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A Space Mapping Method Allowing Models with Dierent Parameter Rank and Physical Meanings or Coarse and Fine Model Horst Bilzer, Florian Frank and Wolgang Menzel Microwave Techniques, University o Ulm, D-89081 Ulm, Germany w with EADS Ewation GmbH, Wörthstraße 85, D-89077 Ulm, Germany w with Institute or Electronics Engineering, University o Erlangen-Nuremberg, D-91058 Erlangen, Germany Abstract Automated optimization o complex microwave circuits is still a critical task in wadays. This contribution demonstrates two new approaches in space mapping techlogy. One substantial beneit is the improvement o calculation eiciency o the TRASM algorithm by enhancing the multiparameter extraction step. Furthermore, the possibility o mapping coarse and ine models o dierent degrees o reedom, i.e. with dierent parameter rank, and unequal physical parameter meanings has been introduced to the method. Thereore, arbitrary microwave circuits o high complexness with well-matching coarse models can be eiciently and automatically optimized. I. INTRODUCTION In recent years speciication or microwave components have been set harder and more restrictive. For example, requency bands are allocated more densely and the signal to ise ratio has to be higher. This inluences severely the requirements o the subcircuits o the components. To ensure a short development time o these high end circuits resulting in short time to market and cost reduction, at least partially automated optimization o subcircuits is a critical task. However, optimization o complex miniaturized subcircuits, or example microwave ilters with numerous tuning parameter, with local optimization strategies like gradient method or Powell s method is t possible, as they oten converge in a local optimum next to the starting point. I there possibly exists a lot o local optima in an objective unction due to the great number o optimization parameters, the chance to hit the global optimum is very small. Thereore, global optimization strategies have been developed in the last decade to handle complex microwave circuits, which are based upon local optimization or the search o local optima and evaluate the results aterwards. Although these methods guarantee accurate results, the procedure is very time consuming. In this paper new developments to the trust region aggressive space mapping optimization (TRASM) method [1] will be presented. This technique uses two models with dierent accuracy, a coarse and a ine model o the treated circuit. With a mathematical mapping, the parameter o the coarse model can be transormed to the ine model, and vice versa, leading to the same simulation response. Thus, the search o the optimum parameters can be perormed with the coarse model, and the corresponding parameter set can be ound aterwards by transormation to the ine model. Thereore, most o the simulation eort is shited to the coarse model, speeding up the optimization process compared to classical global methods as explained above. In this contribution, new mathematical strategies extend the technique in a way that irstly, the method can be used with coarse models being physically t related to the ine model (e. g. coarse model: LC model; ine model: 2.5D ull wave-model). Secondly, the extended algorithm can handle dierent degrees o reedom o the models, so that the number o optimization parameters o both models do t have to be equal. The mathematical explanation to these enhancements is given in chapter II. The validation o the algorithms is given by experiments in chapter III. Chapter IV concludes the paper and looks out to uture improvements. II. EXTENDED TRUST REGION AGGRESSIVE SPACE MAPPING ALGORITHM A. Overview o the original TRASM-methodology In order to explain the base algorithm, a brie repetition will be given beore introducing the new approaches. A more detailed explanation is given in [2]. Generally, space mapping (SM) methods require two dierent models o an electrical circuit. The irst one is the so called coarse model, which or example consists o LC components or mathematical descriptions o physical eects. The coarse model solver used in the examples in this contribution is PSpice [3]. The second one is the so called ine model, which employs ield theory and is simulated using a 2.5D solver like Sonnet em [4], which is used here. The n c -dimensional parameter vector o the coarse model is deted by x c, and the n -dimensional parameter vector o the ine model is deted by x, respectively. Up to w in all published examples the models had the same parameter rank (n c =n ). Let R c (x c ) dete the coarse model simulation response o x c. Similarly, let R (x ) dete the ine model response o x. The goal o the used TRASM algorithm is to determine a linear approximation o a mapping unction x c = P(x ) so that R (x ) R c (P(x )) ǫ (1) 0-7803-8846-1/05/$20.00 (C) 2005 IEEE
is valid or a bounded region. In this equation ǫ is a given small positive constant and indicates the Euclidean rm. With the approximation o P it is possible to map the optimal parameter set xc o the coarse model to the ine parameter space and thus, get the desired solution o the optimization problem. Thereore, an objective unction g(x ) = P(x ) x c (2) is deined which has to be minimized using a quasi- Newton method. With the linear approximation P(x (k+1) ) = P(x (k) ) + B(k)h (k), (3) where B is the approximation o the Jacobian matrix, equation (2) leads to g(x (k+1) ) = P(x (k) ) + B (k) h (k) xc! = 0, (4) B (k) h (k) = g (k). (5) The approximation o the Jacobian matrix is updated by the classic Broyden ormula [2]. Step A: Initialization Step B: δ (k+1) = δ (k) Step C: Calculation o h (k) Step D: h (k) >δ min Step E: Calculation o x (k+1) n and V = x (k+1) o Step F: multi parameter extraction Step G: success criterion satisied? Step L: Broydon update Step M: (9) satisied? Step N: (10) satisied? Step O: k := k + 1 stop Fig. 1. Increasing o δ (k+1) Decreasing o δ (k+1) Step K: Calculation o a temporary point x (i),t Step J: Step H: V = 1 Step I: g < min V = n New calculation o h (k) Flow chart or the TRASM algorithm. For a better understanding o the used TRASM-method [1], which is an extension o the basic algorithm, a short summarization is given here. The low chart o this algorithm is shown in Fig. 1. A: Initialize B (k), x (k) and δ (k) as described in the ollowing chapter. B: Let δ (k+1) := δ (k). C: Find the value h (k) with h (k) δ (k+1). This is done using the algorithm described in [5]. D: Stop i the Euclidean rm o the new step h (k) is smaller than a constant minimal value h min. E: Calculate the{ new point } x (k+1) = h (k) +x (k) and the new set V = x (k+1). All points in this set are used in the parameter extraction step. F: Evaluate the corresponding point x c (k+1) = P(x (k+1) ) using the single parameter extraction and calculate the new value o the objective unction g (k+1). G: I g (k+1) ulills the success criterion ǫ 1 g (k) g (k+1) g (k) g (k) + B (k) h (k), (6) where or example ǫ 1 0.01, go to step L. H: I there is only one element within V, go to step K. I: Compare the value o the objective unction g (k+1) obtained using V elements with the previous value obtained using V 1 elements ( V detes the cardinality o the set V). I there is signiicant dierence, decrease the trust region size δ (k+1) and go to step C. J: Check whether there are less than n elements within the set V. In this case go to step K. Otherwise obtain an approximation o the Jacobian J o the ine model response, decrease the trust region size and calculate a new step h (k) by solving ( J T J + λe ) h (k) = J T b(k). (7) Aterwards go to step D. K: Obtain a temporary point x (k+1) t solving the system o equations ( ) B (k)t B (k) + λe = h (k) t + x (k+1) by h (k) t = B (k)t g (k+1) (8) and add it to the set V. Go to step F. L: Update the matrix B (k) to B (k+1) using Broyden s update. M: Stop, i N: I the success criterion R (x (k+1) ) R (x (k) ) ε. (9) ǫ 2 g (k) g (k+1) g (k) g (k) + B (k) h (k) (10) is satisied, increase the trust region size δ (k+1). In this equation ǫ 2 detes a small positive number with ǫ 2 0.8. O: Let k := k + 1 and go to step B.
B. New initialization approach or handling parameters with dierent physical meanings in the coarse and the ine model s space Due to dierent rank and physical meaning o the coarse and ine model parameters, several new aspects have to be considered during the initialization phase o the algorithm. First, it is necessary to initialize the Broyden matrix with values representing an approximation o the correlation o the two parameter spaces. This means that the elements o the Broyden matrix must be chosen in a way which enables mapping a parameter vector o the ine model to the parameter vector o the coarse model so that (1) is satisied. For deining the correct values it is necessary to use kwn relations between the two model spaces as mentioned in [6] or to consider the changes o the simulation responses caused by variations o particular parameters. Secondly, the start value o the vector x has to be chosen in a way that R (x (0) ) R c (xc ) is as small as possible. Furthermore, the initial trust region size depends on the number and dimensions o the ine model parameters and has to be chosen in a suitable orm. C. An improved parameter extraction technique The crucial step in every space mapping algorithm is the so called parameter extraction (PE). In this step, the parameter set o the coarse model whose response matches a kwn ine model response is obtained. Thereore, x (k) c = arg min R (x (k) ) R c (x c ) (11) x c D has to be solved, using a local optimization method. In many cases this method leads to wrong results because o local convergence. To improve the results o the parameter extraction step a multiparameter extraction (MPE) concept was proposed [7]. In this extension o the simple PE the two models (coarse and ine) are simultaneously matched at a number o points. Thus, more ine model simulations are needed and thereby, the optimization time is increased. A reduction in required optimization time can be obtained with a new improvement o the MPE, which enables parameter extraction without calculating the gradient o the MPE-objective unction. Thereore, beside the error vectors mentioned in [7], a urther error vector ê = min R c (x c, k + x c (k) ) R (x (k) ), (12) has to be evaluated, where k is the iteration index o the MPE optimization method. This new vector detes the minimal distance between the ine model response at x (k) and the kwn coarse model responses at the points x c, k+ x c (i). I ê < R c (x c ) R (x (k) ) (13) is satisied, x c, k+1 = x c, k + x c is the new point. It minimizes the MPE-objective unction with x c being taken as the result o (12). Thus, the new point x c, k+1 is obtained without analyses o the coarse model and without additionally calculating the gradient o the MPE-objective unction, respectively. Because o this improvement it is possible to reduce the number o required simulations o the coarse model rom V (n c + 1) to V per iteration. III. EXPERIMENTS A. Optimization o a planar MSL rectangular resonator In order to check the unction o the algorithm with a well kwn example, the optimization o a planar microstrip line (MSL) rectangular resonator is considered. The corresponding coarse model simply consists o a capacitor C and a serial inductor L, as shown in the inset o Fig. 2. The values o these lumped elements are s21 in db -60-70 -80 1 C 1 l -90 7.5 8 8.5 9 9.5 10 10.5 Fig. 2. The transmission coeicient o the optimal coarse model ( ), and o the ine model at the starting point ( ) and at the ending point (- - -) o the optimization. The simulation models are shown in the inset. the coarse model s parameters, and the parameter o the ine model is the length o the resonator l. Thus, in this simple example models with dierent parameter rank and dierent physical parameter meanings are used. The goal o the optimization is to move the resonant requency o the ine model to 9.6 GHz. The initial value in ine model s space is x,start = [ 19.7 mm ] or the line length and the optimal parameter vector, which is obtained by optimization o the coarse model is xc = [ 0.07 F 3.85 nh ] T or the LC circuit, respectively. The simulation responses or these values are shown in Fig. 2. The initial value o the Broyden-matrix is B start = ( 1 1 ) T. The optimization is inished ater our iterations, which include seven simulations o the ine model. = [ 17.08 mm ] is the optimal parameter vector in ine model s space. This basic example with low complexity but with dierent parameter rank demonstrates the great advantage o the SM-optimization technique over a direct optimization o the ine model with respect to the number x L 1 2
o time consuming analyses o the 2.5D ullwave model. In comparison, direct optimization would require about twenty or more simulations o the physical structure. B. Optimization o a miniature three resonator suspended stripline ilter. The second example is the design and optimization o a three resonator suspended stripline (SSL) bandpass ilter. This circuit has been chosen to demonstrate the unction o the extended SM technique using models with many parameters and high complexity. The ine model is shown in Fig. 3(a), and the corresponding coarse model in Fig. 3(b). This type o microwave ilter and the used design method has been proposed in [6]. The given design speciications or this ilter are center requency 0 = 8 GHz and bandwidth b = 1 GHz. d 01 w 1 w 1 d 12 w 2 w 2 d 12 d 01 x 2 1 2 x 1 x 1 (a) (b) C 01 C 12 C 1 L 1C 2 L 2 C 1 w 1 w 1 C 12 C 01 Fig. 3. The physical structure (a) and the LC-circuit-model (b) o the three resonator SSL-ilter. ((a): dark gray: ront side metalization, light gray: back side). The optimal coarse parameter vector is calculated using classical ilter theory [8]. With approximated start values or the correlations between the ine model s space and the coarse model s space, it is possible to deine the initial values o the Broyden matrix and the vector x. The scattering parameter amplitudes s 11 and s 21 responses o both models at the points x (0) and xc are shown in Fig. 4(a) and (b), respectively. The optimal parameter values o the ine model are obtained ater 11 simulations o the 2.5D ull wave model, including the MPE. The absolute scattering parameter values at the optimal point x are shown in Fig. 4, too. In contrast, optimizing the ine model directly using a gradient method was t possible because o local minima o the objective unction. IV. CONCLUSION The proposed improvements o the TRASM method oer the possibility to map models with dierent degrees L 1 s11 in db s21 in db 0-10 6 6.5 7 7.5 8 8.5 9 9.5 10 (a) 0-10 -60 6 6.5 7 7.5 8 8.5 9 9.5 10 (b) Fig. 4. The optimal coarse model response ( ) and the ine model responses at the starting point ( ) and the ending point (- - -) o the optimization. o reedom and physical parameter meanings. Thus, nearly arbitrary microwave circuits with equivalent lumped element models can be optimized. The new approaches make simulation more eicient by decreasing the number o necessary simulation steps or the MPE algorithm. In order to validate the operability o these methods, two dierent circuits were considered. A microstrip line resonator with a LC circuit coarse model and a complex SSL-ilter, again with a LC circuit coarse model, has been optimized. Results o excellent accuracy have been achieved in short simulation time compared to the handling o these problems with classical optimization methods. REFERENCES [1] M. H. Bakr, J. W. Bandler, R. M. Biernacki, S. H. Chen, and K. Madsen, A trust region agressive space mapping algorithm or em optimization, IEEE Transactions on Microwave Theory and Techniques, vol. 46,. 12, pp. 2412 2425, Dec. 1998. [2] J. W. Bandler, R. Biernacki, S. Chen, R. Hemmers, and K. Madsen, Electromagnetic optimization exploiting aggressive space mapping, IEEE Transactions on Microwave Theory and Techniques, vol. MTT-43,. 12, pp. 2874 2882, Dec. 1995. [3] OrCAD, Inc., PSpice A/D, version 10.0. [4] Sonnet Sotware, Inc., SonnetLite, version 9.51. [5] J. J. Moré and D. C. Sorensen, Computing a trust region step, SIAM J. Sci. Stat. Comput., vol. 4,. 3, pp. 553 572, Sept. 1983. [6] W. Menzel, A vel miniature suspended stripline ilter, European Microwave Con., Munich, pp. 1047 1050, Oct. 2003. [7] M. H. Bakr, J. W. Bandler, and N. Georgieva, An aggressive approach to parameter extraction, IEEE Transactions on Microwave Theory and Techniques, vol. 47,. 12, pp. 2428 2439, Dec. 1999. [8] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave ilters, impedance-matching networks, and coupling structures. 685 Canton Street, Norwood, Massachusetts 02026: Artech House, INC., 1980.