Developments in Obtaining Transient Response Using Fourier Transforms

Abstract

The Modified Fourier TransforrnUse has been made of the inverse Fourier transform viz. 00 f(t) = _L JJ(oo)exp (joot) doo 2,.-00 whereJ( 00) is the frequency response and the integration is carried out along the real axis, in the approach of Lego and Sze. (2) For numerical integration the range must be made finite and any resulting Gibbs oscillations can be reduced by incorporating the sigma factor described in Part I. However, the poles of J(00) may, and in many cases do, lie close to the real axis. This causes the integrand to peak over a series of small intervals lying in the neighbourhood of the poles and hence for accurate numerical integration it is necessary to take a very small step length. This limitation can be overcome by using the modified Fourier transformtti and at the same time this secures the additional advantage of avoiding problems associated with the convergence of the integrals involved. The method Lego and Sze devised to overcome this latter difficulty was to introduce an artificial forcing function they denoted by U(a, t) (see Ref. 2, p. 1032, equation 7).In the present approach we consider the problem of finding the response of a linear system to some input starting at time zero. Suppose that the differential equation of the system is known and can be written in the formwheref(t) is the output, g(t) the input both of which are zero for t < 0. In taking the modified Fourier transform equation (10) is multiplied throughout by exp {-(a +joo) t} and integrated w.r.t, tover the range (-00,00), or,sincef(t),g(t) arezerofort < 0, over the range (0, 00); equation (10) then becomes F(a + joo)J+(a + joo) = K+(a + joo)at The University of Iowa Libraries on June 4, 2016 ije.sagepub.com Downloaded from

Electromagnetic Transients Program: History and Future

“…It should be noted that there exists a transient analysis method based on Fourier/Laplace transform in the frequency domain [1,11,39–46].…”