Analytical Synthesis of a Phase-Coherent Subtractive Crossover with Exact Phase Matching

Abstract

An analytical method for the synthesis of an active subtractive crossover is presented, which ensures exact matching of the phase characteristics of the low-pass (LP) and high-pass (HP) bands across the entire operating frequency range. It is shown that the phase identity condition is reduced to the equality of the poles (denominators) of the all-pass and low-pass filter transfer functions. This allows for closed-form equations for calculating component values and eliminates the need for empirical tuning. The method provides a total amplitude-frequency response (AFR) ripple of less than 0.045 dB in the 20 Hz – 20 kHz band and identical group delay for both bands. Implementation examples using standard E24 component values are provided. The work develops a previously proposed subtractive crossover topology, offering a rigorous analytical solution for it.