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.
Equation 2.3: Broadband Noise Equation
We need to remember that our goal is to measure the magnitude of the noise source Vn in Figure 2.3. This noise source includes broadband noise and 1/f noise. We can calculate the broadband noise using Equations 2.2 and 2.3. We should now calculate the 1/f noise, which requires integrating the power spectrum in the 1/f region of the noise frequency density plot (see Figure 2.10). We can obtain the result of the integration using equations 2.4 and 2.5. Equation 2.4 normalizes the noise measurement in the 1/f region to noise at 1 Hz. In some cases, we can read this value directly from the graph, and sometimes it is more convenient to use an equation (see Figure 2.11). Equation 2.5 calculates the 1/f noise using the normalized noise, the upper noise bandwidth, and the lower noise bandwidth. Appendix 2.2 gives the whole calculation process.
When considering 1/f noise, we must choose a low frequency cutoff. This is because the 1/f function is meaningless when the denominator is zero (ie 1/0 is meaningless). In fact, theoretically the noise approaches infinity at 0 Hz. But we should take into account that when the frequency is very low, the corresponding time is also very long. For example, 0.1Hz corresponds to 10 seconds, and 0.001Hz corresponds to 1000 seconds. For very low frequencies, the corresponding time may be years (eg 10nHz corresponds to 3 years). The larger the frequency separation, the more noise the integral will result in. However, we must also remember that very low frequency noise detection takes a long time. We will explore this issue in more detail in a future article. For now, let’s keep this in mind, 1/f calculations usually use 0.1Hz as the low frequency cutoff.
Now that we have the magnitude of the broadband and 1/f noise, add the noise sources using the uncorrelated noise source equations given in section (see Equation 2.6 below and Equation 1.8 in this article series).
When engineers consider an analysis method, a common concern is whether 1/f noise and broadband noise should be integrated in two different regions. In other words, they believe that the error occurs because the 1/f noise and the broadband noise will exceed the 1/f region when added. In fact, the 1/f region, like the broadband region, covers all frequencies. We have to remember that when the noise spectrum is displayed on a logarithmic plot, the 1/f region has little effect after dropping below the broadband curve. The area where the two curves combine clearly is at the 1/f half-power frequency. In this region, we see the same situation as in the mathematical model at the junction of the two regions. Figure 2.12 shows the actual overlap of the two regions and gives the corresponding magnitude.
We now have all the equations we need to convert the noise spectral density curve to a noise source. Note that we have now derived the equations required for voltage noise, but the same approach can be applied to the calculation of current noise. In subsequent articles in this series, we will discuss the equations used to solve noise analysis problems of op amp currents.