диафрагмированные волноводные фильтры / 8f885481-2cf9-4d62-bb4f-fea568a0737a

European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS 2000 Barcelona, 11-14 September 2000c ECCOMAS

AUTOMATED DESIGN OF INDUCTIVELY COUPLED RECTANGULAR WAVEGUIDE FILTERS USING SPACE MAPPING OPTIMIZATION

P. Soto , A. Bergner , J.L. G´omez , V.E. Boria , and H. Esteban

Departamento de Comunicaciones Universidad Polit´ecnica de Valencia Camino de Vera s/n, 46022 Valencia, Spain

Email: vboria@dcom.upv.es, web page: http://www.dcom.upv.es/

Key words: Automated Design, Microwave Filters, Optimization, Modal Methods.

Abstract. In this paper, an e cient and full-automated design procedure for waveguide filters based on the aggressive space mapping technique is described. A novel procedure for the optimum choice of the starting point is proposed. The method implemented for the parameter extraction phase of the space mapping procedure is also detailed. The design technique is successfully applied to inductively coupled rectangular waveguide filters with and without tuning elements. A CPU comparative study is also included, thus confirming the e ciency of the new method proposed.

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P. Soto, A. Bergner, J.L. G´omez, V.E. Boria and H. Esteban

1 INTRODUCTION

Over the past years, the extremely fast development of precise electromagnetic (EM) analysis tools, as well as the increase in the computation capabilities of modern computers, have made possible the accurate simulation of very complex waveguide structures in reduced computation times. Nevertheless, from a designer point of view, more e orts should be devoted to the integration of such fast and accurate simulation tools into automated computer-aided design (CAD) frames. This topic is receiving a considerable attention in the recent literature [1].

The design procedure of microwave filters usually starts with the selection of a suitable ideal transfer function which verifies some previously fixed electrical specifications. Then, after having identified each element of the ideal network with a real waveguide component, an optimization procedure must be performed to find the optimum structural parameters. Such optimization procedure has normally been accomplished acting on all the design parameters simultaneously [2]. However, an alternative solution based on a decomposition technique has been recently proposed in [3, 4]. This technique, which exploits the particular nature of the structure under consideration, divides the design process into a number of simple steps with clearly defined objectives. Each resulting step only requires the optimization of a reduced number of design parameters, thus improving the robustness and e ciency of the whole optimization procedure.

It is widely recognized that automated design procedures are mostly prohibitive from the CPU e ort point of view, since they require the intensive use of very accurate EM simulation tools. To overcome such limitation a novel procedure called space mapping [5], and the further improved aggressive space mapping (ASM) technique [6], have been recently proposed. These techniques employ two simulation tools, a precise and high CPU demanding fine model, and a very fast but also inaccurate coarse model. The key feature of the space mapping procedure is that it successfully directs the computational burden of the optimization procedure to the coarse model (optimization space), while preserving the numerical accuracy provided by the fine simulation tool (validation space). Aggressive space mapping has already been proved with interdigital filters [7] and 3-D structures in rectangular waveguide (mitered bends and multistep transformers) [8]. A first attempt on H-plane filters has also been presented in [9], where the coarse and fine models were based on the finite-element method and the mode matching technique, respectively.

In this paper, the aggressive space mapping procedure is applied to the e cient and automated design of waveguide filters. In this case, both the coarse and the fine model use the same simulation tool, a modal method based on the generalized admittance matrix (GAM) representation proposed in [10], although with a di erent number of modes in the equivalent network (smaller number of modes in the coarse model, and higher in the fine model). Proceeding in this way, the accuracy required in the design procedure is preserved, while e ciency aspects are highly improved.

The new design procedure proposed is successfully applied to the automated design

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P. Soto, A. Bergner, J.L. G´omez, V.E. Boria and H. Esteban

of inductively coupled rectangular waveguide filters with and without tuning elements. These structures are of great interest in several practical applications, such as communication satellites, radio links, and new wireless communication systems [11, 12, 13]. Two band-pass inductively coupled filters, with electrical responses centered at 11 and 13 GHz, have been designed with and without tuning elements. The results obtained reveals both the numerical e ciency and the accuracy of the full automated design method proposed.

The paper is organized as follows. In Section 2 the bases of the aggressive space mapping procedure are briefly outlined, and a detailed explanation of the models used in the optimization space (OS) and the validation space (VS) is also included. The section ends with a discussion on the optimization methods proposed for the parameter extraction phases of the ASM procedure. The application of the basic theory detailed in Section 2 to the particular cases of inductively coupled rectangular waveguide filters with and without tuning elements is presented in Section 3. We describe the strategy employed for obtaining the starting point required by the initial OS optimization, and also the segmentation technique used to improve the robustness and e ciency of the OS parameter extraction phase. In Section 4, the design strategies proposed in this paper are successfully applied to several inductively coupled rectangular waveguide filters. First, two simple inductive structures without tuning elements are considered. Next, the same electrical band-pass responses have been automatically recovered with a basic inductive geometry including now tuning elements. Finally, Section 5 contains some concluding remarks.

2 THEORY

In this section, the basic theory required to implement the new automated design procedure of waveguide filters is included. First, a brief overview of the ASM strategy is o ered. Next, the modal analysis tool used for the coarse and the fine model is described. Finally, the optimization methods required within the ASM procedure are detailed.

2.1Aggressive Space Mapping

In order to describe the electrical behaviour of the structures to be designed, the space mapping technique [5, 6] makes use of two models in two di erent spaces: a highly e cient but inaccurate coarse model in the OS, and a very accurate but slower fine model in the VS. It is assumed that exists a mapping P between both spaces of design parameters, which relates points xos in the OS with points xem in the VS

such that they provide very similar responses in their respective modeling spaces.

It is also assumed that the transformation P is one-to-one defined within some wide enough region containing the final solution. The procedure of obtaining the only xos point

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P. Soto, A. Bergner, J.L. G´omez, V.E. Boria and H. Esteban

in the OS related by P with a previously given xem point in the VS is called parameter extraction.

Let xos be the coarse solution in the OS that provides a transfer function that satisfies the original electrical specifications in some optimal way. Aggressive space mapping [6] reduces the design process in the validation space to solve the following multidimensional nonlinear equation

To solve (2), a quasi-Newton iteration is proposed in [6]. Such approach exploits the Broyden formula (see [14]) to update the approximation to the Jacobian matrix related to each iteration. The result is that the main computational e ort of the design procedure is transferred from the accurate but slow validation space to the parameter extraction phases performed in a highly e cient optimization space.

The most time consuming step of the space mapping procedure is to find the structure in the OS that provides the response which best agrees with the optimal one. As a result, a key point in space mapping relies on the determination of the coarse optimal solution xos. The di culty of this first optimization in the OS can be dramatically reduced if an initial starting guess of xos is available.

2.2Numerical modeling in the OS and VS

For both the coarse and the fine model, an e cient multimode simulation tool based on the GAM representation has been employed. This full-wave analysis tool decomposes the structure under consideration into planar waveguide junctions and the uniform waveguide sections interconnecting them.

The admittance representation of the planar junctions is obtained following the method based on transmission line theory described in [10]. As a result, the admittance matrix elements take the following form

vector mode functions of the p-th mode in region δ (δ = 1 for the waveguide with bigger cross section or δ = 2 for the smaller waveguide in the junction), βp(δ) and Y0(pδ) represent the propagation constant and the characteristic admittance of the p-th mode in region δ, lref is a reference length in the bigger waveguide, and δm,n stands for the Kronecker

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P. Soto, A. Bergner, J.L. G´omez, V.E. Boria and H. Esteban

delta. All the TE and TM modes in (3)-(5) are sorted according to their increasing cuto wavenumber.

As it is evident from equations (3)-(5), the main computational e ort comes from the evaluation of the Ym,n(2,2) elements, because they involve series at each frequency point. This computational e ort can be dramatically reduced following the frequency dependence extraction technique reported in [15].

The expressions of the admittance matrix elements (3)-(5) also reveals that the proposed analysis technique requires to know the modal spectrum of both waveguides involved in any planar junction, as well as the modal coupling coe cients between these two sets of modes. This modal information can be obtained analytically for waveguides with standard cross section, such as the rectangular ones. In the case of waveguides with arbitrary shaped cross sections, like the ridge waveguides present in the inductively coupled filters with tuning elements, the modes and coupling integrals required are obtained e ciently implementing the BI-RME method fully described in [13, 16, 17].

The sections of uniform waveguide connecting the planar waveguide junctions of the structure are also characterized in terms of a GAM representation. According to [18], the admittance parameters are given by the well known expressions

where now lwg means the waveguide section length.

After cascading the admittance matrices which represent the basic building blocks of the structure considered, and applying the corresponding load conditions, a banded linear system is finally obtained. To solve such system, a very e cient iterative technique which fully exploits its banded nature has been implemented [19].

The accuracy and e ciency of the EM simulator in both the validation and optimization space are basically adjusted by a suitable choice of the number of modes in the equivalent network representation of the structure. In this way, the fine simulator takes into account all the modes required to obtain very precise results. On the other hand, the coarse analysis tool only considers a very reduced set of modes, thus increasing the e - ciency of the simulator at the expense of a lower numerical accuracy. Moreover, the CPU requirements and accuracy of the software used to model the arbitrarily shaped waveguides (BI-RME) have also been reduced. As a result, an extremely fast coarse simulator is obtained.

2.3Optimization methods

The parameter extraction phases performed on the OS, as well as the initial search of the coarse optimal solution xos, always imply an optimization procedure that provides the structure whose coarse model response agrees with a given objective function. To

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P. Soto, A. Bergner, J.L. G´omez, V.E. Boria and H. Esteban

accomplish such optimization task, we propose a combination of three di erent techniques: a general decomposition strategy, a direct search method and a gradient based algorithm.

Many passive microwave structures can be split up into several cascaded parts, for instance multi-section filters or multiplexers. The decomposition technique described in [3] fully exploits this property by dividing the design procedure into a number of simple steps. At each step, an additional part of the final structure is added to the design process, and the objective response of the resulting structure is computed. The goal of each design step then consists on obtaining the values of the physical parameters related to the new part that provide the objective ideal response. As a result, a very limited number of physical parameters must be considered at each optimization stage, thus improving both the robustness and e ciency of the whole optimization procedure.

It is a well-known fact that gradient-based optimization methods are only e cient and reliable close to the optimal solution. From a far starting point it is more convenient to employ a direct search strategy [20]. In this case, a slightly modified version of the multidimensional coordinate rotation algorithm proposed in [21] has been used as direct search method. It essentially consists on a multidimensional encapsulating strategy that applies the Gram-Schmidt orthogonalization procedure (see details in [20]) to align the movement axes with the last direction of maximum descent. To determine such direction, an improved version of the one-dimensional Step-Wise (SW) algorithm described in [4] is applied successively to each one of the current coordinate axis.

In the original version of the SW method, a movement of the considered design parameter is performed in the descent direction of the error function, and the step length is only reduced when a change in the descent direction of the error function is produced. The implemented version of the SW algorithm only performs the referred movement if a decrease in the value of the error function is also observed. If this is not the case, the step length is reduced and a new movement in the correct direction is tried. The SW algorithm stops when the step length is lesser than a prescribed value.

Once a point close enough to the optimum solution is obtained, a gradient-based strategy becomes more appropriate. The Broyden-Fletcher-Goldfarb-Shanno (BFGS) method [22] is generally recognized to be one of the most powerful gradient-based algorithms. This method is closely related to Davidon’s variable metric algorithm, and successfully combines some of the more desirable features of the steepest descent algorithm and the

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P. Soto, A. Bergner, J.L. G´omez, V.E. Boria and H. Esteban

In (9), Uj is the gradient of the error function to be minimized, and H−1j is the j-th approximation to the inverse of the Hessian matrix of the error function. Such inverse approximation can be iteratively obtained using the update formulas compiled in [22]. With regard to the movement parameter αj , an appropriate value must be chosen in order to decrease su ciently the value of the error function. This value has been obtained applying the robust line search and backtracing algorithm fully described in [22].

3DESIGN STRATEGIES OF INDUCTIVELY COUPLED RECTANGULAR WAVEGUIDE FILTERS

This section explains how the ASM can be e ectively applied to the automated design of inductive filters with and without tuning elements. First, we introduce the ideal network that provides the objective response of the filters to be designed. Based on such ideal network, a novel procedure to determine a good starting point for the initial optimization in the OS space is next presented. Finally, for both filters with and without tuning elements, a segmentation strategy is explained for the robust and e cient optimization stages that must be performed in the OS space.

3.1Ideal network and starting point choice

The design of a Chebyshev band-pass filter starts with the selection of an ideal transfer function which satisfies some electrical specifications. To obtain such ideal transfer function, the classical procedure fully described in [23] can be followed. First, an ideal equivalent network composed of both impedance inverters and half-wavelength transmission lines is proposed, which can be seen in Figure 1.

Figure 1: Ideal equivalent network of a four-pole symmetric Chebyshev filter.

The final expressions for the impedance inverters are easily computed in the following way

ZIN X1 ω

k0,1 = (10) g0 g1

Xi Xi+1

ki,i+1 = ω (11) gi gi+1

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P. Soto, A. Bergner, J.L. G´omez, V.E. Boria and H. Esteban

where Xi is the slope parameters of the i-th half wavelength resonator, namely

gi are the Chebyshev low-pass prototype elements derived in [23], λgi means the waveguide wavelength of the i-th resonator, λ0 is the free-space wavelength, and Z0i, ZIN and ZOUT are the characteristic impedance of the i-th resonator, the input waveguide and the output waveguide, respectively. Finally, ω represents the fractional bandwidth of the filter, defined as indicated next

where fc1 and fc2 are the lower and upper cut-o frequencies of the filter, and f0 stands for the resonant frequency.

Once the ideal network is available, a single system of equations can be obtained by assembling the impedance or admitance matrix representing each element of the ideal structure. The solution of this system provides the ideal transfer function of the Chebyshev filter, which will be the objective function to be recovered with the initial coarse optimal solution (xos).

The ideal equivalent network representation in terms of impedance inverters and halfwavelength resonators is appropriate only for band-pass filters of narrow or moderate fractional bandwidths (lower than 3%), but in such cases it takes into account e ectively the dispersive behaviour of waveguide components. As a result, the ideal transfer function can be easily recovered with a real structure by substituting the two basic building blocks, i.e. resonators and impedance inverters, for specific waveguide elements.

The ideal resonators can be easily replaced by half-wavelength sections of uniform waveguide or by an equivalent waveguide of lower length with a tuning element [12]. On the other hand, a coupling window with or without tuning element terminated with two waveguide sections of appropriate length can be used to model the electrical behaviour of an impedance inverter on a moderate frequency range. In fact, under fundamental-mode incidence, the normalized inverter parameter K of the proposed real structure can be determined from its reflection coe cient S11 as follows

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P. Soto, A. Bergner, J.L. G´omez, V.E. Boria and H. Esteban

Output Waveguide Fourth section

of the Filter

Third section of the Filter

Second section of the Filter

l1

Figure 2: A four-pole inductively coupled rectangular waveguide filter without tuning elements.

and the lengths of the input and output terminating waveguides must be adjusted to obtain a π radians phase for both the S11 and S22 scattering parameters, since the normalized inverter parameters (K) required in the filters considered are always smaller than 1. These lengths must be added to the total length of the input and output waveguide resonators connected to the real impedance inverter.

In section 2.1, we pointed out the paramount importance of a suitable choice of the starting point required to perform the initial OS optimization that provides xos. For the particular case of inductively coupled waveguide filters without tuning elements (see Figure 2), an initial guess for the optimal structure in the OS space can be simply obtained by using the aforementioned equivalences between ideal network elements and real waveguide components. To compute the scattering parameters of the coupling window considering an extremely low number of modes, the fundamental-mode equivalent circuit of an inductive window of finite thickness detailed in [24] can be employed.

With regard to the more complex case of inductive waveguide filters including tuning elements (see Figure 3), the choice of a suitable starting point needed to perform the first OS optimization becomes harder, and requires the use of the coarse analysis tool to compute the reflection coe cient S11 of the real waveguide components. The depth of the tuning elements included in the coupling windows can be determined again making use of (15). Since a major penetration depth of the tuning element directly implies a higher coupling value (i.e. inverter parameter K) [12], this procedure has been automated implementing the Brent’s unidimensional root finding method [22]. Next, the penetration depths of the tuning elements placed in the cavities must be determined. To obtain such penetration depths, we must simulate the electrical response of each cavity with their corresponding input and output coupling windows (including also the appropriate tuning elements previously determined), and try to recover the central frequency of the filter.

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P. Soto, A. Bergner, J.L. G´omez, V.E. Boria and H. Esteban

Output Waveguide

Fourth section of the Filter

Third section of the Filter

Second section of the Filter

Resonant Cavity

Coupling Window

Input Waveguide

Figure 3: A four-pole inductively coupled rectangular waveguide filter including tuning elements.

In this case, a higher penetration depth of the tuning element of the cavity produces a decrease in the central frequency of the 1-pole resonator electrical response. Therefore, following again the Brent’s formulation, the required penetration depths of the cavity tuning elements can be automatically determined. Unfortunately, there are mutual coupling between the several tuning elements of the structure which are not considered, thus producing a starting point which can be slightly deviated from the desired xos. Nevertheless, such mutual couplings only represent a second order e ect, and the resulting starting point is good enough to e ciently perform the first optimization in the OS space.

3.2Design strategy of inductively coupled rectangular waveguide filters without tuning elements

The ASM design procedure of waveguide structures always involves a sequence of optimizations in the OS, which exploits the e ciency of the non-accurate model. The initial step of any optimization procedure starts with the choice of a suitable starting point. The method described in Section 3.1 can be employed to determine such point for the initial optimization performed in the OS to find the optimal coarse solution xos. On the other hand, the ASM technique guarantees that xos is the best initial guess for the solution of the remaining OS optimizations.

If an excellent starting point is available and the number of sections of the filter is very low, the optimization procedure can be accomplished e ciently acting on all the design parameters at the same time. In such case, the multidimensional coordinate rotation algorithm followed by the BFGS gradient-based method, both explained in Section 2.3, can be successfully employed.

On the other hand, a worse starting point or an structure involving a large number of design parameters require the use of a more complex decomposition strategy as the one

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