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At microwave frequencies, from about 7 to 60 GHz, inductive irises are very often used as coupling networks between halfwavelength cavities in rectangular waveguides to develop very selective low loss bandpass filters. This is due to the fact that symmetric and asymmetric metal inserts, along with small tuning posts, are very easy to manufacture in large production volume. To facilitate the design and optimization process, a simple and very accurate neural architecture is presented, which is easily translated to a standard electrical equivalent circuit that reproduces in a wide range of iris aperture, thickness and frequency. The proposed new models, although they can be embedded into any commercial microwave software, have been easily implemented into MMICAD.^{®} Comparisons have been made for high order high frequency waveguide halfwave filters, showing an excellent agreement with full threedimensional (3D) electromagnetic HPHFSS^{®} simulations along with computation speeds thousands of times faster.
To the authors' knowledge, none of the current commercial microwave aided design (CAD) programs incorporate models for useful discontinuities in rectangular waveguide, while they have models for the standard TE10 waveguide transmission line. This means that, for a microwave designer, the waveguide world belongs to other kinds of simulators based on hard numerical methods  mode matching and finite elements where the design and optimization cycles are still very long.
In the case of planar structures (microstrip, strip and coplanar lines), the microwave designer has a wide range of electrical models available (sometimes based on electromagnetic simulations) for almost any discontinuity in such transmission media, and the designer can verify or finetune his or her final design through the use of accurate 2D and 2.5D planar electromagnetic simulators. The same concept can be extended to the waveguide world for some useful wellknown discontinuities such as symmetric and asymmetric irises for microwave bandpass filter design shown in Figure 1.

The existing circuit models for waveguide inductive irises, starting from Marcuwitz's work,^{1} are published elsewhere; however, most of them are developed in terms of recursive closed form equations coming from electromagnetic pseudo quasistatic and fullwave approaches. Due to the multimode dispersive nature of these electromagnetic discontinuities, the equations are not easily implemented into commercially available circuit simulators. Furthermore, they are rather tedious and computation intensive, thus preventing an easy filter analysis and optimization process. Their frequency accuracy for a single iris is perhaps sufficient, but when using a high order filter structure, the propagation of the individual errors through the filter gives poor results (for example, bandwidth shift and in band attenuation). The second available solution, the use of full 3D electromagnetic simulators such as HPHFSS,^{®2} is accurate, but unacceptable in computation time (several hours/days) when used for filter design and optimization.
Instead of searching for more precise, and therefore more complicated closed form equations, the idea proposed here is to use simple and accurate neural architectures to fit the scattering parameters obtained by using a precise full 3D simulator for single and double inductive irises in rectangular waveguides. Since the electromagnetic discontinuity of a single or double inductive iris of aperture D and thickness T in a TE10 propagating rectangular waveguide behaves like a lossless symmetrical two port reciprocal network at the reference planes P1 and P2, it is enough to adjust a single two port parameter at the output of the neural network. For microwave filter applications, it is convenient to control the forward scattering parameter S_{21} (easily related to the traditional Z or Y parameters), and furthermore to use the wellknown properties of the scattering matrix to derive the other Sij parameters as
S x (S^{°} )^{T } = I (1)
At this point, it is evident that the input parameters for a possible neural structure should be the physical dimensions of the inductive iris, that is, D and T, along with the waveguide dimensions (A and B) and the frequency of operation. This primary strategy exhibits some important disadvantages because the standard waveguide dimensions are defined for precise frequency bands where the TE10 is the dominant propagating mode, and in a first approach a neural topology should be derived for each waveguide band (which is not a general approach). However, if the scaling properties of the waveguide structures are considered, the normalised iris dimensions D/A and T/A can be used as input parameters, as well as the normalised frequency F/Fc, where Fc is the cutoff frequency of the TE10 mode, as shown in Figure 2.

Furthermore, the range of the input parameters should have some constraints regarding the usual waveguide filter utilisation. The iris aperture could vary from 0 to the maximum aperture A, that is, 0<D/A<1, while a reasonable range for the iris thickness T should be given by 0.01<T/A<0.25. Finally, the normalised frequency band should be 1.2<F/F_{c} <2 in order to avoid unwanted propagation modes. In conclusion, this general strategy uses a single neural architecture for the S_{21} parameter, and three normalised parameters as input data. It should be enough for any given frequency band where this kind of filter is applicable.
THE NEURAL ARCHITECTURE
From an intuitive point of view, a neural network^{3} can be viewed as a parallel distributed processor that exhibits a natural ability for storing experimental knowledge and making it available for ulterior use. This knowledge is acquired through dedicated learning algorithms, along with a weighty interneuron connection. The typical neural networks, MLP and RBF families, normally require a relatively large number of neurons for a close fit to the experimental data. Because the objective is to be highly competitive against the pure electromagnetic simulation, a new SPWL^{4} has been chosen for this particular problem.
The proposed SPWL model is an extension of the wellknown canonical piecewise linear model (PWL) described by Chua.^{5} In its basic formulation, the Canonical PWL model performs any general nonlinear mapping F : R^{M } → R^{N} (M inputs and N outputs) by means of the expression
where
X (M)  =  input vector 
Y (N)  =  output vector 
A (N), B (N*M),  =  fitting vectors 
_{k} (M), C _{k} (N) 


ß_{k}  =  scalar 
< _{k} ,X >  =  inner product 
This model divides the input space into different regions by means of several boundaries implemented by hyperplanes of dimension M1. It then constructs the function approximation by means of a combination of hinging hyperplanes of dimension M. Such hinging hyperplanes are the result of joining two linear hyperplanes over the boundaries defined in the input space.
It can be seen that the expression inside the absolute value function defines the boundaries partitioning the domain space. This function controls the transition between linear regimes and, therefore, the Canonical PWL model inherits some properties from the absolute value function; it is continuous but not derivable along the boundaries. Moreover, the second and higher order derivatives are zero except at the boundaries where they are discontinuous, which is critical for circuit optimization purposes. To overcome this drawback, the substitution of the absolute value function is proposed for a derivable function in order to smooth the joint of hyperplanes at the input space boundaries. Several possibilities exist to smooth the absolute value function allowing, at the same time, a parametric control of the "sharpness" of the transition. The smoothing function is chosen as
where
= parameter that allows the smoothness of the transition to be controlled
The advantage is clear when one looks for the derivative of Equation 3: d/d {LCH ( )} = tanh( ) which is the activation function of a universal approximator such as the MLP. Figure 3 shows a descriptive view of the proposed SPWL model.

MODEL VALIDATION
The above description has been applied to the electromagnetic structures, that is, both symmetric and asymmetric irises. This model provides a smooth and derivable approximation that improves considerably the performance of the Canonical PWL model when it is applied to real microwave devices, mainly in the optimization process. Moreover, it requires a much smaller number of parameters and a lower computation burden than other models commonly used. Extensive full 3D electromagnetic simulations have shown that the proposed architecture, shown in Figure 4, is able to reproduce the two port complex S_{21} parameter for a wide range of input data  (0.01 < T/A < 0.25), (0 < D/A < 1) and (1.2 < F/F_{c} < 2.0), thus covering most applications. In this case, a very good individual iris fit by using a sevenorder SPWL is obtained; the maximum error in S_{21} for any individual iris, when compared with HFSS simulation, is less than 0.02 in module and less than 2° in phase. Figure 5 shows as an example the neural fitting parameters for the symmetric iris case along with a comparison between full 3D electromagnetic simulation and the neural model for a relative large iris thickness (T/A = 0.14) and for various iris apertures D/A as a function of the normalised frequency.


Although it is very easy to show how the proposed method accurately fits the frequency behaviour of an individual inductive iris, when designing high order microwave filters, the propagation of the individual errors could be important, especially for very narrow bandpass filters. This fact is a higher level test of the validity of the approximation. For this reason the neural architecture has been implemented easily into MMICAD^{6} by using its MDL capability along with the flexibility in working with electrical model and local variables. The individual irises are joined by using fundamental TE10 waveguides (available in any simulator). At least for symmetric iris structures, and for these halfwave filters, it is not necessary to take into account high order connecting modes. Up to 21 different multisection Chebyshev/Butterworth bandpass filters in different waveguide bands were tested, always showing very good agreement with full 3D electromagnetic simulations and having a computing simulation time more than 1000 times faster than any conventional analysis. Furthermore, the filter optimization process takes only a few seconds. This is due to the fact that the chosen algorithm is not only very fast but also continuous in its high order derivatives.
Figure 6 shows the general structure for microwave halfwave filters that use double inductive irises in a waveguide environment. For validation purposes, WR22 Kaband waveguide (26.5 to 40.0 GHz) is chosen, where two very different N = 5 (6 iris discontinuities) Chebyshev bandpass filters have been designed and optimized. Filter 1 uses symmetrical irises having moderate (0.3 mm) thickness, with a center frequency at f_{o } = 34.55 GHz and having a fractional bandwidth of 5.5 percent. Conversely, Filter 2 is a very narrow band waveguide filter (2.4 percent fractional bandwidth) centered at f_{o } = 29.25 GHz that uses very thick (0.9 mm) irises. For both cases, all the physical dimensions are shown in the figure. In terms of analysis, HPHFSS means full 3D electromagnetic simulation and SPWL means neural electrical equivalent circuit simulation. Note that the model implementation is extremely robust, even for very narrow filters, and it is difficult to distinguish between the two simulations.

HIGH ORDER TE10 + TE20
IRIS CONNECTION
From a general point of view, a waveguide having P modes should be considered the connecting media between successive discontinuities, that is, the discontinuity should be described as an electrical 2Pport characterised by its generalized multiport scattering matrix S(2Px2P), as shown in Figure 7. Since the structure of a symmetric iris exhibits perfect symmetry, only odd modes can be excited at the discontinuity, that is, the first high order mode to be considered is the TE30. Multimode electromagnetic simulations show that for halfwave waveguide filters that use symmetric iris structures, it is enough to consider only the first connecting mode TE10, as can be seen from the results obtained for the filters using symmetric irises.


Unfortunately, this is not the case for halfwave waveguide filters that use asymmetric iris structures where the above approach is not accurate enough. Due to the nonsymmetrical nature of these discontinuities, a nonnegligible contribution of the TE20 mode along with an insignificant contribution of the remaining high order connecting modes can be expected. Extensive filter simulations corroborate this assertion and the final filter structure is shown in Figure 8.
At this point it should be kept in mind that the generalized scattering matrix for the first two modes of a single iris is a 4x4 matrix having some special properties.^{7} The S_{11 } = S_{33} and S_{13 } = S_{31} elements belong to the propagating mode TE10; they have the matrix properties shown in Equation 1. The terms S_{14 } = S_{41 } = S_{23 } = S_{32} and S_{12 } = S_{21 } = S_{34 } = S_{43} relates the propagating TE10 mode with the evanescent TE20 mode. After manipulation of the scattering properties they can be related in a simple manner, as shown in Equation 4.
Finally, mode matching simulations show that the terms S_{22 } = S_{44} and S_{24 } = S_{42} can be made zero without any loss of accuracy because these elements relate to evanescent modes only.
In conclusion, the only need is to develop an extension of the initial neural network that has as outputs the magnitude and phase of S_{31} , along with S_{14} , for example. The remaining elements of the 4 × 4 scattering matrix can be deduced from the above equations. This SPWL architecture is easily converted into a standard electrical equivalent circuit and implemented into any circuit simulator and then used in the same manner as the element "bend" or "step" in microstrip, for example. The final point is to build an electrical model for the evanescent TE20 waveguide mode; however, this is not a problem because its Z matrix is wellknown in the literature.
As a validation example of the aforementioned theory, Figure 9 shows a comparison for an Xband (6 percent BW) halfwave filter that uses asymmetric iris structures. As shown, it is almost impossible to distinguish between the multimode HFSS simulation and the proposed 4port SPWL approach. The differences are in the range of the mechanical tolerances. However, there is a 1 percent frequency shift when using the simple 2port SPWL approach, which is unacceptable for a 6 percent filter bandwidth, thus confirming the dual mode assumption.

CONCLUSION
A very simple and extremely accurate SPWL neural architecture for inductive iris in electromagnetic structures has been presented. In order to cover the whole microwave range, normalized physical dimensions and frequency have been used as input parameters to the network. Model implementation has been accomplished through the use of standard electrical equivalent circuit structures. Model validation has been achieved for very different high order microwave filter applications, always showing excellent agreement when compared with the wellknown accuracy of a full 3D electromagnetic simulator. Since the neural architecture is continuous in its high order derivatives, the filter optimization process can easily be accomplished. Finally, simulations have shown that the proposed strategy is more than 1000 times faster than any commercially available electromagnetic simulator, thus allowing the microwave engineer to really minimize the microwave filter design process without loss of accuracy. *
References
1. N. Marcuwitz, Waveguide Handbook, McGrawHill.
2. HPHFSS "Full 3D Electromagnetic Simulator and Optimizer," Agilent Technologies Product.
3. Simon Haykin, Neural Networks: A Comprehensive Foundation, Macmillan Publishing Co., IEEE Press, 1994.
4. M. Lázaro, I. Santamaría, C. Pantaleón, A. Mediavilla, A. Tazón and C. Navarro, "Smoothing the Canonical Piecewise Linear Model: An Efficient and Derivable Largesignal Model for MESFET/HEMT Transistors," Accepted for publication in IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications.
5. L.O. Chua and A.C. Deng, "Canonical Piecewiselinear Modeling," IEEE Trans. Circuits Syst., Vol. 33, No. 5, 1986,
pp. 511525.
6. MMICAD is a commercially available Microwave Circuit Analysis and Optimization Software from OPTOTEK Ltd.
7. H. Haskal, "Matrix Description of Waveguide Discontinuities in the Presence of Evanescent Modes," IEEE Transactions on Microwave Theory and Techniques, March 1964, pp. 184188.
A. Mediavilla graduated with honors in 1978 and received his doctor of physics degree in 1984, both from the University of Cantabria, Santander, Spain. From 1980 to 1983 he was ingenieur stagiaire at ThomsonCSF, Corbeville, France. He is currently a professor in the communications engineering department at the University of Cantabria. He has wide experience in analysis and optimization of nonlinear microwave active devices and circuits in both hybrid and monolithic technologies. His current research fields are on active microwave circuits, mainly in the area of nonlinear modeling of high power GaAs devices and their application in largesignal computer circuit design. Antonio Tazón Puente graduated in 1978 and received his doctor of physics degree in 1987, both from the University of Cantabria, Santander, Spain. From 1991 to 1995 he was a professor in the department of electronics at the University of Cantabria, and since 1996 he has been a professor in the department of communication engineering, also of the University of Cantabria. In 1985 and 1986 he carried out stages at the IRCOM department (University of Limoges, France), working in nonlinear modeling and loadpull techniques. He has participated in Spanish and European projects in the nonlinear modeling (Esprit project 6050 MANPOWER) and microwave and millimeterwave communication circuits and systems (Spanish Project PlanSAT, European Project CABSINET). He has carried out research on analysis and optimization of nonlinear microwave active devices and circuits in both hybrid and monolithic technologies. Currently his main research interests are the active microwave circuits, mainly in the area of linear and largesignal modeling and smallsignal intermodulation of GaAs and SiGe devices and their applications in nonlinear computer design. José A. Pereda received his licenciado degree in 1989 and his doctoral degree in 1995, both in physics, from the University of Cantabria, Spain. In 1989 he joined the electronics department at the University of Cantabria, and in 1996 he became an assistant professor in electromagnetism in the communications engineering department. His research interests include electromagnetic field theory and numerical methods for solving electromagnetic problems. Marcelino Lázaro received his telecommunication engineer degree from the University of Cantabria, Spain, in 1996. That same year he joined the communications engineering department at the University of Cantabria, where he is currently pursuing his doctoral degree. His research interests include digital signal processing, nonlinear modeling and neural networks. Carlos Pantaleón received his telecommunication engineer degree and his doctoral degree from the Universidad Politécnica de Madrid (UPM), Spain, in 1990 and 1994, respectively. In 1990 he joined the communications engineering department at the University of Cantabria, Spain, where he is currently an associate professor. His research interests include digital signal processing, nonlinear systems and neural networks. Ignacio Santamaría received his telecommunication engineer degree and his doctoral degree from the Universidad Politécnica de Madrid (UPM), Spain, in 1991 and 1995, respectively. In 1992 he joined the communications engineering department at the University of Cantabria, Spain, where he is currently an associate professor. His research interests include digital signal processing, nonlinear systems and neural networks. 
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