Basic Amplifier

Inverting Op Amp

 

The inverting provides a great insight into 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 for your own designs

 

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

For the same magnitiude of gain, which has the larger errors, the Non-Inverting (K = +5) versus the Inverting (K = -5) config?

 

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

Schematic with Error Sources

Error Sources

Description	Initial	Drift
OFFSET ERRORS
voff, Input Offset Voltage ibn, Input Bias Current (Neg) ibp, Input Bias Current (Pos)	1 mV 10 nA 10 nA	10 uV / C 1 nA / C 1 nA / C
GAIN ERRORS
R2_Tol R1_Tol	0.1 % 0.1 %	100 ppm / C 100 ppm / C

 

Conditions and Assumptions

Temperature

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

Amplifier

• Rs = 500, R1=10k, R2=50k
• K = -R2/R1 = -5.0
• Vin = 1V, 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

The Non-Inverting Amplfier nicely showcases the EBA method. You can modify the analysis for the Inverting or other Op Amp configurations.

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 OFFSET VOLTAGE

Because voff is modelled as voltage in series with the pos input, it gets amplified just like the non-inverting gain: 
   Knon = vo / vp = R2/R1+1 = +6.0

Description	Initial Errors	Drift Errors
Error Source: e	voff = 1mV	voff_TC = 10uV/C
Pick Analysis Node: Va	vo	vo
Calc Sensitivity: S How does e impact Va?	S = vo / voff   = R2/R1+1   = 6	S = 6
Calc Offset Error at Analysis Node   Initial:  ∆Voffset = e * S   Drift:   ∆Voffset = e * ∆T * S	∆Voffset  = 1mV * 6  = 6mV	∆Voffset  = 10uV/C * 30C * 6  = 1.8 mV
Calc Gain from Input to Analysis Node:   Ka = Va / Vin	Ka = vin/vo  = -R2/R1  = -5	Ka = -5
Calc Error RTI (Referred-to-Input):   ∆voffset_RTI = ∆voffset / Ka	∆voffset_RTI  = 6mV / -5  = -1.2mV	∆voffset_RTI  = 1.8mV / -5  = -0.36mV

 

INPUT BIAS CURRENT (NEG)

The op amp acts like a transimpedance amplifier converting the current ibn to a voltage vo = ibn*(-R2).

Description	Initial Errors	Drift Errors
Error Source: e	ibn = 10nA	ibn_TC = 1nA/C
Pick Analysis Node: Va	vo	vo
Calc Sensitivity: S How does e impact Va?	S = vo / ibn   = -R2   = -50000	S = -50000
Calc Offset Error at Analysis Node   Initial:  ∆Voffset = e * S   Drift:   ∆Voffset = e * ∆T * S	∆Voffset  = 10nA * -50000   = -0.5mV	∆Voffset  = 1nA/C * 30C * -50000   = -1.5mV
Calc Gain from Input to Analysis Node:   Ka = Va / Vin	Ka = vin/vo  = -R2/R1  = -5	Ka = -5
Calc Error RTI (Referred-to-Input):   ∆voffset_RTI = ∆voffset / Ka	∆voffset_RTI  = -0.500mV / -5  =  0.100mV	∆voffset_RTI  = -1.50mV / -5  =  0.30mV

 

INPUT BIAS CURRENT (POS)

Easy to analyze - ibp flows into Rp creating a voltage, then gets amplified by the non-inverting signal gain.

Description	Initial Errors	Drift Errors
Error Source: e	ibp = 10nA	ibp_TC = 1nA/C
Pick Analysis Node: Va	vo	vo
Calc Sensitivity: S How does e impact Va?	S = vo / ibp   = Rp*(R2/R1+1)   = 500*6   = 3000	S = 3000
Calc Offset Error at Analysis Node   Initial:  ∆Voffset = e * S   Drift:   ∆Voffset = e * ∆T * S	∆Voffset  = 10nA * 3000  = 0.030mV	∆Voffset  = 1nA/C * 30C * 3000  = 0.090mV
Calc Gain from Input to Analysis Node:   Ka = Va / Vin	Ka = vin/vo  = -R2/R1  = -5	Ka = -5
Calc Error RTI (Referred-to-Input):   ∆voffset_RTI = ∆voffset / Ka	∆voffset_RTI  =  0.030mV / -5  = -0.006mV	∆voffset_RTI  =  0.090mV / -5  = -0.018mV

 

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.

RESISTOR R2
 

Description	Initial Errors	Drift Errors
Error Source: e	R2_Tol  = 0.1%	R1_TC   = 100ppm/C   = 0.0001%/C
Pick Analysis Node: Va	vo	vo
Calc Sensitivity: S How does e impact Gain K? Apply Difference Method:  S = (∆K/K) / (∆R/R) where  ∆K/K = (K'-K)/K	K = -R2/R1 R2 = 50k R1 =10k K = -50k/10k    = -5.0 K'= -50k*1.01/10k    = -5.05 ∆R/R = 0.01 S = (∆K/K) / (∆R/R)    = 1.0	S = 1.0
Calc Gain Error at Analysis Node   Initial:  ∆K/K = e * S   Drift:   ∆K/K = e * ∆T * S	∆K/K  = 0.1% * 1  = 0.1%	∆K/K  = 100ppm/C*30C*1.0  = 3000ppm  = 0.30%
Normailzed gain errors can be referred to input as-is, no RTI calc needed.

 

RESISTOR R1
 

Description	Initial Errors	Drift Errors
Error Source: e	R1_Tol  = 0.1%	R1_TC   = 100ppm/C   = 0.0001%/C
Pick Analysis Node: Va	vo	vo
Calc Sensitivity: S How does e impact Gain K? Apply Difference Method:  S = (∆K/K) / (∆R/R) where  ∆K/K = (K'-K)/K	K = -R2/R1 R2 = 50k R1 =10k K = -50k/10k    = -5.0 K'= -50k/(10k*1.01)    = -4.95 ∆R/R = 0.01 S = (∆K/K) / (∆R/R)    = -1.0	S = -1.0
Calc Gain Error at Analysis Node   Initial:  ∆K/K = e * S   Drift:   ∆K/K = e * ∆T * S	∆K/K  = 0.1% * -1  = -0.1%	∆K/K  = 100ppm/C*30C*-1.0  = -3000ppm  = -0.30%
Normalized gain errors can be referred to input as-is, no RTI calc needed.

 

SUMMARY

Let's review the Gain & Offset errors.

Description	Initial (V)	Drift (V)
OFFSET ERRORS
Input Offset Voltage Input Bias Current (Neg) Input Bias Current (Pos)	-1.200 mV 0.100 mV -0.006 mV	-0.360 mV 0.300 mV -0.018 mV
GAIN ERRORS
R2_Tol R1_Tol	0.1 % -0.1 %	0.3 % -0.3 %

 

TOTAL OFFSET ERROR

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 | + ...
        = 2.0 mV

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

•  WCA < 5.0mV?
    Yes - PASS!

 

TOTAL GAIN ERROR

Calculate the total using Worst Case Analysis.

• WCA = | ∆K1 | + | ∆K2 | + ...
        = 0.8 %

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

•  WCA < 1.0%?
     Yes - PASS!

 

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-inv1-basic-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!

 

NOTES, IDEAS...

• For in-depth tutorials, go to EBA series.
• Try the Inverting Amp. Copy to new file and modify Circuit Calc sheet.
• Add another stage. Copy Circuit Calc to a new sheet and modify for new stage. Add new rows to existing Offst / Gain Error sheets.

 

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