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Based on an optically injected semiconductor laser (OISL) operating at period-one (P1) nonlinear dynamical state, tunable and ultrabroadband microwave frequency combs (MFCs) are generated experimentally through further current-modulating the OISL. First, by introducing an injection light with an injection power of <inline-formula> <tex-math notation="LaTeX">$P_{\rm{i}}\,= \,12.42$</tex-math></inline-formula> mW, whose wavelength is identical to the central wavelength of a free-running distributed feedback semiconductor laser, the OISL can be driven into P1 state with a fundamental frequency of <inline-formula><tex-math notation="LaTeX">$f_{{0}}\,= \,26.44$</tex-math> </inline-formula> GHz. Next, further modulating the OISL with a modulation frequency of<inline-formula> <tex-math notation="LaTeX">$f_{\rm{m}}\,= \,3.3$</tex-math></inline-formula>&#x00A0;GHz and a modulation power of <inline-formula><tex-math notation="LaTeX">$P_{\rm{m}}$</tex-math></inline-formula> &#x003D; 22.0 dBm, an MFC with a bandwidth of 59.4 GHz within a 10 dB amplitude variation is experimentally obtained, and the single sideband phase noise at offset frequency 10 kHz for all comb lines contained within the bandwidth is below &#x2212;95.0 dBc&#x002F;Hz. Finally, through varying the modulation frequency, the MFCs with different comb spacing can be obtained, and the influences of relevant operation parameters on the performances of the generated MFCs have been analyzed.