1

Theo

ry a

nd P

ract

ical

Im

plem

enta

tion

of H

arm

onic

Res

onan

t Rai

l Driv

ers

Joon

g-Se

ok M

oon

Will

iam

C. A

thas

*Pe

ter

A. B

eere

lU

nive

rsity

of

Sout

hern

Cal

iforn

ia*A

pple

Com

pute

r In

c.

2

Mot

ivat

ion

Low

-pow

er s

till g

row

ing

in im

port

ance

Cloc

k dr

iver

s an

d no

des

rem

ain

larg

est

pow

er

cons

umer

sLo

w-p

ower

VLS

I te

chni

ques

Cloc

k ga

ting

Dyn

amic

vol

tage

sup

plie

sAs

ynch

rono

usRes

onan

t cl

ock

driv

ers

Rec

ycle

ene

rgy

of c

lock

nod

eO

ur r

esea

rch

focu

s

3

Clas

sific

atio

n of

Res

onan

t Cl

ock

Driv

ers

Sinu

soid

s+

Easy

and

eff

icie

nt t

o ge

nera

te+

Low

ove

rhea

d-

Har

d to

wor

k w

ith, v

ery

“und

igita

l”

Blip

s+

Adva

ntag

es o

f si

nuso

ids

-Po

sitiv

e-bu

mps

onl

y

Har

mon

ics

(Qua

si S

quar

e-W

ave)

+Ea

sy t

o w

ork

with

, ver

y “d

igita

l”-

Des

ign

can

be c

ompl

ex a

nd c

ostly

4

Prio

r Ar

ts –

Blip

s (A

thas

et.

al,

ISCA

S 19

96)

+Al

l-res

onan

t ne

twor

k+

Yiel

ds a

lmos

t no

n-ov

erla

ppin

g tw

o-ph

ase

cloc

k-

Bala

nced

load

cap

acita

nces

req

uire

d (n

ot o

ften

ava

ilabl

e)-

Freq

uenc

y is

sen

sitiv

e to

var

iatio

ns in

clo

ck lo

ads

C L1,

CL2

V dc

L 1L 2

C L1

C L2

ϕ 1ϕ 2

ϕ 1 ϕ 2

5

+Ca

n be

tun

ed f

or a

ny s

hape

, tra

nsiti

on t

ime,

fre

quen

cy-

Requ

ires

mul

tiple

vol

tage

sup

plie

s-

Res

ultin

g fr

eque

ncy

is s

ensi

tive

to v

aria

tion

in c

lock

load

CL

C1

R1

L1

V1

C2

R2

L2

V2

C3

R3

L3

V3

C LRL

Prio

r Ar

ts –

Har

mon

ics

(You

nis

and

Knig

ht, A

VLSI

199

6)

(3rd

Ord

er)

6

Prop

osed

Har

mon

ic R

eson

ant

Rai

l Driv

er

Back

grou

ndCu

rren

t-Fe

d Pu

lse

Form

ing

Net

wor

k (C

PFN

)

Des

ign

Algo

rithm

Fo

rm li

near

sys

tem

of

equa

tions

to

calc

ulat

e ne

eded

co

mpo

nent

val

ues

Impl

emen

tatio

n Is

sues

Lab

Mea

sure

men

tsCo

nclu

sion

s

7

Def

initi

on:

Nth

-ord

er s

quar

e-w

ave

Nth

-ord

er s

quar

e-w

ave

puls

e ge

nera

tor

Circ

uit

tem

plat

e de

rived

fro

m d

efin

ition

Issu

es-

Casc

aded

con

nect

ion:

no

plac

e fo

r cl

ock

load

-Vo

ltage

sw

ings

neg

ativ

e (t

hus,

nee

d D

C of

fset

) -

Cons

tant

cur

rent

sou

rce

requ

ired

∑−

=

=1

2

,3,1'

'''

sin

)(

n

kk

kkk

DC

CLt

CLI

tv

K

Tt

nnV

Tt

VTt

VTt

Vt

v)1

2(2

sin

)12(

210

sin

526

sin

322

sin

2)

(0

00

0−

−+

++

+=

ππ

ππ

ππ

ππ

L

C'1

L'1

C'3

L'3

v(t)

C'k

L'k

C'2n

-1

L'2n

-1

I DC

S

v k(t)

Back

grou

nd:

Curr

ent-

Fed

Puls

e Fo

rmin

g N

etw

ork

(CPF

N)

8

Back

grou

nd:

Mod

ified

CPF

NN

etw

ork

Impl

emen

tatio

n:

Sam

e im

peda

nce/

adm

ittan

ce f

unct

ion

as o

rigin

al C

PFN

Prov

ides

loca

tion

for

cloc

k lo

ad c

apac

itanc

e

Rem

aini

ng I

ssue

s-

Volta

ge s

win

gs n

egat

ive

(thu

s, n

eed

DC

offs

et)

-Co

nsta

nt c

urre

nt s

ourc

e re

quire

d

New

Iss

ue-

Mor

e di

ffic

ult

to d

eter

min

e va

lue

of e

ach

com

pone

nt

C LL 0

L 1L 2

L n-1

C 1C 2

C n-1

v(t)

I DC

Si C

0i L0

i 1i 2

i n-1

9

Des

ign

Algo

rithm

Step

1:

Conv

ert

to F

requ

ency

Dom

ain

Giv

en:

N-t

hor

der

squa

re-w

ave

equa

tion

Conv

ert

to f

requ

ency

dom

ain

(Lap

lace

tran

sfor

m)

Iden

tify

all b

ranc

h cu

rren

ts

Tt

nnV

Tt

VTt

VTt

Vt

v)1

2(2

sin

)12(

210

sin

526

sin

322

sin

2)

(0

00

0−

−+

++

+=

ππ

ππ

ππ

ππ

L

−+

++

+=

2 02

22 0

20

0

)12(

11

2)

(ω

ωπω

ns

sV

sV

L

−+

++

+=

2 02

22 0

20

00

)12(

2)

(ω

ωπ

ωn

ss

ss

CV

sI CL

L

−+

++

+=

))1

2((

1)

(1

2)

(2 0

22

2 02

0

00

0ω

ωπ

ωn

ss

ss

LVs

I LL

−+

++

+Ω

+=

+=

2 02

22 0

22

20

0

2)1

2(1

2)

(1

1)

(ω

ωπ

ωn

ss

ss

sLV

sV

CL

s

sL

sI

kk

kk

kk

Lk

kk

CL

/1

whe

re=

Ω

CL

L 0

L 1L 2

Ln-1

C1

C2

Cn-1

v(t

)

I DC

Si CL

i L0

i 1i 2

i n-1

10

Des

ign

Algo

rithm

Step

2:

Sim

plify

Ik(

s)G

iven

Conv

ert

I k(s

) us

ing

part

ial f

ract

ion

expa

nsio

n

By c

ompa

ring

each

ter

m,

−+

++

+Ω

+=

2 02

22 0

22

20

0

)12(

12

)(

ωω

πω

ns

sss

sLV

sI

kk

kL

−+

+Ω

++

++

+Ω

+=

2 02

22

22 0

21

22

10

0

)12(

2)

(ω

ωπ

ωn

ss

Bs

sA

ss

Bs

sA

LVs

Ikn

k

knk

k

k

kk

L

1)1

2(

02

2 02

=Ω

+−

=+

kkj

kj

kjkj

Bj

A

BA

ω

2 02

22

22 0

22

22

)12(

))1

2()(

(ω

ω−

++

Ω+

=−

+Ω

+j

ss

Bs

sA

js

ss

kj

k

kj

k

sB

jA

sB

As

kkj

kjkj

kj)

)12(

()

(2

2 03

Ω+

−+

+=

ω

11

Des

ign

Algo

rithm

Step

3:

Find

Ω2 k=

1/L k

C kG

iven

Rew

rite

for Ω

k

By a

pply

ing

KCL,

ter

m w

ith Ω

kha

s to

be

zero

Com

bini

ng c

ondi

tions

yie

lds

char

acte

ristic

equ

atio

n

01

=∑ =n j

kjA

2

01

22 0

1

22 0

2

22 0

2

whe

re,0

)12(

11

)12(

1

1)

)12

((

Ω

==

−−

=∴

Ω−

−=

=Ω

−−

∑∑

==

ωω

ω

ω

kn j

n jkj

kkj

kkj

xx

jA

jA

jA

−+

−+

Ω+

++

+−

+Ω

+=

2 02

22

22 0

21

22

10

0

)12(

2)

(ω

ωπ

ωn

ss

As

sA

ss

As

sA

LVs

Ikn

k

knk

k

k

kk

L

−+

−+

++

−+

Ω+

++

+=

2 02

22 0

21

222

10

0

)12(

)(

2)

(ω

ωπ

ωn

ss

As

sA

ss

AA

ALV

sI

knk

k

knk

k

kk

LL

12

Des

ign

Algo

rithm

Step

4:

Subs

titut

e ro

ots

into

KCL

Equ

atio

nG

iven

: n-

1 ro

ots

from

cha

ract

eris

tic e

quat

ion

Coef

ficie

nt A

kjis

We

rew

rite

bran

ch c

urre

nts

2 01

21

2 02

2 22 0

12 1

, ,

,ω

αω

αω

α−

−=

Ω=

Ω=

Ωn

nK

2 02

21

21

02

00

2 02

11

11

000

22

00

0

2 02

22 0

20

00

1 10

0

)12(

))1

2((

1)1

2(1

2

)1(1

)1(1

12

1)1

2(1

311

2

)12(

2

)(

)(

)(

ωα

πω

ωα

απω

ωπ

ωω

πω

−+

−−

++

−−−

+

−

++

−+

−

−+

++

=

−+

++

+−

+=

−

−−

−−

− =∑

ns

sn

LL

nV

ss

LL

LV

sn

LV

ns

sss

VC

sI

sI

sI

sI

sI

nn

nn

DC

n kk

LC

DC

L

L

L

L

L

kk

kjj

jA

αω

ω−

−=

Ω−

−=

22 0

22 0

2)1

2(1

1)1

2(1

13

By c

ompa

ring

both

sid

es o

f cu

rren

t eq

uatio

n, f

ind

L k

From

Ω2

k=1/

L kC k

, fin

d C k

=

−−

−−

−

−−

−−

−−

−−

2 00

2 00

2 00

110

12

12

2

12

12

2

11

111

)12(

1)1

2(1

)12(

1

31

31

311

11

11

ωωω

αα

αα

αα

CCC

LLL

nn

nn

n

nn

LL

L

LL

LL

LL

Des

ign

Algo

rithm

Step

5:

Find

Lk,

Ck

14

Impl

emen

tatio

n:Is

sues

with

Mod

ified

CPF

N

We

foun

d th

e re

cipe

to

iden

tify

all c

ompo

nent

val

ues

Still

Rem

aini

ng I

ssue

s-

Volta

ge s

win

g :

+V

~ -

V-

Cons

tant

cur

rent

sou

rce

requ

ired

CL

L 0

L 1L 2

L n-1

C1

C2

Cn-1

v(t)

I DC

Si CL

i L0

i 1i 2

i n-1

15

Impl

emen

tatio

n:Pr

opos

ed N

etw

ork

Driv

e ne

twor

k us

ing

squa

re-w

ave

volta

ge s

ourc

eAp

prox

imat

es T

heve

nin’

seq

uiva

lent

net

wor

kIn

trod

uces

Vdc

/2 D

C of

fset

Add

larg

e ta

nk c

apac

itanc

e C T

in s

erie

s w

ith L

0Ab

sorb

s D

C of

fset

fro

m in

put

puls

es

Intr

oduc

e se

ries

resi

stan

ce R

Abso

rbs

unw

ante

d hi

gher

ord

er h

arm

onic

s

Puls

eG

ener

ator

R

L 0L 1

L n-1

C TC 1

C n-1

C L

V iV O

16

Impl

emen

tatio

n:Fr

eque

ncy

Res

pons

e vs

. R (

1MH

z, 2

nd-o

rder

)

F(s)

=V o

(s)/

V i(s

)|F

(s)|

=1,

pha

se(F

(s))

=0

at e

ach

harm

onic

fre

quen

cies

Larg

er R

abs

orbs

hig

her

orde

r ha

rmon

ics

muc

h be

tter

Freq

uenc

y (ra

d/se

c)

Phase (deg)Magnitude (dB)

106

107

108

-90

-4504590-80

-60

-40

-200

R=1

00R

=500

R=1

000

R=1

0000

0

17

Lab

Mea

sure

men

t:Te

st B

oard

Set

up OOXXS3

OXXXS4

XX

Conv

entio

nal

OO

4th -

orde

r

OO

3rd -

orde

r

OO

2nd -

orde

r

S2S1

Mea

sure

cur

rent

flo

win

g in

to

74FC

T244

C CM

OS

buff

erCa

paci

tors

and

indu

ctor

s ar

e al

l tu

nabl

e ex

cept

tan

k ca

paci

tor

C TIn

con

vent

iona

l mod

e, R

is s

et t

o 0

HP8

110A

Puls

eGen

50Ω

74FC

T244

C

Vdd

R

C LC TL 0

C 1L 1

C 2L 2

S3 S2 S1

I

C 3L 3

S4

18

Lab

Mea

sure

men

t:Th

eore

tical

& M

easu

red

Com

pone

nt V

alue

26.0

0

19.5

8

14.6

6

14.8

8

14.5

3

14.2

9

P/fC

V2

0.05

60.

360.

360.

670.

6456

.562

.697

.815

.0

0.12

00.

790.

811.

501.

4456

.062

.697

.810

.0

0.49

32.

963.

245.

65.

7659

.362

.697

.85.

0

1.18

318

.820

.234

.636

.059

.062

.697

.82.

0

2.01

575

.680

.913

5.9

143.

959

.862

.697

.81.

0

3.15

111

9.6

126.

521

5.0

224.

859

.862

.697

.80.

8

2nd

V H=

3V L

=0

Mea

sure

dTh

eory

Mea

sure

dTh

eory

Mea

sure

dTh

eory

RL1

L0C1

C Lfc

lkM

Hz

Indu

ctor

s an

d ca

paci

tors

are

tun

ed f

or m

inim

al p

ower

con

sum

ptio

n,

then

mea

sure

d us

ing

LRC-

met

erM

easu

red

valu

es a

re w

ithin

10%

of

theo

retic

al v

alue

s

19

Lab

Mea

sure

men

t:Va

ryin

g R

500

1000

1500

2000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

R(o

hm)

P/fCV2

500

1000

1500

2000

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

R(o

hm)

Normalized transition time

Mot

ivat

ion

How

R a

ffec

ts p

ower

co

nsum

ptio

n an

d ris

e/fa

ll tim

e?

Expe

rimen

tR:

100

~ 2

000 Ω

2nd -

orde

r, 1

MH

z, C

L=10

0pF

Resu

ltRis

e/fa

ll tim

e de

crea

ses

at

the

expe

nse

of p

ower

co

nsum

ptio

nD

on’t

have

to

desi

gn h

ighe

r or

der

netw

ork

for

fast

ris

e/fa

ll tim

e

20

Lab

Mea

sure

men

t:Va

ryin

g C L

Mot

ivat

ion

In p

ract

ice,

load

cap

acita

nce

C Lm

ay v

ary

from

nom

inal

val

ue

Expe

rimen

tC L

: N

omin

al 1

00pF

(Var

ies

+30

% ~

-30

%)

2nd -

orde

r, 1

MH

zAl

l oth

er v

alue

s ar

e fix

ed

Resu

ltO

utpu

t fr

eque

ncy

is s

tabl

eN

orm

aliz

ed p

ower

con

sum

ptio

n in

crea

ses

due

to d

isto

rtio

n

7080

9010

011

012

013

00

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.91

C0(

pF)

P/fCV2

21

Lab

Mea

sure

men

t:1M

Hz,

2nd

-/3r

d -or

der

Wav

efor

m

f=1M

Hz

C L=

100p

F2n

d -or

der

R=2.

015k

P/fC

V2=

14.5

3%

f=1M

Hz

C L=

100p

F3r

d -or

der

R=1.

671k

P/fC

V2=

16.6

1%

22

Lab

Mea

sure

men

t:1M

Hz,

4th-o

rder

/10M

Hz,

2nd

-ord

er f=1M

Hz

C0=

100p

F4t

h -or

der

R=0.

958k

P/fC

V2=

28.0

3%

f=10

MH

zC0

=10

0pF

2nd -

orde

rR=

0.12

0kP/

fCV2

=19

.58%

23

Conc

lusi

onPr

actic

al s

olut

ion

for

harm

onic

res

onan

t sq

uare

-wav

e si

gnal

gen

erat

or is

pro

pose

dPr

opos

ed a

lgor

ithm

: Si

mpl

e so

lutio

n to

det

erm

ine

(L, C

) co

mpo

nent

s’va

lues

70-8

5% e

nerg

y ef

ficie

ncy

mea

sure

d in

fre

quen

cy r

ange

s 0.

8MH

z~15

MH

zH

ighe

r fr

eque

ncy

rang

e fe

asib

le

Out

put

freq

uenc

y is

rel

ativ

ely

inse

nsiti

ve t

o lo

ad

capa

cita

nce

varia

tion

Can

be a

pplie

d to

Tru

e Si

ngle

Pha

se C

lock

ing

syst

em

with

out

cloc

k dr

iver

s (w

ide

met

al f

or c

lock

nod

e)