Advances in Analog Circuits Part 6 pptx

After the lifetime yield optimization, optimized design parameters are obtained such thatany decreasing worst-case distances during lifetime are increased again as much as possible.The d

Fresh YieldOptimization

by WiCkeD

Circuit SimulationwithFresh Netlist

Lifetime YieldOptimization

by WiCkeD

Circuit SimulationwithDegraded NetlistRelXpert

initial d

d for optimal yield

Fresh sizing rules

d for optimal yield and reliability

Fig 5 The new design flow with reliability optimization

the degraded netlist generated by RelXpert Note that for each internal optimization loop,

an updated netlist from WiCkeD will be given to RelXpert to obtain a renewed version ofdegraded netlist During this step, both the fresh and degraded sizing rules are checked toensure the correct functionality of the circuit both at fresh time and after degradation The

final obtained design parameters d for optimal yield and reliability are the resulting solution

of the design flow

Fresh yield optimization step ensures that the smaller worst-case distances will be increased,thus the fresh design is centered such that it is already less sensitive to parameter drift Thisprovides a reasonable starting point for lifetime yield optimization, since the influence ofparameter degradation on the performance and yield is kept at minimum level

After the lifetime yield optimization, optimized design parameters are obtained such thatany decreasing worst-case distances during lifetime are increased again as much as possible.The design is centered now such that the most degradation-sensitive worst-case distance will

be kept maximum The resulting design solution is thus optimal considering both processvariations and lifetime degradations

8 Prediction: Speed up theβ w( t)evaluation

In this section, a prediction model of lifetime worst-case distance in time domain is presented

to speed up the analysis of lifetime yield value Only performance and statistical parametersensitivity analysis are needed, in comparison to the Monte-Carlo simulation method andnumerical optimization solutions It is based on the linear performance model as follows The

index i of ith performance in vector f is left out for simplicity Without loss of generality, only

upper bound f uis considered hereafter

139Lifetime Yield Optimization of Analog Circuits

Considering Process Variations and Parameter Degradations

8.1 Linear performance model

At any time t during the lifetime, the first-order Taylor expansion of performance f(t)with

respect to s(t)from worst-case point sw,uin s space is

f(s(t )) ≡ f(t ) ≈ f(sw,u(t )) + ∇ f(sw,u(t))T · (s(t ) −sw,u(t)) (19)

By assuming a linear performance model, the sensitivity of performance over statisticalparameters keeps constant, i.e.,

is constant over the entire s space at any time Thus the level contours of f in s space are

equidistant lines as illustrated in dashed lines in Figure 6 f(sw,u)in (19) is the upper bound

value f u So from (19) the linear performance model at t can be formulated as

sw,u(t0)

sw,u(t)g

Fig 6 Linear performance model during lifetime degradation in statistical parameter space

(dashed lines are equidistant level contours of f , ellipsoids are level contours of statistical

parameters)

sw,u(t)is called worst-case statistical parameter vector at t It is the statistical parameter vector where the corresponding performance f reaches its boundary value f u at t It corresponds

to the position in s space where the probability of occurrence reaches it s maximum in the

non-acceptance region (slashed area in Figure 6) A robust design indicates that such a

probability of occurrence should be kept minimum, i.e., sw,u should be positioned furthest

away from s0(t) so that it is least sensitive to the s changes which may cause it fall into

where (23) is constant over time Taking the process variation as second order effects on the

sensitivity towards degradation, C(t)is assumed to be constant, i.e., C(t) = C (Sobe et al.,

2009)

Considering parameter degradation from t0to t, a first-order Taylor approximation of μ(f(t))

with respect to t from t0can be expressed as

μ(f(t)) =μ(f(t0)) +∂μ f

From (22) we have

μ(f(t0)) = f u+gT · (s0(t0) −sw,u(t0)) (25)and

remains zero, since the two vectors g and∂s w,u (t)

∂t | t0are orthogonal to each other This is easy to

understand because during the degradation of s parameters, the worst-case point sw,umoves

along the performance boundary f u, as can be observed in Figure 6, while the performance

gradient g always points to the direction that is vertical to that boundary in the performance

To predictβ w,u(t), a first-order Taylor expansion ofβ w,u(t)with respect to t from t0is

β w,u(t) =β w,u(t0) +dβ w,u(t)

where the sensitivity part,dβ w,u (t)

dt | t0can be derived using results from Section 8.1 as follows

Since at the worst-case point sw,u(t), the corresponding level contour of s(t)is

β2

w,u(t) = (sw,u(t ) −s0(t))T ·C−1 · (sw,u(t ) −s0(t)) (31)

It touches the performance boundary at sw,u(t), which means the orthogonal on (31) is parallel

to g:

C−1 · (sw,u(t ) −s0(t)) =λ ·g (32)Inserting (32) into (31) we have

Considering Process Variations and Parameter Degradations

Then (34) is taken back into (22):

g remain constant, requiring an one-time evaluation only The sensitivity of s0(t)over t is

calculated by finite-difference approximation The values of s0(t)at respective time points areobtained from exemplary aging simulator in our experiment described in Section 7, then thecorresponding sensitivity and the worst-case distance degradation rate can be evaluated.Thus, by taking (37) back into (30), the values of β w,u(t) at time t can be predicted

efficiently without searching for the worst-case statistical parameters sw,u(t)through iterativeoptimization method

9 Experimental results

Vin+

Ibias

Fig 7 Circuit topology of Miller OpAmp used in the experiment

The circuit structure of the two stage Miller OpAmp used in the experiment is shown in Figure

7 The first stage is the differential stage, with the input differential pair, consisting of PMOSMP1 and MP2, and its active load, a current mirror consisting of NMOS MN1 and MN2 Thesecond stage is a CMOS inverter with an NMOS MN3 as driver and a PMOS MP5 as its activeload

It is clear from the circuit structure that certain sizing constraints on transistors concerning thenode voltages impose certain stress levels of each transistor

9.1 Results of the new design flow

yield-optimal reliability-optimalfresh 10 years 10 years fresh

Table 2 Experimental results of the new design flow with reliability optimization

We apply the new design flow in Figure 5 to the Miller OpAmp as introduced above One

of the stop criteria of the tool WiCkeD during fresh or lifetime yield optimization process,the maximum yield difference between two consecutive iterations, is set to 0.1% That is,the fresh or lifetime yield optimization stops if the improvement of the yield value betweentwo consecutive iterations is smaller than 0.1% A 180nm technology is used with a supply

voltage of 1.7V The circuit is degraded to time t=10 years with example AgeMOS degradation

model parameters inside RelXpert The covariance matrix of statistical parameters is assumed

to be constant over time Table 2 shows the simulation results Six of the performances areconsidered here, namely, DC Gain, Rising Slew Rate (SR), Gain-Bandwidth Product(GBW),Phase Margin, Power and Common-Mode Rejection Ratio (CMRR)

From result of fresh yield optimization we can see that the fresh circuit design is centered with

99.96% fresh yield, the corresponding design parameters are initial d at t0 After degradation

to 10 years with the same design parameters, all of the performances and worst-case distanceswill degrade, as well as the lifetime yield, which is only 94.50% now Then a design centering

on the degraded circuit is performed during lifetime yield optimization step The resultshows that the degraded circuit will have a lifetime yield of 99.93% with increased worst-case

distances Thus a design solution d for optimal yield and reliability is found.

Verification result on last column shows that with this optimized design, fresh circuit at t0will

be centered to a better position in terms of both fresh yield and lifetime yield The fresh yield

is 99.99%, and almost all of the worst-case distances here are much bigger compared to thefresh design where no degradation is considered

For the price we pay for the more robust circuit, the approximated total area of the circuitlayout is evaluated For the area of a transistor, it is simply the product of the width and thelength For the area of the Miller capacitor, it is transformed into the corresponding area by aconstant The results in Table 2 show that 7% more relative layout area is needed for the morerobust circuit

143Lifetime Yield Optimization of Analog Circuits

Considering Process Variations and Parameter Degradations

9.2 Results of the prediction model

To verify the prediction model of (30), the lifetime worst-case distance values obtained fromthe tool WiCkeD and the prediction model are compared for two performances, SR and

CMRR The comparison results and relative errors at different t’s are plotted in Figure 8 and

Figure 9 It is clear from the results that the prediction model can track theβ w(t)degradationwith an acceptable error For the simulation time, it takes five minutes on average for WiCkeD

to evaluate oneβ w(t)for one performance at t, while using the prediction model it takes only

about forty seconds A clear speedup about eight times is observed

(a) Comparison of the accurate and

predicted values ofβ w( t)at different t

0.00 0.01 0.03 0.05 0.07

(b) Relative error of the prediction

Fig 8 Prediction results on SR

(a) Comparison of the accurate and

predicted values ofβ w( t)at different t

0.00 0.02 0.04 0.06 0.08 0.10 0.12

(b) Relative error of the prediction

Fig 9 Prediction results on CMRR

10 Conclusion

As semiconductor technology continuously scales, the joint effects of manufacturing processvariations and parameter lifetime degradations have been a major concern for analog circuitdesigners, since the deviation of performance values from the nominal ones will impact boththe fresh yield and lifetime yield

In this chapter, a new analog design flow with reliability optimization is presented The effect

of both process-induced parameter variation and time-dependent parameter degradationcan be analyzed automatically The remaining lifetime yield of the designed circuit can

be predicted and optimized early in the design phase After lifetime yield optimization,simulation results show that a more reliable design is achieved, tolerant of both processvariation and lifetime degradation

A prediction model for the lifetime worst-case distances is proposed to speed up the analysis

of lifetime worst-case distance values The experimental results show that the model can

effectively evaluate during design phase the remaining lifetime yield of the circuits afterdegradation occurs in their lifetime.

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The structure of the chapter is the following: section 2 explains an interval analysis techniques for linear analog tolerance problem In that approach we are interested in calculation tolerances (the range of values) for real and imaginary part of transfer function with respect to change of one parameter of the circuit Section 3 deals with the problem of computing the frequency response of an uncertain transfer function whose numerator and denominator are interval polynomials Studying a solution set of corresponding 2×2 linear interval equation one can obtain bounds on the frequency response Using Kharitonov polynomials family and complex interval division it’s also possible to evaluate the bounds

In this section we compare results obtained by applying presented approaches Numerical studies are also reported in order to illustrate presented methods

2 Evaluation of linear circuits tolerances

The objective of this section is to develop the interval analysis techniques for linear analog circuit tolerance problem In that approach we can compute effectively tolerances for real and imaginary parts of the transfer function with respect to change of one parameter of a circuit

2.1 Bilinear and biquadratic form of a circuit function

The functional dependence of circuit performance on the designable parameters is known

implicitly through the circuit transfer function If the dependence on the R, L, C elements

and on the controlled sources is investigated, the transfer function is a quotient of two linear

polynomials, i.e., a bilinear relation, is arrived at We have the following well-known result:

)s(xD)s(C

)s(xB)s(A)x,s(M)x,s(L)x,s(F

+

+

=

In the above equation the symbol x denotes dependence on the network element parameter

(R or L or C or gain of the controlled source) A(s), B(s), C(s) and D(s) are functions of the

complex frequency s They depend on kind of transfer function and on the structure of a

circuit examined A similar biquadratic relation was derived for the dependence on the ideal

transformer ratio n, on the ideal gyrator resistance r and on the conversion factor k of the

ideal negative impedance converter (Geher, 1971) The transfer function has the following

form:

)s(Gx)s(xE)s(D

)s(Cx)s(xB)s(A)x,s(M)x,s(L)x,s(

++++

=

A(s), B(s), etc are depending on the type of the transfer function and the topology of the

circuit For some fixed frequency transfer function can be represented by its real and

imaginary part, i.e

)x,(M

)x,(Lj)x,(M

)x,(L)x,j(F)x(F

1 2 1

1

ω

ω+ω

ω

=ω

Here L1(ω,x), L2(ω,x), M1(ω,x) denote polynomials in x of second order and fourth order

(maximally) for bilinear and biquadratic transfer functions, respectively We are interested

in calculation tolerance (the range of values) for real and imaginary part of the transfer

function caused by some parameter x ranging in known interval, i.e x ∈ x = [ x, x ]

This one-parameter tolerance problem can be solved by means of the well-known circle

diagram method for bilinear transfer function, unfortunately biquadratic transfer function is

more difficult problem Here we propose a unified approach to tolerance problem for

bilinear and biquadratic transfer function based on the range evaluation of a rational

function by means of interval analysis techniques

2.2 Range values of a rational function

Let L(x) be a polynomial of degree n and M(x) a polynomial of degree m so that

f(x) = L(x)/M(x) is a rational function We want to expand f(x) into its Taylor series

i 0

i(x x )c)x

For computing the first k Taylor coefficients of f(x) at some point x0 where M(x0) ≠ 0, we

start by developing the polynomial L(x) into its Taylor series about the point x0

Linear Analog Circuits Problems by Means of Interval Analysis Techniques 149

i 0

i(x x )a)x(

i 0

i(x x )b)x(

Note that max(m,n) = 2 or 4

Coefficients ai and bi are obtained directly as

ai=L(i)(x0)/i! , bi=M(i)(x0)/i! , (7)

i = 1,2, ,m(n) More effectively we can compute them by using the extended Horner scheme (Elden &

Wittmeyer-Koch, 1990)

It was derived in (Garczarczyk, 1995) that one can compute the values of the first k Taylor

coefficients of a rational function by solving a (k + 1)×(k + 1) lower triangular Toeplitz

system of the form:

c

aaa

0 1 2

Note that for the case k > m(n), the lower triangular Toeplitz system is lower banded

To compute the values of the Taylor coefficients of a rational function the main work is to

solve the lower triangular Toeplitz system (8) Special structure of Toeplitz systems leads to

the variety of solving algorithms, so they belong to more elaborated linear systems Because

system (8) is lower triangular for a small k, we can use the usual forward substitution method

for its solving For large k more efficient method is a variant of Trench algorithm for Toeplitz

band matrices (Trench, 1985) Inversion of a nonsingular Toeplitz matrix of the form

band or not may be computed following:

Let (without loss of generality) b0 = 1, then

is the matrix given by

hrs = -ψr-s-1 , r = 0,1, ,k , s = 0,1, ,r (11) with ψj = 0 if j < -1, ψ-1 = -1, and

j b b , 0 ≤ j ≤ k – 1 (12) Note that matrix T-1 is also lower triangle Toeplitz matrix and is uniquely determined by its

first column (h00, ,hk0)t = (-ψ-1,-ψ0, ,-ψk-1)t The solution

[c0,c1, ,ck]t = T -1 [a0,a1, ,ak]t (13)

of (8) can be calculated by using the fast Fourier transform

For any function f(x) which has an interval arithmetic evaluation the range of values of f

over the interval x

R(f,x) := {f(x)⏐ x ∈ x} (14)

is contained in the interval arithmetic evaluation f(x), i.e

R(f,x) ⊆ f(x) (15)

Additionally, it is strongly dependent on the arithmetic expression which is used for the

interval evaluation of the function (Neumeier, 1990; Moore et al., 2009)

Exact Taylor expansion for a rational function f(x) is following

i

ix)

x(

i r 0 r

Linear Analog Circuits Problems by Means of Interval Analysis Techniques 151

where R(p,x) is the exact range of the polynomial p(x) over x, and

q(R,V) = max(|R - V|,|R - V|) means distance between intervals R = [ R , R] and V = [ V , V]

Relation (19) gives the way of range values evaluation: we need to calculate the range of

polynomial and the range of remainder term It’s seen from (20) that the overestimation of

R(f,x) by V(f,x) decreases with a power k + 1 of w(x) (width of x), so if f(k+1)(x) is bounded we

can omit the remainder term in V(f,x) and then

R(f,x) ≈ R(p,x) (21)

2.3 Bernstein polynomials

Estimates for the maximum, resp the minimum, of the polynomial over x are obtained by

computing Bernstein coefficients

For some order v of Bernstein polynomial we have (Ratschek & Rokne, 1984)

min Bj ≤ min p(x) ≤ max p(x) ≤ max Bj , (22)

0 ≤ j ≤ v , x ∈ x ,

where v ≥ k and

(wxs

t

t j

0 s

k

s t

j)s , j = 0,1, ,v (23) The coefficients Bj are computed using a following finite difference table

0

2 2 0

0

1 2 2

1 0

B

BB

BB

BBBB

BB

ν

− ν

− ν ν

− ν

− ν

Δ

ΔΔ

ΔΔ

r l l r

0

r

lA

r l l r

r

r

lA

where

1 r

rB =Δ−B+ −Δ−B

For example

0 0 1 0

)10

) )

0

B

B)(B

B)(B)(

B

Δ

↑

Δ+

→Δ

↑

↑

+

→+

→

+

+ +

B)(B)(B

↑

↑

←+

←+

(30)

It’s seen we can develop the algorithm of a parallel computation of Bernstein coefficients

starting from slanted entries We note that since αl = 0 for l > k there is no need to compute

entries ΔrBj for r > k ; a triangle table turns into trapezium one In the trapezium table a

bottom row has all entries equal, i.e

s,BB

1 s 0

s =Δ = =Δ ν>

Realisation of scheme (30) leads to the three cases of parallel computation slightly different

according to the value of ν (Garczarczyk, 2002)

2.3 Numerical examples

To illustrate the basic ideas of our approach two examples are considered The first example

refers to the bilinear transfer function and the second to the biquadratic one Taylor

coefficients ai and bi i = 0,1, ,k, k = 2 or 4, were computed by means of extended Horner

scheme For example, polynomial L(x) was developed by the algorithm written in

Pascal-like code as:

EXAMPLE 1 Consider a second-order low-pass filter section of Fig.1, originally proposed

by Sallen and Key

Linear Analog Circuits Problems by Means of Interval Analysis Techniques 153

Fig 1 Second-order low-pass filter section

Bilinear transfer function considered here is following

2 1 2 1 1

2 1 2

2 2

2 1 2 1

1 2

CCGGs)CGG)x1(C

G(s

CCGGxU

U)x,s(F

+++

−+

ωω

−ω+ωω

−

=ω

=

where M(ω,x) = 1+7ω2+ω4-6ω2x+ω2x2

We have applied relation (21) for Taylor expansion of degree k = 5 and Bernstein coefficients

of degree v = 10 were used For x ∈ x = A0[1-ε, 1+ε] with A0 = 1, ε = 0.01 we obtained results presented in the Table 1 In the second column there are values of the ranges for real and imaginary part of the transfer function, the third column contains their nominal values

ω x ∈ x X = A0

0.2 [0.878692,0.899103] + j[-0.372772,-0.367971] 0.888889 - j0.370370

2.0 [-0.127097,-0.122930] + j[-0.168624,-0.164735] -0.125005 - j0.166667

20.0 [-0.024009,-0.023489] + j[-0.002395,-0.002367] -0.023749 - j0.002381

Table 1 Range values of transfer function of Sallen-Key low-pass section

EXAMPLE 2 Consider the gyrator circuit with feedback shown in Fig.2

u 2

u 1

r Y

Z

Fig 2 Gyrator circuit