MICROSYSTEMS LABORATORYDEPARTMENT OF ELECTRICAL &

COMPUTER ENGINEERING

Microelectronic Hysteresis Robert W. Newcomb

Talk for FICAMC: Plovdiv August, 16, 2008

(Fifth International Conference of Applied Mathematics and Computing – Bulgaria’2008)

The Old Town of Plovdiv by Ivan Theofilov

• Your ancient floors float among the stars.Blue donkeys graze the silence around.The Roman road leads down along matrimonial chandeliers.A cry out of woman's flesh calls in the clock.Violet-colored philistines go to bed in the deep houses,they hear the pig, the hens, the train, the mouse.The darkness dawns with quick sensual pupils.The bridal veil flies away with the chimney's breath.Blue donkeys run on the moonlit roofs.Saints take off in a cloud from whitewashed churches,with blood-soaked lambs they welcome the bridal veil.Leopards gaze with amber eyes from the doorsteps.Among box trees bacchantes with satin bandspour fragrant myrrh out of bronze rhytons ...

Ivan Theofilov was born in Plovdiv in 1931 and graduated from the Theatre Academy in Sofia. He is an honorary citizen of Plovdiv. The poem translated here is from Geometry of the Spirit, published by Free Poetic Society, Sofia, 1996, and translated by Zdravka Mihaylova

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Main topic of talkThe mathematics for the design of VLSI CMOS circuits for hysteresis controlled by a voltage or current. Possible uses: Chaotic circuits, robust oscillators, memory,debouncing, pixel holding, emulation of chemical reactions, artificial neural networks,buildings in earth quakes

Items for discussion: Microlectronic hysteresis conceptThe main idea, curve with movable load lineRepresentation via semistate equationsKey circuits usedSome examples

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The concept for microelectronics Hysteresis (ancient Greek = to lag behind)

a) Static: piecewise multi/single valued [reason Spice won't run DC analysis on hysteresis]along withb) Dynamic: single valued given initial conditions

Typical (static): binary bent

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Binary hysteresis curves

5

Example of use for holding pixels in presence of noise6

Main Idea

Slide a load line, which depends upon the hysteresis input parameter, across a nonlinearfunction to give two or more intersections inone region.

==>

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CMOS circuit and bent V-I hysteresisusing inverters

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CMOS bent hysteresis design curves

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Designing all CMOS V-I hysteresis

10

CMOS inverter bent hysteresis

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Hysteresis use in chaos generation

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Multilevel hysteresis13

Typical binary hysteresis circuitOTA = operational transconductance amplifier

= voltage controlled current source

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Variable hystereses15

4 quadrant current mode hysteresis

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Variable hysteresis from last circuit17

Hysteresis in several dimensions

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From UMCP dissertation of Yu Jiang

Circuit to realize 2D hysteresis 19

Dynamics via Semistate Equations

Edx/dt = A(x) + Bu y = Cx

u = input, y = output, x = semistate B, C, E constant matrices, E may be singular A(x) nonlinear to generate hysteresis

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Op-amp circuit for hysteresis21

Semistate equations for op-amp example

1xyu12k)x212k(11x0

)2f(xokσ1xoσdt1dx

)invd(v)2

r1r

(1outv

dtds step,unit 1(x) ));d1(v21(

oσs okσoutv

are equations semistate then the2/r1r12k 1(x)),2(-1f(x)

,dv2 x,outv1 x,invulet

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OTA circuit for hysteresis

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OTA example semistate equations

2xy)1f(xTI2x0

2x-ug1xgdt

1dx

gsC

are equations semistate theuin vy,2xouti ,1xdWith v

)invdvg(dt)dvd(

gsCouti

)df(vTIouti

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OTA CMOS circuit25

OTA curves

5 4 3 2 1 0 1 2 3 4 51.1 10 4

8.8 10 5

6.6 10 5

4.4 10 5

2.2 10 5

0

2.2 10 5

4.4 10 5

6.6 10 5

8.8 10 5

1.1 10 4

IT epsI

IT epsI

iOTA vd( )

0

vdmaxvdmin vd

gimax 2IT

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Sliding load line on OTA curve

5 4 3 2 1 0 1 2 3 4 51.1 10 4

8.8 10 5

6.6 10 5

4.4 10 5

2.2 10 5

0

2.2 10 5

4.4 10 5

6.6 10 5

8.8 10 5

1.1 10 4

IT epsI

IT epsI

iOTA vd( )

0

igi go 2 vi2 go( )( ) vd[ ]

igi go vi2 go( ) vd( )

igi go vi2 go( ) vd( )

igi go 2 vi2 go( ) vd( )

vdmaxvdmin vdHysteresis for case of last curves. Uses two curves, upper & lower

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Resulting OTA hysteresis, Iout vs Vin

5 4 3 2 1 0 1 2 3 4 51.1 10 4

8.8 10 5

6.6 10 5

4.4 10 5

2.2 10 5

0

2.2 10 5

4.4 10 5

6.6 10 5

8.8 10 5

1.1 10 4

IT epsI

IT epsI

hu vi go( )

hl vi go( )

vimaxvimin vi

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OTA VLSI layout29

Neural type cell circuit

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Neural type cell hysteresis31

CMOS resistor from OTA

92,12,1 2)(1

21

KKVVAKIAR

ssadjadj

M9 Vadj

M1

M3 M4

M2V+ V-

M5 M6

M7 M8

Io(IR+) -Io(IR-)M10

M11

Floating resistor; can bepositive or negative (by reversing green Leads to M1 & M2)

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PSpice Simulation: Positive Floating Resistor Circuit

• Observations– Linear I-V region centered at V+ – V- = 0V– IR+ and IR- show good symmetry– Vadj modulates I-V linearity range

IR+

IR-

Vadj=-

2.4v …

-3.4vVadj=-

3.4v …

-2.4v

0.6

0.0

-0.6

-0.3

0.3

0.0 0.75 1.5-0.75-1.5V+ – V- [V]

-6.0 -4.0 -2.0 0.0 2.0 4.0 6.0

V+ – V- [V]

IR+,

IR- [

A

]

IR+,

IR- [

A

]

1.0

0.0

-1.0

-0.5

0.5

magnified

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Reference: V. Petrov, M. Peifer, J. Timmer, “Structural Stability Analysis of a Cell Cycle Control Model,” Comptes rendus de l’Academie bulgare des Sciences, Tome 58, No. 1, 2005, pp. 19 – 24.

u)1)(k2k

1(2z

)2σ)(x1σ)x(xok1(1z

0:d(.)/dtfor uz2kx1kdt

dzzokx21σ2)21(3xdt

dx

x

1.3 1.13 0.95 0.78 0.61 0.43 0.26 0.08670.0867 0.26 0.43 0.61 0.78 0.95 1.13 1.31

0.90.80.70.60.50.40.30.20.1

00.10.20.30.40.50.60.70.80.9

10.922

0.922

z1 x( )

0

z2 x 2 u3( )

z2 x u3( )

z2 x u3( )

z2 x 2 u3( )

xmaxxmin x

Plot for σ1=- σ2=1, ko=1=k2=2k1

u3=0.136 at = slopes

Kinetic cells

Circuit to give Iout=(Ix+Iy)2/(2Iw)

Subtract Ix2/(2Iw) &

Iy2/(2Iw)to get

I=IxIy/Iw

Iterate (= cascadeconnections) to getcubic, etc.

W. Gai, H. Chen, E. Seevinck, “Quadratic-translinear CMOS Multiplier divider circuit,” Electronic Letters, May 1997, p. 860

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Configuration to give cubic products

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NPN differential pair

Apply differential voltage Vd and tail current Io thenIout=I2-I1=Io·tanh(Vd/(2VT)), VT=thermal voltage=KT/Q I2+I1=Io => 2·I2=Io(1+tanh(Vd/2VT)

2·I1=Io(1-tanh(Vd/2VT)

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Multiplier via npn differential pairIout=(I4+I6)-(I3+I5)

=Io·tanh(Vx/2VT)tanh(Vy/2VT)

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Cubic via npn differential pair: Iout=Io·tanh(vx)tanh(vy)tanh(vz)

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<=from previous=>

<=take diff=>

VLSI Transistors

Two basic types with two complementary of each: MOSFET: NMOS & PMOS [piecewise square law]

BJT: npn & pnp [exponential law]

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NMOS LawFor VGS ≤ Vth ID=0 off

For VGS>Vth

ID=(VGS-Vth)2 if VGS-VthVDS saturation

=(2(VGS-Vth)VDS-VDS2) if VGS-Vth<VDS Ohmic

=(Cox/2)(W/L)

If ≠ 0 multiply by (1+VDS); for now Vth>0

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Useful CMOS current circuits42

Setting up semistate equationsUse graph theory: vb & ib = branch voltages and currents

vt & il = tree voltages and link (cotree) currents KCL: 0t = Cib KVL: 0l= Tvb

==> vb = CTvt ib = TTil

ib=idevice+isource=id+is ; vb=vd+vs

by equivalences for devices id=Y(vd)

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Useful equivalences44

Example of setting up equations

T5b

i4b

i3b

i2b

iin

i1biib;bi11010

0110100

Cutset equations = KCL at nodes I and II

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Device characterization

)V(vI)(vI0

vv

CCCC

s0i

i0101001101

0i

00

vv

v;

)V(vI)vsC(v

0)(vI

0

iiiii

i

dd2S21D12

1in

din

2

1t

dd2S2

21

1D1

5d

4d

3d

2d

1d

d

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Final semistate equations

tv01outv

ini01

)ddV2(vS2I)1(vD1I0

dttdv

1111

C

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Idea of an extension

In terms of binary hysteresis canset up a Preisach’s type of theory:

N

1n(u)nhnw

u

ouw(x)dh(x)y(u)

Where h is binary hysteresis and w is a weight

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Alternate CMOS OTA hysteresis

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CMOS OTAVout/Vin hysteresis circuit50

Vout/Vin OTA hysteresis51

VLSI Layout for 1.2U AMI fabrication 526 main transistors10ux10u, cap 38ux32u

Vdd

Gnd

In

OutVr

Vb