Amplifer and ADC

Op Amp, 12-Bit ADC, +5V Ref

 

A basic amplfier and ADC provide great insight into a multi-stage Error Budget Analysis! We'll do a teardown of this critical and challenging analysis. You'll get

• a step-by-step approach
  • walk thru each error source
  • derive the circuit and sensitivity equations
• an Excel file
  • create, review and reuse with 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

 

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 = 5mV
• Gain Error:  ∆K = 1% of Reading

AMPLIFIER AND ADC

Schematic with Errors

Three devices make up this signal-chain.

• U1, Op Amp
• U2, 12-bit ADC
• U3, +5V Reference

Device Errors

The errors below reflect a medium precision of accuracy grade for the devices.

Description	Initial	Temp Drift
OFFSET ERRORS
U1, voff, Input Offset Voltage U1, ibp, Input Bias Current (Pos) U1, ibn, Input Bias Current (Neg) U2, voff, Offset Error U2, vres, Resolution Error U2, vinl, Integral-Non Linearity Error	1 mV 10 nA 10 nA 3 LSB 1/2 LSB 1 LSB	10 uV / C 1 nA / C 1 nA / C 0.05 LSB/C
GAIN ERRORS
R2 Tolerance, Tempco R1 Tolerance, Tempco U2, Gain Tol, Tempco U3, Vref Tol, Tempco	0.1 % 0.1 % 2 LSB 0.5%	100 ppm / C 100 ppm / C 0.08 LSB/C 50 ppm/C

 

Conditions and Assumptions

Temperature

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

Amplifier

• Rs = 500, R1=10k, R2=40k
• Vin = 1V, Vo = 5V

ADC

• Bits = 12
• Vref = 5.0V

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

AMPLIFIER ERRORS

The Amplfier errors have already been covered!
Check out the steps in the Amp1 Error Analysis.

 

ADC OFFSET ERROR

The ADC's offset errors are typically characterized in units of digital output LSB's (Least Significant Bits). It's easy to convert output LSBs to input Volts using the ADC's resolution Vlsb.

 

Description	Initial Errors	Drift Errors
Error Source: e	offset  = 3 LSBs Convert to V: voff = 3 LSB x     x 0.00122 (V/LSB)       = 3.66 mV	offset_TC  = 0.04 LSB/C Convert to V: voff_TC = 0.04 LSB /C     x 0.00122 (V/LSB)       = 49 uV / C
Pick Analysis Node: Va	vadc	vadc
Calc Sensitivity: S How does e impact Va?	S = vadc / voff   = 1	S = 1
Calc Offset Error at Analysis Node   Initial:  ∆Voffset = e * S   Drift:   ∆Voffset = e * ∆T * S	∆Voffset  = 3.7 mV * 1  = 3.7 mV	∆Voffset  = 49uV/C * 30C * 1  = 1.5 mV
Calc Gain from Input to Analysis Node:   Ka = Va / Vin	Ka = vadc/vin  = R2/R1+1  = 5	Ka = 5
Calc Error RTI (Referred-to-Input):   ∆voffset_RTI = ∆voffset / Ka	∆voffset_RTI  = 3.7 mV / 5  = 0.73 mV	∆voffset_RTI  = 1.5 mV / 5  = 0.3 mV

 

ADC RES & INL ERRORS

The Resolution Errors and Integral Non-Linearity (INL) Errors are considered part of the offset errors. Why? Mainly because they don't scale directly with the signal level (which would imply a gain term).

• Resolution Error - also called Quantization error, creates an error of 1/2 LSB.
• INL Error - varies with signal level, but does not scale with the signal level.

The error analysis for both of these ADC errors follow the same steps as the Initial Offset Error shown above. (See Excel file link below.)

 

GAIN ERRORS

Similar to Offset Errors, the Gain Errors can be converted from LSBs to % or ppm.

 

ADC GAIN ERROR

You can write the ADC gain as

Because Kadc directly defines gain of this block, you can assume the Sensitivity is unity, or S = 1.0.

Description	Initial Errors	Drift Errors
Error Source: e	Kadc_Tol  = 2 LSB Convert to %: Kadc_Tol  = 2LSB/4095*100%  = 0.05%	Kadc_TC  = 0.08LSB/C Convert to ppm/C: Kadc_TC  = 0.08LSB/4095*1e6  = 20ppm/C
Pick Analysis Node: Va	ADCword	ADCword
Calc Sensitivity: S How does e impact Gain K?	S = 1.0 (see discussion above)	S = 1.0
Calc Gain Error at Analysis Node   Initial:  ∆K/K = e * S   Drift:   ∆K/K = e * ∆T * S	∆K/K  = 0.05% * 1.0  = 0.05%	∆K/K  = 20ppm/C*30C*1.0  = 600ppm  = 0.06%
Normailzed gain errors can be referred to input as-is, no RTI calc needed.

 

VREF ERROR

The Vref term appears in the denominator of the ADC gain.

Intuition tells us the Sensitivity S should be negative (Kadc goes down as Vref goes up.) We can also guess that the magnitude of S should be 1 because Vref is a direct multiplier (numerator or denominator) of the gain. We'll calculate S anyway below just to confirm.

Description	Initial Errors	Drift Errors
Error Source: e	Vref_Tol  = 0.5%	Vref_TC  = 50 ppm/C
Pick Analysis Node: Va	ADCword	ADCword
Calc Sensitivity: S How does e impact Gain K? Apply Difference Method:  S = (∆K/K)/(∆V/V) where  ∆K = (K'-K)	Kadc  = ADCword/vadc  = (2N-1)/Vref Vref = 5.0 2N-1 = 4095 Kadc = 4095/5  = 819 Kadc' = 4095/(5*1.01)  = 811 ∆Vref/Vref = 0.01 S = (∆Kadc/Kadc) /        (∆Vref/Vref)    = -1.0	S = -1.0
Calc Gain Error at Analysis Node   Initial:  ∆K/K = e * S   Drift:   ∆K/K = e * ∆T * S	∆K/K  = 0.5% * -1.0  = -0.5%	∆K/K  = 50ppm/C*30C*-1.0  = -1500ppm  = -0.15%
Normailzed gain errors can be referred to input as-is, no RTI calc needed.

 

SUMMARY & EXCEL FILE

See Excel file:  amp-ref-ADC-1-EBA.xlsx
Right Click on the filename, select "Save link as...".

OFFSET ERRORS

 

Calculate the total using Worst Case Analysis. WCA assumes the most unfavorable conditions: all errors at their maximum limit AND in the same polarity.

• WCA = | ∆Voffset1 | + | ∆Voffset2 | + ...
        = 2.1mV

Does the Total Error fly under the Max Error Budget (Requirements)?

•  WCA < 5.0mV?
    Yes - PASS!

 

GAIN ERRORS

 

Calculate the total using Worst Case Analysis.

• WCA = | ∆K1 | + | ∆K2 | + ...
        = 1.4 %

Does the Total Error fly under the Max Error Budget (Requirements)?

•  WCA < 1.0%?
      No - FAIL!

Yikes! Can we improve the design? See "Try It" below.

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!

 

Try the hands-on spreadsheet!

• Excel file:  amp-ref-ADC-1-EBA.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!

• The Total Gain Error did NOT meet spec! Can we improve the design?
• Open the Excel file and check the Gain Error Sheet to identify the largest error contributors.
• Try changing the R1 and R2 tempcos from 100 ppm/C to a higher precision - a 50, 25 or 10 ppm/C tempco.
• Try changing the +5V Ref tolerance from 0.5% to 0.1%.

 

For in-depth tutorials and more circuits, go to EBA Series