Precise derivation for the equivalent circuit parameters of a crystal resonator with series capacitor

An exact formula for the equivalent circuit of a crystal resonator in series with a capacitor is derived. Network analysis is used to obtain an exact formula for the equivalent circuit parameters without applying any approximations. The result is an expansion of that obtained by those who used assumptions regarding high frequency, high quality factor, high capacitance ratio, and so on. Hence, this formula can be used for instances of low quality factor or low capacitance ratio, and even for the actual inductor-capacitor network as long as these devices have the same equivalent circuits. The enhanced accuracy of the new formula extends itself to oscillator frequency calibration, temperature compensation, and electronic frequency control.