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An innovative high-precision digital instrument for position measurement of particle accelerators collimators is proposed. The instrument relies on Linear Variable Differential Transformer by exploiting the three-parameter sine fit to estimate the amplitudes of their secondary signals. Moreover, the crucial problem of interfering magnetic fields is thoroughly analyzed by means of two different models: an analytical one aimed at capturing the main physics of the phenomenon and a predictive one based on Finite Elements Method. Both models have been validated by means of simulations and experimental measurements. They provide a basis for design guidelines to an application-independent solution of the interference problem. The experimental validation of the models comprises the development of a measurement method and a measurement system for the general characterization of external magnetic fields as influence quantities for the position measurement by means of Linear Variable Differential Transformer sensors. Preliminary on-field results, gathered during Large Hadron Collider commissioning and operation at CERN, confirming the laboratory results are reported.