consisting of an op amp and three resistors (R3 represents the source resistance at node A), we can find six separate noise sources: the Johnson noise of the three resistors, the op amp voltage noise, and the current noise in each input of the op amp. Each source has its own contribution to the noise at the amplifier output. Noise is generally specified RTI, or referred to the input, but it is often simpler to calculate the noise referred to the output (RTO) and then divide it by the noise gain (not the signal gain) of the amplifier to obtain the RTI noise.

Figure 1-74: Op amp noise model for single pole system

Figure 1-75 (opposite) is a detailed analysis of how each of the noise sources in Fig. 1-74 is reflected to the output of the op amp. Some further discussion regarding the effect of the current noise at the inverting input is warranted. This current, I

_{N–}, does not flow in R1, as might be expected— the negative feedback around the amplifier works to keep the potential at the inverting input unchanged, so that a current flowing from that pin is forced, by negative feedback, to flow in R2 only, resulting in a voltage at the output of I_{N–}R2. We could equally well consider the voltage caused by I_{N–}flowing in the parallel combination of R1 and R2 and then amplified by the noise gain of the amplifier, but the results are identical— only the calculations are more involved.
Notice that the Johnson noise voltage associated with the three resistors has been included in the expressions of Fig. 1-75. All resistors have a Johnson noise of √(4kTBR), where k is Boltzmann's Constant (1.38×10

^{–23}J/K), T is the absolute temperature, B is the bandwidth in Hz, and R is the resistance in Ω. A simple relationship which is easy to remember is that a 1000Ω resistor generates a Johnson noise of 4nV/√Hz at 25ºC.
The analysis so far assumes that the feedback network is purely resistive and that the noise gain versus frequency is flat. This applies to most applications, but if the feedback network contains reactive elements (usually capacitors) the noise gain is not constant over the bandwidth of interest, and more complex techniques must be used to calculate the total noise.

Figure 1-75: Noise sources referred to the output (RTO)

The circuit shown in Figure 1-76 below represents a second-order system, where capacitor C1 represents the source capacitance, stray capacitance on the inverting input, the input capacitance of the op amp, or any combination of these. C1 causes a breakpoint in the noise gain, and C2 is the capacitor that must be added to obtain stability.

Figure 1-76: Op amp noise model with reactive elements (second-order system)

Because of C1 and C2, the noise gain is a function of frequency, and has peaking at the higher frequencies (assuming C2 is selected to make the second-order system critically damped). Textbooks state that a flat noise gain can be achieved if one simply makes R1C1 = R2C2.

But in the case of current-to-voltage converters, however, R1 is typically a high impedance, and the method doesn't work. Maximizing the signal bandwidth in these situations is somewhat complex.

A DC signal applied to input A (B being grounded) sees a gain of 1 + R2/R1, the low frequency noise gain. At higher frequencies, the gain from input A to the output becomes 1 + C1/C2 (the high frequency noise gain).

The closed-loop bandwidth fcl is the point at which the noise gain intersects the open-loop gain. A DC signal applied to B (A being grounded) sees a gain of –R2/R1, with a high frequency cutoff determined by R2-C2. Bandwidth from B to the output is 1/2πR2C2.

The current noise of the non-inverting input, I

_{N+}, flows in R3 and gives rise to a noise voltage of I_{N+}R3, which is amplified by the frequency-dependent noise gain, as are the op amp noise voltage, V_{N}, and the Johnson noise of R3, which is √(4kTR3). The Johnson noise of R1 is amplified by –R2/R1 over a bandwidth of 1/2πR2C2, and the Johnson noise of R2 is not amplified at all but is connected directly to the output over a bandwidth of 1/2πR2C2. The current noise of the inverting input, I_{N–}, flows in R2 only, resulting in a voltage at the amplifier output of I_{N–}R2 over a bandwidth of 1/2πR2C2.
If we consider these six noise contributions, we see that if R1, R2, and R3 are low, then the effect of current noise and Johnson noise will be minimized, and the dominant noise will be the op amp's voltage noise. As we increase resistance, both Johnson noise and the voltage noise produced by noise currents will rise.

If noise currents are low, then Johnson noise will take over from voltage noise as the dominant contributor. Johnson noise, however, rises with the square root of the resistance, while the current noise voltage rises linearly with resistance, so ultimately, as the resistance continues to rise, the voltage due to noise currents will become dominant.

These noise contributions we have analyzed are not affected by whether the input is connected to node A or node B (the other being grounded or connected to some other low-impedance voltage source), which is why the non-inverting gain (1 + Z2/Z1), which is seen by the voltage noise of the op amp, V

_{N}, is known as the "noise gain".
Calculating the total output RMS noise of the second-order op amp system requires multiplying each of the six noise voltages by the appropriate gain and integrating over the appropriate frequency as shown in Figure 1-77 (opposite).

The root-sum-square of all the output contributions then represents the total RMS output noise. Fortunately, this cumbersome exercise may be greatly simplified in most cases by making the appropriate assumptions and identifying the chief contributors.

Figure 1-77: Noise sources referred to the output for a second-order system

Although shown before, the noise gain for a typical second-order system is repeated in Figure 1-78 below. It is quite easy to perform the voltage noise integration in two steps, but notice that because of peaking, the majority of the output noise due to the input voltage noise will be determined by the high frequency portion where the noise gain is 1 + C1/C2. This type of response is typical of second-order systems.

Figure 1-78: Noise gain of a typical second-order system

The noise due to the inverting input current noise, R1, and R2 is only integrated over the bandwidth 1/2πR2C2.

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