Electronic

### Noise Model and Spectral Density Curve for Op Amp

One of the important properties of noise is its spectral density. Voltage noise spectral density refers to the effective (RMS) noise voltage per square root Hertz (usually in nV/rt-Hz). The unit of power spectral density is W/Hz. In the previous article, we learned that the thermal noise of a resistor can be calculated using Equation 2.1. The formula can also be adapted for spectral density with modifications. One of the important properties of thermal noise is that the spectral density plot is flat (that is, the energy is the same at all frequencies). Therefore, thermal noise is sometimes referred to as broadband noise. Op amps also have broadband noise.

One of the important properties of noise is its spectral density. Voltage noise spectral density refers to the effective (RMS) noise voltage per square root Hertz (usually in nV/rt-Hz). The unit of power spectral density is W/Hz. In the previous article, we learned that the thermal noise of a resistor can be calculated using Equation 2.1. The formula can also be adapted for spectral density with modifications. One of the important properties of thermal noise is that the spectral density plot is flat (that is, the energy is the same at all frequencies). Therefore, thermal noise is sometimes referred to as broadband noise. Op amps also have broadband noise. Broadband noise is noise with a relatively flat spectral density plot.

Noise Model and Spectral Density Curve for Op Amp

In addition to broadband noise, op amps often have a low-frequency noise region where the spectral density plot is not flat. This noise is called 1/f noise, or flicker noise, or low frequency noise. Generally speaking, the power spectrum of 1/f noise decreases at a rate of 1/f. That is, the voltage spectrum will drop at a rate of 1/f(1/2). In practice, however, the exponent of the 1/f function is slightly off. Figure 2.1 shows the spectrum of a typical op amp in the 1/f and broadband regions. Note that the spectral density plot also shows current noise in fA/rt-Hz.

It is also important to note that 1/f noise can also be represented by a normal distribution curve, so the math described in the section still applies. Figure 2.2 shows the time domain for 1/f noise. Note that the x-axis of this graph is in seconds, and that slow changes over time are typical of 1/f noise.

Figure 2.2: 1/f noise and statistical analysis results corresponding to the time domain

Figure 2.3 depicts the standard model of op amp noise, which includes two uncorrelated current noise sources and one voltage noise source connected to the op amp’s input. We can think of the voltage noise source as the time-varying input offset voltage component, and the current noise source as the time-varying bias current component.

Figure 2.3: Noise model of an op amp

Operational Amplifier Noise Analysis Methods

The operational amplifier noise analysis method is to calculate the peak-to-peak output noise of the operational amplifier circuit based on the data on the operational amplifier data sheet. In introducing the method, we used the formulas that apply to simple op amp circuits. For more complex circuits, these equations also help us get a rough idea of ​​the noise output that can be expected. We can also provide more accurate formulas for these more complex circuits, but the math involved will be more complex. For more complex circuits, perhaps we should take a three-step approach. First, make a rough estimate with the formula; then, use the spice simulation program to make a more accurate estimate; confirm the result by measurement.

We will use the simple non-inverting amplifier of the TI OPA277 as an example to illustrate the circuit (see Figure 2.4). Our goal is to determine the peak-to-peak output noise. To do this, we should consider the op amp’s current noise, voltage noise, and resistor thermal noise. We will determine the size of the above noise sources based on the spectral density curve in the product specification. In addition, we have to consider circuit gain and bandwidth issues.

Figure 2.4: Example of a noise analysis circuit

First, we should understand how to convert a noise spectral density curve to a noise source. To achieve this, we need to perform calculus operations. As a quick reminder, the integral function determines the area under the curve. Figure 2.5 shows that we can obtain the integral of a constant function by simply multiplying the length and width (ie, the area of ​​the rectangular region). This relationship of converting the spectral density curve to a noise source is relatively simple.

Figure 2.5: Calculate the area under the curve by integrating

It is often said that the total noise value can only be obtained by integrating the voltage spectral density curve. In fact, we have to integrate the power spectral density curve. This curve actually reflects the square of the voltage or current spectral density (remember: P = V2/R and P=I2R). Figure 2.6 shows the strange result of integrating the voltage spectral density curve. Figure 2.7 shows that you can integrate the power spectral density and convert it back to voltage by taking the square root of the result. Note that we get reasonable results from this.

Figure 2.6: Improper way to calculate noise

Figure 2.7: The correct way to calculate noise

By integrating the power spectral density curves of the voltage and current spectra, we can obtain the RMS amplitude of the op amp model signal source (Figure 2.3). However, the spectral density curve will be distributed in the 1/f region and the broadband region with the low pass filter (see Figure 2.8). To calculate the total noise in the above two regions, we need to use the formula derived from calculus calculation. Then, according to the method of dealing with uncorrelated signal sources discussed in section , do the square root of sum (RSS) operation on the results of the above two calculations, corresponding to the uncorrelated signal sources mentioned in section .

First, we want to integrate the broadband region with the low pass filter. Ideally, the low-pass filter portion of the curve is a vertical straight line, which we call a brick wall filter. Since the area under the curve in the case of the brick-wall filter is a rectangle, the problem in this area is easier to solve by multiplying the length by the width. In a practical case, we cannot implement a brick wall filter. However, we can use a set of constants to convert the actual filter bandwidth to the equivalent brick-wall filter bandwidth for noise calculations. Figure 2.9 compares a theoretical brick-wall filter with first-, second-, and third-order filters.

Figure 2.8: Broadband region with filter

Figure 2.9: Brickwall filter compared to actual filter

We can use Equation 2.2 to convert the actual filter or make a brick wall filter equivalent. Table 2.1 lists the scaling factors (Kn) for each filter order. For example, the first-order filter bandwidth multiplied by 1.57 is the brick-wall filter bandwidth. The adjusted bandwidth is also sometimes referred to as the noise bandwidth. Note that the scaling factor gets closer and closer to 1 as the filter order increases. In other words, the higher the filter order, the closer it is to a brick-wall filter.

Now that we have the formula to convert the actual filter to a brick-wall filter, we can easily integrate the power spectrum. Remember that power is integrated as the square of the voltage spectrum. We need to square root the result of the integration to convert it back to a voltage. Equation 2.3 is derived from this (see Appendix 2.1). Therefore, the broadband noise can be calculated by applying Equation 2.2 and Equation 2.3 according to the data in the product specification.