ETC AB-103

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by Michael Steffes
As system bandwidths have increased, an accurate estimate
of the noise contribution for each element in the signal
channel has become increasingly important. Many designers are not, however, particularly comfortable with the
calculations required to predict the total noise for an op
amp, or in the conversions between the different descriptions of noise. Considerable inconsistency between manufacturers in describing noise and, in some cases, incomplete
specifications, have contributed to this confusion. A thorough description of the op amp noise model will be developed here with a detailed discussion of the key differences
between current and voltage feedback amplifiers. The conversions between several different measures for noise used
in the industry will also be described. Broadband effects will
be covered for both low frequencies (the 1/f region) and high
frequencies (noise power bandwidth).
noise power density at low frequencies. This is often called
the 1/f region since the noise power density will often
increase as the inverse of frequency. “Popcorn” noise is a
random long term shift in voltage or current (that sounds like
popcorn if fed into an audio speaker). This phenomena
doesn’t fit well into a frequency domain description and is
best observed over time.
To analyze noise in the frequency domain, equivalent noise
voltage or current generators are introduced into the circuit
that represent the noise density over frequency for that
element. These voltages or currents are the square root of the
noise power densities. We work with voltages and current in
order to avail ourselves of standard circuit analysis techniques. One key caveat to using noise voltages and currents
is that they do not add algebraically. Each individual noise
source has a random phase to any other (with the exception
of correlated sources). This means that, although we can use
superposition to get the contribution of each noise term to a
particular point in the circuit, the voltages or currents themselves cannot simply be added at that point. It is the powers
that are added to get to a total noise power.
Random electrical noise (either a current or a voltage) is
present in almost every type of component used in a circuit.
This noise may be considered as either a frequency domain
phenomena or something that occurs over time. The most
common models approach noise from the frequency domain
first then convert to the time domain using the shape of the
noise density curve combined with a noise power bandwidth
analysis. This is the approach that will be used here.
Figure 1 shows the analysis circuit that will act as a starting
point for the subsequent noise analysis. This schematic
includes the three equivalent input noise terms for the op
amp and the three resistor noise terms that must always be
considered for a complete op amp noise analysis. Any
particular op amp application circuit can typically be reduced to that of Figure 1 by shorting any input voltage
sources and/or opening any input current sources that might
be driving the circuit and reducing the remaining impedances to the three elements shown in Figure 1. Reactive
elements (capacitors, inductors, transformers) are normally
considered to be noiseless. They can, however, strongly
influence the frequency response for the noise generators in
a circuit. Examples of this will be shown later. At this point
consider the elements around the op amp to be purely
resistive. Remember that the Johnson noise of the resistor
may be represented as either a current or a voltage. The
analysis circuit of Figure 1 uses both forms (current noise for
RG and voltage noise for RF) to simplify later computations.
Another useful view of op amp noise is to consider the input
voltage and current noises to be the time varying component
of the input offset voltage and bias currents respectively.
Working from the frequency domain to the time domain, as
will be done here, develops the required tools to predict the
amplitude of this time varying component. Only the random
electrical noise generated by the components themselves
will be considered. Other sources of “noise” that will not be
considered here, (but are nevertheless of interest to the
system designer), include conducted noise through the power
supplies that appear at the output due to finite PSRR, various
sources of radiated emission pickup (EMI), micro-phonic
effects due to system vibration, and high narrowband noise
that is in fact a parasitic oscillation.
The starting point for a frequency domain analysis of noise
is the noise density. This is the noise power normalized to
1Hz bandwidth (at a particular center frequency) and is
sometimes called the “spot” noise. “White” noise has a flat
(or constant) noise power over frequency. Most amplifiers
and resistors show a flat noise region that extends over many
frequency decades. Most, however, also show an increasing
1996 Burr-Brown Corporation
Most of the discrepancies between different noise analysis
come from neglecting as insignificant some of the noise
sources in Figure 1. In general, there will in fact be a
dominant noise source for a particular op amp in a particular
application circuit. However, to maintain the required genAB-103
Printed in U.S.A. October, 1996
Input Voltage Noise (nV/√Hz)
Input Current Noise (fA/√Hz)
Voltage Noise
Current Noise
4kT = 1.6E − 20J •
, T is the Temperature in Kelvin
290o K
IBN = Op Amp Non-Inverting Input Noise Current
ERS = Source Resistor Noise Voltage =
IRG = Gain Setting Resistor Noise Current =
FIGURE 1. Op Amp Noise Analysis Circuit.
erality to handle any op amp in any configuration, all terms
will initially be considered, then dropped later where appropriate.
The noise voltage and current sources shown in Figure 1 are
treated as spot sources that may have a frequency response
of their own. In particular, the op amp input voltage noise,
ENI, will typically increase at low frequencies due to 1/f
noise effects. The two op amp input current noise terms will
also show increasing noise at low frequencies for bipolar
input stages. FET input stages have a very low and constant
spot input noise currents going to low frequencies, but show
Voltage Noise (nV/√Hz)
Current Noise (pA/√Hz)
Normally, the resistor noise terms are considered to have a
constant noise voltage (or current) density over frequency.
Along with this Johnson noise, there is also a low frequency
increase in resistor noise (called “excess” noise) that is
dependent on the DC voltage across the resistor. Carbon
composition resistors show the highest excess noise with
metal film showing a negligible increase in noise at low
frequencies (Ref.1, page 171). Wirewound resistor show the
lowest excess noise but should never be used in the signal
path for high frequency op amps due to their inductive
impedance. For this discussion, resistor noise density will be
considered flat over frequency (white noise).
Inverting Current Noise
As a starting point, calculate the total output spot noise
voltage (EO) of Figure 1. Each term at the output is simply
the value of each source at a particular frequency taken to
the output by its gain at that frequency. For now, treat all
sources as white noise and neglect any frequency response
effects for the gain terms in Figure 1. If all of the noise
voltage and current sources are uncorrelated (the most
common assumption), their powers will add algebraically at
the output. This means that the total output spot noise power
is the sum of the noise power contributed by each term. Take
Non-Inverting Current Noise
Voltage Noise
an increasing noise at higher frequencies. Figure 2 shows an
example input noise density plot for a high speed bipolar
input op amp while Figure 3 shows an example for a high
speed FET input op amp. Note that Figure 2 shows separate
traces for the two input current noise terms while Figure 3
shows a single trace implying that the two are equal (which
is normally true for a voltage feedback op amp). One key
difference between high speed current feedback op amps
(such as the OPA658) and voltage feedback op amps (such
as the OPA655) are that the two inputs for the current
feedback topology show unequal bias and noise currents.
These three input noise terms are intended to model all of
the actual internal noise sources of the device. By using an
input referred model, the total output noise under any
configuration may be calculated.
ERF = Feedback Resistor Noise Voltage =
FIGURE 3. Example Input Noise Density Plot for a FET
Input Op Amp (OPA655).
IBI = Op Amp Inverting Noise Current
Frequency (Hz)
ENI = Op Amp Input Noise Voltage
Frequency (Hz)
FIGURE 2. Example Input Noise Density Plot for a Bipolar
Input Op Amp (OPA658).
Looking at Equation 2, once ENI, IB, and GN are set, the
output noise can always be reduced by decreasing the value
for RF (and then RG and RS to get the desired gain and source
matching). That approach is limited by the increased loading
presented by the feedback network, and, in the case of a
current feedback op amp, by stability considerations. The
value for RF controls the compensation for a current feedback op amp. Reducing it too much can cause excessive
frequency response peaking and possibly oscillations.
the square root of this power to get the total output spot noise
Compute the total output noise assuming:
(1+RF/RG) ≡ GN = Noise gain (identically equal to the op
amp non-inverting signal gain)
First find the gain to the output for each voltage or current
noise term by superposition:
Noise Term
Equation 1b gives us an expression for a noise that can be
physically measured. Connecting this output into a spectrum
analyzer and converting the measured power at a particular
frequency (and resolution bandwidth) into spot noise voltage will give the value predicted by Equation 1b. Often,
designers would prefer to compare a total equivalent input
spot noise voltage to their input signal. This input noise is
not the same thing as the input spot voltage noise of the op
amp itself. It is instead an abstraction derived by “input
referring” the total output spot noise expression (Equation
1b). To “input refer” the total spot output noise, divide it by
the gain from the desired signal input point to the output.
The “input” can in fact be anywhere in the system. It could
be to an inverting input signal—or, to the input of a prior
stage (where it would be combined with the input referred
noise of that stage to see how much is being added by this
2nd stage). Most often, the input referred spot noise voltage
is taken to be the total input referred noise at the noninverting input, EN. Dividing Equation 1b by GN will “input
refer” to the non-inverting input.
Equation 3a:
Now, substitute in for the resistor noises (Figure 1), multiply
each noise term by its gain, square each term (to get to a
power), sum them all, and take the square root of that sum
to develop the total output spot noise expression.
Equation 1a:
EO =
( E NIG N )2 + ( I BN RSG N )2 + 4kTRSG N 2 + ( I BI R F )2 + 4kTR F + 4kT
Combining terms at the non-inverting input and recognizing
4kT 2
4kTR F +
R F = 4kTR F • G N
will give the total output spot noise voltage for any op amp
at a selected frequency.
Equation 1b:
EO =
+ ( I BN R S ) + 4kTR S G N 2 + ( I BI R F ) + 4kTR F G N
To consider the frequency response effects for any of the
gain terms in Equation 1b, substitute in the magnitude of the
gain (for RS, GN, and/or RF) at the frequency of interest.
Parasitic, or intentional, capacitances across the resistors can
strongly influence the spot noise gain for the terms of
Equation 1b. One common trick in an inverting op amp
configuration is to source match with RS set to get DC bias
current error cancellation (by setting RS = RF II RG). To limit
the noise added by this DC matching resistor, add a large
capacitor across RS to filter the 4kTRS•GN and (IBN•RS•GN)^2
terms in the output spot noise expression.
Several simplifications of Equation 1b are possible. For
example, RS is often set to = RF II RG to get bias current
cancellation when using a voltage feedback op amp. This is
seldom done for current feedback op amps since their input
bias currents do not match. Assuming the two input noise
currents are equal in the voltage feedback case
(IBN = IBI = IB), and making the substitution for RS will
reduce Equation 1b to:
E N = E NI + ( I BN R S )
Equation 3b:
( (
E N = E NI 2 + ( I BN R S ) + 4kTR S + I BI R F R G
+ 4kT R F R G
Imposing the source matching condition (RS = RF II RG) and
assuming that IBN = IBI on Equation 3b will allow a further
simplification to—
Equation 4:
( (
E N = E NI 2 + 2 I B R F R G
E O = E NI G N + 2( I BR F ) + 2( 4kTR F G N )
I R 
4kTR F
+ 4kTR S +  BI F  +
 GN 
Equation 3a explicitly shows an RF in the last two terms.
Since RF is relatively fixed for a current feedback op amp,
this shows clearly that the input referred contribution of the
noise terms that are physically on the inverting side of the
circuit will decrease with increasing gain. An equivalent
expression, that is more common for voltage feedback amplifiers, substitutes RF/GN = RF II RG to get:
Equation 2:
+ 2( 4kT ) R F R G
response that each term has separately to the output may
be neglected and the frequency response of this last filter
will be the limiting factor.
Most frequency domain oriented applications deal only in
powers. Either the input or output spot noise expressions
may be converted to dBm. Equation 5 shows this for the
output spot noise voltage.
Most designers consider a frequency response from a
gain and phase standpoint for the V/V transfer function.
For noise, it is really the power gain that is important
since we can only add noise powers directly. This means
that whatever V/V transfer function we have, it must be
squared prior to noise integration. There are a couple of
simple conversions between standard low pass transfer
functions and their equivalent NPB. For a first order low
pass filter, the –3dB bandwidth (F–3dB) is related to the
noise power bandwidth by:
Equation 6:
Equation 5:
E O2
P N = 10 • log 
 = 13 + 20 log( E O ) dBm
 50Ω( 0.001) 
Again, this is the noise power in a 1Hz bandwidth across
50Ωs. A good point of reference here is that a 50Ω resistor
at 290ºK will have a spot noise power of –168dBm.
• F −3dB Hz (Single Pole Low−Pass)
For a 2nd order low pass, the NPB may also be related to the
V/V transfer function (in WO and Q terms) by:
Equation 7:
The spot noise over frequency is a useful means of comparing devices and interpreting spectrum analyzer measurements. It is also the most fundamentally useful starting point
for noise analysis. However, very few systems actually have
a signal that is defined in 1Hz bandwidths and a conversion
from spot noise to the noise over some bandwidth is required. Computing the “integrated noise” is that process of
summing all of the noise power over the frequency band of
interest. Again, since noise really only adds algebraically as
a power, the integration is first done using noise voltages
squared, then converted back to voltage by taking the square
root. Considering first the output integrated noise, there are
really two frequency dependent parts to this calculation.
First, the frequency response of the input spot noise term
itself and then the frequency response of the gain for that
term to the output. Even assuming these are known, there are
a couple of interesting nuances here:
1. Starting at Frequency = 0 will not work. First of all, 1/f
models go to infinite noise density at F = 0 and secondly,
at zero frequency—we must be talking about the beginning of time—so at T = 0 we have infinite noise (big bang
• F O • QHz (2nd Order Low−Pass)
1 
1 
F −3dB = F O •  1 −
 + 1 Hz
2  + 1 −
2Q 2 
FO =
Both of these are calculating the equivalent brick wall
bandwidth that will integrate the same “power” as the
original V/V transfer function. WO and Q may be estimated
in several ways from the measured 2nd order frequency
response. As one example, a maximally flat Butterworth 2nd
order response has a Q = 0.707 and an FO = F–3dB yielding
a NPB = 1.11•F–3dB. The results of Equation 7 show that the
equivalent NPB increases linearly with Q. This is saying that
an increasingly peaked frequency response has the same
effect as a broader NPB. Physically, for a flat input spot
noise, this peaked response is giving more noise gain (in the
peaked region). This can be equivalently accounted for by
increasing the NPB. All this effort to get a single number for
NPB allows a simple calculation for integrated noise. If the
spot output noise is considered flat, the integrated noise is
2. What high frequency limit should be used? Often the
frequency response of the amplifier itself is used (or more
precisely, the frequency response for each term in Equation 1b). This can be pretty painful (from an analysis
standpoint) if the separate frequency response shapes for
each noise term are meticulously applied. Most actual
systems that care about noise impose a bandlimit that is
set just prior to the detection stage. It is this bandlimit,
either an IF filter for frequency domain applications or a
passive low pass for time domain, that sets the system
Noise Power Bandwidth (NPB).
Equation 8:
This is converting from a frequency domain noise description to a time domain. Multiplying the square root of the
measurement channel’s noise power bandwidth times the
spot noise (assumed flat) will give the RMS noise voltage in
that bandwidth. One final conversion that is often of interest
is to take this RMS noise voltage to a peak voltage number.
The most common “crest” factor to convert RMS to VPP is
The NPB is often set so that the output spot noise is flat
over that frequency (i.e., none of the individual noise
sources are allowed to self limit). In an IF chain, the last
IF or baseband filter is much lower bandwidth than any
of the preceding stages. In a pulse domain application,
this last low pass filter is lower bandwidth than the
preceding amplifier stages. In these cases, the frequency
that for a sinusoid (e.g. 2√2 = 2.8). Noise is distinctly not
sinusoidal. Using 6•Vrms will give a VPP limit rarely exceeded.
F 
F 3dB
ln 2 
F 2 − F1  F1 
Considering noise power bandwidth in a frequency domain
application is slightly different. Most IF filters are multiple
pole so that their NPB approach their F–3dB bandwidths. To
convert to integrated noise power, simple add 10•log (NPB)
to the spot noise power of Equation 5 where the NPB is
usually the same as the F–3dB bandwidth.
Now consider the frequency response for the input noise
terms themselves. If the bandwidth of interest for the system
includes the low frequency 1/f noise region, a significant
increase in integrated noise is possible due to this increased
spot noise at lower frequencies. Equation 9 shows the spot
noise equation for 1/f noise.
As Table 1 shows, when F2 (= NPB) far exceeds the 1/f
noise corner frequency, the effects of this increasing noise at
low frequency may be safely neglected. When this is not the
case, each of the three input noise terms for the op amp
should be recomputed using Equation 10c to include the
integrated effects of low frequency noise. These results are
then placed into the total output spot noise expression of
Equation 1b. This result may then be multiplied by the
√NPB to arrive at an integrated noise voltage that correctly
includes the low frequency effects.
Equation 9:
ET = EN • 1 +
, spot noise voltage over frequency
where :
E N = Flatband Spot Noise Voltage
Noise figure is a common description of noise effects for RF
and IF amplifiers. It is defined as 10•log (Signal/Noise at the
input divided by the Signal/Noise at the output). All terms
are spot powers (noise and signal). The input noise power is
defined as the noise power delivered from some source
impedance to the input—often a matching impedance termination. This is somewhat complicated for an op amp since
the input termination is set by the user where it is often fixed
at a matching value for an RF or IF amplifier. Figure 4
shows a non-inverting op amp with an arbitrary input termination (RT) showing the noise sources and definition points
for noise figure analysis
F3dB = Frequency Where Total Spot Noise Power Has Doubled
F = Frequency
Squaring Equation 9 and taking the integral from some
arbitrarily low frequency up to the NPB will give the total
power contributed over that band. Dividing by the frequency
band of integration and taking the square root will give us an
equivalent white noise that will integrate to the same power
as the actual source described by Equation 9. The integral is
set up in Equation 10a, solved in 10b, and taken back to an
equivalent spot input voltage noise in Equation 10c.
Equation 10a:
E EQ 2 =
F 3dB 
∫ 2 E N 1 +
F 
F 2 − F1 F1
F 2 → NPB
Equation 10b:
 F2  
EN2 
F 
ln 2 
( F 2 − F1 ) + F3dB ln   = E N 1+
F 2 − F1 
 F1  
 F 2 − F1 F1 
F1 → Lower Frequency Limit
E EQ 2 =
Equation 10c:
E EQ = E N 1 +
ln 2 , equivalent white noise voltage
F2 − F1 F1
FIGURE 4. Non-Inverting Op Amp Noise Figure Analysis
As the maximum frequency of interest far exceeds the 1/f
noise corner (F3dB), Equation 10c approaches EN.
To illustrate this, let F1 be an arbitrarily low 10Hz (remember we can’t use F1 = 0 here) with F3dB = 10kHz and sweep
F2 to yield the following result for the radical in Equation
The noise figure for an op amp operated in the inverting
mode is considerably more complicated. Figure 5 shows the
noise figure analysis circuit for the inverting op amp topology. This circuit includes a resistor to ground on the noninverting input, RT, which should be set to a relatively low
value for the lowest noise. It also includes a matching
resistor to ground at the input to allow the input impedance
to be set separately from the signal gain. The input impedance looking into this inverting op amp configuration is the
parallel combination of RG II RM.
Letting RT remain an arbitrary input termination yields a
very general noise figure expression for the non-inverting
op amp configuration (NF+) shown in Equation 11 (where
Equation 11:
 R
R  E 
4kTR F  
2  I R 
NF + = 10 log 1 + S + S  NI  + ( I BN ) +  BI F  +
GNRP2 
 R T 4kT  R P 
 R PG N 
Two special cases for RT greatly simplify Equation 11.
Letting RT = RS to get an input impedance match will
produce Equation 12.
Equation 12:
Since the input impedance cannot be infinite, the main
application for this circuit would be providing an input
impedance matched to RS. If RM is constrained (given an RG
and RS) to give an input impedance matched to RS, an
inverting op amp noise figure expression (NF–) can be
derived as shown in Equation 14.
Equation 14:
4kTR F 
R 2 I R 
E NI 2 +  I BN S  +  BI F  +
2 
GN 
 GN 
NF = 10 log  2 +
with R S = R T
I R 
4kTR F 
E NI 2 + ( I BN R T ) + 4kTR T A 2T +  BI F  +
 GI 
  RG
( G I )2 
NF = 10 log 2  2
− 1 +
  RS
Letting RT = infinity will not provide an input match, but
will provide a lower noise figure as shown by Equation 13.
Equation 13:
4kTR F 
2 I R 
 E NI 2 + ( I BN R S ) +  BI F  +
GN 
 GN 
NF + = 10 log 1 +
4kTR S
with R T = ∞
Generally speaking, anything that will reduce the input
referred voltage noise of the op amp (Equation 3a) will
decrease the noise figure (an ideal noise figure = 0dB when
the output SNR = input SNR). Changing RS will trade off
the effects of source resistor noise and op amp produced
noise. It is sometimes suggested to operate at an optimum
source resistance where the noise figure is minimized. This
is effective when using reactive impedance transformations
(such as a transformer) but is not really helping the noise if
the increased source resistance is a real (noisy) resistor
where A T ≡
R 
1 + GI 1 − S 
2R G 
= non−inverting gain
Since RM has been constrained by the values for RG and RS,
it does not appear in the noise figure expression. Using the
same amplifier with a fixed RF in either a non-inverting or
inverting configuration will typically yield a NF– > NF+ at
low gains with the inverting noise figure dropping below the
non-inverting noise figure at higher gains. The non-inverting
noise figure vs. gain asymptotically approaches a limit set by
the noise terms that are physically on the non-inverting
input. Those same noise terms are actually attenuated at
higher inverting gains when input referred to the inverting
input. Taking a simple example: let RS = RG = 50Ω with
RF = 200Ω (and RM = infinity). Looking just at the op-amp’s
non-inverting input noise voltage, it will have a gain to the
output that is (1 + 400/100) = 5. The magnitude of the
ENI, IBN, IBI → Op Amp Noise Sources
and ERX → Resistor Voltage Noise Sources
and GI = RF/RG Inverting Gain
FIGURE 5. Inverting Op Amp Noise Figure Analysis Circuit.
To input refer this voltage noise, simple divide by the
voltage gain to get 23nV/6.32 = 3.64nV/√Hz. To convert
this spot input noise voltage to a spot noise power, use
Equation 5 to get –156dBm. To get a noise floor for signal
detection at the input, add 10•log ((50-5)MHz) to this to get
–79dBm. Normally, to compute a minimum detectable signal, RF engineers add 3dB to this number. Notice that it was
the system defining noise power bandwidth that is physically after this stage that is used for this calculation. The last
common measure of noise used in an IF application would
be the Noise Figure. Substituting into Equation 12 we can
compute that:
Equation 16:
inverting gain from the input reference point of Figure 3 is
400/50 = 8. Input referring the non-inverting input noise
voltage to the inverting input yields a gain of 5/8 = 0.625 in
this case. This effect gives a lower achievable noise figure in
the inverting configuration than in the non-inverting for a
matched input impedance case at higher gains.
To show how the analysis described here can be applied,
let’s consider two widely different applications using the
OPA658 and OPA655 noise characteristics shown in Figure
2 and 3 respectively. First, apply the wideband current
feedback OPA658 to an IF amplifier application where I/O
impedances are matched to 50Ω and a gain of 10dB to the
load over a frequency band from 5 to 50MHz is desired. A
non-inverting circuit that achieves this is shown in Figure 6.
5 to 50MHz
(3.64nV )2 
= 18.3dB
NF + = 10 log 2 +
(4E − 21) • 50 
Notice that the noise term used in this calculation is simply
the total input referred voltage noise calculated earlier using
a source impedance equal to RS/2. How would this noise
figure change for the inverting configuration? Holding
RG = 50Ω, and increasing RF to 316Ω to get the same gain
magnitude, and holding RT = 25Ω on the non-inverting input
will yield an inverting noise figure equal to (using Equation
Equation 17:
32pA•316Ω  16E −21•316Ω 
(3.2nV)2 + (12.6pA•25Ω)2 +16E −21•25Ω (.66)2 + 
6.32 
(6.32)2 
NF− =10log 2(2−1) +
(4E −21)50Ω
FIGURE 6. IF Amplifier Example Circuit.
which is slightly less than the non-inverting configuration
noise figure.
Since we are only interested in a frequency band well above
the 1/f corner frequencies for any of the noise terms shown
in Figure 2, those low frequency effects may be neglected.
The OPA658 is operating at a voltage gain of 6.32 to get the
desired 10dB gain to the matched load. At this gain (and
with the feedback resistor reduced from the recommended
402Ω to 266Ω) the OPA658 will have over 200MHz bandwidth. The IF filter will therefore set the noise power
bandwidth independently of the amplifiers frequency response.
The first step will be to compute the total output spot noise
using Equation 1b. Note that the source resistor should be set
to 25Ω for this calculation. Using the flatband numbers from
Figure 2, Equation 15 shows this calculation of total output
spot noise voltage (at the output pin of the amplifier).
Now consider a DC coupled time domain example using the
OPA655 FET input op amp. In this case, a very good pulse
response is desired with minimal integrated noise over the
bandwidth necessary to achieve the pulse settling times. Set
the OPA655 up for a gain of +5 (using an RF = 10kΩ) and
follow it by a 5MHz cutoff 2nd order Butterworth filter.
Figure 7 shows this application.
Equation 15:
EO =
[(3.2nV) + (12.6pA • 25Ω) + 16E − 21• 25Ω](6.32) + (32pA • 266Ω) + 16E − 21• 266Ω • 6.32
= 23nV/ Hz
The resistor values were chosen here to add an almost equal
noise power at the output as the op amp’s input noise
voltage. The total integrated noise at the output of the filter
is then:
2nd Order
Butterworth Filter
RS = 0
Equation 20:
Integrated E O = 40nV/ Hz
5.55MHz = 94µV rms
And finally, the peak-peak noise excursion will almost
always be less than 6•94µV = 0.56mV due to the noise
added by this stage to the original source.
Once the complete output spot noise equation for an op amp
is developed (Equation 1b), all other descriptions or simplifications may be derived. This equation is also important in
that it describes what is actually being measured in any
noise measurement. Other numbers that may be reported are
simply computations from this measurement. A pedantic
output noise computation would also include the individual
frequency responses to the output for each noise source. In
many cases, these can be ignored if the stage in question is
followed by a bandlimiting filter. Low frequency 1/f effects
can be handled by computing an equivalent white noise
source that will integrate to the same power over the band
of interest (Equation 10c). This increasing low frequency
noise makes a negligible contribution to integrated noise
when the high frequency bandlimit is 100X the 1/f noise
corner frequency. The noise figure for an op amp may be
predicted using the equations developed here (Equation 11
to 14). Resistively source matching for optimum noise
figure will lose the signal to noise ratio battle. Shifting the
source impedance reactively can, however, effectively improve the noise figure. The inverting op amp configuration
can provide a lower noise figure at higher gains. In general,
start a noise analysis including every term, then drop those
that are clearly contributing negligible noise in the application.
FIGURE 7. Low Frequency Pulse Amplifier.
This voltage feedback op amp has a gain bandwidth product
of 240MHz. At a non-inverting signal gain of +5, we will
achieve > 48MHz single pole bandwidth to the output of the
op amp. The system bandwidth will be set by the low pass
Butterworth filter following this stage. First calculate the
noise power bandwidth for this filter. Using Equation 7 (and
a Q = 0.707 and FO = F–3dB = 5MHz):
NPB = 1.11*5MHz = 5.55MHz
Now include 1/f effect by computing the equivalent flatband
noise for the non-inverting input noise voltage density
shown in Figure 3. Using Equation 10c with F3dB = 2kHz,
F1 = 10Hz, and F2 = 5.55MHz, and EN = 5.6nV/√Hz yields
an equivalent constant input voltage noise :
Equation 18:
E EQ = 5.6nV 1 +
= 5.613nV
5.55MHz − 10Hz
Even at this relatively limited frequency band, the contribution of the noise in the 1/f region is negligible. Figure 3
shows the bias current noise increasing with frequency.
Multiplying the 10MHz value of 0.1pA times the feedback
resistor will still only yield a 1nV contribution at the output
well out of band. As is normally the case, the current noise
contribution to the total noise may be neglected for this FET
input op amp. Equation 19 computes the total output spot
noise considering only the voltage and resistor noise terms.
1. Low Noise Electronic Design; Motchenbacher & Fitchen,
Wiley 1973.
2. National Semiconductor Application Note AN104-1.
Equation 19:
EO =
(5.6nV • 5)2 + 16E − 21•10kΩ • 5 = 40nV/ Hz
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