Study of Helix Functions and Multi-Source Rauch Filters

Ph.D. Candidate: Tri Minh Tran*,

Prof. Anna Kuwana, and Prof. Haruo Kobayashi

Gunma

Kobayashi

Laboratory

Gunma University, Japan

Study of Helix Functions and

Multi-Source Rauch Filters

1. Research Background

• Motivation, objectives and achievements

• Helix functions and superposition formula

2. Ringing Test for Rauch Low-Pass Filter

• Behaviors of fully differential op amp

• Stability test for fully differential Rauch LPF

3. Analysis of Rauch Complex Filter

• Spectrum of polyphase signals

• Design of 4th-order Rauch complex filter

4. Conclusions

Outline

Spectrum?


Motivation of Helix Waves and Multi-Source Circuits

A general formula for

multi-source networks?

Definition of Helix functions? Complex filterPolyphase filter

2

Power

lost

Undefined concepts of multi-source signals

• Negative and positive frequencies?

• Negative and positive complex signals?

• Negative and positive polyphase signals?

• Spectrum of multi-source signals?

High

cost

• Fully differential Rauch LPF

• Rauch complex filter

2 2

(High power consumption)(Very complicated)

New architectures

3


Stability Test for Electronic Systems

(Technology limitations)

Nyquist plot of loop gain

Re

Im

0

-1

Nichols plot of loop gain

Nichols chart in Network Analyzer?

(Very complicated)(Unclear operating region)

Overshoot

Undershoot

Ringing in electronic systems

LAB

tool

LAB

tool

Unused

tool

Stability

test


Innovation of This Work

o Easiest selection of components

o Lowest power consumption

o Simplest design in fully differential

forms and complex topologies

Merits of Rauch filters

o Use of complex functions

Fundamental model of motion in both

time and frequency domains

o Superposition formula for multi-source

networks

o Nichols chart of self-loop function

A useful tool for stability test

Easily integrated in network analyzers

Merits of complex numbers

4

Nichols plot of self-loop function

106o

112o

121oPM

59o

PM

74o

PM

68o

Rauch complex filter

Simple

Low cost


Objectives of Study

• Definitions of helix waves and polyphase signals

• Derivation of transfer functions in multi-source

networks using superposition formula

o Investigation of operating regions of 4th-order

fully differential Rauch low-pass filter

Over-damping (high delay in rising time)

Critical damping (max power propagation)

Under-damping (overshoot and ringing)

• Derivation of transfer function and image

rejection ratio for 4th-order Rauch complex filter

5

6


Achievements of Study

Superposition formula for multi-source networks

1

1( ) ( )

n n ni

O O ai gi

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

k pik

V (t)1V (t) V (t) = I t I t

1Z ZZ

1

Z=

+ + −

+

( ) ( )0 0+= +heV t Ahe tω θ( ) ( )0 0−

= − −heV t Ahe tω θ

Stability Test for a 4th-order Rauch LPF

Definitions of helix waves

Analysis of a 4th-order Rauch complex filter

IRR = 32 dB

22

7


Helix and Sinusoidal Waves

Positive helix function Negative helix function

( ) ( )

( ) ( )

0 0 0( )

0 0

0 0 0 0

2

cos sin

+

+= + =

= + + +

j T

heV t Ahe t A e

A t jA t

ω θω θ

ω θ ω θ

( ) ( )

( ) ( )

0 0 0( )

0 0

0 0 0 0

2

cos sin

− −

−= − − =

= − − + − −

j T

heV t Ahe t A e

A t jA t

ω θω θ

ω θ ω θ

Spectrum of helix waves Spectrum of sine wave

2 2 2

2

2

2

8


Superposition Theorem for Multi-Source Systems

1

1( ) ( ) ( )

=

+ + −

+

n n ni

O ai gi

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

k pik

V (t)1V (t) = I t I t

1Z ZZ

1

Z

Superposition formula:

VO(t) : Voltage at one node

Vi(t) : Input voltage sources

Iai(t) : Ahead-toward current sources

Igi(t) : Ground-toward current sources

Zi, si, pi,(t): Impedances at each branch

o Multi-source systems, feedback

networks (op amps, amplifiers,

polyphase filters, complex filters…)

A general

multi-source

network










4. Conclusions

Outline

10


Self-loop Function in A Transfer Function

( ( )

(( )

)

)

)

1

(out

in

HV

V

A

L

ω= =

+

ωω

ωω

Transfer function

( )L ω

( )A ω

( )H ω

: Numerator function

: Transfer function

: Self-loop function

Linear system

Input Output( )H ω

( )ωinV ( )ωoutV

0 1

0 1

( ) ... ( )

()

) ... ((

)

n

n n

n

n n

Hb j b j b

a j a j a

ω ω

ω ωω −

−

+ + +=

+ + +

Model of a linear system

oMagnitude-frequency plot

oAngular-frequency plot

oPolar chart Nyquist chart

oMagnitude-angular diagram Nichols diagram

Bode plots

Variable: angular frequency (ω)

11


Operating Regions of 4th-Order System

Pascal’s Triangle

•Critical damping:

•Under-damping:

•Over-damping:

( ) ( ) ( )31 4 2

1)

2 3 2(

1ω + ω + ω + ω +ω =H

j j j j

( ) ( ) ( )32 4 2

1)

4 6 4(

1ω + ω + ω + ω +ω =H

j j j j

( ) ( ) ( )33 4 2

9 10

1( )

19ω + ω + ω +ω

+=

ωH

j j j j

1 : 4 : 6 : 4 : 1

1 : 2 : 3 : 2 : 1

1 : 9 : 10 : 9 : 1

Nichols plot of self-loop function

Bode plot of transfer function

1dB

-15dB

-9dB

98o

103.7o 132o

PM

48o

PM

92oPM

76.3o

n = 2 1 2 1

n = 3 1 3 3 1

n = 4 1 4 6 4 1

n = 5 1 5 10 10 5 1

12

Fully differential two-stage op amp

Small signal model Open-loop function A(ω)

0

4 3 2 1

40 1 2

4

5 4 3 2

3

1 2 3

( ) ( ) ( ) ( );

( ) ( ))

( ) ( ) 1(

+

ω + ω + ω + ω +=

ω + ω + ω ω + ωω

+

b b b b

aA

j j j j b

aj aja ja j j


Behaviors of Fully Differential Op Amp

Bode plot of open-loop function A(ω)

55 dB

Nichols plot of self-loop function L(ω)

90o

Phase margin = 90 degrees

Self-loop function L(ω)2

0

4

1 2 3 4

5 3( ) ( ) ( ) ( ) ;( ) = ω + ω + ω + ω +ω ωa a a jaj aj j jL

( )

( )

( )

( )

1 1 1

1 1 2 2

1

1 1 2

2 2 2 2

2

1;

1

1

;1

1

1 1

+ ω + = + ω

ωω + + + + +

+ ω

ω= ω − + ω +

+ ω

ω ωω + + + = ω + −

+ ω + ω

ina GS GD b GD

S S

Cb GD DB GS GD

D C C

Ca GD m out GD

C C

C Cout GD DB b GD m

C C D C C

VV j C C V j C

R R

j CV j C C C C

R j R C

j CV j C g V j C

j R C

j C j CV j C C V j C g

j R C R j R C;

Applying superposition formula at each node

13

Transfer function H(ω) and self-loop function L(ω)

( )( ) ( ) ( )

( ) ( ) ( ) ( )

0

0 1 2 3

0 1

3

2

4 2

4 3

3

2

;1

;

=+ ω + ω + ω + ω

= ω + ω + ω +ω

ω

ω

j j jH

j

j

b

a a a a

a a j aL jaj


Analysis of Fully Differential Rauch Low-Pass Filter

5 6 2 3 2 3 5 62 3 5 6 1 2 4 5 6 2 3 2 3 4 2 3 2 5 6 4

4 1 1 4

2 3 5 6 3 62 3 1 2 5 6 3 4 2 3 5 6 2 4 2 3 5 6 1 2 3 4

1

1 3

2 0 0

4 1 4

; ;

; ; ;

= + + + + + = + + + + +

= + + + + + + = =

aR R R R R R R R

R R R R C C C R R R R C C C R R C R R CR R R R

R R R R R RR R C C R R C C R R R R C C R R R R C

R

a

a b C CR

a CR R

( )( )

( ) ( )( )

( )( )

( )

2

2

1 1

1 2 3 1 2 3

2 2

2 2

2 2

2

2

3 3

4 5 6 4 5 6

2

4 4

5 5

1 1 1;

1; ( );

1 1 1;

1;

+ + ω + + = + + + + ω

+ ω = + + ω = − + ω

+ + ω + + = + + + + ω

+ ω = + + ω = − +

cin ba a

ab c c b b

outc ed d

de out out e

j VV VV j C j V j C

R R R R R R

VV j C j V j C V V j V A

R R

j VV VV j C j V j C

R R R R R R

VV j C j V j C V V

R R( )( )2

( );ωej V A

Applying superposition formula at each nodeFully differential Rauch LPF

Here, parameters are given as

• Over-damping (C3 = 0.1 nF),

• Critical damping (C3 = 0.5 nF),

• Under-damping (C3 = 1.4 nF).

Operating regions

14


Behaviors of Fully Differential Rauch Low-Pass Filter

Nichols plot of self-loop function Over-damping:

Phase margin is 74 degrees.

Critical damping:


Under-damping:


15dB

10dB12dB

Bode plot of transfer function Transient response

106o

112o121o

PM

59o

PM

74o

PM

68o










4. Conclusions

Outline

16


Characteristics of Low-IF Receiver

LNA

MIXER

QuadGen .

4th order Poly-phase Filter

Limiting

amplifier

ADC

VCO

RSSI

Antenna

PLL

BPF

This Work

Bandgap

Reference .

Block diagram of Low-IF receiver

MIXER

cos ω tLO

sin ω tLO

Vsig

-jV sig

+Vsig

+jVsig

-Vsig

-jV im

+jVim

-Vim

+Vim

Vim

ω ω

LOω

sigω

im

+

1

3

4

2

image

signals

wanted

signals

0 f

frequency

amplitude

( )2dB

H j fπ

Wanted

signalsImage

signals

1

anti-clockwise

2

3

4

4V jV= −

1V V=3V V=−

2V jV=+

clockwise

3

1

2

4

1V V=

2V jV=−

3V V=−

4V jV= +

Wanted SignalsImage SignalsStep-down frequency converter

Spectrum of received signals

17


Positive Polyphase Signals on Frequency Domain

( ) ( ) ( ) ( ){ }

( ) ( ){ } ( )

2 31_

2 3

4; ; ;

1; ; ; p

p

o

oly

s

Pos

t

VV tS Vt

j

V t

j j

t

V= + + +

Positive polyphase signals

18


Negative Polyphase Signals on Frequency Domain

( ) ( ) ( ) ( ){ }

( ) ( ){ } ( )

2 31

2

_

3

4; ; ;

1; ; ; n

p

e

oly

g

Neg

t

VV VS t

j

t

j

V t

j

t

V− − −=

Negative polyphase signals

19


Polyphase Signals in All Frequency Domains

Wanted signalsImage signals

( ) ( ) ( ) ( ){ }

( ) ( ){ } ( )

2 31

2

_

3

4; ; ;

1; ; ; n

p

e

oly

g

Neg

t

VV VS t

j

t

j

V t

j

t

V− − −=

( ) ( ) ( ) ( ){ }

( ) ( ){ } ( )

2 31_

2 3

4; ; ;

1; ; ; p

p

o

oly

s

Pos

t

VV tS Vt

j

V t

j j

t

V= + + +

Negative polyphase signals Positive polyphase signals

Imaginary unit j

= 90o phase shift

( )2

1+= = −+

jej

π

( )2

1−= =− −

jej

π

Complex number

20


Design Principle for Complex Filter Networks

Frequency shifting of real low-pass filter in all frequency

domains an active complex filter

( ) 12

1ω

ω+

ω =

−

cu

L

t

PF

j

AH

3 1

1cr

CR=ω

( ) 21

1

ωω−

+

= ω ω

−CF

c

cut

rj

AH

ω-ωcr

Low-pass filter (LPF) Complex filter (CF)

ωcr : cross angular frequency

21


Analysis of 4th-Order Rauch Complex Filter

4th-order Rauch complex filter

Positive transfer function

( )( ) ( )

0 1

2 2

0 1 2 3 4 5

; ω = ω + ω + ω + ω +

P

P P P P

b bH

a j a j a a j a j a

( )( ) ( )

0 1

2 2

0 1 2 3 4 5

; ω = ω + ω+ ω + ω +

N

N N N N

b bH

a j a j a a j a j a( )

( )

( )

( )

( )

2 2

0 1 2 3 4 5

2 2

0 1 2 3 4 5

; ω + ω + ω + ω + ω = ω + ω + ω + ω +

N N N N

P P P P

a j a j a a j a j aIRR

a j a j a a j a j a

( )( )

( )( )

( ) ( )

( )( )

2

2

1 1

1 2 3 1 2 3

3

2

2 2

2 4 2 4

2 2

2

2

3 3

5 6 7 5 6 7

6

1 1 1;

1 1;

( ); ( );

1 1 1;

1

+ + ω + + = + + ω + +

+ + ω + = + + ω +

= − + ω = − + ω

+ + ω + + = + + ω + +

cin ba a

cab c

c b b out e e

outc ed d

e

j VV VV j C j V j C

R R R R R R

j VVV j C j V j C

R R R R

V V j V A V V j V A

j VV VV j C j V j C

R R R R R R

VR

( )( )

3

2

4 4

8 6 8

1;

+ + ω + = + + ω +

outdout

j VVj C j V j C

R R R

Negative transfer functionImage rejection ratio

Applying superposition formula at each node

22


Behaviors of 4th-Order Rauch Complex Filter

Wanted signalsImage signals

( ) ( ) ( ) ( ){ }

( ) ( ){ } ( )

2 31

2

_

3

4; ; ;

1; ; ; n

p

e

oly

g

Neg

t

VV VS t

j

t

j

V t

j

t

V− − −=

( ) ( ) ( ) ( ){ }

( ) ( ){ } ( )

2 31_

2 3

4; ; ;

1; ; ;p

p

o

oly

s

Pos

t

VV tS Vt

j

V t

j j

t

V= + + +

Bode plot of transfer function

IRR = 32 dB

Negative

frequency

domain

Positive

frequency

domain

Pass-band gain

= 13 dBBandwidth

= 200 kHz










4. Conclusions

Outline

4. Comparison

FeaturesProposed

formula

Conventional

Superposition

Millan’s

theorem

Effects of all

actuating sourcesAt one time Several times At one time

Transfer function

accuracyYes No No

Single-input

network analysisYes Yes Yes

Polyphase

network analysisYes No No

Complex network

analysisYes No No

Image rejection

ratio accuracyYes No No

4. Discussions

• Ringing test for electronic networks using Nichols chart

of self-loop function

Observation of phase margin can help us determine

the operating regions of high-order systems.

• Transfer function and image rejection ratio give useful

information about the behaviors of polyphase filters

and complex filters.

• Superposition formula: fundamental network analysis

theory for multi-source systems

Compute the effects of all sources at one time,

Get the transfer function faster, and

Reduce the network complexity.

This work:

• Definitions of helix functions and polyphase signals

• Proposal of superposition formula for deriving transfer

function in multi-source networks

• Stability test for 4th-order fully differential Rauch LPF

• Derivation of transfer function and image rejection ratio

for 4th-order Rauch complex filter

Theoretical concepts of stability test are verified by

laboratory simulations.

Future work:

• Stability test for parasitic components in transmission

lines, printed circuit boards, physical layout layers

• Investigation of DC offset and IQ mismatches in

polyphase filters and complex filters

4. Conclusions

References[1] 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.

[4] M. Tran, A. Kuwana, H. Kobayashi "Ringing Test for Negative Feedback Amplifiers" 11th IEEE Annual Information Technology,

Electronics and Mobile Communication Conference (IEMCON 2020), Vancouver, Canada, Nov. 2020.

[6] M. Tran, A. Kuwana, H. Kobayashi, "Design of Active Inductor and Stability Test for Passive RLC Low Pass Filter", 10th Int. Conf.

on Computer Science, Engineering and Applications (CCSEA 2020), London, United Kingdom, Jul. 2020.

[7] M. Tran, A. Kuwana, H. Kobayashi, "Ringing Test for Tow-Thomas Low-Pass Filters", Int. Conf. on Promising Electronic

Technologies (ICPET 2020), Jerusalem, Palestine, 16-17, Dec. 2020.

[9] M. Tran, A. Kuwana, H. Kobayashi, "Study of Behaviors of Electronic Amplifiers using Nichols Chart", IEEE the 3rd Int. Conf. on

Electronics and Communication Engineering (ICECE 2020), Xi'an, China, 14-16, Dec. 2020.

[10] M. Tran, "Study of Multiphase Networks, Noise Reduction for DC-DC Converters, and Stability Test for Electronic Systems", ATS

Doctoral Dissertation Award Contest, 29th IEEE Asian Test Symposium, Penang, Malaysia, Nov. 2020.

[12] M. Tran, A. Kuwana, H. Kobayashi, "Ringing Test for 2nd-order Sallen-Key Low-Pass Filters", IEEE 2nd Int. Conf. on Circuits and

Systems (ICCS 2020), Chengdu, China, 10-13, Dec. 2020.

[17] M. Tran, A. Hatta, A. Kuwana, H. Kobayashi, "Derivation of Image Rejection Ratio for High-Order Complex Filters," 6th Taiwan

and Japan Conf. on Circuits and Systems (TJCAS 2020), Nov. 2020.

[18] M. Tran, A. Kuwana, H. Kobayashi, "Ringing Test for 3rd-order Ladder Low-Pass Filters", 11th IEEE Annual Ubiquitous

Computing, Electronics & Mobile Communication Conference (UEMCON 2020), NY, USA, Oct. 2020.

[20] 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 (CCSEA 2020), United Kingdom, Jul. 2020.

[21] M. Tran, N. Kushita, A. Kuwana, H. Kobayashi, "Mathematical Analysis and Design of 4-Stage Passive RC Network in RF Front-

End System", 3rd Int. Conf. on Technology and Social Science (ICTSS 2019), Kiryu, Japan, May, 2019.

[22] M. Tran, A. Hatta, A. Kuwana, H. Kobayashi, "Design of 6th-order passive quadrature signal generation network based on

polyphase filter", IEEE 15th Int. Conf. on Solid-State and Integrated Circuit Technology (ICSICT2020), Kunming, China, Nov. 2020.

[23] M. Tran, N. Kushita, A. Kuwana, H. Kobayashi, "Mathematical Model and Analysis of 4-Stage Passive RC Polyphase Filter for

Low-IF Receiver", Journal of Mechanical and Electrical Intelligent System, vol. 3, no. 2, May 2020.

[25] M. Tran, N. Kushita, A. Kuwana, H. Kobayashi, "Flat Pass-Band Method with Two RC Band-Stop Filters for 4-Stage Passive RC

Polyphase Filter in Low-IF Receiver Systems", 3th IEEE Int. Conf. on ASIC (ASICON 2019), Oct. 2019.

Gunma

Kobayashi

Laboratory

Thank you very much!

Documents

Study of Helix Functions and Multi-Source Rauch Filters