Design optimization of a current mirror amplifier integrated circuit using a computational statistics technique

(1)

Available online at www.sciencedirect.com

Mathematics and Computers in Simulation 79 (2008) 1165–1177

Design optimization of a current mirror amplifier integrated

circuit using a computational statistics technique

Yiming Li

a

, Yih-Lang Li

b

, Shao-Ming Yu

b,∗

a_{Department of Communication Engineering, National Chiao Tung University, Hsinchu, Taiwan} b_{Department of Computer Science, National Chiao Tung University, Hsinchu, Taiwan}

Received 7 September 2007; received in revised form 2 October 2007; accepted 2 November 2007 Available online 22 November 2007

Abstract

In this work, we implement a computational statistics technique for design optimization of integrated circuits (ICs). Integration of a well-known circuit simulation software and central composite design method enables us to construct a second-order response surface model (RSM) for each concerned constraint. After construction of RSMs, we verify the adequacy and accuracy using the normal residual plots and their residual of squares. The constructed models are further employed for design optimization of current mirror amplifier ICs with 0.18␮m CMOS devices. By considering the voltage gain, cut-off frequency, phase margin, common-mode rejection ratio and slew-rate, six designing parameters including the width and length of different transistors are selected and optimized to fit the targets.

Keywords: Integrated circuits; Current mirror amplifier; Central composite design; Response surface model; Optimization

1. Introduction

It is known that integrated circuits (ICs) design nowadays plays a crucial role for microelectronics industry; in particular, for highly competitive consumer products [8,9,11–13,15,16,18,19,30]. In modern ICs design flow and chip implementation, IC designers perform a series of functional examination and analysis of the characteristics by circuit simulation tools to match specifications. To meet specified electrical characteristics and performance of designed product, designers in general have to tune parameters of the passive and active devices ranging from resistors, capacitors, inductors, line width, line length, to transistor size, etc.[11–16,18]. It thus requires experienced designers to accomplish such complicated works. Diverse approaches have been proposed to reduce this designing cycle, which includes numerical optimization techniques and evolutionary algorithms; and have demonstrated their merit and validity

[1,2,11–16,18,20,21,24–26,29,31]. Furthermore, integration of circuit simulation tool, design of experiment[4,27], and response surface methodology may also provide a cost-effective way to advanced IC design optimization and sensitivity analysis of performance.

In this paper, a computational statistics technique for the design optimization of ICs is developed. Based on HSPICE circuit simulator[11,12,18,28], a central composite design (CCD), and a second-order response surface model (RSM),

∗_{Corresponding author at: P.O. Box 25-90, Hsinchu 300, Taiwan.}

E-mail address:[email protected](S.-M. Yu).

(2)

the circuit performances can be systematically optimized with respect to different specified constraints. The investigated current mirror amplifier IC with 0.18␮m CMOS devices has the specifications that includes the voltage gain within 50–100 db, the cut-off frequency (FT) within 20–70 MHz, the common-mode rejection ratio (CMRR) within 60–85 db, the slew-rate (SR)+ within 20–80 V/␮s, and the range of SR − is 20–70 V/ ␮s. We firstly use the circuit simulator and the central composite design to construct the second-order response surface models. Seventy-seven experimental runs of circuit simulations are completed to generate the necessary data for construction of the quadratic response models. We notice that, for validating the constructed model, the model adequacy checking and the accuracy verification are necessary[3,5–7,20,23]. With the second-order RSMs[3,5,14,20,23], we apply optimization approaches, such as the least squares method and desirability function approach, to extract the optimal parameters to fit the specifications. In the examined current mirror amplifier IC, six parameters including the width and length of different transistors are selected and optimized to satisfy the circuit specifications. The examined results verify the usability and efficiency of the proposed method. We believe that this approach benefits the design optimization for more diverse ICs.

The rest of this paper is organized as follows. Section2introduces the proposed computational statistics approach. Section3discusses the achieved results. Conclusions are drawn in Section4, along with recommendations for future research.

2. The computational statistics technique

The procedure of the proposed methodology is given by

The computational statistics approach{

Step 1. Variables selection by screening design or empirical check Step 2. Central composite design

Step 3. Model construction

Step 4. Accuracy and adequacy veriﬁcations

Step 5. If not satisﬁes the veriﬁcations in Step 4 perform transformation of the parameters, then go to Step 1 Step 6. Optimization of characteristics

Step 7. Check the results with target

If results not achieve the target, repeat Steps 1–6

Step 8. Output the optimized results

}

The first step in our approach is variables selection by using a screening design or empirical check so that we can determine the significant designing parameters. After determining the important factors, we will execute the central composite design by running the HSPICE circuit simulator. A second-order response surface model is then established between circuit performance (i.e., responses) and circuit parameters (i.e., factors), and the optimization for the circuit design then can be followed. In the following subsections, we state the details of some steps in the proposed approach.

2.1. Variables selection

Variables selection is a step to find the few significant factors from a list of many potential ones. Conventionally, we can use a screening design or empirical check to identify significant main effects, rather than interaction effects, the latter being assumed an order of magnitude less important. To determine factor’s significance, two-level fractional factorial design or Plackett–Burman design is ideally suited for screening design[25]. Two-level fractional factorial design can reasonably assume that high-order interactions are negligible. We can run only a fraction of the complete factorial experiment to obtain information on the main effects and low-order interactions. For example, in one-half fraction of the 23_{design (2}3−1_{design), A and BC are aliases, B and AC are aliases, C and AB are aliases, where A, B,}

and C are factors. When designs with resolution III, main effects are aliases with two-factor interactions and two-factor interactions may be aliased with each other. Sometimes designs with resolution IV are also used for screening designs. In this design main effects are aliased with, at worst, three-factor interactions. This is better from the confounding viewpoint, but the designs require more runs than a resolution III design.

Plackett–Burman design is two-level fractional designs[31]for studying up to k = N − 1 variables in N runs, where

(3)

in general. For example, N = 12, every main effect is partially aliased with every two-factor interaction. Each main effect is partially aliased with 45 two-factor interactions. And the plus and minus signs are

K = 11, N = 12 + + − + + + − − − + − . (1)

When we analyze data from screening designs, the use of an error mean square obtained by pooling high-order interactions is inappropriate occasionally. To overcome this problem a half-normal probability plot of the estimates of the effects is suggested. The half-normal plot consists of the point:

Φ−1 0.5 +0.5[i − 0.5] I , |θ|(i)| , (2)

for i = 1, . . . , I. The Φ is the cumulated density function of the standard normal distribution. If factors are unimportant, the effects with mean zero and variance σ2will tend to fall along a straight line on this plot, whereas important factors will not lie along the straight line[5,20,23].

In our current mirror amplifier IC, six variables are selected from eight parameters according to the empirical check. The selected parameters include the width or length of different transistors.

2.2. Central composite design

Response models are constructed relating the characteristics of circuits using data generated from statistical exper-imentation. Mathematically, response surface models may be represented as second-order polynomials:

Y = β0+ k i=1 βixi+ k i=1 βiix2i + k i=1 k i /= j βijxixj+ ε, (3)

where k is the number of input factors, xiis the ith input factor, βiis the ith regression coefficient, and ε represents

model error.

Many applications of response surface models involve constructing and checking the adequacy of a second-order model. The central composite design (CCD) is perhaps the most common experimental design used to generate second-order response models. These designs combine a two-level full factorial or fractional factorial design of nfruns with

2k axial runs and nccenter runs, where k represents the number of control factors[5,20,23]. The axial points represent

new extreme values for each factor in the design. There are three varieties of CCD which include central composite circumscribed (CCC), central composite inscribed (CCI), and face-centered cube (CCF) design[25].

Central composite designs include five input levels for each control factor (0;±1; ±α). Level 0, the nominal factor level, represents the base processing conditions. The cube levels (±1) are selected to reflect the design space of interest. These values are typically set to a multiple of the factor’s standard deviation or a percentage of its nominal value. The precise value of α depends on certain properties desired for the design and on the number of factors involved. To maintain rotatability, the value of α depends on the number of experimental runs in the factorial portion of the central composite design:

α = [nc]1/4. (4)

2.3. Response surface model construction

It is necessary to develop an approximate model for the true response surface. If n observations are collected in an experiment, the model for them takes the form:

(4)

where y = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ y1 y2 .. . yn ⎞ ⎟ ⎟ ⎟ ⎟ ⎠, X = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 1 x11 x12· · · x1k 1 x21 x22· · · x2k .. . ... ... ... 1 xn1 xn2· · · xnk ⎞ ⎟ ⎟ ⎟ ⎟ ⎠, β = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ β0 β1 .. . βk ⎞ ⎟ ⎟ ⎟ ⎟ ⎠, and ε = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ ε1 ε2 .. . εn ⎞ ⎟ ⎟ ⎟ ⎟ ⎠.

In general, y is an n × 1 vector of the observations, X is an n × k matrix of the levels of the independent variables, β is a k × 1 vector of the regression coefficients, and ε is an n × 1 vector of random errors.

We want to find the least squares estimators, ˆβ, that minimizes

L =

n

i=1

ε2_i = εTε = (y − Xβ)T(y − Xβ). (6)

As the result of our calculation, the least squares estimator of β is ˆβ = (XT

X)−1XTy. (7)

The fitted regression model is

ˆy = X ˆβ. (8)

The difference between the responses yiand the fitted value ˆyiis a residual, say ei= yi− ˆyi. The vector of residual is

denoted by

e= y − ˆy. (9)

To check the normality assumption, we prepare a normal probability plot of the residual values. If the assumption holds, this plot will resemble a straight line. If the assumption is violated, a non-linear data transformation (e.g., y= log(y)) may be applied and new models are generated in an attempt to improve model adequacy[23]. A second plot showing the residual values versus the predicted response values is used to verify if the variance of the original observation is constant. A random scattering of the residual values indicates that no correlation exists between the observed variance and the mean level of the response[5,14,20].

To develop an estimator of this parameter, we consider the sum of squares of the residuals, say SSE= n i=1 (yi− ˆyi)2= n i=1 e2_i = eTe. (10)

Eq.(10)is called the error or residual of squares, and it has n − k degrees of freedom associated with it. It can be shown that

E(SSE)= σ2(n − k), (11)

so an unbiased estimator of σ2is given by ˆσ2= SSE

n − k. (12)

To determine if there is a linear relationship between the response variable y and a subset of the regressor variables

x1, x2, . . . , xkwe test for significance of regression. The appropriate hypotheses are[23]:

H0: β1= β2= · · · = βk= 0, H1: βj= 0 for at least one j./ (13)

If we reject H0, it implies that at least one of the regressor variables x1, x2, . . . , x_k contributes significantly to the

model. The test procedure involves partitioning the total sum of squares due to residual:

(5)

where SSRis the sum of squares due to regression. A relatively simple procedure is performed to check for model

significance in relation to random error. This test involves calculating the test statistic:

F0= MSR MSE = SSR/k SSE/(n − k − 1)= (1/k) n j=1 (ˆyi− ¯y)2 (1/(n − k − 1)) n j=1 (yi− ˆyi)2 , (15)

where ¯y is the average of measured response values. yi, ˆyi, and n are the ith measured response, the ith predicted

response, and the number of simulated runs, respectively[23]. If this statistic exceeds the corresponding value of the

F distribution value (Fα,k,n−k−1), the response model is considered significant in relation to random error.

A second statistic, the coefficient of multiple determination R2is defined as

R2= SSR SST = 1 −SSE SST = 1 − n i=1 (yi− ˆyi)2 n i=1 (yi− ¯y)2 . (16)

R2measures the amount of reduction in variability of the response y achieved, using the input factors x1, x2, . . . , x_k.

From Eq.(14)we see that R2varies from zero to one[5,14,20,23]. However, a large value of R2does not necessarily imply that the regression model is good one. Adding a variable to the model will always increase R2, regardless of whether the additional variable is statistically significant or not. About this problem, some regression model builders prefer to use an adjusted R2statistic defined as

R2adj= 1 − SSE/(n − k − 1) SST/(n − 1) = 1 − n − 1 n − k − 1(1− R 2 ). (17)

In general, the adjusted R2statistic will not always increase as variables are added to the model. In fact, if unnecessary terms are added, the value of R2adjwill often decrease.

2.4. Optimization of characteristics

After the construction of models, we can use several techniques, such as the normality assumption and plot of residuals versus predicted value, to verify the adequacy of the response surface model. We further apply some numerical methods to perform the design optimization of our circuits. A useful approach to optimization of multiple responses is to use the simultaneous optimization technique popularized by Derringer and Suich[32]. Their procedure makes use of desirability functions. The general approach is to first convert each response yi into an individual desirability

function dithat varies over the range 0≤ di ≤ 1, where if the response yiis at its goal or target, then di = 1, and if the

response is outside an acceptable region, di = 0. When multiple response are transformed into individual desirabilities,

the individual desirabilities are then combined using geometric mean to maximize the overall desirability D:

D = (d1× d2× · · · × d_m)1/m, (18)

where m is the number of responses[22]. By the equation, if any diis equal to zero, then the overall desirability is zero.

According to the specification for the responses, response is to be maximized, minimized, or achieved a target value. For the ith response yiis a maximum value:

di= ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 0, ˆyi< Li ˆyi− Li Ti− Li _s , Li≤ ˆyi≤ Ti 1, ˆyi> Ti . (19)

(6)

For the response yiis a minimum value: di = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 1, ˆyi< Ti Ui− ˆyi Ui− Ti _s , Ti ≤ ˆyi ≤ Ui 0, ˆyi> Ui . (20)

For the response is achieved a target value:

di = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0, ˆyi < Li ˆyi− Li Ti− Li _s , Li ≤ ˆyi ≤ Ti Ui− ˆyi Ui− Ti _t , Ti≤ ˆyi ≤ Ui 0, ˆyi > Ui , (21)

where the weight s and t determine how important it is close to the target value. When the weight s = 1, t = 1 the desirability function is linear. Choosing s > 1, t > 1 means more important to close the target value with the function that is concave, and choosing 0 < s < 1, 0 < t < 1 means this less important with the function that is convex. Li, Ui,

and Tiare the lower, upper, and target value, respectively.

3. Numerical results and discussion

Operational amplifier plays an indispensable role of analog integrated circuits. We explore the design optimization of a current mirror amplifier[8–10,19]with 0.18␮m CMOS devices to examine the validity of the proposed method.

Fig. 1shows the studied current mirror amplifier circuit. In this experiment, the specifications are including the ranges of voltage gain, cut-off frequency, phase margin, common-mode rejection ratio and slew-rate; and the adjustable parameters are related to the size of transistors used in the designed IC. The gain is a measure of the circuit’s ability, and it is defined as the mean ratio of the signal output to the signal input of a system. The term cut-off frequency represents a boundary in the system response. CMRR measures the tendency of the device to reject input signals. The slew-rate represents the maximum rate of change of signal at any point in a circuit. The phase margin is the additional phase required to bring the phase of the loop gain to−180◦. These are important factors to measure the ability and performance of the current mirror amplifier IC. According to the results of the variable selection, we selected six factors which includes width of transistor M1 (W1), width and length of M3 (W3 and L3), width and length of M7 (W7 and L7) and width of MS (Ws) for our next design, and the face-centered cube design (CCF) is also used[5,14].

Fig. 1. The explored current mirror amplifier circuit. The transistors MB and MS provide the current source for the differential amplifier. The transistors M1 and M2 form the differential amplifier, and transistors M3–M6 make up the current mirror.

(7)

Table 1

List of the residual of squares in the six response surface models

Response Residual of squares

GAIN 0.9993 1.0/Sqrt(FT) 0.9999 PM1.57 _0.9999 CMRR−2.34 0.9997 Sqrt(SR+) 0.9999 1.0/Sqrt(SR−) 0.9984

To generate the necessary data for construction of the quadratic response models, 77 experimental runs (1 center point, 12 axial points, and 26 cube points) are completed by running the HSPICE circuit simulations. We choose the CCF design from the three type of the CCD (CCC, CCI, and CCF), because the CCF design is more suitable for designing the response surface models. Following equations are the responses, and the variables, A, B, C, D, E and F are the factors of W1, W3, L3, W7, L7 and Ws, respectively:

GAIN= 61.3078 − 0.33473 × E2+ 1.415906 × A + 0.285844 × A × F + 0.013121 × B + 0.069125 × B × F + 0.839219 × C − 0.04759 × C × D − 0.16788 × D + 0.240156 × C × E + 1.906719 × E + 0.176656 × C × F − 0.87466 × F + 0.054844 × D × E − 1.09336 × A2_{+ 0.098156} × D × F − 0.14331 × C2_{− 0.23459 × E × F.} (22) 1.0 Sqrt(FT) = 0.0001651 + 8.46 × 10 −8_{× E}2_{− 1.25 × 10}−5_{× A + 3.07 × 10}−8_{× A × B + 5.3 × 10}−7_{× B} + 6.96 × 10−7× A × C + 4.632 × 10−6× C + 3.86 × 10−8× A × E + 2.344 × 10−8 × D − 5.7 × 10−7× A × F + 3.123 × 10−7× E + 3.28 × 10−7× B × C − 2.625 × 10−5 × F − 6.8 × 10−8× B × F + 1.421 × 10−5× A2_{− 5 × 10}−8_{× C × E + 6.348 × 10}−7_{× C}2 − 2.2 × 10−7× C × F. (23)

(8)

Fig. 3. (a) The residual normal probability plot and (b) the residuals vs. predicted plot for the response 1.0/Sqrt(FT). PM1.57 = 721.3923 − 0.80638 × E2− 55.5958 × A − 6.95119 × A × C − 15.3572 × B − 1.8323 × A × E − 132.616 × C − 11.4127 × A × F − 1.42565 × D − 3.27457 × B × C − 18.9915 × E2.38953 × C × E − 27.2108 × F − 4.90147 × C × F + 25.36517 × A2_{− 0.91534 × D × E + 0.880197} × B2_{− 0.82643 × E × F + 9.737562 × C}2 . (24) CMRR−2.34= 6.66626 × 10−5+ 3.713 × 10−7× A × D − 1.0922 × 10−5× A − 8.542 × 10−7× A × E − 9.3073 × 10−7× B − 5.454 × 10−7× A × F + 2.13336 × 10−6× C − 3.928 × 10−7 × B × C − 5.7487 × 10−7× D − 1.065 × 10−7× B × E + 1.36801 × 10−6× E − 3.767 × 10−7× B × F + 1.61471 × 10−5× F − 1.027 × 10−7× C × D + 1.08167 × 10−7× B2_{+ 2.357 × 10}−7_{× C × E − 1.1113 × 10}−7_{× C}2_{+ 7.426 × 10}−7_{× C × F}

(9)

Fig. 5. (a) The residual normal probability plot and (b) the residuals vs. predicted plot for the response CMRR−2.34. − 2.2466 × 10−7× E2_{− 4.531 × 10}−7_{× D × F − 1.4178 × 10}−7_{× A × B} + 1.025 × 10−6× E × F + 3.63478 × 10−7× A × C. (25) Sqrt(SR+) = 6498.221 + 22.16759 × C2_{− 15.1833 × A − 6.28594 × A × C + 4.293962 × B} − 3.11049 × A × F − 90.0353 × C + 7.492849 × B × C + 1805.042 × F + 8.133772 × B × F + 59.15891 × A2_{− 83.5258 × C × F.} (26) 1.0 Sqrt(SR−) = 0.000163577 − 4.3 × 10 −7_{× A × B + 9.28055 × 10}−7_{× A + 4.58 × 10}−7_{× A × C} − 6.0608 × 10−7× B + 3.17 × 10−7× A × F + 2.31514 × 10−6× C − 9.3 × 10−7× B × C − 3.5869 × 10−5× F − 1 × 10−6× B × F + 1.44356 × 10−5× A2_{+ 9.77 × 10}−7_{× C × F.} (27)

(10)

Fig. 7. (a) The residual normal probability plot and (b) the residuals vs. predicted plot for the response 1.0/Sqrt(SR−).

Fig. 8. Scatter plot calculated from the response surface model vs. values obtained from circuit simulator for (a) GAIN and (b) 1.0/Sqrt(FT).

The SR+ and SR− in Eqs.(26) and(27)are the positive and negative slew-rate. After we construct the response models, we need to verify the adequacy of the models. We notice that the transformation of FT, PM, CMRR and SR− by BOX-COX transformation or experience[22]can improve the model adequacy.Table 1shows the residual of squares of each response surface model.Fig. 2shows the (a) residual normal probability plot and (b) the residuals versus predicted plot for the response GAIN, andFigs. 3–7show the model adequacy checking for the 1.0/Sqrt(FT),

Table 2

List of the range of designing parameters and the optimized parameters for the current mirror amplifier circuit

Parameter Range Extracted result

A: W1 (␮m) 40–100 100 B: W3 (␮m) 60–80 80 C: L3 (␮m) 1.2–2.5 1.21 D: W7 (␮m) 14.4–20 18.24 E: L7 (␮m) 1.2–2.5 2.47 F: Ws (␮m) 60–160 66.16

(11)

Fig. 9. Scatter plot calculated from the response surface model vs. values obtained from circuit simulator for (a) PM1.57_{and (b) CMRR}−2.34_.

Fig. 10. Scatter plot calculated from the response surface model vs. values obtained from circuit simulator for (a) Sqrt(SR+) and (b) 1.0/Sqrt(SR−).

PM1.57, CMRR−2.34, Sqrt(SR+), and 1.0/Sqrt(SR−).Figs. 8–10show the scatter plots of values calculated from the response surface models versus values that obtained from the HSPICE circuit simulator for the models of all responses. The results show that there is a high linearity between actual and predicted values and thus confirm the accuracy of the constructed models. Based on the response models, we extract the factors to satisfy the targets.Table 2shows the list of the range of designing parameters and the optimized parameters.Table 3shows the list of targets and the extracted

Table 3

List of the specified targets and the extracted results for the current mirror amplifier circuit

Specification Goal Range Result

Gain (db) Maximize 50–100 61.54 FT (MHz) Maximize 20–70 43.87 PM (◦) In the range 55–70 70 CMRR (db) Maximize 60–85 64.01 SR+ (V/ ␮s) Maximize 20–80 56.27 SR− (V/ ␮s) Maximize 20–70 39.40

(12)

results for the current mirror amplifier circuit. It shows that the extracted parameters can satisfy all the specified targets, and the results confirm the validity of the proposed method. The Design-Expert software is employed in this work, and the proposed method is now developed in our unified optimization framework.

4. Conclusions

In this work, we are proposed an computational statistics technique for design optimization of ICs. Based on circuit simulation and central composite design method, second-order response surface models have been constructed for IC design optimization. The current mirror amplifier IC examined in our experiment has verified the efficiency of the proposed method. The integration of circuit simulation tools, design of experiment and response surface models indicate a cost-effective way to IC design. This statistics computational approach is now embed in our unified optimization framework (UOF)[17]for more applications, and we believe that it benefits the field of ICs design optimization and engineering automation.

Acknowledgments

The author Shao-Ming Yu would like to thank Ta-Ching Yeh for the help of models’ construction. This work was supported in part by the National Science Council of Taiwan under contract NSC-95-2221-E-009-336, contract NSC-95-2752-E-009-003-PAE, and by the MoE ATU Program, Taiwan, under a 2006–2007 grant.

References

[1] A. Alvarez, B. Abdi, D. Young, H. Weed, J. Teplik, E. Herald, Application of statistical design and response surface methods to computer-aided vlsi device design, IEEE Trans. Comput. Aided Design 7 (2) (1988) 272–288.

[2] D. Boning, P. Mozumder, Doe/opt: a system for design of experiments, response surface modeling, and optimization using process and device simulation, IEEE Trans. Semiconduct. Manuf. 7 (2) (1994) 233–244.

[3] G.E.P. Box, N.R. Draper, Empirical Model-Building and Response Surfaces, Wiley, New York, 1987.

[4] J. Carroll, K. Chang, Statistical computer-aided design for microwave circuits, IEEE Trans. Microw. Theory Techn. 44 (1996) 24–32.

[5] Y.-S. Chou, A Novel Statistical Methodology for Sub-100 nm MOSFET Fabrication Optimization and Sensitivity Analysis, Masters Thesis, National Chiao Tung University, Hsinchu, Taiwan, 2005.

[6] C. Daniel, Use of half-normal plots in interpreting factorial two-level experiments, Technometrics 1 (1959) 311–341.

[7] J.H. Dodgson, A graphical method for assessing mean squares in saturated fractional designs, J. Qual. Technol. 35 (2003) 206–212. [8] N. Dye, H. Granberg, Radio Frequency Transistors Principles and Practical Applications, Butterworth-Heinemann, 1993, pp. 204–231. [9] G. Gonzalez, Microwave Transistor Amplifiers, Prentice-Hall, 1984, pp. 139–193.

[10] H. Lee, P. Mok, A cmos current-mirror amplifier with compact slew rate enhancement circuit for large capacitive load applications, in: Proceedings of IEEE International Symposium on Circuits and Systems, 2001, pp. 220–223.

[11] Y. Li, An automatic parameter extraction technique for advanced cmos device modeling using genetic algorithm, Microelectron. Eng. 84 (2007) 260–272.

[12] Y. Li, Y.-Y. Cho, Intelligent bsim4 model parameter extraction for sub-100 nm mosfet era, Jpn. J. Appl. Phys. 43 (2004) 1717–1722. [13] Y. Li, Y.-Y. Cho, C.-S. Wang, K.-Y. Huang, A genetic algorithm approach to ingap/gaas hbt parameters extraction and rf characterization, Jpn.

J. Appl. Phys. 42 (2003) 2371–2374.

[14] Y. Li, Y.-S. Chou, A novel statistical methodology for sub-100 nm mosfet fabrication optimization and sensitivity analysis, in: Extended Abstract of the 2005 International Conference on Solid State Devices and Materials, 2005, pp. 622–623.

[15] Y. Li, C.-S. Lu, Characteristic comparison of sram cells with 20 nm planar mosfet, omega finfet and nanowire finfet, in: Proceedings of IEEE Nanotechnology Conference, 2006, pp. 339–342.

[16] Y. Li, C.-S. Lu, W.-W. Lo, M.-J. Tsai, T.-Y. Wu, Sensitivity analysis of static noise margin in sram cells with 65 nm cmos devices, in: Proceedings Conference on Scientific Computing in Electrical Engineering, 2006, pp. 21–22.

[17] Y. Li, S.-M. Yu, A unified optimization framework for real world problems, in: Lecture Series on Computer and Computational Sciences, vol. 7, Brill Academic Publishers, 2006, pp. 816–819.

[18] Y. Li, S.-M. Yu, J.-W. Lee, Quantum mechanical corrected simulation program with integrated circuit emphasis model for simulation of ultrathin oxide metal-oxide-semiconductor field effect transistor gate tunneling current, Jpn. J. Appl. Phys. 44 (2005) 2132–2136.

[19] S. Liao, Microwave Circuit Analysis and Amplifier Design, Prentice-Hall, 1988, pp. 123–160.

[20] W.-W. Lo, Computational Statistics Approach to Integrated Circuit Design Optimization and Sensitivity Analysis, Masters Thesis, National Chiao Tung University, Hsinchu, Taiwan, 2006.

[21] A.N. Lokanathan, J.B. Brockman, A methodology for concurrent process circuit optimization, IEEE Trans. Comput. Aided Design 18 (1999) 889–902.

(13)

[23] R.H. Myers, D.C. Montgomery, Response Surface Methodology: Process and Product Optimization Using Designed Experiments, John Wiley & Sons, Inc., New York, 2002.

[24] A.V. Nenarokomov, T.D. Fadale, Uncertainties in parameter estimation: the optimal experiment design, Int. J. Heat Mass Transfer 43 (2000) 3331–3339.

[25] NIST/SEMATECH e-Handbook of Statistical Methods,http://www.itl.nist.gov/div898/handbook/. [26] M.S. Phadke, Quality Engineering Using Robust Design, AT&T Bell Laboratories, 1989.

[27] J.S. Ramberg, S.M. Sanchez, P.J. Sanchez, L.J. Hollick, Designing simulation experiments: Taguchi methods and response surface metamodels, in: Proceedings of Winter Simulation Conference, 1991, pp. 167–176.

[28] G.W. Roberts, A.S. Sedra, Spice for Microelectronic circuits, Saunders College Pub., 1992.

[29] S. Williams, K. Varahramyan, W. Maszara, Statistical optimization and manufacturing sensitivity analysis of 0.18␮m soi mosfets, Microelectron. Eng. 49 (1999) 245–261.

[30] P. Vancorenland, G.V. der Plas, M. Steyaert, G. Gielen, W. Sansen, A layout-aware synthesis methodology for rf circuits, in: Proceedings of IEEE/ACM International Conference on CAD, 2001, pp. 358–362.

[31] C.F.J. Wu, M. Hamada, Experiments: Planning, Analysis, and Parameter Design Optimization, John Wiley & Sons Inc., New York, 2000. [32] G. Derringer, R. Suich, Simultaneous Optimization of Several Response Variables, J. Quality Technol. 12 (1980) 214–219.