This paper describes a 23-GHz digital bang-bang phase-locked loop (PLL) fabricated in 65-nm CMOS for millimeter-wave frequency-modulated continuous-wave radars. The presented circuit aims to generate a fast sawtooth chirp signal that grants significant advantages with respect to the more conventional triangular waveform. Such a signal, however, features a very large bandwidth that requires the adoption of a two-point injection scheme. This paper, after intuitively discussing how the nonlinearity of the digitally controlled oscillator affects the accuracy of frequency modulation, presents a novel automatic pre-distortion engine, operating fully in background, which linearizes the tuning characteristic. The 19.7-mA fractional-N PLL having an rms jitter of 213 fs and an in-band fractional spur of -58 dBc is capable of synthesizing fast chirps with 173-MHz/μs maximum slope and an idle time of less than 200 ns after an abrupt frequency step with no over or undershoot.