The diagnosis of induction machines using Fourier transform relies on tracking the frequency signature of each type of fault in the current's spectrum, but this signature depends on the machine's slip and the supply frequency, so it must be recomputed for each working condition by trained personnel or by diagnostic software. Besides, sampling the current at high rates during long times is needed to achieve a good spectral resolution, which requires large memory space to store and process the current spectra. In this paper, a novel approach is proposed to solve both problems. It is based on the fact that each type of fault generates a series of harmonics in the current's spectrum, whose frequencies are multiples of a characteristic main fault frequency. The tracking analysis of the fault components using the harmonic order (defined as the frequency in per unit of the main fault frequency) as independent variable instead of the frequency generates a unique fault signature, which is the same for any working condition. Besides, this signature can be concentrated in just a very small set of values, the amplitudes of the components with integer harmonic order. This new approach is introduced theoretically and validated experimentally.