DESIGN OF ACTIVE INDUCTOR AND STABILITY TEST FOR PASSIVE RLC LOW-PASS FILTERS Minh Tri Tran * , Anna Kuwana, Haruo Kobayashi Gunma 6 th International Conference on Signal and Image Processing (SIPRO 2020) July 25-26, 2020, London, United Kingdom Kobayashi Laboratory
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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
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
2
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.
3
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
4
1. Research BackgroundApproaching Methods
Passive RLC Low-pass Filter Active RLC Low-pass Filter
Balun transformerinput
output
Implementation of active RLC LPF
5
1. Research BackgroundReviews of Complex Functions
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
14
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
15
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.
16
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
17
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:
18
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
19
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
20
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:
21
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
22
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
23
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
24
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
25
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
26
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
27
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
28
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
29
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
30
3. Proposed Designs and Experimental ResultsMeasured Transfer Function in Serial RLC LPF
Phase response
Magnitude response
Transient response
31
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)
32
3. Proposed Designs and Experimental ResultsImplementation of Active RLC Low-pass Filter
Device under test
Over-shoot occurred at output port.
input
output
33
3. Proposed Designs and Experimental ResultsMeasured Transfer Function of Active RLC LPF
Phase response
Magnitude response
Transient response
34
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
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
[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
Thank you very much!
Gunma
KobayashiLaboratory
6th International Conference on Signal and Image Processing (SIPRO 2020)