The limitation of the semi-local integration algorithm is that the sampled waveform is required to be a sine wave. When the sampled analog quantity is not a sine wave but a periodic time function, the Fourier transform algorithm can be used. The Fourier transform algorithm comes from the Fourier series, that is, a periodic function I(t) can be expanded by the Fourier series into the sum of the sine and cosine terms of each harmonic, which can be expressed by the following formula
In the formula, n is a natural number, n=0, 1, 2... represents the order of harmonic components. Then the fundamental wave component in current i(t) can be expressed as
i(t) can also be expressed as a general expression
In the formula, I1 is the effective value of the fundamental wave and is the initial phase angle of the fundamental wave component at t=0.
Expand sin(wt+a) using the sum angle formula, and then compare it with equation (2-4) to get the relationship between sum and the same sum.
From equations (2-6) and (2-7), we can see that it is easy to find the effective value and initial phase angle of the fundamental wave by simply finding the amplitudes of the sine and cosine terms of the fundamental wave.
The sum can be obtained according to the principle of inverse transformation of Fourier series.
When using a microcomputer to calculate the sum, it is usually calculated using the finite discrete method, that is, substituting the values of each sampling point at (t), and replacing the integration method with the trapezoidal method for summation. Considering N·, equation (2-8) and equation (2-9) can be expressed as
In the formula, N is the number of sampling points in one week, is the k-th sampling value, and are the sampling values when k=0 and N respectively. When N=12, the sampling interval Ts is generally expressed as 30 in angle.
After calculating the sum, it is not difficult to obtain the effective value and phase angle of the fundamental wave af + bf according to equations (2-6) and (2-7)
Compared with the half-cycle integral algorithm, the Fourier transform algorithm can calculate the periodic time function and the initial phase angle. The integral operation result also has a digital filtering function and the calculation workload is not large. However, when this algorithm is used for transient sampling calculations, it is greatly affected by the non-periodic components in the input analog quantities. The theoretical analysis can produce an error of more than 15% under the most unfavorable conditions, and some compensation measures are usually required to overcome it. At present, many more advanced protection devices adopt Fourier transform algorithm, such as LFP-900 series protection devices.