Error Budget Analysis

Part 2 - Gain Errors

INTRO

In Part 1, the Error Budget Analysis was introduced by finding the Offset Errors of the Digital Voltmeter..

Here in Part 2, we'll apply the same Key Concepts to find the Gain Errors of the DVM. You may find calcuating the Gain Sensitivities a bit more interesting and challenging.

 

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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 error
  • Error = ±∆Voffset  + ( ±∆K x Reading )

 

MAX BUDGET

• Max allowable error from requirements.
  • 4V:    ∆Voffset = 15mV,   ∆K = 1.5% of Reading
  • 20V:  ∆Voffset = 75mV,   ∆K = 1.5% of Reading

 

CIRCUIT WITH ERRORS

Here's a view of the instrument's Offset and Gain errors. We'll meed to find the errors for both the 4V and 20V Ranges.

ATTENUATOR GAINS

• 4V: SW1 On, SW2 Off
    Kdiv = 1.0
• 20V: SW1 Off, SW2 On
    Kdiv = R2/(R1+R2)
           =200k/(800k+200k)
           = 0.2

 

ERROR SOURCES

Offset Errors	Value
ADC Offset (Zero) Error	2 LSBs
ADC Resolution Error	1/2 LSBs
SW1 Off-State Leakage Current	1nA
SW2 Off-State Leakage Current	1nA

Gain Errors	Value
R1 Tolerance	1%
R2 Tolerance	1%
ADC Gain (Span) Error	2 LSBs
Voltage Reference Error	50mV

 

R1 TOLERANCE

Typically, you'll see Resistor Tolerances (∆R/R) and Gain Errors (∆K/K) expressed as normalized ratios

• Identify Error Source
  •  R1_Tol = ∆R / R = 1%
• Choose Analysis Node - a convenient location to
  calculate the error's impact on Gain.
  • Node:  v1
  • Gain:  K = v1 / vin
• Calculate Sensitivty - how does error impact Gain to Analysis Node?
  • 4V:   S = (∆K/K) / (∆R/R) = 0
  • 20V: S = (∆K/K) / (∆R/R) = +0.8
    (see Sensitivty Calc below)
• Calculate Error
  • 4V: ∆K/K = R1_Tol x S
                  = 1% x 0
                 = 0%
  • 20V: ∆K/K = R1_Tol x S
                     = 1% x 0.8
                     = 0.8%
• No RTI error calc needed! Because Gain Errors are multiplicative, they cause the same equivalent error at any node in signal path.

 

R2 TOLERANCE

• Identify Error Source
  •  R2_Tol = ∆R / R = 1%
• Choose Analysis Node - a convenient location to
  calculate the error's impact on Gain.
  • Node:  v1
  • Gain:  K = v1 / vin
• Calculate Sensitivty - how does error impact Gain to Analysis Node?
  • 4V:   S = (∆K/K) / (∆R/R) = 0
  • 20V: S = (∆K/K) / (∆R/R) = -0.8
    (see Sensitivty Calc details below)
• Calculate Error
  • 4V: ∆K/K = R2_Tol x S
                  = 1% x 0
                 = 0%
  • 20V: ∆K/K = R2_Tol x S
                     = 1% x 0.8
                     = -0.8%
• No RTI error calc needed as stated for R1.

 

ADC GAIN ERROR

• Identify Error Source
  • ADC Gain Error:  2 LSBs, 10 Bits (Arduino)
  • Kadc = 1024 / Vref
  • ∆Kadc = 2 LSB / Vref
  • ∆Kadc / Kadc = 2 LSB / 1024 x 100% = 0.2%
• Choose Analysis Node - a convenient location to
  calculate the error's impact on Gain.
  • Node:  ADCword
  • Signal Gain:  K = ADCword / vadc
• Calculate Sensitivty - how does error impact Gain to Analysis Node?
  •  S = (∆K/K) / (∆Kadc/Kadc) = 1.0
    (see Sensitivty Calc details below)
• Calculate Error
  •  ∆K/K = ∆Kadc/Kadc x S
             = 0.2% x 1.0
             = 0.2%
• No RTI error calc needed as stated for R1.

 

VREF (ADC) ERROR

• Identify Error Source
  • Arduino VREF (VCC):    ∆Vref = 0.050V
• Choose Analysis Node - a convenient location to
  calculate the error's impact on Gain.
  • Node:  ADCword
  • Signal Gain:  K = ADCword / vadc = (1024 / Vref)
• Calculate Sensitivty - how does error impact Gain to Analysis Node?
  •  S = (∆K/K) / (∆Vref) = -0.2
    (see Sensitivty Calc details below)
• Calculate Error
  •  ∆K/K = ∆Vref x S
             = 0.050V x -0.2 x 100%
             = -1.0%
• No RTI error calc needed as stated for R1.

 

GAIN SENSITIVITY CALCS

A walk-through of calculations below showcases the simple, yet powerful Delta Method for calculating Sensitvities!

NOTE! Sensitivities can also be calculated by the Derivative (Calculus) Method. However, the formulas become so unwiedly (even for smmple circuits) that the marginal accuracy gained is not worth the time or risk of mistakes / typos.

 

R1 TOLERANCE

Find the Gain Sensitivity

• Analysis Node:  v1
• Gain:  K = v1/vin = R2/(R1+R2)
• Sensitivity Equation (Delta Method)
  •  S = (∆K/K) / (∆R1/R1)
• Pick any small value of ∆R1/R1 (0.01, 0.001, ...) to find S.
• How does the gain change for, say, a 1% change in R1?
  • ∆K = K' -  K
  • K = R2 / (R1+R2)
  • K' = R2 / (R1*1.01 + R2)
• Evaluate S for R1 = 800k, R2 = 200k, ∆R1/R1 = 0.01
  • K = 200k/(800k+200k) = 0.200
  • K' = 200k/(800k*1.01+200k) = 0.198
  • S =  (∆K/K) / (∆R1/R1)
       = ( (0.198 - 0.200)/0.200 ) / (0.01)
       = -0.8
• This tells us that for any R tolerance Tol, the Gain Error will be Tol x -0.8.

 

R2 TOLERANCE

Find the Gain Sensitivity

• Analysis Node:  v1
• Gain:  K = v1/vin = R2/(R1+R2)
• Sensitivity Equation (Delta Method)
  •  S = (∆K/K) / (∆R2/R2)
• Pick any small value of ∆R2/R2 (0.01, 0.001, ...) to find S.
• How does the gain change for a 1% (for example) change in R2?
  • ∆K = K' -  K
  • K = R2 / (R1+R2)
  • K' = R2*1.01 / (R1 + R2*1.01)
• Evaluate S for R1 = 800k, R2 = 200k, ∆R2/R2 = 0.01
  • K = 200k/(800k+200k) = 0.200
  • K' = 200k*1.01/(800k+200k*1.01) = 0.204
  • S =  (∆K/K) / (∆R2/R2)
       = ( (0.202 - 0.200)/0.200 ) / (0.01)
       = +0.8
• This tells us that for any R tolerance Tol, the Gain Error will be Tol x +0.8.

 

ADC GAIN ERROR

Find the Gain Sensitivity

• Analysis Node:  ADCword
• Signal Gain:  K = ADCword / vadc
• Sensitivity (by inspection)
  •  S =  (∆K/K) / (∆Kadc/Kadc) = 1.0
• No calculation needed. By definition, the ADC Gain Error directly scales the slope of ADCword versus vadc.

 

VREF (ADC) ERROR

Find the Gain Sensitivity

• Analysis Node:  ADCword
• Signal Gain:  K = ADCword/vadc = (1024/Vref)
• Sensitivity Equation (Delta Method)
  •  S = (∆K/K) / (∆Vref)
• Pick any small value of ∆Vref (0.01, 0.001, ...) to find S.
• How does the gain change for a 0.001V (for example) change in Vref?
  • ∆K = K' -  K
  • K = 1024 / Vref
  • K' = 1024 / (Vref + 0.001)
• Evaluate S
  • K = 1024 / 5.0 = 204.80
  • K' = 1024 / (5.0+0.001) = 204.76
  • S =  (∆K/K) / (∆Vref)
       = ( (204.76 - 204.8) / 204.8 ) / (0.001)
       = -0.2
• This tells us that for any ∆Vref, the Gain Error will be ∆Vref x -0.2.

 

TOTAL GAIN ERRORS

Worst Case Analysis estimates the total error under the most unfavoable conditions: all errors at their maximum limit AND in the same polaritiy.

• WCA = | ∆K1/K1 | + | ∆K2/K2 | + ...
         = |Gain_Err_R1| + |Gain_Err_R2| + |Gain_Err_ADC| + |Gain_Err_Vref|
• 4V:  WCA = | 0% | +  | 0% | + | 0.2% | + | -1.0% |
                = 1.2%
• 20V:  WCA = | -0.8% | +  | 0.8% | + | 0.2% | + | -1.0% | 
                  = 2.8%

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

• 4V:  WCA = 1.2% < 1.5%?
     Yes - PASS! 
• 20V: WCA = 2.8% < 1.5%?
     No - FAIL! 

The GOOD News - We've completed the Gain Error Analysis, congratulations!

The BAD News (that's really GOOD News) - We've uncovered that the design does NOT MEET SPEC on the 20V Range!

 

HANDS-ON REDESIGN

Play in the Excel file - modify values, see what happens!

• Excel File:  DVM1_-_EBA_Offset_Gain_Errors1.xlsx
• Right Click on the filename, select "Save link as...".

Redesign 20V Range

• Ask yourself - which are the biggest gain error contributors to the 20V Gain Range?
• Looks like the R1 and R2 tolerances (1%) standout above the others.
• The next highter precision grades are 0.5%, 0.1% and 0.05%. Which precision grade is needed to meet the Gain Error Spec?
• Open the Excel file above and choose a new tolerance for R1, R2 to get the total Gain Error < 1.5%.

 

NEXT UP

We completed the Hardware Design, so now let's tackle the Sofware Design.

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