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Microelectronics Journal 39 (2008) 1761–1769 www.elsevier.com/locate/mejo

An effective parameter extraction method based on memetic differential evolution algorithm Bo Liu, Jing Lu, Yan Wang, Yang Tang Institute of Microelectronics, Tsinghua University, Beijing, China Received 19 August 2007; received in revised form 7 February 2008; accepted 12 February 2008 Available online 18 April 2008

Abstract This paper provides an effective method for parameter extraction of microelectronic devices and elements. A novel method, memetic differential evolution (MDE) algorithm, is proposed in this paper. By combining differential evolution (DE) algorithm, mutations in immune algorithm (IA), and special operators for parameter extraction, MDE possesses characteristics of high accuracy, stability, generality, and efficiency. The effectiveness of the method has been shown by two typical examples, including small-signal equivalent circuit models for an AlGaN/GaN HEMT device up to 40 GHz, as well as an equivalent circuit model for on-chip differential spiral inductors. In both cases, the initial values and parameter ranges of the elements in the equivalent circuits are hard to determine in optimization. The results and comparisons with Levenberg–Marquardt (LM) algorithm, genetic algorithm (GA), particle swarm optimization (PSO) algorithm and canonical DE algorithm, demonstrate the superiority of MDE in terms of accuracy and generality. r 2008 Elsevier Ltd. All rights reserved. Keywords: Parameter extraction; Electronic device; Evolutionary algorithm; Optimization

1. Introduction It is well known that the robust extraction method for an accurate device model is vital for designing a circuit, evaluating the process technology, and optimizing device performances. Among various approaches to parameter extraction are physics-based calculation, numerical simulation, and measurement-based curve fitting and optimization. Numerical simulation can provide high accurate results theoretically, whereas the lack of a physical meaning and long computing time constitute major drawbacks for practical applications. Calculations based on analytical compact models are clear in physical meanings; however, very complicated formulations and proper parameters are needed to produce a high-precision fit to the measurement. On the other hand, measurement-based curve fitting and optimization approach is most practical and convenient to circuit design. Corresponding authors.

E-mail addresses: [email protected] (B. Liu), [email protected] (Y. Wang). 0026-2692/$ - see front matter r 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2008.02.021

Measurement-based parameter extraction methods mainly include direct extraction methods and optimization-based methods. The direct extraction methods are simple but usually need some empirical presumptions [1,2], so a satisfactory accuracy can hardly be achieved, especially in high-frequency ranges where more parasitic effects have to be considered. In addition, with the scaling down of devices, more complicated lumped-element equivalent circuit is a big challenge to direct extraction methods. In contrast, the optimization-based methods are competent for high accuracy in these cases. Techniques for optimization-based extraction methods can be broadly classified into two main categories: traditional numerical optimization algorithms and stochastic search algorithms (evolutionary computation algorithms). Traditional numerical optimization methods mainly include Leverberg–Marquardt (LM) algorithm and exponential gradient algorithm. The drawbacks of traditional numerical optimization algorithms are mainly in the following four aspects: (1) They require a good starting point to initiate the iteration. (2) The resulted solutions are often far from the optimal ones as they

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B. Liu et al. / Microelectronics Journal 39 (2008) 1761–1769

normally represent a local optimum. (3) They often require continuity and differentiability of the objective function. (4) Negative resistors and capacitors may be obtained. Therefore, stochastic search algorithms have potential to replace traditional numerical optimization algorithms because of their global optimization and parallel search characters. Research efforts on stochastic search algorithms, especially works driven by evolutionary computation (EC), have begun to appear in the literature in recent years [3–7]. Due to the ability and efficiency to find a satisfactory solution, genetic algorithm (GA) has been employed as an optimization routine for EC-based parameter extraction. Though these works are important, the evolutionary algorithms for parameter extraction remain an active research area because of the following reasons: (1) GA is the most popular evolutionary algorithm, but the diversities of populations in GA are not very satisfactory, which may cause repeated search and premature convergence. The search ability and convergence rate of GA have been criticized and it is proved that canonical GA cannot converge to the global optimum [8]. (2) High accuracy is the requirement of parameter extraction, but the local search abilities of GA need improvements. (3) Previous knowledge of the problem is destroyed once the population changes, so knowledge of good solutions cannot be retained in the current population of GA. According to the above considerations, a new algorithm, memetic differential evolution (MDE) algorithm, is proposed for parameter extraction in this paper. MDE is constructed by differential evolution (DE) algorithm, selfadaptive mutations in immune algorithm (IA), and operators specially designed for parameter extraction. MDE has several novel features. The most important one is it possesses high-accuracy characteristic, which is proved to have distinct advantages compared to GA and particle swarm optimization (PSO) algorithm. Two typical and difficult examples—small signal models for the AlGaN/ GaN HEMT device up to 40 GHz and an equivalent circuit model for on-chip differential spiral inductors are employed to demonstrate this parameter extraction methodology. The structure of the paper is as follows: Section 2 formulates the investigated problem and provides our goals and motivations; the structure of the proposed optimization method is also shown in this section. MDE algorithm is elaborated in Section 3. Section 4 provides practical examples to prove the efficiency and effectiveness of the proposed optimization algorithm. The concluding remarks are given at the end. 2. Problem formulation and goals Parameter extraction problem is actually an optimization problem aiming to minimize the deviations between the simulated data and experimental data. In addition, some restrictions usually have to be considered, such as the ranges of the parameters.

Model & undetermined parameters

SPICE engine Candidate solutions performances Link

candidates MDE operations

Solution Fig. 1. Architecture of the parameter extraction method.

In real practice, accuracy, ease of use, generality, robustness, and reasonable run-time are necessary for a parameter extraction system to gain acceptance. Other than these, another important motivation of our work is to solve ‘‘hard to extract’’ cases, such as complicated lumpedelement equivalent circuit models with several tens of parameters but initial values in the models are hard to decide, and extraction problems with numerous local optimums. The flow diagram of the system architecture is depicted in Fig. 1. The inputs to the system are the established device or element models, the undetermined parameters and their ranges. Then, the core algorithm MDE will achieve the optimization task. The outputs are extracted parameters. A link between HSPICE and the optimization algorithm is developed to transmit data from the optimization system to the HSPICE engine, and the behaviors of the constructed model by HSPICE simulation are returned to the optimization system. A special point is tens of candidate solutions can simulate simultaneously in one HSPICE simulation through our link, which will take much less time than call HSPICE and simulate one by one. 3. MDE for parameter extraction DE algorithm has been proved to be an efficient and effective solution technique for complex functional optimization problems [9]. Due to the simple concept, easy implementation, and quick convergence, nowadays DE has attracted much attention and wide applications in different fields [10,11]. Various modifications to DE also appeared in the literature [12–14]. However, very few scholars use DE algorithm and its modifications in parameter optimization problems. High accuracy is very important in parameter extraction problems. Though DE algorithm has its advantage on accuracy in local search, it has a drawback of premature convergence for complex problems. Therefore, we borrowed mutation operators in IA to avoid premature convergence by increasing the population

ARTICLE IN PRESS B. Liu et al. / Microelectronics Journal 39 (2008) 1761–1769

diversity. This is the general idea of constructing MDE algorithm for parameter extraction. In MDE, canonical DE is the general framework, and some important operators are combined with canonical DE as modifications, including the self-adaptive mutation operators in IA (real coded hypermutation and stochastic uniform mutation), ranges transformation, and self-adaptive boundary expansion operators. In the following, we will first provide brief introductions to the basic elements of MDE. 3.1. Canonical DE In DE [9,15], it starts with the random initialization of a population of individuals in the search space and works on the cooperative behaviors of the individuals in the population. Therefore, it finds the global best solution by utilizing the distribution of solutions in the search space and differences between pairs of solutions as search directions. However, the searching behavior of each individual in the search space is adjusted by dynamically altering the differentiation’s direction and step length in which this differentiation performs. The ith individual in the d-dimensional search space at generation t can be represented as X(t)=[xi,1, xi,2, y, xi,d] (i=1, 2, y, NP, where NP denotes the size of the population). At each generation t, the mutation and crossover operators are applied to the individuals, and a new population arises. Then, selection takes place, and the corresponding individuals from both populations compete to comprise the next generation. For each target individual, according to the mutation operator, a mutant vector Vt(t+1)=[vi,1(t+1), y, vi,d(t+1)] is generated by adding the weighted difference between a defined number of individuals randomly selected from the previous population to another individual, which is described by the following equation: V t ðt þ 1Þ ¼ X best ðtÞ þ F ðX r1 ðtÞ X r2 ðtÞÞ

(1)

where r1, r2A{1, 2, y, N} are randomly chosen and mutually different, and also different from the current index i. FA(0, 1+) (FA(0, 2) is commonly used) is a constant called scaling factor, which controls amplification of the differential variation Xr1(t)Xr2(t), and NP is at least four so that the mutation can be applied. Xbest(t), the base vector to be perturbed, is the best member of the current population so that the best information could be shared among the individuals. After the mutation phase, the crossover operator is applied to increase the diversity of the population. Thus, for each target individual, a trial vector Ui(t+1)= [ui,1(t+1), y, ui,d(t+1)] is generated by the following equation ( vi;j ðt þ 1Þ; ifðran dðjÞpCRÞ or j ¼ ran dnðiÞ ui;j ðt þ 1Þ ¼ xi;j ðtÞ; otherwise j ¼ 1; 2; . . . ; d

(2)

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where rand(j) is the jth independent random number uniformly distributed in the range of [0, 1]. randn(i) is a randomly chosen index from the set {1, 2, y, d}. CRA [0, 1] is a constant called crossover parameter that controls the diversity of the population. Following the crossover operation, the selection arises to decide whether the trial vector Ui(t+1) would be a member of the population of the next generation t+1. For a minimum optimization problem, Ui(t+1) is compared to the initial target individual Xt(t) by the following one to one based greedy selection criterion: ( U i ðt þ 1Þ; if f ðU i ðt þ 1ÞÞof ðX i ðtÞÞ (3) X i ðt þ 1Þ ¼ X i ðtÞ; otherwise where Xi(t+1) is the individual of the new population, and f(x) is the cost function. The procedure described above is considered as the standard version of DE, and is denoted as DE/best/1/bin. Several strategies of DE have been proposed, depending on the selection of the base vector to be perturbed, the number and selection of the differentiation vectors, and the type of crossover operators [15]. The key parameters in DE are NP (size of population), F (scaling factor), and CR (crossover parameter). Proper configuration of the above parameters would achieve good tradeoff between the global exploration and the local exploitation so as to increase the convergence velocity and robustness of the search process. Some basic principles have been given for selecting appropriate parameters for DE [15]. In general, the population size NP is choosen from 5 to 10.d (number of dimension). F and CR lie in the range of [0.4, 1.0] and [0.1, 1.0], respectively. 3.2. Important operators to construct MDE 3.2.1. Mutation in IA IAs are mutation-based stochastic search algorithms inspired by immunology observed in nature [16]. The biological immune system is a massively parallel and selfadaptive system, which can defend the invading antigen effectively and keep various antigens coexist. IA becomes a valuable research area in operational research because it has characters of diversity, distributivity, dynamics, selfadaptability, and robustness [17]. The algorithm flow and details of IA are elaborated in [16]. The core of IA to exploit the search space is its various mutations. In this paper, real coded hypermutation and stochastic uniform mutation [18] are used. Real coded hypermutation focuses on global exploration, whose procedure is as follows: 1. Calculate mutation rate a: a ¼ 1 expðjjAbi AgjjÞ

(4)

where Abi is the ith antibody (candidate solution), and Ag is the antigen (the best solution of the current population).


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2. Obtain the new antibody Abi: Abi

Abi þ bðAg Abi Þ

(5)

where b is a random number between 0 and a. Stochastic uniform mutation is effective in increasing the diversity of local search, whose procedure is as follows: 1. Calculate mutation rate a0 : a0 ¼ 1 l expðjjAg Abi jjÞ; 0olo1 (6) 2. The new decimal numbers of a candidate solution are randomly chosen from 0 to 9 based on a0 to replace the original decimal numbers. 3.2.2. Normalization In this process, we normalize the parameter ranges to [0, 100] by multiplying ½10n1 ; 10n2 ; ; 10nd . [n1,n2,y,nd] can be any integers that can transform the parameter ranges to [0, 100]. Without this action, the parameter ranges may cover from 106 to 105 due to different units in capacitors, inductors, and resistors, and that is unfavorable for search because of the broad ranges or limitation of decimal digits. 3.2.3. Self-adaptive boundary expansion In canonical DE, nothing will be done to the newly generated individuals if they exceed the predetermined bound values after mutation and crossover. In some DEbased practices, the overstepped individuals will be set to the nearest bounds. In MDE, the designer can choose whether the bound is restricted. If the range of a parameter is precisely determined, it will be set as its nearest bound value. If the range is roughly estimated, we may do nothing to individual that exceeds its upper bound. Sometimes, better solutions exist out of the roughly estimated predetermined bounds, and MDE can find them by the same cost of computational efforts. The algorithm can also avoid impractical parameters, such as negative resistors by strictly restricting the lower bounds. This advantage is proved by Example 1 in Section 4.

high accuracy, so the premature convergence problem must be solved. As mentioned above, the most distinct advantage of IA is its population diversity. Therefore, mutation in IA is included in MDE. For a given population, three clones will be generated. The first sub-population also uses the DE mutation operator, the second sub-population uses real coded hypermutation in IA, and the third sub-population uses stochastic uniform mutation in IA. The first subpopulation is the basic element of DE. The function of the second sub-population is to reinforce global exploration, to avoid premature convergence. The third sub-population aims to enhance local search ability. The selection operator to generate the new population (the same size as the original DE population) should also be changed to adapt to the proposed mutation method. First, one-to-one competition of canonical DE is executed to generate the winner pool. The size of the winner pool is 3 population size of the original DE population. Second, 80% of the individuals of the new population are selected based on their rank in the winner pool. Third, other parts of the new population are selected by random selection of the winner pool. In the optimization process, ranges transformation and self-adaptive boundary expansion operators are executed. The framework of MDE is shown in Fig. 2. 4. Examples and analysis In this section, two typical examples are provided. The first one is the parameter extraction of a small signal model of AlGaN/GaN III-nitride-based heterostructure field

Parameter Normalization

DE&IA Mutation

Initialization

Crossover

Evaluate the Best Member

Selection

Update Parameters

3.3. The framework of MDE Clone

As we know, DE possesses several advantages than PSO and the most widely used GA. The drawbacks have to be considered. The selection operation in DE is one-to-one competition between the parent and the offspring. Thus, it can enhance the speed of DE, but may lead to a high probability to obtain premature solutions. Owing to this, the DE procedure may cause all the candidate solutions to gradually cluster around the best individual, and the diversity of the population is decreased. The crossover operator is designed for increasing diversity, but it is relatively weak. The parameter extraction problem requires

No

Reached Generations?

Yes Output Fig. 2. Flow diagram of MDE.


effect transistor (HEMT). The second example is parameter extraction of an on-chip differential spiral inductor. The difficulties of Example 1 are: (1) GaN-based HEMT is a special kind of device, (2) the range of the frequency of the measured data is up to 40 GHz. The difficulties in Example 2 are: (1) the initial values of on-chip differential spiral inductors are very hard to decide, (2) numerous local optimums exist in the derived optimization problem. In both examples, the population size of MDE is set to be 20, the scaling factor F is 0.8, the crossover probability is 0.8, and l in stochastic uniform mutation is 0.5. Example 1. III-Nitride-based heterostructure field effect transistor has been the focus of intense research in the last few years due to potential high-power applications at RF, microwave, and millimeter-wave frequencies [19–21]. We have proposed a 20-element distributed small-signal equivalent circuit model [1], which is shown in Fig. 3. To verify the proposed method, we adopted LM optimization and canonical DE to extract parameters of the model in Fig. 3 under 11 bias conditions for comparisons. The measured device in the experiment has a channel width of 200 mm and a length of 1 mm. In this work, the scalar error es between the simulated and measured S-parameter of AlGaN/GaN HEMT is chosen as the optimization goal, which is defined as [20] follows: ReðdS ij;n Þ þ ImðdS ij;n Þ ij;n ¼ , W ij i; j ¼ 1; 2;

n ¼ 1; 2; . . . ; N

where

W ij ¼ max S ij ; W ii ¼ 1 þ jS ii j;

i; j ¼ 1; 2iaj i ¼ 1; 2

N is the total number of data points. And dS ¼ Smeasured S simulated

(7)

Intrinsic part

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The total scalar error can be expressed as: s ¼

N 1X n N n¼1 n 1

where n

n ¼

11;n

12;n

21;n

22;n

! (8)

We found that the direct extraction can achieve a good agreement between the simulated and measured S-parameter within 20 GHz [1]. The error in this frequency region is within 10%. However, in the higher frequency range up to 40 GHz, the direct extraction method is not competent. The scalar error between simulation and measurement is larger than 30%, which is unacceptable. Then, three methods are compared. First, LM optimization in ICCAP (Agilent) [22] is adopted using the starting point provided by direct extraction. The 2nd and 3rd methods are directly using canonical DE and MDE with wide but reasonable parameter ranges. Error analysis is practiced over wide bias conditions. The results are shown in the following. On account of the device gain being important for the design of matching networks, especially in power amplifier design, the gain error eG [20] is considered separately. In Table 1, the extracted values for minimization of scalar error at the bias condition of Vgs ¼ 0 V, Vds ¼ 10 V by MDE are listed as a representative. Fig. 4 shows the corresponding S-parameter comparison between measurement and simulation of MDE optimization as well as LM optimization. The presumed ranges of parameters are shown in Table 2. In this example, the recommended bounds are set by extending 100% range around the direct extraction values. Please note that the above process is only one of the methods to determine the recommended bounds, and does not mean that MDE depends on initial values of direct extraction. The recommended bounds can be determined by other ways, which will be shown in Example 2. The error and computation time of LM, canonical DE and MDE at the bias condition of Vgs ¼ 0 V, Vds ¼ 10 V

Rlgd G

Lg

Rg

Cgd Rgd Vi Cgs Rlgs

Cgsi Cgp

Ids

Rd

Rds

Ld

D

Cds Cdsi

Ri

Cdp Rs

Ids=Vi*gmexp(-jωt)

Ls S Fig. 3. Twenty-element equivalent circuit of AlGaN/GaN HEMT used in this work.

Table 1 Extracted results (Vgs ¼ 0 V, Vds ¼ 10 V) Extrinsic parameters

Intrinsic parameters

Lg Ls Ld Rg Rd Rs Cdp Cgp Cgsi Cdsi

Rlgs Rlgd Rgd Rds Ri Cds Cgs Cgd gm tau

63.51pH 19.18pH 0.010pH 13.14 O 12.50 O 15.27 O 1.195 fF 30.53 fF 41.31 fF 26.39 fF

21.02 kO 60.12 kO 0.424 O 147.6 O 1.177 O 5.689 fF 492.2 fF 60.75 fF 26.60 mS 20.61 ps


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Table 3 Results of direct extraction, LM, canonical DE and MDE (Vgs ¼ 0 V, Vds ¼ 10 V) Result method

es (%)

Time (min)

eG (%)

Time (min)

Direct extraction LM Canonical DE MDE

35.14 17.93 8.99 5.43

o1 32 21 23

5.72 4.27 1.14 0.94

o1 13 6 6

18 35

16

scalar error by MDE opt gain error by MDE opt scalar error by ICCAPLM gain error by ICCAPLM

30

14 12

error(%)

error(%)

25

Fig. 4. S-parameter. Comparison between measurement (circle) and result of MDE optimization (solid line) and result of LM optimization (dashed) for a 20-element model (Vgs ¼ 0 V, Vds ¼ 10 V, Freq: 50 MHz–40 GHz).

20 15

Extrinsic parameters

Intrinsic parameters

Lg Ls Ld Rg Rd Rs Cdp Cgp Cgsi Cdsi

Rlgs Rlgd Rgd Rds Ri Cds Cgs Cgd gm tau

0–100.68pH 0–6.014pH 0–87.34pH 0–8.652 O 0–10.708 O 0–10.55 O 0–6.156 fF 0–6.156 fF 0–36.2 fF 0–18.466 fF

0–94.812 kO 0–261.28 kO 0–151.23 O 0–637.53 O 0–34.183 O 0–80.27 fF 0–794.2 fF 0–77.335 fF 0–37.234 mS 0–5.518 ps

are shown in Table 3. Direct extraction results are also shown as a comparison. From the comparisons, the following conclusions can be reached: (1) The direct extraction method is adapt to comparatively simple circuit topology. (2) The LM optimization method can improve the result of direct extraction, but the accuracy strongly depends on the initial value that limits its application, especially when the circuit topology is complicated. (3) Both DE and MDE can achieve satisfactory results, and MDE is more effective. In the above example, the obtained Ls, Rg, Rd, Rs, Cgp, Cgsi, Cdsi are out of the presumed upper bounds, but the selfadaptive bound extension process in MDE helps to find their optimal values by the same cost of computational efforts. The lower bounds are strictly restricted, so no unacceptable parameters are obtained. The advantages of MDE optimization can be acquired from Fig. 5, in which the comparison of errors in different bias conditions between MDE and LM is shown. At a wide bias and frequency range (Vgs: 5 to 0 V, Vds: 2–10 V,

10 8 6

10

4

5

2

0

0

-5

Table 2 Recommended ranges of parameters (Vgs ¼ 0 V, Vds ¼ 10 V)

scalar error by MDE opt gain error by MDE opt scalar error by ICCAPLM gain error by ICCAPLM

-4

-3

-2

-1

Vgs(V), at Vds=10V

0

1 2 3 4 5 6 7 8 9 10 11

Vgs(V), at Vds=-2V

Fig. 5. Comparison of scalar (square) and gain (triangle) errors between MDE optimization (solid) and IC-CAP L-M optimization (dashed) at different bias frequencies from 50 MHz to 40 GHz.

frequency: 50–40 GHz), the average scalar error of the 11 bias points is about 8.42% as optimized by MDE compare with 15.36% as done by the LM optimizer. Also, the average gain error by MDE is only 1.64%, which is much less than 5.96% gained by the LM optimizer. Another character that should be described is the flexibility of the MDE-based parameter extraction method. In Table 1, the values of Ls and Ld are quite different from Rs and Rd, which is caused by the trade-off of accuracy and physical meaning in curve fitting and parameter optimization-based extraction methodology. However, if we set Ls ¼ Ld and Rs ¼ Rd, (the model becomes an 18-parameter model), the proposed extraction method can also handle it. In the 11 bias conditions, the scalar errors of 10 of them are within 10%, and all of the scalar errors are within 11%, while by LM optimization-based methodology, all of the results are more than 25%. Therefore, not only does the MDE-based method possesses high accuracy, it can also try to maintain physical meanings by appropriate settings. Example 2. The RF and mixed-signal IC design processes require accurate inductor models that can be included in the circuit simulation along with the entire IC design. Symmetric spiral inductors are needed in differential circuits such as VCO, LNA, and Mixer. Besides, it has been discovered as an easy method for enhancing performances without major changes to current available technologies. However, there is still lack of extensive study on the symmetric inductor with sufficient experimental


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data. In this paper, we have designed 17 symmetrical spiral inductors in single-layer metal to verify the characteristics in the differential mode. The Equivalent circuit model for differential spiral inductors in double-p topology is shown in Fig. 6.

Rdc, Rt can be determined from Re(Zij), Ldc and Lt can be extracted from the slope of o Im(Zij) versus o2 curves according to Eqs. (11, 12). However, there are only two equations but three unknown inductors—Ldc, Lt, M, an additional condition is needed. Here we assume:

The modeling methodology can be divided into two general steps. First, the number of parameters can be reduced according to the equality constraints relationship among them in the low-frequency region. Second, we optimize the parameters in different frequency ranges according to their influences. At the first step, we consider the low-frequency range (below 3 GHz). The conductor skin effect and distributed capacitance of inductors can be neglected at this frequency range, and the equivalent circuit in Fig. 6 can be reduced to Fig. 7 after decoupling. The Z-parameters can be expressed as follows: Rdc þ Rt þ joðLdc þ Lt Þ Rt þ joðLt MÞ Zdc ¼ Rt þ joðLt MÞ Rdc þ Rt þ joðLdc þ Lt Þ

M ¼ K 12 Ldc

We can obtain the following equations under an extremely low-frequency range: ReðZ 11 Þ ¼ Rdc þ Rt

(9)

ReðZ 12 Þ ¼ Rt

(10)

ImðZ 11 Þ ¼ Ldc þ Lt

(11)

ImðZ 12 Þ ¼ Lt M

(12)

Cp

Rp

M

Lp

Lp

Rp

Port1

Port2 Cox

Rs

Ls

Rs

Lt Rsub

Csub

Ls

Cox

Csub

Rt

Rsub

Fig. 6. Equivalent circuit model for differential spiral inductors in doublep topology.

Port1

Port2 Rdc

Ldc + M

Ldc + M

Rdc

Lt - M Rt

Fig. 7. Decoupled equivalent lumped circuit model for the differential inductor after being reduced at low frequency.

(13)

Then we get ImðZ 12 Þ ¼ Lt K 12 L

(14)

Here, we presume K12=0.5, now we have two parameters—Ldc, Lt to be determined by Eqs. (11, 14). The final value of K12 is determined by the optimized process. In fact, Rdc and Ldc are the DC limits of resistor and inductor of the series branches in Fig. 5. We introduce two factors KIp and Krp [23,24]: 1 Rp ¼ K rp Rs ; Rs ¼ Rdc 1 þ (15) K rp Lp ¼ K lp Ls ;

Ls ¼

Ldc 1 þ K lp =ð1 þ K rp Þ2

(16)

So in the low-frequency range, the three parameters Krp, Klp and K12 are obtained according to the minimization of error between the measured and simulated S-parameters. Then all elements Rs, Rp, Ls, Lp, Rt, Lt can be achieved according to Eqs. (9–11, 14–16). Therefore, the parameters are reduced from 11 inter-dependent variables to 7 independent ones. The second step is to optimize the parameters. Since the elements Cp, Cox, Rsub and Csub have little influence on the characteristics at the low frequency, we can adopt twostage optimization. The parameters Krp, Klp and K12 are optimized in the low frequency range from 300 MHz to 3 GHz. Then, all the parasitic parameters Cp, Cox, Rsub and Csub are all determined by the second optimization in the high-frequency range. To demonstrate the modeling methodology, 17 differential spiral inductors based on a standard 0.18 mm 1P6M RF CMOS process has been fabricated. The root mean square (RMS) error between the simulated data and measured data is defined as the optimization goal. In Table 4, the extracted values for the inductor ]7 by MDE are listed as a representative. The presumed ranges of parameters are shown in Table 5. In this example, the recommended ranges are decided empirically. The error of

Table 4 Extracted model parameters for the inductor #7 Element Value

Ls (nH) 1.031

Rs (O) 2.931

Lp (nH) 1.452

Rp (O) 10.43

Element Value

Cp (fF) 22.71

Cox (fF) 61.59

Rsub (O) 823.7

Csub (fF) 135.6

Element Value

Lt (nH) 0.3448

Rt (O) 2.823

K12 0.82


1768 Table 5 Ranges of parameters for inductor ]7 K12 Klp Cp Rsub

0.4–0.95 0.5–2 0.2–200 fF 10–1500 O

Krp

0.5–2

Cox Csub

1–80 fF 10–700 fF

typical evolutionary algorithms, GA and PSO. In addition, the method has characters of generality and efficiency. The proposed method is an improvement and supplement of the current parameter extraction systems, especially for special kind of devices, complicated circuit topology, as well as cases that have many local optimums. Acknowledgments

Ind. exceeding error distribution 100

This work was supported by National Natural Science Foundation Committee of China, No. 90307016. This work was also subsidized by Special Funds for Major State Basic Research Projects, No. 2006CB302705.

S11 S12 S21 S22

80

%

60 40

References

20 0 0

1

2

3

4 5 RMS Error %

6

7

8

9

Fig. 8. Number of inductors (in percentage) exceeding the RMS error of S-parameters among the 17 inductors.

MDE, PSO, and GA are 4.2%, 11.9%, and 24.7%, respectively. The RMS error distributions of S-parameters of all those 17 spirals are plotted in Fig. 8. The total average error of S-parameters is approximately 4.7%, and the maximal RMS error is below 9% for all 17 samples by the proposed method. The close match between simulation results and measurements, as well as between as-extracted parameters and optimized parameters, demonstrates the validity of the present method. For a comparison, parameter extractions based on conventional fitting optimization using LM in IC-CAP, GA, and PSO have been carried out. In GA, the population size and crossover probability are the same as MDE and PSO. The mutation rate in GA is 0.7. In PSO, the inertia weight decreases linearly form 1.2 to 0.2 as the evolution proceeds. The reason is a large inertia weight is favorable to jump out of the local optimums, and a small inertia weight is beneficial for convergence. The acceleration constants c1 and c2 are 2. When the input initial values were set with a large deviation from the extracted values, the iteration cannot converge to the optimized parameters by LM optimization. As for GA and PSO optimization methods, the corresponding RMS deviations for S-parameters are about 26.8% and 14%, respectively. 5. Conclusions This paper presents a parameter extraction method based on the proposed MDE algorithm. High accuracy is the most important feature of the MDE-based method. Practical examples show that MDE possesses advantages over the traditional LM optimization framework and

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