Non-linear transmission lines for pulse shaping in silicon - Custom Integrated Circuits Conference, 2003. Proceedings of the IEEE 2003

Non-Linear

Transmission

Lines for

Pulse Shaping

in Silicon

Ehsan Afshari

and

Ali

Hajimii

California Institute

of

Technology (Caltech)

Pasadena, CA,

USA

Caltech MC 136-93,1200

E.

California

blvd.,

Pasadena, CA

91

125

Abstract

Non-linear

transmission

limes

(NLTL)

are used for

pulse

shaping.

We

developed

the

theory

of

pulse

propagation through the

NLTL. 

The

problem

of

a wide pulse

degenerating into multiple pulses rather

than

a single

pulse

is solved

by

using

a novel gradually scaled

NLTL. 

We

exploit certain favorable properties

of

accumulation mode

MOS

vaiactors

to

design

an

NLTL

that can sharpen

both

rising and

Falling

edges,

simultaneously.

There

is

a

good agreement among the theory,

simulations, and

measurements.

Introduction

The

wncept

of

a solitary wave was introduced

to

science

by

John

Swtt

Russell more

than

a centwy ago

[I].

In 1834

he

observed

a

wave

formed when a boat which

was rapidly drawn along

a narrow

channel

by

a

pair

of

horses

suddenly stopped. According

to his

diary,

this

wave

wntinued,

“at 

great velocity, assuming the

form

of

a

large

solitary elevation,

a well-defined heap

of

water that

wntinued

its

course along the channel apparently without change

of

form

or

diminution

of

speed”.

These solitary waves,

now

called

‘solifom’,

have become

important

items

of

research

in

diverse fields

of

physics

and engineering. There

is a wnsiderable body

of

work

on

solitons

in

applied mathematics

(e.g.,

[ZJ),

applied physics, especially

in

optics

(e.g.

[3J

and

[4]),

and few works

in

electronics

[5].

The ability

of

solitons

to propagate with

small dispersion can be

used

as

an

effective

means

to

transmit

data

modulated

as

short pulses over

long

distances.

An imponant related application

is

pulse sharpening

for

the more

traditional

non-rem-to-zero

(NRZ)

data

transmission

in

digital

circuits

by

improving the rise and

fall

times of

pulses. Improving the

bansitions

by

shrinking the rise and

fall

times

of

pulses

can

be

useful

in other applications, such

as

high-speed sampling and timing

systems.

Non-hear

transmission lines

(NLTLs)

sharpening either

of

the

rising or the

Falling

edge

of

a

pulse have been demonstrated on

a

GaAs 

technology

1-51,

However,

to

the

best

of

our

knowledge, there

has

been

no

demonstration

of

simultaneous reduction

of

both

rise

and

fall

times

in

an

NLTL

to

this date. Neither are we aware

of

any

demonstration

of

such

NLTLs

in

silicon-based CMOS

process

technologies.

In this

work,

we

first

show

a soliton line on

a wnventional

siliwn

technology

which

can

achieve very narrow pulses,

with

a

bandwidth

in

excess

of

the

cut-off

frequency,

fT,

of the

fastest transistor

in

this

process.

Next,

we

demonstrate

that 

using

a favorable characteristic

of

MOS varactors leading

to

a different kind

of

non-linearity, we

can

improve

both

the rise and

fall

times, simultaneously.

This

is not

possible

with

the nonlinear elements commonly

used

in NLTLs

(e.g.,

reverse

bias

PN

junctions). Neither

can

it

be

done using

transistors,

as

they

are

limited

by

their

fT.

In

this paper, the

propagation

of

soliton waves

in

a non-linear

transmission

line

will

be

studied. Then,

we

introduce

two

different

types

of non-linear

transmission lines

to

generate narrow pulses and

to

sharpen

pulse

bansitions, respectively. Finally, we show the

experimental

results

verifying

the

agreement between the theory and

the

measurement.

The

Theory

of

Non-linear

Transmission Line

In

this section, we review the basic theory

behind

non-linear

transmission

lines

and their

use

for

pulse

narrowing

and

edge

sharpening

in subsections

A

and

B,

respectively.

Fig.

1

shows

an

example

of

a

non-linear

transmission

line

using

inductors,

I,

and voltage dependent (and

hence

non-linear)

capacitors,

C(V.

vi

TI

1

.,

I

=

Figure

I,

A non-linear

transmission

line

By

applying

KCL at

node

n,

whose voltage with

respect

to

ground is

V,,,

and applying

KVL

across the

two

inductors

connected to

this

node,

one can easily show the voltages

of

adjacent

nodes

on

this

NLTL 

are related

via

[5]:

The

right-hand

side

of

(I)

can

be

approximated

with partial

derivatives

with

respect

to

distance,

x,

eom

the beginning

of

the

line,

assuming that the spacing between

two

adjacent sections

is

6

(i.e.,

x.=n6.)

An

approximate continuous

partial

differential equation can

be

obtained

by

using

the

Taylor expansions

of

V(x-6).

I+), 

and

V(x+s)

to

evaluate the right

hand

side

of

(1).

Assuming

a small

6,

and ignoring

the

high

order terms, we obtain:

-

..

(2)

a

av

aLv

s’

a4v

L

-

[C(

V)

-1

=

-

+

-

at

&2

12

&4

where

C

and

L

are

the

capacitance and inductance

per

unit length,

respectively.

It is noteworthy that for

a

continuous transmission line

(8

+

O),

(2)

reduces to:

(3)

A.

Pulse

Narrowing

Non-Linear Transmission Lines

In

this

subsection,

we approximate the

capacitor’s

voltage

dependence using the following first-order linear relationship

C(Y)=C,(I-bY)

(4)

where

CO

and

b

are constants.

In

this case,

(2)

reduces

to

a2v

I

a2v

s2

I

a4v

ba2(v2)

=

_--

+--

(5)

a,2

Le,,

,2

12

Le,,

&4

2

at2

where

the

left-hand side

is the classic wave equation

and

the terms

on

the

right-hand side represent dispersion

and

non-linearity,

respectively.

If the effect

of the

dispersive and

non-linear

terms

in

(5)

are on

the

same order

of

magnitude

it

is wssible

to

have

a single pulse solution

~

..

for

(5)

with a

profile that does not change

as

it propagates

with

velocity,

Y.

A

propagating mode solution

can

be

obtained

by

6-2-1

converting

the

partial differential equation

@‘DE)

of

(5)

to

an

91

0-7803-7842-3/03/$17.00

0

2003

IEEE

2003

CUSTOM INTEGRATED

CIRCUITS

CONFERENCE

ordinary

differential

equation

(ODE) 

by

a simple change of

variable:

where

v

is the propagation velocity

of the

pulse and

vo

=

I

m.

This

solution is

shown

in Fig.

2

for

three 

different values

ofL

and

C,

and

hence

different

6.

Note

that

this solution

is not a

function of

the

input

waveform, and thus any

arbitrary input

will

eventually

tums

into

(6)

going through a line

which

is

long

enough,

if it has

enough

enera.

L=lnH

and

0

InF

(b)

L=ZnH 

and

C=ZnF

(c)

L4nH

and

MnF

As

can

be

seen

from

(6).

the peak

amplitude

is a

fiction

of

the

velocity. Defining

an

effective capacitance,

C,

so

that

=

I

la,

the

pulse height

is

given by:

22

v,,, 

b

vz

b

CO

(7)

Using

(4),

we

can

relate Cefto an

effective

voltage

Vep

It is easy

to

show

that

So

it

is the

capacitance at

one-third the

peak

amplitude

that

determines

the

effective

propagation velocity. Using

(6)-(8)

we can

easily calculate

the

half-height width of the pulse to

be:

WE-

6

"0

(9)

qpq

As

can

be

seen,

in

a weakly

dispersive

and

non-linear transmission

line, the non-linearity

can

counteract the normally

present

dispersive

properties

of

the line

maintaining

solitary

waves

that

propagate

without

dispersion.

This

behavior can

be explained using

the

following

intuitive argument:

The

instantaneous

propagation velocity

at any given point

in

time and

space

is

given

by

1

I

m.

In

the

presence

of

a non-linear

capacitor

with a

characteristic given

by

(4),

the

instantaneous

capacitance

is

smaller for higher

voltages.

Therefore,

the

pints

closer to the crest

of

the voltage

waveform

experience

a faster propagation velocity and produce a

shock-wave

front

due

to

the nonlinearity,

as

shown

symbolically

in

the

upper part

of

Fig.

3.

Note

that

this

is

not a real waveform and

more

a fictitious

representation

of

how

each

pint

on

the curve tends

to

evolve.

On

the

other

hand, dispersion

of the

lie

causes the

waveform

to

spread out.

as

shown

in

the lower half

of

Fig.

3.

For

a proper

non-linearity

determined

by

(9,

these

two

effects can cancel each other

out.

Non

linearity

n

Dispersion

Input

pulse

Figure 

3.

Dispenion

and

non-linear effects

in

he

NLTZ

A

few important

observations in this line

are:

1)

the velocity

of

the

solimy

wave

increases

with

its

amplitude,

2)

pulse

width

decreases

with

increasing

pulse

velocity,

3)

the width

shrinks

for

higher

amplitudes,

4)

the sign

of solution depends

on

si5

of

non-linearity

factor,

b;

For

a

capacitor

with a

positive

voltage

dependence

(e.g.,

an

nMOS

varactor

in

accumulation mode)

we have

resulting

in

upside

down

pulses.

Based

on

these

results,

to achieve

large-amplitude narrow pulses,

inductance and capacitance of the

NLTL

must

be

as

small

as

possible

and non-linearity

factor,

b,

should

be

large

enough

to

compensate

the

dispersion

ofthe

line.

It

is

also important to

know

the characteristic

impedance

of

these

lies

(for impedance

matching,

etc.)

As

in

a

NLTL 

the capacitance

is

a function

of

voltage,

we can

only define an effective

semi-empirical

value for

the characteristic

impedance. Simulations results

indicate

that

one

can

approximate

2,using

the capacitance

at

Vefdefined

in

C(v) 

=

Co(l+bV)

(IO) 

(E),

i.e.,

=

B.

Edge

Sharpening Lines

It is possible

to

design

NLTLs 

to

sharpen the pulse transitions.

This

is

particularly

useful

for digital

transmission

(e.g.,

NRZ

data).

Unfortunately, all

the efforts

in

the past [6] have resulted

in

sharpening

of

only

one of the rising

and

falling

edges.

This, however,

has

very

little

practical

value,

as

both

hansitions

are

equally

important

in

common

NRZ

digital

systems.

This

problem can

be

traced

back to the monotonic dependence of the

non-linear

capacitive

elements

used

in

NLTL

on

the

voltage

(e.g.,

reverse

biased

PN

junction,

or

the

ideal

behavior

of

(4)

and

(I

0)).

Fortunately,

CMOS 

processes offer different

characteristics

for

non-

linear

capacitors

that

can be exploited to achieve

simultaneous

edge

sharpening for both rising and falling edges.

More specifically,

accumulation mode

MOS

varactors

[7]

(an

nMOS

capacitor

in

an

n-

well)

offer

non-monotonic

voltage

dependence. Particularly,

the

secondary

reduction

of

capacitance

shown

in

Fig.

4

due

to

poly-

silicon

depletion

[S,

91

and short-channel

charge quantization [9]

effects

can

be

used

for edge sharpening.

'V

v,

v,

v,

Figure

4.

Capacitance

versus

voltage

for

a

MOSVAR

Fig.

5

shows symbolically how one can use

the behavior

of

Fig.

4

to

sharpen

both edges. First,

let

us

focus

on

the

risptime

reduction.

92

6-2-2 

Consider the

rising edge shown

in

the upper

part 

of

Fig.

5.

Initially

the voltage is low, which

corresponds to

a smaller

capacitance

per

Fig.

4,

and hence a faster

instantaneous propagation

velocity

for

the

lower end

of

the pulse.

As

the

voltage

goes

up,

the capacitance

increases, resulting

in

an

increase

in

the instantaneous

propagation

velocity. This pushes the lower end

of the

transition forward

in

time

and results

in

sharpening

of the rising edge. This effect

is

symbolically shown

in the fictitious

middle

waveform

of

Fig.

5.

The

fall time reduction can

be

explained using

the

lower

part

of

Fig.

5.

This

is where the

non-monotonic

behavior of

Fig.

4

plays its mle.

The

upper part

of

the transition

(voltages

above

V,)

will

be

accelerated

due

to the reduction

of

the

capacitance

and will

create

an

advancing

front

as

symbolically

shown in

the middle

waveform of

Fig.

5.

The

lower capacitance at the very

low

voltages

can

generate a

leading

!ail,

which

will

be

partially

dissipated

by

the

line'.

Figure

5.

How

rise

and

fall

time

vary

within

the

NLTI.

while

the above explanation based on a simplified memory-less

description

of

the

line

provides a

basic intuition for

its

operation, a

complete description

can only

be

obtained

by

solving the

differential

equation

in

(2)

to

account

for

the

memory of

the system.

Our

numerically solution

of

(2)

also confirms that

as

long

as

the

input

voltage

range

exceeds voltages,

V,

and

V,,

for

a range

of

L's

and

Cs,

the line sharpens

both

rising and

falling edges,

simultaneously.

Gradually

Scaled

NLTL 

One

problem

in

pulse

"owing

NLTLs

is that if the input pulse is

wider

than

a certain minimum

related to the

natural pulse

width of

the line

in

(9), 

it is incapable

of concentrating

all

that energy

into

one

pulse and

instead

the input pulse

degenerates

into multiple soliton

pulses,

as

shown

in the simulated

upper waveforms of

Fig.7.

This

is

an

undesirable effect that

cannot

be

avoided

in a standard

line.

We have found a solution

for

this 

problem

by

using gradually scaled

non-linear transmission

lines. 

We

notice that

the

characteristic pulse

width of

the line is controlled

by

the

node

spacing

6,

and the

propagation velocity,

Y,

which

is

in

hum

contmlled

by,

L

and

C.

Thus,

we

use

a gradual

lie

consisting

of

several sections

that

are

gradually scaled

to

have

smaller characteristic

pulse

width,

as

shown

in Fig.

6.

The

first

few

sections have the

widest

characteristic

pulse,

meaning

that

their

output is wider and

has

smaller

amplitude.

As

a result, the

input

pulse will

cause just one

pulse

at the

output

of these

sections.

The

following

sections

have a

narmwer response and

the last section

has

the narmwest

one.

This

will

guarantee

the gradual narrowing

of

the pulses and

avoids

degeneration.

Each section

has

to

be

long

enough

so

that the pulse

can

reach

the section's

steady-state response

'

We

hypothesize

that

other

dynamic

effects

in

the

MOS

varactor

help

edge

sharpening, 

e.g.,

the

processes

of

charge being

amacted

from

the

n+

diffusions

to

the

channel

and

repelling

them

are

not

exact inverses

of

each

other

over

shon

time

intervals.

Some

of

the repelled accumulation charges

will

be

absorbed inside

the

well.

This

changes

the

response

time

of

the

capacitor

and

keeps 

it

higher

for

a

longer

period

of

time

for

the falling

edm

before entering next

section.

Figured

Schematic

ofthe

gradually

scaled

non-linear transmission

1;";

The

waveforms

of this

gradually scaled

NLTL

are

shown

in

lower

prut

of

Fig.

7,

demonstrating

the

effectiveness

of

this

technique.

It is

notewotihy

that

this

gradually

scaling

technique

is also applicable

to

the

edge sharpening lines

and

does

improve

their

performance,

tm.

I

Figure

7

Output

waveform

of

the

normal 

and

dual

diton

line

Simulation

A

Pulse

Narrowing

Lines

We

have

designed a pulse narrowing

NLTL

using

MOS

varactors

and

metal 

mino-ship

transmission lines

in

a

0.181"

BiCMOS

pmcess. We simulated the transmission lines

in

Sonnet and the

complete

NLTL

in ADS.

The

output waveform

of

the line

to

a 65ps

input pulse

is

shown

in

Fig.

8.

This

silicon-based

NLTL

produces

pulses

as

narrow

as

2.5~~

(half ampliNde width)

with

a

0.8V

amplitude. Active

devices

in this process

are

incapable

of

producing

pulses nearly

as

narrow

as

these.

V

1

OO".'.'',.

$8,

1'1'1'

%a

370

SW

t".

Paec

Figure

8

Simulated

output

waveform

afthe

pulse

mwng

line

B.

Edge

ShDrpening

Lines

We

have also

designed

an edge sharpening

non-linear transmission

line using

MOS

varactors and gradual

scaling

of

the lines

in the

same

process.

Fig.

9

shows

the

simulated

input and

the

output waveforms

of this

line.

6-2-3 

93

-

9:

3-

Fig&

IO.

Output

ofpulse

mwing'line

'

The

rise time

of

two

cascaded

systems can be

estimated using

[IO]:

:rise

=

7'.

trrse~

+Irise2

(12)

Using

this

equation and the simulated rise and fall times,

we

should

expect

9.lps

and 22ps at the output

of

a

fust-order system with

a

bandwidth

of

40GHz. Measurement waveforms

of

Fig.

1

I

show

measured output rise and

fall

times

of

I

Ips

and

25ps.

respectively.

These

numbers

art

in

agreement with

the

simulated results

considering

the

additional bandwidth degradation due to

the

pads and

the

multi-pole nature

of

the system.

The

rise

and

fall

times

do

not

change

with the input amplitude which verifies

the

non-linear

behavior

of

the

line.

*",

:Ir-Wk,

-

Conclusion

We have analyzed pulse

narmwing and

edge

sharpening

passive

non-linear

tm"ission

lines, using MOS varactors and

the

novel

gradual scaling lines,

showing

simultaneous edge sharpening

for

both

rising

and

falling edges

in

a

silicon subsbate.

The

experimental

results show considerable improvement

in

the

rise and

fall

times

of

be

(66PSddiV)

Figure

I

I.

hput

and

output

waveforms

of

the

edge

ming

line

Acknowledgements

Authors would like

to

thank 

helpful discussions with

D.

Ham,

H.

Wu,

A.

Komijani, C.

White,

J.

Buckalter,

M.

Taghivand,

H.

Hashemi,

S.

Kee,

B.

Analui,

and

A.

Natarajan

of

Caltech and Pmf.

M.

Homwitz

of

Stanford University.

lhey

also

acknowledge

the

generous

suppon

of

Lee

Center, IBM

Corp., 

Agilent

Tech.,

and NSF.

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J.

S.

Russell,

"Repon

on Waves,"

Repon

of the

founeenth

reetin8

ofthe

Britirh

Associationfor

the

Ahcement

ofSeience,

pp.

31

I-

90.

Plater

XLVII-LVII,

York 

Sepl

1844

(Londq

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G.

Drazin,

and

R.

S.

John,

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PES,

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1989.

J.

R

Tailor,

0pt;colSolitom

-

ThemyondErperiment,

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PM,

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1992.

E.

Infdd,

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G.

Rowhds.

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Ww,

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Chm,

Cambridge University Press, Cambridge,

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M.

RemaisSenet,

Wm

died

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Eqwh",

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