群馬大学 小林研究室

Gunma University Kobayashi Lab

Finite Aperture Time Effects in Sampling Circuit

M. Arai I. Shimizu H. Kobayashi K. Kurihara S. Sasaki S. Shibuya

K. Niitsu K. Kubo

Gunma University

Nagoya University

Oyama National College of Technology

C2-4 16:45-17:00 Nov. 4, 2015 (Wed)

Contents

● Research Objective ● Waveform Sampling and Sampling Circuit ● Finite Aperture Time - Problem Formulation - Formula Derivation - Effective Finite Aperture Time ● Uncertainty Relationship in Sampling Circuit ● Summary 2/34

Contents

● Research Objective ● Waveform Sampling and sampling Circuit ● Finite Aperture Time - Problem Formulation - Formula Derivation - Effective Finite Aperture Time ● Uncertainty Relationship ● Summary 3/34

Research Objective

● To establish fundamental theory of sampling circuit for high-frequency and high-precision waveform acquisition ● Especially, to clarify finite aperture time effect.

4/34

Contents

● Research Objective ● Waveform Sampling and Sampling Circuit ● Finite Aperture Time - Problem Formulation - Formula Derivation - Effective Finite Aperture Time ● Uncertainty Relationship ● Summary 5/34

Waveform Sampling

Waveform acquisition Sampling High-frequency signal sampling - Finite aperture time (non-zero turn-off time) - Aperture jitter

time

― analog signal

● Sampled point

Ts = 1 / fs

suffers from

volt

age

6/34

Sampling Circuit

7

時間

電圧

時間

電圧

時間

電圧

時間

電圧

CSW

Vin Vout

CSW

Vin Vout

CSW

Vin Vout

• SW: ON

•Vout(t) = Vin(t)

Track mode

•SW: OFF

•Vout(t) = Vin(tOFF)

Hold mode

Track Hold

time

time

time time

volt

age

volt

age

volt

age

vo

ltag

e

7/34

Contents

● Research Objective ● Waveform Sampling ● Finite Aperture Time - Problem Formulation - Formula Derivation - Effective Finite Aperture Time ● Uncertainty Relationship ● Summary 8/34

Finite Aperture Time

9

時間

電圧

時間

電圧

CSW

Vin Vout

CSW

Vin Vout

• SW: ON

•Vout(t) = Vin(t)

Track mode

•SW: OFF

•Vout(t) = Vin(tOFF)

Hold mode time

time

volt

age

vo

ltag

e

Finite transition time from track to hold modes

9/34

Analogy with Camera Shutter Speed

Camera: Finite Shutter Speed Sampling Circuit: Finite Aperture Time

Moving Object

Blurred

Input signal

Acquired signal

High frequency

Low pass filtered

10/34

Signal Frequency and Aperture Time

Higher frequency signal ⇒ More affected by finite aperture time

Ｈ T

time

time

Low frequency

Aperture time

Ｈ T

time

time

High frequency

Aperture time

Inp

ut

volt

age

Inp

ut

volt

age

Ou

tpu

t vo

ltag

e

Ou

tpu

t vo

ltag

e

11/34

Contents

● Research Objective ● Waveform Sampling ● Finite Aperture Time - Problem Formulation - Formula Derivation - Effective Finite Aperture Time ● Uncertainty Relationship ● Summary 12/34

VgVinVc

Transfer Function Derivation

Track Hold Circuit

+

-

Voltage

Time

Obtain values of ●

Equivalent time sampling

Obtain gain, phase for each frequency Frequency transfer function

SW

13/34

Contents

● Research Objective ● Waveform Sampling ● Finite Aperture Time - Problem Formulation - Formula Derivation - Effective Finite Aperture Time ● Uncertainty Relationship ● Summary 14/34

Derived Transfer Function

Transfer function in case of finite aperture time

Track Hold Circuit

+

-

SW

τ1 = R C

[1] A. Abidi, M. Arai, K. Niitsu, H. Kobayashi, “Finite Aperture Time Effects in Sampling Circuits,” 24th IEICE Workshop on Circuits and Systems, Awaji Island, Japan (Aug. 2011)

The formula derivation was done by Prof. Asad Abidi.

15/34

Consistency with Zero Aperture Time Case

Transfer function in case of finite aperture time

Transfer function in case of zero aperture time

16/34

τ1, τ2 Effects to Bandwidth

τ1 (= R C)

τ2 (aperture time) : varied : fixed

Bandwidth starts to decrease at τ2 / τ1= 1

τ1, τ2 effects to bandwidth are comparable.

Numerical calculation from the derived transfer function

17/34

Contents

● Research Objective ● Waveform Sampling ● Finite Aperture Time - Problem Formulation - Formula Derivation - Effective Finite Aperture Time ● Uncertainty Relationship ● Summary 18/34

VgVinVc

SPICE Simulation Verification

Track Hold Circuit

+

-

Voltage

Time

Obtain values of ●

Equivalent time sampling

Obtain gain, phase for each frequency Frequency transfer function

19/34

00.50.711.5234567

τ2[ns]0351015202530354050

τ2[ns]

SPICE Simulation Conditions

Results

SPICE Simulation Results

Gai

n [

dB

]

ω [rad/s]

Theory (Derived Transfer Function)

ω [rad/s]

Gai

n [

dB

]

Results

+

―

TSMC 0.18um

20/34

simulationtheory

Comparison of -3dB Bandwidth

W=20um

ω(-

3d

B)[

rad

/s]

14

12

0

2

4

6

8

10

0 5 15 10 τ2[ns]

simulationtheory

W=200um

ω(-

3d

B)[

rad

/s]

20

16

0

4

8

12

τ2[ns] 0 10 50 40 20 30

ω(-

3d

B)[

rad

/s]

simulationtheory

W=2um

τ2[ns] 0 10 50 40 20 30

35

30

0

5

10

15

20

25

Large discrepancies !

21/34

NMOS ON-Conductance Nonlinearity

Effective aperture time

W=20um

1/R

on

[S]

4

3

0

1

2

Vg[V] 0 0.5 2.0 1.0 1.5

W=2um

1/R

on

[S]

4

3

0

1

2

Vg[V] 0 0.5 2.0 1.0 1.5

W=200um

1/R

on

[S]

4

3

0

1

2

Vg[V] 0 0.5 2.0 1.0 1.5

○ region

Strong nonlinearity of 1/Ron

Define effective aperture time

22/34

ON-Conductance and Effective Aperture Time

Region Effective aperture time

1/R

on

(lo

g sc

ale

) [S

]

Vg [V]

200um

20um

2um

23/34

Empirical Effective Aperture Time Derivation

B A

0.43

Vg[V] 0 0.5 2.0 1.0 1.5

1/R

on

(lo

g sc

ale

) [S

]

20um

24/34

simulation*0.44/1.8theory

ω(-

3d

B)[

rad

/s]

τ2[ns]

20

0

5

10

15

0 5 15 10 τ2[ns]

ω(-

3d

B)[

rad

/s]

14 12

0 2 4 6 8

10

0 5 15 10 τ2[ns]

ω(-

3d

B)[

rad

/s]

35 30

0 5

10 15 20 25

0 5 15 10

simulation*0.43/1.8theory

simulation*0.37/1.8theory

simulationtheory

simulationtheory

simulationtheory

Discussion Again W=20um W=2um W=200um

τ2[ns]

ω(-

3d

B)[

rad

/s]

20

0

5

10

15

0 20 40 30 10

X 0.44/1.8 X 0.37/1.8 X 0.43/1.8

0 5 15 10

ω(-

3d

B)[

rad

/s]

14 12

0 2 4 6 8

10

τ2[ns]

ω(-

3d

B)[

rad

/s]

35 30

0 5

10 15 20 25

τ2[ns] 0 20 40 30 10

25/34

simulation*0.44/1.8theory

simulation*0.43/1.8theory

simulation*0.43/1.8theory

simulation*0.37/1.8theory

Various Values for RC, W

200um 20um

20um 2um

R=50Ω, C=10pF

R=50Ω, C=0.1pF

τ2[ns]

ω(-

3d

B)[

rad

/s]

20

0

5

10

15

0 4 10 6 2 8

ω(-

3d

B)[

rad

/s]

15

0

5

10

τ2[ns] 0 5 15 10

τ2[ns]

ω(-

3d

B)[

rad

/s]

15

0

5

10

0 0.5 1.5 1.0 τ2[ns]

ω(-

3d

B)[

rad

/s]

35 30

0 5

10 15 20 25

0 0.5 1.5 1.0

26/34

Contents

● Research Objective ● Waveform Sampling and Sampling Circuit ● Finite Aperture Time - Problem Formulation - Formula Derivation - Effective Finite Aperture Time ● Uncertainty Relationship in Sampling Circuit ● Summary 27/34

Trade-off of Time Constant and Bandwidth

Time

Short

Long

Wide

Narrow

Aperture time Bandwidth

RC time constant and bandwidth Aperture time and bandwidth

Band

RC bandwidth

28/34

-3dB Bandwidth

aperture time

In case of -3dB bandwidth

Transfer function

29/34

Uncertainty Relationship

2 4 6 8 10

2.0

1.5

1.0

0.5

0.5

1.0

Taylor expansion of Sinc function

Uncertainty Relationship Formula

bandwidth RC time constant

aperture time 30/34

Uncertainty Principle and Relationship

● Uncertainty Principle - Quantum mechanics

● Uncertainty Relationship - Sampling circuit

Can NOT be proved

Can be proved

Impossible to know exactly and simultaneously - Where the object is - How fast it is moving

Prof. W. K. Heisenberg

31/34

Contents

● Research Objective ● Waveform Sampling ● Finite Aperture Time - Problem Formulation - Formula Derivation - Effective Finite Aperture Time ● Uncertainty Relationship ● Summary 32/34

Summary

● Derived explicit transfer function of sampling circuit with finite aperture time effect. ● Verified it with SPICE simulation ● Introduced concept of effective finite aperture time ● Showed uncertainty relationship between time constants and bandwidth in sampling circuit. 33/34

Thank you for your kind attention

謝謝

34/34