The four-frequency process for coupled-duals using error-corrected S-parameter measurements

Although the four-frequency measurement process has been used successfully on coupled dual crystals from 9.4 MHz to 21.4 MHz, discrepancies between the crystal data and filter data at 45 MHz and 57.5 MHz suggested an error problem associated with the measurement system. To eliminate the errors, a twelve-term error model was chosen and the equations were derived for use with an automatic network analyzer. The details of the direct deviation of the four-frequency measurement process are given in which the two resonator frequencies and the synchronous peak separation frequency (SPSF) of a coupled dual crystal are calculated from the two frequencies of the zeros of the short-circuit driving point impedance, and the two frequencies of the zeros of the open-circuit driving point impedance. Determination of the four frequencies from the error-corrected S parameters is discussed, as well as the analysis of an external capacitor placed across the output circuit to obtain a more convenient distribution of the four frequencies. A comparison is made between data taken on the original fixture and data taken on an automatic network analyzer using error-corrected S parameters.