Advanced Op Amp

ib, iboff, Aol

 

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

• ib, Input Bias Current
• iboff, Input Offset Current
• Aol, Open-Loop Gain

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

• a step-by-step approach
  • walk thru each error source
  • derive the circuit and sensitivity equations
• an Excel file
  • enter errors in a systematic format
  • vary parameters, see their effect on total error
  • customize & expand template for your own designs

 

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?

• Ideal Amplifier with Gain K
  • Vo = Vin*K
• Actual Amplifier with errors
  •  Vo = Vin* (K+∆K) + ∆Voffset
•  ∆K  - Gain Error
  • change in Gain from ideal (K).
  • also called Slope, Span error
•  ∆Voffset  - Offset Error
  • change in offset from ideal (0V)
  • also called Intercept, Zero Error
• Combined Offset and Gain Errors
  • Error = ±∆Voffset  + ( ±∆K x Reading )

 

MAX ERROR BUDGET

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

• Offset Error:  ∆Voffset = 1.5mV
• Gain Error:  ∆K/K = 0.5% of Reading

AMPLIFIER

Schematic with Error Sources

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

• Ambient: Ta = 25C
• Max change: ∆T = 30C

Amplifier

• Rs = 1k, R1=10k, R2=90k
• K = R2/R1+1 = +10.0
• Vin = 0.5V, Vo = 5V

Errors

• All errors can be either polarity +/-.
• Errors will be Referred to Input (RTI)
• Errors totaled as Worst Case (sum absolute values)

 

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.

• Organizes complex analyses into smaller managable sections
• Easier to create, review and debug
• Customizable and expandable to more complex circuits
• Modular for reuse in other designs

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.

 

Explore the hands-on spreadsheet!

• Excel file:  amp1-advanced-a.xlsx
    Right Click on the filename, select "Save link as...".
• Open file, explore the Calc, Offset and Gain Sheets
• Play in the sandbox, modify values, see the impact on errors.
• Copy to a new file - experiment!

TRY IT!

• Suppose the offset spec was tightened from 1.5mV to 1.0 mV.
• How can you meet the new spec? Try lowering the R1,R2 values, by 5x or 10x. However, keep their ratio the same to mainitain the same gain.

 

For tutorials and other examples, goto EBA Series