The Back-n white neutron source at the China Spallation Neutron Source (CSNS) was put into operation in 2018, providing a good platform for the cross-section measurement of neutron induced light charged particle $\left(lcp\right)$ emission $(n,lcp)$ reactions. Currently, in order to detect and identify light charged particles produced in $(n,lcp)$ reactions at Back-n, a silicon detector array has been developed as a part of Light-charged Particle Detector Array. Composed of 15 silicon detectors, the array can detect the emitted particles ranging from 19.2°to 160.8°. The energy resolution of each silicon detector unit is better than 1% for 5.499 MeV $\alpha$ particles. Adopting this silicon detector array, several experimental measurements have been completed. Differential cross sections and angle-integrated cross sections in the neutron energy region ranging from the eV to the MeV range are extracted. Compared with previous measurements and evaluations, new results are obtained in the present measurements. These experiments indicate the feasibility of the silicon detector array in $(n,lcp)$ reaction cross-section measurements.