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Error Budget Analysis

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Advanced Op Amp

ib, iboff, Aol


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The advanced op amp tackles errors beyond the basic that can be critical for your amplifier design.


We'll perform an analysis for each error. You'll get

 

For tutorials on Key Concepts and other circuits, goto EBA Series

Get a refresh of the Basic Amplifier.

 

OFFSET AND GAIN ERRORS

We'll start with basic error definitions of an amplifier block. What are Offset and Gain Errors?

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MAX ERROR BUDGET

The max budget (target spec) for amplifier has been chosen as:

AMPLIFIER

Schematic with Error Sources

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Error Sources

This topic walks through critical errors highlighted in yellow below.

Note: For errors voff, R1_Tol & R2_Tol, check out analysis already covered in the Basic Amplifier.

Description Initial Drift
OFFSET ERRORS    
voff, Input Offset Voltage
ib, Input Bias Current
iboff, Input Offset Current
0.1 mV
5 nA
5 nA
10 uV / C
1 nA / C
1 nA / C
GAIN ERRORS    
R2_Tol, Resistor Tolerance
R1_Tol, Resistor Tolerance
U1 Aol, Open-Loop Gain
0.1 %
0.1 %
80dB, 10000 (V/V)
100 ppm / C
100 ppm / C

 

Conditions and Assumptions

Temperature

Amplifier

Errors

 

OFFSET ERRORS

While the steps may seem more detailed than needed for simpler errors, the value of creating a systematic approach will pay off when analyzing more complex, multi-stage designs.

 

INPUT BIAS CURRENT - Theory Refresh

Modern op amps achieve low bias currents using a technique called "input bias current cancellation". The current into the pos / neg inputs has two components

 ibp = ib + iboff / 2
 ibn = ib - iboff / 2

where ib is the average of two currents

  ib = (ibp + ibn) / 2

and iboff is the offset between the two currents

  iboff = ibp - ibn

The output due to both ibp and ibn can be written

  vo = ibp*Rs*(R2/R1+1) - ibn*R2

Applying some math, we can find the Gain (Sensitivity) of vo to ib only

   S = vo/ib = Rs*(R2/R1+1) - R2

Note, if you can make Rs = R1||R2, then S = 0 and vo = 0V. Yes that's good news for older devices where ib >> iboff.

However, for newer devices where ib has a similar magnitude to iboff, the output error could be worse for Rs = R1||R2. Best to keep Rs small as possible.

Similary, we can find the Gain (Sensitivity) of vo to iboff only

   S = vo/ib = ½ [ Rs*(R2/R1+1) + R2 ]

Note that R2 carries a + sign. No way to get S = 0 or vo = 0V.

 

INPUT BIAS CURRENT (ib)

The quick theory refresh (above) showed the Gain (Sensitivity) of vo to ib only

   S = vo/ib = Rs*(R2/R1+1) - R2

Let's walk through the error analysis.

Description Initial Errors Drift Errors
Error Source: e ib = 5nA ib_TC = 1nA/C
Pick Analysis Node: Va vo vo
Calc Sensitivity: S
How does e impact Va?
S = vo / ib
 = Rs*(R2/R1+1)-R2
 = -8000
S = -8000
Calc Offset Error at Analysis Node
  Initial:  Voffset = e * S
  Drift:   Voffset = e * T * S
Voffset
 = 5nA * -8000 
 = -0.4mV
Voffset
 = 1nA/C*30C*-8000
 = -2.4mV
Calc Gain from
Input to Analysis Node:
  Ka = Va / Vin
Ka = vin/vo
     = R2/R1+1
     = 10
 
Ka = 10
Calc Error RTI
(Referred-to-Input):
  voffset_RTI = voffset / Ka
voffset_RTI
 = -0.40mV / 10
 = -0.04mV
voffset_RTI
 = -2.4mV / 10
 = -0.24mV

 

INPUT OFFSET CURRENT (iboff)

The quick theory refresh (above) showed the Gain (Sensitivity) of vo to iboff only

 S = vo/iboff = ½ [ Rs*(R2/R1+1) + R2 ]

Let's walk through the error analysis.

Description Initial Errors Drift Errors
Error Source: e iboff = 5nA ibp_TC = 1nA/C
Pick Analysis Node: Va vo vo
Calc Sensitivity: S
How does e impact Va?
S = vo / ib
= ½ [Rs*(R2/R1+1) + R2]
= 50000
S = 50000
Calc Offset Error at Analysis Node
  Initial:  Voffset = e * S
  Drift:   Voffset = e * T * S
Voffset
 = 5nA * 50000
 = 0.25mV
Voffset
 = 1nA/C*30C*50k
 = 1.5mV
Calc Gain from
Input to Analysis Node:
  Ka = Va / Vin
Ka = vin/vo
     = R2/R1+1
     = 10
 
Ka = 10
Calc Error RTI
(Referred-to-Input):
  voffset_RTI = voffset / Ka
voffset_RTI
 = 0.25mV / 10
 = 0.025mV
voffset_RTI
 = 1.5mV / 10
 = 0.15mV

 

GAIN ERRORS

Gain errors often require more effort when calculating the Sensitivity S. You need to write the gain equation and then apply calculus (Difference Method) to find S.

OPEN-LOOP GAIN - Theory Refresh

Ideally, we expect the gain of the Non-Inverting Amplfier to be

  K = vo/vin = R2/R1+1

However, the actual gain can be found by

  K' = Aol / (1+Aol*B)

where Aol is the Open Loop Gain of the Op Amp and B is the feedback factor

  B = vn/vo = R1 / (R1+R2) 

If A is large, then the equation approaches the familiar equation

  K ≈ 1/B = (R1+R2) / R1

What is the error between the actual K' and ideal K? With some basic math, you can find the error as

  ∆K/K = (K' - K) / K ≈ 1 / (Aol*B)

 

OPEN-LOOP GAIN (Aol)

Let's apply our theory refresh (above) to finding the impact of Aol on Gain Error.

Description Initial Errors
Error Source: e

Aol = 80 (dB)
 = 1080/20 (V/V)
 = 10000 (V/V)

B = vn/vo
 = R1/(R1+R2)
 = 10k/(10k+90k)
 = 0.1
1/(Aol*B)
 = 1/(10000*0.1)
 = 0.001
Pick Analysis Node: Va vo
Calc Sensitivity: S
How does e impact Gain K?
S = 1.0

The op amp gain, directly impacts the gain at vo
Calc Gain Error
at Analysis Node
  ∆K/K = e * S
∆K/K
 = 0.001*1.0
 = 0.001
 = 0.1%
Normailzed gain errors can be referred to input as-is, no RTI calc needed.  

 

EBA WITH EXCEL

An Excel file was created to implement the error budget analysis.

3 Worksheets

Worksheet Enter Calculate
CIRCUIT CALC Circuit values Signal gains, levels and error Sensitivities
OFFSET Offset error sources Offset errors and totals
GAIN Gain error Sources Gain errors and totals

While 3 worksheets seems over-the-top for smaller circuits, you'll find a big advantage when analyzing more complex circuits or multi-stage systems!

Check out the easy entry (BLU col) and calculations (RED col) on the Offset Error sheet.

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Explore the hands-on spreadsheet!

TRY IT!

 

For tutorials and other examples, goto EBA Series