Operational Amplifier Noise: Techniques and Tips for Analyzing and Reducing Noise
248
Operational Amplifier Noise: Techniques and Tips for Analyzing and Reducing Noise
248eBook
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Overview
Arthur Kay’s exciting new publication is a must have for practicing, professional electrical engineers. This comprehensive guide shows engineers how to design amplifiers and associated electronics to minimize noise, providing tricks, rules-of-thumb, and analysis to create successful low noise circuits. Forget the classical textbook traps of equations, virtual grounds, and a lot of double-speak, the novel but educational presentation used here uses definition-by -example and straight-forward analysis. This is the ultimate reference book for engineers who don't have the time to read, since the concepts are presented in detailed pictures and then repeated in the text for those who like both.
Operational amplifiers play a vital role in modern electronics design. Today, op amps serve as the interfaces between the digital world of microprocessors, microcontrollers, and other digital circuits and the analog "real world". If an analog signal must be amplified, conditioned, filtered, or converted to be used by a digital system, an op amp is almost always involved. Noise is an unwanted signal that will corrupt or distort the desired signal, and veteran engineers as well as new college graduates are often faced with a lack of experience in noise analysis for operational amplifiers. The author has created a publication that is packed with essential information, while still being accessible to all readers.
- Clear, definition-by-example presentation allows for immediate use of techniques introduced
- Tricks and rules-of-thumb, derived from author's decades of experience
- Extreme use of figures for rapid absorption of concepts
- Concise text explains the key points in all figures
- Accessible to all types of readers
- Analysis and design of low-noise circuits using op amps, including design tradeoffs for low-noise
- Desktop reference for designing low-noise op amp circuits for novice to experienced engineers
- Accurate measurement and prediction of intrinsic noise levels, using analysis by hand and SPICE simulation
Product Details
| ISBN-13: | 9780080942438 |
|---|---|
| Publisher: | Elsevier Science |
| Publication date: | 02/23/2012 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 248 |
| File size: | 9 MB |
Read an Excerpt
Operational Amplifier Noise
Techniques and Tips for Analyzing and Reducing NoiseBy Art Kay
Newnes
Copyright © 2012 Elsevier Inc.All right reserved.
ISBN: 978-0-08-094243-8
Chapter One
Introduction and Review of Statistics
Noise analysis can be done in the time domain, in the frequency domain, or by using statistical analysis. This chapter introduces these three analysis methods. These methods will be utilized throughout this book.
1.1 Time Domain View of Noise
Noise is most commonly viewed in the time domain. A typical plot of time domain noise is shown in Figure 1.1. In the time domain, noise voltage is on the y-axis and time is on the x-axis. Noise can be viewed in the time domain using an oscilloscope. Figure 1.1 also shows that if you look at this random signal statistically, it can be represented as a Gaussian distribution. The distribution is drawn sideways to help show its relationship with the time domain signal. In Section 1.2, the statistical view is discussed in detail.
Figure 1.1 shows thermal noise in the time domain. Thermal noise is generated by the random motion of electrons in a conductor. Because this motion increases with temperature, the magnitude of thermal noise increases with temperature. Thermal noise can be viewed as a random variation in the voltage present across a component (e.g., resistor). Figure 1.2 gives the equation for finding the root mean square (RMS) thermal noise given resistance temperature and bandwidth.
The important thing to know about the thermal noise equation is that it allows you to find an RMS noise voltage. In many cases, engineers want to know the peak-to-peak noise voltage. In Section 1.2, we will learn some statistical methods that can be used to estimate peak-to-peak noise voltage given the RMS noise value. Section 1.2 also covers other basic statistical methods that are used in noise analysis.
1.2 Statistical View of Noise
Most forms of intrinsic noise have a Gaussian distribution and can be analyzed using statistical methods. For example, statistical methods must be used to calculate the sum of two noise signals or estimate peak-to-peak amplitude. This section gives a short review of some basic statistical methods required to carry out noise analysis.
1.2.1 Probability Density Function
The mathematical equation that forms the normal distribution function is called the "probability density function" (see Figure 1.3). Plotting a histogram of noise voltage measured over a time interval will approximate the shape of this function. Figure 1.4 illustrates a measured noise histogram with the probability distribution function superimposed on it.
1.2.2 Probability Distribution Function
The probability distribution function is the integral of the probability density function. This function is very useful because it tells us about the probability of an event that will occur in a given interval (see Figures 1.5 and 1.6. For example, assume that Figure 1.6 is a noise probability distribution function. The function tells us that there is a 30% chance that you will measure a noise voltage between -1V and +1V [i.e., the interval (-1, 1)] at any instant in time.
This probability distribution function is instrumental in helping us translate RMS to peak-to-peak voltage or current noise. Note that the tails of the Gaussian distribution are infinite. This implies that any noise voltage is possible. While this is theoretically true, in practical terms the probability that extremely large instantaneous noise voltages are generated is very small. For example, the probability that we measure a noise voltage between -3σ and +3σ is 99.7%. In other words, there is only a 0.3% chance of measuring a voltage outside of this interval. For this reason, ±3σ (i.e., 6σ) is often used to estimate the peak-to-peak value for a noise signal. Note that some engineers use 6.6 to estimate the peak-to-peak value of noise. There is no agreed-upon standard for this estimation. Figure 1.7 graphically shows how 2σ will catch 68% of the noise. Table 1.1 summarizes the relationship between standard deviation and probability of measuring a noise voltage.
Thus, we have a relationship that allows us to estimate peak-to-peak noise given the standard deviation. In general, however, we want to convert RMS to peak-to-peak amplitude. Often, people assume that the RMS and standard deviation are the same. This is not always the case. The two values are equal only when there is no DC component (the DC component is the average value). In the case of thermal noise, there is no DC component, so the standard deviation and RMS values are equal.
One way of computing the RMS noise voltage is to measure a large number of discrete points and use statistics to estimate the standard deviation. For example, if you have a large number of samples from an analog-to-digital (A/D) converter, you could use Eqs. (1.4)–(1.6) to compute the mean, standard deviation and RMS of the noise signal. Many software packages such as Microsoft Excel can be used to compute these functions. In fact, Excel has built-in functions for computing standard deviation and mean value. Some test equipment will include these and other built-in mathematical functions. For example, many oscilloscopes include RMS, mean and standard deviation functions.
In general, it is best to use the standard deviation function as apposed to RMS when doing noise computations. Intrinsic noise should not have a DC component. In some practical cases, the instrument measuring the noise may have a DC component. The DC component should not be included in the noise computation because it is not really part of the noise signal. When using the RMS formula on a noise signal with a DC component, the results will be affected by the DC component. The standard deviation formula, however, will eliminate the effects of the DC component.
One final statistical concept to cover is the addition of noise signals. To add two noise signals, we must know if the signals are correlated or uncorrelated. Noise signals from two independent sources are uncorrelated. For example, the noise from two independent resistors or two operational amplifiers (op-amps) is uncorrelated. A noise source can become correlated through a feedback mechanism. Noise-canceling headphones are a good example of the addition of correlated noise sources. They cancel acoustic noise by summing inversely correlated noise. Eq. (1.4) shows how to add correlated noise signals. Note that in the case of the noise-canceling headphones, the correlation factor would be C = - 1. (see Figure 1.8).
In most cases, we will add uncorrelated noise sources (see Eq. (1.5)). Adding noise in this form will effectively sum two vectors using the Pythagorean theorem. Figure 1.9 shows the addition graphically. A useful approximation is that if one of these sources is one-third the amplitude of the other, the smaller source can be ignored.
1.3 Frequency Domain View of Noise
An important characteristic of noise is its spectral density. Voltage noise spectral density is a measurement of RMS noise voltage per square root hertz (or commonly nV/ Hz). Power spectral density is given in W/Hz. A random noise signal can be thought of as an infinite summation of sine waves at different frequencies. The right side of Figure 1.10 shows how several signals at different frequency add to form a random noise signal. The left side of Figure 1.10 shows the same signals in the frequency domain. Note that each sine wave creates an impulse or "spike" in the frequency domain. The example shows just five sine waves combining to form a "random" signal. In reality, noise signals have infinite frequency components. You can imagine an infinite number of impulses in the frequency domain combine to form the spectral density curve.
Earlier, we have learned that the thermal noise of a resistor can be computed using Eq. (1.1). This equation can be rearranged into a spectral density form (see Figure 1.11). One important characteristic of this noise is that it has a flat spectral density plot (i.e., it has uniform energy at all frequencies). For this reason, thermal noise is sometimes called broadband noise or white noise (Figure 1.12). The word "white" is used to describe noise with a uniform energy at all frequencies because white light is generated by mixing all colors (wavelengths) with uniform energy. Op-amps also have broadband noise associated with them. Broadband noise is defined as noise that has a flat spectral density plot. Figure 1.13 shows the noise spectral density of a resistor graphed vs. resistance. This plot can be used as a quick way to determine the spectral density of a resistance. Also note that temperature has a very small effect on overall noise.
Figure 1.14 shows two common regions in spectral density curves. In the broadband region the noise spectral density is flat, so the contribution of all the different frequency components is equal. Op-amps also may have a low-frequency noise region that does not have a flat spectral density plot. This noise is called 1/f noise, flicker noise, or low-frequency noise. Typically, the power spectrum of 1/f noise falls at a rate of 1/f. This means that the voltage spectrum falls at a rate of 1/f(1/2). In practice, however, the exponent of the 1/f function may deviate slightly. Figure 1.14 shows a typical op-amp spectrum with both a 1/f region and a broadband region. Note that the spectral density plot also shows current noise (given in fA/√Hz).
Note that 1/f noise also has a normal distribution, and consequently the mathematics described earlier still applies. Figure 1.15 shows the time domain description of 1/f noise. Note that the x-axis of this graph is given in seconds; this slow change with time is typical for 1/f noise.
1.4 Converting Spectral Density to RMS Noise
A very common noise calculation is to convert spectral density to RMS noise. This is used extensively in op-amp noise calculations. There are three different types of spectral densities to consider: noise power spectral density (unit W/Hz), noise voltage spectral density (unit V/√Hz), and noise current spectral density (unit V/√Hz). Noise power spectral density can be converted to RMS power by integrating the spectral density. Power spectral density is defined as voltage or current spectral density squared. Thus, to convert voltage or current noise spectral density to RMS noise, you convert to power (v2n or i2n), integrate, and convert back to voltage or current (square root). See Figure 1.16 for details.
A common error that people make when converting voltage spectral density to RMS noise voltage is integrating the voltage spectral density instead of the power spectral density.
In Figures 1.17 and 1.18, we will do a dimensional analysis to demonstrate why this does not work. Before doing this, however, we will review the integral. As a quick reminder, the integral function will give the area under a curve. Figure 1.19 shows how a constant function can be integrated by simply multiplying the height by the width (i.e., the area of a rectangle). Considering the integral to be the area of a rectangle simplifies the conversion of spectral density curves to RMS noise values.
Figure 1.17 shows the strange units that result when you attempt to integrate the voltage spectral density curve. Figure 1.18 shows how you can integrate the power spectral density and convert back to voltage by taking the square root of the result. Note that we get the proper units. Also note that power spectral density is simply the voltage or current spectral density squared (remember P V2/R and P I2R). Thus, by looking at Figures 1.17 and 1.18, you can see why power spectral density must be integrated rather than voltage or current spectral density.
Chapter Summary
• Oscilloscope measurements show noise in the time domain.
• x-axis is time and y-axis is voltage or current.
• A Gaussian distribution is a statistical view of noise.
• Standard deviation and RMS are the same if noise does not have a DC component (average value is zero).
• Six times the standard deviation is a good estimate of peak-to-peak noise.
• There is a 99.7% probability that noise will be less than six times the standard deviation.
• Most noise is uncorrelated.
• Uncorrelated noise is added with the root sum of the square of each noise component.
• The noise spectral density is a frequency domain view of noise.
• Spectral density has units of V/√Hz or A/√Hz.
• White noise is composed of an infinite number of different frequency components with equal energy.
• Two key regions in a noise spectral density curve are the 1/f region and the broadband region.
• Noise spectral density can be converted to RMS noise by taking the square root of the integral of the noise signal squared.
(Continues...)
Excerpted from Operational Amplifier Noise by Art Kay Copyright © 2012 by Elsevier Inc. . Excerpted by permission of Newnes. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents
Introduction to Noise and Review of StatisticsIntroduction to Op-Amp NoiseOp-Amp Noise Analysis Using Tina SpiceNoise Measurement TechniquesDesign ConsiderationsNoise PhenomenaDesign ExamplesADC Noise AnalysisWhat People are Saying About This
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