design of active inductor and stability test for ... - …

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

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)

6

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

7

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

8

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:

9

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

10

1. Research BackgroundProposed Alternating Current Conservation (2)

One voltage source

One current source Two splitting current sources

Two splitting voltage sources

11

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)

12

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…

13

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)

July 25-26, 2020, London, United Kingdom