Consider a LTI system with input f(t) and output y(t) that can be represented by the strictly-proper first-order differential equation dy/dt + 1/tau y(t) = K/tau f(t), where K and r are tau constant parameters.
State the characteristic equation and find its characteristic root lambda. A characteristic root lambda maybe real or complex valued. If the real part of lambda Ls strictly negative, then the corresponding natural (or characteristic) mode e^lambda t exponentially decays to zero as t increases. In this case the time constant associated with the natural mode is defined as lambda = -1/Re(lambda). The time constant tau is a measure of how quickly a natural mode relaxes towards zero. The smaller tau is, the faster the mode decays.