Mathematical Models of System (original) (raw)
• Dynamic systems are described by differential equations. Laplace work only in case of linear systems. • Based upon the analysis of relation between system variables. • Linear Approximtions of Physical Systems: Within some range of variables, a great range of physical systems are linear. A system is defined as linear in terms of excitation and response • Linear systems follow superposition principle: when the excitation is x1(t), response is y1(t); again when excitation is x2(t), response is y2(t). Now when excitation is x1(t) + x2(t), response is y1(t) + y2(t). Or, the responses are directly mappable to the respective excitations—the excitations do not affect eachother in such a way that responses might be influenced. Homogeniety is also followed—ax(t) gives ay(t) in response. • Although y = mx + b doesn't satisfy the condition of homogeniety, Δy = mΔx satisfies because of constant offset. Here Δy must be seen as a fuction itself, and not a differential change. • Small signal analysis: the slope at the operating point is a good approximation of the function in small interval about the deviation (x – x0). Or, y = g(x0) + dg/dx(x – x0) => y = y0 + m(x – x0) => y – y0 = m(x – x0) which is nothing but Δy = mΔx. • To use laplace transform, the linearized differential equation is used. • The lower limit of the integral in the condition of convergence of f for it to have a mathematically valid Laplace Transform takes care of any discontinuity—one like delta function. The σ1 is abscissa of absolute convergence. • |f(t)| < Mexp(αt), which implies the function converges for all σ1 > α, also gives the region of convergence. • Poles and zeroes are critical frequencies. • Steady state/final value is found by final value theorem stated as • Simple pole of Y(s) at origin is allowed, but repeated poles at origin, in right half plane and on imaginary axis are excluded. • Damping Ratio ζ: is always in multiplication with natural frequency and arises from the coefficient of y' (degree and order 1). The equations below is laplace transform of second order equation of oscillations of y(t).