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DESIGN OF ACTIVE INDUCTOR AND STABILITY TEST FOR PASSIVE RLC
LOW-PASS FILTERS
Minh Tri Tran*, Anna Kuwana, Haruo Kobayashi
Gunma
6th International Conference on Signal and Image Processing (SIPRO 2020)
July 25-26, 2020, London, United Kingdom
KobayashiLaboratory
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1. Research Background• Reviews of Complex Functions• Transfer Function and Its Self-loop Function• Limitations of Conventional Methods2. Analysis of High-Order Transfer Functions• Behaviors of High-order Passive Transmission Spaces• Numerical Examples and Design of Active Inductor3. Experimental Results• Measurements of Self-loop Functions in RLC networks• Operating region of Active Serial RLC Low-pass Filter4. Conclusions
Outline
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1. Research BackgroundMotivation of Study
Large overshoots + ringing + unwanted voltage transients Damped oscillation noise Unstable system
Overshoot
Undershoot
STABILITY TEST
o Ringing occurs in both with and withoutfeedback systems.
o Ringing affects both input and output signals.
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o Investigation of operating region of high-order systems in both time and frequency domains
Over-damping (high delay in rising time) Critical damping (max power propagation) Under-damping (overshoot and ringing)
1. Research BackgroundObjectives and Achievements
o Design of active inductor and measurement of self-loop function in active serial RLC LPF
Achievements
Objectives
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1. Research BackgroundApproaching Methods
Passive RLC Low-pass Filter Active RLC Low-pass Filter
Balun transformerinput
output
Implementation of active RLC LPF
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1. Research BackgroundReviews of Complex Functions
Re Im Real Imag H j H j H
Complex function with frequency variable
In complex plane domain
2 2Re Im
Imarctan
Re
jH H e
H H H
H
H
In spectrum domain
Re Real
Im Imag
Fre
H
H H
angular frequency
oPolar chart (Nyquist chart)oMagnitude-frequency, angular-frequency plots (Bode plots)oMagnitude-angular diagrams (Nicholas diagrams)
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1. Research BackgroundTransfer Function and Its Self-loop Function
( ) ( )( )( ) 1 ( )
out
in
V AHV L
Transfer function
( )L
( )A( )H
: Open loop function : Transfer function
: Self-loop function
((0
)) AH
Unstable system
1 ( ) 0 L
Constraint for oscillation
( 180( ) 1) o
LL
Linear systemInput Output
( )H( )inV ( )outV
PHASE MARGIN AT UNITY GAIN
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1. Research BackgroundSignal Flow Graph for Transfer Function
( )( ) ( ) ( )
out in out
LV A V VA
( ) ( )( )( ) 1 ( )
out
in
V AHV L
Transfer function
Output voltage
Signal flow graph
To meet the specified requirements○ High stability ○ Fast transient response, and ○ Good steady-state performance.
Negative feedback Network
STABILITY TEST
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1. Research BackgroundProposed Comparison Measurement Technique
( ) ( )( )( ) 1 ( )
out
in
V AHV L
Open loop function
Self-loop function
Sequence of steps: (i) Measurement of open loop function A(ω), (ii) Measurement of transfer function H(ω), and(iii) Derivation of self-loop function.
( )( ) 1( )
ALH
Transfer function
( )( )
( )
i
o
VA
VStep1:
Step2:
Step3:
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1. Research BackgroundProposed Alternating Current Conservation (1)
( ) inc
trans
VL
V
Idea: Alternating current is conserved.
Incident current = Transmitted current
Incident current
Transmitted current
Self-loop function:
( )( ) ( )
inc
transV L VA A
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1. Research BackgroundProposed Alternating Current Conservation (2)
One voltage source
One current source Two splitting current sources
Two splitting voltage sources
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1. Research BackgroundProposed Alternating Current Conservation (3)Alternating current conservation using balun transformer
( ) inc
trans
VL
V
Self-loop function:
Balun transformer(10 mH inductance)
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1. Research BackgroundProposed Widened Superposition Principle
n n
i=1 i
i
i i=1A
1 =d d
(t)( EE t) 1d
2d
3d4d
5d
nd …
1E (t)
2E (t)
3E (t)
4E (t)
5E (t)nE (t)
AE (t)
Widened Superposition Principle:
EA(t) : Energy at one placeEi(t) : Input sourcesdi(t) : Resistance distances
o Multi-source systems, feedback networks (op amps, amplifiers), polyphase filters, complex filters…
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1. Research Background Limitations of Conventional Methods (1)
Current injection method Voltage injection method
Current injection Voltage injection
[7] Middlebrook, R.D., "Measurement of Loop Gain in Feedback Systems", Int. J. Electronics, vol 38, No. 4, pp. 485-512, 1975.
Measurement of loop gain( )
( )( )
oA
iA
VL
v
Difficult to measure self-loop function in analog circuits
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1. Research Background Limitations of Conventional Methods (2)
Replica measurementMeasurement of loop gain
[9] A. S. Sedra and K. C. Smith, “Microelectronic Circuits,” 6th ed. Oxford University Press, New York, 2010.
( )( )( )
r
t
VLV
Difficult to measure two real different circuits
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1. Research Background Limitations of Conventional Methods (3)
o Conventional Superposition:Solving for every source voltage and current, perhaps several times.
o Conventional measurement of loop gain (Middle Brook’s)Applying only in feedback systems (switching DC-DC converters).
o Conventional replica measurement of loop gainUsing two identical networks (difficult in practical measurement).
oConventional Nyquist’s stability condition Using in theoretical analysis for feedback systems (Lab simulation).
o Conventional concepts, analysis and measurement of loop gain are not unique.
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2. Analysis of High-Order Transfer FunctionsSecond-order Parallel RLC Low-pass Filter
1 1 1 ;
inout
C L L
VV
R Z Z Z
20 1
20 1
1( ) ;1
( ) ;
out
in
VHV a j a j
L a j a j
21 / 22
L CR Z Z R
LC L21 / 2
2
L CR Z Z R
LC L21 / 2
2
L CR Z Z R
LC L
Operating regions
•Over-damping:
•Critical damping:
•Under-damping:
.
Parallel RLC low-pass filter Apply superposition principle at Vout
Transfer function & self-loop function:
0 1; ; La LC aR
Where:
0
0 0
1 ;; 1 ;
L C
LCZ L Z C
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2. Analysis of High-Order Transfer FunctionsBehaviors of Second-order Transfer Function
Case Over-damped Critically damped Under-dampedDelta
Module
Angular
( )2
211 0
0 0
1 4 02
aa a
a a
221
1 00 0
1 4 02
a a aa a
221
1 00 0
1 4 02
a a aa a
( )H
0
2 22 2
2 21 1 1 1
0 0 0 0 0 0
1
1 12 2 2 2
a
a a a aa a a a a a
02
2 1
0
1
2
a
aa
0
2 22 2 2 2
1 1 1 1
0 0 0 0 0 0
1
1 12 2 2 2
a
a a a aa a a a a a
( ) 2 2
1 1 1 1
0 0 0 0 0 0
arctan arctan1 1
2 2 2 2
a a a aa a a a a a
0
1
22 arctan
aa
2 2
1 1
0 0 0 0
1 1
0 0
1 12 2
arctan arctan
2 2
a aa a a aa aa a
1
02 cut
aa
0
1
2( ) cut
aH
a( )
2
cut0
1
2( ) cut
aH
a ( )2
cut0
1
2( ) cut
aH
a ( )2
cut
20 1
1( )1 ( )
Ha j a j
Second-order transfer function:
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2. Analysis of High-Order Transfer FunctionsExample of Second-order Transfer Function
•Under-damping:
•Critical damping:
•Over-damping:
Magnitude of transfer function
1 12 2 2
1 1
1 1( ) ( )1 3 1 3 1
2 4 2 4
3 1 3 1 3 1( ) 1 12 2 2
cut
H Hj j
H
3 32 2 2
2 2
3 1
1 1( ) ( )3 1 3 5 3 5
2 2
3 5 3 5 3 5( ) 1 12 2 2
cut
H Hj j
H
2 22 2
1 1( ) ( ) 112 1
cutH Hj j
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2. Analysis of High-Order Transfer FunctionsSimulations of Second-order Transfer Function
1 2
2 2
3 2
1( ) ;1
1( ) ;2 1
1( ) ;3 1
Hj j
Hj j
Hj j
•Under-damping:
•Critical damping:
•Over-damping:
Phase response
Magnitude response
Polar chart of transfer function
Nyquist chart
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2. Analysis of High-Order Transfer FunctionsBehaviors of Second-order Self-loop Function
Case Over-damped Critically damped Under-damped
Delta ( ) 21 04 0 a a 2
1 04 0 a a 21 04 0 a a
( )L 2 20 1 a a 2 2
0 1 a a 2 20 1 a a
( ) 0
1arctan2
aa
0
1arctan2
aa
0
1arctan2
aa
11
0
5 22aa
1( ) 1 L 1( ) 76.3 o
1( ) 1 L 1( ) 76.3 o1( ) 1 L 1( ) 76.3 o
12
02aa
2( ) 5 L 2( ) 63.4 o
2( ) 5 L 2( ) 63.4 o2( ) 5 L 2( ) 63.4 o
13
0
aa
3( ) 4 2 L 3( ) 45 o3( ) 4 2 L 3( ) 45 o
3( ) 4 2 L 3( ) 45 o
0 1( ) L j a j aSecond-order self-loop function:
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2. Analysis of High-Order Transfer FunctionsBehaviors of Second-order Self-loop Function
0 1( ) L j a j aSecond-order self-loop function:
•Under-damping:
•Critical damping:
•Over-damping:
Phase margin at unity gain of self-loop function
1Phase margin ( ) 76.3 o
1Phase margin ( ) 76.3 o
1Phase margin ( ) 76.3 o
11
0
5 22aa
2 20 1( ) 1 L a aUnity gain of self-loop function
Angular frequency at unity gain
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2. Analysis of High-Order Transfer FunctionsSimulations of Second-order Self-loop Function
21
22
23
( ) ;
( ) 2 ;
( ) 3 ;
L j j
L j j
L j j
•Under-damping:
•Critical damping:
•Over-damping:
Phase response
Magnitude response
Polar chart of self-loop function
92o
103.7o
128o
Nyquist chart
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2. Analysis of High-Order Transfer FunctionsSummary of Second-order System
Over-damping:Phase margin is 88 degrees.Critical damping:Phase margin is 76.3 degrees.Under-damping: Phase margin is 52 degrees.
92o
103.7o 128o
Magnitude-angular response of self-loop functionMagnitude response of transfer function
Transient response
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2. Analysis of High-Order Transfer FunctionsMathematical Model of Ideal Op Amp
0( )1
out
in in
bw
V AA jV V
5510( ) ; ( ) 10 1
2001200
in
out
VA L jVj
Here, GBW =10 MHz, DC gain Ao = 100000
00
bw bwGBWGBW A f fA
Ideal op amp
Equivalent model of op amp
Gain-bandwidth (GBW), bandwidth fbw
Open-loop function A(ω) of op amp
Open-loop function and self-loop function
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2. Analysis of High-Order Transfer FunctionsBehavior of Open-loop Function of Ideal Op Amp
510( )1
200
Aj
Open-loop function A(ω) Bode plots of open-loop function
Nyquist plot of open-loop function
Nyquist chart
BW =100 Hz
GBW =10 MHz
-45o
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2. Analysis of High-Order Transfer FunctionsBehavior of Self-loop Function of Ideal Op Amp
5( ) 10 1200
in
out
VL jV
Bode plots of self-loop functions
Magnitude-angular plot of self-loop function
Self-loop function L(ω)
90o
Phase margin = 90 degrees
Nicholas diagram
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2. Analysis of High-Order Transfer FunctionsAnalysis of Active InductorGeneral impedance converter
2 43
2 2
1 1
C C
V VV
R Z R Z
3 2 32
1 1L out out
C
R R RRZ Z sCZR Z R
1 3 5 V V V
Approximated value of active inductor
Apply superposition principle at V3
Here,
…….
outZ
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2. Analysis of High-Order Transfer FunctionsSimulations of Passive & Active RLC Low-pass Filters
Passive serial RLC Low-pass Filter
Active serial RLC Low-pass Filter
Magnitude response of transfer function
Magnitude response of transfer function
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3. Proposed Designs and Experimental ResultsImplementation of Second-order Serial RLC LPF
Device under test
Under-shoot occurred at both input and output ports.
input
output
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3. Proposed Designs and Experimental ResultsMeasured Transfer Function in Serial RLC LPF
Phase response
Magnitude response
Transient response
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3. Proposed Designs and Experimental ResultsMeasured Self-loop Function in Serial RLC LPF
Over-damping:Phase margin is 80 degrees.Nearly Critical damping:Phase margin is 75 degrees.Under-damping: Phase margin is 55 degrees.
55o
75o
80o
Phase responseMagnitude response
Phas
e (d
eg)
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3. Proposed Designs and Experimental ResultsImplementation of Active RLC Low-pass Filter
Device under test
Over-shoot occurred at output port.
input
output
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3. Proposed Designs and Experimental ResultsMeasured Transfer Function of Active RLC LPF
Phase response
Magnitude response
Transient response
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3. Proposed Designs and Experimental ResultsMeasured Self-loop Function of Active RLC LPF
Over-damping:Phase margin is 84 degrees.Nearly Critical damping:Phase margin is 76 degrees.Under-damping: Phase margin is 65 degrees.
65o
76o84o
Phase responseMagnitude response
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This work:• Reviews of complex functions and stability test • Proposed methods for derivation of transfer function
and measurement of self-loop function• Implementations and measurements of self-loop
functions for passive and active second-order RLC low-pass filters
• Theoretically, if phase margin is smaller than 76.3-degrees, overshoot occurs in second-order systems.
Future of work:• Stability test for polyphase filters & complex filters
4. Conclusions
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[1] H. Kobayashi, N. Kushita, M. Tran, K. Asami, H. San, A. Kuwana "Analog - Mixed-Signal - RF Circuits for Complex Signal Processing", IEEE 13th International Conference on ASIC (ASICON 2019) Chongqing, China (Nov, 2019). [2] M. Tran, C. Huynh, "A Design of RF Front-End for ZigBee Receiver using Low-IF Architecture with Poly-phase Filter for Image Rejection", M.S. thesis, University of Technology Ho Chi Minh City – Vietnam (Dec. 2014).[3] H. Kobayashi, M. Tran, K. Asami, A. Kuwana, H. San, "Complex Signal Processing in Analog, Mixed - Signal Circuits", Proceedings of International Conference on Technology and Social Science 2019, Kiryu, Japan (May. 2019).[4] M. Tran, N. Kushita, A. Kuwana, H. Kobayashi "Flat Pass-Band Method with Two RC Band-Stop Filters for 4-Stage Passive RC Quadratic Filter in Low-IF Receiver Systems", IEEE 13th ASICON 2019 Chongqing, China (Nov. 2019). [5] M. Tran, Y. Sun, N. Oiwa, Y. Kobori, A. Kuwana, H. Kobayashi, "Mathematical Analysis and Design of Parallel RLC Network in Step-down Switching Power Conversion System", Proceedings of International Conference on Technology and Social Science (ICTSS 2019) Kiryu, Japan (May. 2019). [6] M. Tran, "Damped Oscillation Noise Test for Feedback Circuit Based on Comparison Measurement Technique", 73rd System LSI Joint Seminar, Tokyo Institute of Technology, Tokyo, Japan (Oct. 2019). [7] R. Middlebrook, "Measurement of Loop Gain in Feedback Systems", Int. J. Electronics, Vol 38, No. 4, pp. 485-512, (1975). [8] M. Tran, Y. Sun, Y. Kobori, A. Kuwana, H. Kobayashi, "Overshoot Cancelation Based on Balanced Charge-Discharge Time Condition for Buck Converter in Mobile Applications", IEEE 13th ASICON 2019 Chongqing, China (Nov, 2019). [9] A. Sedra, K. Smith (2010) Microelectronic Circuits 6th ed. Oxford University Press, New York. [10] R. Schaumann, M. Valkenberg, ( 2001) Design of Analog Filters, Oxford University Press.[11] B. Razavi, (2016) Design of Analog CMOS Integrated Circuits, 2nd Edition McGraw-Hill.[12] M. Tran, N. Miki, Y. Sun, Y. Kobori, H. Kobayashi, "EMI Reduction and Output Ripple Improvement of Switching DC-DC Converters with Linear Swept Frequency Modulation", IEEE 14th International Conference on Solid-State and Integrated Circuit Technology, Qingdao, China (Nov. 2018). [13] J. Wang, G. Adhikari, N. Tsukiji, M. Hirano, H. Kobayashi, K. Kurihara, A. Nagahama, I. Noda, K. Yoshii, "Equivalence Between Nyquist and Routh-Hurwitz Stability Criteria for Operational Amplifier Design", IEEE International Symposium on Intelligent Signal Processing and Communication Systems (ISPACS), Xiamen, China (Nov. 2017).
References
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Thank you very much!
Gunma
KobayashiLaboratory
6th International Conference on Signal and Image Processing (SIPRO 2020)
July 25-26, 2020, London, United Kingdom