Broadband hyperspectral cameras can comprehensively record the spectral information of a target, which is currently a significant research direction in studies on hyperspectral cameras. However, the broadband is bound to cause problems with excessive chromatic aberration and secondary spectra of the system, affecting the image quality. Therefore, based on an analysis method called the Buchdahl vector dispersion model, an optical system of broadband hyperspectral camera based on linear variable filter is proposed in this study. An image side telecentric transmitted optical system with a focal length of 100 mm, F-number of 5, field of view of 14.2°, and spectral range of 400-1000 nm was designed. The hyperspectral camera based on this system can capture images with a spatial resolution of 21.5 m, spectral resolution of 10 nm, and swath width of 125 km at an altitude of 500 km. The image quality evaluation and tolerance analysis show that the system has an excellent image quality and satisfies fabrication and alignment requirements. The modulation transfer function test results show that the system can satisfy actual application requirements.