study of behaviors of electronic amplifiers using nichols

Gunma

Kobayashi

Laboratory

Gunma University, Japan

Study of Behaviors of Electronic

Amplifiers using Nichols Chart

PHD. Candidate. MinhTri Tran*,

Prof. Anna Kuwana, Prof. Haruo Kobayashi

1. Research Background• Motivation, objectives and achievements

• Self-loop function in a transfer function

2. Analysis of Behaviors of High-order Systems• Operating regions of high-order systems

3. Ringing Test for Feedback Amplifiers• Stability test for shunt-shunt feedback amplifiers

• Stability test for unity-gain and inverting amplifiers

4. Ringing Test for High-order Low-Pass Filters• Stability test for passive and active RLC circuits

• Stability test for Deboo low-pass filters

5. Conclusions

Outline

2

1. Research BackgroundNoise in Electronic Systems

Ringing does the following things:

• Causes EMI noise,

• Increases current flow,

• Decreases the performance, and

• Damages the devices.

STABILITY TEST

Unstable system

Signal to

Noise Ratio:

Performance of a devicePerformance of a system

Figure of

Merit:=

Signal powerSNR

Noise power=

Output SNR

Input SNRF

Common types of noise:

• Electronic noise, thermal noise, intermodulation noise,

cross-talk, flicker noise, thermal noise...

3

o Derivation of self-loop function based on the

proposed comparison measurement

o Investigation of operating regions of linear

negative feedback networks

o Observation of phase margin at unity gain on

the Nichols chart

à Over-damping (high delay in rising time)

à Critical damping (max power propagation)

à Under-damping (overshoot and ringing)

1. Research BackgroundObjectives of Study

4

1. Research BackgroundAchievements of Study

Comparison measurement

• Feedback amplifiers

• High-order low-pass filters

Alternating current conservationIncident

current

Transmitted

current

2nd-order Deboo low-pass LPF

input

output

Implemented circuit

1()

)(

( )ww = -

wA

HL

Self-loop

function

5

1. Research BackgroundSelf-loop Function in A Transfer Function

( ( )

(( )

)

)

)

1

(out

in

HV

V

A

L

w= =

+w

www

Transfer function

( )L w

( )A w( )H w

: Numerator function

: Transfer function

: Self-loop function

Linear system

Input Output( )H w

( )winV ( )woutV

0 1

0 1

( ) ... ( )

()

) ... ((

)

n

n n

n

n n

Hb j b j b

a j a j a

w w

w ww

-

-

+ + +=

+ + +

Model of a linear system

oMagnitude-frequency plot

oAngular-frequency plot

oPolar chart à Nyquist chart

oMagnitude-angular diagram à Nichols diagram

Bode plots

Variable: angular frequency (ω)

6

1. Research BackgroundComparison Measurement

Transfer functionSelf-loop function

1()

)(

( )ww = -

wA

HL

Sequence of steps:

(i) Measurement of

numerator function A(ω),

(ii) Measurement of transfer

function H(ω), and

(iii) Derivation of self-loop

function.

1 (

( )

(

(

)

)

)( )w =

w=

+www

out

in

A

L

V

VH

Linear system

Input Output

( )H w( )winV ( )woutV

0 1

0 1

( ) ... ( )

()

) ... ((

)

n

n n

n

n n

Hb j b j b

a j a j a

w w

w ww

-

-

+ + +=

+ + +

Model of a linear system

7

1. Research BackgroundAlternating Current Conservation

( ) 1

( )1

;(

( )

)

in

o

i

ut

in

o

ou

n

ut

tHZ

Z

Z

V

L

V

Z

w= =

w +

Þ =

w

w

Transfer function

Simplified linear system

Self-loop function

( )inc trans in

t

c

tra

in

i u nsn o t ou

V ZL

Z Z

V

VZ

VwÞ-= = - =

Incident

current

Transmitted

current

10 mH

inductance

Derivation of self-loop function

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.

à 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

BW =100 Hz GBW =10 MHz

Loop Gain

9

1. Research BackgroundLoop Gain in Feedback Systems

Aβ : loop gain

Transfer function 1

1= »

+ b bA

AHGF : loop gain

Transfer function1

1= »

+GFG

H

G

F

+-

OutputInputA+

-

VoutVin

b

Inverting amplifierAdaptive feedback systems

Nichols plot of loop gain Gain reduction








5. Conclusions

Outline

Case Over-damping Critical damping Under-damping

Delta

Module

Angular

11

2. Analysis of Behaviors of High-order SystemsCharacteristics of 2nd-order Transfer Function

( )D0

2

211 0

0

14 0

2

æ ö< Þ D = - >ç ÷è ø

aa a

a a0

2

211 0

0

14 0

2

æ ö= Þ D = - =ç ÷è ø

aa a

a a0

2

211 0

0

14 0

2

æ ö> ÞD = - <ç ÷è ø

aa a

a a

( )wH2

0

1 1 1

2 22

2 2 1

0 0 0 0 0 0

1

1 1

2 2 2 2

æ ö æ öæ ö æ öç ÷ ç ÷w + - - w + + -ç ÷ ç ÷ç ÷ ç ÷è ø è øè ø è ø

a

a a a a

a a a a a a0

20

2 1

1 1 1

2

2

6= =é ùæ öê úw + ç ÷

-

ê úè øë û

dBa a

a

0

1 1 1 1

0 0 0

2

0 0

22 2 2

0

2

1

1 1

2 2 2 2

æ ö æ öæ ö æ ö æ ö æ öç ÷ ç ÷w- - + w+ - +ç ÷ ç ÷ ç ÷ ç ÷ç ÷ ç ÷è ø è ø è ø è øè ø è ø

a

a a a a

a a a a a a

( )q w 1 1 1 1

0 0

2

0 0

2

0 0

arctan arctan

1 1

2 2 2 2

æ ö æ öç ÷ ç ÷ç ÷ ç ÷w w

- -ç ÷ ç ÷ç ÷ ç ÷æ ö æ ö

- - + -ç ÷ ç ÷ç ÷ ç ÷ç ÷ ç ÷è ø è øè ø è ø

a a a a

a a a a a a

0

1

22 arctan

æ öwç ÷- ç ÷è ø

a

a

1 1

0 0 0 0

1 1

0 0

2 2

1 1

2 2arctan arctan

2 2

æ ö æ öæ ö æ öç ÷ ç ÷w- - w+ -ç ÷ ç ÷ç ÷ ç ÷è ø è ø- -ç ÷ ç ÷ç ÷ ç ÷ç ÷ ç ÷ç ÷ ç ÷è ø è ø

a a

a a a a

a a

a a

1

02w =cut

a

a0

1

( )2

w <cut

a

aH ( )

2

pq w > -cut

1

0( )2

w =cut

a

aH ( )

2

pq w = -cut

0

1

( )2

w >cut

a

aH ( )

2

pq w < -cut

0

2

1

1

1 ( )( ) =

+ w + ww

ja a jHSecond-order transfer function:

Case Over-damping Critical damping Under-damping

Delta

12

2. Analysis of Behaviors of High-order SystemsCharacteristics of 2nd-order Self-loop Function

( )D 2

1 04 0D = - >a a 2

1 04 0D = - =a a2

1 04 0D = - <a a

( )wL ( )2 2

0 1w w +a a ( )2 2

0 1w w +a a ( )2 2

0 1w w +a a

( )q w 0

1

arctan2

wp+

a

a0

1

arctan2

wp+

a

a0

1

arctan2

wp+

a

a

11

0

5 22

w = -a

a1 1( )w >L

1) .3( 76p - q w > o

1 1( )w =L 1) .3( 76p - q w = o1 1( )w <L 1) .3( 76p - q w < o

12

02w =

a

a 25( )w >L 2 ) .4( 63p - q w > o

2 5( )w =L 2 ) .4( 63p-q w = o

2 5( )w <L 2 ) .4( 63p - q w < o

0

31w =a

a 32( 4)w >L

345( )p-q w > o

32( 4)w =L 3

45( )p-q w = o3 2( 4)w <L 3

45( )p - q w < o

[ ]0 1( ) = w w+w aj j aLSecond-order self-loop function:

13

2. Analysis of Behaviors of High-order SystemsOperating 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

=w + +

=w + +

w

w

w+

w

=w +

w

w

•Under-damping:

•Critical damping:

•Over-damping:

( )21 ;( ) w= ww +j jL

( )22 ;( 2) = w w+w j jL

( )23 ;( 3) = w w+w j jL

Phase

margin

92o

Phase

margin

76.3o








5. Conclusions

Outline

15

3.Ringing Test for Feedback AmplifiersAnalysis of Shunt-Shunt Feedback Amplifier

[ ]11

0

2

00

1

;( ) 1

( ( ))w+

= = = w w+w +

w+ w

wout

in

Hj

b baL

V j

V jaj a

aj

( )( ) ( )

0 1 01 1 1 1 1 1 1 1

1 1 1 1 1 11

; ; ;

;

= = - = + +

= + + + +

L GD L m S L GD GS GD DB DB GS

L GD DB S GS GD S L m GD

R C Rb g R R C C C C C C

R C C R C g

a

C R R C

b

a

BJT shunt-shunt feedback amplifier Small signal model

Transfer function and self-loop function

Where,

1 1 1 1 1 1 1 1 1 1 1; ;in out

out m

s C F C s C C CCS C o C F

V VV V V g

R r Z R Z R Z Z Z R r Z Rp p

p p m m m m

æ ö æ ö æ ö+ + + + = + + + + = + -ç ÷ ç ÷ ç ÷ç ÷ ç ÷ ç ÷

è ø è ø è ø

Apply superposition at the nodes Vπ and Vout, we have

16

3.Ringing Test for Feedback AmplifiersCharacteristics of Shunt-Shunt Feedback Amplifier

BJT shunt-shunt feedback amplifier

Nichols plot of self-loop function

Bode plot of transfer function

Transient response

94o

Phase margin

86 degrees

17 dB

Rf = 1 kΩ, RC = 10 kΩ, RS = 950 Ω.

17

Without frequency compensation

Small signal model of 2nd-stage

Transfer function

( ) 1

2

0 1

0

;)1

(w +

=w + w +

wb b

a ja

j

jH

( )( )

( ) ( )

2

0 1

0

1

; ;

;

;

= = -

é ù= + + -ë û

é ù= + + + +ë û

D GD D m

D S GD DB GS GD GD

D GD DB S GS GD D S m GD

a R C R g

R R C C C C C

R C C R

a

C R g C

b

b C R

Where,

3.Ringing Test for Feedback AmplifiersAnalysis of Op Amp without Miller’s Capacitor

Open-loop function

( ) ( )( ) ( ) ( )

0 1 2 3

0

3 2

4 3 2

1 2 3

;( )1

w + w + w+=

w + w + ww

+ w+op

b bA

b b

a a a

j j j

j j j ja

Self-loop function

( ) ( ) ( )4 2

2

3

0 1 3 ;( ) = w + w + w + wwop j j ja a a jaL

( )0 1

2( ) = w + ww ja a jL

Self-loop function

18

Bode plot of transfer function H(ω)

Nichols plot of self-loop function L(ω)

Unity-Gain Amplifier

Transient response

167o

Phase margin = 13 degrees

3.Ringing Test for Feedback AmplifiersUnity-Gain Amplifier without Miller’s Capacitor

15 dB

19

With Miller’s capacitor and resistor

( ) ( )( ) ( ) ( )

2

0 1 2 3

0 3

4

1

3

2

3 2;)1

(w w + w+

=w + w + w + w

+

+w

b b b b

aH

j j j

j a ja j a j

( ) ( ) ( )0 1

2

2

4

3

3( ) = w + w + w + ww j j ja a a a jL

3.Ringing Test for Feedback AmplifiersTwo-stage Op Amp with Frequency Compensation

Open-loop function

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )

5 4 3

5

2

6

0 1 2 3 4

0 1 2 3 4

4

5

5 3 2;(1

)w + w + w + w + w+

=w + w + w + w + w + w+

wopA

j j j j j

j j j j j

b b b b b b

a a a a a ja

Self-loop function

( ) ( ) ( ) ( ) ( )6 5

0 1 2 3 4

4 3 2

5 ;( ) = w + w + w + w ww + + wop j j jL a a a a aj j ja

Transfer function

Self-loop function

Small signal model of 2nd-stage

20

Unity-gain amplifier with Miller’s capacitor

3.Ringing Test for Feedback AmplifiersUnity-Gain Amplifier with Miller’s Capacitor

Under-damping:

R1= 2 kΩ, C1 = 1 pF

Critical damping:

R1 = 3.5 kΩ, C1 = 0.2 pF

Over-damping:

R1 = 3.5 kΩ, C1 = 0.8 pF

( )

( )

( )( )

1 11; ;

11

= » =w

w

w w+

LA

A

H


Simplified model

21

150o

90o

79o


3.Ringing Test for Feedback AmplifiersBehaviors of Unity-Gain Amplifier

Simulated transient responseSimplified model of unity gain amplifier

-9 dB

5 dB

-15dB

PM

30o

PM

90oPM

101o

22

Inverting amplifier

3.Ringing Test for Feedback AmplifiersInverting Amplifier with Miller’s Capacitor

Under-damping:

Rz= 0 kΩ, Cz = 0 pF

Critical damping:

Rz = 0.5 kΩ, Cz = 0.5 pF

Over-damping:

Rz = 4 kΩ, Cz = 0.5 pF( )( )

( )( )

2

1 2 2

1 1

1; 1 ;

1

æ ö= » = +ç ÷+ w

w-

è ø-

ww LH

R

R R

RL A

R

R


Simplified model

23

139o

111o106o


3.Ringing Test for Feedback AmplifiersBehaviors of Inverting Amplifier

Simulated transient responseSimplified model of inverting amplifier

20dB

4dB

8dBPM

41o

PM

74o

PM

69o

5 dB

15 dB

-10 dB








5. Conclusions

Outline

25

4. Ringing Test for High-order Low-Pass FiltersAnalysis of 2nd-Order Passive RLC LPF

Passive RLC Low-pass Filter

( )0 1

2

1;( )1

out

in

VH

jaV ja= =

+ w +w

w

Derivation of self-loop function

( )0

2

1 ;( )L aja j= +w w w

Self-loop function

Implemented circuit

Transfer function

0 1; ;= =a LC RCawhere,

26

4. Ringing Test for High-order Low-Pass FiltersMeasurement Results for 2nd-Order Passive RLC LPF

Transient responses

94o 107o122o

Nichols plot of self-loop function

Bode plot of transfer function

2dB

-10dB

0dB

PM

68o

PM

86o

PM

73o

27

4. Ringing Test for High-order Low-Pass FiltersStability Test for 2nd-Order Active Ladder LPF

Implemented circuit

( )0 1

2

1;( )1

out

in

VH

jaV ja= =

+ w +w

w

Transfer function

( )0

2

1 ;( )L aja j= +w w w

Self-loop function

Active ladder low-pass filter

107o

117o 133o


7dB

-7dB

-3dB

PM

47o

PM

73o

PM

63o

28

4. Ringing Test for High-order Low-Pass FiltersAnalysis of 2nd-Order Deboo low-pass LPF

( )( )

( ) ( )

2

0

1

2

0

0 1

;1

;

=w + w+

= w +

w -

w w

jH

j

b

aL

a

a j

a

j

( )( ) ( )

( ) ( )( )( ) ( )

2 4 7 5 6

1 2 4 5 6 7 4 5 3 6

2 3 4 5 7 1 2

2 4 5 6 7 4 5 3 6

2 7 1 4 5 3 6 3 4 5 7 2

2 4 5 6 7 4 5 3 6

0

0

1

;

;

;

+=

é ù+ + -ë û

=+ + -

- +=

+ + -

R R R R R

R R R R R R R R R R

R R R R R C C

R R R R R R R R R

R R C R R R R R

b

R R R C

R R R R R R Ra

R

a

R

Single ended Deboo low-pass LPF Transfer function & self-loop function

where,

R1 = R3 = R5 = 1 kΩ, R2 = 10 kΩ, R6 = R7 = 5

kΩ, C1 = 1 nF, C2 = 0.5 nF at f0 = 10 kHz.

• Over-damping (R4 = 3 kΩ),

• Critical damping (R4 = 6 kΩ), and

• Under-damping (R4 = 10 kΩ).

Fully differential Deboo low-pass LPF

29

4. Ringing Test for High-order Low-Pass FiltersImplemented Circuit of Deboo low-pass LPF

Schematic of Deboo low-pass LPF

System Under Test Measurement set up

Implemented Circuit

30

4. Ringing Test for High-order Low-Pass FiltersMeasurement Results of Deboo low-pass LPF

Over-damping:

àPhase margin is 81 degrees.

Critical damping:


Under-damping:


Transient response

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

99o 107o 118o

7 dB

-7 dB

0 dB

PM

62o

PM

81o

PM

73o








5. Conclusions

Outline

32

5. Limitations of Conventional Methods

o Middlebrook’s measurement of loop gain

àApplying only in feedback systems (DC-DC converters).

o Replica measurement of loop gain

àUsing two identical networks (not real measurement).

o Nyquist’s stability condition

à Theoretical analysis for feedback systems (Lab tool).

o Nichols chart of loop gain

à Only used in feedback control theory (Lab tool).

5. Comparison

FeaturesComparison

measurement

Alternating

current

conservation

Replica

measurement

Middlebrook’s

method

Main objectiveSelf-loop

function

Self-loop

functionLoop gain Loop gain

Transfer function

accuracyYes Yes No No

Breaking feedback

loopNo Yes Yes Yes

Operating region


Phase margin


Passive networks Yes Yes No No

5. Discussions

• Loop gain is independent of frequency variable.

àLoop 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 comparison measurement for deriving

self-loop function in a transfer function

à Observation of self-loop function can help us

optimize the behavior of a high-order system.

• Implementation of circuit and measurements of self-

loop functions for high-order feedback amplifiers.

àTheoretical 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

5. Conclusions

References[1]L. Fan, Z. Miao, "Admittance-Based Stability Analysis: Bode Plots, Nyquist Diagrams or

Eigenvalue Analysis", IEEE Trans. on Power Systems, vol. 35, no. 4, July 2020.

[2]M. Liu, I. Dassios, G. Tzounas, F. Milano, “Stability Analysis of Power Systems with Inclusion of

Realistic-Modeling of WAMS Delays”, IEEE Trans. on Power Sys., vol.34, no.1, pp. 627-636, 2019.

[3]S. Zhong, Y. Huang, “Comparison of the Phase Margins of Different ADRC Designs,” IEEE Chinese

Control Conf., China, July 2019.

[4]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", IEEE Applied Power Electronics Conference and Exposition, LA, USA, June 2020.

[5]L. Fan, Z. Miao, “Admittance-Based Stability Analysis: Bode Plots, Nyquist Diagrams or

Eigenvalue Analysis?,” IEEE Trans. on Power Systems, vol. 35, no. 4, pp. 3312 – 3315, July 2020.

[6]Y. Ren, X. Wang, L. Chen, Y. Min, G. Li, L. Wang, Y. Zhang, “A Strictly Sufficient Stability Criterion

for Grid-Connected Converters Based on Impedance Models and Gershgorin's Theorem,” IEEE

Trans. on Power Delivery, vol. 35, no. 3, pp. 1606 – 1609, June 2020.

[7]V. Salis, A. Costabeber, S. Cox, F. Tardelli, P. Zanchetta, “Experimental Validation of Harmonic

Impedance Measurement and LTP Nyquist Criterion for Stability Analysis in Power Converter

Networks,” IEEE Trans. on Power Electronics, vol. 34, no. 8, pp. 7972 – 7982, Aug. 2019.

[8]S. Wang, B. Li, Z. Xu, X. Zhao, D. Xu, “A Precise Stability Criterion for Power Hardware-in-the-Loop

Simulation System,” 10th Int. Conf. on Power Electronics and ECCE Asia, Busan, Korea, May 2019.

[9]S. Plesnick, J. Berardino, R. Irwin, “The Generalized Nyquist Criterion Applied to Complex DC

Power System Networks,” IEEE Electric Ship Technologies Symposium, DC, USA, Aug. 2019.

References[10] J. Ardila, E. Roa, “A Novel Loop Gain Adaptation Method for Digital CDRs Based on the Cross-

Correlation Function,” IEEE Int. Symp. on Circuits and Systems, Sapporo, Japan, May 2019.

[11] A. Ochoa, D. Patterson, M. McGuckin, “Driving Point Loop Gain and Return Ratio,” IEEE 62nd

Int. Midwest Symposium on Circuits and Systems, TX, USA, Aug. 2019.

[12] M. Tran, A. Kuwana, H. Kobayashi, "Derivation of Loop Gain and Stability Test for Low Pass

Tow-Thomas Biquad Filter", 10th Int. Conf. on Computer Science, Engineering and Applications,

London, UK, July 2020.

[13] M. Tran, A. Kuwana, H. Kobayashi, "Ringing Test for Third-Order Ladder Low-Pass Filters", 11th

IEEE Annual Ubiquitous Computing, Electronics & Mobile Communication Conference, USA, 2020.

[14] N. Sayyaf, M. Tavazoei, “Frequency Data-Based Procedure to Adjust Gain and Phase Margins

and Guarantee the Uniqueness of Crossover Frequencies,” IEEE Trans. on Industrial Electronics,

vol. 67, no. 3, pp. 2176 – 2185, March 2020.

[15] M. Tran, A. Kuwana, H. Kobayashi, “Derivation of Loop Gain and Stability Test for Multiple

Feedback Low Pass Filter Using Deboo Integrator”, The 8th IIAE Int. Conf. on Industrial

Application Engineering, Shimane Japan, March, 2020.

[16] M. Liu, I. Dassios, F. Milano “On the Stability Analysis of Systems of Neutral Delay Differential

Equations”, Circuits, Systems, and Signal Processing, vol. 38, no. 4, pp. 1639-1653, 2019.

[17] M. Tran, A. Kuwana, H. Kobayashi, “Design of Active Inductor and Stability Test for Ladder RLC

Low Pass Filter Based on Widened Superposition and Voltage Injection”, The 8th IIAE Int. Conf. on

Industrial Application Engineering, Shimane Japan, March 2020.

Gunma

Kobayashi

Laboratory

Thank you very much!

study of behaviors of electronic amplifiers using nichols

Documents