Inverter

Chapter 5Chapter 5

The InverterThe InverterV1. April 10, 03V1.1 April 25, 03V2.1 Nov.12 03

Inverter

Objective of This ChapterObjective of This Chapter

Use Inverter to know basic CMOS Circuits OperationsWatch for performance Index such as

Speed (Delay calculation)Optimal Transistor Sizing for speed and EnergyPower Consumption and Dissipation

Inverter

The CMOS Inverter: A First GlanceThe CMOS Inverter: A First Glance

Vin Vout

CL

VDD

Inverter

CMOS InverterCMOS Inverter

Polysilicon

In Out

VDD

GND

PMOS

Metal 1

NMOS

OutIn

VDD

PMOS

NMOS

Contacts

N Well

A=WxL

Inverter

Two InvertersTwo Inverters

Connect in Metal

Share power and ground

Abut cells

VDD

Vin Vout

Vin

Vout

Inverter

CMOS InverterCMOS InverterFirstFirst--Order DC AnalysisOrder DC Analysis

VOL = 0VOH = VDD

VDD VDD

Vin = VDD Vin = 0

VoutVout

Rn

Rp

Inverter

Delay Definitions (circuit speed)Delay Definitions (circuit speed)

Vout

tf

tpHL tpLH

trt

Vin

t

90%

10%

50%

50%

Inverter

CMOS Inverter: Transient ResponseCMOS Inverter: Transient Response

tpHL = f(Ron.CL)= 0.69 RonCL

V outVout

R n

R p

V DDV DD

V in = V DDV in = 0

(a) Low-to-high (b) High-to-low

CLCL

ln(2)=0.69

Inverter

Voltage TransferVoltage TransferCharacteristicCharacteristic

Inverter

PMOS Load LinesPMOS Load Lines

VDS,p

IDp

VGSp=-2.5

VGSp=-1VDS,p

IDnVin=0

Vin=1.5

Vout

IDnVin=0

Vin=1.5

Vout

IDn

(Vdd = 2.5V in 0.25um CMOS Process)(Vt = 0.4V as shown in Table 3-2)

pDSDDout

pDnD

pGSDDin

VVV

II

VVV

,

,,

,

+=

−=

+=

pDnD

pGSDDin

II

VVV

,,

,

−=

+=pDSDDout VVV ,+=

Inverter

CMOS Inverter Load CharacteristicsCMOS Inverter Load Characteristics

IDn

Vout

Vin = 2.5

Vin = 2

Vin = 1.5

Vin = 0

Vin = 0.5

Vin = 1

NMOS

Vin = 0

Vin = 0.5

Vin = 1Vin = 1.5

Vin = 2

Vin = 2.5

Vin = 1Vin = 1.5

PMOS

Inverter

CMOS Inverter VTCCMOS Inverter VTC

Vout

Vin0.5 1 1.5 2 2.5

0.5

11.

52

2.5

NMOS resPMOS off

NMOS satPMOS sat

NMOS offPMOS res

NMOS satPMOS res

NMOS resPMOS sat

VM: Vin = VoutSwitching Threshold Voltage

Inverter

Switching Threshold as a Function of Switching Threshold as a Function of Transistor RatioTransistor Ratio

NMOS and PMOS are in Saturation Modes

For r = 1, and saturated velocity NMOS = 2 PMOS, Wp = 2Wn

),,when (1 TpTnDSATDD

DDM VVVV

rrVV >>

+≈

Inverter

Switching Threshold as a Function of Switching Threshold as a Function of Transistor RatioTransistor Ratio

100

101

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8M

V(V

)

W p/W n

2 3 4

Inverter

Simulated VTCSimulated VTC

0 0 .5 1 1 .5 2 2 .50

0 .5

1

1 .5

2

2 .5

Vin

(V )

Vou

t(V)

Inverter

Impact of Process VariationsImpact of Process Variations

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

Vin (V)

V out(V

)

Good PMOSBad NMOS

Good NMOSBad PMOS

Nominal

Good definition: Smaller oxide thickness, smaller L, higher W, smaller VT

Inverter

Propagation DelayPropagation Delay

Inverter

CMOS InvertersCMOS Inverters

Polysilicon

InOut

Metal1

VDD

GND

PMOS

NMOS

1.2 µm=2λ

Inverter

CMOS Inverter Propagation DelayCMOS Inverter Propagation Delay

VDD

Vout

Vin = VDD

Ron

CL

tpHL = f(Ron.CL)= 0.69 RonCL

t

Vout

VDD

RonCL

1

0.5

ln(0.5)

0.36

Inverter

The Transistor as a SwitchThe Transistor as a SwitchVG S ≥ V T

RonS D

ID

VDS

VGS = VD D

VDD/2 VDD

R0

Rmid

Inverter

The Transistor as a SwitchThe Transistor as a Switch

0 .5 1 1 .5 2 2 .50

1

2

3

4

5

6

7x 1 0

5

VD D

(V )

Req

(Ohm

)

Inverter

The Transistor as a SwitchThe Transistor as a Switch

Inverter

0 0 .5 1 1 .5 2 2 .5

x 1 0- 1 0

-0 .5

0

0 .5

1

1 .5

2

2 .5

3

t (s e c )

Vou

t(V)

Transient ResponseTransient Response

tp = 0.69 CL (Reqn+Reqp)/2

?

tpLHtpHL

Inverter

Delay (speed degrade) as a function of VDelay (speed degrade) as a function of VDDDD

0 .8 1 1 .2 1 .4 1 .6 1 .8 2 2 .2 2 .41

1 .5

2

2 .5

3

3 .5

4

4 .5

5

5 .5

VD D

(V )

t p(nor

mal

ized

)

DSATnn

LpHL

DSATnTnDD

VkLWCt

VVV

')/(52.0

2/when

≈

+>>

Similar to Rn curve!Sharp change at 2Vt

Inverter

Design for Speed PerformanceDesign for Speed Performance

Keep loading capacitances (CL) smallIncrease transistor ratio (W/L) (adding CMOS gain)

Watch out for self-loading (for the previous stage)!

Increase Vdd! Trade power/energy dissipation for performance!

Inverter

Propagation delay v.s. Transistor sizePropagation delay v.s. Transistor sizeNMOS-to-PMOS Ratio:

Symmetrical tpHL and tpLH PMOS is 2.5~3.5 wider than NMOS in width under same LIs there better propagation delay (tp), or a better N-to-P ratio for overall tp can be found?

Consider two identical cascaded CMOS inverters. The approximated load cap of the 1st gate is

WgngpdndpL CCCCCC ++++= )()( 2211

11 , dndp CC Is drain capacitance of PMOS and NMOS of 1st stage

22 , gngp CC Is gate capacitance of PMOS and NMOS of 2nd stage

Inverter

Propagation delay v.s. Transistor sizePropagation delay v.s. Transistor size

When the PMOS device is made β times larger than the NMOS ThenCL becomes From (5.20), we have

where is the resistance ratio of equal-size NMOS

and PMOS

n

p

LWLW

)/()/(

=β

2211 and gngpdndp CCCC ββ ≈≈

WgndnL CCCC +++= ))(1( 21β

[ ]

[ ] )1())(1(345.0

)())(1(269.0

21

21

βγβ

ββ

++++=

++++=

eqnWgndn

eqpeqnWgndnp

RCCC

RRCCCt

eqn

eqp

RR

=γ

Inverter

1 1 .5 2 2 .5 3 3 .5 4 4 .5 53

3 .5

4

4 .5

5x 1 0

- 1 1

β

t p(sec

)

NMOS/PMOS ratioNMOS/PMOS ratio

tpLH tpHL

tp

β = Wp/Wn

Fig. 5-18

)(1(21 gndn

Wopt CC

C+

+= γβ

γβ =

>>+

opt

Wgndn CCC 21when 1.9

From Table 3.3β= 31Κ/13Κ = 2.4

Inverter

Inverter SizingInverter Sizing

Inverter

Inverter ChainInverter Chain

CL

If CL is given:- How many stages are needed to minimize the delay?- How to size the inverters?

May need some additional constraints.

In Out

1 f2f1

Inverter

Notation Definition Notation Definition

unitR

•Unit-size NMOS Transistor: the NMOS with minimumLmin and Wmin that meets the layout design rule (assume L is fixed, and W is varied)

• : Intrinsic Cap. of unit-size NMOS transistor• : Channel resistance of unit-size NMOS transistor• : Gate cap of unit-size NMOS transistor• : Channel resistance of W-sized NMOS transistor• : Self-loading or intrinsic cap of the inverter (diffusion cap and gate-drain overlap (Miller) cap)

gC

unitC

WR

intC

Inverter

Inverter DelayInverter Delay• Minimum length devices, L=0.25µm• Assume RP = 2RN and WP = 2WN =2W

• same pull-up and pull-down currents• approx. equal resistances RN = RP

• approx. equal rise tpLH and fall tpHL delays

• Analyze as an RC network

WNN

unitunit

P

unitunitP RR

WWR

WWRR ==

≈

= )2(

tpHL = (ln 2) RNCL tpLH = (ln 2) RPCLDelay (D):

2W

W

unitunit

gin CWWC 3=Load for the next stage:

(R of unit size NMOS)

Inverter

Inverter with LoadInverter with Load

Load (CL)

Delay

CL

tp = k RWCL

RP

RW

•k is a constant, equal to 0.69•Assumptions: no load zero delay

2W

W

tpHL = (ln 2) RNCL tpLH = (ln 2) RPCL

Inverter

Inverter with Load and Para. Cap.Inverter with Load and Para. Cap.

Load

Delay

Cint CL

Delay = kRW (Cint + CL) = kRWCint + kRWCL

= Delay (Internal) + Delay (Load)= kRW Cint(1+ CL /Cint)

CN = Cunit

CP = 2Cunit

2W

W

Inverter

Delay FormulaDelay Formula( )

( ) ( )γ/1/1

~

0int ftCCCkRt

CCRDelay

pintLWp

LintW

+=+=

+

Cint = γCg,in with γ ≈ 1f = CL/Cg,in: Effective fanout

RW = Runit / W ; Cint =WCunit

tp0 = 0.69RunitCunit (Intrinsic or unloaded delay)Not function of transistor size!!

Inverter

Apply to Inverter ChainApply to Inverter Chain

CL

In Out

1 2 N

tp = tp1 + tp2 + …+ tpN

+ +

jgin

jginunitunitpj C

CCRt

,

1,1~γ

LNgin

N

i jgin

jginp

N

jjpp CC

CC

ttt =

+== +

=

+

=∑∑ 1,

1 ,

1,0

1, ,1

γ

Inverter

Optimal Tapering for Given NOptimal Tapering for Given NDelay equation has (N-1) unknowns, Cgin,2 ~ Cgin,N

Minimize the delay, find (N – 1) partial derivatives

Result: Cgin,j+1/Cgin,j = Cgin,j/Cgin,j-1

Size of each stage is the geometric mean of two neighbors

- Each stage has the same effective fanout (Cout/Cin)- Each stage has the same delay

1,1,, +−= jginjginjgin CCC

Inverter

Optimum Delay and Number of StagesOptimum Delay and Number of Stages

1,/ ginLN CCFf ==

When each stage is sized by f and has same effective fanout f

N Ff =

( )γ/10N

pp FNtt +=

Minimum path delay

Effective fanout of each stage:

Inverter

ExampleExample

CL= 8 C1

In Out

C11 f f2

283 ==f

CL/C1 has to be evenly distributed across N = 3 stages:

Inverter

Optimum Number of StagesOptimum Number of StagesGiven load, CL and given input capacitance CinFind optimal sizing f

( )

+=+=

fffFt

FNtt ppp lnln

ln1/ 0

0γ

γγ

0ln

1lnln2

0 =−−

⋅=∂

∂

fffFt

ft pp γ

γ

For γ = 0, f = e, N = lnF

fFNCfCFC in

NinL ln

ln with ==⋅=

( )ff γ+= 1exp

Inverter

Optimum Effective Optimum Effective Fanout Fanout ff( )ff γ+= 1exp fopt = 3.6 for γ=1, fopt = 2.718 for γ=0

Inverter

Normalized delay function of Normalized delay function of FF( )γ/10

Npp FNtt +=

Inverter

Buffer DesignBuffer Design

1

1

1

1

8

64

64

64

64

4

2.8 8

16

22.6

N f tp

1 64 65

2 8 18

3 4 15

4 2.8 15.3

Without considering the internal capacitance

Inverter

Power DissipationPower Dissipation

Inverter

Where Does Power Go in CMOS?Where Does Power Go in CMOS?

• Dynamic Power Consumption

• Short Circuit Currents

• Leakage

Charging and Discharging Capacitors

Short Circuit Path between Supply Rails during Switching

Leaking diodes and transistors

Inverter

Dynamic Power ConsumptionDynamic Power Consumption

2

000

)( DDLoutLDDout

LDDDDVDDVDD VCdvCVdtdt

dvCVdtVtiE ∫∫∫∞∞∞

====

Inverter

Dynamic Power DissipationDynamic Power Dissipation

Energy/transition = CL * Vdd2

Power = Energy/transition * f = CL * Vdd2 * f

Need to reduce CL, Vdd, and f to reduce power.

Vin Vout

CL

Vdd

Not a function of transistor sizes!

2)(

2

00

DDLout

outLoutVDDC

VCdtvdt

dvCdtvtiE ∫∫∞∞

=== Energy in CL

Inverter

Node Transition Activity and PowerNode Transition Activity and PowerConsider switching a CMOS gate for N clock cycles

EN CL Vdd• 2 n N( )•=

n(N): the number of 0->1 transition in N clock cycles

EN : the energy consumed for N clock cycles

Pavg N ∞→lim

ENN-------- fclk•= n N( )

N------------N ∞→

lim C•

LVdd•

2 fclk•=

α0 1→n N( )

N------------N ∞→

lim=

Pavg = α0 1→ C• LVdd• 2 fclk•

e)Capacitanc Effective:(

)( 2210

Eff

CLKDDEffCLKDDLAVG

C

fVCfVCP ⋅⋅=⋅⋅⋅= →α

Inverter

Switching Activity (Example 5.12)Switching Activity (Example 5.12)

25.08/210 ==→α

Inverter

Transistor Sizing for Minimum EnergyTransistor Sizing for Minimum EnergyGoal: Minimize Energy of whole circuit while maintaining the speed speed performance

Design parameters: f and VDD

tp ≤ tp,ref of referenced circuit with f=1 and Vdd =Vref

1Cg1

In

fCext

Out

TEDD

DDp

pp

VVVt

fFftt

−∝

++

+=

0

0 11γγ

)/( 1gext CCF =

)2/( DSATTTE VVV +=

Inverter

Transistor Sizing (2)Transistor Sizing (2)Performance Constraint (γ=1) Vdd(f)

Energy for single transition

Energy ratio of the design and reference circuit

( ) ( ) 13

2

3

2

0

0 =+

++

−

−=

+

++

=F

fFf

VVVV

VV

FfFf

tt

tt

TEDD

TEref

ref

DD

refp

p

pref

p

+++

=

FFf

VfV

EE

ref

DD

ref 422)(

2

[ ] ])1)(1[(1 12

12 FfCVFffCVE gDDgDD +++=++++= γγγ

Inverter

Transistor Sizing (4)Transistor Sizing (4)VDD=f(f) E/Eref=f(f)

Required Supply Voltage Energy v.s. Sizing factor

Inverter

Sizing factor for Speed and EnergySizing factor for Speed and Energy

Device sizing, combined with supply voltage reduction, is a very effective way in reducing energy consumption of a logic network.

The gain can be up to 10 for large fanout.

Oversizing beyond the optimal value comes at a hefty price in energy.Optimal size for energy is smaller than the optimal sizing for performance.

For example, f(energy) = 3.53, f(performance) = 4.47= , for F=20

20

Inverter

Short Circuit Currents (during switching)Short Circuit Currents (during switching)

Vin Vout

CL

Vdd

I VD

D (m

A)

0 .15

0.10

0.05

Vin (V)5 .04.03.02. 01.00.0

Inverter

Minimizing ShortMinimizing Short--Circuit PowerCircuit Power

Inverter

Neil Weste Textbook

Inverter

Leakage CurrentLeakage Current

Sub-threshold current is one of most compelling issuesin low-energy circuit design!!

DDstatstat VIP =

Inverter

ReverseReverse--Biased Diode LeakageBiased Diode Leakage

Np+ p+

Reverse Leakage Current

+

-Vdd

GATE

IDL = JS × A

JS = 10-100 pA/µm2 at 25 deg C for 0.25µm CMOSJS doubles for every 9 deg C!

Inverter

SubthresholdSubthreshold Leakage ComponentLeakage Component

Inverter

SubthresholdSubthreshold Leakage Component (2)Leakage Component (2)

Inverter

Putting All TogetherPutting All Together

•In a typical CMOS circuits, the capacitive dissipation is by far the dominant factor.

•Leakage is ignorable at present, but will be major issue in deep-submicron CMOS circuits.

leakDDspeakDDDDL

statdpdynatotal

IVftIVVC

PPPP

++=

++=

→102 )(

Inverter

Principles for Power ReductionPrinciples for Power ReductionPrime choice: Reduce voltage!

Recent years have seen an acceleration in supply voltage reductionDesign at very low voltages still open question (0.6, … , 0.9 V by 2010!)

Reduce switching activity (at different levels)Reduce physical capacitance

Device Sizing: for example, for F = 20fopt(energy)=3.53, fopt(performance)=4.47.

Inverter

PowerPower--Delay Product (PDP)Delay Product (PDP)

PDP stands for the average energy consumed per switching event (0 1, 1 0)

pavtPPDP =)2/(1max ptf =

2

2

max2 DDL

pDDLVCtfVCPDP ==

Inverter

EnergyEnergy--Delay Product (EDP)Delay Product (EDP)Measure of both Performance and Energy

The value of supply voltage that simultaneously optimizes performance and energy. For Vt=0.5V, the VDD is around 1V.

pDDL

pavp tVCtPtPDPEDP2

22 ==×=

)21.5(2/, DSATTTETEDD

DDLp VVV

VVVCt +=−

≈α

)59.5(23,

)(2 ,

32

TEoptDDTEDD

DDL VVVV

VCEDP =−

=α

Inverter

EnergyEnergy--Delay Product (EDP)Delay Product (EDP)

VVVVVVV

VVVVVVVVVVVV

optDDpTEnTETE

pTEpDsatTp

nTEnDsatTn

2.18.0)2/3(8.02/)(

9.0,1,4.074.0,63.0,42.0

,,,

,,

,,

=×=⇒=+=

−=−=−=

===

Note:

Vdd for minimum EDPMay not be the Optimal Vdd for a given design problem (speed contraint)

Inverter

SummarySummary

Inverter Speed (delay), sizing, and power are discussed.The concept can be extended to complex gates in next chapter and future discussionsVery important for the 1st-order guess/approximation for designers in considering power/area/speed of the target CMOS circuits