![\"Fig.1](\"https:\/\/butis-m.ru\/wp-content\/uploads\/2014\/05\/Pic39.jpg\")
<\/a>Fig.1 Transformation of split-finger structure (b)
to regular DWSF structures (a and c)<\/p><\/div>\n
MODELING AND SYNTHESIS ROUTINE<\/h2>\n
Smith\u2019s equivalent circuit model is used for RSPUDT\u00a0analysis and synthesis. The model is based on a\u00a0representation of topological elements (finger or gap) by a\u00a06-pole electro-acoustical equivalent circuit. This allows\u00a0modeling of arbitrary structures of transducer cells in a\u00a0synthesis routine. An equivalent circuit can include the\u00a0modeling of a variety of second order effects such as\u00a0reflection, resistance, propagation loss and storage energy\u00a0effect.<\/p>\n
The accuracy of simulation depends on the accurate\u00a0description of these parameters and their dependency on\u00a0relative metal thickness and frequency. The model we used \u00a0included the measurement results of test structures and\u00a0simulations adjusted to experimental results.\u00a0The synthesis routine includes the following steps:\u00a0local variation of a section of the transducer structure, and\u00a0the simulation of the frequency response and the\u00a0comparison with the filter requirement. As a criterion the\u00a0general error is estimated. The error is simulated as: The first part corresponds to out-of-band attenuation\u00a0of amplitude response, the second part is for passband\u00a0amplitude response and the third part is for other\u00a0parameters such as insertion loss, phase or group delay\u00a0deviation, etc. Examples of the simulation of errors for outof-\u00a0band and passband amplitude responses are shown in\u00a0Fig.2. Errors for both cases are shown as the grey areas.\u00a0Only the response points that are above the required levels\u00a0for rejection band and the response points that are below\u00a0the required levels for passband amplitude are included in\u00a0the general error.<\/p>\n Fig.2 Error simulation for rejection (a) and passband (b)<\/p><\/div>\n Variations of a transducer structure are done within a\u00a0section. It includes all allowed topological variants of\u00a0DWSF cells. For every variant, frequency response for the\u00a0matched filter is simulated and the error Eg is estimated.\u00a0The error is compared with an error Ego, which\u00a0corresponds to a base transducer structure. If the current\u00a0structure variant provides a error Eg smaller than Ego thecurrent structure becomes a new base structure. Then the\u00a0following section is optimized using the same algorithm.\u00a0By varying weights Ak,l, a designer is able to operate the\u00a0synthesis routine and finalize the design. A successful\u00a0choice of the initial section can affect the speed and the\u00a0final result of the synthesis.<\/p>\n It should be noted that the second transducer is fixed\u00a0under the optimization. When the synthesis of the first\u00a0transducer is done the algorithm is applied to the second\u00a0transducer or grating. The procedure can be repeated\u00a0several times until the minimum error is reached.<\/p>\n Local minimum value and design quality of RSPUDT\u00a0filter depend on the accurate choice and description of\u00a0optimization parameters and their weights and initial\u00a0approximation of the filter, e.g., length of transducer and\u00a0grating, metal thickness etc., which can be estimated from\u00a0other simple synthesis programs or by the designer\u2019s\u00a0experience.<\/p>\n The estimation of simulated responses is then done for\u00a0the matched filter. Because the transducer structure is\u00a0changed with each iteration, matching circuits providing an\u00a0optimal match should be changed accordingly. The\u00a0algorithm determining these matching circuits includes the\u00a0estimation of the average impedance of transducer within\u00a0the filter passband and the simulation of values of\u00a0matching components. These matching circuits provide complex-conjugate impedance for the current transducer structure and account for source and load impedances and certain parasitic elements.<\/p>\n The optimization routine is applied to the synthesis of RSPUDT filters based on fixed DWSF structures as in the first step of the synthesis procedure.<\/p>\n The capability of a typical synthesis algorithm is\u00a0limited because a fixed SPUDT structure is not flexible\u00a0enough. In order to reduce the general error and to improve\u00a0the design result a new synthesis approach was proposed. This approach is based on representing a transducer cell as\u00a0a grid including topological elements having fixed width. These base elements can be either gaps or fingers. Polarities of the fingers can be arbitrary.<\/p>\n Such grids with corresponding element width of base elements can describe every SPUDT structure. The maximum grid width for a SPUDT is equaled to the minimal common multiple for the width of topological elements of the structure. For instance, the maximum grid width for DART-SPUDT is 1\/8 of wavelength and a reflector can be represented as 3 identical base element positioned side by side. The maximum grid width for Hanma-Hunsinger structure is 1\/16 of wavelength. A transducer structure, synthesized for a fixed DWSF, is transformed into the grid description and used as the initial structure for further optimization. Such a two step approach allows minimizing simulation time needed for DWSF-RSPUDT structure synthesis. In order to control a topology of a synthesized RSPUDT, a designer is able to set some limitations, e.g., maxima of finger and gap width can be limited.<\/p>\n As simulation shown, the adjustment approach can significantly reduce the error Eg achieved by a fixed DWSF structure under RSPUDT filter synthesis.<\/p>\n Final DWSF structures comprise electrodes with \u00a0irregular placement and width of fingers and gaps. In order to simulate filter response more accurately, the task of charge distribution should be solved for these electrode structures and included in the synthesis algorithm.<\/p>\n \u00a0As an example, simulated response for RSPUDT with fixed DWSF structure and response synthesized by the additional adjustment procedure are shown in Fig.3.<\/p>\n Fig.3 Simulated responses for fixed DWSR (dashed line) Minimal general errors Eg were achieved for both RSPUDT filters. The simulation example shows that the application of the adjustment procedure allows to achieve better results for both out-of-band rejection and insertion loss.<\/p>\n IF RSPUDT filters based on DWSF were designed both on quartz and 112o-LiTaO3 substrates. The experimental response of a 71 MHz filter is shown in Fig.4, with a bandwidth of about 1.1 MHz. The filter is based on H-H RSPUDT structure, packaged in 13.3 x 6.5 mm SMD package, and has a insertion loss of 10.3 dB and out-of-band rejection of more than 40 dB. An identical filter was designed as DART-RSPUDT. Insertion loss of the DART-RSPUDT is about 13.8 dB, i.e., about 3.5 dB higher than that of the equivalent H-H RSPUDT filter.<\/p>\n Fig.4. Response of a 71MHz H-H RSPUDT filter on quartz<\/p><\/div>\n Most of narrow band RSPUDT filters are designed on quartz substrates. Because of the large value of the storage\u00a0energy effect, central frequencies of quartz filters are very\u00a0sensitive to metal thickness variations, especially when the\u00a0relative thickness is more than 1.5-2.0%. Filters on 112o-\u00a0LiTaO3 substrate provide wider fractional bandwidth or\u00a0better insertion loss for the same relative bandwidth than\u00a0filters on quartz substrate. Besides, as the energy storage\u00a0effect is about half of quartz, manufacturing variation of\u00a0such filters is better than quartz filters.<\/p>\n Experimental response of an RSPUDT filter on 112o-\u00a0LiTaO3 substrate is shown in Fig.5. The relative 3dB\u00a0bandwidth is 3.3% and insertion loss is about 5.5dB.\u00a07mmx5 mm SMD package is used for this filter.<\/p>\n
\n<\/a>where Rs^k(fi) is the value of simulated amplitude\u00a0response at frequency fi for the k frequency range, Rr\u00a0k is the\u00a0required attenuation value for this range, Ps\u00a0l is the\u00a0simulated value of a certain parameter, Pr\u00a0l is the required\u00a0value of that parameter, Ak,l is an error weight for a\u00a0specific frequency range or a parameter.<\/p>\n<\/a>ADJUSTMENT OF DWSF-RSPUDT<\/h2>\n
<\/a>
and adjusted DWSF structures (solid)<\/p><\/div>\nEXPERIMENTAL RESULTS<\/h2>\n
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