Op-Amp Noise
Calculation and Measurement
Art Kay
Senior Applications Engineer
Texas Instruments Inc – Tucson
kay_art@ti.com
1
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
2
What is Intrinsic Noise
Why do I Care?
Vin vs Time
1.5
R1 1k
Vin (mV)
1
R2 2k
0.5
0
0
2
4
6
8
10
12
8
10
12
10
12
-0.5
-1
-1.5
Time (mS)
V1 2.5
Ideal
+
+
U1 OPA335
VG1
4
VF1
V2 2.5
3
Vout (mV)
+
Vout Ideal vs Time
2
1
0
-1 0
2
4
6
-2
-3
-4
Time (mS)
Real
• The op-amp itself generates noise
Vout with Noise vs Time
5
4
• Noise acts as an error—it corrupts the signal
• Calculate, simulate, and measure this noise
Vout (mV)
3
2
1
0
-1 0
2
4
6
8
-2
-3
-4
Time (mS)
3
White noise or broadband noise
normal distribution
Noise Voltage
Distribution of Noise
0.02
-0.04
Counts Recorded During
Measurement Period.
0.04
-0.04
0.00E+00
Voltage
(Volts)
Voltage
(Volts)
numb of occ
-0.02
250
-0.02
200
0.00
150
0
0
0.02
100
-0.02
0.02
50
0
-0.04
0.04
0.04
Voltage (Volts)
Thermal Noise Measured In Time Domain
5.00E-04
Time (sec.)
Time (Sec)
1.00E-03
4
1/f or pink noise
normal distribution
1/f Noise vs Time
1/f Noise Measured
in Time Domain
1/f Noise vs Time
Distribution of Noise
-0. 1
0.1
0.08
0.08
0.06
0.06
0.04
0.04
-0. 08
-0. 06
0.02
0
0. 02
-0.02
0.02
0
0
-0. 02
Voltage (V)
-0. 04
Voltage (V)
15
10
5
0
0.1
0. 04
-0.02
-0.04
-0.06
-0.06
0. 06
-0.04
0. 08
0. 1
-0.08
-0.08
0
1
2
3
4
5
Time
(sec.)
Time
(sec)
6
7
8
9
10
0
1
2
3
Counts Recorded During
Measurement Period
4
5
Time (sec
5
<-30.00
Count
Num = 15000
0
0 5
178
<-24.00
500
<-18.00
438
342
851
703
<-6.00
641
[uV rti]
302
1174
991
943
130 46
434
961
2034
1874
Vn RTI (uV)
<-12.00
1000
1330
<0.00
1500
<6.00
1611
<12.00
2000
<18.00
2500
<24.00
Bad INA128 (fc=300Hz)
10 2
<30.00
Mean = 0.00
Std. dev. = 12.67
(Burst) Popcorn Noise
Bimodal (or multi-modal) distribution
Popcorn
Noise(fc=300Hz)
(fc = 300Hz)
Bad INA118
40
30
20
10
0
-10
-20
-30
-40
0
10
20
30
40
50
60
time (sec)
6
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.
7
Statistics Review – PDF
0.4
Probability Density function for Normal (Gaussian) distribution
f( x)
1
 2
Outline of Gaussian Curve
f( x)
  ( x ) 2
 22 

 e
0
Probability Distribution function for Normal (Gaussian) distribution
b
P( a  x  b )

 f( x) d x

a






b
a
1
 2
  ( x ) 2
 22 
 dx
 e
f(x)
0.2
5
a 0 b
x
5
P(a<x<b)
Where
P(a < x < b) -- the probability that x will be in the interval (a, b)
x-- the random variable. In this case noise voltage.
 -- the mean value
 -- the standard deviation
Probability an event will
occur within interval
For example, if P(-1<x<+1) = 0.3 then there is a 30% chance that x is between
-1 and 1.
8
The Probability Distribution Function P(a<x<b) gives the
probability that an event happens between a and b.
b

 f( x) d x

P( a  x  b )
STDEV Relationship to Peak-to-Peak
for a Gaussian PDF
a
f( x)
+/-3 STD Deviations = 6 sigma 99.7%
Let 
Voltage
1
 2
Gaussian PDF
  ( x ) 2
 22 

 e
0 because noise has no mean value (dc component).
Distribution of Noise
sured In Time Domain
250
200
150
100
50
0
-0.04
0.04
3σ
P   x  
b

 f( x) d x

-0.02
0.02
σ
0
0.02
-0.02
-0.04
-3σ
0.04
68% chance in time
that the
5.00E-04
measurement will
me (sec.)
be between ±σ
me (Sec)
-σ
numb of occ
{
0.00
Voltage (Volts)
a
1.00E-03
Counts Recorded During
Measurement Period.
P   x  







1
 2
  ( x) 2
 22 
 dx
 e








1
 2
  ( x) 2
 22 
 d x  0.683
 e

This can be translated to mean that there is a 68.3% probability that
you will measure a noise voltage between - and  at any instant in time.
9
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?
10


n
(14) Discretenform
 xi  
2 1
2

x


i1


i
n 
RMS vs STDEV i  1
(12) Discrete f
Root when
Mean Squared
(RMS)
defined
for DC
a Probability
Distribution
Function
Stdev = RMS
the Mean
is zero
(No
component).
For
all for
thea Probability
noise
Standard
deviation
defined
Distributio
This is the same as  if  = 0
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
 b

set μ = 0 (zero average).
information in appendix.
b
1   See2 further



1
2
RMS
g ( t) d t
  g ( t)    d t
(13) Continuou


b  a  a
(15) Continuous
form


b  a Standard
a
RMS
deviation
RMS
n
n 
1
2
xi
i1
Where
xi – data samples
n – number of samples
1


(16) Discrete form
n
n
2

xi  

2
(14) Discrete f
i1
Where
Root Mean Squared (RMS) defined for a Probability D
xi the
– data
This is
samesamples
as  if  = 0
μ – average of all samples
b

1

2

 gof
RMSn – number
( t) samples
dt

b  a  a
(15) Continuou


DC component will create  reading error
q  RMS
RMS
n
n 
1
i1
2
xi
11
(16) Discrete f
Add Noise As Vectors (RMS Sum)
Sum of two Random Uncorrelated Noise Sources
Vn1
Σ
Vn2
5mVrms
VnT
2
Vn1
Vn2
Example
3mVrms
2
V n1  V n2
VnT
Vn2
Vn1
V nT
2
( 3mVrms )  ( 5 mVrms )
2
5.83mVrms
12
Thermal Noise
The mean- square open- circuit voltage (e) across a resistor (R) is:
en = √ (4kTKRΔf)
where:
TK is Temperature (ºK)
R is Resistance (Ω)
f is frequency (Hz)
k is Boltzmann’s constant
(1.381E-23 joule/ºK)
en is volts (VRMS)
To convert Temperature Kelvin to
TK = 273.15oC + TC
Random motion of charges generate noise
13
0
1  10
468.916 1000
3
en density = √ (4kTKR)
1 10
468.916
3
100
 23
  ( 25  273.15 )  X   10 9
 23
  ( 125  273.15 )  X   10 9
 23
  (  55  273.15 )  X   10 9
nV/rt-Hz
0
Noise Spectral Density vs. Resistance
Noise Spectral Density vs. Resistance
0
Thermal Noise
100
10
 23
 4 1.3806510
 9

 ( 25 273.15)  X  10

25C
 23
9
 4 1.3806510

 ( 125 273.15)  X  10

10
125C
1
 23
9
 4 1.3806510

 (  55 273.15)  X  10

-55C
0.347
1
0.1
10
10
100
1  10
3
1  10
X
4
1  10
5
Resistance (Ohms)
Resistance versus Spectral Density
1  10
6
1  10
7
10
7
0.347 0.1
14
10
10
Op-Amp Noise Model
Noise Model
OPA277 Data
(IN+ and IN- are not correlated)
VN
IN+
IN-
IOP1
Tina Simplified Model
IN
*
+
nV
*
VN
-
fA
U1
15
16
Noise Analysis for Simple Op-Amp Circuit
R3 1k
Noise Sources
Op-Amp Voltage Noise Source
V2 2.5
Op-Amp Current Noise Sources
R2 1k
-
+
+
VG1
VF2
+
U1 OPA277
V1 2.5
Resistor Noise Sources
Calculation Considerations
Convert Noise Spectrum to Noise Voltage
- External Filter Bandwidth Limit
- Op-Amp Closed Loop Bandwidth
Noise Gain
17
Calculus Reminder
9

 4 dx

4 9
36
Height x Width
0
4
Height
Area
9
Width
Integral = Area under the curve
18
Convert Noise Spectrum to Noise Voltage
(Broadband Only – Simple Case)
You can’t integrate the Voltage
spectral density curve to get noise
5V / rtHz
Voltage
Spectral
Density
(V/rt-Hz)



10
0
0
Power
Spectral
Density
(V2/Hz)
Freq (Hz)
Freq (Hz)
5
V
 10 Hz
50
Hz
V Hz
Hz
Wrong
10
You integrate the Power spectral
density curve to get noise
25V2 / Hz
0
V_spec_dens d f
10
Correct
19
Convert Noise Spectrum to Noise Voltage
(Broadband Only – Simple Case)
You integrate the Power spectral
density curve to get noise
Correct
Noise Power = V2 * BW (Hz)
Hz
Noise Voltage = V * BW (Hz)
Hz
20
Noise Gain for Voltage Noise Source
Noise Gain – Gain seen by the noise source.
Example:
Noise_Gain = (R2/R1) + 1 = 2
R2 1k
Signal_Gain = -R2/R1 = -1
Referred to
Output
Output_Noise = Vn*(Noise_Gain)
V1 5
R1 1k
U1 OPA277
Signal
Vs
Source
+
+
*
Vn
Referred to Input
+
VF1
+
V2 5
Noise
Source
21
Understanding The Spectrum:
Total Noise Equation (Current or Voltage)
1/f Noise Region
(Pink Noise Region)
enT = √[(en1/f)2 + (enBB)2]
100k
Voltage Noise (nV/ Hz )
where:
enT = Total rms Voltage Noise in volts rms
en1/f = 1/f voltage noise in volts rms
enBB = Broadband voltage noise in volts rms
White Noise Region
(Broadband Noise Region)
10k
1k
100
10
1
0.1
1
10
100
1k
10k
Frequency (Hz)
fL
enBB calculation
fH
en1/f calculation
22
Low Pass Filter Shapes the Spectrum
RMS
How do we convert this plot to noise?
Broadband Region
Low pass filter
1/f Region
23
Real Filter Correction vs Brickwall Filter
where:
fP = roll-off frequency of pole or poles
fBF = equivalent brickwall filter frequency
Noise BW
Small Signal BW
0
Filter Attenuation (dB)
Skirt of
1-Pole Filter
Response
Skirt of
2-Pole Filter
Response
-20
Skirt of
3-Pole Filter
Response
-40
Brickwall
-80
0.1fP
fP fBF
Frequency (f)
10fP
24
AC Noise Bandwidth Ratios for nth Order Low-Pass Filters
BWn = (fH)(Kn) Effective Noise Bandwidth
Real Filter Correction vs Brickwall Filter
Number of Poles
in Filter
Kn
AC Noise Bandwidth Ratio
1
1.57
2
1.22
3
1.16
4
1.13
5
1.12
25
Broadband Noise Equation
eBB
BWn = (fH)(Kn)
where:
BWn = noise bandwidth for a given system
fH = upper frequency of frequency range of operation
Kn = “Brickwall” filter multiplier to include the “skirt” effects of a low pass filter
enBB = (eBB)(√[BWn])
where:
enBB = Broadband voltage noise in volts rms
eBB = Broadband voltage noise density ; usually in nV/√Hz
BWn = Noise bandwidth for a given system
26
1/f Noise Equation
e1/f@1Hz
(see appendix for derivation)
e1/f@1Hz = (e1/f@f)(√[f])
where:
e1/f@1Hz = normalized noise at 1Hz (usually in nV)
e1/f@f = voltage noise density at f ; (usually in nV/√Hz)
f = a frequency in the 1/f region where noise voltage density is known
en1/f = (e1/f@1Hz)(√[ln(fH/fL)])
where:
en1/f = 1/f voltage noise in volts rms over frequency range of operation
e1/f@1Hz = voltage noise density at 1Hz; (usually in nV)
fH = upper frequency of frequency range of operation
(Use BWn as an approximation for fH)
fL = lower frequency of frequency range of operation
27
Example Noise Calculation
R2 1k
Given:
OPA627
Noise Gain of 101
R1 100k
V1 15
-
+
+
VG1
VF1
+
U1 OPA627/BB
Find (RTI, RTO):
Voltage Noise
Current Noise
Resistor Noise
V2 15
28
Voltage Noise Spectrum and Noise Bandwidth
50nV/rt-Hz
5nV/rt-Hz
Unity Gain Bandwidth = 16MHz
Closed Loop Bandwidth = 16MHz / 101 = 158kHz
29
Example Voltage Noise Calculation
Voltage Noise Calculation:
Broadband Voltage Noise Component:
BWn ≈ (fH)(Kn)
(note Kn = 1.57 for single pole)
BWn ≈ (158kHz)(1.57) =248kHz
enBB = (eBB)(√BWn)
enBB = (5nV/√Hz)(√248kHz) = 2490nV rms
1/f Voltage Noise Component:
e1/f@1Hz = (e1/f@f)(√f)
e1/f@1Hz = (50nV/√Hz)(√1Hz) = 50nV
en1/f = (e1/f@1Hz)(√[ln(fH/fL)]) Use fH = BWn
en1/f = (50nV)(√[ln(248kHz/1Hz)]) = 176nV rms
Total Voltage Noise (referred to the input of the amplifier):
enT = √[(en1/f)2 + (enBB)2]
enT = √[(176nV rms)2 + (2490nV rms)2] = 2496nV rms
30
Example Current Noise Calculation
Note: This example amp doesn’t have 1/f component for current noise.
en-in= (in)x(Req)
R1 1k
en-out= Gain x (in)x(Req)
Rf 3k
Gain
IOP1
U2
-
fA
*
VF1
Req = R1 || Rf
+
*
31
Example Current Noise Calculation
Broadband Current Noise Component:
BWn ≈ (fH)(Kn)
BWn ≈ (158kHz)(1.57) =248kHz
inBB = (iBB)(√BWn)
inBB = (2.5fA/√Hz)(√248kHz) = 1.244pA rms
Req = Rf || R1 = 100k || 1k = 0.99k
eni = (In)( Req) = (1.244pA)(0.99k) = 1.23nV rms
neglect
Since the Total Voltage noise is envt = 2496nV rms
the current noise can be neglected.
32
Example Resistor Noise Calculation
enr = √(4kTKRΔf)
where:
R = Req = R1||Rf
Δf = BWn
enr = √(4 (1.38E-23) (273 + 25) (0.99k)(248kHz)) = 2010nV rms
* U1
en-out= Gain x (√(4kTRΔf))
Gain
nV
R1Rf
2k
nV
R2R1
1k
* U1
en-in= √(4kTRΔf)
IOP1
-
VF1
Req = R1 || Rf
+
*
33
Total Noise Calculation
Voltage Noise From Op-Amp RTI:
env = 2510nV rms
Current Noise From Op-Amp RTI (as a voltage):
eni = 1.24nV rms
Resistor Noise RTI:
enr = 2020nV rms
Total Noise RTI:
en in = √((2510nV)2 + ((1.2nV)2 + ((2010nV)2) = 3216nV rms
Total Noise RTO:
en out = en in x gain = (3216nV)(101) = 325uV rms
34
Calculating Noise Vpp from Noise Vrms
Relation of Peak-to-Peak Value of AC Noise Voltage to rms Value
Peak-to-Peak
Amplitude
Probability of Having
a Larger Amplitude
2 X rms
32%
3 X rms
13%
4 X rms
4.6%
5 X rms
1.2%
6 X rms *
0.3%
6.6 X rms
0.1%
*Common Practice is to use:
Peak-to-Peak Amplitude = 6 X rms
35
Peak to Peak Output For our Example
R2 1k
R1 100k
V1 15
-
+
+
VG1
VF1
+
U1 OPA627/BB
V2 15
en out = 325uV rms
en out p-p = (325uV rms )x6 = 1.95mVp-p
36
37
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 www.ti.com
– Download free
– Application circuits available
38
Tina Spice Analysis
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
39
Noise Model Test procedure:
Is the Tina Noise Model Accurate?
40
Translate Current Noise to Voltage Nose
Set Gain to 1.
41
Generate Spectral Noise Plots
“Output Noise” diagram gives
the output voltage noise spectral
density measured at each volt
meter.
42
Separate Curves
43
Click on Axis to Scale
44
OPA227 Matches Tina Model Matches the Data Sheet
45
The OPA627 Tina Model Does NOT Match the Data Sheet
Wrong magnitude
No 1/f
5nV/rtHz
46
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 www.ti.com
(search for “noise sources”).
47
Right Click on Noise Source to Edit the Macro
Enter magnitude of 1/f and
broadband noise into the
macro.
48
1/f Region
Look for a point in
the 1/f region. Enter
the frequency and
magnitude at this
point
(1Hz, 50nV/rtHz)
49
Broadband
Region
Look for a point in
the broad band
region. Enter the
magnitude at this
point
(100kHz, 5nV/rtHz)
50
Compile
the Macro
After macro is
compiled press
“file > close” and
return to
schematic editor.
51
Same Procedure for Current Noise
Source
Follow the same procedure for
current noise. This example
has no 1/f component (set
FLWF = 0.001).
52
Important Op-Amp Characteristics
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
Dominant Pole
53
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
54
Verify the Noise Model is Correct Using the Test Procedure
2.5fA/rtHz
50nV/rtHz
2.5nV/rtHz
50nV/rtHz
2.5nV/rtHz
55
Let’s 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.
56
Let’s Analyze the Circuit that We did Hand Analysis on
10.00u
Output noise (V/Hz?)
Hand Analysis:
en out = 325uV rms
100.00n
Tina Analysis:
en out = 323uV rms
R2 100k
1.00n
1
10
400.00u
100
1k
10k
100k
Frequency (Hz)
1M
10M
100M
323uV rms
R1 1k
V1 15
OP1 !OPAMP
VG1
-
*
+
300.00u
V627
+
V2 15
Total noise (V)
+
U1
nV
*
fA
U2
200.00u
100.00u
0.00
1
10
100
1k
10k
100k
Frequency (Hz)
1M
10M
57
100M
Use Tina to Analyze this Common Topology
C1 1n
R2 10k
Gain vs Frequency
30.00
AC Analysis > Transfer Characteristic
20.00
R1 1k
OP1 !OPAMP
-
*
+
nV
*
fA
U2
Gain (dB)
V1 15
V627
+
0.00
V2 15
U1
+
10.00
-10.00
VG1
-20.00
1
10
100
1k
10k
100k
Frequency (Hz)
1M
10M
100M
1.00u
Output Noise Spectrum
25.00u
Output noise (V/Hz?)
Total noise (V)
Total Noise
25uV with BWn = 100MHz
100.00n
12.50u
10.00n
1.00n
0.00
1
10
100
1k
10k
100k
Frequency (Hz)
1M
10M
100M
1
10
100
1k
10k
100k
Frequency (Hz)
1M
10M
58
100M
External Filter Improves Noise Performance
Gain vs Frequency (V627 / VG1)
30.00
Cf Filter
Gain (dB)
R2 10k
R1 1k
V1 15
*
+
R2 10k
V627
C1 1n
V2 15
U1
+
External
Filter
+
nV
*
fA
U2
OP1 !OPAMP
-
-35.00
VG1
-100.00
1
Cf Filter
25.00u
10
1.00u
100
1k
10k
100k
Frequency (Hz)
1M
Output noise (V/Hz?)
Total noise (V)
12.50u
Cf Filter
1.00n
External
Filter
External
Filter
0.00
1
10
100
1k
10k
100k
Frequency (Hz)
1M
10M
100M
100M
Output Noise Spectrum
Total Noise
25uV with BWn = 100MHz
10M
1.00p
1
10
100
1k
10k
100k
Frequency (Hz)
1M
10M
59
100M
60
Instruments For Noise Measurements
1. True RMS Meter
2. Oscilloscope
3. Spectrum Analyzer / Signal Analyzer
61
Standard deviation defined for a Probabilit
True RMS Meter – How it works
b

Circuit
under test
HP3458A
Input
terminals



b  a a
1

 g ( t)    2 d t
(1
n
2
x




n  i
1
2
(1
Digitizer
i1
ac
Display
Thousands
coupling of Samples Compute
RMS
RMSSquared (RMS) defined for a P
Root Mean
This is the same as  if  = 0
RMS
RMS
Good Broadband Noise Measurement
b

  g ( t) 2 d t 

b  a  a


1
n
n 
1
(1
2
xi
(1
i1
62
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 . . . 1000V
• See Appendix for Setup
63
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
64
TDS460A Digitizing Oscilloscope Example
Best Noise Floor = 0.2mV
BW Limit = 20MHz
BNC Shorting Cap
Noise Floor = 0.8mV
BW FULL = 400MHz
BNC Shorting Cap
Noise measurements use
BNC cables
Worst Noise Floor = 8mV
BW FULL = 400MHz
10x Scope Probe
65
Oscilloscope Noise Measurement
Ground connection
Measurement Point
The GND lead on the
scope probe can act as
a loop antenna!
BNC
Shorting
Cap
66
Spectrum Analyzer -- Convert Result to nV/rt-Hz
Vnoise
0.1Hz to 100MHz
Noise Components
1MHz
10kHz
Notch
Filter
1MHz +/-5kHz
Noise Component
Spectrum
Analyzer
V
10kHz
100nV
Spectrum Analyzer Plot
f
1MHz
10MHz
To convert to nV/ Hz plot at 1MHz data point we divide 100nV by 10kHz which gives the Spectral Noise Density (nV/ Hz) at 1MHz
67
Effect of Changing the measurement
Bandwidth
Measuring 67kHz and 72kHz
-10dBm @
72kHz
-10dBm @
72kHz
-26dBm @
67kHz
-26dBm @
67kHz
Lower
harmonics
obscured
-77dBm @
74kHz
Noise floor
-80dBm
Noise floor
-87dBm
BW set to 150Hz
BW set to 1200Hz
Narrow BW increases resolution
& lowers noise floor
Wider BW reduces resolution
& raises noise floor
Narrow BW  Longer Sweep Time
68
dBm to Spectral Density
Convert dBm to nV/rt-Hz
 NdBm 

10 
10

  ( 1mW )  R
Vspect_anal
Vspect_den
( 5.4)
Vspect_anal
Kn  RBW
( 5.5)
Where
NdBm -- the noise magnitude in dBm from the spectrum analyzer
R -- the reference impedance used for the dBm calculation
Vspect_anal -- noise voltage meas ured by spectrum analyzer per resolution bandwidth
RBW -- resolution bandwidth setting on spectrum analyzer
Vspect_den -- spectral density in (nV/rt-Hz)
Kn -- conversion factor that changes the resolution bandwidth to a noise bandwidth
Filter Type
Application
Kn
4-pole sync
Most Spectrum Analyzers Analog
1.128
5-pole sync
Some Spectrum Analyzers Analog
1.111
Typical FFT
FFT-based Spectrum Analyzers
1.056
69
Effect of Changing Averaging
No Averaging.
Averaging = 49.
Increase Averaging to Reduce Noise Floor  Increase Measurement Time
70
71
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
72
0.1Hz Second Order 10.0Hz Fourth Order 1/f Filter
0.1Hz HPF
Gain = 10
Gain = 1001
10Hz LPF
Gain = 10
R1 2M
R2 402k
R11 2M
R13 100k
V7 5
C4 22n
V1 5
R12 100
C2 1u
-
C3 1u
V4 5
+
+
-
VG1
V8 5
R6 40.2k
-
R3 634k
+
U1 OPA227
V6 5
C1 470n
+
U2 OPA735
+
R7 97.6k
+
+
U3 OPA735
V2 5
Device Under Test
R4 1k
R5 9.09k
DUT
R8 178k
C6 10n
V5 5
1/f Filter
0.1Hz to 10Hz
C5 470n
-
R9 178k
VF1
R10 226k
+
+
U4 OPA735
V3 5
10Hz LPF
Gain = 1
73
The Circuit Generates the Data Sheet 1/f Plots
(Example OPA227)
Tektronix TDS460A Measurement
OPA227
PDS 0.1Hz to 10Hz Noise Curve
OPA227
Data Sheet Curve
Y-Axis scale must be adjusted to account
for the gain to match data sheet.
5mV/(10x10x1000) = 50nV/div
74
1/f Filter Measured vs Tina
(Example OPA227)
Tektronix TDS460A Measurement
Tina Simulation
1.84m
OPA227
Total noise (V)
OPA227
917.84u
0.00
10m
100m
1
10
Frequency (Hz)
Measured Output is approximately
Tina Simulation
10mVp-p
1.84mV rms
100
1k
(1.84mV)(6) = 11mVp-p
75
Measure Noise Floor of 1/f Filter
What Op-Amp will give us the lowest noise Floor?
R1 2M
R2 2M
C2 1u
C3 22n
V1 15
C3 1u
+
+
V4 15
U3 OPA227
+
-
VG1
R3 634k
V3 15
R2 40.2k
C3 470n
Replace
DUT with
Short
R1 402k
R3 97.6k
+
+
U1 OPA227
V1 15
R4 1k
R5 9.09k
R6 178k
Large Input
Impedance
C4 10n
V5 15
C1 470n
R7 178k
VF1
R8 226k
+
+
U2 OPA227
V2 15
76
Measure Noise Floor of 1/f Filter
What Op-Amp will give us the lowest noise Floor?
Op-Amp
General Description
Vn
(nV/rt-Hz)
In
(fA/rt-Hz)
OPA227
Low noise Bipolar
3.5 @10Hz
6
@1Hz
20 @0.1Hz
2,000 @10Hz
6,000 @1Hz
20,000 @0.1Hz
OPA132
Low noise CMOS
23 @10Hz
80 @1Hz
228 @0.1Hz
3
OPA735
Auto Zero CMOS
135
40
77
OPA227
OPA132
(15mVp-p paint can)
(0.6mVp-p in paint can)
Large Current noise
Low noise in thermally
stable environment
OPA734
OPA132
(0.8mVp-p in free air)
(3mVp-p in free air)
Auto zero No 1/f Noise
Low Vosi Drift
“Noise” from Vosi Drift
Note: measurements in paint can minimize thermal drift.
78
Some General Measurement Precautions
Linear Supply
or Battery
Copper box or
paint can as RFI /
EMI Shield. This
also minimizes
thermal drift
BNC From
Circuit
Under Test
BNC To
Scope
Use BNC Male
Shorting cap
for noise floor
79
A paint can makes a good
shielded environment
Scope or true
RMS DVM
Floating Linear
Supply or
Batteries
20mV p-p
Out
Low Noise Boost Amp 1000x
In
20uV p-p
Out
Circuit under test
Steel Paint Can
80
Measure The OPA627 Example Using A Scope
Copper box
Rf R2100k
1k
R1 1k
R1 1k
V1 9
+
Output noise
To_Scope
+
U1 OPA627/BB
BNC
V2 9
Scope
BNC
Calculated (Previous Example):
Vn = 325uV rms
Measured:
Vn = √((476uV)2 – (148uV)2) = 452uV rms
Noise floor
Note: peak to peak reading is roughly 2mVp-p
81
Measure The OPA627 Example Using A True RMS Meter
(HP3458a)
Copper box
Rf R2100k
1k
R1 1k
R1 1k
V1 9
BNC
+
To_Scope
Coax
HP3458A True RMS
+
U1 OPA627/BB
SETACV RNDM
V2 9
Banana
Calculated (Previous Example):
en = 325uV rms
Measured HP3458A DVM:
DVM_READING =346uV
Noise Floor = 18uV
Noise_Meas = √((346uV)2 – (18uV)2) = 346uV rms
82
Spectrum Analyzer Measurement of OPA627
Low Req to reduce
effect of in and
thermal noise
(Voltage Noise Spectrum)
Agilent 35770a Dynamic Signal Analyzer
R2 10k
• Has nV/rt-Hz Mode
• Bandwidth 0Hz to 100kHz
V2 5
R1 100
U1 OPA627/BB
+
V1 5
R3 1M
+
C1 20u
1M Input
Impedance
20mV dc offset
0.2mVp-p noise
(Bad SNR)
0.008Hz HPF removes dc
component, but still allows
1/f measurement
83
Spectrum Analyzer Measurement of OPA627
The low frequency run
is time consuming.
Approximately 12 min.
OPA627 OutputSpectral Density (Gain=100)
1.00E+00
1.00E-01
Spectral Density (V/rt-Hz)
1.00E-02
Spect
1.00E-03
spect3
spect2
1.00E-04
spect4
spect5
1.00E-05
noise floor
1.00E-06
1.00E-07
1.00E-08
1.00E-02
1.00E-01
1.00E+00
1.00E+01
1.00E+02
1.00E+03
1.00E+04
1.00E+05
1.00E+06
Freq (Hz)
Noise floor verification
Data was collected over five
different frequency ranges
84
Spectrum Analyzer Measurement of OPA627
60Hz noise pickup
OPA627 OutputSpectral Density (Gain=100)
1.00E+00
1.00E-01
Low frequency tail on
each run must be
discarded.
Spectral Density (V/rt-Hz)
1.00E-02
Spect
1.00E-03
spect3
spect2
1.00E-04
spect4
spect5
1.00E-05
noise floor
1.00E-06
1.00E-07
1.00E-08
1.00E-02
1.00E-01
1.00E+00
1.00E+01
1.00E+02
1.00E+03
1.00E+04
1.00E+05
1.00E+06
Freq (Hz)
1. Combine to one curve
2. Discard bad information
3. Divide by gain of 100
85
The “Tail Error” is from relatively wide
measurement
bandwidth
at low frequency.
OPA627 OutputSpectral
Density (Gain=100)
1.00E+00
Spectral Density (V/rt-Hz)
1.00E-01
1.00E-02
1.00E-03
1Hz - Band pass filter
is very wide near 1Hz
(dc and 1/f noise introduce errors)
1Hz - Band pass filter
is very narrow at 1kHz
1.00E-04
1.00E-05
1.00E-06
1.00E-07
1.00E-01
Actual
Noise
1.00E+00
Measured
Noise
1.00E+01
1.00E+02
1.00E+03
1.00E+04
Freq (Hz)
86
Spectrum Analyzer Measurement of OPA627
OPA627 PDS
Voltage Noise Spectral Density for OPA627
Noise Spectral Density (nV/rt-Hz)
1000.0
100.0
10.0
1.0
1.00E-01
1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05
Frequency (Hz)
Measured 1/f noise corner
is better then data sheet.
87
References
1.
2.
3.
Robert V. Hogg, and Elliot A Tanis, Probability and Statistical Inference, 3rd Edition, Macmillan Publishing Co
C. D. Motchenbacher, and J. A. Connelly, Low-Noise Electronic System Design, A Wiley-Interscience Publication
Henry W. Ott, Noise Reduction Techniques in Electronics Systems, John Wiley and Sons
Acknowledgments:
1.
2.
3.
4.
8.
R. Burt, Technique for Computing Noise based on Data Sheet Curves, General Noise Information
N. Albaugh, General Noise Information, AFA Deep-Dive Seminar
T. Green, General Information
B. Trump, General Information
B. Sands, Noise Models
Noise Article Series (www.en-genius.net)
http://www.en-genius.net/site/zones/audiovideoZONE/technical_notes/avt_022508
88
Thank You
for
Your Interest
in
Noise – Calculation and Measurement
Comments, Questions, Technical Discussions Welcome:
Art Kay
520-746-6072
kay_art@ti.com
89
90
Mean defined for a Probability Distribution Function




Mean defined for a Discrete Statistical Population

(1) Continuous form
( x)f( x) d x

b

 g ( t) dt
b  a a
1

(9) Continuous form



( x)  f( x)
(2) Discrete form
Variance defined for a Probability Distribution Function

2




 x   2 f( x) dx
(3) Continuous form

(Formulas)
Statistics Summary
x

2


 x   2 f( x)
(4) Discrete form
x
1

n
n

xi
(10) Discrete form
i1
Variance defined for a Probability Distribution Function
b


2
2


b  a a
 g ( t)    2 d t
1
1
n
(11) Continuous form
n
2
 xi  
(12) Discrete form
i1
Standard deviation defined for a Probability Distribution Function
Standard deviation defined for a Probability Distribution Function





2

 x   2 f( x) dx
(5) Continuous form





2
 x   2 f( x)
b



b  a a


 g ( t)    2 d t
1
(13) Continuous form
(6) Discrete form
x
n
x  2
n  i
1
2
(14) Discrete form
i1
Root Mean Squared (RMS) defined for a Probability Distribution Function
This is the same as  if  = 0
RMS




2
( x) f( x) d x
(7) Continuous form


RMS

x
Root Mean Squared (RMS) defined for a Probability Distribution Function
This is the same as  if  = 0
RMS
2
b

  g ( t) 2 d t 

b  a  a


1
(15) Continuous form
(8) Discrete form
( x)  f( x)
RMS
n
n 
1
i1
2
xi
(16) Discrete form
91
Statistics Summary
(PDF vs Discrete Population)
Example: Statistics on the Probability Distribution Function
Example: Statistics on the Discrete Statistical Population
  1
x  rnorm( 1001  0  1)
  0
f( x) 
  ( x  ) 2
 22 

 e
1
  2
i  1  1000
Probability Distribution Function
for Normal Curve
.5
0.4
f( x)
5
xi  0
0
0.2
5
500
1000
i
0
5
0
5
 
x

  

1000
1
1000

( x)f( x) d x
0

var  

var 
x  
2
f( x) d x

3
  4.181  10
x
i 0
Mean
i1
Mean



var  1
1000
1
1000
Variance
 xi 0  
2
i1
var  1.021
Variance

  var
  var
1

RMS  



2
( x) f( x) d x
  1.01
Standard Deviation
Standard Deviation
RMS 
RMS  1
Root Mean Squared
1
1000
1000
2
 xi 0
i1
RMS  1.01
Root Mean Squared
92
Statistics Summary
(RMS and STDEV)
Example where RMS  STDEV
Example where RMS = STDEV
g ( t)  sin ( t)  0.3
g ( t )  sin ( t)
 


2  0 
1
2
g( t) dt
0
 
  0.3


var 
2  0 
2
 g( t)    2 d t
0
2
g ( t) dt
0
Variance for a Discrete Statistical Population
1


2  0 
1
Variance defined for a Probability Distribution Function


var 
2  0 
2
1
var  0.5
0
 g ( t)    2 d t
0
var  0.5
Standard deviation for a Discrete Statistical Population
Standard deviation defined for a Probability Distribution Function
  var
  var
  0.707
  0.707
Root Mean Squared (RMS) for a Discrete Statistical Population
This is the same as  if  = 0
Root Mean Squared (RMS) defined for a Probability Distribution Function
This is the same as  if  = 0
  2

  g( t) 2 dt 
RMS 

2  0  0


  2

  g( t) 2 dt 
RMS 

2  0  0


1
1
RMS
2
2
2
2
 
  RMS  
So

  0.707
RMS  0.768
2
RMS  
RMS  0.707
2
93
RMS vs STDEV
Stdev = RMS when the Mean is zero (No DC component). Tina gives the true
RMS result in AC calculations. See mathematical proof in appendix.
RMS = STDEV
RMS ≠ STDEV
1Vpk
sinusoidal
94
Useful Numerical Methods
Method for generating a time domain approximation of 1/f noise
For i = 1 To 32768
white = Rnd(1) - 0.5
buf0 = 0.997 * old_buf0 + 0.029591 * white
buf1 = 0.985 * old_buf1 + 0.032534 * white
buf2 = 0.95 * old_buf2 + 0.048056 * white
buf3 = 0.85 * old_buf3 + 0.090579 * white
buf4 = 0.62 * old_buf4 + 0.10899 * white
buf5 = 0.25 * old_buf5 + 0.255784 * white
pink = buf0 + buf1 + buf2 + buf3 + buf4 + buf5
Next i
95
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
96
f
erms
2
2 2
 e   G 2 df
n

f
1
Brick Wall Factor Calculation
for first order filter.
where
erms -- total rms noise from f1 to f2 in Vrms
en -- magnatitude of noise spectral density at f1 in V/ Hz
G -- gain function for a single pole filter
1
G
1

1
G
j
f
2
1 
p
G
2
2
fp
1
1
2
f
2
fp
2
f
erms
erms
2
2
f2






f
1
2
en 
1
2
2

fp
2
 en 
df
2
2

fp  f

f
df
f
2
fp
2
1
1
 f2 
 f1 
2
2
en  fp  atan    en  fp  atan  
 fp 
 fp 
Let f1 = 0, f2 = 
erms
erms
erms
 f1 

2
 en  fp  2
fp
 
2
en  fp  atan     en  fp  atan 
2

2
en  fp 
2
2
2

2
en  fp 
2
97
1/f Noise Derivation
en
f
erms
2
enormal
0.5





2
en
2
2
enormal
enormal
f 0.5
f
2
Units for
en = V / Hz
enormal = V
f = Hz
b
2
enormal
f
df
2
enormal  ln ( f)
b
erms units of V rms
a
a
erms
erms
2
enormal  ln ( b )  enormal  ln ( a)
2
2
enormal  ln 
2
2
enormal  ln 
2
b

a
b

a
b
erms enormal ln  
a
ln 
b
 has no units
a
enormal has units of V
98
Current Noise
- 0V +
+ (In x Rf) -
R1 1k
Rf 3k
IOP1
U2 In
-
fA
*
VF1
+
Vo = (In x Rf)
99
Current Noise
R2 1k
nV
* U1
R11k
R2
Rf2k
R1
-
+
Equ_Input_Noise
Vo
R1 2k
IOP1
Equ. Input Noise
Equ_Input_Noise
IOP1
-
VF1
+
Gain
In  Rf
Rf
1
Equ_Input_Noise = In (Rf || R1)
*
nV
* U1
Vo = (In x Rf)
VF1
R1


In  

Rf  R1


Rf R1
U1
nV
100
Solve for Resistor Noise Components
* U1
nV
R1Rf
2k
nV
R2R1
1k
* U1
Noise Source with Each Resistor
IOP1
-
VF1
+
101
nV
Rf 2k
R1
* U1
* U1
en1
nV
R2R1
1k
IOP1
-
VF1
+
Vo = en1 x Rf/R1
102
* U1
* U1
en2
nV
Rf2k
R1
nV
R11k
R2
IOP1
-
VF1
+
Vo = en2
103
R2 1k
Add noise components and refer to the input.
Note: the input noise is equivalent to Rf || R1 (proof on next page)
nV
* U1
R2 1k
1
U1
nV
VF1
 Rf R1 
4 kT BW  

Rf  R1


Equivalent to noise of Rf || R1
R1 2k
Rf
IOP1
IOP1
Output_noise
R1
Input_noise
-
+
Input_noise
R1 2k
-
VF1
+
*
nV
* U1
2
Output_noise
Let

Rf 

2
e


e
 n1 
n2
R1


4 kT BW
104
2
Output_Noise
Let

2
en1  en2
2
Output_noise
Rf 

   R1 R  
1

2
R2 1k
nV
Rf 

   R1  
R1


Input_noise
Rf
R1
* U1
R2 1k

-
+
Input_noise
IOP1
R1 2k
IOP1
U1
nV



  Rf
2
2
  Rf
1
2
R1 2k
-
Rf 

2
 en1 R   en2
1

2
4 kT BW
Proof: Simple Amp Resistor Noise
Input Noise is equivalent to noise from
parallel combination of Rf and R1.
2
VF1
2
Rf
R1
   Rf
2
 Rf R1 2
R1
2
  Rf  R1    Rf R1
Rf  R12
2
+
*
nV
* U1
2
Input_noise
2
  Rf  R1    Rf R1
Rf  R12

Rf R1
Rf  R1
VF1
Input_noise
 Rf R1 

 Rf  R1 
4 kT BW  
Equ to noise of Rf || R1
105