Input Divider and ADC

R-Divider, 10-Bit ADC, +5V Ref

A basic input divider and ADC provide a easy introduction to the errors of input scaling and data conversion! 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
- enter errors in a systematic format
- vary parameters, see their effect on total error
- customize & expand template for your own designs

For Tutorials and more examples, see EBA Series

OFFSET AND GAIN ERRORS

What are the basic definitions of an amplifier's 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 = 0.25V (1% of 25V Full Scale)
Gain Error: ∆K = 1% of Reading

AMPLIFIER AND ADC

Schematic with Errors

Device Errors

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

Description Initial Temp Drift

OFFSET ERRORS

U1, voff, ADC Offset Error
U1, vres, ADC Resolution Error
U1, vinl, ADC Integral-Non Linearity Error 3 LSB
1/2 LSB
1 LSB
0.04 LSB/C

GAIN ERRORS

R1 Tolerance, Tempco
R2 Tolerance, Tempco

U1, ADC Gain Tol, Tempco
U2, Vref Tol, Tempco 0.1 %
0.1 %

2 LSB
0.5% 100 ppm / C
100 ppm / C

0.08 LSB/C
50 ppm/C

Description	Initial	Temp Drift
OFFSET ERRORS
U1, voff, ADC Offset Error U1, vres, ADC Resolution Error U1, vinl, ADC Integral-Non Linearity Error	3 LSB 1/2 LSB 1 LSB	0.04 LSB/C
GAIN ERRORS
R1 Tolerance, Tempco R2 Tolerance, Tempco U1, ADC Gain Tol, Tempco U2, 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

R-Divider

R1=800k, R2=200k
Vin = 25V, Vo = Vadc = 5V

ADC

Bits = 10
Vref = 5.0V

Errors

All errors can be either polarity +/-.
Errors will be Referred to Input (RTI)
Errors totalled 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.

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 0.00488 (V/LSB) = 14.6 mV	offset_TC = 0.04 LSB/C Convert to V: voff_TC = 0.04 LSB /C x 0.00488 (V/LSB) = 195 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 = 14.7 mV * 1 = 14.7 mV	∆Voffset = 195V/C * 30C * 1 = 5.86 mV
Calc Gain from Input to Analysis Node: Ka = Va / Vin	Ka = vadc/vin = R2/(R1+R2) = 0.2	Ka = 0.2
Calc Error RTI (Referred-to-Input): ∆voffset_RTI = ∆voffset / Ka	∆voffset_RTI = 14.6 mV / 0.2 = 73 mV	∆voffset_RTI = 5.86 mV / 0.2 = 29.3 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/1024*100% = 0.20%	Kadc_TC = 0.08LSB/C Convert to ppm/C: Kadc_TC = 0.08LSB/1024*1e6 = 78 ppm/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 = 0.20% * 1.0 = 0.20%	∆K/K = 78ppm/C30C1.0 = 2300ppm = 0.23%
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 (as Vref goes up, Kadc goes down.) 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 = (2^N-1)/Vref Vref = 5.0 2^N-1 = 1023 Kadc = 1023/5 = 204.6 Kadc' = 1023/(5*1.01) = 202.6 ∆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/C30C-1.0 = -1500ppm = -0.15%
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) R1 = 800k R2 = 200k K = 800k/(200k+800k) = 0.200 K' = 800k/(200k*1.01+800k) = 0.198 ∆R/R = 0.01 S = (∆K/K) / (∆R/R) = -0.8	S = -0.8
Calc Gain Error at Analysis Node Initial: ∆K/K = e * S Drift: ∆K/K = e * ∆T * S	∆K/K = 0.1%*-0.8 = -0.08%	∆K/K = 100ppm/C30C-0.8 = -2400ppm = -0.24%
Normalized gain errors can be referred to input as-is, no RTI calc needed.

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) R1 = 800k R2 = 200k K = 800k/(200k+800k) = 0.200 K' = 800k1.01/(200k+800k1.01) = 0.202 ∆R/R = 0.01 S = (∆K/K) / (∆R/R) = +0.8	S = 0.8
Calc Gain Error at Analysis Node Initial: ∆K/K = e * S Drift: ∆K/K = e * ∆T * S	∆K/K = 0.1% * 0.8 = 0.08%	∆K/K = 100ppm/C30C0.8 = 2400ppm = 0.24%
Normailzed gain errors can be referred to input as-is, no RTI calc needed.

EXCEL FILE

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

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

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 | + ...
= 139 mV

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

WCA < 250 mV?
Yes - PASS!

GAIN ERRORS

Calculate the total using Worst Case Analysis.

WCA = | ∆K1 | + | ∆K2 | + ...
= 1.0 %

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

WCA < 1.17%?
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 Sensitivities (S)

OFFSET Offset error sources Offset errors and totals

GAIN Gain error sources Gain errors and totals

While 3 worksheets may seem 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: divider-ref-ADC-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 Gain Error did NOT meet spec! Can we improve the design?
Open the Excel file. 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

Worksheet	Enter	Calculate
CIRCUIT CALC	Circuit values	Signal gains / levels and Sensitivities (S)
OFFSET	Offset error sources	Offset errors and totals
GAIN	Gain error sources	Gain errors and totals