On the four-frequency measurement process for coupled dual resonator crystals

A simplified derivation of the basic four-frequency measurement process for a coupled dual resonator crystal is given which shows a more direct path than previously published for obtaining the two resonator frequencies either from the poles and zeros of 1/y₁₁ or from the zeros of 1/y₁₁ and the zeros of z₁₁ the latter of which tends to be the more useful. Using the resonator frequencies obtained from either of the two cases, the synchronous peak separation frequency (SPSF) is obtained