We report a wavelength converter based on linearly chirped gratings in a lithium niobate crystal through cascaded second-order nonlinear processes. The influence of the chirp coefficient on the conversion bandwidth and the efficiency is analysed. The exact phase matching point is optimized at the 0.33 times of the waveguide total length. When the waveguide length is 3 cm, the bandwidth of the linearly chirped grating increases 42% with the efficiency penalty of 1.7dB compared with the periodic grating. At the same mean conversion efficiency of -1.4 dB via adjusting the waveguide total lengths, the bandwidth of the linearly chirped grating is 36% broader than the periodic grating.