Differential Amp - Part 2

Common-Mode Offset Errors:
vcmr, R1, R2, R3, R4

 

Ideally, the differential amplifier rejects the common-mode level while amplifying the differential signal. However, the tolerance of resistors (R1-R4) as well as the U1's common-mode error voltage (vcmr) creates unwanted Common-Mode Rejection Offset Errors. How can you predict these errors? Part 2 breaks down the following offset errors into manageable concepts and steps.

• vcmr, Input Common-Mode Rejection Error
• R1-R4, Resistor Tolerance

Part 1 tackles the basic Differential Offset and Gain Errors.

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

 

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 = 2.5mV
• Gain Error:  ∆K/K = 0.5% of Reading

AMPLIFIER

Schematic with Error Sources

Here's a color map of the Offset and Gain Errors. Note how resistors R1-R4 contribute to both!

Error Sources

Part 2 examines the Offset Errors cause by the Common-Mode errors shown in yellow below.

Description	Initial	Drift
OFFSET ERRORS
voff, Input Offset Voltage ib, Input Bias Current iboff, Input Offset Current vcmr, Common-Mode Rejection Error R1, Resistor Tolerance R2, Resistor Tolerance R3, Resistor Tolerance R4, Resistor Tolerance	0.1 mV 5 nA 5 nA 80 dB 0.1 % 0.1 % 0.1 % 0.1 %	10 uV / C 1 nA / C 1 nA / C 25 ppm / C 25 ppm / C 25 ppm / C 25 ppm / C
GAIN ERRORS
R1, Resistor Tolerance R2, Resistor Tolerance R3, Resistor Tolerance R4, Resistor Tolerance	0.1 % 0.1 % 0.1 % 0.1 %

 

Conditions and Assumptions

Temperature

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

Amplifier

• vin = 100mV
  •  equal pos & neg signals
         vin+ = 50mV, vin- = -50mV
  • see theory refresh for "Three Signal Types" below.
• vcm = 5V
• R1=100k, R2=500k,  R3=100k, R4=500k
• K = vo/vin = R2/R1 = +5.0
• vo = 100mV * 5 = 500mV

Errors

• All errors can be either polarity +/-.
• Errors will be Referred to Input (RTI)
• Errors totaled as Root Sum Square (RSS)

 

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 OFFSET VOLTAGE

U1's input common-mode rejection error produces an offset voltage modelled at the positive input. The error vcmr gets amplified by the non-invertng gain.

Description	Initial Errors
Error Source: e	vcmr = vcm_vp / CMRR = 4.17V / 1080/20  = 0.416mV
Pick Analysis Node: Va	vo
Calc Sensitivity: S How does e impact Va?	S = vo / voff   = R2/R1+1   = 6
Calc Offset Error at Analysis Node   Initial:  ∆Voffset = e * S   Drift:   ∆Voffset = e * ∆T * S	∆Voffset  = 0.416mV * 6  = 2.5mV
Calc Gain from Input to Analysis Node:   Ka = Va / Vin	Ka = vin/vo  = R2/R1  = 5
Calc Error RTI (Referred-to-Input):   ∆voffset_RTI = ∆voffset / Ka	∆voffset_RTI  = 2.5mV / 5  = 0.5mV

 

COMMON-MODE RESISTORS R1-R4 - Theory Refresh

How can you quantify the Offset Error due to resistor tolerances? Calculating the Sensitivities requires applying the Difference Method to the common-mode output equation.

What is the output due to vcm? By setting vin = 0V, you can write vo as

To make the calculations of Sensitivity (S) easier, let's rewrite vo with fewer variables

By setting equal ratios R2/R1 = R4/R3, we get the familiar ideal output

        vo = 0V

 

RESISTOR R3

As an example, we'll walk through the Common-Mode Rejection contributed by R3.

To calculate Sensitivity, start with the ideal vo

Increment R3 by a small ratio (say 1.01) to find it's impact on vo.

Calculate the Sensitivity of vo to R3.

Finally calculate the actual output error given R3's tolerance

Let's jump in with some numbers

Description	Initial Errors	Drift Errors
Error Source: e	R3_Tol  = 0.1%	R4_TC   = 25ppm/C   = 25e-6/C
Pick Analysis Node: Va	vo	vo
Calc Sensitivity: S How does e impact Va? Apply Difference Method:  S = (∆vo) / (∆R/R) where  ∆vo = vo'-vo	R1=R3=100k R2=R4=500k vo = 0 vo' = -0.0416 ∆R/R = 0.01 S = ∆vo / (∆R/R)    = -4.16	S = -4.16
Calc Offset Error at Analysis Node   Initial:  ∆voffset = e * S   Drift:   ∆voffset = e * ∆T * S	∆voffset  = 0.1% * -4.16  = -4.16mV	∆K/K  = 25e-6/C*30C*-4.16  = -3.12mV
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  = -4.16mV / 5  = -0.832mV	∆voffset_RTI  = -3.12mV / 5  = -0.624mV

 

RESISTORS R1, R2, R4

For the remaining resistors, simply follow the basic method shown above for R3. Take each resistor, increment its value by 1.01 to find vo', then calculate the Sensitivity as

   S = ∆vo / (∆R/R)
      = (vo'-vo) / 0.01

For the error results of all resistors, check out the hands-on Excel file (see link 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!

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

 

Explore the hands-on spreadsheet!

• Excel file:  diff-amp1-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!

• The total Offset Error (RSS) does not meet our spec of 2.5mV!
• How can you meet spec? Try lowering the R1-R4 tempcos from 25ppm/C to 10ppm/C or 5ppm/C.

 

For tutorials and other examples, goto EBA Series