![\"Fig.1](\"https:\/\/butis-m.ru\/wp-content\/uploads\/2014\/04\/Pic681.jpg\")
<\/a>Fig.1 Two-grid representation of IDT (a) and BE (b)<\/p><\/div>\n
For such a layout representation there are three basic\u00a0topology elements (BE), shown in Fig.1-b, to form a filter.\u00a0These elements are strips connected to upper or lower bars\u00a0of transducers, and free space (or gap). The width of the\u00a0BE is a unit of the width of electrodes and gaps. The\u00a0length of a strip is changed with the quantization step\u00a0equal to the spatial period along the Y-axis. So each\u00a0electrode can be described by several adjoining strips with\u00a0the same length, and gaps can be described as the free\u00a0space elements.<\/p>\n
The synthesis routine includes the analysis of arbitrary\u00a0combinations of the basic elements and the selection of\u00a0structure, which gives a reasonable optimization criteria.\u00a0Normally the number of simulation variants is N=3Nx\u00a0(Ny+1)Nx, where Nx is the number of BE along X-axis\u00a0within the range for optimization, Ny is the number of\u00a0layers along the filter aperture.<\/p>\n
It is obvious that the optimization of a filter as\u00a0described above is an unreal task because of the huge\u00a0number of variants. However, the number can be\u00a0dramatically reduced for practical application by structure\u00a0restrictions and design approaches.<\/p>\n
The fragment length for structure variations is much\u00a0shorter than a transducer length. As a rule the fragment\u00a0does not exceed the wave length. Structure constraints that\u00a0can be readily applied are as follows:<\/p>\n
- \n
- Variants including neighbor strips connected to bars with opposite polarity are excluded because in this case the transducer is shorted;<\/li>\n
- An electrode is composed of a few elementary strips connected to a common bar, a gap is the sum of free space elements.<\/li>\n
- Maximal and minimal widths of electrodes and gaps are limited. Restrictions of widths are determined by the \u00a0limitation of fabrication technology and SAW properties.<\/li>\n
- Simulation time also depends on the model, the software structure and the optimization algorithm.<\/li>\n<\/ul>\n
THE MODEL<\/h2>\n
Most phenomenological models such as COM or P\u00a0matrix are well suited to periodic or near periodic\u00a0structures [4]. Because electrode structures synthesized by\u00a0this TDG method are not fixed, using these models will\u00a0cause difficulties. Instead, equivalent network model [5] is\u00a0more suitable for this application. The advantage of this\u00a0model is its flexibility with variations of finger structures\u00a0because parameters of cells are determined by edge\u00a0positions of electrodes.<\/p>\n
The basic cell of a transducer is shown in Fig.2. It\u00a0extends between centers of neighboring fingers and
\nincludes three elements connected in the electrical circuit\u00a0by transformers r. Every cell can be described by the 3x3\u00a0matrix as follows:<\/p>\n![\"Pic69\"](\"https:\/\/butis-m.ru\/wp-content\/uploads\/2014\/04\/Pic691.jpg\")
<\/a><\/p>\nwhere Z=Zo or Zm are the acoustic impedances for free or\u00a0metalized surface, respectively; L v i \u03b8 =\u03c9 \/ is the\u00a0angle shift under wave propagation in the cell.<\/p>\n
![\"Fig.2](\"https:\/\/butis-m.ru\/wp-content\/uploads\/2014\/04\/Pic701.jpg\")
<\/a>Fig.2 Equivalent circuit model representation<\/p><\/div>\n
The shunt susceptance B superposing the electrode\u00a0edges is a complex number. It describes the second order\u00a0effects such as propagation loss Re(B) and energy storage\u00a0Im(B). These elements can incorporate asymmetry\u00a0associated with NSPUDT orientation. The static\u00a0capacitance per period is modeled by Cp, and
\nelectromechanical coupling is embodied in the ideal\u00a0transformer with turns ratio r. The turns ratio r depends on\u00a0charge distribution on the electrodes. Charge distribution\u00a0is determined by solving the electrostatic problem for the\u00a0arbitrary electrode configuration.<\/p>\nIn order to improve the simulation accuracy, physical\u00a0parameters for a given material and film properties\u00a0(thickness, pitch ratio etc.) were obtained experimentally\u00a0using specially designed test devices.<\/p>\n
Acoustic models of transducers are calculated by\u00a0serial connections of basic cells. The 3-port matrix\u00a0representation for a filter is then transformed to 2-port and\u00a0combined with matrices of matching circuits and electrical\u00a0stray elements in order to generate the filter model as a\u00a0whole. Matrices of separate transducers can be used for the\u00a0analysis of these structure elements under the synthesis\u00a0routine.<\/p>\n
SYNTHESIS ROUTINE<\/h2>\n
The synthesis algorithm is based on step-by-step\u00a0approaching of current responses and parameters to the
\nspecifications. In the optimization routine, the fragment is\u00a0applied to different parts of transducers in accordance with\u00a0a scanning algorithm.<\/p>\nAs initial inputs, the designer should determine the\u00a0substrate material and the model coefficients for such a\u00a0material, the lengths of transducers and gratings, the\u00a0distances between these structure units, the \u00a0fragment\u00a0length and spatial periods of grids along X and Y\u00a0directions. Design constraints and optimization criteria\u00a0should also be set to start with.<\/p>\n
The synthesis algorithm developed is based on the\u00a0published technique used for the synthesis of RSPUDT
\nfilters [7]. Responses of the synthesized filter are\u00a0simulated for every current structure variants and\u00a0compared with the specification of the filter. The error Eg\u00a0between current response and specification is estimated as\u00a0(1):<\/p>\n![\"Pic71\"](\"https:\/\/butis-m.ru\/wp-content\/uploads\/2014\/04\/Pic711.jpg\")
<\/a><\/p>\nwhere Rs\u00a0k(fi) is the value of amplitude response at\u00a0frequency fi for the k frequency range, Rr\u00a0k is the required\u00a0value for the range, Ps\u00a0l and Pr\u00a0l is the simulation and\u00a0required values for l parameter, Ak,l is the weighting\u00a0coefficients.<\/p>\n
The optimization requires Eg to be minimized. This \u00a0criterion estimates the accuracy of a filter as a whole. In\u00a0this case, values of filter parameters are calculated as the\u00a0sum of un-equivalent ones of separate transducers, but\u00a0parameters of transducers can differ significantly.<\/p>\n
The design adequacy can be improved if qualities of\u00a0transducers are estimated by the optimization routine. In\u00a0this case the error E, which has to be minimized, is\u00a0represented as E = Eg + Et1 + Et2, where Et1 and Et2 are\u00a0errors of transducers simulated in accordance with (1), but\u00a0for modified requirements and weighting coefficients.<\/p>\n
Based on this approach, the optimization software is\u00a0developed. Depending on the requirement, the synthesis\u00a0can be realized for tuned or un-tuned filter. Optimal values\u00a0of matching elements are simulated automatically for\u00a0current transducer impedances. Parasitic elements are\u00a0included in the model as well.<\/p>\n
DESIGN APPROACHES<\/h2>\n
As mentioned above, a general optimization and\u00a0design technique is limited by speed and robustness of the\u00a0simulations. This method can be applied to practical\u00a0designs or for the improvement of existing SAW filters.\u00a0Some practical recommendations and design approaches\u00a0are outlined as below.<\/p>\n
- \n
- Simulation time will be reduced if the optimization\u00a0procedure is independently applied to X and Y directions.\u00a0At the first synthesis stage, the optimization of the finger\u00a0structure along X-axis is generated. When the minimal\u00a0error is reached, the routine for finger length optimization\u00a0can start. Additional apodization mainly allows the\u00a0improvement of far out-of-band rejection and suppression\u00a0of parasitic modes.<\/li>\n
- An effective approach is to combine the synthesis\u00a0method developed for RSPUDT filters with that of TDG\u00a0method. In this case the preliminary synthesis is first done\u00a0for fixed SPUDT cell structures such as EWC, DART,\u00a0TES, etc. When the response obtained comes close to the\u00a0optimal target, the transducer representations can be\u00a0transformed into a format corresponding to the grid pitch,\u00a0which can be divided into widths of fingers and gaps of\u00a0the SPUDT cells. These data can be used as an initial\u00a0approximation for the final optimization using TDG\u00a0algorithm.<\/li>\n
- The optimization speed for gratings is much faster than\u00a0that for transducers, as only variations of reflectors are\u00a0accounted for grating. If a filter structure is determined,\u00a0various optimization algorithms can be applied to different\u00a0structure units in order to reduce total simulation time. The\u00a0generalized topology of a SAW filter used in the software\u00a0is shown in Fig.3.<\/li>\n<\/ul>\n
![\"Fig.3](\"https:\/\/butis-m.ru\/wp-content\/uploads\/2014\/04\/Pic721.jpg\")
<\/a>Fig.3 The Generalized topology of a SAW filter<\/p><\/div>\n
RESULTS<\/h2>\n
Flexibility and effectiveness of the TDG method can\u00a0be shown by applying the method to design different types\u00a0of SAW filters, including filters on substrates that provide\u00a0natural directivity.<\/p>\n
A. Filters on langasite:<\/h3>\n
Some results for advanced filter designs on langasite\u00a0with low insertion loss were published last year [6]. These\u00a0filters were designed with the optimization approach \u00a0applied for X-direction only. The grid pitches were 1\/8\u03bb. The use of TDG algorithm and shorter grid pitch (1\/16\u03bb) allows the improvement of insertion loss and rejection.<\/p>\n
The responses of an RSPUDT filter on LGS orientation yxlt\/48.5o\/26.6o are shown in Fig.4. Film thickness is 0.45 um. The minimal width of fingers and gaps is 1\/8\u03bb. 1dB bandwidth is 1.5 MHz (i.e., 1%), out-ofband rejection is higher than 45 dB and insertion loss is about 6.2 dB. Note the response of the unturned filter (dotted plot) corresponds to the typical response of anRSPUDT filter.<\/p>\n
![\"Fig.4](\"https:\/\/butis-m.ru\/wp-content\/uploads\/2014\/04\/Pic731.jpg\")
<\/a>Fig.4 Responses of RSPUDT filter on langasite: solid line \u2013 tuned filter; dash line \u2013 unturned one<\/p><\/div>\n
B. DMS filters:<\/h3>\n
The problem of optimization of low-loss DMS filters is practical because this filter type is widely used for RF applications [7]. TDG was used in order to estimate the adaptability of the method for the synthesis of such filters. The basic structure of 1-2 mode units (Fig.3) was used as an example. In order to improve the response of a typical structure of 4-pole DMS filter, modeling and synthesis algorithm were adapted for the optimization of responses of two units serially connected. The optimization criterion was applied to the total responses of the pair of identical filters. As an illustration of the capability of this synthesis method, the initial and final responses of the 4-pole DMS filters on YXl-36o LiTaO3 are shown in Fig.5. The width of the basic elements used in the synthesis was 1\/16\u03bb.<\/p>\n