Crystalline Raman amplifier is an important way to obtain high beam quality, high spectral purity and high power Raman laser. The normalized transport equations of the external cavity Raman amplifier were derived by inducing four normalized composite parameters. A set of universal theoretical curves describing the operation of Raman amplifiers were obtained by numerical solving the transport equations, and the influence of the complex normalized variables on the performance of the Raman amplifiers was analyzed. The dependences of the magnification of the Raman pulse peak intensity, pulse shape of the output Raman laser, and the conversion efficiency from pumping laser to output Raman laser on the pumping pulse intensity, the relative width of the pumping pulse and Raman seed pulse, and the time overlap of the pumping pulse and Raman seed pulse were studied in detail. The optimum working condition of crystalline Raman amplifier was explored. The normalized theory was verified with actual experimental data. It is indicated that the theoretical results are consistent with the measured data. The normalized rate-equation model is proved to be precise and feasible.