Further Bounds Existing on the Transient Responses of Various Types of Networks

Abstract

Five new theorems stating bounds on the transient response of certain types of networks are derived and illustrated. The first one states that any system function, whose real part on the positive real-frequency axis decreases monotonically as frequency increases, cannot have an overshoot in its step response greater than eighteen per cent, nor can its rise time, from the time that the input step is applied to the time that the response first crosses the final value line, be less than 1.22(r-K)C where r is the final value of the step response and K and 1/C are the constant term and the coefficient of the 1/s term in the inverse power-seres expansion of the system function. Similarly, the other four theorems show that when the system function is appropriately restricted, the transient response is bounded. These restrictions include never-negative or never-positive conditions on the real or imaginary parts of the system function along the real-frequency axis, while the resulting bounds are determined by certain coefficients occurring in the powerseries expansions of the system function.