Finite Aperture Time Effects in Sampling CircuitH... · 2015. 11. 26. · Derived Transfer Function...
Transcript of Finite Aperture Time Effects in Sampling CircuitH... · 2015. 11. 26. · Derived Transfer Function...
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.
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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
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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
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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
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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
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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
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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
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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.
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Consistency with Zero Aperture Time Case
Transfer function in case of finite aperture time
Transfer function in case of zero aperture time
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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
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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
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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
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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 !
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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
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ON-Conductance and Effective Aperture Time
Region Effective aperture time
1/R
on
(lo
g sc
ale
) [S
]
Vg [V]
200um
20um
2um
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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
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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
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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
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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
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-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
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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
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