In practice, LVDTs are used with carrier demodulator modules that provide the carrier signal (the AC signal to the primary coil) and converts (or demodulates) the induced AC signal to a DC signal.
The phase sensitive demodulator is an AC to DC converter that produces a DC voltage (typically between 0 and 10 volts) proportional to the magnitude of the LVDT output and sensitive to the phase of the output signal relative to the input (carrier) signal.
When the core is displaced to one side of the primary, the LVDT output is in phase with the input and the demodulator produces a positive signal. When the core is displaced to the other side, the LVDT output is 180 degrees out of phase with the signal to the primary. The demodulator output is then a negative voltage proportional to the displacement.