AN1228

AN1228
Op Amp Precision Design: Random Noise
Author:
Kumen Blake
Microchip Technology Inc.
INTRODUCTION
This application note covers the essential background
information and design theory needed to design low
noise, precision op amp circuits. The focus is on
simple, results oriented methods and approximations
useful for circuits with a low-pass response.
The material will be of interest to engineers who design
op amps circuits which need better signal-to-noise ratio
(SNR), and who want to evaluate the design trade-offs
quickly and effectively.
This application note is general enough to cover both
voltage feedback (VFB) (traditional) and current
feedback (CFB) op amps. The examples, however, will
be limited to Microchip’s voltage feedback op amps.
Additional material at the end of this application note
includes references to the literature, vocabulary and
computer design aids.
Key Words and Phrases
•
•
•
•
•
Op Amp
Device Noise
Noise Spectral Density
Integrated Noise
Signal-to-Noise Ratio (SNR)
Prerequisites
The material in this application note will be much easier
to follow after reviewing the following statistical
concepts:
•
•
•
•
•
•
•
Average
Standard Deviation
Variance
Gaussian (normal) probability density function
Histograms
Statistical Independence
Correlation
BACKGROUND INFORMATION
This section covers the basics of low frequency noise
work. It is somewhat theoretical in nature, but has
some numerical examples to illustrate the concepts. It
serves as a foundation for the following sections. See
references [2, 4, 5] for a more in depth theoretical
coverage of these concepts.
The material after this section illustrates these
concepts. For those readers new to this subject matter,
it may be beneficial to read the complete application
note several times, while working all of the examples.
Where Did the Average Go?
The most commonly used statistical concept is the
average. Standard circuit analysis gives a deterministic
value (DC plus AC) at any point in time. Once these
deterministic values are subtracted out, the noise
variables left have an average of zero.
Noise is interpreted as random fluctuations
(a stochastic value) about the average response. We
will deal with linear circuits, so superposition applies;
we can add the average and the random fluctuations to
obtain the correct final result.
Noise Spectral Density
The easiest approach to analyzing random analog
noise starts in the frequency domain (even for
engineers that strongly prefer the time domain).
Stationary noise sources (their statistics do not change
with time) can be represented with a Power Spectral
Density (PSD) function.
Because we are analyzing analog electronic circuits,
the units of power we will deal with are W, V2/Ω and
A2Ω. This noise power is equivalent to statistical
variance (σ2). The variance of the sum of uncorrelated
random variables is:
EQUATION 1:
VARIANCE OF THE SUM OF
UNCORRELATED
VARIABLES
⎛
⎞
var ⎜ ∑ X k⎟ =
⎝ k ⎠
Knowledge of basic circuit analysis is also assumed.
∑ var ( Xk )
k
Where:
© 2008 Microchip Technology Inc.
Xk
=
uncorrelated random variables
var()
=
the variance function
DS01228A-page 1
AN1228
This fact is very important because the various random
noise sources in a circuit are caused by physically
independent phenomena. Circuit noise models that are
based on these physically independent sources
produce uncorrelated statistical quantities.
The PSD is an extension of the concept of variance. It
spreads the variation of any noise power variable
across many frequency bins. The noise in each bin
(power with units of Watts) is statistically independent
of all other bins. The units for PSD are (W/Hz), which is
why it is called a “density” function. The picture in
Figure 1 illustrates these concepts.
Strictly speaking, in passive circuits (RLC circuits), this
conversion needs to be done with a specific resistance
value (P = V2/R = I2R). In most noise work involving
active devices, however, a standard resistance value of
1 Ω is assumed.
Integrated Noise
To make rational design choices, we need to know
what the total noise variation is; this section gives us
that capability. We will convert the PSD to the statistical
variance (or standard deviation squared) using a
definite integral across frequency.
CALCULATIONS
PSD (W/Hz)
Bin Power ≈ PSD(fk) · Δfk
Using Equation 1, and the fact that the power in a
frequency bin is independent of all other bins, we can
add up all of the bin powers together:
EQUATION 2:
Δfk
TOTAL NOISE VARIATION
N ≈ ∑ ( PSD ( f k ) ⋅ Δf k )
k
∞
f (Hz)
0
fk
FIGURE 1:
Power Spectral Density.
In this application note, all PSD plots (and functions)
are one-sided, with the x-axis in units of Hertz. This is
the traditional choice for circuit analysis because this is
the output of (physical) spectrum analyzers.
Note:
•
•
Where:
N
=
∫0 PSD ( f ) df
total noise power (W)
We use the summation approximation for measured
noise data at discrete time points. The integral applies
to continuous time noise; it is useful for deriving
theoretical results.
It is very important, when reading the electronic literature on noise, to determine:
PREFERRED EQUATIONS
Is the PSD one-sided or two-sided?
Is frequency in units of Hertz (Hz) or
Radians per Second (rad/s)?
In circuit analysis, the conversion to integrated noise
(En) usually takes place with the noise voltage density;
see Equation 3. En is the noise’s standard deviation.
In most low frequency circuits, signals and noise are
interpreted and measured as voltages and currents,
not power. For this reason, PSD is usually presented in
two equivalent forms:
• Noise voltage density (en) with units (V/√Hz)
• Noise current density (in) with units (A/√Hz)
The voltage and current units are RMS values; they
could be given as (VRMS/√Hz) and (ARMS/√Hz).
Traditionally, the RMS subscript is understood, but not
shown.
Note:
N =
Many beginners find the √Hz units to be
confusing. It is the natural result, however,
of converting PSD (in units of W/Hz) into
noise voltage or current density via the
square root operation.
DS01228A-page 2
EQUATION 3:
INTEGRATED NOISE
VOLTAGE
∞
En =
∫0 e n ( f ) df
2
Where:
en(f)
=
=
En
noise voltage density (V/√Hz)
PSD ( f ) ⋅ ( 1 Ω )
=
integrated noise voltage (VRMS)
=
standard deviation (VRMS)
© 2008 Microchip Technology Inc.
AN1228
Noise current densities can also be converted to
integrated noise (In):
EQUATION 4:
∞
∫0 i n ( f ) df
2
Where:
in(f)
=
noise current density (A/√Hz)
In
xL
PG(|x| > xL; 0, 1)
1.64
10%
integrated noise current (ARMS)
=
standard deviation (ARMS)
1%
2.58
5.15
3.29
6.58
10-6
4.50
9.00
-9
6.00
12.00
6.80 ×
1.97 × 10
Microchip’s op amp data sheets use
6.6 VP-P/VRMS when reporting Eni (usually
between 0.1 Hz and 10 Hz). This is about
the range of visible noise on an analog
oscilloscope trace.
INTERPRETATION
The principle noise sources within op amps, and
resistors on the PCB, are Gaussian. When they are
combined, they produce a total noise that is also
Gaussian. Figure 2 shows the standard Gaussian
probability density function (mean = 0 and standard
deviation = 1) on a logarithmic y-axis.
0
1.E+00
10
-1
1.E-01
10
-2
1.E-02
10
The integrated noise results in this application note are
independent of frequency and time. They can only be
used to describe noise in a global sense; correlations
between the noise seen at two different time points are
lost after the integration is done.
Filtered Noise
Any time we measure noise, it has been altered from its
original form seen within the physical noise source. The
easiest way to represent these alterations to the noise,
in linear systems, is by the transfer function (in the
frequency domain) from the source to the output. The
resulting output noise has a different spectral shape
than the source.
TRANSFER FUNCTIONS AND NOISE
-3
1.E-03
10
pG(x; 0, 1)
3.29
0.1%
Note 1:
We need to know the probability density function in
order to make informed decisions based on the
integrated (RMS) noise. For the work in this application
note, the noise will have a Gaussian (Normal)
probability density function.
1.64
3.29
6.00
=
Peak-to-Peak
Peak
(VPK/VRMS) (VP-P/VRMS)
2.58
4.50
PSD ( f ) ⁄ ( 1Ω )
=
IMPORTANT TWO-TAILED
PROBABILITIES
Crest Factor (Note 1)
INTEGRATED NOISE
CURRENT
In =
TABLE 1:
It turns out [3, 4, 5] that the noise at the output of a linear operation (represented by the transfer function) is
related to the input noise by the transfer function’s
squared magnitude; see Equation 5. This can be
thought of as a result of the statistical independence
between the PSD’s frequency bins (see Figure 1).
-4
1.E-04
10
1.E-05
10-5
-6
1.E-06
10
1.E-07
10-7
-8
1.E-08
10
-9
1.E-09
10
-6
-5
-4
-3
-2
-1
0
x
1
2
3
4
5
6
EQUATION 5:
FIGURE 2:
Standard Gaussian
Probability Density Function.
Table 1 shows important points on this curve and the
corresponding (two tailed) probability that the random
Gaussian variable is outside of those points. This
information is useful in converting RMS values
(voltages or currents) to either peak or peak-to-peak
values. The column label xL is sometimes called the
number of sigma from the mean.
© 2008 Microchip Technology Inc.
2
e nout
OUTPUT NOISE
V OUT 2 2
= ------------e ni
V IN
Where:
eni
=
noise voltage density at VIN (V/√Hz)
enout
=
noise voltage density at VOUT (V/√Hz)
Example 1 shows the conversion of a simple transfer
function to its squared magnitude. It starts as a Laplace
Transform [2], it is converted to a Fourier Transform
(substituting jω for s) and then converted to its squared
magnitude form (a function of ω2). It is best to do this
last conversion with the transform in factored form.
DS01228A-page 3
AN1228
EXAMPLE 1:
TRANSFER FUNCTION
CONVERSION EXAMPLE
Laplace Transfer Function:
V OUT
1 ------------- = ---------------------V IN
1 + s ⁄ ωP
Conversion to Fourier Transfer Function:
V OUT
1
-, s → jω
------------- = -------------------------V IN
1 + j ω ⁄ ωP
Note:
Conversion to Magnitude Squared:
2
V OUT 2
1
1
- = --------------------------------2
------------- = -------------------------V IN
1 + j ω ⁄ ωP
1 + ( ω ⁄ ωP )
1
= ---------------------------2 , ω → 2πf
1 + ( f ⁄ fP )
Where:
s
=
Laplace frequency (1/s)
=
σ + jω
ω
=
Radian frequency (rad/s)
ωP
=
Pole (rad/s)
f
=
Frequency (Hz)
fP
=
Pole frequency (Hz)
In the physical world, however, brick wall filters would
have horrible behavior. They cannot be realized with a
finite number of circuit elements. Physical filters that try
to approach this ideal show three basic problems: their
step response exhibits Gibbs phenomenon (overshoot
and ringing that decays slowly), they suffer from noise
enhancement (due to high pole quality factors) and
they are very difficult to implement.
Comments in the literature (e.g., in filter
textbooks) about “ideal” brick wall filters
should be viewed with skepticism.
The integrated noise voltage integrals (Equation 3 and
Equation 4) are in their most simple terms when a brick
wall filter is used. Equation 6 shows that, in this case,
the brick wall filter’s frequencies fL and fH become the
new integration limits. The integrated current noise is
treated similarly.
EQUATION 6:
INTEGRATED NOISE WITH
BRICK WALL FILTER
∞
E nout =
∫0 e nout ( f ) df
∞
=
2
V
OUT
∫0 e ni ( f ) ------------V
2
2
df
IN
BRICK WALL FILTERS
The transfer function that is easiest to manipulate
mathematically is the brick wall filter. It has infinite
attenuation (zero gain) in its stop bands, and constant
gain (HM) in its pass band; see Figure 3.
|H(j2πf)| (V/V)
Where:
fH
∫fL e ni ( f ) df
2
fL
=
Lower cutoff frequency (Hz)
fH
=
Upper cutoff frequency (Hz)
HM
=
Pass band gain (V/V)
See Appendix B: “Computer Aids” for popular circuit
simulators and symbolic mathematics packages that
help in these calculations.
HM
0
= HM ⋅
0
fH
fL
FIGURE 3:
f (Hz)
Brick Wall Filter.
We will use three variations of the brick wall filter (refer
to Figure 3):
• Low-pass (fL is at zero)
- fL = 0 < fH < ∞
• Band-pass (as shown)
- 0 < fL < fH < ∞
• High-pass (fH is at infinity)
- 0 < fL < ∞ = fH
Brick wall filters are a mathematical convenience that
simplifies our noise calculations.
DS01228A-page 4
White Noise
White noise has a PSD that is constant over frequency.
It received its name from the fact that white light has an
equal mixture of all visible wavelengths (or
frequencies). This is a mathematical abstraction of real
world noise phenomena.
A truly white noise PSD would produce an infinite integrated noise. Physically, this is not a concern because
all circuits and physical materials have limited
bandwidth.
We start with white noise because it is the easiest to
manipulate mathematically. Other spectral shapes will
be addressed in subsequent sections.
© 2008 Microchip Technology Inc.
AN1228
NOISE POWER BANDWIDTH
When white noise is passed through a brick wall filter
(see Figure 3), the integrated noise becomes a very
simple calculation. Equation 6 is simplified to:
EQUATION 7:
INTEGRATED WHITE NOISE
WITH BRICK WALL FILTER
E nout = H M e ni f H – f L
The shot noise current density’s magnitude depends
on the diode’s DC current (ID) and the electron charge
(q). It is usually modeled as white noise; see
Equation 9.
EQUATION 9:
i nd =
q
=
Input noise voltage density (V/√Hz)
enout
=
Output noise voltage density (V/√Hz)
This equation is usually represented by what is called
the Noise Power Bandwidth (NPBW). NPBW is the
bandwidth (under the square root sign) that converts a
white noise density into the correct integrated noise
value. For the case of brick wall filters, we can use
Equation 8.
EQUATION 8:
2q I D
Where:
Where:
eni
DIODE SHOT NOISE
ID
=
Electron charge
=
1.602 × 10-19 (C)
=
Diode Current (A)
Let’s look at a specific example:
EXAMPLE 2:
A DIODE SHOT NOISE
CALCULATION
Given:
ID = 1 mA
INTEGRATED WHITE NOISE
WITH NPBW
Calculate:
i nd =
2 ( 1.602 × 10
– 19
C ) ( 1 mA )
= 17.9 pA/√Hz
E nout = H M e ni NPBW
Where:
NPBW
=
fH –fL,for brick wall filters
The high-pass filter appears to cause infinite integrated
noise. In real circuits, however, the bandwidth is
limited, so fH is finite (a band-pass response).
Note:
NPBW applies to white noise only; other
noise spectral shapes require more
sophisticated formulas or computer
simulations.
Circuit Noise Sources
This section discusses circuit noise sources for
different circuit components and transfer functions
between sources and the output.
DIODE SHOT NOISE
Diodes and bipolar transistors exhibit shot noise, which
is the effect of the electrons crossing a potential barrier
at random arrival times. The equivalent circuit model
for a diode is shown in Figure 4.
Note:
All of the calculation results in this
application note show more decimal
places than necessary; two places are
usually good enough. This is done to help
the reader verify his or her calculations.
RESISTOR THERMAL NOISE
The thermal noise present in a resistor is usually
modeled as white noise (for the frequencies and
temperatures we are concerned with). This noise
depends on the resistor’s temperature, not on its DC
current. Any resistive material exhibits this
phenomenon, including conductors and CMOS
transistors’ channel.
Figure 5 shows the models for resistor thermal noise
voltage and current densities. The sources are shown
with a polarity for convenience in circuit analysis.
enr
R
inr
R
ID
D
FIGURE 4:
Model for Diodes.
ind
FIGURE 5:
Physically Based Noise
Model for Resistors.
Physically Based Noise
© 2008 Microchip Technology Inc.
DS01228A-page 5
AN1228
The equivalent noise voltage and current spectral
densities are (remember that 273.15 K = 0°C):
EQUATION 10:
RESISTOR THERMAL NOISE
DENSITY
e nr =
4kT A R
i nr =
4kT A ⁄ R
VDD
eni
VP
VM
=
Boltzmann constant
=
1.381 × 10-23 (J/K)
TA
=
Ambient temperature (K)
R
=
Resistance (Ω)
4kTA represents a resistor’s internal power. The
maximum available power to another resistor is kTA
(when they are equal). Many times the maximum
available power is shown as kTA/2 because physicists
prefer using two-sided noise spectra.
Let’s use a 1 kΩ resistor as an example.
EXAMPLE 3:
A THERMAL NOISE
DENSITY CALCULATION
Given:
R = 1 kΩ
TA = 25°C = 298.15 K
Calculate the noise voltage density:
4 ( 1.381 × 10
e nr =
– 23
J/K ) ( 298.15 K )
= 4.06 nV/√Hz
4 ( 1.381 × 10
– 23
J/K ) ⁄ ( 298.15 K )
= 4.06 pA/√Hz
OP AMP NOISE
An op amp’s noise is modeled with three noise sources:
one for the input noise voltage density (eni) and two for
the input noise current density (ibn and ibi). All three
noise sources are physically independent, so they are
statistically uncorrelated. Figure 6 shows this model; it
is similar to the DC error model covered in [1].
VOUT
ibi
FIGURE 6:
Physically Based Noise
Model for Op Amps.
The noise voltage source can also be placed at the
other input of the op amp, with its negative pin is
connected to VI and its positive pin to VM. This alternate
connection gives the same output voltage (VOUT).
For voltage feedback (VFB) op amps, both noise
current sources have the same magnitude. This
magnitude is shown in Microchip’s op amp data sheets
with the symbol ini; it has units of fA/√Hz (f stands for
femto, or 10-15).
For now, we will discuss the white noise part of these
spectral densities. We will defer a discussion on 1/f
noise until later.
The literature sometimes shows an amplifier noise
model that has only one noise current source. In these
cases, the second noise current’s power has been
combined into the noise voltage magnitude.
Note:
Keep in mind that op amps have two
physically independent noise current
sources.
For current feedback (CFB) op amps, the two noise
current sources (ibn and ibi) are different in magnitude
because the two input bias currents (IBN and IBI) are
different in magnitude. They are produced by physically
independent and statistically uncorrelated processes.
CFB op amps are typically used in wide bandwidth
applications (e.g., above 100 MHz).
Microchip’s CMOS input op amps have a noise current
density based on the input pins’ ESD diode leakage
current (specified as the input bias current, IB). Table 2
gives the MCP6241 op amp’s white noise current
values across temperature.
TABLE 2:
MCP6241 (CMOS INPUT) NOISE
CURRENT DENSITY
IB
(pA)
ini
(fA/√Hz)
25
1
0.57
85
20
2.5
125
1100
19
TA
(°C)
DS01228A-page 6
AOL
VSS
Calculate the noise current density:
i nr =
VI
ibn
Where:
k
VN
© 2008 Microchip Technology Inc.
AN1228
Table 3 gives the MCP616 op amp’s white input noise
current density across temperature. This part has a
bipolar (PNP) input; the base current is the input bias
current, which decreases with temperature.
TABLE 3:
TA
(°C)
MCP616 (BIPOLAR INPUT)
NOISE CURRENT DENSITY
IB
(nA)
ini
(fA/√Hz)
-40
-21
82
25
-15
69
85
-12
62
NOISE ANALYSIS PROCESS
This section goes through the analysis process
normally followed in noise design. It uses a very simple
noise design problem to make this process clear.
Simple Example
The circuit shown in Figure 8 uses an op amp and a
lowpass brick wall filter (fL = 0). The filter’s bandwidth
(fH) is 10 kHz and its gain (HM) is 1 V/V. The op amp’s
input noise voltage density (eni) is 100 nV/√Hz, and its
gain bandwidth product is much higher than fH.
The input noise voltage density (eni) typically does not
change much with temperature.
Note:
Noise current density (ini) usually changes
significantly with temperature (TA).
Note:
Most of the time, you can use IB vs. TA and
the shot noise formula to calculate ini vs.
TA. One exception to this rule is op amps
with input bias current cancellation circuitry.
U1
Brick Wall
Low-pass Filter
VIN
VOUT
Op Amp Circuit.
FIGURE 8:
Figure 9 shows both the op amp noise voltage density
(eni) and the output noise voltage density (enout). Notice
that enout is simply eni multiplied by the low-pass brick
wall’s pass-band gain (HM).
TRANSFER FUNCTIONS
The transfer function from each noise source in a circuit
to the output is needed. This may be obtained with
SPICE simulations (see Appendix B: “Computer
Aids”) or with analysis by hand. This application note
emphasizes the manual approach more in order to
build understanding and to derive handy design
approximations.
The most convenient manual approach is circuit analysis using the Laplace frequency variable (s). Figure 7
shows a resistor, inductor and capacitor with their
corresponding impedances (using s).
R
sL
1
sC
FIGURE 7:
Impedance Models for
Common Passive Components.
Noise Voltage Density (nV/√Hz)
100
enout
0
0
eni
f (Hz)
10k
Noise Voltage Densities.
FIGURE 9:
The noise current densities ibn and ibi can be ignored in
this circuit because they flow into a voltage source and
the op amp output, which present zero impedance.
Now we can calculate the integrated noise at the output
(Enout). The result is shown in three different units
(RMS, peak and peak-to-peak):
EXAMPLE 4:
AN INTEGRATED NOISE
CALCULATION
∞
E nout =
∫0 e nout ( f ) df
2
10 kHz
=
∫0
2
( 100 nV/ Hz ) df
= ( 100 nV/ Hz ) 10 kHz
= 10 µVRMS = 33 µVPK= 66 µVP-P
Note:
© 2008 Microchip Technology Inc.
This application note uses the crest factor
3.3 VPK/VRMS (or 6.6 VP-P/VRMS).
DS01228A-page 7
AN1228
Enout(t) (µV)
Figure 10 shows numerical simulation results of the
output noise over time. Enout describes the variation of
this noise. This same data is plotted in histogram form
in Figure 11; the curve represents the ideal Gaussian
probability density function (with the same average and
variation).
50
40
30
20
10
0
-10
-20
-30
-40
-50
FILTERED NOISE
This section covers the op amp circuits that have filters
at their output. The discussion focuses on filters with
real poles to develop insight and useful design
formulas.
The effect that reactive circuit components have on
noise is deferred to a later section. Noise generated by
the filters is ignored for now.
fSAM = 10 kSPS
Low-pass Filter With Single Real Pole
0
10
20
30
40
50 60
t (ms)
70
80
90 100
Figure 12 shows an op amp circuit with a low-pass filter
at the output, which has a single real pole (fP). We do
not need to worry about the noise current densities
because the ibn and ibi sources see zero impedance
(like Figure 8). We will assume that the op amp BW can
be neglected because fP is much lower.
U1
Output Noise vs. Time.
FIGURE 10:
Real Pole
Low-pass Filter
VIN
Percentage of Occurrences
9%
8%
VOUT
1024 Samples
6%
Enout
Gaussian
5%
We need the filter’s transfer function in order to
calculate the output integrated noise; it needs to be in
squared magnitude form (see Example 1 for the
derivation of these results):
4%
3%
2%
1%
40
30
20
10
0
-10
-20
-30
-40
0%
EQUATION 11:
Enout (µV)
FIGURE 11:
Op Amp Circuit With Low-
FIGURE 12:
pass Filter.
7%
V OUT
1
------------- = -------------------------V IN
1 + j ω ⁄ ωP
Output Noise Histogram.
Review of the Process
V OUT
------------V IN
The basic process we have followed can be described
as follows.
2
1
= ---------------------------2
1 + ( f ⁄ fP )
Figure 13 shows the transfer function magnitude in
decibels.
0
-5
| H(j2πf) | (dB)
• Collect noise and filter information
• Convert noise at the sources to noise at the
output
• Combine and integrate the output noise terms
• Evaluate impact on the output signal
LOW-PASS TRANSFER
FUNCTION
-10
-15
-20
-25
-30
-35
-40
0.01
FIGURE 13:
DS01228A-page 8
0.1
1
f / fP
10
100
Filter Magnitude Response.
© 2008 Microchip Technology Inc.
AN1228
Now we can obtain the integrated noise, assuming the
op amp’s input noise voltage density (eni) is white:
EQUATION 12:
INTEGRATED NOISE
DERIVATION
∞
E nout =
∫0
e
2
nout ( f ) df =
∞
∫0
2
e ni df
--------------------2 2
1 + f ⁄ fP
Low-pass Filter With Two Real Poles
The low-pass filter in Figure 14 has two real poles (fP1
and fP2). We do not need to worry about the noise
current densities because the ibn and ibi sources see
zero impedance (like Figure 8). We assume that fP1
and fP2 are much lower than the op amp BW, so the op
amp BW can be neglected.
∞
U1
f P [ atan ( f ⁄ f P ) ] 0
= e ni
= e ni ( π ⁄ 2 ) ⋅ f P
Two Real Pole
Low-pass Filter
VIN
Thus, the NPBW for this filter is (see Equation 8):
EQUATION 13:
NOISE POWER BANDWIDTH
NPBW = ( π ⁄ 2 ) ⋅ f P
We can always reduce the integrated output noise by
reducing NPBW, but the signal response may suffer if
we go too far. We need to keep the filter’s -3 dB
bandwidth (BW) at least as large as the desired signal
BW (fP is this filter’s BW).
Op Amp Circuit With Low-
FIGURE 14:
pass Filter.
The filter’s transfer function and the magnitude
squared transfer function (a function of ω2), in factored
form, are:
EQUATION 15:
LOW-PASS TRANSFER
FUNCTION
V OUT
1
1
- ⋅ ----------------------------------------- = ----------------------------V IN
1 + j ω ⁄ ω P1 1 + j ω ⁄ ω P2
For low-pass filters, we can also select the BW based
on the maximum allowable step response rise time [6]
(this applies to any reasonable low-pass filter):
V OUT 2
1
1
------------- = -----------------------------2- ⋅ -----------------------------2V IN
1 + ( f ⁄ f P1 ) 1 + ( f ⁄ fP2 )
RISE TIME VS. BANDWIDTH
t R ≈ 0.35 ⁄ BW
Where:
fP1
=
First pole frequency (Hz)
BW = Any low-pass filter’s -3 dB bandwidth (Hz)
fP2
=
Second pole frequency (Hz)
Where:
tR = 10% to 90% Rise time (s)
Let’s try a numerical example with reasonably wide
bandwidth; the noise is limited by the filter’s bandwidth.
EXAMPLE 5:
Figure 15 shows the transfer function magnitude in
decibels for the specific case where fP2 is double fP1.
0
AN INTEGRATED NOISE
CALCULATION
Filter Specifications:
fP = BW = 10 kHz
Gain = 1 V/V
Op Amp Specifications:
eni = 100 nV/√Hz
BW = 1 MHz
Filter Rise Time:
t R ≈ 35 μs
fP2/fP1 = 2
-10
| H(j2πf) | (dB)
EQUATION 14:
VOUT
-20
-30
-40
-50
-60
-70
-80
0.01
FIGURE 15:
0.1
1
f / fP1
10
100
Filter Magnitude Response.
Integrated Noise Calculations:
fP << Op amp’s bandwidth
NPBW = ( π ⁄ 2 ) ⋅ ( 10 kHz ) = 15.8 kHz
E nout = ( 100 nV/ Hz ) ⋅ ( 15.8 kHz )
= 12.6 µVRMS = 41.4 µVPK = 82.9 µVP-P
© 2008 Microchip Technology Inc.
DS01228A-page 9
AN1228
We can follow the same process as before to calculate
NPBW.
EQUATION 16:
NPBW
π
1
1
NPBW = ⎛ ---⎞ ⁄ ⎛ ------- + -------⎞
⎝ 2⎠ ⎝ f P1 f P2⎠
As before, NPBW and BW are similar and BW can be
traded-off with rise time (see Equation 14).
EQUATION 17:
BW
f P1
1 f P1 2
BW = ----------------------------------- , X = 1 + --- ⋅ ⎛⎝ -------⎞⎠
2 f P2
2
X+ 1+X
EXAMPLE 7:
AN INTEGRATED NOISE
CALCULATION
Change in Filter Specifications:
fP1 = fP2 = 15.5 kHz
Filter Bandwidth and Rise Time:
BW = 9.98 kHz
t R ≈ 35 μ s
Integrated Noise Calculations:
fP2 << Op amp’s bandwidth
NPBW = 12.2 kHz
E nout = ( 100 nV/ Hz ) ⋅ ( 12.2 kHz )
= 11.0 µVRMS = 36.4 µVPK = 72.9 µVP-P
Where:
fP1
≤
High-pass Filter With Single Real Pole
fP2
Let’s go through a numerical example where the op
amp’s bandwidth can be neglected.
EXAMPLE 6:
AN INTEGRATED NOISE
CALCULATION
Filter Specifications:
Figure 16 shows an op amp circuit with a high-pass filter with a single real pole (fP). We do not need to worry
about the noise current densities because the ibn and
ibi sources see zero impedance (like Figure 8). For
practical circuits, there needs to be a low-pass filter at
a frequency much higher than fP (at fH); the integrated
noise would be infinite otherwise. If nothing else, the op
amp BW may be used to limit the NPBW.
fP1 = 13.4 kHz
fP2 = 26.8 kHz
U1
Real Pole
High-pass Filter
Gain = 1 V/V
Op Amp Specifications:
VIN
VOUT
eni = 100 nV/√Hz
BW = 1 MHz
Filter Bandwidth and Rise Time:
BW = 9.98 kHz
t R ≈ 35 μs
Op Amp Circuit With High-
FIGURE 16:
pass Filter.
The filter’s transfer function and the magnitude
squared transfer function (a function of ω2), in factored
form, are:
Integrated Noise Calculations:
fP2 << Op amp’s bandwidth
EQUATION 18:
HIGH-PASS TRANSFER
FUNCTION
NPBW = 14.0 kHz
jω ⁄ ω P
V OUT
- , ω < ωH
------------- = ------------------------1 + jω ⁄ ω P
V IN
E nout = ( 100 nV/ Hz ) ⋅ ( 14.0 kHz )
= 11.8 µVRMS = 39.0 µVPK = 78.1 µVP-P
ω ≥ ωH
= 0,
V OUT
------------V IN
Let’s redo this example with equal poles at 15.5 kHz.
2
2
( f ⁄ fP )
= --------------------------2- , f < f H
1 + ( f ⁄ fP )
= 0,
f ≥ fH
Where:
DS01228A-page 10
fP
=
Pole frequency (Hz)
fH
=
Low-pass NPBW (Hz)
© 2008 Microchip Technology Inc.
AN1228
Figure 17 shows the transfer function magnitude in
decibels (fH is not shown).
0
| H(j2πf) | (dB)
-5
-10
-15
-20
Band-pass Filter With Two Real Poles
Figure 18 shows an op amp circuit with a band-pass
filter with two real poles (highpass pole fP1 and lowpass pole fP2). We do not need to worry about the noise
current densities because the ibn and ibi sources see
zero impedance (like Figure 8). The op amp BW is
neglected because we assume that it is much higher
than fP1 and fP2.
-25
U1
-30
-35
Two Real Pole
Band-pass Filter
VIN
-40
0.01
0.1
1
f / fP
10
100
Filter Magnitude Response.
FIGURE 17:
VOUT
Op Amp Circuit With Band-
FIGURE 18:
pass Filter.
We can follow the same process as before to calculate
NPBW (fH acts like the upper integration limit in the
integrated noise equation).
The filter’s transfer function and the magnitude
squared transfer function (a function of ω2), in factored
form, are:
EQUATION 19:
EQUATION 20:
NPBW
NPBW = f H – ( π ⁄ 2 ) ⋅ f P
Where:
fP
<<
j ω ⁄ ω P1
V OUT
1
- ⋅ ----------------------------------------- = ----------------------------V IN
1 + j ω ⁄ ω P1 1 + j ω ⁄ ω P2
fH
V OUT
------------V IN
Let do a numerical example with the op amp bandwidth
much higher than the filter pole (this is very common).
EXAMPLE 8:
AN INTEGRATED NOISE
CALCULATION
fH = Op amp’s NPBW
2
2
( f ⁄ f P1 )
1
= -----------------------------2- ⋅ -----------------------------21 + ( f ⁄ f P1 ) 1 + ( f ⁄ f P2 )
Where:
Filter Specifications:
fP = 10 kHz
BAND-PASS TRANSFER
FUNCTION
fP1
=
High-pass pole frequency (Hz)
fP2
=
Low-pass pole frequency (Hz)
Figure 19 shows the transfer function magnitude in
decibels, with fP2 = 100 fP1.
Gain = 1 V/V
0
Op Amp Specifications:
BW = 1 MHz
NPBW ≈ ( π ⁄ 2 ) ⋅ BW = 1.57 MHz
Integrated Noise Calculations:
fP << Op amp’s bandwidth
NPBW = ( 1.57 MHz ) – ( 15.8 kHz ) = 1.55 MHz
E nout = ( 100 nV/ Hz ) ⋅ ( 1.55 MHz )
= 124 µVRMS = 411 µVPK = 822 µVP-P
Note:
A high-pass filter’s NPBW has little effect
on the integrated noise, unless fH is near
fP (but that would be a band-pass filter).
© 2008 Microchip Technology Inc.
| H(j2πf) | (dB)
eni = 100 nV/√Hz
fP2/fP1 = 100
-5
-10
-15
-20
-25
-30
-35
-40
10m
0.01
100m
0.1
FIGURE 19:
1
1
10
10
f / fP1
100
100
1k
1000
10k
10000
Filter Magnitude Response.
Using a symbolic solver to derive NPBW is a big help.
EQUATION 21:
NPBW
1
NPBW = ( π ⁄ 2 ) ⋅ f P2 ⋅ ---------------------------1 + f P1 ⁄ f P2
DS01228A-page 11
AN1228
Let’s do another numerical example.
EXAMPLE 9:
AN INTEGRATED NOISE
CALCULATION
Filter Specifications:
fP1 = 100 Hz
fP2 = 10 kHz
Gain = 1 V/V
FILTERS WITH GREATER SELECTIVITY
There are other filters with a sharper transition region,
when n > 1, such as: Chebyshev, Inverse Chebyshev
and Elliptic filters. Their NPBW to BW ratios are closer
to 1 because they have a smaller transition region
(between pass-band and stop-band). This smaller
transition region reduces the integrated noise at the
output. Their step response, however, tends to have
more ringing and slower decay.
Again, NPBW can be approximated with the -3 dB
bandwidth. More exact results can be obtained with
simulations (see Appendix B: “Computer Aids”).
Op Amp Specifications:
eni = 100 nV/√Hz
BW = 1 MHz
NOISE INTERNAL TO FILTERS
Integrated Noise Calculations:
fP2 << Op amp’s bandwidth
NPBW = ( 15.7 kHz ) ⁄ 1.01 = 15.5 kHz
E nout = ( 100 nV/ Hz ) ⋅ ( 15.5 kHz )
= 12.5 µVRMS = 41.2 µVPK = 82.3 µVP-P
Comments on Other Filters
As will be shown later (see Figure 25), active filters
may produce much more noise than first expected. The
op amps inside the filter produce a noise voltage
density at the filter's output that has a wider bandwidth
than the filter; it may be as wide as the op amp bandwidths. The resistors and op amp noise contributions
tend to show a peak at the edges of the filter passband
(noise enhancement), which increases the integrated
output noise.
This section discusses other filters and how they affect
the output integrated noise. It gives a very simple
approximation to NPBW when the filter order is greater
than n = 1. It then discusses noise generated interal to
a filter.
SOME SIMPLE LOW-PASS FILTERS
Table 4 shows the NPBW to BW ratio for some lowpass filters up to order 5.
TABLE 4:
NPBW FOR SOME LOW-PASS
FILTERS
NPBW / BW
Low-pass
Filter Type
n=1 n=2 n=3 n=4 n=5
Identical Real Poles
1.571 1.220 1.155 1.128 1.114
Bessel
1.571 1.153 1.071 1.046 1.038
Butterworth
1.571
Note:
1.111
1.047 1.026 1.017
The -3 dB bandwidth is a rough estimate
of NPBW for almost all filters (the main
exception is when n = 1).
DS01228A-page 12
© 2008 Microchip Technology Inc.
AN1228
MULTIPLE NOISE SOURCES
R-C Low-pass Filter
This section covers two approaches to combining
multiple noise sources into one output integrated noise
result. This knowledge is applied to a simple R-C lowpass filter and a non-inverting gain circuit.
Figure 21 shows a circuit with a R-C low-pass filter with
a real pole (fP). We do not need to worry about the
noise current densities because the ibn and ibi sources
see zero impedance (like Figure 8). We will assume
that the op amp BW can be neglected because fP is
much lower.
Combining Noise Outputs
When we combine noise results, at the output, we take
advantage of the statistical independence of:
• PSD noise in separate frequency bins
• Physically independent noise sources
We can integrate the output noise densities one at a
time, then combine the results using a Sum of Squares
approach (see Equation 1). We can also combine all of
the noise densities using a Sum of Squares approach
first, then integrate the resulting noise density.
… , Enok
2
Sum of Squares
R1
VOUT
C1
Circuit With R-C Low-pass
FIGURE 21:
Filter.
We will integrate the noise densities first because this
will give us important insight into this R-C low-pass filter. This circuit is like the one we already saw in
Figure 12, but we have added R1’s thermal noise.
The filter’s transfer function and the magnitude
squared transfer function (a function of ω2), in factored
form, are in Equation 22 (Figure 13 shows the transfer
function magnitude in decibels).
eno12, … , enok2
Eno12,
enr1
VIN
This independence simplifies our work, since we do not
need to worry about correlations.
Integrate Noise Densities
(over frequency)
U1
Sum of Squares
(at each frequency)
eno
2
Integrate Noise Density
(over frequency)
Eno2
EQUATION 22:
1 ⁄ ( sC 1 )
V OUT
1
1
- = ------------------------------------- = --------------------------------→ --------------------------V IN
R 1 + 1 ⁄ ( sC1 )
1 + sR 1 C 1
1 + j ω ⁄ ωP
V OUT 2
V OUT
= ------------------------V IN
e nr1
Each approach has its advantages. Integrating first
helps determine which noise source dominates; it is
handy for hand designs. Finding the output noise
density first helps to adjust frequency shaping
elements in the design; it is easier with computer
simulations.
2
1
= ---------------------------2
1 + ( f ⁄ fP )
Where:
fP
ωP
FIGURE 20:
Approaches to Combining
Output Noise Terms.
R-C LOW-PASS FILTER
TRANSFER FUNCTION
=
R-C filter’s pole frequency (Hz)
1
⁄
= ( 2πR1 C1 )
=
2πfP
We can follow the same process as before to calculate
NPBW. The trade-offs between NPBW (or BW) and tR
shown in Equation 14 apply to this filter.
EQUATION 23:
NPBW
NPBW = ( π ⁄ 2 ) ⋅ f P
The next section (“R-C Low-pass Filter”) demonstrates the approach on the left of Figure 20. The section following that one (“Non-inverting Gain Circuit”)
demonstrates the approach on the right of Figure 20.
© 2008 Microchip Technology Inc.
DS01228A-page 13
AN1228
The integrated noise becomes:
Non-inverting Gain Circuit
EQUATION 24:
Figure 22 is a complete model for a non-inverting gain
circuit. R1 and R3 use the series noise voltage density
sources because their transfer function to VOUT is simpler in that form. R2 uses the shunt noise current
density source because we can use the same transfer
function to VOUT that ibi uses; this reduces our work.
INTEGRATED NOISE
E noU1 = e ni ⋅ NPBW
E noR1 = e nr1 ⋅ NPBW
=
4kT A R 1 ⋅ ( π ⁄ 2 ) ⁄ ( 2 π R 1 C 1 )
=
kT A ⁄ C 1
2
E noU1
E nout =
+
2
E noRC
VIN
R1
enr1
eni
EnoU1
=
U1’s output integrated noise (VRMS)
EnoR1
=
R1’s output integrated noise (VRMS)
Enout
=
Total output integrated noise (VRMS)
The last expression shown for EnoR1 (sqrt(kTA/C1)) is
popularly called “kT on C noise” (referring to the PSD
inside the square root). This result applies only to this
particular case (integrated thermal noise at the output
of an R-C lowpass filter).
Do not let this equation mislead you; R1
generates the thermal noise, not C1.
Let’s do a numerical example where the op amp and
the filter resistor both contribute to the noise.
EXAMPLE 10:
AN INTEGRATED NOISE
CALCULATION
Ambient Temperature:
TA = 25°C = 298.15 K
Filter Specifications:
R1 = 10 kΩ
C1 = 1.5 nF
Gain = 1 V/V
R2
We will combine the noise densities first to obtain the
output noise density (enout). In this case, because we
have no reactive elements in the circuit, it will be a
simple matter to integrate enout by hand to produce
Enout.
We will start with all of the transfer functions from each
source to VOUT (see reference [1]). The gains will be
assumed constant for now; we will deal with frequency
shaping later on.
EQUATION 25:
V OUT
------------V IN
V OUT
------------e nr3
V OUT
------------i bn
V OUT
------------i bi
= 12.9 µVRMS
e nr1 = 12.8 nV/ Hz
E noRC = ( 12.8 nV/ Hz ) ⋅ ( 16.7 kHz )
= 1.66 µVRMS
= 13.0 µVRMS = 43.0 µVPK = 85.9 µVP-P
DS01228A-page 14
TRANSFER FUNCTIONS
V OUT
V OUT
= ------------- = ------------- = G N
e nr1
e ni
= 1
= –R1 GN
V OUT
= ------------- = R 3
i nr2
Where:
GN
=
Noise Gain (V/V)
= 1 + R3 ⁄ R2
NPBW = ( π ⁄ 2 ) ⋅ ( 10.6 kHz ) = 16.7 kHz
E noU1 = ( 100 nV/ Hz ) ⋅ ( 16.7 kHz )
enr3
ANALYSIS WITH CONSTANT GAINS
BW = 1 MHz
Integrated Noise Calculations:
fP << Op amp’s bandwidth
R3
FIGURE 22:
Non-inverting gain Amplifier,
with multiple noise sources.
eni = 100 nV/√Hz
Filter Pole, Bandwidth and Rise Time:
fP = BW = 10.6 kHz
t R ≈ 33 μ s
ibi
inr2
Op Amp Specifications:
E nout
VOUT
ibn
Where:
Note:
U1
Note that noise gain (GN) is from the non-inverting input
to VOUT, when the op amp is in a closed-loop condition,
and when other (external) energy sources are zero.
Note:
The concept of noise gain is central
to understanding op amp behavior.
It simplifies op amp bandwidth and
stability analyses.
© 2008 Microchip Technology Inc.
AN1228
The magnitude squared transfer functions are simply
the squares of the constant terms in Equation 25. We
will now combine these noise densities into one
equation for the output noise density (using a sum of
squares approach):
Reactive elements in the circuit will require a more
detailed analysis because each noise source may have
a different frequency shape.
EQUATION 26:
EXAMPLE 11:
2
2
COMBINING NOISE
DENSITIES
2
2
2
2
2
2
2
2
e nout = G N ( e nr1 + e ni + i bn R 1 ) + e nr3 + ( i bi + i nr2 )R 3
=
2
G N ( 4kT A R 1
2
2 2
+ e ni + i bn R 1 ) + 4kT A R 3
2
2
+ ( i bi + 4kT A ⁄ R 2 )R 3
While this equation is sufficient to calculate Enout,
converting to an input referred form gives more insight
to the designer. Dividing both sides by GN2, substituting
1 + R3/R2 for GN, and simplifying, gives:
EQUATION 27:
2
2
OP AMP NOISE EQUATION
2
2
2
2
e nout ⁄ G N = e ni + i bn R 1 + i bi ( R 2 ||R 3 )
2
+ 4kT A ( R 1 + ( R 2 ||R 3 ) )
This shows that the output noise density has a very
simple relationship to the resistances seen by the
inputs (R1 and (R2||R3)).
Note:
Equation 27 also applies to inverting
amplifiers (i.e., R2 is driven by VIN and R1
is grounded).
ANALYSIS WITH LIMITED BANDWIDTH
To produce a finite output integrated noise, we need a
filter that limits the NPBW. This filter can be
implemented with the op amp, by reactive elements in
the circuit (e.g., capacitors) or by a filter after the op
amp.
We can use the op amp’s BW to set NPBW. The
response can be approximated with a single real pole
for hand calculations. The Gain Bandwidth Product
(GBWP) specification in VFB op amp data sheets
gives:
EQUATION 28:
NPBW SET BY OP AMP’S
BANDWIDTH
The following example has all of the noise sources at
about the same magnitude.
AN INTEGRATED NOISE
CALCULATION
Ambient Temperature:
TA = 25°C = 298.15 K
Circuit Specifications:
R1 = 100 kΩ
R2 = R3 = 200 kΩ
Op Amp Specifications:
eni = 100 nV/√Hz
ini = 1 pA/√Hz
GBWP = 100 kHz
Preliminary Calculations:
GN = 2.00 V/V
(R2||R3) = 100 kΩ
Input Noise Densities:
2
e ni = ( 100 nV/ Hz )
2
2
2
i bn R 1 = ( 100 nV/ Hz )
2
2
2
i bi ( R 2 ||R 3 ) = ( 100 nV/ Hz )
2
4kT A ( R 1 + ( R 2 ||R 3 ) ) = ( 57.4 nV/ Hz )
2
Output Noise Density:
2
2
e nout ⁄ G N = ( 182 nV/ Hz )
2
e nout = 365 nV/ Hz
Integrated Output Noise:
GBWP = 100 kHz
BW ≈ 50 kHz
NPBW ≈ 78.6 kHz
E nout ≈ ( 365 nV/ Hz ) ⋅ ( 78.6 kHz )
≈ 102 µVRMS = 338 µVPK = 675 µVP-P
Simulated Examples
GBWP
=
Gain Bandwidth Product (Hz)
BW
=
Bandwidth (Hz)
This section covers two filter designs. It uses SPICE
simulations to quickly obtain numerical results. The first
design demonstrates potential issues with op amp
circuits that need good noise performance. The second
design improves the noise performance dramatically
using simple changes.
NPBW
=
Noise Power Bandwidth (Hz)
SECOND ORDER FILTER
BW ≈ GBWP ⁄ G N
NPBW ≈ ( π ⁄ 2 ) ⋅ BW
Where:
Note:
CFB op amp data sheets specify BW
instead of GBWP.
© 2008 Microchip Technology Inc.
Figure 23 shows a second order Butterworth filter with
a bandwidth of 1 kHz. It uses the MCP616 for the op
amp; we will assume that it has no 1/f noise for now.
DS01228A-page 15
AN1228
The resistor R3 balances the resistances seen by the
op amp inputs, which minimizes the output offset due
to input bias currents [1]. It uses the Sallen-Key
topology.
The hump in the noise curves, seen at 1 kHz, is caused
by the feedback action of the filter. The noise due to R3
and R2 is significant compared to the noise due to eni
(the op amp’s input noise voltage density).
THIRD ORDER FILTER
C1
4.7n
R1
R2
U1
38.3k 64.9k MCP616
VIN
VOUT
C2
2.2n
R3
102k
Butterworth Lowpass Filter.
FIGURE 23:
There are some obvious improvements we should
make to this filter. Reducing the resistor values will
reduce the thermal noise densities. Adding a filter at
the output will significantly reduce the integrated noise
at the output.
The circuit in Figure 26 is the result of making these
improvements. The resistors are about four times
smaller; this reduction was limited to avoid output
loading concerns. The filter design was changed to a
3rd order Butterworth to take maximum advantage of
the additional filter stage (R4 and C4).
C1
27n
|VOUT/VIN| (dB)
Figure 24 shows the simulated transfer function for
Figure 23.
R1
8.35k
10
0
-10
-20
-30
-40
-50
-60
-70
-80
-90
-100
-110
-120
100
1.E+2
VOUT
C4
10n
C2
5.6n
R3
28.7k
FIGURE 26:
pass Filter.
1k
1.E+3
f (Hz)
10k
1.E+4
100k
1.E+5
Filter Transfer Function.
Figure 25 shows the output noise voltage densities; the
labels indicate the source of a particular output density.
enr1, enr2 and enr3 represent R1, R2 and R3’s thermal
noise, while eni, ibn and ibi represent the op amp’s noise
sources. The combined output noise density is labeled
“total.”
1000
total
enr3
eni
ibi
100
10
enr1
ibn
100
1.E+2
FIGURE 25:
1k
10k
1.E+3
1.E+4
f (Hz)
enr2
100k
1.E+5
1M
1.E+6
Output Noise Densities.
It is possible to reduce R3’s noise contribution more by
adding a capacitor (C3, which isn’t shown) in parallel to
R3. SPICE simulations will help determine if the
reduction in noise is worth the additional cost.
Figure 27 shows the simulated transfer function for
Figure 26; notice the improved attenuation in the stopband compared to that shown previously (see
Figure 24).
10
0
-10
-20
-30
-40
-50
-60
-70
-80
-90
-100
-110
-120
100
1.E+2
FIGURE 27:
DS01228A-page 16
Improved Butterworth Low-
A buffer placed after R4 and C4 would have a wide
NPBW, so its noise contribution would be significant.
For this reason, the output has no output buffer.
|VOUT/VIN| (dB)
eno (nV/Hz)
R4
15.8k
VIN
FIGURE 24:
1
10
1.E+1
R2
U1
20.0k MCP616
1k
1.E+3
f (Hz)
10k
1.E+4
100k
1.E+5
Filter Transfer Function.
© 2008 Microchip Technology Inc.
AN1228
Figure 28 shows the output noise voltage densities for
Figure 26; the labels indicate the source of a particular
output density. enr1, enr2, enr3 and enr4 represent R1,
R2, R3 and R4’s thermal noise, while eni, ibn and ibi
represent the op amp’s noise sources. The combined
output noise density is labeled “total”.
eno (nV/Hz)
total
eni
enr3
enr2
enr4
ibn
100
1
10
1.E+1
Flicker noise (also called 1/f noise or pink noise) can be
important in low frequency applications (e.g., below
1 kHz). This noise increases the output variation above
what the white noise predictions give.
Note:
1000
10
FLICKER NOISE
1/f noise is caused by defects, at the atomic level, in
semiconductor and resistive devices. These defects
affect the DC current flowing through these devices.
With many defects operating simultaneously, each with
a different time constant, 1/f noise typically results.
enr1
ibi
100
1.E+2
1k
10k
1.E+3
1.E+4
f (Hz)
100k
1.E+5
1M
1.E+6
Output Noise Densities.
FIGURE 28:
Auto-zeroed op amps have such low 1/f
noise that it can be neglected.
Comparing Figure 25 to Figure 28 shows that we have
been successful in reducing the low frequency (i.e.,
below 200 Hz) output noise density. We have also
reduced the overall NPBW significantly.
Components with high 1/f noise include carbon
resistors and semiconductor devices (diodes and
transistors). All conductors, however, exhibit 1/f noise
at some level.
This section discusses 1/f noise, its impact on output
variability and how to find relevant information in data
sheets. A low frequency design example illustrates an
approach to these designs.
1/f Noise Basics
Table 5 compares the integrated output noise for these
two designs. It summarizes the information found in
Figure 25 and Figure 28 in convenient form.
1/f noise derives its name from its PSD shape (with
units of VRMS2/Hz). The noise power increases at low
frequencies as the reciprocal of frequency:
TABLE 5:
EQUATION 29:
COMPARISON OF DESIGNS
Noise Source
Thermal
Op Amp
2
3rd Order
R1
5.4
2.5
R2
10.6
5.4
R3
154.4
9.1
R4
—
4.2
eni
120.6
13.4
ibn
1.9
0.5
ibi
26.9
0.9
Total
198.1
17.8
2
e nf ( f ) = e nf ( 1 Hz ) ⁄ f
Eno (µVP-P)
2nd Order
1/F NOISE
Where:
enf(f)
Note:
=
1/f noise voltage density
at the frequency f (nV/√Hz)
The noise voltage density (enf) varies
as 1/√f (10 dB/decade).
Notice how the 1/f noise was specified at one
frequency point in Equation 29 (at 1 Hz); this is for
convenience in our later work.
A DC current needs to flow for 1/f noise to be present.
For instance, the PSpice diode noise model uses the
following equation:
EQUATION 30:
2
DIODE 1/F NOISE
i nd ( f ) = KF ⋅ I D
AF
⁄f
Where:
© 2008 Microchip Technology Inc.
inf(f)
=
Diode’s 1/f noise current density
at the frequency f (A/√Hz)
KF
=
PSpice noise parameter (A2 – AF);
default is 0 (usually around 10-15)
AF
=
PSpice noise exponent; default is 1
DS01228A-page 17
AN1228
Noise Voltage Density (nV/√Hz),
log scale
1/f Noise
Figure 31 shows a histogram of the same noise data.
The curve is the ideal Gaussian distribution with the
same mean (0 µV) and standard deviation (3.55 µV).
7%
Percentage of Occurrences
1/f noise is sometimes specified with a corner
frequency. This happens when a noise source has both
white and 1/f noise. The corner frequency occurs
where the white noise density equals the 1/f noise
density; see Figure 29. As we will see later, the combination of these two noise types produces a smooth
bend in the region of fcorner, not the sharp corner
depicted here.
Gaussian
3%
2%
1%
With the white noise density and the corner frequency,
it is easy to calculate the 1/f noise voltage density
(enf (1 Hz)):
e nf ( 1 Hz ) = e nw f corner ⁄ ( 1 Hz )
enw
=
White noise voltage density (nV/√Hz)
=
corner frequency (Hz)
The local average of 1/f noise wanders
enough to be a concern in applications.
fSAM = 1 SPS
1m
1.E-03
10m
1.E-02
f (Hz)
EQUATION 32:
FIGURE 30:
DS01228A-page 18
12
8
INTEGRATED 1/F NOISE
fH
∫ fL
2
e nf ( f ) df =
2
e nf ( 1
40
1/f Noise vs. Time.
45
50
fH
∫ fL
2
e nf ( 1 Hz )
---------------------- df
f
Hz ) ⋅ ln ( f H ⁄ f L )
E nf = e nf ( 1 Hz ) ⋅
35
10
In order to keep this analysis simple, we’ll use a bandpass brick wall filter with cutoff frequencies fL and fH
(see Figure 3). This gives:
=
25 30
t (min)
1
1.E+00
Integrated 1/f Noise
2
20
100m
1.E-01
1/f Noise; FFT (first 2048
FIGURE 32:
points).
E nf =
15
6
fSAM = 1 SPS
100n
1.E+02
100µ
1.E-04
Figure 30 plots 1/f noise data (from bench evaluation
work) that shows typical 1/f noise behavior. The data
was adjusted to have zero mean and was sampled at
one sample per second (1 SPS). The local average
wanders over time (compare to the white noise shown
in Figure 10).
10
4
100µ
1.E+05
1µ
1.E+03
fcorner
5
2
The first 2048 data points were converted to the noise
density plot in Figure 32 (the blue curve) using a FFT
routine. The red curve is the best fit 1/f noise curve (it
has the same integrated noise power).
10µ
1.E+04
Where:
0
0
-2
-4
1/f Noise Histogram.
FIGURE 31:
1/f Noise (V/√Hz)
CONVERSION FROM
CORNER FREQUENCY
12
10
8
6
4
2
0
-2
-4
-6
-8
-10
-12
-6
-8
-10
-12
f (Hz),
log scale
FIGURE 29:
Conceptual Diagram of the
Corner Frequency.
1/f Noise (µV)
4%
1/f Noise (µV)
fcorner
Note:
Enf
5%
0%
White Noise
EQUATION 31:
3000 Samples
6%
ln ( f H ⁄ f L )
In other words, the integrated power (statistical
variance) is proportional to the number of decades (or
octaves) encompassed by the brick wall filter.
© 2008 Microchip Technology Inc.
AN1228
GROWTH OF 1/F NOISE
(NOTE 1, NOTE 2)
fH/fL
No. Decades
Enf (µVP-P)
1/fL
1.259
0.10
3.7
0.13 s
101
1
11.6
1.00 s
2
10
2
16.4
10 s
103
3
20.1
100 s
104
4
23.2
1000 s
105
5
26.0
2.78 hr
106
6
28.5
27.8 hr
107
7
30.7
11.6 day
108
8
32.9
116 day
109
9
34.9
3.17 year
9.50
35.8
10.0 year
3.16 × 109
Note 1:
2:
Note:
These numbers are based on fH = 10 Hz
and enf(1 Hz) = 1160 nV/√Hz.
The last entry was limited to a reasonable
design lifetime for a PCB circuit.
Changing the band-pass filter’s ratio of fH
to fL has little impact on 1/f noise variability
when fH/fL > 10 and fH << fcorner.
Information in Data Sheets
Table 7 shows the noise specifications in the MCP616/
7/8/9 Data Sheet. This op amp family has a bipolar
(PNP) input, so the noise current is higher than the
CMOS input op amps.
TABLE 7:
MCP616/7/8/9 NOISE
SPECIFICATIONS
Parameters
Sym Typ
Units
Conditions
µVP-P
f = 0.1 to
10 Hz
The Input Noise Current Density (ini) is usually given at
a frequency where the white noise dominates (1 kHz in
this case). This specification helps select an op amp
where the resistances are high. Remember that this
curve describes both input noise current sources,
which are statistically independent.
Figure 33 shows the noise density plot in the MCP616/
7/8/9 Data Sheet. The noise specifications describe
this data. Note that our CMOS input op amps do not
show ini in this plot because it is low enough to not
affect most designs.
10,000
10,000
1,000
1,000
ini
100
100
eni
Input Noise Current
Density (fA/Hz)
TABLE 6:
The Input Noise Voltage Density (eni) is usually given at
a frequency where the white noise dominates (1 kHz in
this case). This specification helps select an op amp for
high frequency work.
Input Noise Voltage
Density (nV/Hz)
Table 6 shows the growth that 1/f noise would exhibit
with different ratios of fH to fL. Mathematically, Enf has
unbounded growth as f approaches zero. Practically
speaking, however, that growth is so slow that it does
not affect most applications. The numerical values are
based on the data shown in Figure 32.
10
10
0.1 1.E+0
1
10 1.E+0
100 1.E+0
1k 1.E+0
10k
1.E1.E+0
01
0 Frequency
1
2 (Hz) 3
4
FIGURE 33:
MCP616/7/8/9 Input Noise
Voltage Density Plot.
Refer to Table 2 and Table 3 for examples of how the
white noise portion of ini changes with temperature.
Design Example
This design example is a simple modification to the
filter shown in Figure 26. The goal is to show a low
frequency circuit that is dominated by 1/f noise. To
obtain a cut-off frequency of 100 Hz, the capacitors
have all been increased by a factor of 10. Figure 34
shows the result; this is still a 3rd order Butterworth
filter.
Noise
Input Noise
Voltage
Eni
2.2
Input Noise
Voltage Density
eni
32
nV/√Hz f = 1 kHz
Input Noise
Current Density
ini
70
fA/√Hz f = 1 kHz
The Input Noise Voltage (Eni) is the integrated noise
voltage between 0.1 Hz and 10 Hz, with units of
(µVP-P). It helps select an op amp for low frequency
work. Typically, it is dominated by 1/f noise; autozeroed op amps are the main exception to this rule.
© 2008 Microchip Technology Inc.
C1
270n
R1
8.35k
R2
U1
20.0k MCP616
R4
15.8k
VIN
VOUT
C2
56n
C4
100n
R3
28.7k
FIGURE 34:
Butterworth Low-pass Filter.
DS01228A-page 19
AN1228
A buffer placed after R4 and C4 would have a wide
NPBW, so its noise contribution would be significant.
For this reason, the output has no output buffer.
|VOUT/VIN| (dB)
Figure 35 shows the simulated transfer function
(compare to Figure 27).
10
0
-10
-20
-30
-40
-50
-60
-70
-80
-90
-100
10
1.E+1
Table 8 summarizes the information found in Figure 36
in convenient form. It is instructive to compare these
results with those shown in Figure 25, Figure 28 and
Table 5.
TABLE 8:
NOISE VOLTAGE
CONTRIBUTIONS TO THE
OUTPUT
Noise Source
Thermal
Op Amp
100
1.E+2
f (Hz)
1k
1.E+3
10k
1.E+4
Filter Transfer Function.
FIGURE 35:
Eno (µVP-P)
R1
0.78
R2
1.72
R3
2.89
R4
1.32
eni
5.31
ibn
0.36
ibi
0.45
Total
6.49
Figure 36 shows the output noise voltage densities; the
labels indicate the source of a particular output density.
enr1, enr2, enr3 and enr4 represent R1, R2, R3 and R4’s
thermal noise, while eni, ibn and ibi represent the op
amp’s noise sources. The combined output noise
density is labeled “total.”
eno (nV/Hz)
1000
total
eni
enr3
enr2
enr4
100
10
1
0.1
1.E-1
enr1
ibi
ibn
1
1.E+0
FIGURE 36:
10
100
1.E+1
1.E+2
f (Hz)
1k
1.E+3
10k
1.E+4
Output Noise Densities.
Comparing Figure 28 to Figure 36 shows that the white
noise has been reduced. We also see the 1/f noise
effect below 30 Hz.
DS01228A-page 20
© 2008 Microchip Technology Inc.
AN1228
DESIGN OPTIMIZATION
FIND THE DOMINANT NOISE SOURCES
With the basics of noise analysis and design under
your belt, it is time to learn how to quickly and
effectively optimize the noise performance of an op
amp circuit.
Any noise source that is at least half as large (in VRMS)
as the largest source should be considered to be a
dominant source. This may appear to be a very loose
requirement at first glance, but works very well in
practice.
Signal-to-Noise Ratio
To illustrate this point, Table 9 illustrates how a larger
noise source (Enout1) and a smaller noise source
(Enout2) contribute to the total noise (Enout). The ratio
Enout2/Enout1 represents Enout2's magnitude relative to
Enout1. The ratio Enout/Enout1 represents how much
larger Enout is, compared to Enout1, due to the contribution from Enout2. When Enout2/Enout1 is ½, or smaller,
we can ignore Enout2's contribution within engineering
accuracy (error less than 12%). Remember, the noise
terms are the result of a Sum of Squares (followed by
a square root operation).
The Signal-to-Noise Ratio (SNR) is one of the most
common ways to decide if the noise in a circuit meets
its design requirements. Usually, it is defined as the
ratio of signal power (of a sine wave) to integrated
noise power in decibels:
EQUATION 33:
OUTPUT SNR
SNR = 20 ⋅ log 10 ( V OUT ⁄ E nout )
Where:
VOUT = Sinusoidal output signal (VRMS)
Enout = Integrated output noise voltage (VRMS)
SNR = Signal-to-Noise Ratio (dB)
In some applications, VOUT is expressed in relation to
its full scale range (VPK or VP-P). This will not be done
in this application note.
Select a SNR value that supports the required
accuracy for your design. Modify your circuit until it
meets this SNR requirement. For a fixed output
voltage, this is the same as minimizing the output
noise.
Note:
Make your signal’s full scale range as
large as possible; this minimizes the cost
and effort in reducing the output noise.
Reduce the Noise
When done properly, a design’s noise performance will
depend on only a couple of critical components. All of
the other components can be chosen for other design
goals.
PLACE THE GAIN UP FRONT
Place your high gain amplifier as close to the signal
source (e.g., a sensor) as possible. Any noise sources
after this gain will be divided by this gain; they should
have little impact on the output noise. Noise sources
before this gain, and in the high gain amplifier, will be
critical to your design’s success.
All of the amplifiers and other circuit elements after the
high gain amplifier should be set to a gain as close to
1 V/V as possible.
© 2008 Microchip Technology Inc.
TABLE 9:
EFFECT OF SECOND NOISE
SOURCE
Enout2/Enout1
Enout/Enout1
1/1
1.414
1/2
1.118
1/3
1.054
1/5
1.020
1/7
1.010
1/10
1.005
FILTER THE NOISE
Filter any noise with the lowest NPBW possible.
Place simple filters as close to dominant noise sources
as possible; this helps when testing your design on the
bench. A single real pole filter, using a resistor and a
capacitor, is usually enough for most purposes.
Place more sophisticated filters further away from the
source. This has the benefits of using one complex
filter for many noise sources. This reduces the overall
cost of active filter designs with low component
sensitivities (changes in capacitors, resistors and op
amp bandwidth have little impact).
Noise can alias into an ADC’s baseband. Select an
anti-aliasing filter with a BW much slower than the
sample rate (e.g., ten times slower) to minimize this
effect.
As we have seen before, a simple R-C lowpass filter at
the very end, without a buffer, can minimize the op
amps’ contribution to the total noise. This filter can be
placed at the input of an Analog-to-Digital Converter
(ADC) as long as the last capacitor is much larger than
the ADC's input sampling capacitor (to minimize gain
error).
DS01228A-page 21
AN1228
Component Selection
SUMMARY
There are a few simple rules that make it easy to select
components that will meet your design goals.
This application note gives a simple overview of the
noise theory used in circuit design. It is presented in a
way to help like the reader’s knowledge of statistics and
circuit design to circuit noise design.
RESISTORS
Resistors are usually chosen to be as small as possible
at critical points in the design. The exception to this rule
happens when the resistor acts like a current source in
a circuit (e.g., the gain resistor in a transimpedance
amplifier); the noise current is reduced by increasing
the resistance (see Equation 10).
Avoid resistors that use carbon as the resistive
material. They generate high levels of 1/f noise.
Use resistors with metal for the resistive material. Wire
wound resistors typically have the best 1/f noise, but
can cause high frequency circuit problems due to their
parasitic inductance and capacitance. Metal film
resistors have low 1/f noise and have good high
frequency characteristics.
OP AMPS
Start your design with a general purpose part. Look at
lower noise parts only after optimizing the rest of the
circuit.
For high frequency applications (e.g., above 1 kHz), or
applications that use auto-zeroed op amps, select the
op amp based on its white noise (eni and ini).
For low frequency applications (e.g., below 1 kHz), also
compare the 1/f noise performance (eni and ini).
Compare the integrated noise between 0.1 Hz and
10 Hz (the Noise Voltage spec (µVP-P) in Microchip’s
op amp data sheets). If that specification is not
available in a data sheet, the noise spectrum plot will
give the needed information. Compare the op amps’
noise density at the same frequency in the 1/f noise
region.
Many examples help build the reader’s knowledge of
the design process, how filters affect noise, how to
combine multiple noise terms at the circuit’s output,
and optimizing a circuit’s noise performance.
The topics cover what is needed in the majority of noise
designs. Both white and 1/f noise are discussed.
Manual analysis and computer simulations are used
many times. Computer aided analysis is mentioned as
a labor saving device.
After the body of this application note, there are
selected references to the literature to help the reader
find background material that covers this material well.
Appendices with additional vocabulary and an
overview of computer aids completes this application
note.
REFERENCES
Related Application Notes
[1]
AN1177, “Op Amp Precision Design: DC
Errors,” Kumen Blake; Microchip Technology
Inc., DS01177, 2008.
Noise
[2]
Paul R. Gray and Robert G. Meyer, “Analysis
and Design of Analog Integrated Circuits,” 2nd
Ed., John Wiley & Sons, 1984.
[3]
Jerry D. Gibson, “Principles of Digital and
Analog Communications,” Macmillan, 1989.
[4]
A. Bruce Carlson, “Communication Systems: An
Introduction to Signals and Noise in Electrical
Communication,” 3rd Ed., McGraw-Hill, 1986.
[5]
Athanasios Papoulis, “Probability, Random
Variables, and Stochastic Processes,” 3rd Ed.,
McGraw-Hill,1991.
Miscellaneous
[6]
DS01228A-page 22
Howard Johnson and Martin Graham, HighSpeed Digital Design: A Handbook of Black
Magic,” Prentice Hall, 1993.
© 2008 Microchip Technology Inc.
AN1228
APPENDIX A:
VOCABULARY
This appendix gives a brief list of common terms used
in amplifier noise work. They are organized by topic so
that their context is easier to grasp.
A.1
Spectral Densities
Power Spectral Density (PSD) is the frequency domain
description of a noise source’s statistical variation. Its
units are (W/Hz) (sometimes converted to dBm/Hz). It
is also related the noise’s auto correlation function. It is
also called Noise Power Density.
Noise Voltage Density (en) is the square root of PSD,
normalized to a standard resistance (usually 1Ω). It has
units of (V/√Hz). It is also called spot noise or noise per
root Hertz.
Noise Current Density (in) is the square root of PSD,
normalized to a standard resistance (usually 1Ω). It has
units of (A/√Hz). It is also called spot noise or noise per
root Hertz.
A.2
Spectral Shapes
White noise is a PSD that has a constant value over
frequency. It is a mathematical convenience used to
make system noise calculations simpler.
Broadband noise describes a noise source that is
(nearly) white over a circuit’s frequency range of
interest. It isn’t white, but appears to be white to that
circuit.
Noise Power Bandwidth (NPBW) is mathematically
convenient parameter used to describe how a circuit
processes white noise. It has units of (Hz). It is the
equivalent bandwidth of a brick wall filter that produces
the same output noise as the actual circuit.
Excess Noise is any noise that exceeds the white noise
level at low frequencies (only 1/f noise is discussed in
this application note):
• 1/f noise, also known as flicker noise or pink noise
• 1/f2 noise, also known as red noise
• Random Telegraph Signal (RTS) noise, also
known as burst noise or popcorn noise (has a
spectral shape reminiscent of white noise filtered
by a lowpass filter with a single real pole)
A.3
Integrated Noise
Noise Current (In) is the square root of Noise Power
normalized by standard resistance (usually 1Ω). It has
units of (ARMS, APK or AP-P). When in units of ARMS, it
is also called the standard deviation.
A.4
Probability Density Functions
Many physical noise sources, but not all, have the
Gaussian (or Normal) probability density function. They
are said to be Gaussian Noise, or sometimes Additive
White Gaussian Noise (AWGN). This noise is usually
associated with random processes that fulfill the
Identical and Independently Distributed (IID)
assumption; it is the sum of a large number of
statistically independent random variables with the
same probability density function. The probability
density function is:
EQUATION A-1:
2
1
–( x – μ )
p ( x ;μ, σ ) = -------------- ⋅ exp ⎛ ----------------------⎞
⎝
2σ ⎠
2 πσ
Analog to Digital Converters (ADC) and Digital to
Analog Converters (DAC) usually have their
quantization errors modeled as random noise with a
Uniform probability density function (the device noise at
the inputs would be Gaussian, however). The
probability density function is:
EQUATION A-2:
p ( x ;μ, σ ) = 1 ⁄ ( 2 3 σ ) , x – μ < 3 σ
= 0, otherwise
A.5
Figures of Merit
Signal-to-Noise Ratio (SNR) is the ratio of the signal
power to the noise power. It is usually shown in units of
(dB), although (VRMS/VRMS) or (ARMS/ARMS) are also
acceptable. Sometimes the signal’s full scale range is
the numerator of the ratio, with units of (VPK or VP-P).
Other figures of merit not covered in this application
note are:
• Noise Figure (NF) (dB)
• Noise Factor (F) (V/V)
• Noise Temperature (TN) (K)
Noise Power (N) is the noise source’s statistical
variation. Its units are (W) (sometimes dBm).
Noise Voltage (En) is the square root of Noise Power
normalized by standard resistance (usually 1Ω). It has
units of (VRMS, VPK or VP-P). When in units of VRMS, it
is also called the standard deviation.
© 2008 Microchip Technology Inc.
DS01228A-page 23
AN1228
APPENDIX B:
COMPUTER AIDS
While this application note emphasizes manual
calculation and formulas, most design work uses
computers.
B.1
Noise Simulations
Circuit noise simulations can be done as part of an AC
simulation in SPICE simulators. The SPICE program
was developed at UC Berkeley. Many SPICE derivative
simulators are used in circuit design; the most popular
for board level design is PSpice® (from Cadence®).
B.1.1
GENERAL REMARKS
The component models need to be correctly defined for
noise simulations to give realistic results. Op amp
macro models from Microchip are set up to work
properly in PSpice. Resistors, diodes and transistors
usually give the correct white noise (when the model is
accurate). 1/f noise in diodes and transistors will not
simulate correctly without special attention to the
relevant parameters. The resistor model does not
include 1/f noise; this can be added to the circuit using
diodes and a dependent source, if needed.
You will need to define the input source and output
circuit node before the noise analysis can run. SPICE
produces an input referred noise vector (across
frequency); it is referred to the chosen input source.
SPICE also produces an output noise vector at the
chosen node.
The noise results in different SPICE simulators come in
two different forms: as noise voltage (current) density
(VRMS/√Hz) or as the square of the noise voltage
(current) density (VRMS2/Hz). Check your simulator
using resistor thermal noise; 1 kΩ of resistance, at
+25°C, will give 4×10-9 VRMS/√Hz in the former case,
and 1.6×10-17 VRMS2/Hz in the latter case.
Note:
PSpice produces noise in units of (VRMS/√Hz).
Other simulators may, or may not.
Use the simulator’s plotting tool to determine which
noise sources dominate and to improve the noise
filtering and shaping.
To obtain the integrated noise (Enout) between fL and fH
(see Figure 3), read En0f’s value at f = fL and f = fH (let’s
call these values EL and EH). The integrated noise is
(for any spectral shape):
EQUATION B-1:
E nout =
2
2
EH – EL
To put this trace’s data into a spreadsheet, click on its
label found at the bottom left of Probe’s screen. This
selects this trace (the label changes color). Copy the
data to Window’s clipboard by typing the key sequence
Ctrl-C. Paste the results into your spreadsheet.
Following this sequence produced two columns of
data; one frequency vector and one noise vector.
B.1.3
ESTIMATING NPBW WITH SPICE
Now that we can extract the integrated noise from our
simulations, we can easily estimate a filter’s NPBW.
The following steps will make this more clear:
• Use a very large resistor (or 2 in parallel) as the
noise voltage source
• Insert a buffer between the noise source and the
filter’s input
• Plot the output noise density (enout)
• Calculate the integrated output noise (Enout) from
DC to infinity (a high enough frequency)
• Choose the enout value that represents the passband (at the chosen gain, HM)
• Calculate the NPBW
EQUATION B-2:
NPBW ESTIMATE
NPBW = ( E nout ⁄ e nout )
B.2
2
Using Symbolic Solver Engines
There are several places where a symbolic solver can
speed up your noise analysis:
To calculate integrated noise in PSpice, open its plotting utility (Probe) and add the following trace:
• Converting node equations to transfer functions
• Factoring transfer functions
• Expanding a magnitude squared transfer function
into its Partial Fraction Expansion form
• Evaluating the definite integrals used for
integrated noise (or NPBW)
EXAMPLE B-1:
Some popular tools are:
B.1.2
CALCULATING INTEGRATED
NOISE WITH PSPICE
PSPICE TRACE
FUNCTION
sqrt(s(v(onoise)*v(onoise)))
This is the running integral (from 0 Hz to f, using the
PSpice function s()) of the output noise (let’s call it
En0f) in units of (VRMS/√Hz).
DS01228A-page 24
• Mathematica® (from Wolfram Research)
• Maple™ (from Waterloo Maple Software)
• Matlab® (from The MathWorks); use the Symbolic
Math Toolbox™
• MathCad® (from Parametric Technology
Corporation)
© 2008 Microchip Technology Inc.
Note the following details of the code protection feature on Microchip devices:
•
Microchip products meet the specification contained in their particular Microchip Data Sheet.
•
Microchip believes that its family of products is one of the most secure families of its kind on the market today, when used in the
intended manner and under normal conditions.
•
There are dishonest and possibly illegal methods used to breach the code protection feature. All of these methods, to our
knowledge, require using the Microchip products in a manner outside the operating specifications contained in Microchip’s Data
Sheets. Most likely, the person doing so is engaged in theft of intellectual property.
•
Microchip is willing to work with the customer who is concerned about the integrity of their code.
•
Neither Microchip nor any other semiconductor manufacturer can guarantee the security of their code. Code protection does not
mean that we are guaranteeing the product as “unbreakable.”
Code protection is constantly evolving. We at Microchip are committed to continuously improving the code protection features of our
products. Attempts to break Microchip’s code protection feature may be a violation of the Digital Millennium Copyright Act. If such acts
allow unauthorized access to your software or other copyrighted work, you may have a right to sue for relief under that Act.
Information contained in this publication regarding device
applications and the like is provided only for your convenience
and may be superseded by updates. It is your responsibility to
ensure that your application meets with your specifications.
MICROCHIP MAKES NO REPRESENTATIONS OR
WARRANTIES OF ANY KIND WHETHER EXPRESS OR
IMPLIED, WRITTEN OR ORAL, STATUTORY OR
OTHERWISE, RELATED TO THE INFORMATION,
INCLUDING BUT NOT LIMITED TO ITS CONDITION,
QUALITY, PERFORMANCE, MERCHANTABILITY OR
FITNESS FOR PURPOSE. Microchip disclaims all liability
arising from this information and its use. Use of Microchip
devices in life support and/or safety applications is entirely at
the buyer’s risk, and the buyer agrees to defend, indemnify and
hold harmless Microchip from any and all damages, claims,
suits, or expenses resulting from such use. No licenses are
conveyed, implicitly or otherwise, under any Microchip
intellectual property rights.
Trademarks
The Microchip name and logo, the Microchip logo, Accuron,
dsPIC, KEELOQ, KEELOQ logo, MPLAB, PIC, PICmicro,
PICSTART, rfPIC and SmartShunt are registered trademarks
of Microchip Technology Incorporated in the U.S.A. and other
countries.
FilterLab, Linear Active Thermistor, MXDEV, MXLAB,
SEEVAL, SmartSensor and The Embedded Control Solutions
Company are registered trademarks of Microchip Technology
Incorporated in the U.S.A.
Analog-for-the-Digital Age, Application Maestro, CodeGuard,
dsPICDEM, dsPICDEM.net, dsPICworks, dsSPEAK, ECAN,
ECONOMONITOR, FanSense, In-Circuit Serial
Programming, ICSP, ICEPIC, Mindi, MiWi, MPASM, MPLAB
Certified logo, MPLIB, MPLINK, mTouch, PICkit, PICDEM,
PICDEM.net, PICtail, PIC32 logo, PowerCal, PowerInfo,
PowerMate, PowerTool, REAL ICE, rfLAB, Select Mode, Total
Endurance, UNI/O, WiperLock and ZENA are trademarks of
Microchip Technology Incorporated in the U.S.A. and other
countries.
SQTP is a service mark of Microchip Technology Incorporated
in the U.S.A.
All other trademarks mentioned herein are property of their
respective companies.
© 2008, Microchip Technology Incorporated, Printed in the
U.S.A., All Rights Reserved.
Printed on recycled paper.
Microchip received ISO/TS-16949:2002 certification for its worldwide
headquarters, design and wafer fabrication facilities in Chandler and
Tempe, Arizona; Gresham, Oregon and design centers in California
and India. The Company’s quality system processes and procedures
are for its PIC® MCUs and dsPIC® DSCs, KEELOQ® code hopping
devices, Serial EEPROMs, microperipherals, nonvolatile memory and
analog products. In addition, Microchip’s quality system for the design
and manufacture of development systems is ISO 9001:2000 certified.
© 2008 Microchip Technology Inc.
DS01228A-page 25
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01/02/08
DS01228A-page 26
© 2008 Microchip Technology Inc.
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