To understand the principles underlying digital protection technology, it is necessary to review briefly the mathematical basis of numerical algorithms. The topics covered in this Chapter therefore include finite differences, numerical differentiation, curve fitting and smoothing, Fourier analysis, Walsh analysis, and the relationship between Fourier and Walsh coefficients. It is not the intention that the material presented should be highly rigorous in the mathematical sense, but rather that it should give a working knowledge of the numerical techniques used and thus provide a basis for the work on specific protection algorithms that is presented in later Chapters.

The algorithms covered in this Chapter assume that the post-fault current and voltage waveforms are sinusoidal. This assumption is not, of course, generally valid, particularly when EHV or UHV network applications are involved. However, in practice, the signals processed are often prefiltered and, in lower voltage distribution systems in particular, the waveforms often very quickly attain a nominally sinusoidal form.

In this Chapter we discuss techniques used to fit faulted current and voltage waveforms, each to a sinusoidal waveform containing a fundamental component, a decaying/constant DC component and/or harmonics. These techniques use the least squares (LSQ) method to minimise the fitting error, and all have the common goal of extracting the fundamental components of voltage and current waveforms, to calculate the impedance to the fault or the comparison of current-based signals in digital differential protection.

In this Chapter, transmission lines are treated as distributed circuits, in order to explain how travelling-wave phenomena are propagated and detected. The basic principles of travelling-wave schemes, the formation of relaying signals, Bergeron's equation and discriminant functions using single-phase line models are also explained. These principles are then extended to three-phase lines by decomposing the line into three equivalent single-phase lines using the modal decomposition approach.

This Chapter gives a brief general review of the principles of transformer differential protection. This is followed by an explanation of the application of digital techniques and the algorithms that have been developed specifically for application to transformer protection. The algorithms covered include finite duration impulse (FIR) filters, least-squares curve fitting, the digital Fourier algorithm and the flux-restrained current differential algorithm. Finally, the basic hardware arrangement for implementing digital techniques to the protection of transformers is described. It is, however, important to note that closely similar techniques can be applied to the protection of generators, although, in this case, the transformation ratio of currents is the same on each side of the protected zone.