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Precision Design: Random Noise
BACKGROUND INFORMATION
This section covers basics frequency noise work. somewhat theoretical nature, some numerical examples illustrate concepts. serves foundation following sections. references more depth theoretical coverage these concepts. material after this section illustrates these concepts. those readers this subject matter, beneficial read complete application note several times, while working examples.
INTRODUCTION
This application note covers essential background information design theory needed design noise, precision circuits. focus simple, results oriented methods approximations useful circuits with low-pass response. material will interest engineers design amps circuits which need better signal-to-noise ratio (SNR), want evaluate design trade-offs quickly effectively. This application note general enough cover both voltage feedback (VFB) (traditional) current feedback (CFB) amps. examples, however, will limited Microchip's voltage feedback amps. Additional material this application note includes references literature, vocabulary computer design aids.
Where Average
most commonly used statistical concept average. Standard circuit analysis gives deterministic value plus point time. Once these deterministic values subtracted out, noise variables left have average zero. Noise interpreted random fluctuations stochastic value) about average response. will deal with linear circuits, superposition applies; average random fluctuations obtain correct final result.
Words Phrases
Device Noise Noise Spectral Density Integrated Noise Signal-to-Noise Ratio (SNR)
Noise Spectral Density
easiest approach analyzing random analog noise starts frequency domain (even engineers that strongly prefer time domain). Stationary noise sources (their statistics change with time) represented with Power Spectral Density (PSD) function. Because analyzing analog electronic circuits, units power will deal with This noise power equivalent statistical variance (2). variance uncorrelated random variables
Prerequisites
material this application note will much easier follow after reviewing following statistical concepts: Average Standard Deviation Variance Gaussian (normal) probability density function Histograms Statistical Independence Correlation
EQUATION
VARIANCE UNCORRELATED VARIABLES
Knowledge basic circuit analysis also assumed. Where: var()
uncorrelated random variables variance function
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This fact very important because various random noise sources circuit caused physically independent phenomena. Circuit noise models that based these physically independent sources produce uncorrelated statistical quantities. extension concept variance. spreads variation noise power variable across many frequency bins. noise each (power with units Watts) statistically independent other bins. units (W/Hz), which called "density" function. picture Figure illustrates these concepts. (W/Hz) Power PSD(fk) Strictly speaking, passive circuits (RLC circuits), this conversion needs done with specific resistance value V2/R I2R). most noise work involving active devices, however, standard resistance value assumed.
Integrated Noise
make rational design choices, need know what total noise variation this section gives that capability. will convert statistical variance standard deviation squared) using definite integral across frequency.
CALCULATIONS
Using Equation fact that power frequency independent other bins, powers together:
EQUATION
(Hz) Where:
TOTAL NOISE VARIATION
FIGURE
Power Spectral Density.
total noise power
this application note, plots (and functions) one-sided, with x-axis units Hertz. This traditional choice circuit analysis because this output (physical) spectrum analyzers. Note: very important, when reading electronic literature noise, determine: one-sided two-sided? frequency units Hertz (Hz) Radians Second (rad/s)?
summation approximation measured noise data discrete time points. integral applies continuous time noise; useful deriving theoretical results.
PREFERRED EQUATIONS
circuit analysis, conversion integrated noise (En) usually takes place with noise voltage density; Equation noise's standard deviation.
most frequency circuits, signals noise interpreted measured voltages currents, power. this reason, usually presented equivalent forms: Noise voltage density (en) with units (V/Hz) Noise current density (in) with units (A/Hz) voltage current units values; they could given (VRMS/Hz) (ARMS/Hz). Traditionally, subscript understood, shown. Note: Many beginners find units confusing. natural result, however, converting units W/Hz) into noise voltage current density square root operation.
EQUATION
INTEGRATED NOISE VOLTAGE
Where: en(f) noise voltage density (V/Hz) integrated noise voltage (VRMS) standard deviation (VRMS)
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Noise current densities also converted integrated noise (In):
TABLE
IMPORTANT TWO-TAILED PROBABILITIES
Crest Factor (Note Peak-to-Peak Peak (VPK/VRMS) (VP-P/VRMS) 1.64 2.58 3.29 4.50 6.00
EQUATION
INTEGRATED NOISE CURRENT
1.64 2.58 3.29 4.50 6.00
PG(|x| 0.1% 6.80 10-6 1.97
3.29 5.15 6.58 9.00 12.00
Where: in(f) noise current density (A/Hz) integrated noise current (ARMS) standard deviation (ARMS)
Note
INTERPRETATION
need know probability density function order make informed decisions based integrated (RMS) noise. work this application note, noise will have Gaussian (Normal) probability density function. principle noise sources within amps, resistors PCB, Gaussian. When they combined, they produce total noise that also Gaussian. Figure shows standard Gaussian probability density function (mean standard deviation logarithmic y-axis.
1.E+00 1.E-01 1.E-02
Microchip's data sheets VP-P/VRMS when reporting (usually between Hz). This about range visible noise analog oscilloscope trace.
integrated noise results this application note independent frequency time. They only used describe noise global sense; correlations between noise seen different time points lost after integration done.
Filtered Noise
time measure noise, been altered from original form seen within physical noise source. easiest represent these alterations noise, linear systems, transfer function frequency domain) from source output. resulting output noise different spectral shape than source.
pG(x;
1.E-03
TRANSFER FUNCTIONS NOISE
turns that noise output linear operation (represented transfer function) related input noise transfer function's squared magnitude; Equation This thought result statistical independence between PSD's frequency bins (see Figure
1.E-04
1.E-05 10-5
1.E-06
1.E-07 10-7
1.E-08 1.E-09
EQUATION
nout
OUTPUT NOISE
FIGURE Standard Gaussian Probability Density Function.
Table shows important points this curve corresponding (two tailed) probability that random Gaussian variable outside those points. This information useful converting values (voltages currents) either peak peak-to-peak values. column label sometimes called number sigma from mean. Where: enout
noise voltage density (V/Hz) noise voltage density VOUT (V/Hz)
Example shows conversion simple transfer function squared magnitude. starts Laplace Transform [2], converted Fourier Transform (substituting then converted squared magnitude form function best this last conversion with transform factored form.
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EXAMPLE TRANSFER FUNCTION CONVERSION EXAMPLE
Laplace Transfer Function: Conversion Fourier Transfer Function: Conversion Magnitude Squared: Where: Laplace frequency (1/s) Radian frequency (rad/s) Pole (rad/s) Frequency (Hz) Pole frequency (Hz) nout physical world, however, brick wall filters would have horrible behavior. They cannot realized with finite number circuit elements. Physical filters that approach this ideal show three basic problems: their step response exhibits Gibbs phenomenon (overshoot ringing that decays slowly), they suffer from noise enhancement (due high pole quality factors) they very difficult implement. Note: Comments literature (e.g., filter textbooks) about "ideal" brick wall filters should viewed with skepticism.
integrated noise voltage integrals (Equation Equation their most simple terms when brick wall filter used. Equation shows that, this case, brick wall filter's frequencies become integration limits. integrated current noise treated similarly.
EQUATION
INTEGRATED NOISE WITH BRICK WALL FILTER
nout
BRICK WALL FILTERS
transfer function that easiest manipulate mathematically brick wall filter. infinite attenuation (zero gain) stop bands, constant gain (HM) pass band; Figure |H(j2f)| (V/V) Where:
Lower cutoff frequency (Hz) Upper cutoff frequency (Hz) Pass band gain (V/V)
Appendix "Computer Aids" popular circuit simulators symbolic mathematics packages that help these calculations. (Hz)
White Noise
White noise that constant over frequency. received name from fact that white light equal mixture visible wavelengths frequencies). This mathematical abstraction real world noise phenomena. truly white noise would produce infinite integrated noise. Physically, this concern because circuits physical materials have limited bandwidth. start with white noise because easiest manipulate mathematically. Other spectral shapes will addressed subsequent sections.
FIGURE
Brick Wall Filter.
will three variations brick wall filter (refer Figure Low-pass zero) Band-pass shown) High-pass infinity) Brick wall filters mathematical convenience that simplifies noise calculations.
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NOISE POWER BANDWIDTH
When white noise passed through brick wall filter (see Figure integrated noise becomes very simple calculation. Equation simplified shot noise current density's magnitude depends diode's current (ID) electron charge (q). usually modeled white noise; Equation
EQUATION
INTEGRATED WHITE NOISE WITH BRICK WALL FILTER
EQUATION
Where:
DIODE SHOT NOISE
nout Where: enout Input noise voltage density (V/Hz) Output noise voltage density (V/Hz)
Electron charge 1.602 10-19 Diode Current
This equation usually represented what called Noise Power Bandwidth (NPBW). NPBW bandwidth (under square root sign) that converts white noise density into correct integrated noise value. case brick wall filters, Equation
Let's look specific example:
EXAMPLE
Given:
DIODE SHOT NOISE CALCULATION
Calculate: 1.602
EQUATION
INTEGRATED WHITE NOISE WITH NPBW
nout NPBW Where: NPBW -fL,for brick wall filters Note:
17.9 pA/Hz
calculation results this application note show more decimal places than necessary; places usually good enough. This done help reader verify calculations.
high-pass filter appears cause infinite integrated noise. real circuits, however, bandwidth limited, finite band-pass response). Note: NPBW applies white noise only; other noise spectral shapes require more sophisticated formulas computer simulations.
RESISTOR THERMAL NOISE
thermal noise present resistor usually modeled white noise (for frequencies temperatures concerned with). This noise depends resistor's temperature, current. resistive material exhibits this phenomenon, including conductors CMOS transistors' channel. Figure shows models resistor thermal noise voltage current densities. sources shown with polarity convenience circuit analysis.
Circuit Noise Sources
This section discusses circuit noise sources different circuit components transfer functions between sources output.
DIODE SHOT NOISE
Diodes bipolar transistors exhibit shot noise, which effect electrons crossing potential barrier random arrival times. equivalent circuit model diode shown Figure
FIGURE Physically Based Noise Model Resistors.
FIGURE Model Diodes.
Physically Based Noise
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equivalent noise voltage current spectral densities (remember that 273.15 0°C):
EQUATION
RESISTOR THERMAL NOISE DENSITY
VOUT
Where:
Boltzmann constant 1.381 10-23 (J/K) Ambient temperature Resistance
FIGURE Physically Based Noise Model Amps.
noise voltage source also placed other input amp, with negative connected positive This alternate connection gives same output voltage (VOUT). voltage feedback (VFB) amps, both noise current sources have same magnitude. This magnitude shown Microchip's data sheets with symbol ini; units fA/Hz stands femto, 10-15). now, will discuss white noise part these spectral densities. will defer discussion noise until later. literature sometimes shows amplifier noise model that only noise current source. these cases, second noise current's power been combined into noise voltage magnitude. Note: Keep mind that amps have physically independent noise current sources.
4kTA represents resistor's internal power. maximum available power another resistor (when they equal). Many times maximum available power shown kTA/2 because physicists prefer using two-sided noise spectra. Let's resistor example.
EXAMPLE
Given:
THERMAL NOISE DENSITY CALCULATION
25°C 298.15 Calculate noise voltage density: 1.381
298.15
4.06 nV/Hz
Calculate noise current density: 1.381
298.15
4.06 pA/Hz
NOISE
amp's noise modeled with three noise sources: input noise voltage density (eni) input noise current density (ibn ibi). three noise sources physically independent, they statistically uncorrelated. Figure shows this model; similar error model covered [1].
current feedback (CFB) amps, noise current sources (ibn ibi) different magnitude because input bias currents (IBN IBI) different magnitude. They produced physically independent statistically uncorrelated processes. amps typically used wide bandwidth applications (e.g., above MHz). Microchip's CMOS input amps have noise current density based input pins' diode leakage current (specified input bias current, IB). Table gives MCP6241 amp's white noise current values across temperature.
TABLE
(°C)
MCP6241 (CMOS INPUT) NOISE CURRENT DENSITY
(pA) 1100 (fA/Hz) 0.57
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Table gives MCP616 amp's white input noise current density across temperature. This part bipolar (PNP) input; base current input bias current, which decreases with temperature.
NOISE ANALYSIS PROCESS
This section goes through analysis process normally followed noise design. uses very simple noise design problem make this process clear.
TABLE
(°C)
MCP616 (BIPOLAR INPUT) NOISE CURRENT DENSITY
(nA) (fA/Hz)
Simple Example
circuit shown Figure uses lowpass brick wall filter filter's bandwidth (fH) gain (HM) V/V. amp's input noise voltage density (eni) nV/Hz, gain bandwidth product much higher than
input noise voltage density (eni) typically does change much with temperature. Note: Note: Noise current density (ini) usually changes significantly with temperature (TA). Most time, shot noise formula calculate exception this rule amps with input bias current cancellation circuitry.
Brick Wall Low-pass Filter
VOUT
FIGURE
Circuit.
Figure shows both noise voltage density (eni) output noise voltage density (enout). Notice that enout simply multiplied low-pass brick wall's pass-band gain (HM). Noise Voltage Density (nV/Hz) enout
TRANSFER FUNCTIONS
transfer function from each noise source circuit output needed. This obtained with SPICE simulations (see Appendix "Computer Aids") with analysis hand. This application note emphasizes manual approach more order build understanding derive handy design approximations. most convenient manual approach circuit analysis using Laplace frequency variable (s). Figure shows resistor, inductor capacitor with their corresponding impedances (using
(Hz)
FIGURE
Noise Voltage Densities.
noise current densities ignored this circuit because they flow into voltage source output, which present zero impedance. calculate integrated noise output (Enout). result shown three different units (RMS, peak peak-to-peak):
FIGURE Impedance Models Common Passive Components.
EXAMPLE
INTEGRATED NOISE CALCULATION
nout
nout
µVRMS µVPK= µVP-P
Note: This application note uses crest factor VPK/VRMS VP-P/VRMS).
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Figure shows numerical simulation results output noise over time. Enout describes variation this noise. This same data plotted histogram form Figure curve represents ideal Gaussian probability density function (with same average variation).
FILTERED NOISE
This section covers circuits that have filters their output. discussion focuses filters with real poles develop insight useful design formulas. effect that reactive circuit components have noise deferred later section. Noise generated filters ignored now.
fSAM kSPS
Enout(t) (µV)
Low-pass Filter With Single Real Pole
Figure shows circuit with low-pass filter output, which single real pole (fP). need worry about noise current densities because sources zero impedance (like Figure will assume that neglected because much lower.
(ms)
FIGURE
Percentage Occurrences
Enout Gaussian 1024 Samples
Output Noise Time.
Real Pole Low-pass Filter
VOUT
FIGURE pass Filter.
Circuit With Low-
need filter's transfer function order calculate output integrated noise; needs squared magnitude form (see Example derivation these results):
EQUATION
Enout (µV)
LOW-PASS TRANSFER FUNCTION
FIGURE
Output Noise Histogram.
Review Process
basic process have followed described follows. Collect noise filter information Convert noise sources noise output Combine integrate output noise terms Evaluate impact output signal
Figure shows transfer function magnitude decibels.
H(j2f) (dB) 0.01
FIGURE
Filter Magnitude Response.
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obtain integrated noise, assuming amp's input noise voltage density (eni) white:
Low-pass Filter With Real Poles
low-pass filter Figure real poles (fP1 fP2). need worry about noise current densities because sources zero impedance (like Figure assume that much lower than neglected.
EQUATION
INTEGRATED NOISE DERIVATION
nout
nout
atan
Thus, NPBW this filter (see Equation
Real Pole Low-pass Filter
VOUT
EQUATION
NOISE POWER BANDWIDTH
FIGURE pass Filter.
Circuit With Low-
NPBW always reduce integrated output noise reducing NPBW, signal response suffer far. need keep filter's bandwidth (BW) least large desired signal this filter's BW). low-pass filters, also select based maximum allowable step response rise time (this applies reasonable low-pass filter):
filter's transfer function magnitude squared transfer function function factored form, are:
EQUATION
LOW-PASS TRANSFER FUNCTION
Where: First pole frequency (Hz) Second pole frequency (Hz)
EQUATION
Where:
RISE TIME BANDWIDTH 0.35
low-pass filter's bandwidth (Hz) Rise time Let's numerical example with reasonably wide bandwidth; noise limited filter's bandwidth.
Figure shows transfer function magnitude decibels specific case where double fP1.
H(j2f) (dB) 0.01
fP2/fP1
EXAMPLE
INTEGRATED NOISE CALCULATION
Filter Specifications: Gain Specifications: nV/Hz Filter Rise Time: Integrated Noise Calculations: amp's bandwidth NPBW 15.8 nout 15.8
FIGURE
Filter Magnitude Response.
12.6 µVRMS 41.4 µVPK 82.9 µVP-P
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follow same process before calculate NPBW.
EXAMPLE
INTEGRATED NOISE CALCULATION
EQUATION
NPBW
Change Filter Specifications: 15.5 Filter Bandwidth Rise Time: 9.98 Integrated Noise Calculations: amp's bandwidth NPBW 12.2 nout 12.2
11NPBW
before, NPBW similar traded-off with rise time (see Equation 14).
EQUATION
Where:
11.0 µVRMS 36.4 µVPK 72.9 µVP-P
High-pass Filter With Single Real Pole
Figure shows circuit with high-pass filter with single real pole (fP). need worry about noise current densities because sources zero impedance (like Figure practical circuits, there needs low-pass filter frequency much higher than fH); integrated noise would infinite otherwise. nothing else, used limit NPBW.
Let's through numerical example where amp's bandwidth neglected.
EXAMPLE
INTEGRATED NOISE CALCULATION
Filter Specifications: 13.4 26.8 Gain Specifications: nV/Hz Filter Bandwidth Rise Time: 9.98 Integrated Noise Calculations: amp's bandwidth NPBW 14.0 nout 14.0
Real Pole High-pass Filter
VOUT
FIGURE pass Filter.
Circuit With High-
filter's transfer function magnitude squared transfer function function factored form, are:
EQUATION
HIGH-PASS TRANSFER FUNCTION
11.8 µVRMS 39.0 µVPK 78.1 µVP-P
Let's redo this example with equal poles 15.5 kHz.
Where: Pole frequency (Hz) Low-pass NPBW (Hz)
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Figure shows transfer function magnitude decibels shown).
H(j2f) (dB) 0.01
Band-pass Filter With Real Poles
Figure shows circuit with band-pass filter with real poles (highpass pole lowpass pole fP2). need worry about noise current densities because sources zero impedance (like Figure neglected because assume that much higher than fP2.
Real Pole Band-pass Filter
VOUT
FIGURE
Filter Magnitude Response.
FIGURE pass Filter.
Circuit With Band-
follow same process before calculate NPBW acts like upper integration limit integrated noise equation).
filter's transfer function magnitude squared transfer function function factored form, are:
EQUATION
NPBW
NPBW Where:
EQUATION
BAND-PASS TRANSFER FUNCTION
Where: High-pass pole frequency (Hz) Low-pass pole frequency (Hz)
numerical example with bandwidth much higher than filter pole (this very common).
EXAMPLE
INTEGRATED NOISE CALCULATION
Filter Specifications: amp's NPBW Gain Specifications: NPBW 1.57 Integrated Noise Calculations: amp's bandwidth NPBW 1.57 15.8 1.55 nout 1.55
H(j2f) (dB)
Figure shows transfer function magnitude decibels, with fP1.
fP2/fP1
nV/Hz
0.01
100m
µVRMS µVPK µVP-P
Note: high-pass filter's NPBW little effect integrated noise, unless near (but that would band-pass filter).
1000
10000
FIGURE
Filter Magnitude Response.
Using symbolic solver derive NPBW help.
EQUATION
NPBW
NPBW
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Let's another numerical example.
FILTERS WITH GREATER SELECTIVITY
There other filters with sharper transition region, when such Chebyshev, Inverse Chebyshev Elliptic filters. Their NPBW ratios closer because they have smaller transition region (between pass-band stop-band). This smaller transition region reduces integrated noise output. Their step response, however, tends have more ringing slower decay. Again, NPBW approximated with bandwidth. More exact results obtained with simulations (see Appendix "Computer Aids").
EXAMPLE
INTEGRATED NOISE CALCULATION
Filter Specifications: Gain Specifications: nV/Hz Integrated Noise Calculations: amp's bandwidth NPBW 15.7 1.01 15.5 nout 15.5
NOISE INTERNAL FILTERS
will shown later (see Figure 25), active filters produce much more noise than first expected. amps inside filter produce noise voltage density filter's output that wider bandwidth than filter; wide bandwidths. resistors noise contributions tend show peak edges filter passband (noise enhancement), which increases integrated output noise.
12.5 µVRMS 41.2 µVPK 82.3 µVP-P
Comments Other Filters
This section discusses other filters they affect output integrated noise. gives very simple approximation NPBW when filter order greater than then discusses noise generated interal filter.
SOME SIMPLE LOW-PASS FILTERS
Table shows NPBW ratio some lowpass filters order
TABLE
Low-pass Filter Type
NPBW SOME LOW-PASS FILTERS
NPBW
1.571 1.220 1.155 1.128 1.114 1.571 1.153 1.071 1.046 1.038 1.571 1.111 1.047 1.026 1.017
Identical Real Poles Bessel Butterworth
Note:
bandwidth rough estimate NPBW almost filters (the main exception when
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MULTIPLE NOISE SOURCES
This section covers approaches combining multiple noise sources into output integrated noise result. This knowledge applied simple lowpass filter non-inverting gain circuit.
Low-pass Filter
Figure shows circuit with low-pass filter with real pole (fP). need worry about noise current densities because sources zero impedance (like Figure will assume that neglected because much lower. enr1 VOUT
Combining Noise Outputs
When combine noise results, output, take advantage statistical independence noise separate frequency bins Physically independent noise sources This independence simplifies work, since need worry about correlations. integrate output noise densities time, then combine results using Squares approach (see Equation also combine noise densities using Squares approach first, then integrate resulting noise density.
FIGURE Filter.
Circuit With Low-pass
will integrate noise densities first because this will give important insight into this low-pass filter. This circuit like already Figure have added R1's thermal noise. filter's transfer function magnitude squared transfer function function factored form, Equation (Figure shows transfer function magnitude decibels).
eno12, enok2 Integrate Noise Densities (over frequency) Eno12, Enok
Squares each frequency)
EQUATION
LOW-PASS FILTER TRANSFER FUNCTION
Where: filter's pole frequency (Hz)
Squares
Integrate Noise Density (over frequency) Eno2
FIGURE Approaches Combining Output Noise Terms.
Each approach advantages. Integrating first helps determine which noise source dominates; handy hand designs. Finding output noise density first helps adjust frequency shaping elements design; easier with computer simulations. next section ("R-C Low-pass Filter") demonstrates approach left Figure section following that ("Non-inverting Gain Circuit") demonstrates approach right Figure
follow same process before calculate NPBW. trade-offs between NPBW shown Equation apply this filter.
EQUATION
NPBW
NPBW
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integrated noise becomes:
Non-inverting Gain Circuit
Figure complete model non-inverting gain circuit. series noise voltage density sources because their transfer function VOUT simpler that form. uses shunt noise current density source because same transfer function VOUT that uses; this reduces work. enr1 VOUT
EQUATION
INTEGRATED NOISE
noU1 NPBW noR1 NPBW nout Where: EnoU1 EnoR1 Enout U1's output integrated noise (VRMS) R1's output integrated noise (VRMS) Total output integrated noise (VRMS)
noU1
noRC
enr3
last expression shown EnoR1 (sqrt(kTA/C1)) popularly called noise" (referring inside square root). This result applies only this particular case (integrated thermal noise output lowpass filter). Note: this equation mislead you; generates thermal noise,
inr2
FIGURE Non-inverting gain Amplifier, with multiple noise sources. ANALYSIS WITH CONSTANT GAINS
will combine noise densities first obtain output noise density (enout). this case, because have reactive elements circuit, will simple matter integrate enout hand produce Enout. will start with transfer functions from each source VOUT (see reference [1]). gains will assumed constant now; will deal with frequency shaping later
Let's numerical example where filter resistor both contribute noise.
EXAMPLE
INTEGRATED NOISE CALCULATION
Ambient Temperature: 25°C 298.15 Filter Specifications: Gain Specifications: nV/Hz Filter Pole, Bandwidth Rise Time: 10.6 Integrated Noise Calculations: amp's bandwidth NPBW 10.6 16.7 noU1 16.7
EQUATION
Where:
TRANSFER FUNCTIONS
Noise Gain (V/V)
Note that noise gain (GN) from non-inverting input VOUT, when closed-loop condition, when other (external) energy sources zero. Note: concept noise gain central understanding behavior. simplifies bandwidth stability analyses.
12.9 µVRMS
12.8 noRC 12.8 16.7 nout 1.66 µVRMS 13.0 µVRMS 43.0 µVPK 85.9 µVP-P
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magnitude squared transfer functions simply squares constant terms Equation will combine these noise densities into equation output noise density (using squares approach): Reactive elements circuit will require more detailed analysis because each noise source have different frequency shape. following example noise sources about same magnitude.
EQUATION
COMBINING NOISE DENSITIES
EXAMPLE
INTEGRATED NOISE CALCULATION
nout
Ambient Temperature: 25°C 298.15 Circuit Specifications: Specifications: nV/Hz pA/Hz GBWP Preliminary Calculations: 2.00 (R2||R3) Input Noise Densities:
While this equation sufficient calculate Enout, converting input referred form gives more insight designer. Dividing both sides GN2, substituting R3/R2 simplifying, gives:
EQUATION
NOISE EQUATION
nout
This shows that output noise density very simple relationship resistances seen inputs (R2||R3)). Note: Equation also applies inverting amplifiers (i.e., driven grounded).
57.4 Output Noise Density: nout nout Integrated Output Noise: GBWP
ANALYSIS WITH LIMITED BANDWIDTH
produce finite output integrated noise, need filter that limits NPBW. This filter implemented with amp, reactive elements circuit (e.g., capacitors) filter after amp. amp's NPBW. response approximated with single real pole hand calculations. Gain Bandwidth Product (GBWP) specification data sheets gives:
NPBW 78.6 nout 78.6 µVRMS µVPK µVP-P
EQUATION
NPBW AMP'S BANDWIDTH
Simulated Examples
This section covers filter designs. uses SPICE simulations quickly obtain numerical results. first design demonstrates potential issues with circuits that need good noise performance. second design improves noise performance dramatically using simple changes.
GBWP NPBW Where: GBWP NPBW Note: Gain Bandwidth Product (Hz) Bandwidth (Hz) Noise Power Bandwidth (Hz)
SECOND ORDER FILTER
Figure shows second order Butterworth filter with bandwidth kHz. uses MCP616 amp; will assume that noise now.
data sheets specify instead GBWP.
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resistor balances resistances seen inputs, which minimizes output offset input bias currents [1]. uses Sallen-Key topology. 4.7n 38.3k 64.9k MCP616 2.2n 102k VOUT hump noise curves, seen kHz, caused feedback action filter. noise significant compared noise (the amp's input noise voltage density).
THIRD ORDER FILTER
There some obvious improvements should make this filter. Reducing resistor values will reduce thermal noise densities. Adding filter output will significantly reduce integrated noise output. circuit Figure result making these improvements. resistors about four times smaller; this reduction limited avoid output loading concerns. filter design changed order Butterworth take maximum advantage additional filter stage C4). 8.35k 5.6n 28.7k 20.0k MCP616 15.8k VOUT
FIGURE
Butterworth Lowpass Filter.
Figure shows simulated transfer function Figure
-100 -110 -120 1.E+2
|VOUT/VIN| (dB)
FIGURE pass Filter.
1.E+3 (Hz) 1.E+4 100k 1.E+5
Improved Butterworth Low-
FIGURE
Filter Transfer Function.
buffer placed after would have wide NPBW, noise contribution would significant. this reason, output output buffer. possible reduce R3's noise contribution more adding capacitor (C3, which isn't shown) parallel SPICE simulations will help determine reduction noise worth additional cost. Figure shows simulated transfer function Figure notice improved attenuation stopband compared that shown previously (see Figure 24).
-100 -110 -120 1.E+2
Figure shows output noise voltage densities; labels indicate source particular output density. enr1, enr2 enr3 represent R3's thermal noise, while eni, represent amp's noise sources. combined output noise density labeled "total."
1000 total enr3
enr1 1.E+2 1.E+3 1.E+4 (Hz) enr2 100k 1.E+5 1.E+6
1.E+1
|VOUT/VIN| (dB)
FIGURE
Output Noise Densities.
1.E+3
(Hz)
1.E+4
100k 1.E+5
FIGURE
Filter Transfer Function.
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Figure shows output noise voltage densities Figure labels indicate source particular output density. enr1, enr2, enr3 enr4 represent R4's thermal noise, while eni, represent amp's noise sources. combined output noise density labeled "total".
1000 total enr3 enr2 enr4
enr1
FLICKER NOISE
Flicker noise (also called noise pink noise) important frequency applications (e.g., below kHz). This noise increases output variation above what white noise predictions give. Note: Auto-zeroed amps have such noise that neglected.
noise caused defects, atomic level, semiconductor resistive devices. These defects affect current flowing through these devices. With many defects operating simultaneously, each with different time constant, noise typically results. Components with high noise include carbon resistors semiconductor devices (diodes transistors). conductors, however, exhibit noise some level. This section discusses noise, impact output variability find relevant information data sheets. frequency design example illustrates approach these designs.
1.E+1
1.E+2
1.E+3 1.E+4 (Hz)
100k 1.E+5
1.E+6
FIGURE
Output Noise Densities.
Comparing Figure Figure shows that have been successful reducing frequency (i.e., below output noise density. have also reduced overall NPBW significantly. Table compares integrated output noise these designs. summarizes information found Figure Figure convenient form.
Noise Basics
noise derives name from shape (with units VRMS2/Hz). noise power increases frequencies reciprocal frequency:
TABLE
COMPARISON DESIGNS
(µVP-P) Order 10.6 154.4 120.6 26.9 198.1 Order 13.4 17.8
EQUATION
NOISE
Noise Source Thermal Total
Where: enf(f) noise voltage density frequency (nV/Hz)
Note:
noise voltage density (enf) varies dB/decade).
Notice noise specified frequency point Equation Hz); this convenience later work. current needs flow noise present. instance, PSpice diode noise model uses following equation:
EQUATION
DIODE NOISE
Where: inf(f)
Diode's noise current density frequency (A/Hz) PSpice noise parameter AF); default (usually around 10-15) PSpice noise exponent; default
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noise sometimes specified with corner frequency. This happens when noise source both white noise. corner frequency occurs where white noise density equals noise density; Figure will later, combination these noise types produces smooth bend region fcorner, sharp corner depicted here. Noise Voltage Density (nV/Hz), scale Noise White Noise (Hz), scale Figure shows histogram same noise data. curve ideal Gaussian distribution with same mean standard deviation (3.55 µV).
Percentage Occurrences Noise (µV)
3000 Samples
Gaussian
FIGURE
Noise Histogram.
fcorner
FIGURE Conceptual Diagram Corner Frequency.
With white noise density corner frequency, easy calculate noise voltage density (enf Hz)):
first 2048 data points were converted noise density plot Figure (the blue curve) using routine. curve best noise curve same integrated noise power).
100µ 1.E+05
fSAM
corner Where: fcorner White noise voltage density (nV/Hz) corner frequency (Hz)
Noise (V/Hz)
EQUATION
CONVERSION FROM CORNER FREQUENCY
1.E+04
1.E+03
100n 1.E+02 100µ 1.E-04
1.E-03
Figure plots noise data (from bench evaluation work) that shows typical noise behavior. data adjusted have zero mean sampled sample second SPS). local average wanders over time (compare white noise shown Figure 10). Note: local average noise wanders enough concern applications.
fSAM
1.E-02 (Hz)
100m 1.E-01
1.E+00
FIGURE points).
Noise; (first 2048
Integrated Noise
order keep this analysis simple, we'll bandpass brick wall filter with cutoff frequencies (see Figure This gives:
(min)
EQUATION
INTEGRATED NOISE
Noise (µV)
FIGURE
Noise Time.
other words, integrated power (statistical variance) proportional number decades octaves) encompassed brick wall filter.
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Table shows growth that noise would exhibit with different ratios Mathematically, unbounded growth approaches zero. Practically speaking, however, that growth slow that does affect most applications. numerical values based data shown Figure Input Noise Voltage Density (eni) usually given frequency where white noise dominates this case). This specification helps select high frequency work. Input Noise Current Density (ini) usually given frequency where white noise dominates this case). This specification helps select where resistances high. Remember that this curve describes both input noise current sources, which statistically independent. Figure shows noise density plot MCP616/ 7/8/9 Data Sheet. noise specifications describe this data. Note that CMOS input amps show this plot because enough affect most designs.
10,000 Input Noise Voltage Density 10,000 Input Noise Current Density
TABLE
fH/fL 1.259
GROWTH NOISE (NOTE NOTE
Decades 0.10 9.50 (µVP-P) 11.6 16.4 20.1 23.2 26.0 28.5 30.7 32.9 34.9 35.8 1/fL 0.13 1.00 1000 2.78 27.8 11.6 3.17 year 10.0 year
3.16 Note Note:
1,000
1,000
These numbers based enf(1 1160 nV/Hz. last entry limited reasonable design lifetime circuit. Changing band-pass filter's ratio little impact noise variability when fH/fL fcorner.
1.E- 1.E+0 1.E+0 1.E+0 1.E+0 1.E+0 Frequency (Hz)
FIGURE MCP616/7/8/9 Input Noise Voltage Density Plot.
Refer Table Table examples white noise portion changes with temperature.
Information Data Sheets
Table shows noise specifications MCP616/ 7/8/9 Data Sheet. This family bipolar (PNP) input, noise current higher than CMOS input amps.
Design Example
This design example simple modification filter shown Figure goal show frequency circuit that dominated noise. obtain cut-off frequency capacitors have been increased factor Figure shows result; this still order Butterworth filter. 270n 8.35k 28.7k 100n 20.0k MCP616 15.8k VOUT
TABLE
Parameters Noise Input Noise Voltage
MCP616/7/8/9 NOISE SPECIFICATIONS
Units µVP-P Conditions
Input Noise Voltage Density Input Noise Current Density
nV/Hz fA/Hz
Input Noise Voltage (Eni) integrated noise voltage between with units (µVP-P). helps select frequency work. Typically, dominated noise; autozeroed amps main exception this rule.
FIGURE
Butterworth Low-pass Filter.
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buffer placed after would have wide NPBW, noise contribution would significant. this reason, output output buffer. Figure shows simulated transfer function (compare Figure 27).
-100 1.E+1
Table summarizes information found Figure convenient form. instructive compare these results with those shown Figure Figure Table
TABLE
NOISE VOLTAGE CONTRIBUTIONS OUTPUT
(µVP-P) 0.78 1.72 2.89 1.32 5.31 0.36 0.45 6.49
Noise Source Thermal
|VOUT/VIN| (dB)
1.E+2 (Hz) 1.E+3 1.E+4
Total
FIGURE
Filter Transfer Function.
Figure shows output noise voltage densities; labels indicate source particular output density. enr1, enr2, enr3 enr4 represent R4's thermal noise, while eni, represent amp's noise sources. combined output noise density labeled "total."
1000
total enr3 enr2 enr4
enr1
1.E-1
1.E+0
1.E+1 1.E+2 (Hz)
1.E+3
1.E+4
FIGURE
Output Noise Densities.
Comparing Figure Figure shows that white noise been reduced. also noise effect below
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DESIGN OPTIMIZATION
With basics noise analysis design under your belt, time learn quickly effectively optimize noise performance circuit.
FIND DOMINANT NOISE SOURCES
noise source that least half large VRMS) largest source should considered dominant source. This appear very loose requirement first glance, works very well practice. illustrate this point, Table illustrates larger noise source (Enout1) smaller noise source (Enout2) contribute total noise (Enout). ratio Enout2/Enout1 represents Enout2's magnitude relative Enout1. ratio Enout/Enout1 represents much larger Enout compared Enout1, contribution from Enout2. When Enout2/Enout1 smaller, ignore Enout2's contribution within engineering accuracy (error less than 12%). Remember, noise terms result Squares (followed square root operation).
Signal-to-Noise Ratio
Signal-to-Noise Ratio (SNR) most common ways decide noise circuit meets design requirements. Usually, defined ratio signal power sine wave) integrated noise power decibels:
EQUATION
OUTPUT
nout Where: VOUT Sinusoidal output signal (VRMS) Enout Integrated output noise voltage (VRMS) Signal-to-Noise Ratio (dB) some applications, VOUT expressed relation full scale range (VPK VP-P). This will done this application note. Select value that supports required accuracy your design. Modify your circuit until meets this requirement. fixed output voltage, this same minimizing output noise. Note: Make your signal's full scale range large possible; this minimizes cost effort reducing output noise.
TABLE
EFFECT SECOND NOISE SOURCE
Enout/Enout1 1.414 1.118 1.054 1.020 1.010 1.005 1/10
Enout2/Enout1
FILTER NOISE
Filter noise with lowest NPBW possible. Place simple filters close dominant noise sources possible; this helps when testing your design bench. single real pole filter, using resistor capacitor, usually enough most purposes. Place more sophisticated filters further away from source. This benefits using complex filter many noise sources. This reduces overall cost active filter designs with component sensitivities (changes capacitors, resistors bandwidth have little impact). Noise alias into ADC's baseband. Select anti-aliasing filter with much slower than sample rate (e.g., times slower) minimize this effect. have seen before, simple lowpass filter very end, without buffer, minimize amps' contribution total noise. This filter placed input Analog-to-Digital Converter (ADC) long last capacitor much larger than ADC's input sampling capacitor minimize gain error).
Reduce Noise
When done properly, design's noise performance will depend only couple critical components. other components chosen other design goals.
PLACE GAIN FRONT
Place your high gain amplifier close signal source (e.g., sensor) possible. noise sources after this gain will divided this gain; they should have little impact output noise. Noise sources before this gain, high gain amplifier, will critical your design's success. amplifiers other circuit elements after high gain amplifier should gain close possible.
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Component Selection
There simple rules that make easy select components that will meet your design goals.
SUMMARY
This application note gives simple overview noise theory used circuit design. presented help like reader's knowledge statistics circuit design circuit noise design. Many examples help build reader's knowledge design process, filters affect noise, combine multiple noise terms circuit's output, optimizing circuit's noise performance. topics cover what needed majority noise designs. Both white noise discussed. Manual analysis computer simulations used many times. Computer aided analysis mentioned labor saving device. After body this application note, there selected references literature help reader find background material that covers this material well. Appendices with additional vocabulary overview computer aids completes this application note.
RESISTORS
Resistors usually chosen small possible critical points design. exception this rule happens when resistor acts like current source circuit (e.g., gain resistor transimpedance amplifier); noise current reduced increasing resistance (see Equation 10). Avoid resistors that carbon resistive material. They generate high levels noise. resistors with metal resistive material. Wire wound resistors typically have best noise, cause high frequency circuit problems their parasitic inductance capacitance. Metal film resistors have noise have good high frequency characteristics.
AMPS
Start your design with general purpose part. Look lower noise parts only after optimizing rest circuit. high frequency applications (e.g., above kHz), applications that auto-zeroed amps, select based white noise (eni ini). frequency applications (e.g., below kHz), also compare noise performance (eni ini). Compare integrated noise between (the Noise Voltage spec (µVP-P) Microchip's data sheets). that specification available data sheet, noise spectrum plot will give needed information. Compare amps' noise density same frequency noise region.
REFERENCES
Related Application Notes
AN1177, Precision Design: Errors," Kumen Blake; Microchip Technology Inc., DS01177, 2008.
Noise
Paul Gray Robert Meyer, "Analysis Design Analog Integrated Circuits," Ed., John Wiley Sons, 1984. Jerry Gibson, "Principles Digital Analog Communications," Macmillan, 1989. Bruce Carlson, "Communication Systems: Introduction Signals Noise Electrical Communication," Ed., McGraw-Hill, 1986. Athanasios Papoulis, "Probability, Random Variables, Stochastic Processes," Ed., McGraw-Hill,1991.
Miscellaneous
Howard Johnson Martin Graham, HighSpeed Digital Design: Handbook Black Magic," Prentice Hall, 1993.
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APPENDIX VOCABULARY
This appendix gives brief list common terms used amplifier noise work. They organized topic that their context easier grasp. Noise Current (In) square root Noise Power normalized standard resistance (usually units (ARMS, AP-P). When units ARMS, also called standard deviation.
Spectral Densities
Probability Density Functions
Power Spectral Density (PSD) frequency domain description noise source's statistical variation. units (W/Hz) (sometimes converted dBm/Hz). also related noise's auto correlation function. also called Noise Power Density. Noise Voltage Density (en) square root PSD, normalized standard resistance (usually units (V/Hz). also called spot noise noise root Hertz. Noise Current Density (in) square root PSD, normalized standard resistance (usually units (A/Hz). also called spot noise noise root Hertz.
Many physical noise sources, all, have Gaussian Normal) probability density function. They said Gaussian Noise, sometimes Additive White Gaussian Noise (AWGN). This noise usually associated with random processes that fulfill Identical Independently Distributed (IID) assumption; large number statistically independent random variables with same probability density function. probability density function
EQUATION A-1:
Analog Digital Converters (ADC) Digital Analog Converters (DAC) usually have their quantization errors modeled random noise with Uniform probability density function (the device noise inputs would Gaussian, however). probability density function
Spectral Shapes
White noise that constant value over frequency. mathematical convenience used make system noise calculations simpler. Broadband noise describes noise source that (nearly) white over circuit's frequency range interest. isn't white, appears white that circuit. Noise Power Bandwidth (NPBW) mathematically convenient parameter used describe circuit processes white noise. units (Hz). equivalent bandwidth brick wall filter that produces same output noise actual circuit. Excess Noise noise that exceeds white noise level frequencies (only noise discussed this application note): noise, also known flicker noise pink noise 1/f2 noise, also known noise Random Telegraph Signal (RTS) noise, also known burst noise popcorn noise (has spectral shape reminiscent white noise filtered lowpass filter with single real pole)
EQUATION A-2:
otherwise
Figures Merit
Signal-to-Noise Ratio (SNR) ratio signal power noise power. usually shown units (dB), although (VRMS/VRMS) (ARMS/ARMS) also acceptable. Sometimes signal's full scale range numerator ratio, with units (VPK VP-P). Other figures merit covered this application note are: Noise Figure (NF) (dB) Noise Factor (V/V) Noise Temperature (TN)
Integrated Noise
Noise Power noise source's statistical variation. units (sometimes dBm). Noise Voltage (En) square root Noise Power normalized standard resistance (usually units (VRMS, VP-P). When units VRMS, also called standard deviation.
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APPENDIX COMPUTER AIDS
While this application note emphasizes manual calculation formulas, most design work uses computers. obtain integrated noise (Enout) between (see Figure read En0f's value (let's call these values EH). integrated noise (for spectral shape):
Noise Simulations
EQUATION B-1:
nout
Circuit noise simulations done part simulation SPICE simulators. SPICE program developed Berkeley. Many SPICE derivative simulators used circuit design; most popular board level design PSpice® (from Cadence®).
B.1.1
GENERAL REMARKS
component models need correctly defined noise simulations give realistic results. macro models from Microchip work properly PSpice. Resistors, diodes transistors usually give correct white noise (when model accurate). noise diodes transistors will simulate correctly without special attention relevant parameters. resistor model does include noise; this added circuit using diodes dependent source, needed. will need define input source output circuit node before noise analysis run. SPICE produces input referred noise vector (across frequency); referred chosen input source. SPICE also produces output noise vector chosen node. noise results different SPICE simulators come different forms: noise voltage (current) density (VRMS/Hz) square noise voltage (current) density (VRMS2/Hz). Check your simulator using resistor thermal noise; resistance, +25°C, will give VRMS/Hz former case, VRMS2/Hz latter case. Note: PSpice produces noise units (VRMS/Hz). Other simulators may, not.
this trace's data into spreadsheet, click label found bottom left Probe's screen. This selects this trace (the label changes color). Copy data Window's clipboard typing sequence Ctrl-C. Paste results into your spreadsheet. Following this sequence produced columns data; frequency vector noise vector.
B.1.3
ESTIMATING NPBW WITH SPICE
that extract integrated noise from simulations, easily estimate filter's NPBW. following steps will make this more clear: very large resistor parallel) noise voltage source Insert buffer between noise source filter's input Plot output noise density (enout) Calculate integrated output noise (Enout) from infinity high enough frequency) Choose enout value that represents passband chosen gain, Calculate NPBW
EQUATION B-2:
NPBW ESTIMATE
NPBW nout nout
Using Symbolic Solver Engines
simulator's plotting tool determine which noise sources dominate improve noise filtering shaping.
There several places where symbolic solver speed your noise analysis: Converting node equations transfer functions Factoring transfer functions Expanding magnitude squared transfer function into Partial Fraction Expansion form Evaluating definite integrals used integrated noise NPBW) Some popular tools are: Mathematica® (from Wolfram Research) Maple(from Waterloo Maple Software) Matlab® (from MathWorks); Symbolic Math MathCad® (from Parametric Technology Corporation)
B.1.2
CALCULATING INTEGRATED NOISE WITH PSPICE
calculate integrated noise PSpice, open plotting utility (Probe) following trace:
EXAMPLE B-1:
PSPICE TRACE FUNCTION
sqrt(s(v(onoise)*v(onoise))) This running integral (from using PSpice function s()) output noise (let's call En0f) units (VRMS/Hz).
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