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AN-539
APPLICATION NOTE
One Technology Way • P.O. Box 9106 • Norwood, MA 02062-9106 • 781/329-4700 • World Wide Web Site: http://www.analog.com
Errors and Error Budget Analysis in Instrumentation Amplifier Applications
by Eamon Nash
+VS
350V
VNOISE
350V
INOISE+
350V
INOISE–
IBIAS
VOSIN
IOFS
VOSOUT
RGAIN
VOUT
REFERENCE
IBIAS
VCM = +VS/2
Figure 1. Error Sources in a Typical Instrumentation Amplifier
INTRODUCTION
This application note describes a systematic approach
to calculating the overall error in an instrumentation
amplifier (in amp) application. We will begin by describing the primary sources of error (e.g., offset voltage,
CMRR, etc.) in an in amp. Then, using data sheet specifications and practical examples, we will compare the
accuracy of various in amp solutions (e.g., discrete vs.
integrated, three op amp integrated vs. two op amp
integrated).
Because instrumentation amplifiers are most often used
in low speed precision applications, we generally focus
on dc errors such as offset voltage, bias current and low
frequency noise (primarily at harmonics of the line frequency of either 50 Hz or 60 Hz). We must also estimate
the errors that will result from sizable changes in temperature due the rugged and noisy environment in
which many in amps find themselves.
It is also important to remember that the effect of particular error sources will vary from application to application. In thermocouple applications, for example, the
source impedance of the sensor is very low (typically
not greater than a few ohms even when there is a long
cable between sensor and amplifier). As a result, errors
due to bias current and noise current can be neglected
when compared to input offset voltage errors.
RTO and RTI
Before we consider individual error sources, understanding of what we mean by RTO and RTI is important.
In any device that can operate with a gain greater than
unity (e.g., any op amp or in amp), the absolute size of
an error will be greater at the output than at the input.
For example, the noise at the output will be the gain
times the specified input noise. We must, therefore,
specify whether an error is referred to the input (RTI) or
referred to the output (RTO). For example, if we wanted
to refer output offset voltage to the input, we would simply divide the error by the gain, i.e.,
Output Offset Error (RTI) = VOSOUT/Gain
Referring all errors to the input, as is common practice,
allows easy comparison between error sizes and the size
of the input signal.
Parts per Million—PPM
Parts per million or ppm is a popular way of specifying
errors that are quite small. PPM is dimensionless so we
must make the error relative to something. In these
examples, it is appropriate to compare to the full-scale
input signal. For example, the input offset voltage,
expressed in ppm, is given by the equation:
Input Offset Error (ppm) = (VOS/VIN FULL SCALE) × 106
AN-539
current times the source impedance. Because either of
the bias currents can be the greater, the offset current
can be of either polarity.
Error Sources in Discrete and Integrated Instrumentation
Amplifiers
Figure 1 shows the most common and prevalent error
sources in discrete and integrated in amps. These error
sources are detailed individually below.
Common-Mode Rejection
An ideal in amp will amplify the differential voltage between its inverting and noninverting inputs regardless
of any dc offsets appearing on both inputs. So any dc
offset appearing on both inputs (+VS/2 in Figure 1) will
be rejected by the in amp. This dc or common-mode
component is present in many applications. Indeed,
removing this common-mode component is often the
primary function of an instrumentation amplifier in an
application.
Offset Voltage
Offset voltage results from a mismatch between transistor VBEs in an amplifier’s input stage. This voltage can be
modeled as a small dc voltage in series with the input
signal, as shown in Figure 1. Like the input signal, it will
be amplified by the gain of the in amp. In the case of in
amps with more than one stage (e.g., the classic three
op amp in amp) the input transistors of the output stage
will also contribute an offset component. However, as
long as the output stage has gain of unity, as is generally
the practice, the in amp’s programmed gain will have no
effect on the absolute size of the output offset error.
However, for error computation, this error is usually
referred back to the input so that its effect can be compared to the size of the input signal. This yields the
equation:
In practice, not all of the input common-mode signal
will be rejected and some will appear at the output.
Common-mode rejection ratio is a measure of how well
the instrumentation amplifier rejects common-mode
signals. It is defined by the formula:
Gain ×VCM 
CMRR (dB) = 20 × log 

 VOUT 
Total Offset Error (RTI) = VOS_IN + VOS_OUT/Gain
We can rewrite this equation to allow calculation of
the output voltage that results from a particular input
common-mode voltage.
From this equation, it is clear that the effect of output
offset voltage will decrease as the in amp’s programmed gain increases.
Offset and Bias Currents
Bias currents flow into or out of the in amp’s inputs.
These are usually the base currents of npn or pnp transistors. These small currents will therefore have a defined polarity for a particular type of in amp.
VOUT =
Gain ×VCM
CMRR 
log–1 

 20 
AC and DC Common-Mode Rejection
Poor common-mode rejection at dc will result in a dc offset at the output. While this can be calibrated away (just
like offset voltage), poor common-mode rejection of ac
signals is much more troublesome. If, for example, the
input circuit picks up 50 Hz or 60 Hz interference from
the mains, an ac voltage will result at the output. Its
presence will reduce resolution. Filtering is a solution
only in very slow applications where the maximum frequency is much less than 50 Hz/60 Hz.
Bias currents generate error voltages when they flow
through source impedances. The bias current times the
source impedance generates a small dc voltage which
appears in series with the input offset voltage. However
if both inputs of the in amp are looking at the same
source impedance, equal bias currents will generate a
small common-mode input voltage (typically in the µV
range) that will be well suppressed by any device which
has reasonable common-mode rejection. If the source
impedances of the in amp’s inverting and noninverting inputs are not equal, the resulting error will be
greater by the bias current times the difference in the
impedances.
Table I shows the output voltages of two in amps, the
AD623 and the INA126, that result when a 60 Hz
common-mode voltage of 100 mV amplitude is picked
up by the input.
We must also consider the offset current, which is the
difference between the two bias currents. This difference will generate an offset type error equal to the offset
Table I. Effect of CMRR on Output Voltage of AD623 and INA126 for a 60 Hz, 100 mV Amplitude Common-Mode Input
Gain
VIN (cm)
CMRR–INA126
VOUT–INA126
10
100
1000
100 mV @ 60 Hz
100 mV @ 60 Hz
100 mV @ 60 Hz
83 dB
83 dB
83 dB
100 dB
110 dB
110 dB
70.7 µV
707 µV
7.07 mV
10 µV
31.6 µV
316 µV
–2–
AN-539
Because voltage and current noise are uncorrelated (i.e.,
are random and bear no relationship to one another),
the overall error due to noise should not be simply the
sum of all noise errors. It is more accurate to do a rootsum-square calculation of the total noise error.
Noise
While offset voltages and bias currents ultimately lead
to offset errors at the output, noise sources will degrade
the resolution of a circuit. There are two noise sources in
most amplifiers, voltage noise and current noise. As is
the case with offset voltage and bias current, the degree
to which these sources affect the resolution varies with
the application.
Total noise = Voltage Noise 2 + RSOURCE × Current Noise 2
Linearity
This error will generally be specified in ppm (for a particular input span) in the data sheets of integrated instrumentation amplifiers. In the case of discrete in
amps, built from op amps, the nonlinearity is more difficult to calculate. Op amp data sheets generally do not
specify linearity. Furthermore, even if the linearity of
one op amp is known, it is necessary to estimate how
the two or three op amps will interact to give an overall
linearity specification. In many cases, the only option is
to measure the linearity of the circuit by doing a
dc sweep. The linearity in ppm will be given by the
expression.
VOLTAGE NOISE – nV/ Hz
1000
GAIN = 1
100
GAIN = 10
10
1
10
100
FREQUENCY – Hz
Nonlinearity (ppm) = (Maximum deviation of output
voltage from ideal/gain/full-scale input) × 106
1k
Figure 2. Voltage Noise Spectral Density of a Typical In
Amp
Gain Error
The gain error of an integrated instrumentation amplifier will have two components, internal gain error and
error due to the tolerance of the external gain setting
resistor. While a precision external gain resistor will prevent the overall gain from degrading, there is little point
in wasting cost on an external resistor which is much
more accurate than the in amp’s accuracy. It is also generally difficult to achieve exactly the desired gain (e.g.,
10 or 100) when using standard resistor sizes.
The voltage noise spectral density of a typical in amp
is shown in Figure 2 (a plot of current noise spectral density would have a similar characteristic). While the
response is flat at higher frequencies (above about
100 Hz, the so-called 1/f frequency), the noise spectral
density increases as the frequency approaches dc. To
calculate the resulting RTI rms noise, the noise spectral
density is multiplied by the square root of the bandwidth
of interest. The bandwidth may be the bandwidth of the
in amp at that particular gain or it may be something
less. For example, if the output signal from the in amp is
low pass filtered, the corner frequency of the filter will
define the bandwidth of interest. Note that if the output
of the instrumentation amplifier is being digitized in an
analog-to-digital converter (ADC), any post filtering
should also be factored into calculating the bandwidth
of interest.
It should be noted however that the choice of gain resistor can help to reduce the overall gain drift of the circuit.
As an example, let’s consider the AD623. Its gain is
given by the equation:
Gain = 1 + (100 kΩ/RG)
The 100 kΩ value in this equation stems from two internal 50 kΩ resistors. These resistors have a temperature
coefficient of –50 ppm/°C. By choosing an external gain
resistor, which also has a negative temperature coefficient, the gain drift can be reduced.
In high frequency applications, low frequency noise
generally can be neglected. So the RTI rms noise would
simply be the product of the “flat” noise spectral density
and the square root of the bandwidth. Note that the calculated rms noise must be converted to peak-to-peak by
multiplying the rms value by 6.61. For low frequency applications, data sheets usually specify peak-to-peak
noise in the 0.1 Hz to 10 Hz frequency band. If high
frequency noise is filtered at some point in the system, it
can be neglected so that only the 0.1 Hz to 10 Hz noise is
counted.
Error Budget Analysis of Two Integrated In Amps: AD623
vs. INA126
Figure 3 shows the popular resistive bridge application.
The bridge consists of four variable resistors. The bridge
is excited by a single +5 V supply. Any change in the
resistance of the variable resistors will generate a differential voltage (±20 mV full scale) which appears across
the input of the in amp. The common-mode voltage of
the differential signal is +2.5 V. This follows from the
+5 V excitation voltage.
–3–
AN-539
Table II shows the error budget calculations using the
integrated in amps AD623 and INA126. All errors are referred to the input (i.e., are compared to the full-scale
input voltage of 20 mV) and are then converted to parts
per million (ppm) by multiplying the fractional error by
1 × 106. Alternatively, percentage errors are converted
to ppm by multiplying by 1 × 104. Conversion between
fractional, percentage and ppm is shown in Table III.
The gain of the in amp has been set so that the output
signal swing is close to its maximum level but is still not
clipping. Care should be taken not to saturate any of the
internal nodes of the in amp. This is a function of the
differential input voltage, the programmed gain and the
common-mode level2.
+5V
+5V
350V
350V
350V
350V 620 mV
RG 1.13kV
0.1% TOL
+10ppm/8C
+5V
VOUT
RG 953V
0.1% TOL
+10ppm/8C
VOUT
INA126P
REFERENCE
REFERENCE
+2.5V
AD623B GAIN = 89.5 (1+100kV/RG)
+2.5V
INA125P GAIN = 89 (5+80kV/RG)
Figure 3. Amplifying the Differential Voltage from a Resistive Bridge Transducer
Table II. Error Budget Analysis: AD623 vs. INA126
Error Source
ABSOLUTE ACCURACY at TA = +25°C
Input Offset Voltage, mV
Output Offset Voltage, mV
Input Offset Current, nA
CMR, dB
Gain
DRIFT TO +85°C
Gain Drift, ppm/°C
Input Offset Voltage, mV/°C
Input Offset Current, pA/°C
Output Offset Voltage Drift, mV/°C
RESOLUTION
Gain Nonlinearity, ppm of Full Scale
Typ 0.1 Hz–10 Hz Voltage Noise, mV p-p
Circuit Calculation
INA126P
Error Calculation
Total Error
Total Error
INA126 (ppm)
100 µV/20 mV
500 µV/89.5/20 mV
2 nA × 350 Ω/20 mV
105 dBv5.6 ppm ×
2.5 V/20 mV
0.35% + 0.1%
250 µV/20 mV
Not Applicable
2 nA × 350 Ω/20 mV
83 dBv71 ppm ×
2.5 V/20 mV
0.5% + 0.1%
5,000
279
35
12,500
Not Applicable
35
700
4500
8875
6000
Total Absolute Error
10514
27410
(100 + 10) ppm/°C × 60°C
3 µV/°C × 60°C/20 mV
10 pA/°C × 350 Ω ×
60°C/20 mV
3600
3000
6600
9000
5.25
10.5
(50 + 10) ppm/°C × 60°C
1 µV/°C × 60°C/20 mV
5 pA/°C × 350 Ω ×
60°C/20 mV
10 µV/°C × 60°C/
89.5/20 mV
50 ppm
1.5 µV p-p/20 mV
Table III. Conversion Between ppm, Fractional Error and
Percentage Error
% Error
Fractional Error
ppm Error
10
1
0.1
0.01
0.001
0.0001
0.1
0.01
0.001
0.0001
0.00001
0.000001
100000
10000
1000
100
10
1
Not Applicable
335
Not Applicable
Total Drift Error
6940
15610
20 ppm
0.7 µV p-p/20 mV
50
75
20
35
Total Resolution Error
125
55
Grand Total Error
17579
43075
Table II shows that the predominant error source is
static errors (e.g., offset voltage, etc.). In many applications where some form of calibration is available, these
errors can be removed. With the addition of some kind
of ambient temperature measurement, this calibration
can be extended to compensate for drift of static errors.
It is more difficult to compensate for errors in resolution
caused by the nonlinearity and noise of the in amp. Note
that errors due to current noise have been neglected.
–4–
AN-539
These errors are quite small and become insignificant
when they are quadratically summed with the voltage
noise.
Make vs. Buy: A Typical Application Error Budget
The example in Figure 5 serves as a good comparison
between the errors associated with an integrated and
a discrete in amp implementation. Again, we have a
±20 mV signal we want to amplify. Using a dual op amp
and a precision resistor network, a two op amp in amp
can be implemented.
Additional resolution errors that occur due to external
interference cannot be characterized here. Significant in
this area is the degradation in resolution that will be
caused by common-mode pickup on the differential inputs of 50 Hz or 60 Hz interference (from lights or any
equipment running on the mains). This will result in the
50 Hz/60 Hz hum being visible on the in amp’s output.
Obviously, high common-mode rejection, not just at dc
but also over frequency, will help to minimize this interference. The common-mode rejection over frequency of
the AD623 is shown in Figure 4. This shows for example,
that the CMRR at 1 kHz, for a gain of 10, is still over
80 dB, more than sufficient for most applications.
The errors associated with each implementation are detailed in Table IV and show the integrated in amp to be
more precise, both at ambient and over temperature. It
should be noted that the discrete implementation is
quite a bit more expensive (by about 100% in this example). This is primarily due to the cost of the low drift
precision resistor network.
Note, the input offset current of the discrete in amp
implementation is the maximum difference in the bias
currents of the two op amps, not the offset currents of
the individual op amps. Also, while the values of the
resistor network are chosen so that the inverting and
noninverting inputs of each op amp see the same
impedance (about 350 Ω), the offset current of each
op amp will add an additional error that must be
characterized.
120
110
31000
100
CMR – dB
90
3100
80
310
70
60
31
50
40
30
1
10
100
1k
FREQUENCY – Hz
10k
100k
Figure 4. AD623 CMR vs. Frequency, +5 V Single
Supply, VREF = 2.5 V, Gain = 1, 10, 100, 1000
+5V
+5V
350V
350V
350V
350V 620mV
+5V
VOUT
RG 1.13kV
0.1% TOL
+10ppm/8C
VOUT
1/2
OP296g
1/2
OP296g
+2.5V
AD623A GAIN = 89.5 (1+100kV/RG)
31.5kV*
350V*
350V*
31.5kV*
+2.5V
“HOMEBREW” IN AMP, G = 90
*0.1% RESISTOR MATCH, 50ppm/8C TRACKING
Figure 5. Make vs. Buy
–5–
AN-539
Error Source
ABSOLUTE ACCURACY at TA = +25°C
Total RTI Offset Voltage, mV
Input Offset Current, nA
Internal Offset Current
(Homebrew Only)
CMR, dB
Gain
DRIFT TO +85°C
Gain Drift, ppm/°C
Total RTI Offset Voltage, mV/°C
Input Offset Current, pA/°C
Internal Offset Current
(Homebrew Only)
RESOLUTION
Gain Nonlinearity, ppm of Full Scale
Typ 0.1 Hz–10 Hz Voltage Noise,
mV p-p
“Homebrew”
Circuit Calculation
Total Error
Total Error
Homebrew–ppm
(200 µV+[1000 µV/89.5])/20 mV
2 nA × 350 Ω/20 mV
(300 µV × 2)/20 mV
100 nA × 350 Ω/20 mV*
10,559
35
30,000
1750
Not Applicable
16 nA × 350 Ω/20 mV*
105 dBv5.6 ppm × 2.5 V/20 mV (0.1% Match × 2.5 V)/
90/20 mV
0.35% + 0.1%
0.1% Match
280
700
4500
1388
1000
Total Absolute Error
15794
34418
(50 + 10) ppm/°C × 60°C
(2 µV/°C+[10 µV/°C/89.5])
× 60°C/20 mV
5 pA/°C × 350 Ω × 60°C/20 mV
50 ppm/°C × 60°C
3600
3000
(7 µV/°C × 60°C)/20 mV
Not Applicable*
6335
5.25
21,000
Not Applicable
(120 pA/°C × 350 Ω × 2)
× 60°C/20 mV
50 ppm
1.5 µV p-p/20 mV
E3316a–0–8/99
Table IV. Make vs. Buy Error Budget
252
Total Drift Error
9940
24252
20 ppm
50
20
(0.8 µV p-p × √2)/20 mV
75
56.57
Total Resolution Error
125
77
Grand Total Error
25859
58747
*Error over temperature. bias current of OP296 is only specified as a maximum value over temperature (i.e., no value specified at + 25°C).
REFERENCES
1. Analog-Digital Conversion Handbook, Third Edition, pp 550–553, by the Engineering Staff of Analog Devices, Inc., edited
by Daniel H. Sheingold, Prentice Hall, Englewood Cliffs, NJ 07632.
PRINTED IN U.S.A.
2. AD623 Single Supply, Rail-to-Rail, Low Cost Instrumentation Amplifier, data sheet, p 15.
–6–
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