A simulation-based evolutionary approach to LNA circuit design optimization

(1)

A simulation-based evolutionary approach to LNA circuit

design optimization

Yiming Li

Department of Communication Engineering, National Chaio Tung University, Hsinchu 300, Taiwan

a r t i c l e

i n f o

Keywords: Simulation-based evolutionary methodology Nonlinear ODEs Genetic algorithm Levenberg–Marquardt method Computational efﬁciency Circuit simulation Design optimization

a b s t r a c t

In this paper, we propose a simulation-based evolutionary approach for designing low noise amplifier (LNA) integrated circuits (ICs). Based on a genetic algorithm (GA), the Levenberg–Marquardt (LM) method, and a circuit simulator, the simulation-based evolu-tionary approach is developed for design optimization of LNA circuits. For a given LNA cir-cuit, the simulation-based evolutionary approach simultaneously optimizes the electrical specifications, such as S11, S12, S21, S22, K factor, the noise figure, and the input third-order intercept point in the process between simulation and optimization. First of all, the neces-sary parameters of the LNA circuit for circuit simulation are loaded. By solving a set of non-linear ordinary differential equations, the circuit simulator will then be performed for the circuit simulation and specification evaluation. Once the specification meets the aforemen-tioned seven constraints, we output the optimized parameters. Otherwise, we activate the GA for the global optimization; in the meanwhile, the LM method searches the local optima according to the results of the GA. We then call circuit simulator to compute and evaluate newer results until the specification is matched. In numerical experiment, 10 parameters of the LNA circuit including device configuration and biasing condition are optimized with respect to the constraints. The design of LNA circuit is with 0.18lm metal-oxide-silicon filed effect transistors. Benchmark results also computationally confirm the robustness and efficiency of the proposed method. This simulation-based evolutionary approach, in general can be applied to optimal design of other analog and radio frequency circuits. We believe that this systematical approach will help IC design optimization, and benefit computer-aided design of wireless communication system-on-a-chip.

1. Introduction

Low noise amplifier (LNA) circuit plays an important role in radio frequency (RF) circuit design[1–6]. In modern inte-grated circuit (IC) design flow and chip implementation, designers perform a series of functional examination and analysis of electrical characteristics of a designed circuit by circuit simulation and electronic computer-aided design (ECAD) software to match specifications[7,8]. In order to achieve the specification, designers must continuously and repeatedly tune the de-sign coefficients and perform the circuit simulation to get a set of optimized active device model parameters, passive device parameters, device size, circuit layout, width of wires, and biasing condition. It is in general requires experienced designers to accomplish such complicated works. Circuit simulation tool including ECAD software has manually been used in perform-ing IC designs in the last decades, yet proper usage of optimization techniques will have a positive contribution to the com-munities of fabrication and design[6,9,10].

E-mail address:[email protected]

Contents lists available atScienceDirect

Applied Mathematics and Computation

(2)

In this paper, based on a genetic algorithm[6,9–14](GA), the Levenberg–Marquardt[6,10,15–17](LM) method, and a cir-cuit simulator[18,19], we propose a simulation-based evolutionary approach for optimal design of LNA circuits. The basic concept of the simulation-based evolutionary approach has been proposed for model parameter extraction of sub-100 nm metal-oxide-silicon filed effect transistors (MOSFETs) and optimal characterization of heterojunction bipolar transistor in our recent works[9,10]. We in this work successfully generalize this approach to IC design optimization; in particular, for analog and RF circuits. For a given LNA circuit, the simulation-based evolutionary approach simultaneously optimizes the electrical specifications[1,6]such as S11, S12, S21, S22, K factor, the noise figure, and the input third-order intercept point

in the optimization process. First of all, the necessary parameters of the LNA circuit for circuit simulation, such as the mac-romodel of RF MOSFETs[18]and the netlist of explored LNA circuit are loaded[6,18]. A circuit simulator will then be per-formed for the circuit simulation and specification evaluation. In the circuit simulation, a set of nonlinear ordinary differential equations (ODEs) corresponding to the LNA circuit will be solved. Once the specification meets the aforemen-tioned seven constraints, optimized parameters associated with the specified LNA circuit are then outputted. Otherwise, we activate the GA for the global optimization; in the meanwhile, the LM method searches the local optima according to the evolutionary results of the GA. This numerical optimization method does significantly accelerate the evolution process. We then call circuit simulator to compute and evaluate newer results until the specification is matched. We note that the well-known circuit simulation tool, HSPICE[18], is successfully integrated in our numerical implementation. To verify the validity of the proposed methodology for LNA circuits design optimization, more than 10 parameters including capacitance, inductance, resistance, and biasing conditions are optimized with respect to the aforementioned seven constraints. The de-sign of LNA circuit is focus on the usage of the 0.18

l

m MOSFETs, but it can be applied to other technology nodes. Benchmark results including the convergence property and the sensitivity of optimized parameters also computationally confirm the robustness and efficiency of the proposed simulation-based evolutionary approach. This approach not only works at the lev-els of device and a single circuit module but also can be applied to other analog and RF circuits; even an electronic system which is with a small number of transistors. It is because the proposed approach is mainly based upon a simulation-based technique. Therefore, once the netlist of circuit simulation for any specified circuits is generated the optimization technique works with the similar operation mechanism.

This paper is organized as follows: in Section2, we introduce the proposed simulation-based optimization technique. In Sec-tion3, the numerical experiment for this work is introduced. The achieved simulation results are discussed in Section4. We also verify the validity, robustness, and efﬁciency of the method. Finally, we draw conclusions and suggest some future works.

2. The simulation-based evolutionary approach

It is difficult to find the solution of multidimensional global optimization problems in modern IC design by using either conventional numerical method or soft computing techniques. Typical genetic searching methods are plagued by problems such as rapid decrease in the population diversity and disproportionate exploitation and exploration of the solution space with multiple dimensions. The results are frequent premature convergence and inefficient search. Newton-based numerical methods find a solution rapidly compared with approaches of GA, but they are still a local method and are often trapped into local optimum. A basic idea of simulation-based evolutionary approach proposed in this paper mainly takes a GA to perform global search, and while the evolution seems to be saturated, the LM method is then enhancing the searching behavior to perform the local search.

For a circuit to be optimized, such as a LNA circuit, we automatically parse and generate the corresponding netlist of the circuit[19]. The generated script file will be inputted into an adopted circuit simulator for simulation and evaluation of re-sults, where a set of circuit ODEs are solved. If the results meet the target, we then output the final optimized data. If the dif-ference of errors between the target and result does not meet the convergence criterion, the established optimization kernel will enable the circuit parameter extraction in a global sense. The number of circuit parameters to be extracted depends upon the specification that we want to achieve. For examples, they could include active, passive, design window, and biasing con-ditions. The optimized results are used for automatically modifying the netlist and the next newer optimization process is performed. We note that the optimization technique implemented in this work is so-called the simulation-based evolutionary approach because an adopted circuit simulator is preformed repeatedly to calculate the cost function in the evolution loop. An architecture of the optimization kernel for the proposed simulation-based evolutionary approach is shown below: Initialize parameters extraction environment

Initialize GA

while ﬁtness score of the best chromosome > tolerated score GA searches for better solution

if the evolution seems to be saturated LM searches for currently local solution End while

According to the procedure shown above, the GA ﬁrstly searches the entire problem space. During this period, the can-didates that GA searched are passed to certain adopted circuit simulator[18]to retrieve the results of circuit simulation. For

(3)

the speciﬁed desired targets, the simulated results are then passed to the evaluation to measure the ﬁtness score. The eval-uated score is provided for the global optimization of GA. After a rough solution is obtained, the LM method simultaneously performs local searches and sets the local optimum as the initial values for the GA performing further optimizations. In the following sub-sections, the computational details of the GA and LM methods are described.

2.1. The proposed genetic algorithm

GA is a global search optimization method based on the mechanics of natural selection and natural genetics. It has been applied to different domain [6,9–14]. It works with a coded of parameters string called chromosome instead of the solutions themselves. Each chromosome represents a solution set, and the ﬁtness functions used to measure the sur-vival scores of all chromosomes in the population. Then the GA will accord its selection scheme to select several chro-mosomes for copulation, discard unwanted chrochro-mosomes, and adopt the crossover scheme to produce the new generation. Because the new chromosomes are made by better chromosomes, they may have higher probability to achieve better result. After crossover, one may apply mutation to change some genes in the new chromosomes to achieve higher diversity. Then the GA will apply ﬁtness function for the new population again and loop this cycle until certain stop criteria is achieved.

In this work, the problem is deﬁned as follows:

f ðSp;Vin;~pÞ ¼ Oresult; ð1Þ

where the function f can be regarded as a circuit simulator which solves a system of circuit nonlinear ODEs, the Spis a netlist

required by the circuit simulator, the Vinis input bias, and the ~p is the parameters needs to be extracted. By feeding the three

components into f, a series of result under different frequencies is generated in Oresult, such as S11, S12, S21, S22, . . . The GA has

to optimize the ~p under several Spand Vin.

The typical GA has basic ﬁve genetic operators to perform the solution evolvement. It includes the gene encoding, the ﬁtness evaluation, the chromosome selection, the sexual crossover, and the gene mutation. In the following, we describe each operator step by step.

2.1.1. Gene encoding

The gene encoding method strongly depends on the problem to be optimized. Encoding operator is the procedure that encodes the solution of the problem into the coded string format, and others genetic operators operate on the coded string directly instead of the solution itself. The design of gene encoding strategy strongly depends on the properties of the prob-lem. There are 10 parameters in the LNA circuit. In the proposed GA for the simulation-based evolutionary algorithm, we transform these continuous ﬂoating-point numbers into discrete steps through step function as shown in the Eq.(2)instead of real numbers, and we encode the discrete steps as genes on chromosomes. The discrete steps show the strongly combi-natorial properties, and we have found this representation has better results in crossover:

Pvalue¼ Pminþ

Pmax Pmin

Resolution: ð2Þ

2.1.2. Fitness evaluation

The fitness evaluation evaluates the fitness score for each chromosome in the population. The fitness score can be seen as the accommodation status of each chromosome in current environment, and it is also an important reference for the selec-tion procedure to judge the suitability of each chromosome. We consider the fitness funcselec-tion:

F ¼ X

all target

ðWðsim speÞÞ; ð3Þ

where the sim and the spe indicate the simulated result and target speciﬁcation, respectively. The W is a weight function, when the sim is not matched with the spe, the weight increase to emphasize this problem.

2.1.3. Selection

After obtaining the ﬁtness score of each chromosome, the selection method selects the better chromosomes by the spe-ciﬁc schemes, such as the roulette wheel selection, the tournament selection, and the ranking selection, and different selec-tion schemes may lead to the different convergence behavior. In this work, the tournament selecselec-tion method is applied. The pseudocode of a tournament selection is shown below:

Set the number of contest up according to the selection rate for each contest

Randomly pick 2 competitors

Select the competitor with better ﬁtness score End for

(4)

2.1.4. Crossover

Crossover combines the features of two parent chromosomes to form two similar off-springs by swapping corresponding segment of the parents. It is intuitive that the crossover operator is exchanging information between different potential solu-tions. We take a uniform crossover scheme in our developed GA; and based on our simulation experience, it is more effective than single and two-point cuts crossover schemes[9,10]. A pseudocode is shown below:

Randomly selects 2 parent

for each category of parameters of the new chromosome Randomly pick a parent and named it as p

Parameters in this category of the chromosome =parameters in this category of p

End for 2.1.5. Mutation

Mutation arbitrarily changes one or more genes of a selected chromosome by a random variation with a probability factor so called the mutation rate. The mutation procedure may cause the entity unable to accommodate with the environment; however, a successful mutation may lead the evolutionary trend to achieve the better situation. A pseudoprocedure for each step in GA is described as:

GeneEncoding through Eq.(2) do Chromosomes selection Offsprings reproduce Genes mutation Fitness evaluation if BestChromosome.error < Threshold then IfAchieveGoal = true

else IfAchieveGoal = false while IfAchieveGoal is false

2.2. The adopted Levenberg–Marquardt method

The LM method is a quasi-Newton method to accelerate the Gauss–Newton method[6,10,15–17]. The Gauss–Newton method is the basic algorithm for solving the nonlinear optimization problem. Due to the nonlinear property of the problem, a gradient for each variable can be obtained. It starts from an initial guess, and follows the direction of the normal of the gradient to ﬁnd the optimal solution. Therefore, the initial guess must be chosen carefully, or the solution may fell into a local optima. Unlike the Gauss–Newton method has the ﬁxed steps toward the solution, LM optimization method detects that some regions with monotonic variation property can be speed up by increasing the step size. On the other hand, when the optimization process encounters a sensitive region, the step should be shorten to avoid skipping the optimum. The pro-cedure of the LM method is shown below:

Given: a; IM D; I

E

D;where ‘‘a” is parameters set, ‘‘I M

D” is the optimization target, and ‘‘I E

D” is the initial guess of the solution.

x2_{ðaÞ ¼}P IM DIEDðaÞ

r

2

, where

r

is the mean of the measured data. k= 0.0001.

Compute

a

0

jj

a

jjð1 þ kÞ;

a

0jk

a

jk ðj–kÞ

while not converge

Solve linear matrix problem: PM l¼1

a

0kldal¼ bk, where

a

0kl o2v2 2oakoal; bk ov2 2oak if

v

2_{(a + da) >}

_v

2_(a), knþ1¼ kn_þv2_ðaþdaÞ_v2_ðaÞ a ; else if

v

2 (a + da) <

v

2 (a), knþ1¼ kn aþ1; an+1_{= a}n_{+ da,} End while

3. Application to LNA circuit design

The explored LNA circuit, shown inFig. 1, focuses on the working frequencies ranging from 2.11 to 2.17 GHz. The LNA circuit is with two cascaded MOSFETs. We note that the Lloadand Rloadare the nonlinear functions in our LNA circuit[6].

(5)

The choke inductor Lchokeworking at high frequency is assumed to be ﬁxed at 1

l

H. Cinis an external signal couple capacitor

which is ﬁxed at 20 pF. According to the KCL conservation law[3–6], a set of ODEs can be formulated for the LNA circuit, shown inFig. 1: Cmatch1 dðVP1 VAÞ dt ¼ 1 Lmatch1 Z VAdt þ Cin dðVA VBÞ dt ; ð4Þ 1 Lbond Z ðVB VG1Þdt ¼ 1 Lchoke Z ðVG1 VB1Þdt þ IG; ð5Þ and Cmatch2 dðVP2 VD2Þ dt þ 1 Lload Z ðVDD VD2Þdt þ VDD VD2 Rload ¼ ID2 : ð6Þ

The unknowns to be solved in Eqs.(4)–(6)are VA, VG1, and VD2. We note that the IGand ID2are current models[6]. To solve

the system of nonlinear ODEs, the simulation program with integrated circuit emphasis (SPICE) is applied. The corresponding netlist for the circuit simulation of the LNA circuit is shown below:

CCIN N_10 N_9 20P CCMATCH1 N_10 IN 610F CCMATCH2 N_5 OUT 1.7P CCMATCH3 GND_L OUT 4.5P LLBOND N_9 N_6 1N LLCHOKE VB1 N_6 1U LLDEG N_8 GND_L 1N LLLOAD VDD_L N_5 2.8N LLMATCH1 N_10 GND_L 4.6N XM1 N_7 N_6 N_8 GND_L NMOS_RFW5 LR=0.24U NR=64 XM2 N_5 VB2 N_7 GND_L NMOS_RFW5 LR=0.24U NR=64 RRLOAD VDD_L N_5 1K

we note that the format and grammar of the netlist can be found in[19].

S-parameters (S11, S12, S21, and S22) will aid in the stability analysis of the LNA circuit. The S-parameters are deﬁned as

follows: S11¼ V 1 Vþ 1 ; S21¼ V 2 Vþ 1 ; S12¼ V 1 Vþ 2 ; and S22¼ V 2 Vþ 2 ; ð7Þ where the V 1and V

2are the output waveforms from the port 1 and the port 2, and V þ 1and V

þ

2are the input waveforms from

the port 1 and the port 2, respectively. For the stability analysis of S-parameters, the Rollett stability factor (K) is numerically calculated. An intermediate quantity called delta (D) should be calculated ﬁrst to simplify the ﬁnal equation for the K factor:

V_DD V_DD Lchoke Cin Lload Rload Cmatch3 Port 2 Port 1 Ldeg Cmatch2 Lbond Cmatch1 Lmatch1 W2/L2 W1/L 1 V_B1 V_B2 VP1 VA VG1 V_D2 VP2 VB IG I_D2

(6)

D

¼ S11 S22 S21 S12; ð8Þ then K ¼1 þ j

D

j 2 jS11j2 jS12j2 2 jS11j jS12j : ð9Þ

To evaluate the noise performance of a low noise amplifier, by the definition of noise factor F, the equation for noise figure (NF) is given by NF ¼ 10 logðFÞ ¼ 10 log SNRin SNRout ¼ 10 log Signal Noisein Signal Noiseout !

¼ 10 log Noiseinþ Noiseamp Noisein

¼ 10 log 1 þNoiseamp Noisein

; ð10Þ

where SNRinis the signal-to-noise ratio at the input and SNRoutis the signal-to-noise ratio at the output. Noiseinis the noise

from the previous stage, Noiseoutis the noise at the output which consists of the noise from ampliﬁer (Noiseamp) plus the

noise from Noisein.

The deﬁnition of the third-order input intermodulation distortion (IIP3) is the input power in dBm where the fundamen-tal output power and the third-order intermodulation output power are the same.

4. Results and discussion

By calculating several interested speciﬁcations, we ﬁrstly verify the feasibility of the proposed method.Fig. 2shows the initial state (dash-dot) and an optimized result (line) of S11parameter. The acceptable result is when S11< 10 dB within the

working frequency range. It is clearly thatFig. 2states that the result has achieved to this goal. We note that the amplitude of the input sinusoidal signal within the working frequency range is with 1.0 V, VB1= 0.75 V, and VB2= 2.7 V, shown inFig. 1.

Fig. 3shows the initial state and an optimized result of S12parameter. The acceptable result is when S12< 25 dB within

the working frequency range, whereFig. 3clearly conﬁrms the achieved results with much more improvements than the original one.Fig. 4shows a comparison between the initial state and an optimized result of S21parameter, where a larger

S21is expected in the optimization process. Typically, we do not deﬁne an engineering speciﬁcation for S21in this testing

case. However, a large value of S21is good for optimal design of LNA circuits. Compared with the initial data, the obtained

optimized result is improved. This phenomenon is due to a compromise among all physical constraints so that all character-istics can meet their targets at the same time. However, it can be further improved by performing more evolution genera-tions.Fig. 5shows the initial state and an optimized result of S22parameter. The goal is the same with the parameter S11, i.e.,

the result is acceptable if S22< 10 dB within the working frequency range. Very good result, 20 dB is achieved when we

refer to the setting of standard goal. Once the S parameters are optimized, with Eq.(9), we also calculate the K factor accord-ingly[1].

Fig. 6shows a comparison between the initial state and an optimized result of the noise figure. The desired specification of the noise figure is that NF < 2 within the working frequency range. Both the initial state and an optimized result meet the target, but the obtained optimized result is a little bit shifted away due to the same reason of a global compromise among all electrical characteristics.Fig. 7indicates the initial state and an optimized result of the input third-order intercept point. For the optimization criterion of IIP3, we do hope that the amplitude of output >20 dB and is as large as possible. As shown in Fig. 7, the optimized IIP3 = 26.Table 1shows the optimized parameters of the investigated experiment and theTable 2 shows the corresponding optimized characteristics for the experiment. Results shown in both tables confirm the validity of the method.

Frequency (GHz)

2.10

2.12

2.14

2.16

2.18 S

11

(dB)

-15

-12

-9

-6

-3

(7)

Inspecting the sensitivities of extracted parameters is an important work in IC design and performance analysis. The sen-sitivity examination of circuit parameters can point out which parameters affect behavior of the performance the most and which ones barely make effect. This experiment is designed as follow. The proposed simulation-based evolutionary approach optimizes single parameters category meanwhile locks other parameters. The LNA circuit parameters to be optimized are classiﬁed into three categories illustrated inTable 3. As shown inFig. 8, it reveals that the geometry parameters would make the most improvement, while the input and output categories makes little improvement after 120 generations. This phe-nomenon indicates that the geometry of the active devices is more difﬁcult to be optimized than passive device parameters.

Frequency (GHz)

2.10

2.12

2.14

2.16

2.18 S

12

(dB)

-48

-45

-42

-39

-36

Fig. 3. The initial state (dash-dot) and an optimized result (line) for the parameter of S12.

Frequency (GHz)

2.10

2.12

2.14

2.16

2.18 S

21

(dB)

4

8

12

16

20

Fig. 4. The initial state (dash-dot) and an optimized result (line) for the parameter of S21.

Frequency (GHz)

2.10

2.12

2.14

2.16

2.18 S

22

(dB)

-30

-20

-10

0

10

(8)

Frequency (GHz)

2.10

2.12

2.14

2.16

2.18 NF

0.8

0.9

1.0

1.1

1.2

Fig. 6. The initial state (dash-dot) and an optimized result (line) of the parameter NF.

Frequency (GHz)

2.12

2.14

2.16

2.18 Volts (dB)

-100

-75

-50

-25

0

Fig. 7. The initial state (dash-dot) and an optimized result (line) of the parameter IIP3.

Table 1

A list of the optimized parameters for the testing experiment

Element Unit Range Result

Cmatch1 fF 300–800 512.132 Cmatch2 pF 1–10 4.6104 Cmatch3 pF 1–10 4.5511 Lbond nH 1–10 1.0782 Ldeg nH 0.1–5 1.145 Lmatch1 nH 1–10 6.202 Rload X 1.5–5.5 3.5 Lload H 1.5–5.5 3.5 VB1 V 0.5–1.5 0.75 VB2 V 0.5–5 2.7 Table 2

The ﬁnal achieved result

Speciﬁcation Target Result

S11 <10 dB 14.1 dB S22 <10 dB 22.6 dB S12 <25 dB 39.3 dB S21 As large as possible 12.7 dB K <1 10.7 NF <2 0.979 IIP3 <10 1.3

(9)

By considering the experiment,Fig. 9shows a comparison of the score convergence behavior among population sizes, where the mutation rate is fixed at 0.5. The fitness score versus the number of generation suggests that the score conver-gence behavior does not have a satisfied result if the population size is too small. According to our experience, the population size = 50 is good for the optimal design of LNA circuit.

In addition,Fig. 10shows the ﬁtness score convergence behavior for the circuit optimization with different mutation rate, where the population size = 50. The results suggest that the mutation = 0.5 keeps the population diversity and ﬁnally has better evolutionary results.

Finally, the computational efficiency of the proposed simulation-based evolutionary approach is investigated.Fig. 11 shows the score convergence behavior comparison of the standard GA and the simulation-based evolutionary approach. The setting is with the population size = 50 and mutation = 0.5. As shown in this figure, the proposed methodology is supe-rior to the pure GA after 60 generations. The proposed method shows no significant advantage at the beginning because the LM method has not been triggered yet. Once the LM method is activated, based on the result of GA to perform local optimi-zation, the GA follows the local optima obtained by the LM method to keep evolving. Under this mechanism, our simulation-based evolutionary approach shows better trend of convergence and the robustness of our proposed methodology hence is held. We note that the LM method encounters divergent results when the optimization problem is solved.

Table 3

Three categories of the circuit parameter of the LNA circuit

Category Parameters

Geometry L1, W1, L2, and W2

Input Cmatch1, Lmatch1, Lbond, Lchoke, Cin, Ldeg, VB1, and VB2

Output Lload, Rload, Cmatch2, and Cmatch3

Number of generation

0

100

200

300

400

500 Fitness score

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

Geometry Input Output

Fig. 8. A veriﬁcation of the sensitivity analysis for the three cataloged parameters.

Number of generation

0

100

200

300

400

500 Fitness score

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Population size: 20 Population size: 50 Population size: 100

(10)

5. Conclusions

In this paper, a simulation-based evolutionary approach for optimal design of LNA circuit has been reported. Based on the GA, the LM, and a well-known circuit simulator, the engineering optimization problem has successfully been solved accord-ing to the simulation-based evolutionary approach. Electrical characteristics of the explored LNA circuit considered in the optimization process are S11, S12, S21, S22, K factor, the noise ﬁgure, and the input third-order intercept point. Testing

exam-ples of LNA circuits with the 0.18

l

m MOSFETs have been examined to show the validity, efficiency, and robustness of the method. To explore the realistic feasibility of optimized designs of the LNA circuit, we are currently fabricating and testing the functionality of the corresponding IC chips. We note that this simulation-based evolutionary approach can also be ap-plied to optimal design of other circuit modules with more advanced technology nodes. Developed algorithms that based on this simulation-based evolutionary approach can be directly incorporated into any existed electronic computer-aided de-sign software which benefits the communities of dede-sign and fabrication. Currently, we apply the method to optimal dede-sign of operation amplifier and phase-locked loop circuits, and pattern optimization of antenna problems. Furthermore, to enhance the efficiency, distributed computing technique will be considered[20].

Acknowledgements

This work was supported in part by the National Science Council of Taiwan under 96-2221-E-009-210, Contract NSC-95-2221-E-009-336, and Contract NSC-95-2752-E-009-003-PAE, and by the MoE ATU Program, Taiwan, under a 2006 Grant. References

[1] D.K. Misra, Radio-Frequency and Microwave Communication Circuits Analysis and Design, John Wiley & Sons, 2004. [2] O. Mitrea, M. Glesner, Microelectron. Reliab. 44 (2004) 877.

Number of generation

0

100

200

300

400

500 Fitness score

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Mutation rate: 0.5 Mutation rate: 0.3 Mutation rate: 0.1

Fig. 10. The ﬁtness score versus the number of generations with respect to different mutation rates.

Number of generation

0

40

80

120

160

200 Fitness score

0.0

0.2

0.4

0.6

0.8

Pure GA

The simulation-based evolutionary approach

(11)

[3] Y. Li, K.-Y. Huang, Comput. Phys. Commun. 152 (2003) 307.

[4] K.-Y. Huang, Y. Li, C.-P. Lee, IEEE Trans. Microwave Theory Tech. 51 (2003) 2055.

[5] J.G. Fijnvandraat, S.H.M.J. Houben, E.J.W. ter Maten, J.M.F. Peters, J. Comput. Appl. Math. 185 (2006) 441. [6] H.-M. Chou, Masters Thesis, National Chiao Tung University, Hsinchu, Taiwan, 2005.

[7] R. Gupta, D.J. Allstot, Dig. IEEE MTT-S Int. Microwave Symp. (1998) 1867.

[8] P. Vancorenland, G. Van der Plas, M. Steyaert, G. Gielen, W. Sansen, in: Proceedings of the IEEE/ACM International Conference on Computer Aided Design, 2001, p. 358.

[9] Y. Li, Y.-Y. Cho, C.-S. Wang, K.-Y. Huang, Jpn. J. Appl. Phys. 42 (2003) 2371. [10] Y. Li, Y.-Y. Cho, Jpn. J. Appl. Phys. 43 (2004) 1717.

[11] L.M. Schmitt, Theor. Comput. Sci. 259 (2001) 1. [12] J. He, L. Kang, Theor. Comput. Sci. 229 (1999) 23. [13] J. McCall, J. Comput. Appl. Math. 184 (2005) 205. [14] P. Xu, J. Comput. Appl. Math. 155 (2003) 243.

[15] J.Z. Chang, L.H. Chen, J. Optim. Theory Appl. 92 (1997) 393. [16] Q.M. Han, J. Numer. Methods Comput. Appl. 19 (1998) 99.

[17] C. Kanzow, N. Yamashita, M. Fukushima, J. Comput. Appl. Math. 173 (2005) 321.

[18] M.S. Shur, T.A. Fjeldly, Silicon and Beyond: Advanced Circuit Simulators and Device Models, World Scientiﬁc Publishing Co., Singapore, 2000. [19] T.L. Quarles, The SPICE3 Implementation Guide, Tech. Rep. No. UCB/ERL M89/44, 1989.