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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

INTRODUCTION

This application note describes a systematic approach

to calculating the overall error in an instrumentation

amplifier (in amp) application. We will begin by describ-

ing the primary sources of error (e.g., offset voltage,

CMRR, etc.) in an in amp. Then, using data sheet specifi-

cations 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 fre-

quency of either 50 Hz or 60 Hz). We must also estimate

the errors that will result from sizable changes in tem-

perature due the rugged and noisy environment in

which many in amps find themselves.

Its also important to remember that the effect of particu-

lar error sources will vary from application to applica-

tion. 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, under-

standing 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 sim-

ply 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

AD623

VOSOUT

VOUT

REFERENCE

VOSIN

RGAIN

IOFS

INOISE+

350?

350?

+VS

INOISE–

VCM = +VS/2

VNOISE

350?

IBIAS

IBIAS

Figure 1. Error Sources in a Typical Instrumentation Amplifier

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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.

Offset Voltage

Offset voltage results from a mismatch between transis-

tor 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 com-

pared to the size of the input signal. This yields the

equation:

Total Offset Error (RTI) = VOS_IN + VOS_OUT/Gain

From this equation, it is clear that the effect of output

offset voltage will decrease as the in amp’s pro-

grammed 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 tran-

sistors. These small currents will therefore have a de-

fined polarity for a particular type of in amp.

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 noninvert-

ing inputs are not equal, the resulting error will be

greater by the bias current times the difference in the

impedances.

We must also consider the offset current, which is the

difference between the two bias currents. This differ-

ence will generate an offset type error equal to the offset

current times the source impedance. Because either of

the bias currents can be the greater, the offset current

can be of either polarity.

Common-Mode Rejection

An ideal in amp will amplify the differential voltage be-

tween 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.

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:

CMRR(dB)20

Gain V

V

OUT

CM

=×

×

log

We can rewrite this equation to allow calculation of

the output voltage that results from a particular input

common-mode voltage.

V

Gain V

CMRR

20

OUT

CM

–1

=

×

log

AC and DC Common-Mode Rejection

Poor common-mode rejection at dc will result in a dc off-

set 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 fre-

quency is much less than 50 Hz/60 Hz.

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.

Table I. Effect of CMRR on Output Voltage of AD623 and INA126 for a 60 Hz, 100 mV Amplitude Common-Mode Input

GainVIN (cm) CMRR–INA126 CMRR–AD623VOUT–INA126VOUT–AD623

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

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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.

FREQUENCY – Hz

1000

100

101

1k 10

VOLTAGE NOISE – nV/ Hz

100

GAIN = 1

GAIN = 10

Figure 2. Voltage Noise Spectral Density of a Typical In

Amp

The voltage noise spectral density of a typical in amp

is shown in Figure 2 (a plot of current noise spectral den-

sity 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.

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 cal-

culated rms noise must be converted to peak-to-peak by

multiplying the rms value by 6.61. For low frequency ap-

plications, 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.

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 root-

sum-square calculation of the total noise error.

Total noise = Voltage Noise2+RSOURCE×Current Noise2

Linearity

This error will generally be specified in ppm (for a par-

ticular input span) in the data sheets of integrated in-

strumentation 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.

Nonlinearity (ppm) = (Maximum deviation of output

voltage from ideal/gain/full-scale input) × 106

Gain Error

The gain error of an integrated instrumentation ampli-

fier 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 pre-

vent 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 gen-

erally difficult to achieve exactly the desired gain (e.g.,

10 or 100) when using standard resistor sizes.

It should be noted however that the choice of gain resis-

tor 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 inter-

nal 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 coeffi-

cient, the gain drift can be reduced.

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 differ-

ential 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.

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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.

Table II shows the error budget calculations using the

integrated in amps AD623 and INA126. All errors are re-

ferred 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.

AD623B

VOUT

REFERENCE

+2.5V

+5V

RG 1.13k?

0.1% TOL

+10ppm/?C

AD623B GAIN = 89.5 (1+100k?/RG)

350?

350?

350?

INA126P

VOUT

REFERENCE

+2.5V

+5V

RG 953?

0.1% TOL

+10ppm/?C

INA125P GAIN = 89 (5+80k?/RG)

350?

+5V

?20 mV

Figure 3. Amplifying the Differential Voltage from a Resistive Bridge Transducer

Table II. Error Budget Analysis: AD623 vs. INA126

AD623B

Circuit Calculation

INA126P

Error Calculation

Total Error

AD623 (ppm)

Total Error

INA126 (ppm)Error Source

ABSOLUTE ACCURACY at TA = +25°C

Input Offset Voltage, mV

Output Offset Voltage, mV

Input Offset Current, nA

CMR, dB

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

6000Gain

Total Absolute Error1051427410

DRIFT TO +85°C

Gain Drift, ppm/°C

Input Offset Voltage, mV/°C

Input Offset Current, pA/°C

(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

(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

5.2510.5

Output Offset Voltage Drift, mV/°C

Not Applicable335Not Applicable

Total Drift Error6940 15610

RESOLUTION

Gain Nonlinearity, ppm of Full Scale

Typ 0.1 Hz–10 Hz Voltage Noise, mV p-p

50 ppm

1.5 µV p-p/20 mV

20 ppm

0.7 µV p-p/20 mV

50

75

20

35

Total Resolution Error12555

Grand Total Error2585943075

Table III. Conversion Between ppm, Fractional Error and

Percentage Error

% ErrorFractional Errorppm 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

Table II shows that the predominant error source is

static errors (e.g., offset voltage, etc.). In many applica-

tions 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.

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These errors are quite small and become insignificant

when they are quadratically summed with the voltage

noise.

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 in-

puts 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 inter-

ference. 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.

FREQUENCY – Hz

1100k 10

CMR – dB

1001k10k

120

30

110

100

90

80

70

60

50

40

?1000

?100

?10

?1

Figure 4. AD623 CMR vs. Frequency, +5 V Single

Supply, VREF = 2.5 V, Gain = 1, 10, 100, 1000

AD623A

VOUT

+2.5V

+5V

RG 1.13k?

0.1% TOL

+10ppm/?C

AD623A GAIN = 89.5 (1+100k?/RG)

350?

350?

350?

VOUT

+5V

350?

+5V

?20mV

1/2

OP296g

1/2

OP296g

350?*350?*31.5k?* 31.5k?*

“HOMEBREW” IN AMP, G = 90

*0.1% RESISTOR MATCH, 50ppm/?C TRACKING

+2.5V

Figure 5. Make vs. Buy

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.

The errors associated with each implementation are de-

tailed 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 ex-

ample). 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 re-

sistor network are chosen so that the inverting and non-

inverting inputs of each op amp see the same

impedance (about 350 Ω), the offset current of each

op amp will add an additional error which must be

characterized.