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1 Op-Amp Noise Calculation and Measurement Art Kay Senior Applications Engineer Texas Instruments Inc – Tucson

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2 Noise Presentation Contents Review of white noise and 1/f noise Noise Hand Calculations Tina Spice Noise Analysis Noise Measurement Appendix 1 – Measurement Example Appendix 2 – Analysis Details

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3 What is Intrinsic Noise Why do I Care? The op-amp itself generates noise Noise acts as an errorit corrupts the signal Calculate, simulate, and measure this noise Ideal Real

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4 White noise or broadband noise normal distribution

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5 1/f or pink noise normal distribution

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6 (Burst) Popcorn Noise Bimodal (or multi-modal) distribution

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7 Synonyms Broadband Noise – White Noise, Johnson Noise, Thermal Noise 1/f Noise – Pink Noise, Flicker Noise, Low Frequency Noise, Excess Noise Burst Noise – Popcorn Noise, Red Noise random telegraph signals (RTS). Strictly speaking, these terms are not 100% synonymous. For example, broadband noise on an op-amp may be a combination of thermal noise and shot noise.

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8 For example, if P(-1

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9 STDEV Relationship to Peak-to-Peak for a Gaussian PDF Gaussian PDF +/-3 STD Deviations = 6 sigma 99.7%

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10 STDEV Relationship to Peak-to-Peak Number of Standard Deviations Percent chance of measuring voltage 2σ (same as ±σ)68.3% 3σ (same as ±1.5σ)86.6% 4σ (same as ±2σ)95.4% 5σ (same as ±2.5σ)98.8% 6σ (same as ±3σ)99.7% 6.6σ (same as ±3.3σ)99.9% Is standard deviation the same as RMS?

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11 Stdev = RMS when the Mean is zero (No DC component). For all the noise analysis we do this will be the case. The noise signals we consider are Gaussian signals with zero mean. Note that the two formulas are equal to each other if you set μ = 0 (zero average). See further information in appendix. RMS vs STDEV RMS Standard deviation Where x i – data samples μ – average of all samples n – number of samples Where x i – data samples n – number of samples DC component will create reading error RMS

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12 V n1 V n2 Σ V n1 V n2 V nT Add Noise As Vectors (RMS Sum) Sum of two Random Uncorrelated Noise Sources

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13 The mean- square open- circuit voltage (e) across a resistor (R) is: e n = (4kT K RΔf) where: T K is Temperature (ºK) R is Resistance (Ω) f is frequency (Hz) k is Boltzmanns constant (1.381E-23 joule/ºK) e n is volts (V RMS ) To convert Temperature Kelvin to T K = o C + T C Thermal Noise Random motion of charges generate noise

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14 Noise Spectral Density vs. Resistance Resistance (Ohms)Noise Spectral Density vs. Resistance nV/rt-Hz Thermal Noise e n density = (4kT K R) Resistance versus Spectral Density

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15 Op-Amp Noise Model OPA277 Data VNVNVNVN IN-IN-IN-IN- IN+IN+IN+IN+ Noise Model (I N + and I N - are not correlated) Tina Simplified Model ININ VNVN

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17 Noise Analysis for Simple Op-Amp Circuit Noise Sources Op-Amp Voltage Noise Source Op-Amp Current Noise Sources Resistor Noise Sources Calculation Considerations Convert Noise Spectrum to Noise Voltage - External Filter Bandwidth Limit - Op-Amp Closed Loop Bandwidth Noise Gain

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18 Calculus Reminder Height Width 4 9 Area Integral = Area under the curve

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19 Freq (Hz)010 Freq (Hz) 010 Voltage Spectral Density (V/rt-Hz) Power Spectral Density (V 2 /Hz) 25V 2 / Hz 5V / rtHzYou cant integrate the Voltage spectral density curve to get noise You integrate the Power spectral density curve to get noise Convert Noise Spectrum to Noise Voltage (Broadband Only – Simple Case) Wrong Correct

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20 Convert Noise Spectrum to Noise Voltage (Broadband Only – Simple Case) Noise Power = V 2 * BW (Hz) Noise Voltage = V * BW (Hz) Hz You integrate the Power spectral density curve to get noise Correct

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21 Noise Gain for Voltage Noise Source Noise Gain – Gain seen by the noise source. Example: Noise_Gain = (R2/R1) + 1 = 2 Signal_Gain = -R2/R1 = -1 Output_Noise = Vn*(Noise_Gain) Referred to Input Referred to Output Signal Source Noise Source *

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22 Understanding The Spectrum: Total Noise Equation (Current or Voltage) e nT = [(e n1/f ) 2 + (e nBB ) 2 ] where: e nT = Total rms Voltage Noise in volts rms e n1/f = 1/f voltage noise in volts rms e nBB = Broadband voltage noise in volts rms

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23 RMS Low Pass Filter Shapes the Spectrum How do we convert this plot to noise? Broadband Region 1/f Region Low pass filter

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24 Real Filter Correction vs Brickwall Filter where: f P = roll-off frequency of pole or poles f BF = equivalent brickwall filter frequency

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25 Number of Poles in Filter Kn AC Noise Bandwidth Ratio AC Noise Bandwidth Ratios for n th Order Low-Pass Filters Real Filter Correction vs Brickwall Filter BW n = (f H )(K n ) Effective Noise Bandwidth

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26 Broadband Noise Equation en BB = (e BB )([BW n ]) where: en BB = Broadband voltage noise in volts rms e BB = Broadband voltage noise density ; usually in nV/Hz BW n = Noise bandwidth for a given system BW n = (f H )(K n ) where: BW n = noise bandwidth for a given system f H = upper frequency of frequency range of operation K n = Brickwall filter multiplier to include the skirt effects of a low pass filter e BB

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27 1/f Noise Equation (see appendix for derivation) en 1/f = (e )([ln(f H /f L )]) where: en 1/f = 1/f voltage noise in volts rms over frequency range of operation e = voltage noise density at 1Hz; (usually in nV) f H = upper frequency of frequency range of operation (Use BW n as an approximation for f H ) f L = lower frequency of frequency range of operation e = (e )([f]) where: e = normalized noise at 1Hz (usually in nV) e = voltage noise density at f ; (usually in nV/Hz) f = a frequency in the 1/f region where noise voltage density is known e

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28 Example Noise Calculation Given: OPA627 Noise Gain of 101 Find (RTI, RTO): Voltage Noise Current Noise Resistor Noise

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29 Unity Gain Bandwidth = 16MHz Closed Loop Bandwidth = 16MHz / 101 = 158kHz Voltage Noise Spectrum and Noise Bandwidth 50nV/rt-Hz 5nV/rt-Hz

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30 Example Voltage Noise Calculation Voltage Noise Calculation: Broadband Voltage Noise Component: BW n (f H )(K n )(note Kn = 1.57 for single pole) BW n (158kHz)(1.57) =248kHz en BB = (e BB )(BW n ) en BB = (5nV/Hz)(248kHz) = 2490nV rms 1/f Voltage Noise Component: e = (e )(f) e = (50nV/Hz)(1Hz) = 50nV en 1/f = (e )([ln(f H /f L )]) Use f H = BW n en 1/f = (50nV)([ln(248kHz/1Hz)]) = 176nV rms Total Voltage Noise (referred to the input of the amplifier): en T = [(en 1/f ) 2 + (en BB ) 2 ] en T = [(176nV rms) 2 + (2490nV rms) 2 ] = 2496nV rms

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31 Example Current Noise Calculation Note: This example amp doesnt have 1/f component for current noise. e n-out = Gain x (i n )x(R eq )e n-in = (i n )x(R eq )

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32 Example Current Noise Calculation Broadband Current Noise Component: BWn (f H )(K n ) BWn (158kHz)(1.57) =248kHz i nBB = (i BB )(BW n ) i nBB = (2.5fA/Hz)(248kHz) = 1.244pA rms R eq = R f || R 1 = 100k || 1k = 0.99k e ni = (I n )( R eq ) = (1.244pA)(0.99k) = 1.23nV rms Since the Total Voltage noise is e nvt = 2496nV rms the current noise can be neglected. neglect

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33 e = (4kT K RΔf) e nr = (4kT K RΔf)where: R = Req = R1||Rf Δf = BW n e = (4 (( ) (0.99k)( e nr = (4 (1.38E-23) ( ) (0.99k)(248kHz)) = 2010nV rms Example Resistor Noise Calculation (4kTRΔf) e n-out = Gain x ((4kTRΔf)) (4kTRΔf) e n-in = (4kTRΔf)

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34 Total Noise Calculation Voltage Noise From Op-Amp RTI: e nv = 2510nV rms Current Noise From Op-Amp RTI (as a voltage): e ni = 1.24nV rms Resistor Noise RTI: e nr = 2020nV rms Total Noise RTI: e n in = ((2510nV) 2 + ((1.2nV) 2 + ((2010nV) 2 ) = 3216nV rms Total Noise RTO: e n out = e n in x gain = (3216nV)(101) = 325uV rms

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35 Calculating Noise Vpp from Noise Vrms Peak-to-Peak Amplitude Probability of Having a Larger Amplitude 2 X rms32% 3 X rms13% 4 X rms4.6% 5 X rms1.2% 6 X rms *0.3% 6.6 X rms0.1% Relation of Peak-to-Peak Value of AC Noise Voltage to rms Value *Common Practice is to use: Peak-to-Peak Amplitude = 6 X rms

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36 Peak to Peak Output For our Example e n out = 325uV rms e n out p-p = (325uV rms )x6 = 1.95mVp-p

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38 Tina Spice – Free simulation software DC, AC, Transient, and Noise simulation Includes all Texas Instruments op-amps Unlimited Nodes Does not include some options (e.g. Monte Carlo analysis) Search for Tina Spice on –Download free –Application circuits available

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39 1.How to Verify that the Tina Model is Accurate 2.How to Build Your Own Model 3.How to Compute Input and Output Noise Tina Spice Analysis

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40 Noise Model Test procedure: Is the Tina Noise Model Accurate?

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41 Translate Current Noise to Voltage Nose Set Gain to 1.

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42 Output Noise diagram gives the output voltage noise spectral density measured at each volt meter. Generate Spectral Noise Plots

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43 Separate Curves

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44 Click on Axis to Scale

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45 OPA227 Matches Tina Model Matches the Data Sheet

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46 The OPA627 Tina Model Does NOT Match the Data Sheet 5nV/rtHz No 1/f Wrong magnitude

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47 Build Your Own Noise Model Using Burr-Brown Macro Model Noise Sources and Generic Op-Amp The voltage and current noise source is available at (search for noise sources).

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48 Right Click on Noise Source to Edit the Macro Enter magnitude of 1/f and broadband noise into the macro.

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49 Look for a point in the 1/f region. Enter the frequency and magnitude at this point (1Hz, 50nV/rtHz) 1/f Region

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50 (100kHz, 5nV/rtHz) Look for a point in the broad band region. Enter the magnitude at this point Broadband Region

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51 Compile the Macro After macro is compiled press file > close and return to schematic editor.

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52 Follow the same procedure for current noise. This example has no 1/f component (set FLWF = 0.001). Same Procedure for Current Noise Source

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53 OLG = 10 (Ndb/20) = 1E6 (From Data Sheet) GBW = 16MHz (From Data Sheet) Dominant Pole = GBW / OLG = (16MHz) / (1E6) = 16Hz where: GBW – Unity Gain-Bandwidth Product OLG – Open Loop Gain Important Op-Amp Characteristics OPA627 Data Sheet Dominant Pole

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54 Edit Generic Op-Amp Macro-model 1.Double Click on Op-Amp 2.Press Type Button 3.Edit Open loop gain and Dominant Pole according to Op-Amp data sheet

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55 2.5fA/rtHz 2.5nV/rtHz 50nV/rtHz2.5nV/rtHz 50nV/rtHz Verify the Noise Model is Correct Using the Test Procedure

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56 Lets Use Tina on the Hand Analysis Circuit Output Noise will give the noise Spectrum at all output meters (V627 in this example). Total Noise will give the integrated total RMS noise at all output meters.

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57 323uV rms Lets Analyze the Circuit that We did Hand Analysis on Hand Analysis: e n out = 325uV rms Tina Analysis: e n out = 323uV rms

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58 Gain vs Frequency AC Analysis > Transfer Characteristic Output Noise Spectrum Total Noise 25uV with BW n = 100MHz Use Tina to Analyze this Common Topology

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59 Gain vs Frequency (V627 / VG1) Output Noise Spectrum Total Noise 25uV with BW n = 100MHz External Filter Cf Filter External Filter Improves Noise Performance

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61 Instruments For Noise Measurements 1.True RMS Meter 2.Oscilloscope 3.Spectrum Analyzer / Signal Analyzer

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62 True RMS Meter – How it works ac coupling Input terminals Digitizer Thousands of Samples Compute RMS HP3458A Display RMS Circuit under test Good Broadband Noise Measurement

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63 True RMS Meter – HP3458A 3 True RMS modes (Synchronous, Analog, Random) Random – optimal mode for broadband measurements - Specified Bandwidth (BW = 20Hz to 10MHz) - Accuracy 0.1% for Specified Bandwidth - Noise Floor 17uVrms (on 10mV range) - Ranges: 10mV, 100mV V See Appendix for Setup

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64 Oscilloscope Noise Measurements Do NOT use 10x Probes for low noise measurements Use direct BNC Connection (10 times better noise floor) Use Male BNC Shorting Cap to Measure Noise Floor Use BW Limiting if Appropriate Use digital scope in dc coupling for 1/f noise measurements (ac coupling has a 60Hz high pass) Use AC coupling for broadband measurements if necessary

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65 TDS460A Digitizing Oscilloscope Example Best Noise Floor = 0.2mV BW Limit = 20MHz BNC Shorting Cap Noise measurements use BNC cables Worst Noise Floor = 8mV BW FULL = 400MHz 10x Scope Probe Noise Floor = 0.8mV BW FULL = 400MHz BNC Shorting Cap

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66 Oscilloscope Noise Measurement BNC Shorting Cap

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67 Spectrum Analyzer -- Convert Result to nV/rt-Hz

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68 Effect of Changing the measurement Bandwidth BW set to 150Hz Measuring 67kHz and 72kHz BW set to 1200Hz Wider BW reduces resolution & raises noise floor Narrow BW increases resolution & lowers noise floor Narrow BW Longer Sweep Time

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69 Filter TypeApplicationKn 4-pole syncMost Spectrum Analyzers Analog pole syncSome Spectrum Analyzers Analog1.111 Typical FFTFFT-based Spectrum Analyzers1.056 dBm to Spectral Density Convert dBm to nV/rt-Hz

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70 Effect of Changing Averaging No Averaging.Averaging = 49. Increase Averaging to Reduce Noise Floor Increase Measurement Time

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72 Noise Measurement Circuits 1.1/f Measurement Filter Circuit Measurement results Noise floor (choosing the best amp) 2.Example Circuit Noise Measurement Broadband with scope and true RMS meter 3.Voltage Noise Spectral Density Measurement Look at typical spectrum analyzer errors

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73 0.1Hz Second Order 10.0Hz Fourth Order 1/f Filter Device Under Test DUT 1/f Filter 0.1Hz to 10Hz 0.1Hz HPF Gain = 10 10Hz LPF Gain = 10 10Hz LPF Gain = 1 Gain = 1001

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74 Y-Axis scale must be adjusted to account for the gain to match data sheet. 5mV/(10x10x1000) = 50nV/div PDS 0.1Hz to 10Hz Noise Curve Tektronix TDS460A Measurement OPA227 The Circuit Generates the Data Sheet 1/f Plots (Example OPA227) Data Sheet Curve

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75 Tektronix TDS460A Measurement OPA227 1/f Filter Measured vs Tina (Example OPA227) Measured Output is approximately 10mVp-p OPA227 Tina Simulation 1.84mV rms (1.84mV)(6) = 11mVp-p Tina Simulation

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76 Measure Noise Floor of 1/f Filter What Op-Amp will give us the lowest noise Floor? Replace DUT with Short Large Input Impedance

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77 Op-AmpGeneral DescriptionV n (nV/rt-Hz) I n (fA/rt-Hz) OPA227Low noise OPA132Low noise 3 OPA735Auto Zero CMOS13540 Measure Noise Floor of 1/f Filter What Op-Amp will give us the lowest noise Floor?

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78 OPA132 (0.6mVp-p in paint can) OPA132 (3mVp-p in free air) OPA227 (15mVp-p paint can) OPA734 (0.8mVp-p in free air) Large Current noise Auto zero No 1/f Noise Low Vosi Drift Noise from Vosi Drift Low noise in thermally stable environment Note: measurements in paint can minimize thermal drift.

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79 Linear Supply or Battery BNC From Circuit Under Test BNC To Scope Copper box or paint can as RFI / EMI Shield. This also minimizes thermal drift Use BNC Male Shorting cap for noise floor Some General Measurement Precautions

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80 A paint can makes a good shielded environment

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81 Calculated (Previous Example): Vn = 325uV rms Measured: Vn = ((476uV) 2 – (148uV) 2 ) = 452uV rms Note: peak to peak reading is roughly 2mVp-p Noise floor Output noise Measure The OPA627 Example Using A Scope Rf 100k R1 1k Copper box Scope BNC BNC

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82 Calculated (Previous Example): e n = 325uV rms Measured HP3458A DVM: DVM_READING =346uV Noise Floor = 18uV Noise_Meas = ((346uV) 2 – (18uV) 2 ) = 346uV rms Measure The OPA627 Example Using A True RMS Meter (HP3458a) Rf 100k R1 1k HP3458A True RMS SETACV RNDM Coax BNC Banana Copper box

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83 Spectrum Analyzer Measurement of OPA627 (Voltage Noise Spectrum) 1M Input Impedance Agilent 35770a Dynamic Signal Analyzer Has nV/rt-Hz Mode Bandwidth 0Hz to 100kHz 20mV dc offset 0.2mVp-p noise (Bad SNR) 0.008Hz HPF removes dc component, but still allows 1/f measurement Low Req to reduce effect of i n and thermal noise

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84 Spectrum Analyzer Measurement of OPA627 Data was collected over five different frequency ranges Noise floor verification The low frequency run is time consuming. Approximately 12 min.

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85 Spectrum Analyzer Measurement of OPA627 Low frequency tail on each run must be discarded. 60Hz noise pickup 1.Combine to one curve 2.Discard bad information 3.Divide by gain of 100

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86 The Tail Error is from relatively wide measurement bandwidth at low frequency.

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87 Measured 1/f noise corner is better then data sheet. Spectrum Analyzer Measurement of OPA627 OPA627 PDS

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88 1.Robert V. Hogg, and Elliot A Tanis, Probability and Statistical Inference, 3rd Edition, Macmillan Publishing Co 2.C. D. Motchenbacher, and J. A. Connelly, Low-Noise Electronic System Design, A Wiley-Interscience Publication 3.Henry W. Ott, Noise Reduction Techniques in Electronics Systems, John Wiley and Sons References Noise Article Series (www.en-genius.net) Acknowledgments: 1.R. Burt, Technique for Computing Noise based on Data Sheet Curves, General Noise Information 2.N. Albaugh, General Noise Information, AFA Deep-Dive Seminar 3.T. Green, General Information 4.B. Trump, General Information 8.B. Sands, Noise Models

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89 Thank You for Your Interest in Noise – Calculation and Measurement Comments, Questions, Technical Discussions Welcome: Art Kay

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91 Statistics Summary (Formulas)

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92 Statistics Summary (PDF vs Discrete Population)

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93 Statistics Summary (RMS and STDEV)

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94 Stdev = RMS when the Mean is zero (No DC component). Tina gives the true RMS result in AC calculations. See mathematical proof in appendix. 1Vpk sinusoidal RMS STDEV RMS = STDEV RMS vs STDEV

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95 For i = 1 To white = Rnd(1) buf0 = * old_buf * white buf1 = * old_buf * white buf2 = 0.95 * old_buf * white buf3 = 0.85 * old_buf * white buf4 = 0.62 * old_buf * white buf5 = 0.25 * old_buf * white pink = buf0 + buf1 + buf2 + buf3 + buf4 + buf5 Next i Useful Numerical Methods Method for generating a time domain approximation of 1/f noise

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96 Useful Numerical Methods Method for generating average, standard deviation, and RMS for a discrete population (sampled data) N is the Number of samples, and f(i) is an array of measured data Average=0 Initialize the variables Stdev=0 RMS=0 For i = 1 To N Average=Average + f(i) Next I Average = Average/N For i = 1 To N stdev=(f(i) – Average)^2 Next I Stdev = Stdev/N For i = 1 To N RMS=(f(i))^2 Next I RMS = RMS/N

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97 Brick Wall Factor Calculation for first order filter.

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98 1/f Noise Derivation

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99 Current Noise - 0V ++ (I n x R f ) - Vo = (I n x R f ) InIn

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100 Current Noise Vo = (I n x R f ) Equ. Input Noise

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101 Solve for Resistor Noise Components Noise Source with Each Resistor

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102 Vo = e n1 x R f /R 1 e n1

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103 Vo = e n2 e n2

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104 Add noise components and refer to the input. Note: the input noise is equivalent to Rf || R1 (proof on next page)

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105 Proof: Simple Amp Resistor Noise Input Noise is equivalent to noise from parallel combination of Rf and R1.

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