AN1228 Op Amp Precision Design: Random Noise Author: Kumen Blake Microchip Technology Inc. INTRODUCTION This application note covers the essential background information and design theory needed to design low noise, precision op amp circuits. The focus is on simple, results oriented methods and approximations useful for circuits with a low-pass response. The material will be of interest to engineers who design op amps circuits which need better signal-to-noise ratio (SNR), and who want to evaluate the design trade-offs quickly and effectively. This application note is general enough to cover both voltage feedback (VFB) (traditional) and current feedback (CFB) op amps. The examples, however, will be limited to Microchip’s voltage feedback op amps. Additional material at the end of this application note includes references to the literature, vocabulary and computer design aids. Key Words and Phrases • • • • • Op Amp Device Noise Noise Spectral Density Integrated Noise Signal-to-Noise Ratio (SNR) Prerequisites The material in this application note will be much easier to follow after reviewing the following statistical concepts: • • • • • • • Average Standard Deviation Variance Gaussian (normal) probability density function Histograms Statistical Independence Correlation BACKGROUND INFORMATION This section covers the basics of low frequency noise work. It is somewhat theoretical in nature, but has some numerical examples to illustrate the concepts. It serves as a foundation for the following sections. See references [2, 4, 5] for a more in depth theoretical coverage of these concepts. The material after this section illustrates these concepts. For those readers new to this subject matter, it may be beneficial to read the complete application note several times, while working all of the examples. Where Did the Average Go? The most commonly used statistical concept is the average. Standard circuit analysis gives a deterministic value (DC plus AC) at any point in time. Once these deterministic values are subtracted out, the noise variables left have an average of zero. Noise is interpreted as random fluctuations (a stochastic value) about the average response. We will deal with linear circuits, so superposition applies; we can add the average and the random fluctuations to obtain the correct final result. Noise Spectral Density The easiest approach to analyzing random analog noise starts in the frequency domain (even for engineers that strongly prefer the time domain). Stationary noise sources (their statistics do not change with time) can be represented with a Power Spectral Density (PSD) function. Because we are analyzing analog electronic circuits, the units of power we will deal with are W, V2/Ω and A2Ω. This noise power is equivalent to statistical variance (σ2). The variance of the sum of uncorrelated random variables is: EQUATION 1: VARIANCE OF THE SUM OF UNCORRELATED VARIABLES ⎛ ⎞ var ⎜ ∑ X k⎟ = ⎝ k ⎠ Knowledge of basic circuit analysis is also assumed. ∑ var ( Xk ) k Where: © 2008 Microchip Technology Inc. Xk = uncorrelated random variables var() = the variance function DS01228A-page 1 AN1228 This fact is very important because the various random noise sources in a circuit are caused by physically independent phenomena. Circuit noise models that are based on these physically independent sources produce uncorrelated statistical quantities. The PSD is an extension of the concept of variance. It spreads the variation of any noise power variable across many frequency bins. The noise in each bin (power with units of Watts) is statistically independent of all other bins. The units for PSD are (W/Hz), which is why it is called a “density” function. The picture in Figure 1 illustrates these concepts. Strictly speaking, in passive circuits (RLC circuits), this conversion needs to be done with a specific resistance value (P = V2/R = I2R). In most noise work involving active devices, however, a standard resistance value of 1 Ω is assumed. Integrated Noise To make rational design choices, we need to know what the total noise variation is; this section gives us that capability. We will convert the PSD to the statistical variance (or standard deviation squared) using a definite integral across frequency. CALCULATIONS PSD (W/Hz) Bin Power ≈ PSD(fk) · Δfk Using Equation 1, and the fact that the power in a frequency bin is independent of all other bins, we can add up all of the bin powers together: EQUATION 2: Δfk TOTAL NOISE VARIATION N ≈ ∑ ( PSD ( f k ) ⋅ Δf k ) k ∞ f (Hz) 0 fk FIGURE 1: Power Spectral Density. In this application note, all PSD plots (and functions) are one-sided, with the x-axis in units of Hertz. This is the traditional choice for circuit analysis because this is the output of (physical) spectrum analyzers. Note: • • Where: N = ∫0 PSD ( f ) df total noise power (W) We use the summation approximation for measured noise data at discrete time points. The integral applies to continuous time noise; it is useful for deriving theoretical results. It is very important, when reading the electronic literature on noise, to determine: PREFERRED EQUATIONS Is the PSD one-sided or two-sided? Is frequency in units of Hertz (Hz) or Radians per Second (rad/s)? In circuit analysis, the conversion to integrated noise (En) usually takes place with the noise voltage density; see Equation 3. En is the noise’s standard deviation. In most low frequency circuits, signals and noise are interpreted and measured as voltages and currents, not power. For this reason, PSD is usually presented in two equivalent forms: • Noise voltage density (en) with units (V/√Hz) • Noise current density (in) with units (A/√Hz) The voltage and current units are RMS values; they could be given as (VRMS/√Hz) and (ARMS/√Hz). Traditionally, the RMS subscript is understood, but not shown. Note: N = Many beginners find the √Hz units to be confusing. It is the natural result, however, of converting PSD (in units of W/Hz) into noise voltage or current density via the square root operation. DS01228A-page 2 EQUATION 3: INTEGRATED NOISE VOLTAGE ∞ En = ∫0 e n ( f ) df 2 Where: en(f) = = En noise voltage density (V/√Hz) PSD ( f ) ⋅ ( 1 Ω ) = integrated noise voltage (VRMS) = standard deviation (VRMS) © 2008 Microchip Technology Inc. AN1228 Noise current densities can also be converted to integrated noise (In): EQUATION 4: ∞ ∫0 i n ( f ) df 2 Where: in(f) = noise current density (A/√Hz) In xL PG(|x| > xL; 0, 1) 1.64 10% integrated noise current (ARMS) = standard deviation (ARMS) 1% 2.58 5.15 3.29 6.58 10-6 4.50 9.00 -9 6.00 12.00 6.80 × 1.97 × 10 Microchip’s op amp data sheets use 6.6 VP-P/VRMS when reporting Eni (usually between 0.1 Hz and 10 Hz). This is about the range of visible noise on an analog oscilloscope trace. INTERPRETATION The principle noise sources within op amps, and resistors on the PCB, are Gaussian. When they are combined, they produce a total noise that is also Gaussian. Figure 2 shows the standard Gaussian probability density function (mean = 0 and standard deviation = 1) on a logarithmic y-axis. 0 1.E+00 10 -1 1.E-01 10 -2 1.E-02 10 The integrated noise results in this application note are independent of frequency and time. They can only be used to describe noise in a global sense; correlations between the noise seen at two different time points are lost after the integration is done. Filtered Noise Any time we measure noise, it has been altered from its original form seen within the physical noise source. The easiest way to represent these alterations to the noise, in linear systems, is by the transfer function (in the frequency domain) from the source to the output. The resulting output noise has a different spectral shape than the source. TRANSFER FUNCTIONS AND NOISE -3 1.E-03 10 pG(x; 0, 1) 3.29 0.1% Note 1: We need to know the probability density function in order to make informed decisions based on the integrated (RMS) noise. For the work in this application note, the noise will have a Gaussian (Normal) probability density function. 1.64 3.29 6.00 = Peak-to-Peak Peak (VPK/VRMS) (VP-P/VRMS) 2.58 4.50 PSD ( f ) ⁄ ( 1Ω ) = IMPORTANT TWO-TAILED PROBABILITIES Crest Factor (Note 1) INTEGRATED NOISE CURRENT In = TABLE 1: It turns out [3, 4, 5] that the noise at the output of a linear operation (represented by the transfer function) is related to the input noise by the transfer function’s squared magnitude; see Equation 5. This can be thought of as a result of the statistical independence between the PSD’s frequency bins (see Figure 1). -4 1.E-04 10 1.E-05 10-5 -6 1.E-06 10 1.E-07 10-7 -8 1.E-08 10 -9 1.E-09 10 -6 -5 -4 -3 -2 -1 0 x 1 2 3 4 5 6 EQUATION 5: FIGURE 2: Standard Gaussian Probability Density Function. Table 1 shows important points on this curve and the corresponding (two tailed) probability that the random Gaussian variable is outside of those points. This information is useful in converting RMS values (voltages or currents) to either peak or peak-to-peak values. The column label xL is sometimes called the number of sigma from the mean. © 2008 Microchip Technology Inc. 2 e nout OUTPUT NOISE V OUT 2 2 = ------------e ni V IN Where: eni = noise voltage density at VIN (V/√Hz) enout = noise voltage density at VOUT (V/√Hz) Example 1 shows the conversion of a simple transfer function to its squared magnitude. It starts as a Laplace Transform , it is converted to a Fourier Transform (substituting jω for s) and then converted to its squared magnitude form (a function of ω2). It is best to do this last conversion with the transform in factored form. DS01228A-page 3 AN1228 EXAMPLE 1: TRANSFER FUNCTION CONVERSION EXAMPLE Laplace Transfer Function: V OUT 1 ------------- = ---------------------V IN 1 + s ⁄ ωP Conversion to Fourier Transfer Function: V OUT 1 -, s → jω ------------- = -------------------------V IN 1 + j ω ⁄ ωP Note: Conversion to Magnitude Squared: 2 V OUT 2 1 1 - = --------------------------------2 ------------- = -------------------------V IN 1 + j ω ⁄ ωP 1 + ( ω ⁄ ωP ) 1 = ---------------------------2 , ω → 2πf 1 + ( f ⁄ fP ) Where: s = Laplace frequency (1/s) = σ + jω ω = Radian frequency (rad/s) ωP = Pole (rad/s) f = Frequency (Hz) fP = Pole frequency (Hz) In the physical world, however, brick wall filters would have horrible behavior. They cannot be realized with a finite number of circuit elements. Physical filters that try to approach this ideal show three basic problems: their step response exhibits Gibbs phenomenon (overshoot and ringing that decays slowly), they suffer from noise enhancement (due to high pole quality factors) and they are very difficult to implement. Comments in the literature (e.g., in filter textbooks) about “ideal” brick wall filters should be viewed with skepticism. The integrated noise voltage integrals (Equation 3 and Equation 4) are in their most simple terms when a brick wall filter is used. Equation 6 shows that, in this case, the brick wall filter’s frequencies fL and fH become the new integration limits. The integrated current noise is treated similarly. EQUATION 6: INTEGRATED NOISE WITH BRICK WALL FILTER ∞ E nout = ∫0 e nout ( f ) df ∞ = 2 V OUT ∫0 e ni ( f ) ------------V 2 2 df IN BRICK WALL FILTERS The transfer function that is easiest to manipulate mathematically is the brick wall filter. It has infinite attenuation (zero gain) in its stop bands, and constant gain (HM) in its pass band; see Figure 3. |H(j2πf)| (V/V) Where: fH ∫fL e ni ( f ) df 2 fL = Lower cutoff frequency (Hz) fH = Upper cutoff frequency (Hz) HM = Pass band gain (V/V) See Appendix B: “Computer Aids” for popular circuit simulators and symbolic mathematics packages that help in these calculations. HM 0 = HM ⋅ 0 fH fL FIGURE 3: f (Hz) Brick Wall Filter. We will use three variations of the brick wall filter (refer to Figure 3): • Low-pass (fL is at zero) - fL = 0 < fH < ∞ • Band-pass (as shown) - 0 < fL < fH < ∞ • High-pass (fH is at infinity) - 0 < fL < ∞ = fH Brick wall filters are a mathematical convenience that simplifies our noise calculations. DS01228A-page 4 White Noise White noise has a PSD that is constant over frequency. It received its name from the fact that white light has an equal mixture of all visible wavelengths (or frequencies). This is a mathematical abstraction of real world noise phenomena. A truly white noise PSD would produce an infinite integrated noise. Physically, this is not a concern because all circuits and physical materials have limited bandwidth. We start with white noise because it is the easiest to manipulate mathematically. Other spectral shapes will be addressed in subsequent sections. © 2008 Microchip Technology Inc. AN1228 NOISE POWER BANDWIDTH When white noise is passed through a brick wall filter (see Figure 3), the integrated noise becomes a very simple calculation. Equation 6 is simplified to: EQUATION 7: INTEGRATED WHITE NOISE WITH BRICK WALL FILTER E nout = H M e ni f H – f L The shot noise current density’s magnitude depends on the diode’s DC current (ID) and the electron charge (q). It is usually modeled as white noise; see Equation 9. EQUATION 9: i nd = q = Input noise voltage density (V/√Hz) enout = Output noise voltage density (V/√Hz) This equation is usually represented by what is called the Noise Power Bandwidth (NPBW). NPBW is the bandwidth (under the square root sign) that converts a white noise density into the correct integrated noise value. For the case of brick wall filters, we can use Equation 8. EQUATION 8: 2q I D Where: Where: eni DIODE SHOT NOISE ID = Electron charge = 1.602 × 10-19 (C) = Diode Current (A) Let’s look at a specific example: EXAMPLE 2: A DIODE SHOT NOISE CALCULATION Given: ID = 1 mA INTEGRATED WHITE NOISE WITH NPBW Calculate: i nd = 2 ( 1.602 × 10 – 19 C ) ( 1 mA ) = 17.9 pA/√Hz E nout = H M e ni NPBW Where: NPBW = fH –fL,for brick wall filters The high-pass filter appears to cause infinite integrated noise. In real circuits, however, the bandwidth is limited, so fH is finite (a band-pass response). Note: NPBW applies to white noise only; other noise spectral shapes require more sophisticated formulas or computer simulations. Circuit Noise Sources This section discusses circuit noise sources for different circuit components and transfer functions between sources and the output. DIODE SHOT NOISE Diodes and bipolar transistors exhibit shot noise, which is the effect of the electrons crossing a potential barrier at random arrival times. The equivalent circuit model for a diode is shown in Figure 4. Note: All of the calculation results in this application note show more decimal places than necessary; two places are usually good enough. This is done to help the reader verify his or her calculations. RESISTOR THERMAL NOISE The thermal noise present in a resistor is usually modeled as white noise (for the frequencies and temperatures we are concerned with). This noise depends on the resistor’s temperature, not on its DC current. Any resistive material exhibits this phenomenon, including conductors and CMOS transistors’ channel. Figure 5 shows the models for resistor thermal noise voltage and current densities. The sources are shown with a polarity for convenience in circuit analysis. enr R inr R ID D FIGURE 4: Model for Diodes. ind FIGURE 5: Physically Based Noise Model for Resistors. Physically Based Noise © 2008 Microchip Technology Inc. DS01228A-page 5 AN1228 The equivalent noise voltage and current spectral densities are (remember that 273.15 K = 0°C): EQUATION 10: RESISTOR THERMAL NOISE DENSITY e nr = 4kT A R i nr = 4kT A ⁄ R VDD eni VP VM = Boltzmann constant = 1.381 × 10-23 (J/K) TA = Ambient temperature (K) R = Resistance (Ω) 4kTA represents a resistor’s internal power. The maximum available power to another resistor is kTA (when they are equal). Many times the maximum available power is shown as kTA/2 because physicists prefer using two-sided noise spectra. Let’s use a 1 kΩ resistor as an example. EXAMPLE 3: A THERMAL NOISE DENSITY CALCULATION Given: R = 1 kΩ TA = 25°C = 298.15 K Calculate the noise voltage density: 4 ( 1.381 × 10 e nr = – 23 J/K ) ( 298.15 K ) = 4.06 nV/√Hz 4 ( 1.381 × 10 – 23 J/K ) ⁄ ( 298.15 K ) = 4.06 pA/√Hz OP AMP NOISE An op amp’s noise is modeled with three noise sources: one for the input noise voltage density (eni) and two for the input noise current density (ibn and ibi). All three noise sources are physically independent, so they are statistically uncorrelated. Figure 6 shows this model; it is similar to the DC error model covered in . VOUT ibi FIGURE 6: Physically Based Noise Model for Op Amps. The noise voltage source can also be placed at the other input of the op amp, with its negative pin is connected to VI and its positive pin to VM. This alternate connection gives the same output voltage (VOUT). For voltage feedback (VFB) op amps, both noise current sources have the same magnitude. This magnitude is shown in Microchip’s op amp data sheets with the symbol ini; it has units of fA/√Hz (f stands for femto, or 10-15). For now, we will discuss the white noise part of these spectral densities. We will defer a discussion on 1/f noise until later. The literature sometimes shows an amplifier noise model that has only one noise current source. In these cases, the second noise current’s power has been combined into the noise voltage magnitude. Note: Keep in mind that op amps have two physically independent noise current sources. For current feedback (CFB) op amps, the two noise current sources (ibn and ibi) are different in magnitude because the two input bias currents (IBN and IBI) are different in magnitude. They are produced by physically independent and statistically uncorrelated processes. CFB op amps are typically used in wide bandwidth applications (e.g., above 100 MHz). Microchip’s CMOS input op amps have a noise current density based on the input pins’ ESD diode leakage current (specified as the input bias current, IB). Table 2 gives the MCP6241 op amp’s white noise current values across temperature. TABLE 2: MCP6241 (CMOS INPUT) NOISE CURRENT DENSITY IB (pA) ini (fA/√Hz) 25 1 0.57 85 20 2.5 125 1100 19 TA (°C) DS01228A-page 6 AOL VSS Calculate the noise current density: i nr = VI ibn Where: k VN © 2008 Microchip Technology Inc. AN1228 Table 3 gives the MCP616 op amp’s white input noise current density across temperature. This part has a bipolar (PNP) input; the base current is the input bias current, which decreases with temperature. TABLE 3: TA (°C) MCP616 (BIPOLAR INPUT) NOISE CURRENT DENSITY IB (nA) ini (fA/√Hz) -40 -21 82 25 -15 69 85 -12 62 NOISE ANALYSIS PROCESS This section goes through the analysis process normally followed in noise design. It uses a very simple noise design problem to make this process clear. Simple Example The circuit shown in Figure 8 uses an op amp and a lowpass brick wall filter (fL = 0). The filter’s bandwidth (fH) is 10 kHz and its gain (HM) is 1 V/V. The op amp’s input noise voltage density (eni) is 100 nV/√Hz, and its gain bandwidth product is much higher than fH. The input noise voltage density (eni) typically does not change much with temperature. Note: Noise current density (ini) usually changes significantly with temperature (TA). Note: Most of the time, you can use IB vs. TA and the shot noise formula to calculate ini vs. TA. One exception to this rule is op amps with input bias current cancellation circuitry. U1 Brick Wall Low-pass Filter VIN VOUT Op Amp Circuit. FIGURE 8: Figure 9 shows both the op amp noise voltage density (eni) and the output noise voltage density (enout). Notice that enout is simply eni multiplied by the low-pass brick wall’s pass-band gain (HM). TRANSFER FUNCTIONS The transfer function from each noise source in a circuit to the output is needed. This may be obtained with SPICE simulations (see Appendix B: “Computer Aids”) or with analysis by hand. This application note emphasizes the manual approach more in order to build understanding and to derive handy design approximations. The most convenient manual approach is circuit analysis using the Laplace frequency variable (s). Figure 7 shows a resistor, inductor and capacitor with their corresponding impedances (using s). R sL 1 sC FIGURE 7: Impedance Models for Common Passive Components. Noise Voltage Density (nV/√Hz) 100 enout 0 0 eni f (Hz) 10k Noise Voltage Densities. FIGURE 9: The noise current densities ibn and ibi can be ignored in this circuit because they flow into a voltage source and the op amp output, which present zero impedance. Now we can calculate the integrated noise at the output (Enout). The result is shown in three different units (RMS, peak and peak-to-peak): EXAMPLE 4: AN INTEGRATED NOISE CALCULATION ∞ E nout = ∫0 e nout ( f ) df 2 10 kHz = ∫0 2 ( 100 nV/ Hz ) df = ( 100 nV/ Hz ) 10 kHz = 10 µVRMS = 33 µVPK= 66 µVP-P Note: © 2008 Microchip Technology Inc. This application note uses the crest factor 3.3 VPK/VRMS (or 6.6 VP-P/VRMS). DS01228A-page 7 AN1228 Enout(t) (µV) Figure 10 shows numerical simulation results of the output noise over time. Enout describes the variation of this noise. This same data is plotted in histogram form in Figure 11; the curve represents the ideal Gaussian probability density function (with the same average and variation). 50 40 30 20 10 0 -10 -20 -30 -40 -50 FILTERED NOISE This section covers the op amp circuits that have filters at their output. The discussion focuses on filters with real poles to develop insight and useful design formulas. The effect that reactive circuit components have on noise is deferred to a later section. Noise generated by the filters is ignored for now. fSAM = 10 kSPS Low-pass Filter With Single Real Pole 0 10 20 30 40 50 60 t (ms) 70 80 90 100 Figure 12 shows an op amp circuit with a low-pass filter at the output, which has a single real pole (fP). We do not need to worry about the noise current densities because the ibn and ibi sources see zero impedance (like Figure 8). We will assume that the op amp BW can be neglected because fP is much lower. U1 Output Noise vs. Time. FIGURE 10: Real Pole Low-pass Filter VIN Percentage of Occurrences 9% 8% VOUT 1024 Samples 6% Enout Gaussian 5% We need the filter’s transfer function in order to calculate the output integrated noise; it needs to be in squared magnitude form (see Example 1 for the derivation of these results): 4% 3% 2% 1% 40 30 20 10 0 -10 -20 -30 -40 0% EQUATION 11: Enout (µV) FIGURE 11: Op Amp Circuit With Low- FIGURE 12: pass Filter. 7% V OUT 1 ------------- = -------------------------V IN 1 + j ω ⁄ ωP Output Noise Histogram. Review of the Process V OUT ------------V IN The basic process we have followed can be described as follows. 2 1 = ---------------------------2 1 + ( f ⁄ fP ) Figure 13 shows the transfer function magnitude in decibels. 0 -5 | H(j2πf) | (dB) • Collect noise and filter information • Convert noise at the sources to noise at the output • Combine and integrate the output noise terms • Evaluate impact on the output signal LOW-PASS TRANSFER FUNCTION -10 -15 -20 -25 -30 -35 -40 0.01 FIGURE 13: DS01228A-page 8 0.1 1 f / fP 10 100 Filter Magnitude Response. © 2008 Microchip Technology Inc. AN1228 Now we can obtain the integrated noise, assuming the op amp’s input noise voltage density (eni) is white: EQUATION 12: INTEGRATED NOISE DERIVATION ∞ E nout = ∫0 e 2 nout ( f ) df = ∞ ∫0 2 e ni df --------------------2 2 1 + f ⁄ fP Low-pass Filter With Two Real Poles The low-pass filter in Figure 14 has two real poles (fP1 and fP2). We do not need to worry about the noise current densities because the ibn and ibi sources see zero impedance (like Figure 8). We assume that fP1 and fP2 are much lower than the op amp BW, so the op amp BW can be neglected. ∞ U1 f P [ atan ( f ⁄ f P ) ] 0 = e ni = e ni ( π ⁄ 2 ) ⋅ f P Two Real Pole Low-pass Filter VIN Thus, the NPBW for this filter is (see Equation 8): EQUATION 13: NOISE POWER BANDWIDTH NPBW = ( π ⁄ 2 ) ⋅ f P We can always reduce the integrated output noise by reducing NPBW, but the signal response may suffer if we go too far. We need to keep the filter’s -3 dB bandwidth (BW) at least as large as the desired signal BW (fP is this filter’s BW). Op Amp Circuit With Low- FIGURE 14: pass Filter. The filter’s transfer function and the magnitude squared transfer function (a function of ω2), in factored form, are: EQUATION 15: LOW-PASS TRANSFER FUNCTION V OUT 1 1 - ⋅ ----------------------------------------- = ----------------------------V IN 1 + j ω ⁄ ω P1 1 + j ω ⁄ ω P2 For low-pass filters, we can also select the BW based on the maximum allowable step response rise time  (this applies to any reasonable low-pass filter): V OUT 2 1 1 ------------- = -----------------------------2- ⋅ -----------------------------2V IN 1 + ( f ⁄ f P1 ) 1 + ( f ⁄ fP2 ) RISE TIME VS. BANDWIDTH t R ≈ 0.35 ⁄ BW Where: fP1 = First pole frequency (Hz) BW = Any low-pass filter’s -3 dB bandwidth (Hz) fP2 = Second pole frequency (Hz) Where: tR = 10% to 90% Rise time (s) Let’s try a numerical example with reasonably wide bandwidth; the noise is limited by the filter’s bandwidth. EXAMPLE 5: Figure 15 shows the transfer function magnitude in decibels for the specific case where fP2 is double fP1. 0 AN INTEGRATED NOISE CALCULATION Filter Specifications: fP = BW = 10 kHz Gain = 1 V/V Op Amp Specifications: eni = 100 nV/√Hz BW = 1 MHz Filter Rise Time: t R ≈ 35 μs fP2/fP1 = 2 -10 | H(j2πf) | (dB) EQUATION 14: VOUT -20 -30 -40 -50 -60 -70 -80 0.01 FIGURE 15: 0.1 1 f / fP1 10 100 Filter Magnitude Response. Integrated Noise Calculations: fP << Op amp’s bandwidth NPBW = ( π ⁄ 2 ) ⋅ ( 10 kHz ) = 15.8 kHz E nout = ( 100 nV/ Hz ) ⋅ ( 15.8 kHz ) = 12.6 µVRMS = 41.4 µVPK = 82.9 µVP-P © 2008 Microchip Technology Inc. DS01228A-page 9 AN1228 We can follow the same process as before to calculate NPBW. EQUATION 16: NPBW π 1 1 NPBW = ⎛ ---⎞ ⁄ ⎛ ------- + -------⎞ ⎝ 2⎠ ⎝ f P1 f P2⎠ As before, NPBW and BW are similar and BW can be traded-off with rise time (see Equation 14). EQUATION 17: BW f P1 1 f P1 2 BW = ----------------------------------- , X = 1 + --- ⋅ ⎛⎝ -------⎞⎠ 2 f P2 2 X+ 1+X EXAMPLE 7: AN INTEGRATED NOISE CALCULATION Change in Filter Specifications: fP1 = fP2 = 15.5 kHz Filter Bandwidth and Rise Time: BW = 9.98 kHz t R ≈ 35 μ s Integrated Noise Calculations: fP2 << Op amp’s bandwidth NPBW = 12.2 kHz E nout = ( 100 nV/ Hz ) ⋅ ( 12.2 kHz ) = 11.0 µVRMS = 36.4 µVPK = 72.9 µVP-P Where: fP1 ≤ High-pass Filter With Single Real Pole fP2 Let’s go through a numerical example where the op amp’s bandwidth can be neglected. EXAMPLE 6: AN INTEGRATED NOISE CALCULATION Filter Specifications: Figure 16 shows an op amp circuit with a high-pass filter with a single real pole (fP). We do not need to worry about the noise current densities because the ibn and ibi sources see zero impedance (like Figure 8). For practical circuits, there needs to be a low-pass filter at a frequency much higher than fP (at fH); the integrated noise would be infinite otherwise. If nothing else, the op amp BW may be used to limit the NPBW. fP1 = 13.4 kHz fP2 = 26.8 kHz U1 Real Pole High-pass Filter Gain = 1 V/V Op Amp Specifications: VIN VOUT eni = 100 nV/√Hz BW = 1 MHz Filter Bandwidth and Rise Time: BW = 9.98 kHz t R ≈ 35 μs Op Amp Circuit With High- FIGURE 16: pass Filter. The filter’s transfer function and the magnitude squared transfer function (a function of ω2), in factored form, are: Integrated Noise Calculations: fP2 << Op amp’s bandwidth EQUATION 18: HIGH-PASS TRANSFER FUNCTION NPBW = 14.0 kHz jω ⁄ ω P V OUT - , ω < ωH ------------- = ------------------------1 + jω ⁄ ω P V IN E nout = ( 100 nV/ Hz ) ⋅ ( 14.0 kHz ) = 11.8 µVRMS = 39.0 µVPK = 78.1 µVP-P ω ≥ ωH = 0, V OUT ------------V IN Let’s redo this example with equal poles at 15.5 kHz. 2 2 ( f ⁄ fP ) = --------------------------2- , f < f H 1 + ( f ⁄ fP ) = 0, f ≥ fH Where: DS01228A-page 10 fP = Pole frequency (Hz) fH = Low-pass NPBW (Hz) © 2008 Microchip Technology Inc. AN1228 Figure 17 shows the transfer function magnitude in decibels (fH is not shown). 0 | H(j2πf) | (dB) -5 -10 -15 -20 Band-pass Filter With Two Real Poles Figure 18 shows an op amp circuit with a band-pass filter with two real poles (highpass pole fP1 and lowpass pole fP2). We do not need to worry about the noise current densities because the ibn and ibi sources see zero impedance (like Figure 8). The op amp BW is neglected because we assume that it is much higher than fP1 and fP2. -25 U1 -30 -35 Two Real Pole Band-pass Filter VIN -40 0.01 0.1 1 f / fP 10 100 Filter Magnitude Response. FIGURE 17: VOUT Op Amp Circuit With Band- FIGURE 18: pass Filter. We can follow the same process as before to calculate NPBW (fH acts like the upper integration limit in the integrated noise equation). The filter’s transfer function and the magnitude squared transfer function (a function of ω2), in factored form, are: EQUATION 19: EQUATION 20: NPBW NPBW = f H – ( π ⁄ 2 ) ⋅ f P Where: fP << j ω ⁄ ω P1 V OUT 1 - ⋅ ----------------------------------------- = ----------------------------V IN 1 + j ω ⁄ ω P1 1 + j ω ⁄ ω P2 fH V OUT ------------V IN Let do a numerical example with the op amp bandwidth much higher than the filter pole (this is very common). EXAMPLE 8: AN INTEGRATED NOISE CALCULATION fH = Op amp’s NPBW 2 2 ( f ⁄ f P1 ) 1 = -----------------------------2- ⋅ -----------------------------21 + ( f ⁄ f P1 ) 1 + ( f ⁄ f P2 ) Where: Filter Specifications: fP = 10 kHz BAND-PASS TRANSFER FUNCTION fP1 = High-pass pole frequency (Hz) fP2 = Low-pass pole frequency (Hz) Figure 19 shows the transfer function magnitude in decibels, with fP2 = 100 fP1. Gain = 1 V/V 0 Op Amp Specifications: BW = 1 MHz NPBW ≈ ( π ⁄ 2 ) ⋅ BW = 1.57 MHz Integrated Noise Calculations: fP << Op amp’s bandwidth NPBW = ( 1.57 MHz ) – ( 15.8 kHz ) = 1.55 MHz E nout = ( 100 nV/ Hz ) ⋅ ( 1.55 MHz ) = 124 µVRMS = 411 µVPK = 822 µVP-P Note: A high-pass filter’s NPBW has little effect on the integrated noise, unless fH is near fP (but that would be a band-pass filter). © 2008 Microchip Technology Inc. | H(j2πf) | (dB) eni = 100 nV/√Hz fP2/fP1 = 100 -5 -10 -15 -20 -25 -30 -35 -40 10m 0.01 100m 0.1 FIGURE 19: 1 1 10 10 f / fP1 100 100 1k 1000 10k 10000 Filter Magnitude Response. Using a symbolic solver to derive NPBW is a big help. EQUATION 21: NPBW 1 NPBW = ( π ⁄ 2 ) ⋅ f P2 ⋅ ---------------------------1 + f P1 ⁄ f P2 DS01228A-page 11 AN1228 Let’s do another numerical example. EXAMPLE 9: AN INTEGRATED NOISE CALCULATION Filter Specifications: fP1 = 100 Hz fP2 = 10 kHz Gain = 1 V/V FILTERS WITH GREATER SELECTIVITY There are other filters with a sharper transition region, when n > 1, such as: Chebyshev, Inverse Chebyshev and Elliptic filters. Their NPBW to BW ratios are closer to 1 because they have a smaller transition region (between pass-band and stop-band). This smaller transition region reduces the integrated noise at the output. Their step response, however, tends to have more ringing and slower decay. Again, NPBW can be approximated with the -3 dB bandwidth. More exact results can be obtained with simulations (see Appendix B: “Computer Aids”). Op Amp Specifications: eni = 100 nV/√Hz BW = 1 MHz NOISE INTERNAL TO FILTERS Integrated Noise Calculations: fP2 << Op amp’s bandwidth NPBW = ( 15.7 kHz ) ⁄ 1.01 = 15.5 kHz E nout = ( 100 nV/ Hz ) ⋅ ( 15.5 kHz ) = 12.5 µVRMS = 41.2 µVPK = 82.3 µVP-P Comments on Other Filters As will be shown later (see Figure 25), active filters may produce much more noise than first expected. The op amps inside the filter produce a noise voltage density at the filter's output that has a wider bandwidth than the filter; it may be as wide as the op amp bandwidths. The resistors and op amp noise contributions tend to show a peak at the edges of the filter passband (noise enhancement), which increases the integrated output noise. This section discusses other filters and how they affect the output integrated noise. It gives a very simple approximation to NPBW when the filter order is greater than n = 1. It then discusses noise generated interal to a filter. SOME SIMPLE LOW-PASS FILTERS Table 4 shows the NPBW to BW ratio for some lowpass filters up to order 5. TABLE 4: NPBW FOR SOME LOW-PASS FILTERS NPBW / BW Low-pass Filter Type n=1 n=2 n=3 n=4 n=5 Identical Real Poles 1.571 1.220 1.155 1.128 1.114 Bessel 1.571 1.153 1.071 1.046 1.038 Butterworth 1.571 Note: 1.111 1.047 1.026 1.017 The -3 dB bandwidth is a rough estimate of NPBW for almost all filters (the main exception is when n = 1). DS01228A-page 12 © 2008 Microchip Technology Inc. AN1228 MULTIPLE NOISE SOURCES R-C Low-pass Filter This section covers two approaches to combining multiple noise sources into one output integrated noise result. This knowledge is applied to a simple R-C lowpass filter and a non-inverting gain circuit. Figure 21 shows a circuit with a R-C low-pass filter with a real pole (fP). We do not need to worry about the noise current densities because the ibn and ibi sources see zero impedance (like Figure 8). We will assume that the op amp BW can be neglected because fP is much lower. Combining Noise Outputs When we combine noise results, at the output, we take advantage of the statistical independence of: • PSD noise in separate frequency bins • Physically independent noise sources We can integrate the output noise densities one at a time, then combine the results using a Sum of Squares approach (see Equation 1). We can also combine all of the noise densities using a Sum of Squares approach first, then integrate the resulting noise density. … , Enok 2 Sum of Squares R1 VOUT C1 Circuit With R-C Low-pass FIGURE 21: Filter. We will integrate the noise densities first because this will give us important insight into this R-C low-pass filter. This circuit is like the one we already saw in Figure 12, but we have added R1’s thermal noise. The filter’s transfer function and the magnitude squared transfer function (a function of ω2), in factored form, are in Equation 22 (Figure 13 shows the transfer function magnitude in decibels). eno12, … , enok2 Eno12, enr1 VIN This independence simplifies our work, since we do not need to worry about correlations. Integrate Noise Densities (over frequency) U1 Sum of Squares (at each frequency) eno 2 Integrate Noise Density (over frequency) Eno2 EQUATION 22: 1 ⁄ ( sC 1 ) V OUT 1 1 - = ------------------------------------- = --------------------------------→ --------------------------V IN R 1 + 1 ⁄ ( sC1 ) 1 + sR 1 C 1 1 + j ω ⁄ ωP V OUT 2 V OUT = ------------------------V IN e nr1 Each approach has its advantages. Integrating first helps determine which noise source dominates; it is handy for hand designs. Finding the output noise density first helps to adjust frequency shaping elements in the design; it is easier with computer simulations. 2 1 = ---------------------------2 1 + ( f ⁄ fP ) Where: fP ωP FIGURE 20: Approaches to Combining Output Noise Terms. R-C LOW-PASS FILTER TRANSFER FUNCTION = R-C filter’s pole frequency (Hz) 1 ⁄ = ( 2πR1 C1 ) = 2πfP We can follow the same process as before to calculate NPBW. The trade-offs between NPBW (or BW) and tR shown in Equation 14 apply to this filter. EQUATION 23: NPBW NPBW = ( π ⁄ 2 ) ⋅ f P The next section (“R-C Low-pass Filter”) demonstrates the approach on the left of Figure 20. The section following that one (“Non-inverting Gain Circuit”) demonstrates the approach on the right of Figure 20. © 2008 Microchip Technology Inc. DS01228A-page 13 AN1228 The integrated noise becomes: Non-inverting Gain Circuit EQUATION 24: Figure 22 is a complete model for a non-inverting gain circuit. R1 and R3 use the series noise voltage density sources because their transfer function to VOUT is simpler in that form. R2 uses the shunt noise current density source because we can use the same transfer function to VOUT that ibi uses; this reduces our work. INTEGRATED NOISE E noU1 = e ni ⋅ NPBW E noR1 = e nr1 ⋅ NPBW = 4kT A R 1 ⋅ ( π ⁄ 2 ) ⁄ ( 2 π R 1 C 1 ) = kT A ⁄ C 1 2 E noU1 E nout = + 2 E noRC VIN R1 enr1 eni EnoU1 = U1’s output integrated noise (VRMS) EnoR1 = R1’s output integrated noise (VRMS) Enout = Total output integrated noise (VRMS) The last expression shown for EnoR1 (sqrt(kTA/C1)) is popularly called “kT on C noise” (referring to the PSD inside the square root). This result applies only to this particular case (integrated thermal noise at the output of an R-C lowpass filter). Do not let this equation mislead you; R1 generates the thermal noise, not C1. Let’s do a numerical example where the op amp and the filter resistor both contribute to the noise. EXAMPLE 10: AN INTEGRATED NOISE CALCULATION Ambient Temperature: TA = 25°C = 298.15 K Filter Specifications: R1 = 10 kΩ C1 = 1.5 nF Gain = 1 V/V R2 We will combine the noise densities first to obtain the output noise density (enout). In this case, because we have no reactive elements in the circuit, it will be a simple matter to integrate enout by hand to produce Enout. We will start with all of the transfer functions from each source to VOUT (see reference ). The gains will be assumed constant for now; we will deal with frequency shaping later on. EQUATION 25: V OUT ------------V IN V OUT ------------e nr3 V OUT ------------i bn V OUT ------------i bi = 12.9 µVRMS e nr1 = 12.8 nV/ Hz E noRC = ( 12.8 nV/ Hz ) ⋅ ( 16.7 kHz ) = 1.66 µVRMS = 13.0 µVRMS = 43.0 µVPK = 85.9 µVP-P DS01228A-page 14 TRANSFER FUNCTIONS V OUT V OUT = ------------- = ------------- = G N e nr1 e ni = 1 = –R1 GN V OUT = ------------- = R 3 i nr2 Where: GN = Noise Gain (V/V) = 1 + R3 ⁄ R2 NPBW = ( π ⁄ 2 ) ⋅ ( 10.6 kHz ) = 16.7 kHz E noU1 = ( 100 nV/ Hz ) ⋅ ( 16.7 kHz ) enr3 ANALYSIS WITH CONSTANT GAINS BW = 1 MHz Integrated Noise Calculations: fP << Op amp’s bandwidth R3 FIGURE 22: Non-inverting gain Amplifier, with multiple noise sources. eni = 100 nV/√Hz Filter Pole, Bandwidth and Rise Time: fP = BW = 10.6 kHz t R ≈ 33 μ s ibi inr2 Op Amp Specifications: E nout VOUT ibn Where: Note: U1 Note that noise gain (GN) is from the non-inverting input to VOUT, when the op amp is in a closed-loop condition, and when other (external) energy sources are zero. Note: The concept of noise gain is central to understanding op amp behavior. It simplifies op amp bandwidth and stability analyses. © 2008 Microchip Technology Inc. AN1228 The magnitude squared transfer functions are simply the squares of the constant terms in Equation 25. We will now combine these noise densities into one equation for the output noise density (using a sum of squares approach): Reactive elements in the circuit will require a more detailed analysis because each noise source may have a different frequency shape. EQUATION 26: EXAMPLE 11: 2 2 COMBINING NOISE DENSITIES 2 2 2 2 2 2 2 2 e nout = G N ( e nr1 + e ni + i bn R 1 ) + e nr3 + ( i bi + i nr2 )R 3 = 2 G N ( 4kT A R 1 2 2 2 + e ni + i bn R 1 ) + 4kT A R 3 2 2 + ( i bi + 4kT A ⁄ R 2 )R 3 While this equation is sufficient to calculate Enout, converting to an input referred form gives more insight to the designer. Dividing both sides by GN2, substituting 1 + R3/R2 for GN, and simplifying, gives: EQUATION 27: 2 2 OP AMP NOISE EQUATION 2 2 2 2 e nout ⁄ G N = e ni + i bn R 1 + i bi ( R 2 ||R 3 ) 2 + 4kT A ( R 1 + ( R 2 ||R 3 ) ) This shows that the output noise density has a very simple relationship to the resistances seen by the inputs (R1 and (R2||R3)). Note: Equation 27 also applies to inverting amplifiers (i.e., R2 is driven by VIN and R1 is grounded). ANALYSIS WITH LIMITED BANDWIDTH To produce a finite output integrated noise, we need a filter that limits the NPBW. This filter can be implemented with the op amp, by reactive elements in the circuit (e.g., capacitors) or by a filter after the op amp. We can use the op amp’s BW to set NPBW. The response can be approximated with a single real pole for hand calculations. The Gain Bandwidth Product (GBWP) specification in VFB op amp data sheets gives: EQUATION 28: NPBW SET BY OP AMP’S BANDWIDTH The following example has all of the noise sources at about the same magnitude. AN INTEGRATED NOISE CALCULATION Ambient Temperature: TA = 25°C = 298.15 K Circuit Specifications: R1 = 100 kΩ R2 = R3 = 200 kΩ Op Amp Specifications: eni = 100 nV/√Hz ini = 1 pA/√Hz GBWP = 100 kHz Preliminary Calculations: GN = 2.00 V/V (R2||R3) = 100 kΩ Input Noise Densities: 2 e ni = ( 100 nV/ Hz ) 2 2 2 i bn R 1 = ( 100 nV/ Hz ) 2 2 2 i bi ( R 2 ||R 3 ) = ( 100 nV/ Hz ) 2 4kT A ( R 1 + ( R 2 ||R 3 ) ) = ( 57.4 nV/ Hz ) 2 Output Noise Density: 2 2 e nout ⁄ G N = ( 182 nV/ Hz ) 2 e nout = 365 nV/ Hz Integrated Output Noise: GBWP = 100 kHz BW ≈ 50 kHz NPBW ≈ 78.6 kHz E nout ≈ ( 365 nV/ Hz ) ⋅ ( 78.6 kHz ) ≈ 102 µVRMS = 338 µVPK = 675 µVP-P Simulated Examples GBWP = Gain Bandwidth Product (Hz) BW = Bandwidth (Hz) This section covers two filter designs. It uses SPICE simulations to quickly obtain numerical results. The first design demonstrates potential issues with op amp circuits that need good noise performance. The second design improves the noise performance dramatically using simple changes. NPBW = Noise Power Bandwidth (Hz) SECOND ORDER FILTER BW ≈ GBWP ⁄ G N NPBW ≈ ( π ⁄ 2 ) ⋅ BW Where: Note: CFB op amp data sheets specify BW instead of GBWP. © 2008 Microchip Technology Inc. Figure 23 shows a second order Butterworth filter with a bandwidth of 1 kHz. It uses the MCP616 for the op amp; we will assume that it has no 1/f noise for now. DS01228A-page 15 AN1228 The resistor R3 balances the resistances seen by the op amp inputs, which minimizes the output offset due to input bias currents . It uses the Sallen-Key topology. The hump in the noise curves, seen at 1 kHz, is caused by the feedback action of the filter. The noise due to R3 and R2 is significant compared to the noise due to eni (the op amp’s input noise voltage density). THIRD ORDER FILTER C1 4.7n R1 R2 U1 38.3k 64.9k MCP616 VIN VOUT C2 2.2n R3 102k Butterworth Lowpass Filter. FIGURE 23: There are some obvious improvements we should make to this filter. Reducing the resistor values will reduce the thermal noise densities. Adding a filter at the output will significantly reduce the integrated noise at the output. The circuit in Figure 26 is the result of making these improvements. The resistors are about four times smaller; this reduction was limited to avoid output loading concerns. The filter design was changed to a 3rd order Butterworth to take maximum advantage of the additional filter stage (R4 and C4). C1 27n |VOUT/VIN| (dB) Figure 24 shows the simulated transfer function for Figure 23. R1 8.35k 10 0 -10 -20 -30 -40 -50 -60 -70 -80 -90 -100 -110 -120 100 1.E+2 VOUT C4 10n C2 5.6n R3 28.7k FIGURE 26: pass Filter. 1k 1.E+3 f (Hz) 10k 1.E+4 100k 1.E+5 Filter Transfer Function. Figure 25 shows the output noise voltage densities; the labels indicate the source of a particular output density. enr1, enr2 and enr3 represent R1, R2 and R3’s thermal noise, while eni, ibn and ibi represent the op amp’s noise sources. The combined output noise density is labeled “total.” 1000 total enr3 eni ibi 100 10 enr1 ibn 100 1.E+2 FIGURE 25: 1k 10k 1.E+3 1.E+4 f (Hz) enr2 100k 1.E+5 1M 1.E+6 Output Noise Densities. It is possible to reduce R3’s noise contribution more by adding a capacitor (C3, which isn’t shown) in parallel to R3. SPICE simulations will help determine if the reduction in noise is worth the additional cost. Figure 27 shows the simulated transfer function for Figure 26; notice the improved attenuation in the stopband compared to that shown previously (see Figure 24). 10 0 -10 -20 -30 -40 -50 -60 -70 -80 -90 -100 -110 -120 100 1.E+2 FIGURE 27: DS01228A-page 16 Improved Butterworth Low- A buffer placed after R4 and C4 would have a wide NPBW, so its noise contribution would be significant. For this reason, the output has no output buffer. |VOUT/VIN| (dB) eno (nV/Hz) R4 15.8k VIN FIGURE 24: 1 10 1.E+1 R2 U1 20.0k MCP616 1k 1.E+3 f (Hz) 10k 1.E+4 100k 1.E+5 Filter Transfer Function. © 2008 Microchip Technology Inc. AN1228 Figure 28 shows the output noise voltage densities for Figure 26; the labels indicate the source of a particular output density. enr1, enr2, enr3 and enr4 represent R1, R2, R3 and R4’s thermal noise, while eni, ibn and ibi represent the op amp’s noise sources. The combined output noise density is labeled “total”. eno (nV/Hz) total eni enr3 enr2 enr4 ibn 100 1 10 1.E+1 Flicker noise (also called 1/f noise or pink noise) can be important in low frequency applications (e.g., below 1 kHz). This noise increases the output variation above what the white noise predictions give. Note: 1000 10 FLICKER NOISE 1/f noise is caused by defects, at the atomic level, in semiconductor and resistive devices. These defects affect the DC current flowing through these devices. With many defects operating simultaneously, each with a different time constant, 1/f noise typically results. enr1 ibi 100 1.E+2 1k 10k 1.E+3 1.E+4 f (Hz) 100k 1.E+5 1M 1.E+6 Output Noise Densities. FIGURE 28: Auto-zeroed op amps have such low 1/f noise that it can be neglected. Comparing Figure 25 to Figure 28 shows that we have been successful in reducing the low frequency (i.e., below 200 Hz) output noise density. We have also reduced the overall NPBW significantly. Components with high 1/f noise include carbon resistors and semiconductor devices (diodes and transistors). All conductors, however, exhibit 1/f noise at some level. This section discusses 1/f noise, its impact on output variability and how to find relevant information in data sheets. A low frequency design example illustrates an approach to these designs. 1/f Noise Basics Table 5 compares the integrated output noise for these two designs. It summarizes the information found in Figure 25 and Figure 28 in convenient form. 1/f noise derives its name from its PSD shape (with units of VRMS2/Hz). The noise power increases at low frequencies as the reciprocal of frequency: TABLE 5: EQUATION 29: COMPARISON OF DESIGNS Noise Source Thermal Op Amp 2 3rd Order R1 5.4 2.5 R2 10.6 5.4 R3 154.4 9.1 R4 — 4.2 eni 120.6 13.4 ibn 1.9 0.5 ibi 26.9 0.9 Total 198.1 17.8 2 e nf ( f ) = e nf ( 1 Hz ) ⁄ f Eno (µVP-P) 2nd Order 1/F NOISE Where: enf(f) Note: = 1/f noise voltage density at the frequency f (nV/√Hz) The noise voltage density (enf) varies as 1/√f (10 dB/decade). Notice how the 1/f noise was specified at one frequency point in Equation 29 (at 1 Hz); this is for convenience in our later work. A DC current needs to flow for 1/f noise to be present. For instance, the PSpice diode noise model uses the following equation: EQUATION 30: 2 DIODE 1/F NOISE i nd ( f ) = KF ⋅ I D AF ⁄f Where: © 2008 Microchip Technology Inc. inf(f) = Diode’s 1/f noise current density at the frequency f (A/√Hz) KF = PSpice noise parameter (A2 – AF); default is 0 (usually around 10-15) AF = PSpice noise exponent; default is 1 DS01228A-page 17 AN1228 Noise Voltage Density (nV/√Hz), log scale 1/f Noise Figure 31 shows a histogram of the same noise data. The curve is the ideal Gaussian distribution with the same mean (0 µV) and standard deviation (3.55 µV). 7% Percentage of Occurrences 1/f noise is sometimes specified with a corner frequency. This happens when a noise source has both white and 1/f noise. The corner frequency occurs where the white noise density equals the 1/f noise density; see Figure 29. As we will see later, the combination of these two noise types produces a smooth bend in the region of fcorner, not the sharp corner depicted here. Gaussian 3% 2% 1% With the white noise density and the corner frequency, it is easy to calculate the 1/f noise voltage density (enf (1 Hz)): e nf ( 1 Hz ) = e nw f corner ⁄ ( 1 Hz ) enw = White noise voltage density (nV/√Hz) = corner frequency (Hz) The local average of 1/f noise wanders enough to be a concern in applications. fSAM = 1 SPS 1m 1.E-03 10m 1.E-02 f (Hz) EQUATION 32: FIGURE 30: DS01228A-page 18 12 8 INTEGRATED 1/F NOISE fH ∫ fL 2 e nf ( f ) df = 2 e nf ( 1 40 1/f Noise vs. Time. 45 50 fH ∫ fL 2 e nf ( 1 Hz ) ---------------------- df f Hz ) ⋅ ln ( f H ⁄ f L ) E nf = e nf ( 1 Hz ) ⋅ 35 10 In order to keep this analysis simple, we’ll use a bandpass brick wall filter with cutoff frequencies fL and fH (see Figure 3). This gives: = 25 30 t (min) 1 1.E+00 Integrated 1/f Noise 2 20 100m 1.E-01 1/f Noise; FFT (first 2048 FIGURE 32: points). E nf = 15 6 fSAM = 1 SPS 100n 1.E+02 100µ 1.E-04 Figure 30 plots 1/f noise data (from bench evaluation work) that shows typical 1/f noise behavior. The data was adjusted to have zero mean and was sampled at one sample per second (1 SPS). The local average wanders over time (compare to the white noise shown in Figure 10). 10 4 100µ 1.E+05 1µ 1.E+03 fcorner 5 2 The first 2048 data points were converted to the noise density plot in Figure 32 (the blue curve) using a FFT routine. The red curve is the best fit 1/f noise curve (it has the same integrated noise power). 10µ 1.E+04 Where: 0 0 -2 -4 1/f Noise Histogram. FIGURE 31: 1/f Noise (V/√Hz) CONVERSION FROM CORNER FREQUENCY 12 10 8 6 4 2 0 -2 -4 -6 -8 -10 -12 -6 -8 -10 -12 f (Hz), log scale FIGURE 29: Conceptual Diagram of the Corner Frequency. 1/f Noise (µV) 4% 1/f Noise (µV) fcorner Note: Enf 5% 0% White Noise EQUATION 31: 3000 Samples 6% ln ( f H ⁄ f L ) In other words, the integrated power (statistical variance) is proportional to the number of decades (or octaves) encompassed by the brick wall filter. © 2008 Microchip Technology Inc. AN1228 GROWTH OF 1/F NOISE (NOTE 1, NOTE 2) fH/fL No. Decades Enf (µVP-P) 1/fL 1.259 0.10 3.7 0.13 s 101 1 11.6 1.00 s 2 10 2 16.4 10 s 103 3 20.1 100 s 104 4 23.2 1000 s 105 5 26.0 2.78 hr 106 6 28.5 27.8 hr 107 7 30.7 11.6 day 108 8 32.9 116 day 109 9 34.9 3.17 year 9.50 35.8 10.0 year 3.16 × 109 Note 1: 2: Note: These numbers are based on fH = 10 Hz and enf(1 Hz) = 1160 nV/√Hz. The last entry was limited to a reasonable design lifetime for a PCB circuit. Changing the band-pass filter’s ratio of fH to fL has little impact on 1/f noise variability when fH/fL > 10 and fH << fcorner. Information in Data Sheets Table 7 shows the noise specifications in the MCP616/ 7/8/9 Data Sheet. This op amp family has a bipolar (PNP) input, so the noise current is higher than the CMOS input op amps. TABLE 7: MCP616/7/8/9 NOISE SPECIFICATIONS Parameters Sym Typ Units Conditions µVP-P f = 0.1 to 10 Hz The Input Noise Current Density (ini) is usually given at a frequency where the white noise dominates (1 kHz in this case). This specification helps select an op amp where the resistances are high. Remember that this curve describes both input noise current sources, which are statistically independent. Figure 33 shows the noise density plot in the MCP616/ 7/8/9 Data Sheet. The noise specifications describe this data. Note that our CMOS input op amps do not show ini in this plot because it is low enough to not affect most designs. 10,000 10,000 1,000 1,000 ini 100 100 eni Input Noise Current Density (fA/Hz) TABLE 6: The Input Noise Voltage Density (eni) is usually given at a frequency where the white noise dominates (1 kHz in this case). This specification helps select an op amp for high frequency work. Input Noise Voltage Density (nV/Hz) Table 6 shows the growth that 1/f noise would exhibit with different ratios of fH to fL. Mathematically, Enf has unbounded growth as f approaches zero. Practically speaking, however, that growth is so slow that it does not affect most applications. The numerical values are based on the data shown in Figure 32. 10 10 0.1 1.E+0 1 10 1.E+0 100 1.E+0 1k 1.E+0 10k 1.E1.E+0 01 0 Frequency 1 2 (Hz) 3 4 FIGURE 33: MCP616/7/8/9 Input Noise Voltage Density Plot. Refer to Table 2 and Table 3 for examples of how the white noise portion of ini changes with temperature. Design Example This design example is a simple modification to the filter shown in Figure 26. The goal is to show a low frequency circuit that is dominated by 1/f noise. To obtain a cut-off frequency of 100 Hz, the capacitors have all been increased by a factor of 10. Figure 34 shows the result; this is still a 3rd order Butterworth filter. Noise Input Noise Voltage Eni 2.2 Input Noise Voltage Density eni 32 nV/√Hz f = 1 kHz Input Noise Current Density ini 70 fA/√Hz f = 1 kHz The Input Noise Voltage (Eni) is the integrated noise voltage between 0.1 Hz and 10 Hz, with units of (µVP-P). It helps select an op amp for low frequency work. Typically, it is dominated by 1/f noise; autozeroed op amps are the main exception to this rule. © 2008 Microchip Technology Inc. C1 270n R1 8.35k R2 U1 20.0k MCP616 R4 15.8k VIN VOUT C2 56n C4 100n R3 28.7k FIGURE 34: Butterworth Low-pass Filter. DS01228A-page 19 AN1228 A buffer placed after R4 and C4 would have a wide NPBW, so its noise contribution would be significant. For this reason, the output has no output buffer. |VOUT/VIN| (dB) Figure 35 shows the simulated transfer function (compare to Figure 27). 10 0 -10 -20 -30 -40 -50 -60 -70 -80 -90 -100 10 1.E+1 Table 8 summarizes the information found in Figure 36 in convenient form. It is instructive to compare these results with those shown in Figure 25, Figure 28 and Table 5. TABLE 8: NOISE VOLTAGE CONTRIBUTIONS TO THE OUTPUT Noise Source Thermal Op Amp 100 1.E+2 f (Hz) 1k 1.E+3 10k 1.E+4 Filter Transfer Function. FIGURE 35: Eno (µVP-P) R1 0.78 R2 1.72 R3 2.89 R4 1.32 eni 5.31 ibn 0.36 ibi 0.45 Total 6.49 Figure 36 shows the output noise voltage densities; the labels indicate the source of a particular output density. enr1, enr2, enr3 and enr4 represent R1, R2, R3 and R4’s thermal noise, while eni, ibn and ibi represent the op amp’s noise sources. The combined output noise density is labeled “total.” eno (nV/Hz) 1000 total eni enr3 enr2 enr4 100 10 1 0.1 1.E-1 enr1 ibi ibn 1 1.E+0 FIGURE 36: 10 100 1.E+1 1.E+2 f (Hz) 1k 1.E+3 10k 1.E+4 Output Noise Densities. Comparing Figure 28 to Figure 36 shows that the white noise has been reduced. We also see the 1/f noise effect below 30 Hz. DS01228A-page 20 © 2008 Microchip Technology Inc. AN1228 DESIGN OPTIMIZATION FIND THE DOMINANT NOISE SOURCES With the basics of noise analysis and design under your belt, it is time to learn how to quickly and effectively optimize the noise performance of an op amp circuit. Any noise source that is at least half as large (in VRMS) as the largest source should be considered to be a dominant source. This may appear to be a very loose requirement at first glance, but works very well in practice. Signal-to-Noise Ratio To illustrate this point, Table 9 illustrates how a larger noise source (Enout1) and a smaller noise source (Enout2) contribute to the total noise (Enout). The ratio Enout2/Enout1 represents Enout2's magnitude relative to Enout1. The ratio Enout/Enout1 represents how much larger Enout is, compared to Enout1, due to the contribution from Enout2. When Enout2/Enout1 is ½, or smaller, we can ignore Enout2's contribution within engineering accuracy (error less than 12%). Remember, the noise terms are the result of a Sum of Squares (followed by a square root operation). The Signal-to-Noise Ratio (SNR) is one of the most common ways to decide if the noise in a circuit meets its design requirements. Usually, it is defined as the ratio of signal power (of a sine wave) to integrated noise power in decibels: EQUATION 33: OUTPUT SNR SNR = 20 ⋅ log 10 ( V OUT ⁄ E nout ) Where: VOUT = Sinusoidal output signal (VRMS) Enout = Integrated output noise voltage (VRMS) SNR = Signal-to-Noise Ratio (dB) In some applications, VOUT is expressed in relation to its full scale range (VPK or VP-P). This will not be done in this application note. Select a SNR value that supports the required accuracy for your design. Modify your circuit until it meets this SNR requirement. For a fixed output voltage, this is the same as minimizing the output noise. Note: Make your signal’s full scale range as large as possible; this minimizes the cost and effort in reducing the output noise. Reduce the Noise When done properly, a design’s noise performance will depend on only a couple of critical components. All of the other components can be chosen for other design goals. PLACE THE GAIN UP FRONT Place your high gain amplifier as close to the signal source (e.g., a sensor) as possible. Any noise sources after this gain will be divided by this gain; they should have little impact on the output noise. Noise sources before this gain, and in the high gain amplifier, will be critical to your design’s success. All of the amplifiers and other circuit elements after the high gain amplifier should be set to a gain as close to 1 V/V as possible. © 2008 Microchip Technology Inc. TABLE 9: EFFECT OF SECOND NOISE SOURCE Enout2/Enout1 Enout/Enout1 1/1 1.414 1/2 1.118 1/3 1.054 1/5 1.020 1/7 1.010 1/10 1.005 FILTER THE NOISE Filter any noise with the lowest NPBW possible. Place simple filters as close to dominant noise sources as possible; this helps when testing your design on the bench. A single real pole filter, using a resistor and a capacitor, is usually enough for most purposes. Place more sophisticated filters further away from the source. This has the benefits of using one complex filter for many noise sources. This reduces the overall cost of active filter designs with low component sensitivities (changes in capacitors, resistors and op amp bandwidth have little impact). Noise can alias into an ADC’s baseband. Select an anti-aliasing filter with a BW much slower than the sample rate (e.g., ten times slower) to minimize this effect. As we have seen before, a simple R-C lowpass filter at the very end, without a buffer, can minimize the op amps’ contribution to the total noise. This filter can be placed at the input of an Analog-to-Digital Converter (ADC) as long as the last capacitor is much larger than the ADC's input sampling capacitor (to minimize gain error). DS01228A-page 21 AN1228 Component Selection SUMMARY There are a few simple rules that make it easy to select components that will meet your design goals. This application note gives a simple overview of the noise theory used in circuit design. It is presented in a way to help like the reader’s knowledge of statistics and circuit design to circuit noise design. RESISTORS Resistors are usually chosen to be as small as possible at critical points in the design. The exception to this rule happens when the resistor acts like a current source in a circuit (e.g., the gain resistor in a transimpedance amplifier); the noise current is reduced by increasing the resistance (see Equation 10). Avoid resistors that use carbon as the resistive material. They generate high levels of 1/f noise. Use resistors with metal for the resistive material. Wire wound resistors typically have the best 1/f noise, but can cause high frequency circuit problems due to their parasitic inductance and capacitance. Metal film resistors have low 1/f noise and have good high frequency characteristics. OP AMPS Start your design with a general purpose part. Look at lower noise parts only after optimizing the rest of the circuit. For high frequency applications (e.g., above 1 kHz), or applications that use auto-zeroed op amps, select the op amp based on its white noise (eni and ini). For low frequency applications (e.g., below 1 kHz), also compare the 1/f noise performance (eni and ini). Compare the integrated noise between 0.1 Hz and 10 Hz (the Noise Voltage spec (µVP-P) in Microchip’s op amp data sheets). If that specification is not available in a data sheet, the noise spectrum plot will give the needed information. Compare the op amps’ noise density at the same frequency in the 1/f noise region. Many examples help build the reader’s knowledge of the design process, how filters affect noise, how to combine multiple noise terms at the circuit’s output, and optimizing a circuit’s noise performance. The topics cover what is needed in the majority of noise designs. Both white and 1/f noise are discussed. Manual analysis and computer simulations are used many times. Computer aided analysis is mentioned as a labor saving device. After the body of this application note, there are selected references to the literature to help the reader find background material that covers this material well. Appendices with additional vocabulary and an overview of computer aids completes this application note. REFERENCES Related Application Notes  AN1177, “Op Amp Precision Design: DC Errors,” Kumen Blake; Microchip Technology Inc., DS01177, 2008. Noise  Paul R. Gray and Robert G. Meyer, “Analysis and Design of Analog Integrated Circuits,” 2nd Ed., John Wiley & Sons, 1984.  Jerry D. Gibson, “Principles of Digital and Analog Communications,” Macmillan, 1989.  A. Bruce Carlson, “Communication Systems: An Introduction to Signals and Noise in Electrical Communication,” 3rd Ed., McGraw-Hill, 1986.  Athanasios Papoulis, “Probability, Random Variables, and Stochastic Processes,” 3rd Ed., McGraw-Hill,1991. Miscellaneous  DS01228A-page 22 Howard Johnson and Martin Graham, HighSpeed Digital Design: A Handbook of Black Magic,” Prentice Hall, 1993. © 2008 Microchip Technology Inc. AN1228 APPENDIX A: VOCABULARY This appendix gives a brief list of common terms used in amplifier noise work. They are organized by topic so that their context is easier to grasp. A.1 Spectral Densities Power Spectral Density (PSD) is the frequency domain description of a noise source’s statistical variation. Its units are (W/Hz) (sometimes converted to dBm/Hz). It is also related the noise’s auto correlation function. It is also called Noise Power Density. Noise Voltage Density (en) is the square root of PSD, normalized to a standard resistance (usually 1Ω). It has units of (V/√Hz). It is also called spot noise or noise per root Hertz. Noise Current Density (in) is the square root of PSD, normalized to a standard resistance (usually 1Ω). It has units of (A/√Hz). It is also called spot noise or noise per root Hertz. A.2 Spectral Shapes White noise is a PSD that has a constant value over frequency. It is a mathematical convenience used to make system noise calculations simpler. Broadband noise describes a noise source that is (nearly) white over a circuit’s frequency range of interest. It isn’t white, but appears to be white to that circuit. Noise Power Bandwidth (NPBW) is mathematically convenient parameter used to describe how a circuit processes white noise. It has units of (Hz). It is the equivalent bandwidth of a brick wall filter that produces the same output noise as the actual circuit. Excess Noise is any noise that exceeds the white noise level at low frequencies (only 1/f noise is discussed in this application note): • 1/f noise, also known as flicker noise or pink noise • 1/f2 noise, also known as red noise • Random Telegraph Signal (RTS) noise, also known as burst noise or popcorn noise (has a spectral shape reminiscent of white noise filtered by a lowpass filter with a single real pole) A.3 Integrated Noise Noise Current (In) is the square root of Noise Power normalized by standard resistance (usually 1Ω). It has units of (ARMS, APK or AP-P). When in units of ARMS, it is also called the standard deviation. A.4 Probability Density Functions Many physical noise sources, but not all, have the Gaussian (or Normal) probability density function. They are said to be Gaussian Noise, or sometimes Additive White Gaussian Noise (AWGN). This noise is usually associated with random processes that fulfill the Identical and Independently Distributed (IID) assumption; it is the sum of a large number of statistically independent random variables with the same probability density function. The probability density function is: EQUATION A-1: 2 1 –( x – μ ) p ( x ;μ, σ ) = -------------- ⋅ exp ⎛ ----------------------⎞ ⎝ 2σ ⎠ 2 πσ Analog to Digital Converters (ADC) and Digital to Analog Converters (DAC) usually have their quantization errors modeled as random noise with a Uniform probability density function (the device noise at the inputs would be Gaussian, however). The probability density function is: EQUATION A-2: p ( x ;μ, σ ) = 1 ⁄ ( 2 3 σ ) , x – μ < 3 σ = 0, otherwise A.5 Figures of Merit Signal-to-Noise Ratio (SNR) is the ratio of the signal power to the noise power. It is usually shown in units of (dB), although (VRMS/VRMS) or (ARMS/ARMS) are also acceptable. Sometimes the signal’s full scale range is the numerator of the ratio, with units of (VPK or VP-P). Other figures of merit not covered in this application note are: • Noise Figure (NF) (dB) • Noise Factor (F) (V/V) • Noise Temperature (TN) (K) Noise Power (N) is the noise source’s statistical variation. Its units are (W) (sometimes dBm). Noise Voltage (En) is the square root of Noise Power normalized by standard resistance (usually 1Ω). It has units of (VRMS, VPK or VP-P). When in units of VRMS, it is also called the standard deviation. © 2008 Microchip Technology Inc. DS01228A-page 23 AN1228 APPENDIX B: COMPUTER AIDS While this application note emphasizes manual calculation and formulas, most design work uses computers. B.1 Noise Simulations Circuit noise simulations can be done as part of an AC simulation in SPICE simulators. The SPICE program was developed at UC Berkeley. Many SPICE derivative simulators are used in circuit design; the most popular for board level design is PSpice® (from Cadence®). B.1.1 GENERAL REMARKS The component models need to be correctly defined for noise simulations to give realistic results. Op amp macro models from Microchip are set up to work properly in PSpice. Resistors, diodes and transistors usually give the correct white noise (when the model is accurate). 1/f noise in diodes and transistors will not simulate correctly without special attention to the relevant parameters. The resistor model does not include 1/f noise; this can be added to the circuit using diodes and a dependent source, if needed. You will need to define the input source and output circuit node before the noise analysis can run. SPICE produces an input referred noise vector (across frequency); it is referred to the chosen input source. SPICE also produces an output noise vector at the chosen node. The noise results in different SPICE simulators come in two different forms: as noise voltage (current) density (VRMS/√Hz) or as the square of the noise voltage (current) density (VRMS2/Hz). Check your simulator using resistor thermal noise; 1 kΩ of resistance, at +25°C, will give 4×10-9 VRMS/√Hz in the former case, and 1.6×10-17 VRMS2/Hz in the latter case. Note: PSpice produces noise in units of (VRMS/√Hz). Other simulators may, or may not. Use the simulator’s plotting tool to determine which noise sources dominate and to improve the noise filtering and shaping. To obtain the integrated noise (Enout) between fL and fH (see Figure 3), read En0f’s value at f = fL and f = fH (let’s call these values EL and EH). The integrated noise is (for any spectral shape): EQUATION B-1: E nout = 2 2 EH – EL To put this trace’s data into a spreadsheet, click on its label found at the bottom left of Probe’s screen. This selects this trace (the label changes color). Copy the data to Window’s clipboard by typing the key sequence Ctrl-C. Paste the results into your spreadsheet. Following this sequence produced two columns of data; one frequency vector and one noise vector. B.1.3 ESTIMATING NPBW WITH SPICE Now that we can extract the integrated noise from our simulations, we can easily estimate a filter’s NPBW. The following steps will make this more clear: • Use a very large resistor (or 2 in parallel) as the noise voltage source • Insert a buffer between the noise source and the filter’s input • Plot the output noise density (enout) • Calculate the integrated output noise (Enout) from DC to infinity (a high enough frequency) • Choose the enout value that represents the passband (at the chosen gain, HM) • Calculate the NPBW EQUATION B-2: NPBW ESTIMATE NPBW = ( E nout ⁄ e nout ) B.2 2 Using Symbolic Solver Engines There are several places where a symbolic solver can speed up your noise analysis: To calculate integrated noise in PSpice, open its plotting utility (Probe) and add the following trace: • Converting node equations to transfer functions • Factoring transfer functions • Expanding a magnitude squared transfer function into its Partial Fraction Expansion form • Evaluating the definite integrals used for integrated noise (or NPBW) EXAMPLE B-1: Some popular tools are: B.1.2 CALCULATING INTEGRATED NOISE WITH PSPICE PSPICE TRACE FUNCTION sqrt(s(v(onoise)*v(onoise))) This is the running integral (from 0 Hz to f, using the PSpice function s()) of the output noise (let’s call it En0f) in units of (VRMS/√Hz). DS01228A-page 24 • Mathematica® (from Wolfram Research) • Maple™ (from Waterloo Maple Software) • Matlab® (from The MathWorks); use the Symbolic Math Toolbox™ • MathCad® (from Parametric Technology Corporation) © 2008 Microchip Technology Inc. Note the following details of the code protection feature on Microchip devices: • Microchip products meet the specification contained in their particular Microchip Data Sheet. • Microchip believes that its family of products is one of the most secure families of its kind on the market today, when used in the intended manner and under normal conditions. • There are dishonest and possibly illegal methods used to breach the code protection feature. All of these methods, to our knowledge, require using the Microchip products in a manner outside the operating specifications contained in Microchip’s Data Sheets. Most likely, the person doing so is engaged in theft of intellectual property. • Microchip is willing to work with the customer who is concerned about the integrity of their code. • Neither Microchip nor any other semiconductor manufacturer can guarantee the security of their code. Code protection does not mean that we are guaranteeing the product as “unbreakable.” Code protection is constantly evolving. We at Microchip are committed to continuously improving the code protection features of our products. Attempts to break Microchip’s code protection feature may be a violation of the Digital Millennium Copyright Act. If such acts allow unauthorized access to your software or other copyrighted work, you may have a right to sue for relief under that Act. Information contained in this publication regarding device applications and the like is provided only for your convenience and may be superseded by updates. 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