Dec. 3, 2020

4th International Conference on Technology and Social ScienceAnalysis, Testing and Calibration ofCharge Distribution SAR ADC
Architecture with Split Capacitor
Riho Aoki, Jianglin Wei, Yujie Zhao, Anna Kuwana
Haruo Kobayashi, Takayuki Nakatani, Kazumi Hatayama
Keno Sato, Takashi Ishida,Toshiyuki Okamoto, Tamotsu Ichikawa
Division of Electronics and Informatics
Gunma University
ROHM Semiconductor
Kobayashi Lab.
Gunma University

Outline• Research Background• Split Capacitor Calculation• Split Capacitor Parasitic Capacitance Problem
 In case of parasitic capacitance in left side
 In case of parasitic capacitance in right side How to find parasitic capacitance value• Capacitor Mismatch Effects to SAR ADC
Non-Linearity• Testing and Calibration• Conclusion

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Outline• Research Background• Split Capacitor Calculation• Split Capacitor Parasitic Capacitance Problem
 In case of parasitic capacitance in left side
 In case of parasitic capacitance in right side How to find parasitic capacitance value• Capacitor Mismatch Effects to SAR ADC
Non-Linearity• Testing and Calibration• Conclusion

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Research BackgroundNatural signals are analog

e.g. sound, light,temperature, pressure, etc.

Signals computer can handle are digitale.g. 0 or 1Requires analog-to-digital converter (ADC)

Analog input

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Digital output

SAR, Pipeline, ΔΣ, etc.

SAR ADC Configuration
ObjectBalance Scale

Linearity reference ofthe entire ADC is
the internal DAC linearity

Based onPrinciple of Balance

WeightGenerally use binary weight(1 , 2 , 4 , 8 , 16 , 32, 64 …) 1 2 4

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Problem with Conventional SAR ADC

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Key part of SAR ADC

Small capacity

Achieves high accuracy witha minute capacitance

Capacitance ratio with othercapacitances in t circuit → Largee.g. C≪32C

Difficult tomanufacture!
SAR ADC：
Successive-Approximation-Register
Analog-to-Digital Converter7/46

Improvement using Split Capacitor
Key part of SAR ADC
Split Capacitor

Large capacity

Reduce capacitance ratio
Obtain high-precision output
without using a minute capacitance

Using Split Capacitor

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Research Goal

Examine parasitic capacitance effectcaused by split capacitor

Vout=0Cp

Parasitic capacitance

Vref8C+Δ8C

Examine mismatchesamong split capacitor

VrefOutline• Research Background• Split Capacitor Calculation• Split Capacitor Parasitic Capacitance Problem
 In case of parasitic capacitance in left side
 In case of parasitic capacitance in right side How to find parasitic capacitance value• Capacitor Mismatch Effects to SAR ADC
Non-Linearity• Testing and Calibration• Conclusion

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Combined Capacitance

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Cs

Parallel

8C+16C+32C=56C

Equivalent circuit

Combined capacitance
Series156𝐶𝐶𝑆=1
156𝐶 + 𝐶𝑆+56𝐶 𝐶𝑠

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Split Capacitor Calculation
C+2C+4C=7CC2C

4C

Vout=08C

16C

32C

Vref

56𝐶𝐶𝑆
56𝐶 + 𝐶𝑆 Split Capacitor
8C

16C

32C

8C

16C

Cs=8C
Vout=032C

Vref

Outline• Research Background• Split Capacitor Calculation• Split Capacitor Parasitic Capacitance Problem
 In case of parasitic capacitance in left side
 In case of parasitic capacitance in right side How to find parasitic capacitance value• Capacitor Mismatch Effects to SAR ADC
Non-Linearity• Testing and Calibration• Conclusion

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Parasitic Capacitance Problem
Vout=0CpThe parasitic capacitance

Vref

Advantage

Alleviate problem of capacitance ratio

Disadvantage Affect overall ADC linearity

Derivation of general formula
Obtain parasitic capacitance value

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Outline

14/46• Research Background• Split Capacitor Calculation• Split Capacitor Parasitic Capacitance Problem
 In case of parasitic capacitance in left side
 In case of parasitic capacitance in the right side How to find the parasitic capacitance value• Capacitor Mismatch Effects to SAR ADC
Non-Linearity• Testing and Calibration• Conclusion

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Initialization
Set charge amount to each capacitance to 0

Vout=0Cp

Vref

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Calculation of Vxx

2C+16C=18C

Vxx → V18

Equivalent to 2CVxx
CpVref17/46

Calculation of V01(1)
V01’

V01

Q0 ＋ Q1 ＋ Q2 ＋

Q3 ＋ Q4 ＋ Q5ー1ー0ー0ー0ー0＋ー0

Law of conservation of electric charges
Q0+Q1+Q2=Qs
Qs+Q3+Q4+Q5=0

Vref18/46

Calculation of V01(2)
V01’

V01

Q0 ＋ Q1 ＋ Q2 ＋Q3 ＋ Q4 ＋ Q5ーーーー＋ーーVref

1Q0+Q1+Q2=Qs
Qs+Q3+Q4+Q5=0Q0=8C(V01’-Vref)
Q1=16CV01’
Q2=32CV01’
Q3=8CV01
Q4=16CV01
Q5=32CV01000008C(V01’-Vref)+16CV01’+32CV01’+8CV01+16CV01+32CV01=0
56CV01=8CVref – 56CV01’
V01=(8/56)Vref –V01’?

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Calculation of V01’
V01’

V01

Q0 ＋ Q1 ＋ Q2 ＋Q3 ＋ Q4 ＋ Q5ーーーー＋ーーVref

100000Charge of split capacitor → Sum of charges of 8C, 16C, 32C
8C(V01-V01’)=8C(V01’-Vref)+16CV01’+32CV01’64CV01’=8CVref+8CV01
8Vref + 8V01∴ V01′ =64

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Calculation of V01(3)
V01’

V01

Q0 ＋ Q1 ＋ Q2 ＋Q3 ＋ Q4 ＋ Q5ーーーー＋ーー

Vref

1000

V01=(8/56)Vref – V01’
8Vref 8Vref + 8V01
V01 =−56
64

V01 =

1Vref
63

00
SubstituteV01′ =

8Vref + 8V01
64

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Vxx formula(no parasitic capacitance)
1V01 =
Vref
63
2V02 =
Vref
63xx
Vxx =
Vref
63・・・
5V05 =
Vref
63・・・
29V29 =Vref
63

Vxx is proportional to xx

LinearGraph without Parasitic Capacitance

When Vref=1Linear
0.4xx
Vxx =
630.35Vxx

0.30.25

0.2
0.15
0.1
0.05
00522/46

10152025xx

No parasitic capacitanceLinear characteristics

3 Capacitances to Left

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3 capacitancesCp

xx=00～xx=078xx
Vxx =
Vref
63 × 8 + 8α

xx=08～xx=15Vxx =8xx + αVref
63 × 8 + 8α

xx=16～xx=23Vxx =8xx + 2αVref
63 × 8 + 8α

xx=24～xx=318xx + 3α
Vxx =
Vref
63 × 8 + 8α

Vrefα=Cp/CVxx

Effect of Parasitic Capacitance
10.9
0.8
0.7
0.60.50.4
0.3
0.2
0.1
008162432404856xx

Linearity breaks in multiples of 824/46

Consideration

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CpC2C4CVref

7 is MAX on the left sideof Split Capacitor

The effect appears every 8

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General Formula(Split Capacitor/C) × xx + Bα
Vxx =
Vref
63 × (Split Capacitor/C) + Aα
Sum of total capacitors(1+2+4+8+16+32)
A, B depend on split capacitor position
Left capacity number 3
Split Capacitor=8C

Left capacity number 2
Split Capacitor=4C

Left capacity number 4
Split Capacitor=20CA81617B

Increase by 1

Increase by 1

Increase by 4

General Formula Summary

27/46(Split Capacitor/C) × xx + Bα
Vxx =
Vref
63 × (Split Capacitor/C) + Aα
Sum of total capacitors(1+2+4+8+16+32)
Sum of capacity on right side of Split Capacitor Value of Split Capacitor
A=+Minimum capacity
Minimum capacity
Capacitor weight to right of Split Capacitor
B=
Minimum capacity

If the shape of the circuit is known,
the general formula can be derived

Outline• Research Background• Split Capacitor Calculation• Split Capacitor Parasitic Capacitance Problem
 In case of parasitic capacitance in left side
 In case of parasitic capacitance in right side How to find parasitic capacitance value• Capacitor Mismatch Effects to SAR ADC
Non-Linearity• Testing and Calibration• Conclusion

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3 Capacitances to Right
Qp
Cp2

VrefxxVxx =Vref
63 + β

β=Cp2/C
Cp2=βC

Parasitic capacitance：Right side of split capacitor
Vxx is proportional to xx and no non-linearity appears

Outline• Research Background• Split Capacitor Calculation• Split Capacitor Parasitic Capacitance Problem
 In case of parasitic capacitance in left side
 In case of parasitic capacitance in right side How to find parasitic capacitance value• Capacitor Mismatch Effects to SAR ADC
Non-Linearity• Testing and Calibration• Conclusion

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Measurement of V01
V01’’
Q0

Q1

Q2

Qp

V01
Q4

Q3

Q5

Cp

Unknown value of Cp
Vref

100

Connect only
the least significant bit to Vref
Vxx at this time is V01

000

With internal or external ADC
Measure V01
SupposeV01 = 0.0136986V

Preparation of General Formula (1)
V01’’
Q0

Q1

Q2

Qp

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V01
Q4

Q3

Q5Cp

Unknown value of Cp
Vref

100000Sum of capacity on the right side of Split Capacitor Value of Split Capacitor+Minimum capacity
Minimum capacity
56 8=+ =88 8

A=Preparation of General Formula (2)
V01’’
Q0

Q1

Q2

Qp

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V01
Q4

Q3

Q5Cp

Unknown value of Cp
Vref

100000(Split Capacitor/C) × xx + BαVxx =Vref
63 × (Split Capacitor/C) + Aα
A=88xxVxx =Vref
63 × 8 + 8α

Substitute in General FormulaV01’’
Q0

Q1

Q2

Qp

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V01
Q4Q3

Q5Cp

Unknown value of Cp
Vref

1008xxVxx =Vref
63 × 8 + 8α000

Substitute xx=01, Vxx=0.0136986V8×10.0136986 =×163 × 8 + 8α
Set Vref=1

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Solve α
V01’’
Q0

Q1

Q2

Qp

V01
Q4Q3

Q5

Cp

Unknown value of Cp
Vref

10000010.0136986 =63 + α0.8630118+ 0.0136986α=10.0136986α=0.1369882∴α=10.0001606Divide by CParasitic capacitance can be foundOutline• Research Background• Split Capacitor calculation• Split Capacitor Parasitic Capacitance Problem
 In case of parasitic capacitance in left side
 In case of parasitic capacitance in right side How to find parasitic capacitance value• Capacitor Mismatch Effects to SAR ADC
Non-Linearity• Testing and Calibration• Conclusion

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Split Capacitor Mismatch Effects

Examination in case of mismatch in split capacitor

8C+Δ8C

Vxx

Vref

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Vxx

Graph of Vxx
10.90.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
008

162432xx4048

Mismatch of split capacitor is 15%
Non-linearity at multiples of 856

39/46Graph of INL
0.015
0.01

INL

0.005
008

16

24

32

-0.005-0.01-0.015

xx

40

48

56

Outline• Research Background• Split Capacitor Calculation• Split Capacitor Parasitic Capacitance Problem
 In case of parasitic capacitance in left side
 In case of parasitic capacitance in right side How to find the parasitic capacitance value• Capacitor Mismatch Effects to SAR ADC
Non-Linearity• Testing and Calibration• Conclusion

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Calibration of Mismatch EffectCalibration of mismatch of split capacitor

8C+Δ8CVxx

Vref● Measure the error at mass production test stage
and store its data in Flash memory.
● Then at normal operation stage, use it for correction.

42/46Vxx

Before Correction
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
008162432
xx

404856Store Vxx in memory when there is a mismatch

43/46After Correction
xxVxx × Y =63
Derive Y and multiply Vxx stored in memory by this Y
1Vxx

0.8
0.6
0.4
0.2
008162432xx

404856

44/46Graph of INL
0.02

INL

0.01
0-0.010816

INL is almost 0INL

-0.02

2E-16
1.5E-16
1E-16
5E-17
0-5E-17 0-1E-16

2432

40

4856

xx

816 24 32 40 48 56xx

Outline

45/46• Research Background• Split Capacitor calculation• Split Capacitor Parasitic Capacitance Problem
 In case of parasitic capacitance in the left side
 In case of parasitic capacitance in the right side How to find the parasitic capacitance value• Capacitor Mismatch Effects to SAR ADC
Non-Linearity• Testing and Calibration• Conclusion

ConclusionEffect when split capacitor hasparasitic capacitance is analyzed.
General formula of internal output (Vxx)
when there is parasitic capacitance isderived.
Method ofderiving value of parasitic capacitancefrom the general formula is shown.

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Q&AQ. If there are split capacitor errors andparasitic capacitance at the same time?
A. I haven't verified it yet.
I’ll do future research.
Q. What is the advantage ofusing split capacitor?
A. Since it’s not necessary to use
a small capacity,
the capacity ratio becomes small andit becomes easy to manufacture.

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