ic engineering, fju and interconnection effects switching...

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Lecture 5

MOS Inverter: Switching Characteristics

and Interconnection Effects

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Introduction

Cload = (Cgd,n + Cgd,p + Cdb,n + Cdb,p) + (Cint + Cg)Lumped linear capacitance

intrinsic cap. extrinsic cap.

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JUFirst-Stage CMOS Inverter With Lumped

Load Capacitance

The question of inverter transient response is reduced to finding the charge-up and charge-downtimes of a single capacitance.

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Delay-Time Definitions

V50% = VOL +0.5(VOH – VOL) =0.5(VOL + VOH)τP = ½ (τPLH + τPHL)

High-to-Low tPHL Low-to-High tPLH

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Rise Time and Fall Time

Rise Time trise: output to rise from V10% to V90%Fall Time tfall: output to fall from V90% to V10%

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Propagation Delay Calculation

Estimate the average capacitance current during charge down and charge up, respectively.

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A More Accurate Method

nDout

load

pD

nDpDCout

load

idt

dVC

i

iiidt

dVC

,

,

,,

0

−=

≈

−==

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A More Accurate Method (Cont.)

When the nMOS transistor starts conducting, it initially operates in the saturation region. When the output voltage falls below (VDD – VT,n), the nMOSstarts to conduct in the linear region.

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Saturation region

Linear region

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2,

,01

2,

'

,

)(2

'

)(2

)1(

,

1 ,

nTOHn

nTload

VVV

VV outnTOHn

load

tt

tt

VVV

VV outnD

load

VVkVC

tt

dVVVk

C

dVi

Cdt

dtdVCi

nTOHout

OHout

o

nTOHout

OHout

−=−

−−=

−=∴

=

∫

∫ ∫−=

=

=

=

−=

=

Q


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At t = t1’, the output voltage will be equal to (VDD- VT,n) and the transistor will be at the saturation-linear region boundary.

)]1)(4

ln()(

2[

)(

))(2

ln()(

|))(2

ln()(2

12

)])(2[

1(2

)1(

,

,

,

,

%50

%50,

,

'11

,,

'11

2,

',

%50

,

%50

,

1

1

%50

,

−+−

+−

=

−−=−

−−=−

−−=

−=

−

−

−

=−=

−−

=

−=−

=

=

=

−=

∫

∫ ∫

OLOH

nTOH

nTOH

nT

nTOHn

loadPHL

nTOH

nTOHn

load

VVVVV

outnTOH

out

nTOHn

load

out

VV

VVVoutoutnTOHn

load

tt

tt out

VV

VVVnD

load

VVVV

VVV

VVkC

VVVV

VVkCtt

VVVV

VVkCtt

dVVVVVk

C

dVi

Cdt

out

nTOHout

out

nTOHout

out

nTOHout

τ

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loading

slop

e

Timingmatrix

Consider Inputs With Finite Slopes

Assuming: Input has finite rise tr & fall time tf

A cell’s delay in the Library is often characterized by a table(e.g., 5x5) in terms of two circuit-dependent indexes

- slopes of the input signals- extrinsic loading (related to no. of fanouts)

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Reducing a Cell Delay

Two Major Factors of Cell Delay– The driving current

• The larger current, the shorter propagation delay– The loading capacitance

• The larger output loading, the longer propagation delay

Common Techniques for Speeding Up A Cell– Increase VDD (ID increases)– Use low-VT device (ID increases)

• May introduce a larger sub-threshold current– Gate-sizing: increase the W/L ratio of the gate (ID increases)

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Inverter Design with Delay Constraints

Assumption: ignore the intrinsic capacitance in calculating the loading capacitance.Question: what’s the min. W/L ratio to achieve a given speed ?

By the assumption,Cload is regarded as a constant

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Considering Intrinsic Capacitance

= α0 + αnWn + αpWp

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Asymptotic Speed Limit by Sizing

Due to the drain parasitic capacitance:– Sizing up a transistor (i.e., increasing Wn and Wp) will have a

diminishing effect in reducing the propagation delay.

Limit Delay:

Delay in terms of design parameter Wn and Wp

R = Wp/Wn

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Example: Transient Response

50% reduction in tPHL when Wn = 2 to 3.2 mmBut almost no effect when Wn = 10 to 20 mm

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Example: Delay v.s. Channel Width

Limit value is about 0.2 ns,Determined by technology-specific parameters,Independent of the extrinsic capacitance components

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Optimum Wn for Area-Delay Product

Gate Sizing: is a trade-off for delay reduction by silicon areaQuestion: What’s a good balance between area and delay ?

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Another Side-Effect of Gate Sizing

Sizing Up a Gate– increases its fanin gates’ loading, slightly offseting the speed

advantages

Example– Action: increase the size of gate G3– Side-Effect: the loading capacitances of G1 and G2 will increase,

and therefore the delays across G1 and G2 increase too.

A

B

C

D

E

G1

G2

G3 F

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Three-Stage Ring Oscillator

When n is an odd no. Oscillation frequency

pnTf

τ211

==

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Interconnection Models

τrise (τfall) < 2.5 × (l / v) transmission-line model2.5 × (l / v) < τrise (τfall) < 5 × (l / v) either oneτrise (τfall) > 5 × (l / v) lumped modeling

Time of flight across the line: l/v , l is the length, v is the travel speed Inductance is important when signal rise/fall time comparable to time-of-flight

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An RLCG Interconnection Tree

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Typical Signal Waveforms

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Interconnect in Submicron Design

Interconnect delay begins to dominate the cell delayin the submicron designs

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Typical Interconnection Length

Probability

Wire Length

Chip Diagonal Length

Long running wires need accurate wire models:- inter-module connections- global bus connections- clock distribution networks

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Interconnect Segments

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Interconnect Segment

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Single Line Capacitance

Two basic components(i) Parallel Capacitance: Cpp(ii) Fringing Capacitance

Width-to-height ratio W/H

Fringing effects

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Lateral (Inter-Wire) Capacitance

Lateral cap.

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Inter-Layer Capacitance

Parallel plate cap.

Fringing & overlap cap.Fringing cap.

Lateral cap.

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Double Metal CMOS Structure

Only inter-layer cap. are shown

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Inter-Layer Parasitic Cap. For 0.8 μm

Inter-layer (vertical) capacitances- area cap. (parallel-plate cap.)- perimeter cap. (fringing cap.)

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Summary Of Parasitic Capacitances

Total Cap.

Inter-wire cap. CxInter-layer cap.

Fringing cap.Parallel-plate cap. Cp-p

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Interconnect Resistance Estimation

Rwire = r ·w · t

l

wl

= Rsheet ·

Rsheet = tr

Where r is resistivityt is the thickness

(sheet resistance)

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Interconnect Delay Models

T Model

Distributed RC Model (ladder network)

RC Model

tPLH ～ 0.69 RC

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Uniform RC Ladder Network

assume Rk=(R/N) Cj=(C/N)

For very large N

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Example: Comparison of Models

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Example: Comparison of Models

input

output withdistributed RC model

output withlumped RC model

output with

T model

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Impact of Wire Length to Delay

Assume thatUnit length resistance is RUnit length capacitance is C

Then the delay constant of a wire of k-unit length will be:RC· N· (N+1)/2

i.e., delay is quadratically proportional to the wire lengthFor example, increasing a wire length by 10 timeswill approximately increase the delay by 100 times

Using distributed RC model

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Clock Skew Problem

Definition– Clock skew refers to the maximum clock arrival time difference of the

flop-flops’ clock portsImpact– clock skew may slow down the operating speed

Clock skew minimization– near zero clock-skew can be achieved

in automatic place-and-route (APR) tools via proper buffer insertion– can be done by adopting a regular clock

tree structure (e.g., H-tree) enforced manually

H-treeclock nets

clocksource

clock sink

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Assume that:unit length capacitance is Cunit length resistance is Rthe input gate capacitance of FF’s clock port is Cgthe number labeled with each segment is the length

Example: Clock Skew Calculation

Clock source

FF

FF

A

B

6

6

22

22

1

S

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JUVoltage & Current Outputs of Inverter

Vin Vout

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Energy Transfer at Charge-Up

VDD

0

VDD

0

fVCPTf

VCT

VCCVVdt

dVCT

dtdt

dVCVVdtdt

dVCVT

dttitvT

P

DDloadavg

DDload

TToutloadloadoutDD

Toutload

T

Tout

loadoutDD

T outloadout

T

avg

2

2

2/22/

0

2

2/

2/

0

0

/1

1

]|)21(|)[(1

]))(()([1

)()(1

=∴

=

=

−+−=

−+−=

=

∫∫

∫

Q

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Lecture 6

Combinational MOS Logic Circuits

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Introduction

• nMOS Logic- NOR Gate- NAND Gate

• CMOS Logic• Complex Logic Circuits• Transmission Logic

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Combinational Logic Circuit

Boolean Operations are the basic building block of digital systemsPositive logic convention– Logic ‘1’ represents Vdd– Logic ‘0’ represents a low voltage

Important design concerns:– DC Voltage Transfer Characteristic– VOL, Vth

– Dynamic Characteristics– Silicon Area– Static & Dynamic Power Dissipation

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Two-Input nMOS NOR Gate

Pull-up network

– Pull output voltage to high when both inputs are low

Pull-down network

–Pull output voltage to low when either input is high

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Calculation of VOL

Obviously, VOH = VDD

Three conditions in calculating VOL– VA = VOH , VB = VOL reduced to an inverter – VA = VOL , VB = VOH reduced to an inverter (Equ. 7. 4)– VA = VOH , VB = VOH both drivers are turned on

VOL = Equ. 7. 8

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Design Strategy for nMOS NOR Gate

Set a certain maximum VOL for the worst case– (i.e., output voltage should be less than this value

in normal operation)Worst case happens when only one input is highSet kdriver,A = kdriver,B = kR · kload– This design choice yields two identical drivers– Find the proper channel W and L for each transistor

When both inputs are logic-high,– The output voltage is even lower than the required maximum

VOL because lower equivalent driver-to-load k leads to a lower VOL

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Generalized to Multiple Inputs

Generalized n-input NORAssume that– The input voltages ofall drivers are identicalVGS,k = VGS for k=1,..,n

Reduced to an inverter

No body-effect in any driverSubstrate-bias of depletion-type load is VSB = Vout

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Transient Analysis

Lumped load capacitance is– Cload = Cgd,A + Cgd,B + Cgd,load + Cdb,A + Cdb,B + Csb,load + Cwire

– Valid even for single-input switching– Slower than the equivalent inverter with the same kR

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Two-Input NAND Gate

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Equivalent Driver Transconductance

Consider the only case as pull-down is turned ON– VA = high (VOH) and VB = high (VOH)

Ignore the body-effect of Driver A

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Generalized to Multiple Inputs

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Load Capacitance of NAND Gate (I)

Consider Case (I) • VA = high • VB is from VOH to VOL

Lumped load capacitanceCload = Cgd,load + Csb,load +

Cgd,A + Cgs,A + Cdb,A + Csb,A + Cgd,B + Cdb,B + Cwire

This cap. is conservativeIn reality, only a fraction ofInternal node’s capacitance isReflected into Cload

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Load Capacitance of NAND Gate (II)

Consider Case (II) • VA is from VOH to VOL• VB = VOH

Lumped load capacitanceCload = Cgd,load + Csb,load +

Cgd,A + Cdb,A + Cwire

This cap. is smaller– It means that outputLow-to-high switching isFaster in this case(e.g., 30% faster)– This property could be used for speed optimization

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Example (2-Input nMOS NAND Gate)

Two switching events

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Example: Output Waveforms

For NAND-gate,Turning off the nMOS closer to the output Results in a faster output LOW-to-HIGH transition

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CMOS Two-Input NOR Gate

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CMOS Two-Input NOR Gate

VOH = VDD, VOL = 0VVA = VB = VOUT = Vth → ID = Kn(Vth- VT,n)2

Vth = VT,n + (ID/ Kn)1/2 (7.32)From Fig. 7. 10, M3: linear region, M4: saturation for Vin = Vout

If kn = kp, Vtn = |Vtp|, the switchingthreshold of the CMOS inverter is equalto Vdd/2. the switching threshold of theNOR2 gate is (Vdd+ VT,n)/3, whichis not equal to Vdd/2

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Switching Threshold Voltage

n

p

tpDDn

pTn

th

n

p

tpDDn

pTn

th

p

DtpthDD

DDD

SDtpthDDp

D

SDSDtpthDDp

D

kk

VVkk

VINRV

kk

VVkk

VNORV

kIVVV

III

VVVVk

I

VVVVVk

I

+

−+=

+

−+=

=−−

==

−−−=

−−−=

1

|)|()(

1

|)|()2(

2||

)|)|(2

]|)|(2[2

21

21

43

234

2333

Q

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Equivalent Inverter

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Calculating Switching Threshold Vth

Definition: VA = VB = Vout = Vth

To achieve VDD/2 switching threshold:Set VT,n = | VT,p | and kp = 4kn

NOR2 = Equivalent Inverter (pull-down kdown = 2kn)(pull-down kup = (kp/2)

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Internal Parasitic Capacitances

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CMOS Two-Input NAND Gate

Assume that– (W/L)n,A = (W/L)n,A and (W/L)n,B = (W/L)n,B

(From Eq. 7.37)

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Impact of Body Effects on Series-Connected Transistor Chain

A

B

C

D

A B C D

VDD

OUT

1pFα

β

γ

The discharging currentis limited by theuppermost transistor(the one controlled by A)

because it has a larger Vtdue to Body Effects

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Area Comparison

Transistor Count for n-input logic gate– nMOS logic : (n+1)– CMOS logic : (2n)

Real silicon area is used for– Transistor– Signal Routing– Contact– Therefore, the disadvantage of CMOS in terms of

silicon area may not be as worse as the transistor count suggests

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Sample Layout of CMOS NOR2 Gate

VA VB

GND

VDD

nwell

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Sample Layout of CMOS NAND2 Gate

VA VB

GND

VDD

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nMOS Complex Logic

A Boolean function– Z = A(D+E) + BC– OR operations are done

by parallel-connected drivers– AND operations are done

by series-connected drivers– Inversion is provided by the

nature of MOS circuit

EDACB

equivalent

LW

LW

LW

LW

LW

LW

)()(

1

)(

11

)(

1

)(

11)(

++

++

=

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Worst-Case Logic-Low Voltage VOL

Various Paths from VDD to GND lead to different VOL

– A-D Class 1– A-E Class 1– B-C Class 1– A-D-E Class 2– A-D-B-C Class 3– A-E-B-C Class 3– A-D-E-B-C Class 4

Assume all (W/L) the same

– A larger path resistance leadsto a larger VOL

– VOL1 > VOL2 > VOL3 > VOL4where subscript denotesclass number

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Complex CMOS Logic Gates

Dual Network– Given a Boolean expression– Dual network is obtained via the following operations:

• Each variable is replaced by its complement• Change AND to OR operation, and vice-versa

– De-Morgan Law: f = Dual-function(f)’

Example– f = (A(D+E) + BC)’– Dual-function(f) = [A’ + (D’ E’) ] · (B’ + C’)

CMOS Logic– Use n-network’s dual function as the p-network

to replace the depletion-type nMOS load

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Graphical Method for Dual Network

outVDD

P-network

out

GND

D E

A B

C

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Example: CMOS Complex Gate

Vout = A(D+E) + BC

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Example: Equivalency Rule

5.12

151

151

1

151

151

151

1

)(

1

)(

11

)(

1

)(

1

)(

11)(

12

201

301

1

)()(

1

)()()(

11)(

))((

,

,

=+

+++

=

++

++=

=+

=

++

++

=

+++=

CBAED

eqp

CBAED

eqn

LW

LW

LW

LW

LW

LW

LW

LW

LW

LW

LW

LW

CBAEDZ

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Gate Matrix Layout

Poly-gate ordering is important:

– An improper ordering may result in extra silicon area for diffusion-to-diffusion separation

out

GND

A

D EC

B

Stick-Diagram Layout

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Optimal Gate Ordering

Euler Path– Uninterrupted path that traverses each edge of a graph exactly

onceMinimum Layout (without any diffusion break)– Gate ordering forms a common Euler path for n-net and p-net

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Optimized Stick-Diagram Layout

• Poly column separation Dd- only need to allow for one metal-to-diffusion contact

• Advantages: - smaller area and parasitic cap.

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Exclusive-OR Gate

Total no. of transistors: 12 (including two inverters for A and B)

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And-Or-Inverter (AOI) Gate

Enables sum-of-product (SOP) realizationPull-down net consists of parallel branches of series-connected nMOS driver transistors

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Or-And-Inverter (OAI) Gate

Enables product-of-sum (POS) realizationPull-down net consists of series branches of parallel-connected nMOS driver transistors

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Pseudo-nMOS Gates

Always conducting pMOS for pull-upTo reduce silicon areaNonzero static power dissipationVOL is not zero voltage any moreNoise margin is smaller– Depending on the ratio ofpMOS Load’s kp to the pull-down net’s equivalent kdown,eq

Application:– In design where densityis the no.1 concern (e.g., Memory)

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CMOS Full-Adder Circuit

BCACABoutcarryBCACBACBAABC

CBAoutsum

++=+++=

⊕⊕=

_

_

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Full Adder Schematic

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JULayout Using Minimum-sized

Transistors

co’ SUM’

A B C

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Optimized Layout

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Simulated IO Waveforms

Carry-in

B

A

Carry-out

Sum

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N-bit Binary Adder

Ripple-Carry Adder– Constructed by cascaded-connection of full adders– Speed is limited by the long carry chain

( C0 C1 C2 C3 C4 C5 C6 C7 C8 )

FA FA FA FA…

S0 S1 S2 S7

A0 B0 A1 B1 A2 B2 A7 B7

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Transmission Gate (TG)

TG is a bi-directional switch between A and B which is controlled by signal CnMOS passes perfect 0, but only up to (VDD-VT,n)pMOS passes perfect 1, but only down to |VT,p|

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Operating Regions for TG

Vin=VDD

Transmit logic ‘1’

ID

ISD,p

IDS,nVDD

Vout

0 V

ss dd

Isd,p getting smaller

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Equivalent Resistance of CMOS TG

Equivalent resistance is quite a constant

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TG-Based Logic (Steering Logic)

TG-based logic may be smaller than their standard CMOS counterparts

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JUTG-Based Logic – Arbitrary Function

A B F0 0 C’0 1 C’1 0 C1 1 1

n-well

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Sample Layout

F = AB + A C + A B C

x

F

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Only n-network is used to steer input to outputMight be faster than full TG-based logicBut overall noise immunity is weaker because nMOScannot pass a full logic “1”

CPL NAND2 CPL NOR2

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Common Standard Cell Library

Combinational Cells:– Inverter– Buffer– 2-t0-1, 4-to-1, 8-to-1 Multiplexor– NOR gates– NAND gates– And-Or-Inverter (AOI) gates– Or-And-Inverter (OAI) gates– Half Adder and Full Adder Cell

Flip-Flops– JK Flip-Flops– D Flip-Flops with Set, Reset, Enable, Scan, …, etc.

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Combinational Logic Circuits !

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Lecture 7

Sequential MOS Logic Circuits

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Synchronous Design Model

FFs FFs

clk

Comb.logic

Comb.logic

ABC out1

out2

Sequential CircuitsFFs

CombinationalLogic

ABC

OUT1OUT2

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Cross-Coupled Inverters

Bistable Element

Vi1

Vi2

Vo1

Vo2

Voltage-Transfer Curves

Circuit Diagram

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SR Latch Circuit

Transistor schematicGate-level Schematic

Truth Table

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Capacitances of CMOS SR Latch

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Rise/Fall Times of CMOS SR Latch

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NAND-Based SR Latch Circuit

Gate-level Schematic

Truth Table

Transistor schematic

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Clocked Latch Circuit

Gate-level Schematic

Example Waveforms

AOI-based implementation

shortglitch

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Clocked JK Latch

All NANDimplementation

S

R

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Truth Table of Clocked JK Latch

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AOI Realization of JK Latch

NOR-Based JK Latch

AOI-implementation

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Master-Slave Flip-Flop

Qs

Qs

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Sample I/O Waveforms

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CMOS D-Latch (I)

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CMOS D-latch (II)

Master-SlaveD Flip-Flop Tri-state inverter

A

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Sample Layout of CMOS DFF

Cross-coupled inverter-pair

AQm

Qm

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Setup & Hold Time

Valid data

Setup time is due to D-to-Q delay: violated by long-pathsHold time is due to Clock-to-Q delay: violated by short-paths

FFs FFs

clk

Comb.logic

Comb.logic

ABC out1

out2

D Q

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Common Latch Types in Cell Library

Transparent LatchJK Flip-Flop (Edge-Triggered)D Flip-Flop (Edge-Triggered)– Primitive D Flop-Flop– With Asynchronous Set and Reset– With Extra Asynchronous Enable Signal

Scan Flip-Flop– Mux-Scan Flip-Flops– SI means “scan input”– SC means “scan Control”

D-FF

MUX

clk

DSISC

Q

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JU THE END of

Sequential Logic Circuits !

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Lecture 8

CAD for VLSI Design( Source ref. EECS244, Prof.K. Keutzer, U.C. Berkeley )