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Low-Power Design Techniques in Digital Systems

Prof. Vojin G. Oklobdzija

University of California

November 19, 2003

2

Outline of the Talk

• Power trends in VLSI • Scaling theory and predictions• Research efforts in power reduction• Efficiency measures and design guidelines• Latches and Flip-Flops for Low-Power

– Dual-Edge FFs– SOI

• Conclusion: Low-Power perspective

3

Power trends in VLSI

4

“CMOS Circuits dissipate little power by nature. So believed circuit designers”(Kuroda-Sakurai, 95)

“By the year 2000 power dissipation of high-end ICs will exceed the practical limits of ceramic packages, even if the supply voltage can be feasibly reduced.”

(* Taken from Sakurai’s ISSCC 2001 presentation)

959085800.01

0.1

1

10

100P

ow

er (

W)

x4 / 3years

5

Gloom and Doom predictions

Source: Shekhar Borkar, Intel

6Source: Shekhar Borkar, Intel

7

y = 3E-97e0.1131x

y = 2E-124e0.1442x

y = 6E-109e0.126x

y = 2E-222e0.2574x

0

10

20

30

40

50

60

70

80

1995.5 1996 1996.5 1997 1997.5 1998 1998.5 1999 1999.5 2000 2000.5

Year

Pow

er (W

atts

)

RISC

x86

Consumer

Dec Alpha

Expon. (RISC)

Expon. (x86)

Expon. (Consumer)

Expon. (Dec Alpha)

High-end growing at 25% / year

Consumer (low-end)At 13% / year

X86 @ 15% / yrRISC @ 12% / yr

Power versus Year: taken from ISSCC, uP Report, Hot-Chips

8

Year

Vo

ltag

e [V

]

Po

wer

per

ch

ip [

W]

VD

D c

urr

ent

[A]

VDD, Power and Current Trend

1998 2002 2006 2010 20140

0.5

1

1.5

2

2.5

0 0

200 500

Current

Power

Voltage

International Technology Roadmap for Semiconductors 1999 update sponsored by the Semiconductor Industry Association in cooperation with European Electronic Component Association (EECA) , Electronic Industries Association of Japan (EIAJ), Korea Semiconductor Industry Association (KSIA), and Taiwan Semiconductor Industry Association (TSIA)

(* Taken from Sakurai’s ISSCC 2001 presentation)

9

Power Delivery Problem (not just California)

Source: Shekhar Borkar, Intel

Your carstarter !

10

Trend in L di/dt

• di/dt is roughly proportional to I * f, where I is the chip’s current and f is the clock frequency

or I * Vdd * f / Vdd = P * f / VddP * f / Vdd, where P is the chip’s

power. • The trend is: P f Vdd

on-chip L package L slightly decreasesslightly decreases

• Therefore, L di/dt fluctuation increases significantly.

(* Taken from Norman Chang, HP)

11

ISPEC^2/Watt vs Feature Size (microns)

y = 0.3733x-2.5778

1

10

100

0.00 0.20 0.40 0.60Feature Size (microns)

ISP

EC

^2/

Wat

t

Energy-Delay product is improving more than 2x / generation

Saving Grace !

12

ISPEC^2/Watt vsYear

0102030405060708090

100

1995 1996 1997 1998 1999 2000 2001

Year

Consumerx86Server

X86 efficiency improving dramatically 4X / generation

average improving3X / generationHigh-End

processors efficiency not improving

13

Scaling theory and predictions

14

The power dissipation has increased 1000 times over the 15 yearsand is exceeding 70 Watts

Scaling principles:

1. A “constant field scaling” theory [Dennard] assumes that device voltages as well as device dimensions are scaled by a scaling factor x (>1), resulting in a constant electric field in a device:

power density remains constant circuit performance can be improved in terms of:

density x2

speed x power 1/ x2

power-delay product 1/ x3

Limitless progress in CMOS is promised with this scaling scenario

15

In practice neither a supply voltage nor a threshold voltagehad been scaled till 1990 leading to the theory of:

“Constant voltage scaling” which assumes the constant voltage

This assumption yields:

• speed improvement by x2

• power density increases rapidly by x3

16

The constant field is not realistic, x0.5 is satisfactory - however even with that the power dissipation would exceed ECL by 2001: a new philosophy is required !

(* Taken from Sakurai and Kuroda, IEICE 95 paper)

17

High-Performance View Point on Power*taken from Ron Preston, DEC Alpha

P=k C V2 f :

• Shrinking to the new technology (30% reduction in )– C decreases by 30%

– f increases by 1/0.7 = 43%

– Pnew=0.7 (1/0.7) Pold = Pold (No Change in Power ! )

• New design:– Double the No. of devices

– Pnew=2 x 0.7 (1/0.7) Pold = 2 X Pold (Power Doubles !)

Scale Vdd by 30% in the new design:

– Pnew=2 x 0.7 (1/0.7) (0.7)2Pold = Pold (Power stays constant !)

18

High-Performance View Point on Power*taken from Ron Preston, DEC Alpha

Reality:

Paradigm Changes: More Aggressive Circuits, Toggle rate increasing,

Out of Order, Speculative Execution What to Expect: Power will be limited by the package and cooling techniques

Frequency will be determined by the power - as high as package can take !

Chip Vdd Freq. Power

21164 05u 3.3V 300MHz 50W

21264 0.35u 2.0V 600MHz 72W

Change -30% -39% +100% +44%

19

Research Efforts in Low-Power Design

Psw = k CL V2

cc fCLK

Reduce Switching Activity:•Conditional clock•Conditional precharge•Switching-off inactive blocks•Conditional execution

Run it slower:•Use parallelism•Less pipeline stages•Use double-edge flip-flop

Technology scaling:•The highest win•Thresholds should scale•Leakage starts to byte•Dynamic voltage scaling

Reduce the active load:•Minimize the circuits•Use more efficient design•Charge recycling •More efficient layout

20

Reducing the Power Dissipation

• The power dissipation can be minimized by reducing:

• supply voltage• load capacitance• switching activity

– Reducing the supply voltage brings a quadratic improvement

– Reducing the load capacitance contributes to the improvement of both power dissipation and circuit speed.

21

Voltage Scaling

There are three means to maintain the throughput:

• Reduce Vth to improve circuit speed

• Introduce parallel and pipelined architecture while

using slower device speeds (assumes limitless no. of transistors, in reality the transistor density is

only increasing by 60% per year)

• Prepare multiple supply voltages and for each cluster

of circuits choose the lowest supply voltage that satisfies

the speed. (A good level converter is necessary which exhibits small delay and consumes

little power, small area)

22

23

Is there an optimal design point ?

24

Power Dissipation and Circuit Delay

Power : P = pt •fCLK •CL •VDD + I0 •10 •VDD 2

V th

S

(=1.3)

k • CL • VDD

(VDD - Vth)Delay =

k•Q

I=

12

34

-0.400.4

0.8

0

0.2

0.4

0.6

0.8

1x 10

-4

Vth (V)

VDD(V)

Po

wer

(W

)

A

B

12

34

-0.400.40.8

0

1

2

3

4

5x 10

-10

Del

ay (

s)

Vth (V)

VDD(V)

AB

(* Taken from T. Sakurai)

25

Power-Delay Product, Energy-Delay Product

Lowest Voltage – Highest Threshold –

no optimum

•Power-Delay Product is a misleading measure; it will always favor a processor that operates at lower frequency

•Energy-Delay is more adequate - but Energy-Delay2 should be used

(*from Sakurai, Kuroda, IEICE 95 paper)

26

Power-Delay Product, Energy-Delay Product

Horowitz, Indermaur, Gonzales argue against Power-Delay, SLPE’94

27

Energy-Delay**2

(*courtesy of Prof. T. Sakurai)

28

Energy-Delay Product vs. Energy-Delay**2

Nowka, Hofstee, Carpenter of IBM argue against Energy-Delay as a design efficiency measure (private communication)

29

Energy-Delay Product vs. Energy-Delay**2

Nowka, Hofstee, Carpenter of IBM argue against Energy-Delay as a design efficiency measure (private communication)

The same design should have relatively

the same efficiency

Optimal point: (due to to Vth being fixed ?)

30

Feature 601+ 604 620 Diff.

FrequencyMHz

100 100 133(100)

 same

CMOS Process .5u 5-metal .5u 4-metal .5u 4-metal ~same

Cache Total 32KB Cache 16K+16K Cache

64K ~same

Load/Store Unit No Yes Yes  

Dual Integer Unit No Yes Yes  

Register Renaming No Yes Yes  

Peak Issue 2 + Br 4 Insts 4 Insts ~double

Transistors 2.8 Million 3.6 Million 6.9 Million +30% /+146%

SPECint92 105 160 225(169)

+50% /+61%

SPECfp02 125 165 300(225)

+30% /+80%

Power 4W 13W 30W(22.5W)

+225%/+463%

Spec/Watt 26.5/31.2 12.3/12.7 7.5/10 -115%/-252%

PF=Watt/Freq**3 4.0E-6 13.0E-6 12.8E-6  

(PF/Trans)*E12 1.43 3.61 1.86  

IPC 1.05 1.6 1.69  

PE*IPC**3 (*E6) 4.01 12.98 12.69  

PE=Watt/Spec**3 3.46E-6 3.17E-6 2.63E-6  

Example: PowerPC

31

Feature Digital 21164

MIPS 10000 PowerPC 620

HP 8000 Sun Ultra-Sparc

Freq 500 MHz 200 MHz 200 MHz 180 MHz 250 MHz

Pipeline Stages 7 5-7 5 7-9 6-9

Issue Rate 4 4 4 4 4

Out-of-Order Exec. 6 lds 32 16 56 none

Register Renam. (int/FP) none/8 32/32 8/8 56 none

Transistors/Logic transistors

9.3M/1.8M

5.9M/2.3M

6.9M/2.2M

3.9M*/3.9M

3.8M/2.0M

SPEC95(Intg/FlPt)

12.6/18.3 8.9/17.2 9/9 10.8/18.3 8.5/15

Power 25W 30W 30W 40W 20W

SpecInt/Watt

0.5 0.3 0.3 0.27 0.43

1/Energy*Delay 6.4 2.6 2.7 2.9 3.6

Watt/Freq**3 0.2E-6 3.75E-6 3.75E-6 6.86E-6 1.28E-6

(PF/Trans)*E12 0.022 0.64 0.54 1.76 0.34

(PF/LTrans)*E12 0.11 1.63 1.7 1.76 0.64

Watt/Spec**3 12.5E-3 42.5E-3 41.5E-3 31.7E-3 32.5E-3

32

Sensitivity to Vth fluctuation

VTH (V)

0 0.2 0.4 0.7 1

1.5 V

3.0 V

5.0 V

0.6

1.0

1.4

1.8N

orm

aliz

ed D

elay ± 0.15V

VDD =1.0 V

± 0.05V

ΔVTH =

0.5

(* Taken from T. Sakurai)

33

Use of Different Circuits Families

34

Capacitance Reduction

The load capacitance is the sum of:

• gate capacitance• diffusion capacitance • routing capacitance

Using small number of transistors, or small size of transistorscontributes to the reduction in the gate capacitance and the diffusion capacitance.

Pass transistor logic may have advantage because it comprises fewer transistors and exhibits smaller straycapacitance than conventional static CMOS logic.

35

Pass-Transistor Logic

36

Pass-Transistor Logic: CVSL, CPL, SRPL, DSL, DPL, DCVSPG

37

SAPL:Sense-Amplifying Pass-transistor

Logic

All nodes are first discharged and then evaluated by inputs.Outputs are 100mV above GND

38

Where does the power go ?

39

Power use is different from chip to chip:

MPU1 is a low end microprocessorMPU2 is a high-end CPU with large cacheASSP1 is MPEG-2 decoderASSP2 is an ATM switch

(*from Sakurai, Kuroda, IEICE 95 paper)

40

Design Example: Strong Arm 110Two power modes: idle and sleep

Power:0.5W using 1.1V internal PS: 184 Drystone/MIPS @162MHz

1.1W using 2V internal PS: 245 Drystone/MIPS @ 215MHz

Power Breakdown:I-Cache 27%

D-Cache 16%

I-Unit 18%

Exec-Unit 8%

I-MMU 9%

D-MMU 8%

Clock 10%

Others 4% (PLL < 1%)

*from D. Dobberpuhl

41

Design Example: Strong Arm 110

*from D. Dobberpuhl

42

Design Example: Strong Arm 110

However, leakage currents starts to affect stand-by power

*from D. Dobberpuhl

*from D. Dobberpuhl

43

Controlling both: VDD and VTH for low power

44

Controlling VDD and VTH for low power

Low power Low VDD Low speed Low VTH High leakage VDD-VTH control

Active Stand-byMultiple VTH Dual-VTH MTCMOS

Variable VTH VTH hopping VTCMOS

Multiple VDD Dual-VDD Boosted gate MOS

Variable VDD VDD hopping

*) MTCMOS: Multi-Threshold CMOS*) VTCMOS: Variable Threshold CMOS• Multiple : spatial assignment• Variable : temporal assignment

Software-hardware cooperation

Technology-circuit cooperation

(* from Prof. T. Sakurai)

45

Dual-VTH concept

Low-VTH circuit(High leakage)

High-VTH circuit(Low leakage)

Critical paths

Non-critical paths

(* from Prof. T. Sakurai)

46(* from Prof. T. Sakurai)

Clustered Voltage Scaling for Multiple VDD’s

Lower VDD portion is shown as shaded

CVS StructureConventional Design

Critical Path

Level-Shifting F/F

Critical Path

FF

FF

FF

FF

FF

FF

FF

FF

FF

FF

FF

FF

FF

FF

FF

FF

FF

FF

FF

FF

FF

M.Takahashi et al., “A 60mW MPEG4 Video Codec Using Clustered Voltage Scaling with Variable Supply-Voltage Scheme,” ISSCC, pp.36-37, Feb.1998.

Once VL is applied to a logic gate, VL is applied to subsequent logic gates until F/F’s to eliminate DC current paths. F/F’s restore VH.

47

Energy consumption isproportional tothe square of VDD.

Energy consumption isproportional tothe square of VDD.

VDD should be loweredto the minimum levelwhich ensuresthe real-time operation.

VDD should be loweredto the minimum levelwhich ensuresthe real-time operation.

Normalized workload0.0 0.2 0.4 0.6 0.8 1.0

No

rmal

ized

po

wer

0.0

0.2

0.4

0.6

0.8

1.0

Variable VddFixed Vdd

If you don’t need to hussle,VDD should be as low as possible

(* from Prof. T. Sakurai)

48

Measured voltage waveforms

1 sync frame

200ms

Sleep

V DDmax =8% on average

V DD

V DDmax

V DDmin

Sleep signal

Sleep=6% on average

(* from Prof. T. Sakurai)

49

Measured power characteristics

Total power = 0.8W x 0.08 + 0.16W x 0.86 + 0.07W x 0.06 = 0.2WTotal power = 0.8W x 0.08 + 0.16W x 0.86 + 0.07W x 0.06 = 0.2W

VDD hopping can cut down power consumption to 1/4VDD hopping can cut down power consumption to 1/4

0.8W

0

0.2

0.4

0.6

0.8

1

Supply voltage: VDD [V]

Po

wer

: P

[W

]

0 1 2

ƒ=100MHz

ƒ=200MHz

0.16W

Downto 1/5

Time for sleep: 6% 0.07W

Time for VDDmin : 86%

Time for VDDmax : 8%

(* from Prof. T. Sakurai)

50

Simulation results

0.0 0.2 0.4 0.6 0.8 1.00.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

RPC: 2 levels (f,f/2)RPC: 3 levels (f,f/2,f/3)RPC: 4 levels (f,f/2,f/3,f/4)RPC: infinite levelspost-simulation analysis

0.0 0.2 0.4 0.6 0.8 1.00.00

0.04

0.08

0.12

0.16

0.20

0.24

0.28

0.32

RPC: 2 levels (f,f/2)RPC: 3 levels (f,f/2,f/3)RPC: 4 levels (f,f/2,f/3,f/4)RPC: infinite levelspost-simulation analysis

MPEG-2 video decoding VSELP speech encoding

No

rma

lize

d P

ow

er

P/P

FIX

No

rma

lize

d P

ow

er

P/P

FIX

Transition Delay TTD

(ms)Transition Delay TTD

(ms)

(* from Prof. T. Sakurai)

51

Aggressive Voltage Scaling

If we can dynamically scale Vdd and Vth the advantage is obvious

*Taken from Kuroda

52

Example

53

TransMeta Example

*Taken from Doug Laird’s presentation, January 19 th 2000

54

TransMeta Example

*Taken from Doug Laird’s presentation, January 19 th 2000

55

TransMeta Example

• “Code Morphing” is another contributor to power reduction since it eliminates unnecessary external memory access

*Taken from Doug Laird’s presentation, January 19 th 2000

56

TransMeta Example

57

Latches and Flip-Flops for Low-Power

58

Simulation Condition and TestbenchTiming

Total FF overhead is setup + clock-to-output time

Circuit optimization towards td-q

Clock skew robustness obtained from observing DQ curve

Power-Delay Product Overall performance

parameter at fixed frequency

)(tmint QDt

dCLKD

dissd PtEDPf) fixedPDP(at

D

Q

QSET

CLR

Clk

Data In

Clock

14X m ininv

14X m ininv

14X m ininv

59

Flip-Flop Performance Comparison

• Total power consumed– internal power

– data power

– clock power

• Measured for four cases– no activity (0000… and 1111…)

– maximum activity (0101010..)

– average activity (random sequence)

Test bench

Delay is (minimum D-Q):Clk-Q + Setup time

C lk

D ata

C lo ck

5 0 fF

2 0 0 fF

2 0 0 fFD Q

Q

60

OLD TEST BENCH:

• Total Power = Drivers Power + Test Unit Power

• PDP- Optimized = Equal Trade-off on Power and Delay

• Improper Load on Drivers

NEW TEST BENCH:

• Drivers: Fixed Gain and Driving Test Unit Only

• Data-to-Output Delay

• PD2P Optimized = Best for Constant-Field Scaling

OLD TEST BENCH

NEW TEST BENCH

61

Comparison in terms of speed and EDPtot

Technology: 0.2u, Vdd=2V, T=20oC, measured @ 100MHz

• Delay: below 200ps• SDFF 187ps• HLFF 199ps• K-6 ETL 200ps

– 200-300ps• PowerPC latch 266ps• 21264 Alpha FF 272ps• Strong Arm FF 275ps• mC2MOS latch 292ps

– above 500ps• SSTC latch 592ps• DSTC latch 629ps• SSTC* latch 898ps• DSTC* latch 1060ps

• PDPtot @100MHz

– below 30fJ• PowerPC latch 28fJ

– 30 - 50fJ• HLFF 29fJ• SDFF 39fJ• mC2MOS latch 40fJ• 21264 Alpha FF 43fJ• Strong Arm FF 45fJ

– 50 - 70fJ• K-6 ETL 70fJ

– above 70fJ• SSTC latch 95fJ• DSTC latch 125fJ

62

Delay comparison

• F-F design brings the fastest structures

0

50

100

150

200

250

300

350

SDFF HLFF K6 PowerPC Alpha 21264FF

Strong ArmFF

mC2MOS

De

lay

[p

s]

63

Delay comparison

• F-F design brings the fastest structures

0

50

100

150

200

250

300

350

SDFF HLFF PowerPC mC2MOS

Del

ay [

ps]

0

100

200

300

400

500

600

700

K6 SA-F/F StrongArm SSTC DSTC

Del

ay

[ps

]

64

Overall rankingPDPtot ranges

0

50

100

150

200

250

300

350

HLFF SDFF Pow erPC mC2MOS StrongArm

Alpha21264

K6 SSTC DSTC SSTC* DSTC*

PDPt

otal

[fJ]

Activity=0.25 equaltransition probability

• EDPtot accepted as the overall cost function• Proposed “low-power” latches from Yuan & Svensson, compared with

other presented structures do not show advantage, (the optimization was not properly done - optimization is yet to be repeated under different setup)

@100MHz

65

Overall ranking, zoomed

0

10

20

30

40

50

60

70

80

HLFF SDFF Pow erPC mC2MOS Strong Arm Alpha21264

K6

PD

Pto

t [f

J]

Activity=0.25 equal transition probability

• Real signals have the activity between 0 and 1.0 ()• Precharged hybrid structures are the fastest but their power consumption

strongly depends on the probability of “ones”• More “ones” above the point

66

Overall performance

• Real signals have the activity between 0 and 1.0 ()• Precharged hybrid structures are the fastest but their power

consumption strongly depends on the probability of “ones”

• More “ones” above the point

0

10

20

30

40

50

60

HLFF SDFF PowerPC mC2MOS

PDP

tot [

fJ]

Activity=0.5 equal transition probability

0

20

40

60

80

100

120

140

160

SA-F/F StrongArm110

K6 SSTC DSTC

PD

Pto

t [fJ

]Activity=0.5 equal transition probability

67

Conventional Clk-Q vs. minimum D-Q

• Hidden positive setup time

• Degradation of Clk-Q

0

50

100

150

200

250

300

350

400

150 200 250 300 350 400 450 500 550 600 650

Delay [ps]

To

tal

po

we

r [u

W]

HLFF

PowerPC

Strong Arm FF

Alpha 21264 FF

mC2MOS latch

K6 ETL

SSTC

DSTC

SDFF

0

50

100

150

200

250

300

350

400

100 150 200 250 300 350

Clk-Q delay [ps]

To

tal

Po

we

r [u

W]

HLFF

PowerPC

Strong Arm FF

Alpha 21264 FF

mC2MOS latch

K6 ETL

SSTC

DSTC

SDFF

68

Internal Power distribution

• Four sequences characterize the boundaries for internal power consumption– …010101… maximum

– random, equal transition probability, average

– …111111… precharge activity

– …000000… leakage + internal clock processing

0

50

100

150

200

250

300

350

400

Random,activity=0.5

…01010101…activity=1

…11111111…activity=0

…00000000…activity=0

Data patterns

Inte

rna

l P

ow

er

[uW

]

HLFF SDFF PowerPC 603 latch

mC2MOS latch StrongARM FF Alpha 21264 FF

K6 ETL

69

Comparison of Clock power consumption

0 10 20 30 40 50

Local Clock power consumption [W]

DSTC MS latch

SSTC MS latch

K6 ETL

StrongArm FF

SA-F/F

mC2MOS

PowerPC MS latchSDFF

HLFF

70

Using Dual-Edge Flip-Flop(run at ½ of the frequency

save on the power consumed in clock distribution tree)

71

Dual-Edge vs. Single-Edge Flip-Flops Comparison

0

50

100

150

200

250

300

350

400

DETFF1DETFF2DETFF3SDFFHLFFPowerPC

Delay [ps]

0

50

100

150

200

250

300

350

DETFF1DETFF2DETFF3SDFFHLFFPowerPC

Total Power [W]

•Fujitsu 0.18u process; Clock frequency 500MHz (250MHz for Dual Edge FFs)•Data activity ratio = 0.5•VDD = 1.8V•Temp = 25º

72

Dual-Edge vs. Single-Edge Flip-Flops Comparison

0

50

100

150

200

250

300

DETFF1DETFF2DETFF3SDFFHLFFPowerPC

Internal Power [W]

0

10

20

30

40

50

60

DETFF1DETFF2DETFF3SDFFHLFFPowerPC

Clock Power [W]

0

5

10

15

20

25

DETFF1DETFF2DETFF3SDFFHLFFPowerPC

Data Power [W]•Fujitsu 0.18u process; Clock frequency 500MHz (250MHz for Dual Edge FFs)•Data activity ratio = 0.5•VDD = 1.8V•Temp = 25º

73

Silicon on Insulator (SOI) Technology

74

SOI Comparison

0

10

20

30

40

50

60

70

Delay [ps]0

20

40

60

80

100

120

140

Internal Power [uW]05

10

15202530

35404550

Clock Power [uW]

0

20

40

60

80

100

120

140

160

Total Power [uW]0

1

2

3

4

5

6

EDP [fJ]@500Mhz

HALPowPCHLFFSDFFSAFFSA 110

F= 1GHz, = 0.5, Le = 0.08 m, VDD=1.3V, T = 25C

75

In conclusion….

What can we expect that low power will bring to us ?

76

Wearable Computer

77

Wearable Computer

78

Wearable Computer

79

Digital Ink

80

Implantable Computer

81

Bluetooth

82

Brain Ultra small volumeSmall number of neuron cellsExtremely low power

Real time image processing(Artificial) Intelligence3D flight control

Sensor

InfraredHumidityCO2

Mosquito

Year 2110

Long lifetime by DNA manipulation Bio-computer

Year 2010Extrapolation of the trend with some saturation

Many important interesting applicationHome, Entertainment, Office, Translation , Health care

Year 2020???

More assembly technique: 3D

Combination of bio and semiconductor

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