STABILITY TEST FOR HIGH-ORDER LADDER LOW-PASS FILTERS · 2020. 11. 9. · STABILITY TEST FOR HIGH-ORDER LADDER LOW-PASS FILTERS MinhTriTran*, Anna Kuwana, and Haruo Kobayashi Gunma

STABILITY TEST FOR HIGH-ORDER LADDER LOW-PASS FILTERS

MinhTri Tran*, Anna Kuwana, and Haruo Kobayashi

Gunma

KobayashiLaboratory

Gunma University, Japan(Nov. 7th, 2020)

Virtual TJCAS 2020Taiwan and Japan Conference on

Circuits and Systems

1. Research Background• Motivation, objectives and achievements• Self-loop function in a transfer function2. Ringing Test for Passive Networks• Stability test for RLC low-pass filters3. Ringing Test for Ladder Low-Pass Filters• Stability test for active high-order ladder low-

pass filters4. Conclusions

Outline

2

1. Research BackgroundNoise in Electronic Systems

Common types of noise:• Electronic noise• Thermal noise,• Intermodulation noise,• Cross-talk,• Impulse noise,• Shot noise, and• Transit-time noise.

Linear networks• Overshoot,• Ringing• Oscillation noise

Device noise:• Flicker noise,• Thermal noise,• White noise.

Signal to Noise Ratio:

Performance of a devicePerformance of a system

Figure of Merit:

Signal power

SNRNoise power

Output SNRInput SNR

F

3

1. Research BackgroundEffects of Ringing on Electronic Systems

Ringing represents a distortion of a signal.Ringing is overshoot/undershoot voltage or current when it’s seen on time domain.

Ringing does the following things:• Causes EMI noise,• Increases current flow,• Consumes the power,• Decreases the performance, and• Damages the devices.

STABILITY TEST

Unstable system

4

• Derivation of transfer function in electronic systems using superposition theorem

o Investigation of operating regions of linear negative feedback networks

Over-damping (high delay in rising time) Critical damping (max power propagation) Under-damping (overshoot and ringing)• Stability test for electronic networks based on

alternating current conservation

1. Research BackgroundObjectives of Study

Alternating current conservationIncident current

Transmitted current

5

1. Research BackgroundAchievements of Study

Superposition formula for multi-source networks

1

1 ( ) ( )n n n

iO O ai gi

i=1 i=1 i=1i isi n

k pik

V (t)1V (t) V (t) = I t I t1Z ZZ1Z

( ) inc

trans

VV

L

Self-loop function

10 mHinductance

6

1. Research BackgroundApproaching Methods

3rd-order ladder LPFBalun

transformer

input

output

Implemented circuitDerivation of self-loop function

Baluntransformer

7

1. Research BackgroundSelf-loop Function in A Transfer Function

( ( )(( ) ))

)1

(out

in

H VV

AL

Transfer function

( )L

( )A ( )H

: Open loop function : Transfer function

: Self-loop function

Linear systemInput Output

( )H ( )inV ( )outV

0 1

0 1

( ) ... ( )(

)) ... (

()

nn n

nn n

Hb j b j ba j a j a

Model of a linear system

oMagnitude-frequency plotoAngular-frequency plot

oPolar chart Nyquist chart

oMagnitude-angular diagram Nichols diagram

Bode plots

Variable: angular frequency (ω)

8

1. Research BackgroundCharacteristics of Adaptive Feedback Network

Block diagram of a typical adaptive feedback system

Reference voltage

Adaptive feedback is used to control the output source along with the decision source (DC-DC Buck converter).

Transfer function of an adaptive feedback network is significantly different from transfer function of a linear negative feedback network. Loop gain is independent of frequency variable (referent voltage, feedback voltage, and error voltage are DC voltages).

DC voltage

DC voltage

DC voltage

DC voltage+ Ripple voltage

9

1. Research BackgroundAlternating Current Conservation

( ) 1( ) 1

;(

( )

)

in

o

i

ut

in

o

ou

n

ut

tHZZ

Z

V

L

V

Z

Transfer function

Simplified linear system

Self-loop function

( )inc trans in

t

c

tra

in

i u nsn o t ou

V ZL

Z ZV

VZV

Incident current

Transmitted current

10 mHinductance

Derivation of self-loop function

10

1. Research Background Limitations of Conventional Methods

o Middlebrook’s measurement of loop gainApplying only in feedback systems (DC-DC converters).o Replica measurement of loop gainUsing two identical networks (not real measurement).o Nyquist’s stability condition Theoretical analysis for feedback systems (Lab tool).o Nichols chart of loop gain Only used in feedback control theory (Lab tool).o Conventional superposition Solving for every source (several times).



Outline

12

2. Ringing Test for Passive NetworksCharacteristics of 2nd-order Transfer Function

Case Over-damping Critical damping Under-dampingDelta

Module

Angular

( )2

211 0

0 0

1 4 02

aa a

a a

2211 0

0 0

1 4 02

a a aa a

2211 0

0 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

11 62

2

dBa

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:

13

2. Ringing Test for Passive NetworksCharacteristics of 2nd-order Self-loop Function

Case Over-damping Critical damping Under-damping

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:

14

2. Ringing Test for Passive NetworksOperating Regions of 2nd-Order System

98o103.7o

128o

Nichols plot of self-loop functionBode plot of transfer function

Transient response

0dB

-12dB

-6dB

Phase margin

52o

21

2

3

2

2

1 ;1

1 ;1

1 ;

( )

( )

( )

2

3 1

j

j

j

j

j

j

H

H

H

•Under-damping:

•Critical damping:

•Over-damping:

21 ;( ) j jL

22 ;( 2) j jL

23 ;( 3) j jL

Phase margin 92o

Phase margin 76.3o

15

Passive RLC Low-pass Filter

0 12

1 ;( )1

out

in

VH

jaV ja

Derivation of self-loop function

02

1 ;( )L aja j

Self-loop function

Implemented circuit

Transfer function

2. Ringing Test for Passive NetworksStability Test for Passive 2rd-Order RLC LPF

0 1; ; a LC a RCwhere,

16

Transient responses

94o 107o

122o

2. Ringing Test for Passive NetworksStability Test for 2rd-Order Passive RLC LPF

Phase margin58 degrees

Nichols plot of self-loop function

Bode plot of transfer function

2dB

-10dB0dB



Outline

18

3. Ringing Test for Ladder Low-Pass FiltersAnalysis of Active 3rd-Order Ladder LPF

3 20 1 2

20 1 2

1( ) ;( ) ( ) 1

( ) ( )

outout

in

VHV a j a j a j

L j a j a j a

0 2 2 1 2 2

0 1 1 2 2 1 1 1 2 2 2 2

2 1 1 2 2 2

; ;; ;

;

b L C b R Ca RC L C a RC R C L Ca R C C R C

Passive 3rd-order ladder LPF Transfer function & self-loop function

where,

Active 3rd-order ladder LPFR1 = 100 Ω, R2 = 50 kΩ, R3 = R4 = 50 kΩ, C1 = 5 nF, C2 = 10 nF, C3 = 3.18 nF, at f0 = 100 kHz. • Over-damping (R5 = 0.5 kΩ), • Critical damping (R5 = 1 kΩ), and• Under-damping (R5 = 2 kΩ).

General impedance converter

19

3. Ringing Test for Ladder Low-Pass FiltersSimulation Results of 3nd-Order Ladder LPFBode plot of transfer function Transient response

100o

107o 144o

Nichols plot of self-loop functionOver-damping:Phase margin is 80 degrees.Critical damping:Phase margin is 73 degrees.Under-damping: Phase margin is 36 degrees.


0dB

-12dB

-6dB

20

3. Ringing Test for Ladder Low-Pass FiltersImplemented Circuit of 3rd-Order Ladder LPF

Balun transformer(10 mH inductance)

Measurement of self-loop function

Device under test

21

3. Ringing Test for Ladder Low-Pass FiltersMeasurement Results of 3nd-order Ladder LPF

Over-damping:Phase margin is 77 degrees.Critical damping:Phase margin is 70 degrees.Under-damping: Phase margin is 64 degrees.

Transient response

Bode plot of transfer function Nichols plot of self-loop function

103o

110o116o

1 dB

-6 dB

-3dB




Outline

4. Comparison (Self-loop function)

Features Alternating current conservation

Replicameasurement

Middlebrook’smethod

Main objective Self-loop function Loop gain Loop gain

Transfer function accuracy Yes No No

Ringing Test Yes Yes Yes

Operating region accuracy Yes No No

Phase margin accuracy Yes No No

Passive networks Yes No No

4. Discussions (Self-loop function)

• Loop gain is independent of frequency variable.Loop gain in adaptive feedback network is significantly

different from self-loop function in linear negative feedback network.

https://www.mathworks.com/help/control/ref/nichols.html

Network Analyzer

Nichols chart is only used in MATLAB simulation.

Nichols chart isn’t used widely in practical measurements

(only used in control theory).

(Technology limitations)

This work:• Proposal of alternating current conservation for

deriving self-loop function in a transfer function Observation of self-loop function can help us optimize the behavior of a high-order system.

• Implementations of circuits and measurements of self-loop functions for passive & active low-pass filtersTheoretical concepts of stability test are verified by laboratory simulations and practical experiments.

Future of work:• Stability test for parasitic components in transmission

lines, printed circuit boards, physical layout layers

4. Conclusions

References[1] H. Kobayashi, N. Kushita, M. Tran, K. Asami, H. San, A. Kuwana "Analog - Mixed-Signal - RF Circuits for

Complex Signal Processing", 13th IEEE Int. Conf. ASIC, Chongqing, China, Nov, 2019.[2] M. Liu, I. Dassios, F. Milano, "On the Stability Analysis of Systems of Neutral Delay Differential

Equations", Circuits, Systems, and Signal Processing, Vol. 38(4), 1639-1653, 2019.[3] X. Peng, H. Yang, "Impedance-based stability criterion for the stability evaluation of grid-connected

inverter systems with distributed parameter lines", CSEE J. Power and Energy Systems, pp 1-13, 2020.[4] N. Kumar, V. Mummadi, "Stability Region Based Robust Controller Design for High-gain Boost DC-DC

Converter", IEEE Trans. Industrial Electronics, Feb. 2020.[5] H. Abdollahi, A. Khodamoradi, E. Santi, P. Mattavelli, "Online Bus Impedance Estimation and

Stabilization of DC Power Distribution Systems: A Method Based on Source Converter Loop-Gain Measurement", 2020 IEEE Applied Power Electronics Conference and Exposition, LA, USA, June 2020.

[6] L. Fan, Z. Miao, "Admittance-Based Stability Analysis: Bode Plots, Nyquist Diagrams or Eigenvalue Analysis", IEEE Trans. Power Systems, Vol. 35, Issue 4, July 2020.

[7] P. Wang, S. Feng, P. Liu, N. Jiang, X. Zhang, "Nyquist stability analysis and capacitance selection method of DC current flow controllers for meshed multi-terminal HVDC grids", CSEE J. Power and Energy Syst., pp. 1-13, 2020.

[8] N. Tsukiji, Y. Kobori, H. Kobayashi, "A Study on Loop Gain Measurement Method Using Output Impedance in DC-DC Buck Converter", IEICE Trans. Com., Vol. E101-B(9), pp.1940-1948, 2018.

[9] J. Wang, G. Adhikari, N. Tsukiji, H. Kobayashi, "Analysis and Design of Operational Amplifier Stability Based on Routh-Hurwitz Stability Criterion", IEEJ Trans. Electronics, Information and Systems, Vol. 138(128), pp.1517-1528, Dec. 2018.

[10] M. Tran, "Damped Oscillation Noise Test for Feedback Circuit Based on Comparison Measurement Technique", 73rd System LSI Joint Seminar, Tokyo, Japan, Oct. 2019.

Gunma

KobayashiLaboratory

Thank you very much!

Virtual TJCAS 2020Taiwan and Japan Conference on

Circuits and Systems

Documents

STABILITY TEST FOR HIGH-ORDER LADDER LOW-PASS FILTERS · 2020. 11. 9. · STABILITY TEST FOR HIGH-ORDER LADDER LOW-PASS FILTERS MinhTriTran*, Anna Kuwana, and Haruo Kobayashi Gunma