TOLERANCE ANALYSIS

The problem:

Every parameter of a manufactured

t i it hcomponent or circuit has a

TOLERANCE associated with it.

Consider:

Customer specification

Design

ManufactureManufacture

Test

PASS FAIL

Sell Scrap Rework

1

Profit

Is the yield acceptable?

Could yield be improved?

The design stage should include TOLERANCEg g

ANALYSIS and SENSITIVITY ANALYSIS

Example

Design a potential divider with a voltage

ratio of T = 0.1 ± 1%

Design:Ωk9

Ωk1 Ωk1

O i f d lOr using preferred values:

Ωk.28 Ω820

2

Ωk1

Now %).(.....

T 20099800210

1−==

Using standard metal-film resistors

with ± 1% tolerance

( )%.....

.Tmax

6110160811801188011

011+=

++=

( )%.....

.Tmin

0209800828202828990

990−=

++=

If a large number of these circuits is

manufactured many will not satisfy the

’ ifi icustomer’s specification.

How many?

3

Possible distribution of manufactured circuits

0 +1 +1.6-0.2-1-2

% deviation from nominal

Total area corresponds

t j t i itto reject circuits

4

-0.2-2 -1 +1 +1.6

% deviation from nominal

A number of circuits will fail - proportional to

the total area arrowedthe total area arrowed.

(Note that if the customer specification is ± 2%

then all circuits should pass)

The number of failures could be predicted,

using integration, if the mathematical shape

of the curve is known.

Would it be better to choose R1 and /or R2 with

0.5% tolerance ?

This would require more expensive components

but fewer failures.

TOLERANCE DESIGN i d t i i iTOLERANCE DESIGN is used to minimise

the unwanted effects of component tolerances.

•Is there some optimum choice of component

tolerances which though perhaps leading totolerances which, though perhaps leading to

less than 100% yield, minimises the cost of

acceptable circuits?

•Is the yield particularly sensitive to the nominal

5

Is the yield particularly sensitive to the nominal

value of any particular components? If so,

which ones?

PARAMETER SPACE

A point in parameter space defines a unique

circuit.

If the values of the parameters p1 and p2

describing two components of a circuit are

known, then the corresponding point C in

parameter space with axes p1 and p2 represent

h i ithe circuit.

p2 Tolerance region

RT

C represents

a circuit

C

p1

The tolerance region, RT, represents the

bo nds on all possible samples of a massbounds on all possible samples of a mass

produced circuit.

6

REGION OF ACCEPTABILITY (RA)

The customer’s specification of a circuitThe customer s specification of a circuit

performance transferred onto component

parameter space defines a region of acceptability

- RA.RA.p2

RA

p1

Manufacturing yield corresponds to the

overlap of RA and RT.

If RT is not wholly within RA then yield

is less than 100%.

There are two obvious design improvements;

1. Use DESIGN CENTERING. Adjust the

i l l ith fi d t lnominal values with fixed tolerances

and / or

2. Use TOLERANCE ASSIGNMENT. Choose

parameter tolerances to decrease the cost of pass

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parameter tolerances to decrease the cost of pass

circuits, or to achieve more pass circuits.

The practical problem is that we do not

know in advance the exact locations of theknow in advance the exact locations of the

points describing the manufactured circuits.

This is because the component parameters

are RANDOM VARIABLES.are RANDOM VARIABLES.

PARAMETER DISTRIBUTIONS

Measured data produces HISTOGRAMS.p

It is more convenient to analyse an infinite

number of components with infinitesimally

small class intervals to produce a p

Probability Density Function (p.d.f) [Φ]

( ) 1=∫∞

dxxΦ ( )∫∞−

Manufacturing yield is usually dependent on

the p d f s of some or all components in athe p.d.f.s of some or all components in a

circuit.

8

Examples of p.d.f.s

φ(x)φ(x)

GAUSSIAN (NORMAL)

x

φ(x)

x

TRUNCATED GAUSSIAN

φ(x)

BIMODAL

x

9

Tolerance Analysis

Tolerance analysis

Worst-case Non-worst caseWorst case

analysisNon-worst case

analysis

Non-sampling

methods

Sampling

methods

Statistical Deterministic

MONTE-CARLO REGIONALIZATION /

SIMPLICIAL APPROXIMATION

Worst-case analysis

Try the extreme values of components within

h i ltheir tolerance range.

p2 RTCan be difficult to

identify the worst-case

component values

10p3 p3

Example of worst-case analysis

Consider a parallel combination of 3 resistors

R1 = 100Ω ± 1%

R2 = 200Ω ± 2%2

R3 = 300Ω ± 5%

Determine the nominal value of the equivalent

resistance, R.,

Use worst-case analysis to estimate the

predicted tolerance in R.

Does worst-case analysis really give the y y g

worst-case?

RRRR1

R2

R3 Ω554

321

321 .RRR

RRRR =

++=

R 4 5R3

1

3

5

68

11R2

R1

2 7

R1 R2 R3 RR1 R2 R3 RNominal 100 200 300 54.54Node 1 99 204 315 55.01Node 2 99 204 285 54.02Node 3 99 196 285 53.44 -2%Node 4 99 196 315 54.41Node 5 101 196 315 55.01Node 6 101 196 285 54.02Node 7 101 204 285 54.61Node 8 101 204 315 55.62 +2%

R = 54.5Ω ± 2%

Note:

1. Worst-case response may not be at a vertex.

2 Analysis is not expensive2. Analysis is not expensive.

3. Useful for circuit redesigns.

4. Useful for intermediate stages of design when

it may not be possible to carry out tolerancet ay ot be poss b e to ca y out to e a ce

analysis for the entire circuit.

5. May be pessimistic if parameters are

correlated.

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Non-worst-case analysis

Requires some knowledge of the statistical

distribution of component values.

Non-sampling method

• METHOD OF MOMENTS

• not uncommon - established technique

• used in other engineering fields

• based on a Taylor series representation of

the performance functions

• not valid for every circuit

• suitable for evaluating the effect of

component mismatch on performance

i bilivariability

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Sampling methods

D t i i tiDeterministic

Sample points are selected in a regular

method

(a) Regionalization

Perform a circuit analysis at points in RT.

With this method it is easy to estimate yieldsWith this method it is easy to estimate yields

for uncorrelated component parameter p.d.f.s

eg

RTp2p2 approx. RATp2p2 app o . A

fail

p1 p1

- may be modified to account for

non-uniform, correlated p.d.f.s

- expensive in computer time since

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this increases with the power of the

number of component parameters

(b) Simplicial approximationf l h i li i- more successful than regionalization

- involves approximation to the boundary of RA

in the form of a polyhedron.

Procedure:

1. Identify a pass circuit in RA

2. Search for the RA boundary by varying one

parameter

3. Search for the RA boundary in the opposite3. Search for the RA boundary in the opposite

direction

4. Search for the RA boundary from the initial

pass circuit in the perpendicular direction by

varying the other parametervarying the other parameter.

The polygon (triangle) formed is the first

approximation to RA.

5. Inscribe the largest possible circle in the

polygon

6. Identify the longest side touching the circle -

thi ill b th t i ti t ththis will be the poorest approximation to the

boundary of RA

7. From the mid-point on this side and

perpendicular to it in a direction away from the

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polygon interior a new search direction is defined

to find another point on the boundary of RA

8. The original polygon approximation is now expanded

(using spheres for multi-dimensions)

Yield estimation:

From a knowledge of Φ(p1) and Φ(p2) sample points are

generated in RT (without performing any circuit analysis)generated in RT (without performing any circuit analysis)

[The polygon is called a SIMPLEX - hence “simplicial

approximation”]

Drawbacks:

- RA must be convex

- this method has limited value except for low-dimensional

i icircuits.

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(c) Statistical exploration

The MONTE CARLO approach

- The sample points in parameter space are

t d i d d tgenerated in a pseudo-random manner to

simulate the actual manufacturing process

There is no need to approximate R- There is no need to approximate RA

- This approach directly mimics the process of

random component value selection (includingrandom component value selection (including

correlations) by generating component values

according to known p.d.f.s

- Confidence in yields may be determined

- Useful for medium / large circuitsg

- Produces a performance spread

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- Is dimensionally independent.

Monte Carlo statistical exploration of RT

RT

hhho

o

o

o

o

o

o

o

o

o

o

o

o

RA

hhhh

Say there are 20 “explorations” in R (N)Say there are 20 explorations in RT (N)

giving 7 “pass” circuits (Np)

The yield estimate isN

Yp=The yield estimate is N

Y =

(assuming uniform p.d.f.s)

and the RA boundary is not requiredand the RA boundary is not required.

For non-uniform p.d.f.s each sample is

“weighted”.

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weighted .