System Response Characteristics ISAT 412 -Dynamic Control of Energy Systems (Fall 2005)

Review  We have overed several O.D.E. solution techniques Direct integration Exponential solutions (classical) Laplace transforms  Such techniques allow us to find the time response of systems described by differential equations

Generic 1 st order model  Solution in Laplace domain  Solution comprised of Free Response (homogeneous solution) Forced Response (non-homogeneous solution)

Free response of 1 st order model  Free response means:  Converting back to the time domain:

Time constant  Define the system time constant as  Rewriting the free response or

Free response behavior Unstable Stable Unstable

Meaning of the time constant  When t =   When t = 2 , t = 3 , and t = 2 ,

Transfer Functions and Common Forcing Functions ISAT 412 -Dynamic Control of Energy Systems (Fall 2005)

Forced response of 1 st order system  The forced response corresponds to the case where x(0) = 0  In the Laplace domain, the forced response of a 1 st order system is

Transfer functions  Solve for the ratio X(s)/F(s)  T(s) is the transfer function  Can be used as a multiplier in the Laplace domain to obtain the forced response to any input

Using the transfer function  Now that we know the transfer function for a 1 st order system, we can obtain the forced response to any input if we can express that input in the Laplace domain

Step input  Used to model an abrupt change in input from one constant level to another constant level Example: turning on a light switch

Heaviside (unit) step function  Used to model step inputs

Time shifted unit step function  For a unit step shifted in time,  Using the shifting property of the Laplace transform (property 6)

Step input model  For a step of magnitude b at time D

Pulse input

Pulse input model  Use two step functions

Pulse input model  For a pulse input of magnitude M, starting at time A and ending at time B

Impulse input  Examples: explosion, camera flash, hammer blow

Impulse input model  Unit impulse function  For an impulse input of magnitude M at time A

Ramp input

Ramp input model  For a ramp input beginning at time A with a slope of m

Other input functions  Sinusoidal inputs  Combinations of step, pulse, impulse, and ramp functions

Modeling periodic inputs

Square wave input model  Addition of an infinite number of step functions with amplitudes A and - A

Laplace transform of square wave