Analysis of capacitances of electrical equivalent circuit of quartz-crystal tuning fork in consideration of electric field distribution using conformal mapping method

The formulas of capacitances of the electrical equivalent circuit of a quartz-crystal tuning fork at resonance were derived only for free-free bar model in consideration of the electric field distribution and the potential ratio in the section of its arm using conformal mapping method and Fourier series in Ref. [1]. We analyzed the same problem for both free-free and clamped-free bar models and compared the calculated values with measured values of capacitances of the electrical equivalent circuit of the quartz crystals which were available in practical wrist watch application.